(PDF) Crop evapotranspiration-Guidelines for computing crop water requirements-FAO Irrigation and drainage paper 56

FAO Irrigation and Drainage Paper

No. 56

Crop

Evapotranspiration

(guidelines for computing crop water requirements)

by

Richard G. ALLEN

Utah State University

Logan, Utah, U.S.A.

Luis S. PEREIRA

Instituto Superior de Agronomia

Lisbon, Portugal

Dirk RAES

Katholieke Universiteit Leuven

Leuven, Belgium

Martin SMITH

FAO, Water Resources, Development and Management Service

Rome, Italy

Crop evapotranspiration

iii

Preface

This publication presents an updated procedure for calculating reference and crop

evapotranspiration from meteorological data and crop coefficients. The procedure, first presented

in the FAO Irrigation and Drainage Paper No. 24 'Crop Water Requirements', is termed the 'Kc

ETo' approach, whereby the effect of the climate on crop water requirements is given by the

reference evapotranspiration ETo and the effect of the crop by the crop coefficient Kc. Other

procedures developed in FAO Irrigation and Drainage Paper No. 24 such as the estimation of

dependable and effective rainfall, the calculation of irrigation requirements and the design of

irrigation schedules are not presented in this publication but will be the subject of later papers in

the series.

Since the publication of FAO Irrigation and Drainage Paper No. 24 in 1977, advances in research

and more accurate assessment of crop water use have revealed the need to update the FAO

methodologies for calculating ETo. The FAO Penman method was found to frequently

overestimate ETo while the other FAO recommended equations, namely the radiation, the Blaney-

Criddle, and the pan evaporation methods, showed variable adherence to the grass reference crop

evapotranspiration.

In May 1990, FAO organized a consultation of experts and researchers in collaboration with the

International Commission for Irrigation and Drainage and with the World Meteorological

Organization, to review the FAO methodologies on crop water requirements and to advise on the

revision and update of procedures.

The panel of experts recommended the adoption of the Penman-Monteith combination method as a

new standard for reference evapotranspiration and advised on procedures for calculating the

various parameters. The FAO Penman-Monteith method was developed by defining the reference

crop as a hypothetical crop with an assumed height of 0.12 m, with a surface resistance of 70 s m-1

and an albedo of 0.23, closely resembling the evaporation from an extensive surface of green grass

of uniform height, actively growing and adequately watered. The method overcomes the

shortcomings of the previous FAO Penman method and provides values that are more consistent

with actual crop water use data worldwide. Furthermore, recommendations have been developed

using the FAO Penman-Monteith method with limited climatic data, thereby largely eliminating

the need for any other reference evapotranspiration methods and creating a consistent and

transparent basis for a globally valid standard for crop water requirement calculations.

The FAO Penman-Monteith method uses standard climatic data that can be easily measured or

derived from commonly measured data. All calculation procedures have been standardized

according to the available weather data and the time scale of computation. The calculation methods,

as well as the procedures for estimating missing climatic data, are presented in this publication.

In the 'Kc-ETo' approach, differences in the crop canopy and aerodynamic resistance relative to the

reference crop are accounted for within the crop coefficient. The Kc coefficient serves as an

aggregation of the physical and physiological differences between crops. Two calculation methods

iv

to derive crop evapotranspiration from ETo are presented. The first approach integrates the

relationships between evapotranspiration of the crop and the reference surface into a single Kc

coefficient. In the second approach, Kc is split into two factors that separately describe the

evaporation (Ke) and transpiration (Kcb) components. The selection of the Kc approach depends

on the purpose of the calculation and the time step on which the calculations are to be executed.

The final chapters present procedures that can be used to make adjustments to crop coefficients to

account for deviations from standard conditions, such as water and salinity stress, low plant

density, environmental factors and management practices.

Examples demonstrate the various calculation procedures throughout the publication. Most of the

computations, namely all those required for the reference evapotranspiration and the single crop

coefficient approach, can be performed using a pocket calculator, calculation sheets and the

numerous tables given in the publication. The user may also build computer algorithms, either

using a spreadsheet or any programming language.

These guidelines are intended to provide guidance to project managers, consultants, irrigation

engineers, hydrologists, agronomists, meteorologists and students for the calculation of reference

and crop evapotranspiration. They can be used for computing crop water requirements for both

irrigated and rainfed agriculture, and for computing water consumption by agricultural and natural

vegetation.

Crop evapotranspiration

v

Acknowledgements

These guidelines constitute the efforts of eight years of deliberations and consultations by the

authors, who together formed the working group to pursue the recommendations of the FAO expert

consultation that was held in May 1990 in Rome. The consultation was organized to review the then

current FAO guidelines to determine Crop Water Requirements, published in 1977 as FAO

Irrigation and Drainage paper No. 24 (FAO-24) and authored by J. Doorenbos and W. Pruitt. The

conceptual framework for the revised methodologies introduced in this publication came forth out

of the advice of the group of eminent experts congregated in the 1990 meetings and who have

importantly contributed to the development of the further studies conducted in the framework of

the publication. Members of the 1990 FAO expert consultation included Dr P. Fleming of

Australia, Dr A. Perrier of France, Drs L. Cavazza and L. Tombesi from Italy, Drs R. Feddes and

J. Doorenbos of the Netherlands, Dr L.S. Pereira of Portugal, Drs J.L. Monteith and H. Gunston

from the United Kingdom, Drs R. Allen, M. Jensen and W.O. Pruitt of USA, Dr D. Rijks from the

World Meteorological Organization and various staff of FAO .

Many other experts and persons from different organizations and institutes have provided, in

varying degrees and at different stages, important advice and contributions. Special

acknowledgements for this are due in particular to Prof. W.O. Pruitt (retired) of the University of

California, Davis and J. Doorenbos of FAO (retired) who set the standard and template for this

work in the predecessor FAO-24, and to Prof. J.L. Monteith whose unique work set the scientific

basis for the ETo review. Prof. Pruitt, despite his emeritus status, has consistently contributed in

making essential data available and in advising on critical concepts. Dr James L. Wright of the

USDA, Kimberly, Idaho, further contributed in providing data from the precision lysimeter for

several crops. Further important contributions or reviews at critical stages of the publication were

received from Drs M. Jensen, G. Hargreaves and C. Stockle of USA, Dr B. Itier of France, and

various members of technical working groups of the International Commission on Irrigation and

Drainage (ICID) and the American Societies of Civil and Agricultural Engneers.

The authors thank their respective institutions, Utah State University, Instituto Superior de

Agronomia of Lisbon, Katholieke Universiteit Leuven and FAO for the generous support of faculty

time and staff services during the development of this publication.

The authors wish to express their gratitude to Mr H. Wolter, Director of the Land and Water

Development Division for his encouragement in the preparation of the guidelines and to FAO

colleagues and others who have reviewed the document and made valuable comments.

Special thanks are due to Ms Chrissi Redfern for her patience and valuable assistance in the

preparation and formatting of the text. Mr Julian Plummer further contributed in editing the final

document.

vi

Crop evapotranspiration

vii

Contents

Page

1. INTRODUCTION TO EVAPOTRANSPIRATION 1

Evapotranspiration process 1

Evaporation 1

Transpiration 3

Evapotranspiration 3

Units 3

Factors affecting evapotranspiration 5

Weather parameters 5

Crop factors 5

Management and environmental conditions 5

Evapotranspiration concepts 7

Reference crop evapotranspiration (ETo) 7

Crop evapotranspiration under standard conditions (ETc) 7

Crop evapotranspiration under non-standard conditions (ETc adj) 9

Determining evapotranspiration 9

ET measurement 9

ET computed from meteorological data 13

ET estimated from pan evaporation 13

PART A. REFERENCE EVAPOTRANSPIRATION (ETO) 15

2. FAO PENMAN-MONTEITH EQUATION 17

Need for a standard ETo method 17

Formulation of the Penman-Monteith equation 18

Penman-Monteith equation 18

Aerodynamic resistance (ra) 20

(Bulk) surface resistance (rs) 20

Reference surface 23

FAO Penman-Monteith equation 23

Equation 23

Data 25

Missing climatic data 27

3. METEOROLOGICAL DATA 29

Meteorological factors determining ET 29

Solar radiation 29

Air temperature 29

Air humidity 30

Wind speed 30

Atmospheric parameters 31

Atmospheric pressure (P) 31

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Page

Latent heat of vaporization (λ) 31

Psychrometric constant (γ) 31

Air humidity 33

Concepts 33

Measurement 35

Calculation procedures 36

Radiation 41

Concepts 41

Units 43

Measurement 45

Calculation procedures 45

Wind speed 55

Measurement 55

Wind profile relationship 55

Climatic data acquisition 57

Weather stations 57

Agroclimatic monthly databases 57

Estimating missing climatic data 58

Estimating missing humidity data 58

Estimating missing radiation data 59

Missing wind speed data 63

Minimum data requirements 64

An alternative equation for ETo when weather data are missing 64

4. DETERMINATION OF ETO 65

Penman-Monteith equation 65

Calculation procedure 66

ETo calculated with different time steps 66

Calculation procedures with missing data 76

Pan evaporation method 78

Pan evaporation 78

Pan coefficient (Kp) 79

PART B. CROP EVAPOTRANSPIRATION UNDER STANDARD CONDITIONS 87

5. INTRODUCTION TO CROP EVAPOTRANSPIRATION (ETC) 89

Calculation procedures 89

Direct calculation 89

Crop coefficient approach 90

Factors determining the crop coefficient 91

Crop type 91

Climate 91

Soil evaporation 93

Crop growth stages 95

Crop evapotranspiration

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Page

Crop evapotranspiration (ETc) 97

Single and dual crop coefficient approaches 98

Crop coefficient curve 99

Flow chart of the calculations 101

6. ETC - SINGLE CROP COEFFICIENT (KC) 103

Length of growth stages 103

Crop coefficients 109

Tabulated Kc values 109

Crop coefficient at the initial stage (Kc ini) 114

Crop coefficient at the mid-season stage (Kc mid) 121

Crop coefficient at the end of the late season stage (Kc end) 125

Construction of the Kc curve 127

Annual crops 127

Kc curves for forage crops 127

Fruit trees 129

Calculating ETc 129

Graphical determination of Kc 129

Numerical determination of Kc 132

Alfalfa-based crop coefficients 133

Transferability of previous Kc values 134

7. ETC - DUAL CROP COEFFICIENT (KC = KCB + KE) 135

Transpiration component (Kcb ETo) 135

Basal crop coefficient (Kcb) 135

Determination of daily Kcb values 141

Evaporation component (Ke ETo) 142

Calculation procedure 142

Upper limit Kc max 143

Soil evaporation reduction coefficient (Kr) 144

Exposed and wetted soil fraction (few) 147

Daily calculation of Ke 151

Calculating ETc 156

PART C. CROP EVAPOTRANSPIRATION UNDER NON-STANDARD CONDITIONS 159

8. ETC UNDER SOIL WATER STRESS CONDITIONS 161

Soil water availability 161

Total available water (TAW) 161

Readily available water (RAW) 162

Water stress coefficient (Ks) 167

Soil water balance 169

Forecasting or allocating irrigations 171

Effects of soil salinity 174

Yield-salinity relationship 175

Yield-moisture stress relationship 176

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Page

Combined salinity-ET reduction relationship 176

No water stress (Dr < RAW) 176

With water stress (Dr > RAW) 177

Application 181

9. ETC FOR NATURAL, NON-TYPICAL AND NON-PRISTINE CONDITIONS 183

Calculation approach 183

Initial growth stage 183

Mid and late season stages 183

Water stress conditions 184

Mid-season stage - adjustment for sparse vegetation 184

Adjustment from simple field observations 184

Estimation of Kcb mid from Leaf Area Index (LAI) 185

Estimation of Kcb mid from effective ground cover (fc eff) 187

Estimation of Kcb full 189

Conclusion 190

Mid-season stage - adjustment for stomatal control 191

Late season stage 193

Estimating ETc adj using crop yields 193

10. ETC UNDER VARIOUS MANAGEMENT PRACTICES 195

Effects of surface mulches 195

Plastic mulches 195

Organic mulches 196

Intercropping 198

Contiguous vegetation 199

Overlapping vegetation 199

Border crops 200

Small areas of vegetation 200

Areas surrounded by vegetation having similar roughness and moisture

conditions 200

Clothesline and oasis effects 202

Management induced environmental stress 203

Alfalfa seed 204

Cotton 204

Sugar beets 204

Coffee 204

Tea 204

Olives 205

11. ETC DURING NON-GROWING PERIODS 207

Types of surface conditions 207

Bare soil 207

Surface covered with dead vegetation 207

Surface covered with live vegetation 208

Frozen or snow covered surfaces 209

Crop evapotranspiration

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Page

ANNEX 1. UNITS AND SYMBOLS 211

ANNEX 2. METEOROLOGICAL TABLES 213

ANNEX 3. BACKGROUND ON PHYSICAL PARAMETERS USED IN

EVAPOTRANSPIRATION COMPUTATIONS 223

ANNEX 4. STATISTICAL ANALYSIS OF WEATHER DATA SETS 229

ANNEX 5. MEASURING AND ASSESSING INTEGRITY OF WEATHER DATA 245

ANNEX 6. CORRECTION OF WEATHER DATA OBSERVED AT NON-REFERENCE

WEATHER SITES TO COMPUTE ETO 257

ANNEX 7. BACKGROUND AND COMPUTATIONS FOR KC INI 263

ANNEX 8. CALCULATION EXAMPLE FOR APPLYING THE DUAL KC PROCEDURE

IN IRRIGATION SCHEDULING 269

BIBLIOGRAPHY 281

xii

List of figures

1. Schematic representation of a stoma 2

2. The partitioning of evapotranspiration into evaporation and transpiration over the

growing period for an annual field crop 2

3. Factors affecting evapotranspiration with reference to related ET concepts 4

4. Reference (ETo), crop evapotranspiration under standard (ETc) and non-standard

conditions (ETc adj) 6

5. Schematic presentation of the diurnal variation of the components of the energy

balance above a well-watered transpiring surface on a cloudless day 10

6. Soil water balance of the root zone 12

7. Simplified representation of the (bulk) surface and aerodynamic resistances for

water vapour flow 19

8. Typical presentation of the variation in the green Leaf Area Index over the

growing season for a maize crop 22

9. Characteristics of the hypothetical reference crop 24

10. Illustration of the effect of wind speed on evapotranspiration in hot-dry and humid-

warm weather conditions 30

11. Saturation vapour pressure shown as a function of temperature: e°(T) curve 34

12. Variation of the relative humidity over 24 hours for a constant actual vapour

pressure of 2.4 kPa 34

13. Annual variation in extraterrestrial radiation (Ra) at the equator, 20 and 40° north

and south 41

14. Annual variation of the daylight hours (N) at the equator, 20 and 40° north and

south 42

15. Various components of radiation 44

16. Conversion factor to convert wind speed measured at a certain height above

ground level to wind speed at the standard height (2 m) 56

17. Relationship between the fraction of extraterrestrial radiation that reaches the

earth's surface, Rs/Ra, and the air temperature difference Tmax - Tmin for

interior (kRs = 0.16) and coastal (kRs = 0.19) regions 61

18. ETo computed by CROPWAT 69

19. Two cases of evaporation pan siting and their environment 79

20. Typical Kc for different types of full grown crops 92

21. Extreme ranges expected in Kc for full grown crops as climate and weather change 92

22. The effect of evaporation on Kc. The horizontal line represents Kc when the soil

surface is kept continuously wet. The curved line corresponds to Kc when the

soil surface is kept dry but the crop receives sufficient water to sustain full

transpiration 94

23. Crop growth stages for different types of crops 94

Crop evapotranspiration

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Page

24. Typical ranges expected in Kc for the four growth stages 97

25. Generalized crop coefficient curve for the single crop coefficient approach 100

26. Crop coefficient curves showing the basal Kcb (thick line), soil evaporation Ke)

(thin line) and the corresponding single Kc = Kcb + Ke curve (dashed line) 100

27. General procedure for calculating ETc 102

28. Variation in the length of the growing period of rice (cultivar: Jaya) sown during

various months of the year at different locations along the Senegal River (Africa) 109

29. Average Kc ini as related to the level of ETo and the interval between irrigations

and/or significant rain during the initial growth stage for all soil types when

wetting events are light (about 10 mm per event) 117

30. Average Kc ini as related to the level of ETo and the interval between irrigations

greater than or equal to 40 mm per wetting event, during the initial growth stage

for: a) coarse textured soils; b) medium and fine textured soils 118

31. Partial wetting by irrigation120

32. Adjustment (additive) to the Kc mid values from Table 12 for different crop

heights and mean daily wind speeds (u2) for different humidity conditions 122

33. Ranges expected for Kc end 126

34. Crop coefficient curve 126

35. Constructed curve for Kc for alfalfa hay in southern Idaho, the United States using

values from Tables 11 and 12 and adjusted using Equations 62 and 65 128

36. Kc curve and ten-day values for Kc and ETc derived from the graph for the dry

bean crop example (Box 15) 132

37. Constructed basal crop coefficient (Kcb) curve for a dry bean crop (Example 28)

using growth stage lengths of 25, 25, 30 and 20 days 142

38. Soil evaporation reduction coefficient, Kr 145

39. Determination of variable few as a function of the fraction of ground surface

coverage (fc) and the fraction of the surface wetted (fw) 148

40. Water balance of the topsoil layer 152

41. Depletion factor for different levels of crop evapotranspiration 166

42. Water stress coefficient, Ks 167

43. Water balance of the root zone 169

44. The effect of soil salinity on the water stress coefficient Ks 181

45. Different situations of intercropping 198

46. Kc curves for small areas of vegetation under the oasis effect as a function of the

width of the expanse of vegetation for conditions where RHmin = 30%, u2 =

2 m/s, vegetation height (h) = 2 m and LAI = 3 203

47. Mean evapotranspiration during non-growing, winter periods at Kimberly, Idaho,

measured using precision weighing lysimeters 210

xiv

List of tables

Page

1. Conversion factors for evapotranspiration 4

2. Average ETo for different agroclimatic regions in mm/day 8

3. Conversion factors for radiation 45

4. General classes of monthly wind speed data 63

5. Pan coefficients (Kp) for Class A pan for different pan siting and environment and

different levels of mean relative humidity and wind speed 81

6. Pan coefficients (Kp) for Colorado sunken pan for different pan siting and

environment and different levels of mean relative humidity and wind speed 81

7. Pan coefficients (Kp): regression equations derived from Tables 5 and 6 82

8. Ratios between the evaporation from sunken pans and a Colorado sunken pan for

different climatic conditions and environments 83

9. Approximate values for Kc ini for medium wetting events (10-40 mm) and a

medium textured soil 95

10. General selection criteria for the single and dual crop coefficient approaches 98

11. Lengths of crop development stages for various planting periods and climatic

regions 104

12. Single (time-averaged crop coefficients) Kc and mean maximum plant heights for

non-stressed, well-managed crops in sub-humid climates (RHmin ≈ 45%, u2 ≈

2 m/s) for use with the FAO Penman-Monteith ETo 110

13. Classification of rainfall depths 115

14. Kc ini for rice for various climatic conditions 121

15. Empirical estimates of monthly wind speed data 124

16. Typical values for RHmin compared with RHmean for general climatic

classifications 124

17. Basal crop coefficient Kcb for non-stressed, well-managed crops in sub-humid

climates (RHmin ≈ 45%, u2 ≈ 2 m/s) for use with the FAO Penman-Monteith

ETo 137

18. General guidelines to derive Kcb from the Kc values listed in Table 12 141

19. Typical soil water characteristics for different soil types 144

20. Common values of fraction fw of soil surface wetted by irrigation or precipitation 149

21. Common values of fractions covered by vegetation (fc) and exposed to sunlight (1-

fc) 149

22. Ranges of maximum effective rooting depth (Zr), and soil water depletion fraction

for no stress (p), for common crops 163

23. Salt tolerance of common agricultural crops as a function of the electrical

conductivity of the soil saturation extract at the threshold when crop yield first

reduces below the full yield potential (ECe, threshold) and when crop yields

becomes zero (ECe, no yield). source: FAO Irrigation and Drainage Paper No.

33 178

24. Seasonal yield response functions from FAO Irrigation and Drainage Paper No. 33 181

25. Approximate reductions in Kc and surface evaporation, and increases in

transpiration for various horticultural crops under plastic mulch as compared

with no mulch using trickle irrigation 196

Crop evapotranspiration

xv

List of boxes

1. Chapters concerning the calculation of the reference crop evapotranspiration (ETo) 8

2. Chapters concerning the calculation of crop evapotranspiration under standard

conditions (ETc) 8

3. Chapters concerning the calculation of crop evapotranspiration under non-standard

conditions (ETc adj) 10

4. The aerodynamic resistance for a grass reference surface 21

5. The (bulk) surface resistance for a grass reference crop 22

6. Derivation of the FAO Penman-Monteith equation for the hypothetical grass

reference crop 26

7. Calculation sheet for vapour pressure deficit (es - ea) 40

8. Conversion from energy values to equivalent evaporation 44

9. Calculation sheet for extraterrestrial radiation (Ra) and daylight hours (N) 49

10. Calculation sheet for net radiation (Rn) 53

11. Calculation sheet for ETo (FAO Penman-Monteith equation) 67

12. Description of Class A pan 84

13. Description of Colorado sunken pan 85

14. Demonstration of effect of climate on Kc mid for tomato crop grown in field 123

15. Case study of a dry bean crop at Kimberly, Idaho, the United States (single crop

coefficient) 130

16. Case study of dry bean crop at Kimberly, Idaho, the United States (dual crop

coefficient) 158

17. Measuring and estimating LAI 186

18. Measuring and estimating fc eff 187

xvi

List of examples

Page

1. Converting evaporation from one unit to another 4

2. Determination of atmospheric parameters 32

3. Determination of saturation vapour pressure 36

4. Determination of actual vapour pressure from psychrometric readings 38

5. Determination of actual vapour pressure from relative humidity 39

6. Determination of vapour pressure deficit 39

7. Conversion of latitude in degrees and minutes to radians 46

8. Determination of extraterrestrial radiation 47

9. Determination of daylight hours 49

10. Determination of solar radiation from measured duration of sunshine 50

11. Determination of net longwave radiation 52

12. Determination of net radiation 53

13. Determination of soil heat flux for monthly periods 55

14. Adjusting wind speed data to standard height 56

15. Determination of solar radiation from temperature data 61

16. Determination of net radiation in the absence of radiation data 62

17. Determination of ETo with mean monthly data 70

18. Determination of ETo with daily data 72

19. Determination of ETo with hourly data 75

20. Determination of ETo with missing data 77

21. Determination of ETo from pan evaporation using tables 83

22. Determination of ETo from pan evaporation using equations 86

23. Estimation of interval between wetting events 116

24. Graphical determination of Kc ini 116

25. Interpolation between light and heavy wetting events 119

26. Determination of Kc ini for partial wetting of the soil surface 120

27. Determination of Kc mid 125

28. Numerical determination of Kc 133

29. Selection and adjustment of basal crop coefficients, Kcb 136

30. Determination of daily values for Kcb 141

Crop evapotranspiration

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31. Determination of the evapotranspiration from a bare soil 146

32. Calculation of the crop coefficient (Kcb + Ke) under sprinkler irrigation 150

33. Calculation of the crop coefficient (Kcb + Ke) under furrow irrigation 151

34. Calculation of the crop coefficient (Kcb + Ke) under drip irrigation 151

35. Estimation of crop evapotranspiration with the dual crop coefficient approach 154

36. Determination of readily available soil water for various crops and soil types 166

37. Effect of water stress on crop evapotranspiration 168

38. Irrigation scheduling to avoid crop water stress 172

39. Effect of soil salinity on crop evapotranspiration 182

40. First approximation of the crop coefficient for the mid-season stage for sparse

vegetation 185

41. Estimation of mid-season crop coefficient 190

42. Estimation of mid-season crop coefficient for reduced ground cover 191

43. Estimation of Kcb mid from ground cover with reduction for stomatal control 192

44. approximate estimation of Ks from crop yield data 194

45. Effects of surface mulch 197

46. Intercropped maize and beans 200

47. Overlapping vegetation 201

xviii

List of equations

Page

1. Energy balance equation 11

2. Soil water balance 12

3. Penman-Monteith form of the combination equation 19

4. Aerodynamic resistance (ra) 20

5. (Bulk) surface resistance (rs) 21

6. FAO Penman-Monteith equation for daily, ten-day and monthly time steps 24

7. Atmospheric pressure (P) 31

8. Psychrometric constant (γ) 32

9. Mean air temperature (Tmean) 33

10. Relative humidity (RH) 35

11. Saturation vapour pressure as a function of temperature (e°(T)) 36

12. Saturation vapour pressure (es) 36

13. Slope e°(T) curve (∆) 37

14. Actual vapour pressure derived from dewpoint temperature (ea) 37

15. Actual vapour pressure derived from psychrometric data (ea) 37

16. Psychrometric constant of the (psychrometric) instrument (γpsy) 37

17. Actual vapour pressure derived from RHmax and RHmin (ea) 38

18. Actual vapour pressure derived from RHmax (ea) 39

19. Actual vapour pressure derived from RHmean (ea) 39

20. Conversion form energy to equivalent evaporation 44

21. Extraterrestrial radiation for daily periods (Ra) 46

22. Conversion from decimal degrees to radians 46

23. Inverse relative distance Earth-Sun (dr) 46

24. Solar declination (δ) 46

25. Sunset hour angle - arccos function (ωs) 46

26. Sunset hour angle - arctan function (ωs) 47

27. Parameter X of Equation 26 47

28. Extraterrestrial radiation for hourly or shorter periods (Ra) 47

29. Solar time angle at the beginning of the period (ω1) 48

Crop evapotranspiration

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30. Solar time angle at the end of the period (ω2) 48

31. Solar time angle at midpoint of the period (ω) 48

32. Seasonal correction for solar time (Sc) 48

33. Parameter b of Equation 32 48

34. Daylight hours (N) 48

35. Solar radiation (Rs) 50

36. Clear-sky radiation near sea level (Rso) 51

37. Clear-sky radiation at higher elevations (Rso) 51

38. Net solar or net shortwave radiation (Rns) 51

39. Net longwave radiation (Rnl) 52

40. Net radiation (Rn) 53

41. Soil heat flux (G) 54

42. Soil heat flux for day and ten-day periods (Gday) 54

43. Soil heat flux for monthly periods (Gmonth) 54

44. Soil heat flux for monthly periods if Tmonth,i+1 is unknown (Gmonth) 54

45. Soil heat flux for hourly or shorter periods during daytime (Ghr) 55

46. Soil heat flux for hourly or shorter periods during nighttime (Ghr) 55

47. Adjustment of wind speed to standard height (u2) 56

48. Estimating actual vapour pressure from Tmin (ea) 58

49. Importing solar radiation from a nearby weather station (Rs) 59

50. Estimating solar radiation from temperature differences (Hargreaves’ formula) 60

51. Estimating solar radiation for island locations (Rs) 62

52. 1985 Hargreaves reference evapotranspiration equation 64

53. FAO Penman-Monteith equation for hourly time step 74

54. Actual vapour pressure for hourly time step 74

55. Deriving ETo from pan evaporation 79

56. Crop evapotranspiration (ETc) 90

57. Dual crop coefficient 98

58. Crop evapotranspiration - single crop coefficient (ETc) 103

59. Interpolation for infiltration depths between 10 and 40 mm 117

60. Adjustment of Kc ini for partial wetting by irrigation 119

61. Irrigation depth for the part of the surface that is wetted (Iw) 119

62. Climatic adjustment for Kc mid 121

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63. Minimum relative humidity estimated from e°(Tdew) 123

64. Minimum relative humidity estimated from e°(Tmin) 124

65. Climatic adjustment for Kc end 125

66. Interpolation of Kc for crop development stage and late season stage 132

67. Relation between grass-based and alfalfa-based crop coefficients 133

68. Ratio between grass-based and alfalfa-based Kc for Kimberly, Idaho 134

69. Crop evapotranspiration - dual crop coefficient (ETc) 135

70. Climatic adjustment for Kcb 136

71. Soil evaporation coefficient (Ke) 142

72. Upper limit on evaporation and transpiration from any cropped surface (Kc max) 143

73. Maximum depth of water that can be evaporated from the topsoil (TEW) 144

74. Evaporation reduction coefficient (Kr) 146

75. Exposed and wetted soil fraction (few) 147

76. Effective fraction of soil surface that is covered by vegetation (fc) 149

77. Daily soil water balance for the exposed and wetted soil fraction 152

78. Limits on soil water depletion by evaporation (De) 153

79. Drainage out of topsoil (DPe) 156

80. Crop evapotranspiration adjusted for water stress - dual crop coefficient 161

81. Crop evapotranspiration adjusted for water stress - single crop coefficient 161

82. Total available soil water in the root zone (TAW) 162

83. Readily available soil water in the root zone (RAW) 162

84. Water stress coefficient (Ks) 169

85. Water balance of the root zone 170

86. Limits on root zone depletion by evapotranspiration (Dr) 170

87. Initial depletion (Dr,i-1) 170

88. Deep percolation (DP) 171

89. Relative crop yield (Ya/Ym) determined from soil salinity (ECe) and crop salinity

threshold

90. Yield response to water function (FAO Irrigation and Drainage Paper No. 33) 176

91. Water stress coefficient (Ks) under saline conditions 177

92. Water stress coefficient (Ks) under saline and water stress conditions 177

93. Soil salinity (ECe) predicted from leaching fraction (LF) and irrigation water

quality (ECiw) 180

94. Kc adj for reduced plant coverage 184

Crop evapotranspiration

xxi

95. Adjustment coefficient (from LAI) 185

96. Adjustment coefficient (from fc) 185

97. K(cb mid) adj from Leaf Area Index 186

98. K(cb mid) adj from effective ground cover 187

99. Kcb full for agricultural crops 189

100. Kcb full for natural vegetation 189

101. Kcb h for full cover vegetation 189

102. Adjustment for stomatal control (Fr) 191

103. Water stress coefficient (Ks) estimated from yield response to water function 194

104. Crop coefficient estimate for intercropped field (Kc field) 199

105. Crop coefficient estimate for windbreaks (Kc) 203

xxii

List of principal symbols and acronyms

apsy coefficient of psychrometer [°C-1]

as fraction of extraterrestrial radiation reaching the earth on an overcast day [-]

as+bs fraction of extraterrestrial radiation reaching the earth on a clear day [-]

cp specific heat [MJ kg-1 °C-1]

cs soil heat capacity [MJ m-3 °C-1]

CR capillary rise [mm day-1]

De cumulative depth of evaporation (depletion) from the soil surface layer [mm]

Dr cumulative depth of evapotranspiration (depletion) from the root zone [mm]

d zero plane displacement height [m]

dr inverse relative distance Earth-Sun [-]

DP deep percolation [mm]

DPe deep percolation from the evaporation layer [mm]

E evaporation [mm day-1]

Epan pan evaporation [mm day-1]

e°(T) saturation vapour pressure at air temperature T [kPa]

es saturation vapour pressure for a given time period [kPa]

ea actual vapour pressure [kPa]

es- ea saturation vapour pressure deficit

ECe electrical conductivity of the saturation extract of the soil [dS m-1]

ECe, threshold electrical conductivity of the saturation extract of the soil above which yield

begins to decrease [dS m-1]

ET evapotranspiration [mm day-1]

ETo reference crop evapotranspiration [mm day-1]

ETc crop evapotranspiration under standard conditions [mm day-1]

ETc adj crop evapotranspiration under non-standard conditions [mm day-1]

exp[x] 2.7183 (base of natural logarithm) raised to the power x

Fr resistance correction factor [-]

Crop evapotranspiration

xxiii

fc fraction of soil surface covered by vegetation (as observed from overhead) [-]

fc eff effective fraction of soil surface covered by vegetation [-]

1-fc exposed soil fraction [-]

fw fraction of soil surface wetted by rain or irrigation [-]

few fraction of soil that is both exposed and wetted (from which most evaporation

occurs) [-]

G soil heat flux [MJ m-2 day-1]

Gday soil heat flux for day and ten-day periods [MJ m-2 day-1]

Ghr soil heat flux for hourly or shorter periods [MJ m-2 hour-1]

Gmonth soil heat flux for monthly periods [MJ m-2 day-1]

Gsc solar constant [0.0820 MJ m-2 min-1]

H sensible heat [MJ m-2 day-1]

HWR height to width ratio

h crop height [m]

I irrigation depth [mm]

Iw irrigation depth for that part of the surface wetted [mm]

J number of day in the year [-]

Kc crop coefficient [-]

Kc ini crop coefficient during the initial growth stage [-]

Kc mid crop coefficient during the mid-season growth stage [-]

Kc end crop coefficient at end of the late season growth stage [-]

Kc max maximum value of crop coefficient (following rain or irrigation) [-]

Kc min minimum value of crop coefficient (dry soil with no ground cover) [-]

Kcb basal crop coefficient [-]

Kcb full basal crop coefficient during mid-season (at peak plant size or height) for

vegetation with full ground cover of LAI > 3 [-]

Kcb ini basal crop coefficient during the initial growth stage [-]

Kcb mid basal crop coefficient during the mid-season growth stage [-]

Kcb end basal crop coefficient at end of the late season growth stage [-]

Ke soil evaporation coefficient [-]

Kp pan coefficient [-]

Kr soil evaporation reduction coefficient [-]

xxiv

Ks water stress coefficient [-]

Ky yield response factor [-]

k von Karman's constant [0.41] [-]

kRs adjustment coefficient for the Hargreaves’ radiation formula [°C-0.5]

Lini length of initial growth stage [day]

Ldev length of crop development growth stage [day]

Lmid length of mid-season growth stage [day]

Llate length of late season growth stage [day]

Lz longitude of centre of local time zone [degrees west of Greenwich]

Lm longitude [degrees west of Greenwich]

LAI leaf area index [m2 (leaf area) m-2 (soil surface)]

LAIactive active (sunlit) leaf area index [-]

N maximum possible sunshine duration in a day, daylight hours [hour]

n actual duration of sunshine in a day [hour]

n/N relative sunshine duration [-]

P rainfall [mm], atmospheric pressure [kPa]

p evapotranspiration depletion factor [-]

R specific gas constant [0.287 kJ kg-1 K-1]

Ra extraterrestrial radiation [MJ m-2 day-1]

Rl longwave radiation [MJ m-2 day-1]

Rn net radiation [MJ m-2 day-1]

Rnl net longwave radiation [MJ m-2 day-1]

Rns net solar or shortwave radiation [MJ m-2 day-1]

Rs solar or shortwave radiation [MJ m-2 day-1]

Rso clear-sky solar or clear-sky shortwave radiation [MJ m-2 day-1]

ra aerodynamic resistance [s m-1]

rl bulk stomatal resistance of well-illuminated leaf [s m-1]

rs (bulk) surface or canopy resistance [s m-1]

Rs/Rso relative solar or relative shortwave radiation [-]

RAW readily available soil water of the root zone [mm]

Crop evapotranspiration

xxv

REW readily evaporable water (i.e., maximum depth of water that can be evaporated

from the soil surface layer without restriction during stage 1) [mm]

RH relative humidity [%]

RHhr average hourly relative humidity

RHmax daily maximum relative humidity [%]

RHmean daily mean relative humidity [%]

RHmin daily minimum relative humidity [%]

RO surface runoff [mm]

Sc seasonal correction factor for solar time [hour]

SF subsurface flow [mm]

T air temperature [°C]

TK air temperature [K]

TKv virtual air temperature [K]

Tdew dewpoint temperature [°C]

Tdry temperature of dry bulb [°C]

Tmax daily maximum air temperature [°C]

Tmax,K daily maximum air temperature [K]

Tmean daily mean air temperature [°C]

Tmin daily minimum air temperature [°C]

Tmin,K daily minimum air temperature [K]

Twet temperature of wet bulb [°C]

TAW total available soil water of the root zone [mm]

TEW total evaporable water (i.e., maximum depth of water that can be evaporated from

the soil surface layer)[mm]

t time [hour]

u2 wind speed at 2 m above ground surface [m s-1]

uz wind speed at z m above ground surface [m s-1]

W soil water content [mm]

Ya actual yield of the crop [kg ha-1]

Ym maximum (expected) yield of the crop in absence of environment or water stresses

[kg ha-1]

Ze depth of surface soil layer subjected to drying by evaporation [m]

xxvi

Zr rooting depth [m]

z elevation, height above sea level [m]

zh height of humidity measurements [m]

zm height of wind measurements [m]

zom roughness length governing momentum transfer [m]

zoh roughness length governing heat and vapour transfer [m]

α albedo [-]

γ psychrometric constant [kPa °C-1]

γpsy psychrometric constant of instrument [kPa °C-1]

∆ slope of saturation vapour pressure curve [kPa °C-1]

∆SW variation in soil water content [mm]

∆t length of time interval [day]

∆z effective soil depth [m]

δ solar declination [rad]

ε ratio molecular weight of water vapour/dry air (= 0.622)

η mean angle of the sun above the horizon

θ soil water content [m3(water) m-3(soil)]

θFC soil water content at field capacity [m3(water) m-3(soil)]

θt threshold soil water content below which transpiration is reduced due to water

stress [m3(water) m-3(soil)]

θWP soil water content at wilting point [m3(water) m-3(soil)]

λ latent heat of vaporization [MJ kg-1]

λET latent heat flux [MJ m-2 day-1]

ρa mean air density [kg m-3]

ρw density of water [kg m-3]

σ Stefan-Boltzmann constant [4.903 10-9 MJ K-4 m-2 day-1]

ϕ latitude [rad]

ω solar time angle at midpoint of hourly or shorter period [rad]

ω1 solar time angle at beginning of hourly or shorter period [rad]

ω2 solar time angle at end of hourly or shorter period [rad]

ωs sunset hour angle [rad]

Crop evapotranspiration

1

Chapter 1

Introduction to evapotranspiration

This chapter explains the concepts of and the differences between reference crop

evapotranspiration (ETo) and crop evapotranspiration under standard conditions (ETc) and

various management and environmental conditions (ETc adj). It also examines the factors that

affect evapotranspiration, the units in which it is normally expressed and the way in which it

can be determined.

EVAPOTRANSPIRATION PROCESS

The combination of two separate processes whereby water is lost on the one hand from the

soil surface by evaporation and on the other hand from the crop by transpiration is referred to

as evapotranspiration (ET).

Evaporation

Evaporation is the process whereby liquid water is converted to water vapour (vaporization)

and removed from the evaporating surface (vapour removal). Water evaporates from a variety

of surfaces, such as lakes, rivers, pavements, soils and wet vegetation.

Energy is required to change the state of the molecules of water from liquid to vapour. Direct

solar radiation and, to a lesser extent, the ambient temperature of the air provide this energy.

The driving force to remove water vapour from the evaporating surface is the difference

between the water vapour pressure at the evaporating surface and that of the surrounding

atmosphere. As evaporation proceeds, the surrounding air becomes gradually saturated and

the process will slow down and might stop if the wet air is not transferred to the atmosphere.

The replacement of the saturated air with drier air depends greatly on wind speed. Hence,

solar radiation, air temperature, air humidity and wind speed are climatological parameters to

consider when assessing the evaporation process.

Where the evaporating surface is the soil surface, the degree of shading of the crop

canopy and the amount of water available at the evaporating surface are other factors that

affect the evaporation process. Frequent rains, irrigation and water transported upwards in a

soil from a shallow water table wet the soil surface. Where the soil is able to supply water

fast enough to satisfy the evaporation demand, the evaporation from the soil is determined

only by the meteorological conditions. However, where the interval between rains and

irrigation becomes large and the ability of the soil to conduct moisture to near the surface is

small, the water content in the topsoil drops and the soil surface dries out. Under these

circumstances the limited availability of water exerts a controlling influence on soil

evaporation. In the absence of any supply of water to the soil surface, evaporation decreases

rapidly and may cease almost completely within a few days.

Introduction to evapotranspiration

2

FIGURE 1

Schematic representation of a stoma

water vapour

cuticula

epidermal

cells

mesophyll

cells

intercellular

space

Atmosphere

water

Leaf

FIGURE 2

The partitioning of evapotranspiration into evaporation and transpiration over the growing period

for an annual field crop

0%

20%

40%

60%

80%

100%

time

partitioning of evapotranspiration

leaf area index (LAI)

evaporation

transpiration

sowing harvest

crop

soil

L

A

I

Crop evapotranspiration

3

Transpiration

Transpiration consists of the vaporization of liquid water contained in plant tissues and the

vapour removal to the atmosphere. Crops predominately lose their water through stomata.

These are small openings on the plant leaf through which gases and water vapour pass

(Figure 1). The water, together with some nutrients, is taken up by the roots and transported

through the plant. The vaporization occurs within the leaf, namely in the intercellular spaces,

and the vapour exchange with the atmosphere is controlled by the stomatal aperture. Nearly

all water taken up is lost by transpiration and only a tiny fraction is used within the plant.

Transpiration, like direct evaporation, depends on the energy supply, vapour pressure

gradient and wind. Hence, radiation, air temperature, air humidity and wind terms should be

considered when assessing transpiration. The soil water content and the ability of the soil to

conduct water to the roots also determine the transpiration rate, as do waterlogging and soil

water salinity. The transpiration rate is also influenced by crop characteristics, environmental

aspects and cultivation practices. Different kinds of plants may have different transpiration

rates. Not only the type of crop, but also the crop development, environment and

management should be considered when assessing transpiration.

Evapotranspiration (ET)

Evaporation and transpiration occur simultaneously and there is no easy way of distinguishing

between the two processes. Apart from the water availability in the topsoil, the evaporation

from a cropped soil is mainly determined by the fraction of the solar radiation reaching the

soil surface. This fraction decreases over the growing period as the crop develops and the

crop canopy shades more and more of the ground area. When the crop is small, water is

predominately lost by soil evaporation, but once the crop is well developed and completely

covers the soil, transpiration becomes the main process. In Figure 2 the partitioning of

evapotranspiration into evaporation and transpiration is plotted in correspondence to leaf area

per unit surface of soil below it. At sowing nearly 100% of ET comes from evaporation,

while at full crop cover more than 90% of ET comes from transpiration.

UNITS

The evapotranspiration rate is normally expressed in millimetres (mm) per unit time. The rate

expresses the amount of water lost from a cropped surface in units of water depth. The time

unit can be an hour, day, decade, month or even an entire growing period or year.

As one hectare has a surface of 10 000 m2 and 1 mm is equal to 0.001 m, a loss of 1

mm of water corresponds to a loss of 10 m3 of water per hectare. In other words, 1 mm day-1

is equivalent to 10 m3 ha-1 day-1.

Water depths can also be expressed in terms of energy received per unit area. The

energy refers to the energy or heat required to vaporize free water. This energy, known as

the latent heat of vaporization (λ), is a function of the water temperature. For example, at

20°C, λ is about 2.45 MJ kg-1. In other words, 2.45 MJ are needed to vaporize 1 kg or 0.001

m3 of water. Hence, an energy input of 2.45 MJ per m2 is able to vaporize 0.001 m or 1 mm

of water, and therefore 1 mm of water is equivalent to 2.45 MJ m-2. The evapotranspiration

rate expressed in units of MJ m-2 day-1 is represented by λET, the latent heat flux.

Introduction to evapotranspiration

4

Table 1 summarizes the units used to express the evapotranspiration rate and the

conversion factors.

TABLE 1

Conversion factors for evapotranspiration

depth volume per unit area energy per unit area *

mm day-1 m

3 ha-1 day-1 l s-1 ha-1 MJ m-2 day-1

1 mm day-1 1 10 0.116 2.45

1 m3 ha-1 day-1 0.1 1 0.012 0.245

1 l s-1 ha-1 8.640 86.40 1 21.17

1 MJ m-2 day-1 0.408 4.082 0.047 1

* For water with a density of 1 000 kg m-3 and at 20°C.

EXAMPLE 1

Converting evaporation from one unit to another

On a summer day, net solar energy received at a lake reaches 15 MJ per square metre per day. If 80%

of the energy is used to vaporize water, how large could the depth of evaporation be?

From Table 1:

Therefore:

1 MJ m-2 day-1 =

0.8 x 15 MJ m-2 day-1 = 0.8 x 15 x 0.408 mm d-1 =

0.408

4.9

mm day-1

The evaporation rate could be 4.9 mm/day

FIGURE 3

Factors affecting evapotranspiration with reference to related ET concepts

crop

characteristics

weather

parameters

ET

T

E

ET

o

ET

c

ET

environmental

factors

management

c adj

Crop evapotranspiration

5

FACTORS AFFECTING EVAPOTRANSPIRATION

Weather parameters, crop characteristics, management and environmental aspects are factors

affecting evaporation and transpiration. The related ET concepts presented in Figure 3 are

discussed in the section on evapotranspiration concepts.

Weather parameters

The principal weather parameters affecting evapotranspiration are radiation, air temperature,

humidity and wind speed. Several procedures have been developed to assess the evaporation

rate from these parameters. The evaporation power of the atmosphere is expressed by the

reference crop evapotranspiration (ETo). The reference crop evapotranspiration represents the

evapotranspiration from a standardized vegetated surface. The ETo is described in detail later

in this Chapter and in Chapters 2 and 4.

Crop factors

The crop type, variety and development stage should be considered when assessing the

evapotranspiration from crops grown in large, well-managed fields. Differences in resistance

to transpiration, crop height, crop roughness, reflection, ground cover and crop rooting

characteristics result in different ET levels in different types of crops under identical

environmental conditions. Crop evapotranspiration under standard conditions (ETc) refers to

the evaporating demand from crops that are grown in large fields under optimum soil water,

excellent management and environmental conditions, and achieve full production under the

given climatic conditions.

Management and environmental conditions

Factors such as soil salinity, poor land fertility, limited application of fertilizers, the presence

of hard or impenetrable soil horizons, the absence of control of diseases and pests and poor

soil management may limit the crop development and reduce the evapotranspiration. Other

factors to be considered when assessing ET are ground cover, plant density and the soil water

content. The effect of soil water content on ET is conditioned primarily by the magnitude of

the water deficit and the type of soil. On the other hand, too much water will result in

waterlogging which might damage the root and limit root water uptake by inhibiting

respiration.

When assessing the ET rate, additional consideration should be given to the range of

management practices that act on the climatic and crop factors affecting the ET process.

Cultivation practices and the type of irrigation method can alter the microclimate, affect the

crop characteristics or affect the wetting of the soil and crop surface. A windbreak reduces

wind velocities and decreases the ET rate of the field directly beyond the barrier. The effect

can be significant especially in windy, warm and dry conditions although evapotranspiration

from the trees themselves may offset any reduction in the field. Soil evaporation in a young

orchard, where trees are widely spaced, can be reduced by using a well-designed drip or

trickle irrigation system. The drippers apply water directly to the soil near trees, thereby

leaving the major part of the soil surface dry, and limiting the evaporation losses. The use of

mulches, especially when the crop is small, is another way of substantially reducing soil

evaporation. Anti-transpirants, such as stomata-closing, film-forming or reflecting material,

reduce the water losses from the crop and hence the transpiration rate.

Introduction to evapotranspiration

6

FIGURE 4

Reference (ETo), crop evapotranspiration under standard (ETc) and non-standard conditions (ETc adj)

x=

ET

K

water & environmental

stress

scadjusted c adj

o

x

Radiation

Temperature

Wind speed

Humidity

climate

+=

x=

well watered

grass

reference

crop ET

o

ET

o

ETc

Kcfactor

grass

crop

conditions

optimal agronomic

Crop evapotranspiration

7

Where field conditions differ from the standard conditions, correction factors are

required to adjust ETc. The adjustment reflects the effect on crop evapotranspiration of the

environmental and management conditions in the field.

EVAPOTRANSPIRATION CONCEPTS

Distinctions are made (Figure 4) between reference crop evapotranspiration (ETo), crop

evapotranspiration under standard conditions (ETc) and crop evapotranspiration under non-

standard conditions (ETc adj). ETo is a climatic parameter expressing the evaporation power of

the atmosphere. ETc refers to the evapotranspiration from excellently managed, large, well-

watered fields that achieve full production under the given climatic conditions. Due to sub-

optimal crop management and environmental constraints that affect crop growth and limit

evapotranspiration, ETc under non-standard conditions generally requires a correction.

Reference crop evapotranspiration (ETo)

The evapotranspiration rate from a reference surface, not short of water, is called the

reference crop evapotranspiration or reference evapotranspiration and is denoted as ETo. The

reference surface is a hypothetical grass reference crop with specific characteristics. The use

of other denominations such as potential ET is strongly discouraged due to ambiguities in

their definitions.

The concept of the reference evapotranspiration was introduced to study the

evaporative demand of the atmosphere independently of crop type, crop development and

management practices. As water is abundantly available at the reference evapotranspiring

surface, soil factors do not affect ET. Relating ET to a specific surface provides a reference

to which ET from other surfaces can be related. It obviates the need to define a separate ET

level for each crop and stage of growth. ETo values measured or calculated at different

locations or in different seasons are comparable as they refer to the ET from the same

reference surface.

The only factors affecting ETo are climatic parameters. Consequently, ETo is a climatic

parameter and can be computed from weather data. ETo expresses the evaporating power of

the atmosphere at a specific location and time of the year and does not consider the crop

characteristics and soil factors. The FAO Penman-Monteith method is recommended as the

sole method for determining ETo. The method has been selected because it closely

approximates grass ETo at the location evaluated, is physically based, and explicitly

incorporates both physiological and aerodynamic parameters. Moreover, procedures have

been developed for estimating missing climatic parameters.

Typical ranges for ETo values for different agroclimatic regions are given in Table 2.

These values are intended to familiarize inexperienced users with typical ranges, and are not

intended for direct application. The calculation of the reference crop evapotranspiration is

discussed in Part A of this handbook (Box 1).

Crop evapotranspiration under standard conditions (ETc)

The crop evapotranspiration under standard conditions, denoted as ETc, is the

evapotranspiration from disease-free, well-fertilized crops, grown in large fields, under

optimum soil water conditions, and achieving full production under the given climatic

conditions.

Introduction to evapotranspiration

8

TABLE 2

Average ETo for different agroclimatic regions in mm/day

Regions Mean daily temperature (°C)

Cool

~ 10°C

Moderate

20°C

Warm

> 30°C

Tropics and subtropics

- humid and sub-humid

- arid and semi-arid

2 - 3

2 - 4

3 - 5

4 - 6

5 - 7

6 - 8

Temperate region

- humid and sub-humid

- arid and semi-arid

1 - 2

1 - 3

2 - 4

4 - 7

6 - 9

BOX 1

Chapters concerning the calculation of the reference crop evapotranspiration (ETo)

PART A -----

Chapter 2 - FAO Penman-Monteith equation:

This chapter introduces the user to the need to standardize one method to compute ETo from

meteorological data. The FAO Penman-Monteith method is recommended as the method for

determining reference ETo. The method and the corresponding definition of the reference surface are

described.

Chapter 3 - Meteorological data:

The FAO Penman-Monteith method requires radiation, air temperature, air humidity and wind speed

data. Calculation procedures to derive climatic parameters from the meteorological data are presented.

Procedures to estimate missing meteorological variables required for calculating ETo are outlined. This

allows for estimation of ETo with the FAO Penman-Monteith method under all circumstances, even in

the case of missing climatic data.

Chapter 4 - Determination of ETo:

The calculation of ETo by means of the FAO Penman-Monteith equation, with different time steps, from

the principal weather parameters and with missing data is described. The determination of ETo from

pan evaporation is also presented.

BOX 2

Chapters concerning the calculation of crop evapotranspiration under standard conditions (ETc)

PART B -----

Chapter 5 - Introduction to crop evapotranspiration:

This chapter introduces the user to the 'Kc ETo' approach for calculating crop evapotranspiration. The

effects of characteristics that distinguish field crops from the reference grass crop are integrated into

the crop coefficient Kc. Depending on the purpose of the calculation, the required accuracy, the

available climatic data and the time step with which the calculations have to be executed, a distinction

is made between two calculation methods.

Chapter 6 - ETc - Single crop coefficient (Kc):

This chapter presents the first calculation method for crop evapotranspiration whereby the difference in

evapotranspiration between the cropped and reference grass surface is combined into a single crop

coefficient (Kc).

Chapter 7 - ETc - Dual crop coefficient (Kc = Kcb + Ke):

This chapter presents the other calculation method for crop evapotranspiration. Kc is split into two

separate coefficients, one for crop transpiration (i.e., the basal crop coefficient Kcb) and one for soil

evaporation (Ke).

Crop evapotranspiration

9

The amount of water required to compensate the evapotranspiration loss from the

cropped field is defined as crop water requirement. Although the values for crop

evapotranspiration and crop water requirement are identical, crop water requirement refers to

the amount of water that needs to be supplied, while crop evapotranspiration refers to the

amount of water that is lost through evapotranspiration. The irrigation water requirement

basically represents the difference between the crop water requirement and effective

precipitation. The irrigation water requirement also includes additional water for leaching of

salts and to compensate for non-uniformity of water application. Calculation of the irrigation

water requirement is not covered in this publication, but will be the topic of a future Irrigation

and Drainage Paper.

Crop evapotranspiration can be calculated from climatic data and by integrating directly

the crop resistance, albedo and air resistance factors in the Penman-Monteith approach. As

there is still a considerable lack of information for different crops, the Penman-Monteith

method is used for the estimation of the standard reference crop to determine its

evapotranspiration rate, i.e., ETo. Experimentally determined ratios of ETc/ETo, called crop

coefficients (Kc), are used to relate ETc to ETo or ETc = Kc ETo.

Differences in leaf anatomy, stomatal characteristics, aerodynamic properties and even

albedo cause the crop evapotranspiration to differ from the reference crop evapotranspiration

under the same climatic conditions. Due to variations in the crop characteristics throughout its

growing season, Kc for a given crop changes from sowing till harvest. The calculation of crop

evapotranspiration under standard conditions (ETc) is discussed in Part B of this handbook

(Box 2).

Crop evapotranspiration under non-standard conditions (ETc adj)

The crop evapotranspiration under non-standard conditions (ETc adj) is the evapotranspiration

from crops grown under management and environmental conditions that differ from the

standard conditions. When cultivating crops in fields, the real crop evapotranspiration may

deviate from ETc due to non-optimal conditions such as the presence of pests and diseases,

soil salinity, low soil fertility, water shortage or waterlogging. This may result in scanty plant

growth, low plant density and may reduce the evapotranspiration rate below ETc.

The crop evapotranspiration under non-standard conditions is calculated by using a

water stress coefficient Ks and/or by adjusting Kc for all kinds of other stresses and

environmental constraints on crop evapotranspiration. The adjustment to ETc for water stress,

management and environmental constraints is discussed in Part C of this handbook (Box 3).

DETERMINING EVAPOTRANSPIRATION

ET measurement

Evapotranspiration is not easy to measure. Specific devices and accurate measurements of

various physical parameters or the soil water balance in lysimeters are required to determine

evapotranspiration. The methods are often expensive, demanding in terms of accuracy of

measurement and can only be fully exploited by well-trained research personnel. Although the

methods are inappropriate for routine measurements, they remain important for the evaluation

of ET estimates obtained by more indirect methods.

Introduction to evapotranspiration

10

BOX 3

Chapters concerning the calculation of crop evapotranspiration under non-standard conditions

(ETc adj)

PART C -----

Chapter 8 - ETc under soil water stress conditions:

This chapter discusses the reduction in transpiration induced by soil moisture stress or soil water

salinity. The resulting evapotranspiration will deviate from the crop evapotranspiration under standard

conditions. The evapotranspiration is computed by using a water stress coefficient, Ks, describing the

effect of water stress on crop transpiration.

Chapter 9 - ETc for natural, non-typical and non-pristine vegetation:

Procedures that can be used to make adjustments to the Kc to account for less than perfect growing

conditions or stand characteristics are discussed. The procedures can also be used to determine Kc for

agricultural crops not listed in the tables of Part B.

Chapter 10 - ETc under various management practices:

This chapter discusses various types of management practices that may cause the values for Kc and

ETc to deviate from the standard conditions described in Part B. Adjustment procedures for Kc to

account for surface mulches, intercropping, small areas of vegetation and management induced stress

are presented.

Chapter 11 - ETc during non-growing periods:

This chapter describes procedures for predicting ETc during non-growing periods under various types

of surface conditions.

FIGURE 5

Schematic presentation of the diurnal variation of the components of the energy balance above a

well-watered transpiring surface on a cloudless day

04812162024

time (hour)

energy arriving/leaving the surface

Rn

ET

H

G

Crop evapotranspiration

11

Energy balance and microclimatological methods

Evaporation of water requires relatively large amounts of energy, either in the form of

sensible heat or radiant energy. Therefore the evapotranspiration process is governed by

energy exchange at the vegetation surface and is limited by the amount of energy available.

Because of this limitation, it is possible to predict the evapotranspiration rate by applying the

principle of energy conservation. The energy arriving at the surface must equal the energy

leaving the surface for the same time period.

All fluxes of energy should be considered when deriving an energy balance equation.

The equation for an evaporating surface can be written as:

R

n − G - λET − H = 0 (1)

where Rn is the net radiation, H the sensible heat, G the soil heat flux and λET the latent heat

flux. The various terms can be either positive or negative. Positive Rn supplies energy to the

surface and positive G, λET and H remove energy from the surface (Figure 5).

In Equation 1 only vertical fluxes are considered and the net rate at which energy is

being transferred horizontally, by advection, is ignored. Therefore the equation is to be

applied to large, extensive surfaces of homogeneous vegetation only. The equation is

restricted to the four components: Rn, λET, H and G. Other energy terms, such as heat stored

or released in the plant, or the energy used in metabolic activities, are not considered These

terms account for only a small fraction of the daily net radiation and can be considered

negligible when compared with the other four components.

The latent heat flux (λET) representing the evapotranspiration fraction can be derived

from the energy balance equation if all other components are known. Net radiation (Rn) and

soil heat fluxes (G) can be measured or estimated from climatic parameters. Measurements of

the sensible heat (H) are however complex and cannot be easily obtained. H requires accurate

measurement of temperature gradients above the surface.

Another method of estimating evapotranspiration is the mass transfer method. This

approach considers the vertical movement of small parcels of air (eddies) above a large

homogeneous surface. The eddies transport material (water vapour) and energy (heat,

momentum) from and towards the evaporating surface. By assuming steady state conditions

and that the eddy transfer coefficients for water vapour are proportional to those for heat and

momentum, the evapotranspiration rate can be computed from the vertical gradients of air

temperature and water vapour via the Bowen ratio. Other direct measurement methods use

gradients of wind speed and water vapour. These methods and other methods such as eddy

covariance, require accurate measurement of vapour pressure, and air temperature or wind

speed at different levels above the surface. Therefore, their application is restricted to

primarily research situations.

Soil water balance

Evapotranspiration can also be determined by measuring the various components of the soil

water balance. The method consists of assessing the incoming and outgoing water flux into

the crop root zone over some time period (Figure 6). Irrigation (I) and rainfall (P) add water

to the root zone. Part of I and P might be lost by surface runoff (RO) and by deep percolation

Introduction to evapotranspiration

12

(DP) that will eventually recharge the water table. Water might also be transported upward by

capillary rise (CR) from a shallow water table towards the root zone or even transferred

horizontally by subsurface flow in (SFin) or out of (SFout) the root zone. In many situations,

however, except under condititions with large slopes, SFin and SFout are minor and can be

ignored. Soil evaporation and crop transpiration deplete water from the root zone. If all fluxes

other than evapotranspiration (ET) can be assessed, the evapotranspiration can be deduced

from the change in soil water content (∆SW) over the time period:

ET = I + P − RO − DP + CR ± ∆SF ± ∆SW (2)

Some fluxes such as subsurface flow, deep percolation and capillary rise from a water table

are difficult to assess and short time periods cannot be considered. The soil water balance

method can usually only give ET estimates over long time periods of the order of week-long

or ten-day periods.

FIGURE 6

Soil water balance of the root zone

evaporation

transpiration irrigation

rainfall

root zone

deep

percolation

capillary

rise

in

out

flowflow

runoff

subsurface

Lysimeters

By isolating the crop root zone from its environment and controlling the processes that are

difficult to measure, the different terms in the soil water balance equation can be determined

with greater accuracy. This is done in lysimeters where the crop grows in isolated tanks filled

with either disturbed or undisturbed soil. In precision weighing lysimeters, where the water

loss is directly measured by the change of mass, evapotranspiration can be obtained with an

accuracy of a few hundredths of a millimetre, and small time periods such as an hour can be

Crop evapotranspiration

13

considered. In non-weighing lysimeters the evapotranspiration for a given time period is

determined by deducting the drainage water, collected at the bottom of the lysimeters, from

the total water input.

A requirement of lysimeters is that the vegetation both inside and immediately outside

of the lysimeter be perfectly matched (same height and leaf area index). This requirement has

historically not been closely adhered to in a majority of lysimeter studies and has resulted in

severely erroneous and unrepresentative ETc and Kc data.

As lysimeters are difficult and expensive to construct and as their operation and

maintenance require special care, their use is limited to specific research purposes.

ET computed from meteorological data

Owing to the difficulty of obtaining accurate field measurements, ET is commonly computed

from weather data. A large number of empirical or semi-empirical equations have been

developed for assessing crop or reference crop evapotranspiration from meteorological data.

Some of the methods are only valid under specific climatic and agronomic conditions and

cannot be applied under conditions different from those under which they were originally

developed.

Numerous researchers have analysed the performance of the various calculation

methods for different locations. As a result of an Expert Consultation held in May 1990, the

FAO Penman-Monteith method is now recommended as the standard method for the

definition and computation of the reference evapotranspiration, ETo. The ET from crop

surfaces under standard conditions is determined by crop coefficients (Kc) that relate ETc to

ETo. The ET from crop surfaces under non-standard conditions is adjusted by a water stress

coefficient (Ks) and/or by modifying the crop coefficient.

ET estimated from pan evaporation

Evaporation from an open water surface provides an index of the integrated effect of

radiation, air temperature, air humidity and wind on evapotranspiration. However,

differences in the water and cropped surface produce significant differences in the water loss

from an open water surface and the crop. The pan has proved its practical value and has been

used successfully to estimate reference evapotranspiration by observing the evaporation loss

from a water surface and applying empirical coefficients to relate pan evaporation to ETo.

The procedure is outlined in Chapter 3.

Introduction to evapotranspiration

14

Crop evapotranspiration

15

Part A

Reference evapotranspiration (ETo)

Part A deals with the evapotranspiration from the reference surface, the so-called reference crop

evapotranspiration or reference evapotranspiration, denoted as ETo. The reference surface is a

hypothetical grass reference crop with an assumed crop height of 0.12 m, a fixed surface resistance of

70 s m-1 and an albedo of 0.23. The reference surface closely resembles an extensive surface of green,

well-watered grass of uniform height, actively growing and completely shading the ground. The fixed

surface resistance of 70 s m-1 implies a moderately dry soil surface resulting from about a weekly

irrigation frequency.

ETo can be computed from meteorological data. As a result of an Expert Consultation held in

May 1990, the FAO Penman-Monteith method is now recommended as the sole standard method for

the definition and computation of the reference evapotranspiration. The FAO Penman-Monteith

method requires radiation, air temperature, air humidity and wind speed data. Calculation procedures

to derive climatic parameters from meteorological data and to estimate missing meteorological

variables required for calculating ETo are presented in this Part (Chapter 3). The calculation

procedures in this Publication allow for estimation of ETo with the FAO Penman-Monteith method

under all circumstances, even in the case of missing climatic data.

ETo can also be estimated from pan evaporation. Pans have proved their practical value and

have been used successfully to estimate ETo by observing the water loss from the pan and using

empirical coefficients to relate pan evaporation to ETo. However, special precautions and

management must be applied.

16

Crop evapotranspiration

17

Chapter 2

FAO Penman-Monteith equation

This chapter introduces the user to the need to standardize one method to compute reference

evapotranspiration (ETo) from meteorological data. The FAO Penman-Monteith method is

recommended as the sole ETo method for determining reference evapotranspiration. The

method, its derivation, the required meteorological data and the corresponding definition of

the reference surface are described in this chapter.

NEED FOR A STANDARD ETO METHOD

A large number of more or less empirical methods have been developed over the last 50 years

by numerous scientists and specialists worldwide to estimate evapotranspiration from different

climatic variables. Relationships were often subject to rigorous local calibrations and proved

to have limited global validity. Testing the accuracy of the methods under a new set of

conditions is laborious, time-consuming and costly, and yet evapotranspiration data are

frequently needed at short notice for project planning or irrigation scheduling design. To meet

this need, guidelines were developed and published in the FAO Irrigation and Drainage Paper

No. 24 'Crop water requirements'. To accommodate users with different data availability,

four methods were presented to calculate the reference crop evapotranspiration (ETo): the

Blaney-Criddle, radiation, modified Penman and pan evaporation methods. The modified

Penman method was considered to offer the best results with minimum possible error in

relation to a living grass reference crop. It was expected that the pan method would give

acceptable estimates, depending on the location of the pan. The radiation method was

suggested for areas where available climatic data include measured air temperature and

sunshine, cloudiness or radiation, but not measured wind speed and air humidity. Finally, the

publication proposed the use of the Blaney-Criddle method for areas where available climatic

data cover air temperature data only.

These climatic methods to calculate ETo were all calibrated for ten-day or monthly

calculations, not for daily or hourly calculations. The Blaney-Criddle method was

recommended for periods of one month or longer. For the pan method it was suggested that

calculations should be done for periods of ten days or longer. Users have not always

respected these conditions and calculations have often been done on daily time steps.

Advances in research and the more accurate assessment of crop water use have revealed

weaknesses in the methodologies. Numerous researchers analysed the performance of the

four methods for different locations. Although the results of such analyses could have been

influenced by site or measurement conditions or by bias in weather data collection, it became

evident that the proposed methods do not behave the same way in different locations around

the world. Deviations from computed to observed values were often found to exceed ranges

indicated by FAO. The modified Penman was frequently found to overestimate ETo, even by

FAO Penman-Monteith equation

18

up to 20% for low evaporative conditions. The other FAO recommended equations showed

variable adherence to the reference crop evapotranspiration standard of grass.

To evaluate the performance of these and other estimation procedures under different

climatological conditions, a major study was undertaken under the auspices of the Committee

on Irrigation Water Requirements of the American Society of Civil Engineers (ASCE). The

ASCE study analysed the performance of 20 different methods, using detailed procedures to

assess the validity of the methods compared to a set of carefully screened lysimeter data from

11 locations with variable climatic conditions. The study proved very revealing and showed

the widely varying performance of the methods under different climatic conditions. In a

parallel study commissioned by the European Community, a consortium of European research

institutes evaluated the performance of various evapotranspiration methods using data from

different lysimeter studies in Europe.

The studies confirm the overestimation of the modified Penman introduced in FAO

Irrigation and Drainage Paper No. 24, and the variable performance of the different methods

depending on their adaptation to local conditions. The comparative studies may be

summarized as follows:

• The Penman methods may require local calibration of the wind function to achieve

satisfactory results.

• The radiation methods show good results in humid climates where the aerodynamic term

is relatively small, but performance in arid conditions is erratic and tends to

underestimate evapotranspiration.

• Temperature methods remain empirical and require local calibration in order to achieve

satisfactory results. A possible exception is the 1985 Hargreaves’ method which has

shown reasonable ETo results with a global validity.

• Pan evapotranspiration methods clearly reflect the shortcomings of predicting crop

evapotranspiration from open water evaporation. The methods are susceptible to the

microclimatic conditions under which the pans are operating and the rigour of station

maintenance. Their performance proves erratic.

• The relatively accurate and consistent performance of the Penman-Monteith approach in

both arid and humid climates has been indicated in both the ASCE and European studies.

The analysis of the performance of the various calculation methods reveals the need for

formulating a standard method for the computation of ETo. The FAO Penman-Monteith

method is recommended as the sole standard method. It is a method with strong likelihood of

correctly predicting ETo in a wide range of locations and climates and has provision for

application in data-short situations. The use of older FAO or other reference ET methods is

no longer encouraged.

FORMULATION OF THE PENMAN-MONTEITH EQUATION

Penman-Monteith equation

In 1948, Penman combined the energy balance with the mass transfer method and derived an

equation to compute the evaporation from an open water surface from standard climatological

records of sunshine, temperature, humidity and wind speed. This so-called combination

Crop evapotranspiration

19

method was further developed by many researchers and extended to cropped surfaces by

introducing resistance factors.

The resistance nomenclature distinguishes between aerodynamic resistance and surface

resistance factors (Figure 7). The surface resistance parameters are often combined into one

parameter, the ‘bulk’ surface resistance parameter which operates in series with the

aerodynamic resistance. The surface resistance, rs, describes the resistance of vapour flow

through stomata openings, total leaf area and soil surface. The aerodynamic resistance, ra,

describes the resistance from the vegetation upward and involves friction from air flowing

over vegetative surfaces. Although the exchange process in a vegetation layer is too complex

to be fully described by the two resistance factors, good correlations can be obtained between

measured and calculated evapotranspiration rates, especially for a uniform grass reference

surface.

The Penman-Monteith form of the combination equation is:

÷

ø

ö

ç

è

æ+γ+∆

−

ρ+−∆

=λ

a

s

a

as

pan

r

1

r

)e(e

cG)(R

ET (3)

where Rn is the net radiation, G is the soil heat flux, (es - ea) represents the vapour pressure

deficit of the air, ρa is the mean air density at constant pressure, cp is the specific heat of the

air, ∆ represents the slope of the saturation vapour pressure temperature relationship, γ is the

FIGURE 7

Simplified representation of the (bulk) surface and aerodynamic resistances for water vapour flow

soil

stomatal

air flow

r

s

a

(bulk) surface

resistance

aerodynamic

resistance

reference

level

evaporating

surface

cuticular

FAO Penman-Monteith equation

20

psychrometric constant, and rs and ra are the (bulk) surface and aerodynamic resistances. The

parameters of the equation are defined in Chapter 3.

The Penman-Monteith approach as formulated above includes all parameters that govern

energy exchange and corresponding latent heat flux (evapotranspiration) from uniform

expanses of vegetation. Most of the parameters are measured or can be readily calculated

from weather data. The equation can be utilized for the direct calculation of any crop

evapotranspiration as the surface and aerodynamic resistances are crop specific.

Aerodynamic resistance (ra)

The transfer of heat and water vapour from the evaporating surface into the air above the

canopy is determined by the aerodynamic resistance:

z

2

oh

h

om

m

au

k

z

dz

ln

z

dz

ln

r

ú

û

ù

ê

ë

é−

ú

û

ù

ê

ë

é−

= (4)

where ra aerodynamic resistance [s m-1],

zm height of wind measurements [m],

zh height of humidity measurements [m],

d zero plane displacement height [m],

zom roughness length governing momentum transfer [m],

zoh roughness length governing transfer of heat and vapour [m],

k von Karman's constant, 0.41 [-],

uz wind speed at height z [m s-1].

The equation is restricted for neutral stability conditions, i.e., where temperature,

atmospheric pressure, and wind velocity distributions follow nearly adiabatic conditions (no

heat exchange). The application of the equation for short time periods (hourly or less) may

require the inclusion of corrections for stability. However, when predicting ETo in the well-

watered reference surface, heat exchanged is small, and therefore stability correction is

normally not required.

Many studies have explored the nature of the wind regime in plant canopies. Zero

displacement heights and roughness lengths have to be considered when the surface is

covered by vegetation. The factors depend upon the crop height and architecture. Several

empirical equations for the estimate of d, zom and zoh have been developed. The derivation of

the aerodynamic resistance for the grass reference surface is presented in Box 4.

(Bulk) surface resistance (rs)

The ‘bulk’ surface resistance describes the resistance of vapour flow through the transpiring

crop and evaporating soil surface. Where the vegetation does not completely cover the soil,

the resistance factor should indeed include the effects of the evaporation from the soil surface.

If the crop is not transpiring at a potential rate, the resistance depends also on the water status

of the vegetation. An acceptable approximation to a much more complex relation of the

surface resistance of dense full cover vegetation is:

Crop evapotranspiration

21

BOX 4

The aerodynamic resistance for a grass reference surface

For a wide range of crops the zero plane displacement height, d [m], and the roughness length

governing momentum transfer, zom [m], can be estimated from the crop height h [m] by the

following equations:

d = 2/3 h

zom = 0.123 h

The roughness length governing transfer of heat and vapour, zoh [m], can be approximated by:

zoh = 0.1 zom

Assuming a constant crop height of 0.12 m and a standardized height for wind speed,

temperature and humidity at 2 m (zm = zh = 2 m), the aerodynamic resistance ra [s m-1] for the

grass reference surface becomes (Eq. 4):

2

au

208

u(0.41)

(0.12)0.123(0.1)

(0.12)2/3-2

ln

(0.12)0.123

(0.12)2/3-2

ln

r=

ú

û

ù

ê

ë

é

ú

û

ù

ê

ë

é

=

where u2 is the wind speed [m s-1] at 2 m.

active

1

sLAI

r

r= (5)

where rs (bulk) surface resistance [s m-1],

rl bulk stomatal resistance of the well-illuminated leaf [s m-1],

LAIactive active (sunlit) leaf area index [m2 (leaf area) m-2 (soil

surface)].

The Leaf Area Index (LAI), a dimensionless quantity, is the leaf area (upper side only)

per unit area of soil below it. It is expressed as m2 leaf area per m2 ground area. The active

LAI is the index of the leaf area that actively contributes to the surface heat and vapour

transfer. It is generally the upper, sunlit portion of a dense canopy. The LAI values for

various crops differ widely but values of 3-5 are common for many mature crops. For a given

crop, green LAI changes throughout the season and normally reaches its maximum before or

at flowering (Figure 8). LAI further depends on the plant density and the crop variety.

The bulk stomatal resistance, rl, is the average resistance of an individual leaf. This

resistance is crop specific and differs among crop varieties and crop management. It usually

increases as the crop ages and begins to ripen. There is, however, a lack of consolidated

information on changes in rl over time for the different crops. The information available in

the literature on stomatal conductance or resistance is often oriented toward physiological or

ecophysiological studies.

The stomatal resistance, rl, is influenced by climate and by water availability. However,

influences vary from one crop to another and different varieties can be affected differently.

The resistance increases when the crop is water stressed and the soil water availability limits

FAO Penman-Monteith equation

22

crop evapotranspiration. Some studies indicate that stomatal resistance is influenced to some

extent by radiation intensity, temperature, and vapour pressure deficit. The derivation of the

surface resistance for the grass reference surface is presented in Box 5.

BOX 5

The (bulk) surface resistance for a grass reference crop

A general equation for LAIactive is:

LAIactive = 0.5 LAI

which takes into consideration the fact that generally only the upper half of dense clipped grass is

actively contributing to the surface heat and vapour transfer. For clipped grass a general equation

for LAI is:

LAI = 24 h

where h is the crop height [m].

The stomatal resistance, rl, of a single leaf has a value of about 100 s m-1 under well-watered

conditions. By assuming a crop height of 0.12 m, the surface resistance, rs [s m-1], for the grass

reference surface becomes (Eq. 5):

1

sms70

(0.12)24)(0.5

100

r−

≈=

FIGURE 8

Typical presentation of the variation in the active (green) Leaf Area Index over the growing

season for a maize crop

Harvest

LAI

Sowing

0

1

2

3

4

5

0 20 40 60 80 100 120 140

time after sow ing (days)

green Leaf Area Index (m

2

/m

2

)

Crop evapotranspiration

23

REFERENCE SURFACE

To obviate the need to define unique evaporation parameters for each crop and stage of

growth, the concept of a reference surface was introduced. Evapotranspiration rates of the

various crops are related to the evapotranspiration rate from the reference surface (ETo) by

means of crop coefficients.

In the past, an open water surface has been proposed as a reference surface. However,

the differences in aerodynamic, vegetation control and radiation characteristics present a

strong challenge in relating ET to measurements of free water evaporation. Relating ETo to a

specific crop has the advantage of incorporating the biological and physical processes

involved in ET from cropped surfaces.

Grass, together with alfalfa, is a well-studied crop regarding its aerodynamic and surface

characteristics and is accepted worldwide as a reference surface. Because the resistance to

diffusion of vapour strongly depends on crop height, ground cover, LAI and soil moisture

conditions, the characteristics of the reference crop should be well defined and fixed.

Changes in crop height result in variations in the roughness and LAI. Consequently, the

associated canopy and aerodynamic resistances will vary appreciably with time. Moreover,

water stress and the degree of ground cover have an effect on the resistances and also on the

albedo.

To avoid problems of local calibration which would require demanding and expensive

studies, a hypothetical grass reference has been selected. Difficulties with a living grass

reference result from the fact that the grass variety and morphology can significantly affect

the evapotranspiration rate, especially during peak water use. Large differences may exist

between warm-season and cool-season grass types. Cool-season grasses have a lower degree

of stomatal control and hence higher rates of evapotranspiration. It may be difficult to grow

cool-season grasses in some arid, tropical climates.

The FAO Expert Consultation on Revision of FAO Methodologies for Crop Water

Requirements accepted the following unambiguous definition for the reference surface:

"A hypothetical reference crop with an assumed crop height of 0.12 m, a fixed

surface resistance of 70 s m-1 and an albedo of 0.23."

The reference surface closely resembles an extensive surface of green grass of uniform

height, actively growing, completely shading the ground and with adequate water. The

requirements that the grass surface should be extensive and uniform result from the

assumption that all fluxes are one-dimensional upwards.

The FAO Penman-Monteith method is selected as the method by which the

evapotranspiration of this reference surface (ETo) can be unambiguously determined, and as

the method which provides consistent ETo values in all regions and climates.

FAO PENMAN-MONTEITH EQUATION

Equation

A consultation of experts and researchers was organized by FAO in May 1990, in

collaboration with the International Commission for Irrigation and Drainage and with the

World Meteorological Organization, to review the FAO methodologies on crop water

requirements and to advise on the revision and update of procedures.

FAO Penman-Monteith equation

24

The panel of experts recommended the adoption of the Penman-Monteith combination

method as a new standard for reference evapotranspiration and advised on procedures for

calculation of the various parameters. By defining the reference crop as a hypothetical crop

with an assumed height of 0.12 m having a surface resistance of 70 s m-1 and an albedo of

0.23, closely resembling the evaporation of an extension surface of green grass of uniform

height, actively growing and adequately watered, the FAO Penman-Monteith method was

developed. The method overcomes shortcomings of the previous FAO Penman method and

provides values more consistent with actual crop water use data worldwide.

From the original Penman-Monteith equation (Equation 3) and the equations of the

aerodynamic (Equation 4) and surface resistance (Equation 5), the FAO Penman-Monteith

method to estimate ETo can be derived (Box 6):

)u34.01(

)ee(u

273T

900

)GR(408.0

ET

2

as2n

o+γ+∆

−

+

γ+−∆

= (6)

where ETo reference evapotranspiration [mm day-1],

Rn net radiation at the crop surface [MJ m-2 day-1],

G soil heat flux density [MJ m-2 day-1],

T mean daily air temperature at 2 m height [°C],

u2 wind speed at 2 m height [m s-1],

es saturation vapour pressure [kPa],

ea actual vapour pressure [kPa],

es-ea saturation vapour pressure deficit [kPa],

∆ slope vapour pressure curve [kPa °C-1],

γ psychrometric constant [kPa °C-1].

FIGURE 9

Characteristics of the hypothetical reference crop

weather measurements

2 m

0.12 m

r =

a

208

us/m

r = 70 s/m

s

radiation

solar

αRs

s

2

d + zoh

= 0.23 Rs

R

h =

reference level

Crop evapotranspiration

25

The reference evapotranspiration, ETo, provides a standard to which:

• evapotranspiration at different periods of the year or in other regions can be compared;

• evapotranspiration of other crops can be related.

The equation uses standard climatological records of solar radiation (sunshine), air

temperature, humidity and wind speed. To ensure the integrity of computations, the weather

measurements should be made at 2 m (or converted to that height) above an extensive surface

of green grass, shading the ground and not short of water.

No weather-based evapotranspiration equation can be expected to predict

evapotranspiration perfectly under every climatic situation due to simplification in formulation

and errors in data measurement. It is probable that precision instruments under excellent

environmental and biological management conditions will show the FAO Penman-Monteith

equation to deviate at times from true measurements of grass ETo. However, the Expert

Consultation agreed to use the hypothetical reference definition of the FAO Penman-Monteith

equation as the definition for grass ETo when deriving and expressing crop coefficients.

It is important, when comparing the FAO Penman-Monteith equation to ETo

measurements, that the full Penman-Monteith equation (Equation 3) and associated equations

for ra and rs (Equations 4 and 5) be used to enable accounting for variation in ET due to

variation in height of the grass measured. Variations in measurement height can significantly

change LAI, d and zom and the corresponding ETo measurement and predicted value. When

evaluating results, it should be noted that local environmental and management factors, such

as watering frequency, also affect ETo observations.

The FAO Penman-Monteith equation is a close, simple representation of the physical and

physiological factors governing the evapotranspiration process. By using the FAO Penman-

Monteith definition for ETo, one may calculate crop coefficients at research sites by relating

the measured crop evapotranspiration (ETc) with the calculated ETo, i.e., Kc = ETc/ETo. In

the crop coefficient approach, differences in the crop canopy and aerodynamic resistance

relative to the hypothetical reference crop are accounted for within the crop coefficient. The

Kc factor serves as an aggregation of the physical and physiological differences between crops

and the reference definition.

Data

Apart from the site location, the FAO Penman-Monteith equation requires air temperature,

humidity, radiation and wind speed data for daily, weekly, ten-day or monthly calculations.

The computation of all data required for the calculation of the reference evapotranspiration is

given in Chapter 3. It is important to verify the units in which the weather data are reported.

Factors to convert common units to the standard unit are presented in Annex I.

Location

Altitude above sea level (m) and latitude (degrees north or south) of the location should be

specified. These data are needed to adjust some weather parameters for the local average

value of atmospheric pressure (a function of the site elevation above mean sea level) and to

compute extraterrestrial radiation (Ra) and, in some cases, daylight hours (N). In the

calculation procedures for Ra and N, the latitude is expressed in radian (i.e., decimal degrees

times π/180).

FAO Penman-Monteith equation

26

BOX 6

Derivation of the FAO Penman-Monteith equation for the hypothetical grass reference crop

With standardized height for wind speed, temperature and humidity measurements at 2 m

(zm = zh = 2 m) and the crop height h = 0.12 m, the aerodynamic and surface resistances

become (Boxes 4 & 5):

ra = 208/u2 s m-1, (with u2 wind speed at 2 m height)

rs = 70 s m-1

(1 + rs/ra) = (1 + 0.34 u2)

Rn and G is energy available per unit area and expressed in MJ m-2 day-1. To convert the energy

units for radiation to equivalent water depths (mm) the latent heat of vaporization, λ is used as a

conversion factor (Chapter 1). The conversion from energy values to equivalent depths of water or

vice versa is given by (Eq. 20):

]daymMJ[Radiation408.0

45.2

]daymMJ[Radiation

]daymm[Radiation 12

12

1−−

−−

−=≈

By substituting cp with a rearrangement of Eq. 8:

P

cp

λεγ

=

and considering the ideal gas law for ρa :

R

Kv

T

P

aρ=

where TKv, the virtual temperature, may be substituted by:

)273T(01.1TKv +=

results in:

]sCmMJ[u

)208(R)273T(01.1r

c11o2

2

a

ap −−−

+

λεγ

=

ρ

where cp specific heat at constant pressure [MJ kg-1 °C-1],

ρa mean air density at constant pressure [kg m-3],

r

a aerodynamic resistance [s m-1],

γ psychrometric constant [kPa °C-1],

ε ratio molecular weight of water vapour/dry air = 0.622,

λ latent heat of vaporization [MJ kg-1],

u2 wind speed at 2 m [m s-1],

R specific gas constant = 0.287 kJ kg-1 K-1,

T air temperature [°C],

P atmospheric pressure [kPa],

]dayCmMJ[u

)208()287.0()273T(01.1

)622.0(

86400 11o2

2−−−

+

λγ

=

or, when divided by λ (λ = 2.45),

]dayCmm[u

273T

900 11o

2−−

+

γ≈

Crop evapotranspiration

27

A positive value is used for the northern hemisphere and a negative value for the southern

hemisphere.

Temperature

The (average) daily maximum and minimum air temperatures in degrees Celsius (°C) are

required. Where only (average) mean daily temperatures are available, the calculations can

still be executed but some underestimation of ETo will probably occur due to the non-linearity

of the saturation vapour pressure - temperature relationship (Figure 11). Using mean air

temperature instead of maximum and minimum air temperatures yields a lower saturation

vapour pressure es, and hence a lower vapour pressure difference (es - ea), and a lower

reference evapotranspiration estimate.

Humidity

The (average) daily actual vapour pressure, ea, in kilopascals (kPa) is required. The actual

vapour pressure, where not available, can be derived from maximum and minimum relative

humidity (%), psychrometric data (dry and wet bulb temperatures in °C) or dewpoint

temperature (°C) according to the procedures outlined in Chapter 3.

Radiation

The (average) daily net radiation expressed in megajoules per square metre per day (MJ m-2

day-1) is required. These data are not commonly available but can be derived from the

(average) shortwave radiation measured with a pyranometer or from the (average) daily actual

duration of bright sunshine (hours per day) measured with a (Campbell-Stokes) sunshine

recorder. The calculation procedures are outlined in Chapter 3.

Wind speed

The (average) daily wind speed in metres per second (m s-1) measured at 2 m above the

ground level is required. It is important to verify the height at which wind speed is measured,

as wind speeds measured at different heights above the soil surface differ. The calculation

procedure to adjust wind speed to the standard height of 2 m is presented in Chapter 3.

Missing climatic data

Situations might occur where data for some weather variables are missing. The use of an

alternative ETo calculation procedure, requiring only limited meteorological parameters,

should generally be avoided. It is recommended that one calculate ETo using the standard

FAO Penman-Monteith method after resolving the specific problem of the missing data.

Procedures for estimating missing climatic data are outlined in Chapter 3. Differences

between ETo values obtained with the FAO Penman-Monteith equation with, on the one hand,

a limited data set and, on the other hand, a full data set, are expected to be smaller than or of

similar magnitude to the differences resulting from the use of an alternative ETo equation.

Even where the data set contains only maximum and minimum air temperature it is still

possible to obtain reasonable estimates of ten-day or monthly ETo with the FAO Penman-

Monteith equation. As outlined in Chapter 3, radiation data can be derived from the air

temperature difference, or, along with wind speed and humidity data, can be imported from a

FAO Penman-Monteith equation

28

nearby weather station. Humidity data can also be estimated from daily minimum air

temperature. After evaluating the validity of the use of data from another station, ten-day or

monthly estimates of ETo can be calculated.

The procedures for estimating missing data should be validated at the regional level. This

can be done for weather stations with full data sets by comparing ETo calculated with full and

with limited data sets. The ratio should be close to one. Where the ratio deviates significantly

from one, the ratio can be used as a correction factor for estimates made with the limited data

set. Where the standard error of estimate exceeds 20% of the mean ETo, a sensitivity analysis

should be performed to determine causes (and limits) for the method utilized to import the

missing data. A validation should be completed for each month and variable, for the monthly

as well as for the daily estimates.

Crop evapotranspiration

29

Chapter 3

Meteorological data

The methods for calculating evapotranspiration from meteorological data require various

climatological and physical parameters. Some of the data are measured directly in weather

stations. Other parameters are related to commonly measured data and can be derived with the

help of a direct or empirical relationship. This chapter discusses the source, measurement and

computation of all data required for the calculation of the reference evapotranspiration by

means of the FAO Penman-Monteith method. Different examples illustrate the various

calculation procedures. Appropriate procedures for estimating missing data are also provided.

Meteorological data can be expressed in several units. Conversion factors between

various units and standard S.I. units are given in Annex 1. Climatic parameters, calculated by

means of the equations presented in this chapter are tabulated and displayed for different

meteorological conditions in Annex 2. Only the standardized relationships are presented in this

chapter. The background of certain relationships and more information about certain procedures

are given in Annex 3. Annexes 4, 5 and 6 list procedures for the statistical analysis, assessment,

correction and completion of partial or missing weather data.

METEOROLOGICAL FACTORS DETERMINING ET

The meteorological factors determining evapotranspiration are weather parameters which

provide energy for vaporization and remove water vapour from the evaporating surface. The

principal weather parameters to consider are presented below.

Solar radiation

The evapotranspiration process is determined by the amount of energy available to vaporize

water. Solar radiation is the largest energy source and is able to change large quantities of

liquid water into water vapour. The potential amount of radiation that can reach the evaporating

surface is determined by its location and time of the year. Due to differences in the position of

the sun, the potential radiation differs at various latitudes and in different seasons. The actual

solar radiation reaching the evaporating surface depends on the turbidity of the atmosphere and

the presence of clouds which reflect and absorb major parts of the radiation. When assessing

the effect of solar radiation on evapotranspiration, one should also bear in mind that not all

available energy is used to vaporize water. Part of the solar energy is used to heat up the

atmosphere and the soil profile.

Air temperature

The solar radiation absorbed by the atmosphere and the heat emitted by the earth increase the

air temperature. The sensible heat of the surrounding air transfers energy to the crop and exerts

as such a controlling influence on the rate of evapotranspiration. In sunny, warm weather the

loss of water by evapotranspiration is greater than in cloudy and cool weather.

Meteorological data

30

Air humidity

While the energy supply from the sun and surrounding air is the main driving force for the

vaporization of water, the difference between the water vapour pressure at the evapotranspiring

surface and the surrounding air is the determining factor for the vapour removal. Well-watered

fields in hot dry arid regions consume large amounts of water due to the abundance of energy

and the desiccating power of the atmosphere. In humid tropical regions, notwithstanding the

high energy input, the high humidity of the air will reduce the evapotranspiration demand. In

such an environment, the air is already close to saturation, so that less additional water can be

stored and hence the evapotranspiration rate is lower than in arid regions.

Wind speed

The process of vapour removal depends to a large extent on wind and air turbulence which

transfers large quantities of air over the evaporating surface. When vaporizing water, the air

above the evaporating surface becomes gradually saturated with water vapour. If this air is not

continuously replaced with drier air, the driving force for water vapour removal and the

evapotranspiration rate decreases.

The combined effect of climatic factors affecting evapotranspiration is illustrated in

Figure 10 for two different climatic conditions. The evapotranspiration demand is high in hot

dry weather due to the dryness of the air and the amount of energy available as direct solar

radiation and latent heat. Under these circumstances, much water vapour can be stored in the air

FIGURE 10

Illustration of the effect of wind speed on evapotranspiration in hot-dry and humid-warm

weather conditions

01.52.0

3.0

wind speed (m/s)

reference evapotranspiration

humid and warm

hot and dry

1.5 2.50.5

Crop evapotranspiration

31

while wind may promote the transport of water allowing more water vapour to be taken up. On

the other hand, under humid weather conditions, the high humidity of the air and the presence

of clouds cause the evapotranspiration rate to be lower. The effect on evapotranspiration of

increasing wind speeds for the two different climatic conditions is illustrated by the slope of the

curves in Figure 10. The drier the atmosphere, the larger the effect on ET and the greater the

slope of the curve. For humid conditions, the wind can only replace saturated air with slightly

less saturated air and remove heat energy. Consequently, the wind speed affects the

evapotranspiration rate to a far lesser extent than under arid conditions where small variations

in wind speed may result in larger variations in the evapotranspiration rate.

ATMOSPHERIC PARAMETERS

Several relationships are available to express climatic parameters. The effect of the principal

weather parameters on evapotranspiration can be assessed with the help of these equations.

Some of the relationships require parameters which express a specific characteristic of the

atmosphere. Before studying the four principal weather parameters, some atmospheric

parameters will be discussed.

Atmospheric pressure (P)

The atmospheric pressure, P, is the pressure exerted by the weight of the earth's atmosphere.

Evaporation at high altitudes is promoted due to low atmospheric pressure as expressed in the

psychrometric constant. The effect is, however, small and in the calculation procedures, the

average value for a location is sufficient. A simplification of the ideal gas law, assuming 20°C

for a standard atmosphere, can be employed to calculate P:

26.5

293

z0065.0293

3.101P ÷

ø

ö

ç

è

æ−

= (7)

where P atmospheric pressure [kPa],

z elevation above sea level [m],

Values for atmospheric pressure as a function of altitude are given in Annex 2 (Table 2.1).

Latent heat of vaporization (λ

λλ

λ)

The latent heat of vaporization, λ, expresses the energy required to change a unit mass of water

from liquid to water vapour in a constant pressure and constant temperature process. The value

of the latent heat varies as a function of temperature. At a high temperature, less energy will be

required than at lower temperatures. As λ varies only slightly over normal temperature ranges a

single value of 2.45 MJ kg-1 is taken in the simplification of the FAO Penman-Monteith

equation. This is the latent heat for an air temperature of about 20°C.

Psychrometric constant (γ

γγ

γ)

The psychrometric constant, γ, is given by:

Meteorological data

32

P10x665.0

Pc 3

p−

=

λε

=γ (8)

where γ psychrometric constant [kPa °C-1],

P atmospheric pressure [kPa],

λ latent heat of vaporization, 2.45 [MJ kg-1],

cp specific heat at constant pressure, 1.013 10-3 [MJ kg-1 °C-1],

ε ratio molecular weight of water vapour/dry air = 0.622.

The specific heat at constant pressure is the amount of energy required to increase the

temperature of a unit mass of air by one degree at constant pressure. Its value depends on the

composition of the air, i.e., on its humidity. For average atmospheric conditions a value cp =

1.013 10-3 MJ kg-1 °C-1 can be used. As an average atmospheric pressure is used for each

location (Equation 7), the psychrometric constant is kept constant for each location. Values for

the psychrometric constant as a function of altitude are given in Annex 2 (Table 2.2).

EXAMPLE 2

Determination of atmospheric parameters.

Determine the atmospheric pressure and the psychrometric constant at an elevation of 1 800 m.

With:

From Eq. 7:

From Eq. 8:

z =

P = 101.3 [(293 - (0.0065) 1800)/293]5.26 =

γ = 0.665 10-3 (81.8) =

1 800

81.8

0.054

m

kPa

kPa °C-1

The average value of the atmospheric pressure is 81.8 kPa.

The psychrometric constant, γ, is 0.054 kPa/°C.

AIR TEMPERATURE

Agrometeorology is concerned with the air temperature near the level of the crop canopy. In

traditional and modern automatic weather stations the air temperature is measured inside

shelters (Stevenson screens or ventilated radiation shields) placed in line with World

Meteorological Organization (WMO) standards at 2 m above the ground. The shelters are

designed to protect the instruments from direct exposure to solar heating. The louvered

construction allows free air movement around the instruments. Air temperature is measured

with thermometers, thermistors or thermocouples mounted in the shelter. Minimum and

maximum thermometers record the minimum and maximum air temperature over a 24-hour

period. Thermographs plot the instantaneous temperature over a day or week. Electronic

weather stations often sample air temperature each minute and report hourly averages in

addition to 24-hour maximum and minimum values.

Due to the non-linearity of humidity data required in the FAO Penman-Monteith

equation, the vapour pressure for a certain period should be computed as the mean between the

vapour pressure at the daily maximum and minimum air temperatures of that period. The daily

maximum air temperature (Tmax) and daily minimum air temperature (Tmin) are, respectively,

the maximum and minimum air temperature observed during the 24-hour period, beginning at

midnight. Tmax and Tmin for longer periods such as weeks, 10-day's or months are obtained by

dividing the sum of the respective daily values by the number of days in the period. The mean

Crop evapotranspiration

33

daily air temperature (Tmean) is only employed in the FAO Penman-Monteith equation to

calculate the slope of the saturation vapour pressure curves (∆) and the impact of mean air

density (Pa) as the effect of temperature variations on the value of the climatic parameter is

small in these cases. For standardization, Tmean for 24-hour periods is defined as the mean of

the daily maximum (Tmax) and minimum temperatures (Tmin) rather than as the average of

hourly temperature measurements.

2

TT

Tminmax

mean

+

= (9)

The temperature is given in degrees Celsius (°C) or Fahrenheit (°F). The conversion table

is given in Annex 1. In some calculation procedures, temperature is required in Kelvin (K),

which can be obtained by adding 273.16 to the temperature expressed in degrees Celsius (in

practice K = °C + 273.16). The Kelvin and Celsius scale have the same scale interval.

AIR HUMIDITY

Concepts

The water content of the air can be expressed in several ways. In agrometeorology, vapour

pressure, dewpoint temperature and relative humidity are common expressions to indicate air

humidity.

Vapour pressure

Water vapour is a gas and its pressure contributes to the total atmospheric pressure. The amount

of water in the air is related directly to the partial pressure exerted by the water vapour in the

air and is therefore a direct measure of the air water content.

In standard S.I. units, pressure is no longer expressed in centimetre of water, millimetre

of mercury, bars, atmosphere, etc., but in pascals (Pa). Conversion factors between various

units and Pa are given in Annex 1. As a pascal refers to a relatively small force (1 newton)

applied on a relatively large surface (1 m2), multiples of the basic unit are often used. In this

handbook, vapour pressure is expressed in kilopascals (kPa = 1 000 Pa).

When air is enclosed above an evaporating water surface, an equilibrium is reached

between the water molecules escaping and returning to the water reservoir. At that moment, the

air is said to be saturated since it cannot store any extra water molecules. The corresponding

pressure is called the saturation vapour pressure (eo(T) ). The number of water molecules that

can be stored in the air depends on the temperature (T). The higher the air temperature, the

higher the storage capacity, the higher its saturation vapour pressure (Figure 11).

As can be seen from Figure 11, the slope of the curve changes exponentially with

temperature. At low temperatures, the slope is small and varies only slightly as the temperature

rises. At high temperatures, the slope is large and small changes in T result in large changes in

slope. The slope of the saturation vapour pressure curve, ∆, is an important parameter in

describing vaporization and is required in the equations for calculating ETo from climatic data.

Meteorological data

34

FIGURE 11

Saturation vapour pressure shown as a function of temperature: e°(T) curve

0 5 10 15 20 25 30 35 40 45

temperature (°C)

0

2

4

6

8

10

e (T) saturation vapour pressure (kPa)

o

FIGURE 12

V

ariation of the relative humidity over 24 hours for a constant actual vapour pressure of 2.4

kPa

0 3 6 9 12 15 18 21 24

hour

20

25

30

35

temperature (°C)

40

60

80

100

RH relative humidity (%)

RH

Temperature

e = 2.4 kPa

a

Crop evapotranspiration

35

The actual vapour pressure (ea ) is the vapour pressure exerted by the water in the air.

When the air is not saturated, the actual vapour pressure will be lower than the saturation

vapour pressure. The difference between the saturation and actual vapour pressure is called the

vapour pressure deficit or saturation deficit and is an accurate indicator of the actual

evaporative capacity of the air.

Dewpoint temperature

The dewpoint temperature is the temperature to which the air needs to be cooled to make the air

saturated. The actual vapour pressure of the air is the saturation vapour pressure at the dewpoint

temperature. The drier the air, the larger the difference between the air temperature and

dewpoint temperature.

Relative humidity

The relative humidity (RH) expresses the degree of saturation of the air as a ratio of the actual

(ea) to the saturation (eo(T)) vapour pressure at the same temperature (T):

(T)

o

e

a

e

100RH = (10)

Relative humidity is the ratio between the amount of water the ambient air actually holds

and the amount it could hold at the same temperature. It is dimensionless and is commonly

given as a percentage. Although the actual vapour pressure might be relatively constant

throughout the day, the relative humidity fluctuates between a maximum near sunrise and a

minimum around early afternoon (Figure 12). The variation of the relative humidity is the result

of the fact that the saturation vapour pressure is determined by the air temperature. As the

temperature changes during the day, the relative humidity also changes substantially.

Measurement

It is not possible to directly measure the actual vapour pressure. The vapour pressure is

commonly derived from relative humidity or dewpoint temperature.

Relative humidity is measured directly with hygrometers. The measurement is based on

the nature of some material such as hair, which changes its length in response to changes in air

humidity, or using a capacitance plate, where the electric capacitance changes with RH. Vapour

pressure can be measured indirectly with psychrometers which measure the temperature

difference between two thermometers, the so-called dry and wet bulb thermometers. The dry

bulb thermometer measures the temperature of the air. The bulb of the wet bulb thermometer is

covered with a constantly saturated wick. Evaporation of water from the wick, requiring energy,

lowers the temperature of the thermometer. The drier the air, the larger the evaporative cooling

and the larger the temperature drop. The difference between the dry and wet bulb temperatures

is called the wet bulb depression and is a measure of the air humidity.

The dewpoint temperature is measured with dewpoint meters. The underlying principle

of some types of apparatus is the cooling of the ambient air until dew formation occurs. The

corresponding temperature is the dewpoint temperature.

Meteorological data

36

Relative humidity and dewpoint temperature data are notoriously plagued by

measurement errors. Measurement error is common for both older hygrothermograph types of

instruments and for the more modern electronic instruments. These instruments are described

in Annex 5. Great care should be made to assess the accuracy and integrity of RH and

dewpoint data. The user is encouraged to always compare computed dewpoint temperatures to

daily minimum air temperatures, as described at the end of this chapter and in Annexes 5 and 6.

Frequently, it is better to utilize a dewpoint temperature that is predicted from daily minimum

air temperature, rather than to use unreliable relative humidity measurements. The user is

encouraged to utilize good judgement in this area.

Calculation procedures

Mean saturation vapour pressure (es )

As saturation vapour pressure is related to air temperature, it can be calculated from the air

temperature. The relationship is expressed by:

ú

û

ù

ê

ë

é

+

=3.237T

T27.17

exp6108.0)T(eo (11)

where e°(T) saturation vapour pressure at the air temperature T [kPa],

T air temperature [°C],

exp[..] 2.7183 (base of natural logarithm) raised to the power [..].

Values of saturation vapour pressure as a function of air temperature are given in Annex

2 (Table 2.3). Due to the non-linearity of the above equation, the mean saturation vapour

pressure for a day, week, decade or month should be computed as the mean between the

saturation vapour pressure at the mean daily maximum and minimum air temperatures for that

period:

()()

2

TeTe

emin

o

max

o

s

+

= (12)

Using mean air temperature instead of daily minimum and maximum temperatures results

in lower estimates for the mean saturation vapour pressure. The corresponding vapour pressure

deficit (a parameter expressing the evaporating power of the atmosphere) will also be smaller

and the result will be some underestimation of the reference crop evapotranspiration. Therefore,

the mean saturation vapour pressure should be calculated as the mean between the saturation

vapour pressure at both the daily maximum and minimum air temperature.

EXAMPLE 3

Determination of mean saturation vapour pressure

The daily maximum and minimum air temperature are respectively 24.5 and 15°C.

Determine the saturation vapour pressure for that day.

From Eq. 11 e°(Tmax) = 0.6108 exp[17.27(24.5)/(24.5+237.3)] 3.075 kPa

From Eq. 11 e°(Tmin) = 0.6108 exp[17.27(15)/(15+237.3)] 1.705 kPa

From Eq. 12 es = (3.075 + 1.705)/2

Note that for temperature 19.75°C (which is Tmean), e°(T) =

2.39

2.30

kPa

The mean saturation vapour pressure is 2.39 kPa.

Crop evapotranspiration

37

Slope of saturation vapour pressure curve (

∆

∆∆

∆

)

For the calculation of evapotranspiration, the slope of the relationship between saturation

vapour pressure and temperature, ∆, is required. The slope of the curve (Figure 11) at a given

temperature is given by.

()

2

3.237T

T27.17

exp6108.04098

+

ú

û

ù

ê

ë

é

÷

ø

ö

ç

è

æ

+

=∆ (13)

where ∆ slope of saturation vapour pressure curve at air temperature T [kPa °C-1],

T air temperature [°C],

exp[..] 2.7183 (base of natural logarithm) raised to the power [..].

Values of slope ∆ for different air temperatures are given in Annex 2 (Table 2.4). In the

FAO Penman-Monteith equation, where ∆ occurs in the numerator and denominator, the slope

of the vapour pressure curve is calculated using mean air temperature (Equation 9).

Actual vapour pressure (ea ) derived from dewpoint temperature

As the dewpoint temperature is the temperature to which the air needs to be cooled to make the

air saturated, the actual vapour pressure (ea ) is the saturation vapour pressure at the dewpoint

temperature (Tdew) [ºC], or:

()

ú

û

ù

ê

ë

é

+

== 3.237T

T27.17

exp6108.0Tee

dew

o

a (14)

Actual vapour pressure (ea ) derived from psychrometric data

The actual vapour pressure can be determined from the difference between the dry and wet bulb

temperatures, the so-called wet bulb depression. The relationship is expressed by the following

equation:

()

wetdrypsywet

o

aTTTee −γ−= (15)

where ea actual vapour pressure [kPa],

e°(Twet) saturation vapour pressure at wet bulb temperature [kPa],

γpsy psychrometric constant of the instrument [kPa °C-1],

T

dry-Twet wet bulb depression, with Tdry the dry bulb and Twet the wet bulb

temperature [°C].

The psychrometric constant of the instrument is given by:

Pa psypsy =γ (16)

Meteorological data

38

where apsy is a coefficient depending on the type of ventilation of the wet bulb [°C-1], and P is

the atmospheric pressure [kPa]. The coefficient apsy depends mainly on the design of the

psychrometer and rate of ventilation around the wet bulb. The following values are used:

apsy = 0.000662 for ventilated (Asmann type) psychrometers, with an air movement of

some 5 m/s,

0.000800 for natural ventilated psychrometers (about 1 m/s),

0.001200 for non-ventilated psychrometers installed indoors.

EXAMPLE 4

Determination of actual vapour pressure from psychrometric readings

Determine the vapour pressure from the readings of an aspirated psychrometer in a location at an

elevation of 1 200 m. The temperatures measured by the dry and wet bulb thermometers are 25.6 and

19.5°C respectively.

From Eq. 7 (Table 2.1), at:

Then:

z=

P =

1 200

87.9

m

kPa

From Eq. 11 (Table 2.3), for

Then:

Twet =

e°(Twet) =

19.5

2.267

°C

kPa

Ventilated psychrometer apsy = 0.000662 °C-1

From Eq. 15:

ea = 2.267 - 0.000662 (87.9) (25.6 - 19.5)=

1.91

kPa

The actual vapour pressure is 1.91 kPa.

Actual vapour pressure (ea ) derived from relative humidity data

The actual vapour pressure can also be calculated from the relative humidity. Depending on the

availability of the humidity data, different equations should be used.

• For RHmax and RHmin:

() ()

2

100

RH

Te

100

RH

Te

e

min

max

o

max

min

o

a

+

= (17)

where ea actual vapour pressure [kPa],

e°(Tmin) saturation vapour pressure at daily minimum temperature [kPa],

e°(Tmax) saturation vapour pressure at daily maximum temperature [kPa],

RHmax maximum relative humidity [%],

RHmin minimum relative humidity [%].

For periods of a week, ten days or a month, RHmax and RHmin are obtained by dividing the

sum of the daily values by the number of days in that period.

• For RHmax:

When using equipment where errors in estimating RHmin can be large, or when RH data

integrity are in doubt, then one should use only RHmax:

Crop evapotranspiration

39

()

100

RH

Tee max

min

o

a= (18)

• For RHmean:

In the absence of RHmax and RHmin, another equation can be used to estimate ea:

()()

ú

û

ù

ê

ë

é+

=2

TeTe

100

RH

emin

o

max

o

mean

a (19)

where RHmean is the mean relative humidity, defined as the average between RHmax and

RHmin. However, Equation 19 is less desirable than are Equations 17 or 18.

EXAMPLE 5

Determination of actual vapour pressure from relative humidity

Given the following daily minimum and maximum air temperature and the corresponding relative

humidity data:

Tmin = 18°C and RHmax = 82%

Tmax = 25°C and RHmin = 54%

Determine the actual vapour pressure.

From Eq. 11 (Table 2.3), at:

Then:

Tmin =

e°(Tmin) =

18

2.064

°C

kPa

From Eq. 11 (Table 2.3), at:

Then:

Tmax =

e°(Tmax) =

25

3.168

°C

kPa

From Eq. 17:

Note that when using Eq. 19:

ea = [2.064 (82/100) + 3.168 (54/100)] =

ea =

1.70

1.78

kPa

Vapour pressure deficit (es - ea )

The vapour pressure deficit is the difference between the saturation (es) and actual vapour

pressure (ea) for a given time period. For time periods such as a week, ten days or a month es is

computed from Equation 12 using the Tmax and Tmin averaged over the time period and

similarly the ea is computed with one of the equations 4 to 19, using avrage measurements over

the period. As stated above, using mean air temperature and not Tmax and Tmin in Equation 12

results in a lower estimate of es, thus in a lower vapour pressure deficit and hence an

underestimation of the ETo (see Box 7). When desired, es and ea for long time periods cal also

be calculated as averages of values computed for each day of the period.

EXAMPLE 6

Determination of vapour pressure deficit

Determine the vapour pressure deficit with the data of the previous example (Example 5).

From Example 5: e°(Tmin) =

e°(Tmax) =

ea =

es - ea = (2.064 + 3.168)/2 - 1.70 =

2.064

3.168

1.70

0.91

kPa

The vapour pressure deficit is 0.91 kPa.

Meteorological data

40

BOX 7

Calculation sheet for vapour pressure deficit (es - ea)

Saturation vapour pressure: es (Eq. 11 or Table 2.3)

Tmax

°C

ú

û

ù

ê

ë

é

+

=3.237T

T27.17

exp6108.0)T(e

max

o

kPa

Tmin

°C

ú

û

ù

ê

ë

é

+

=3.237T

T27.17

exp6108.0)T(e

min

o

kPa

saturation vapour pressure es = [e°(Tmax)+e°(Tmin)]/2 Eq.12

kPa

Actual vapour pressure: ea

1. ea derived from dewpoint temperature (Eq. 14 or Table 2.3)

Tdew

°C

ú

û

ù

ê

ë

é

+

=3.237T

T27.17

exp6108.0e

dew

a

kPa

OR 2. ea derived from maximum and minimum relative humidity

RHmax

%

()

100

RH

Te max

min

o

kPa

RHmin

%

()

100

RH

Te min

max

o

kPa

ea = [e°(Tmin) RHmax/100 + e°(Tmax) RHmin/100] / 2 Eq. 17

kPa

OR 3. ea derived from maximum relative humidity (errors in RHmin)

RHmax

%

ea = e°(Tmin) RHmax/100 Eq. 18

kPa

OR 4. ea derived from mean relative humidity (less recommended)

RHmean

%

ea = es (RHmean)/100 Eq. 19

kPa

Vapour pressure deficit: (es - ea)

kPa

Crop evapotranspiration

41

RADIATION

Concepts

Extraterrestrial radiation (Ra )

The radiation striking a surface perpendicular to the sun's rays at the top of the earth's

atmosphere, called the solar constant, is about 0.082 MJ m-2 min-1. The local intensity of

radiation is, however, determined by the angle between the direction of the sun's rays and the

normal to the surface of the atmosphere. This angle will change during the day and will be

different at different latitudes and in different seasons. The solar radiation received at the top of

the earth's atmosphere on a horizontal surface is called the extraterrestrial (solar) radiation, Ra.

If the sun is directly overhead, the angle of incidence is zero and the extraterrestrial

radiation is 0.0820 MJ m-2 min-1. As seasons change, the position of the sun, the length of the

day and, hence, Ra change as well. Extraterrestrial radiation is thus a function of latitude, date

and time of day. Daily values of Ra throughout the year for different latitudes are plotted in

Figure 13.

Solar or shortwave radiation (Rs )

As the radiation penetrates the atmosphere, some of the radiation is scattered, reflected or

absorbed by the atmospheric gases, clouds and dust. The amount of radiation reaching a

horizontal plane is known as the solar radiation, Rs. Because the sun emits energy by means of

electromagnetic waves characterized by short wavelengths, solar radiation is also referred to as

shortwave radiation.

FIGURE 13

Annual variation in extraterrestrial radiation (Ra) at the equator, 20 and 40° north and south

0

10

20

30

40

50

R

a

(MJ m

-2

day

-1

)

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

40°S

20°S

20°N

40°N

Equator

Meteorological data

42

For a cloudless day, Rs is roughly 75% of extraterrestrial radiation. On a cloudy day, the

radiation is scattered in the atmosphere, but even with extremely dense cloud cover, about 25%

of the extraterrestrial radiation may still reach the earth's surface mainly as diffuse sky

radiation. Solar radiation is also known as global radiation, meaning that it is the sum of direct

shortwave radiation from the sun and diffuse sky radiation from all upward angles.

Relative shortwave radiation (Rs /Rso )

The relative shortwave radiation is the ratio of the solar radiation (Rs) to the clear-sky solar

radiation (Rso). Rs is the solar radiation that actually reaches the earth's surface in a given

period, while Rso is the solar radiation that would reach the same surface during the same

period but under cloudless conditions.

The relative shortwave radiation is a way to express the cloudiness of the atmosphere;

the cloudier the sky the smaller the ratio. The ratio varies between about 0.33 (dense cloud

cover) and 1 (clear sky). In the absence of a direct measurement of Rn, the relative shortwave

radiation is used in the computation of the net longwave radiation.

Relative sunshine duration (n/N)

The relative sunshine duration is another ratio that expresses the cloudiness of the atmosphere.

It is the ratio of the actual duration of sunshine, n, to the maximum possible duration of

sunshine or daylight hours N. In the absence of any clouds, the actual duration of sunshine is

equal to the daylight hours (n = N) and the ratio is one, while on cloudy days n and

consequently the ratio may be zero. In the absence of a direct measurement of Rs, the relative

sunshine duration, n/N, is often used to derive solar radiation from extraterrestrial radiation.

As with extraterrestrial radiation, the daylength N depends on the position of the sun and

is hence a function of latitude and date. Daily values of N throughout the year for different

latitudes are plotted in Figure 14.

FIGURE 14

Annual variation of the daylight hours (N) at the equator, 20 and 40° north and south

40 S

20 S

20 N

40 N

o

8

9

10

11

12

13

14

15

16

N daylight hours (hours/day)

Equator

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Crop evapotranspiration

43

Albedo (

α

αα

α

) and net solar radiation (Rns)

A considerable amount of solar radiation reaching the earth's surface is reflected. The fraction,

α, of the solar radiation reflected by the surface is known as the albedo. The albedo is highly

variable for different surfaces and for the angle of incidence or slope of the ground surface. It

may be as large as 0.95 for freshly fallen snow and as small as 0.05 for a wet bare soil. A green

vegetation cover has an albedo of about 0.20-0.25. For the green grass reference crop, α is

assumed to have a value of 0.23.

The net solar radiation, Rns, is the fraction of the solar radiation Rs that is not reflected

from the surface. Its value is (1-α)Rs.

Net longwave radiation (Rnl)

The solar radiation absorbed by the earth is converted to heat energy. By several processes,

including emission of radiation, the earth loses this energy. The earth, which is at a much lower

temperature than the sun, emits radiative energy with wavelengths longer than those from the

sun. Therefore, the terrestrial radiation is referred to as longwave radiation. The emitted

longwave radiation (Rl,up) is absorbed by the atmosphere or is lost into space. The longwave

radiation received by the atmosphere (Rl,down) increases its temperature and, as a

consequence, the atmosphere radiates energy of its own, as illustrated in Figure 15. Part of the

radiation finds it way back to the earth's surface. Consequently, the earth's surface both emits

and receives longwave radiation. The difference between outgoing and incoming longwave

radiation is called the net longwave radiation, Rnl. As the outgoing longwave radiation is

almost always greater than the incoming longwave radiation, Rnl represents an energy loss.

Net radiation (Rn)

The net radiation, Rn, is the difference between incoming and outgoing radiation of both short

and long wavelengths. It is the balance between the energy absorbed, reflected and emitted by

the earth's surface or the difference between the incoming net shortwave (Rns) and the net

outgoing longwave (Rnl) radiation (Figure 15). Rn is normally positive during the daytime and

negative during the nighttime. The total daily value for Rn is almost always positive over a

period of 24 hours, except in extreme conditions at high latitudes.

Soil heat flux (G)

In making estimates of evapotranspiration, all terms of the energy balance (Equation 1) should

be considered. The soil heat flux, G, is the energy that is utilized in heating the soil. G is

positive when the soil is warming and negative when the soil is cooling. Although the soil heat

flux is small compared to Rn and may often be ignored, the amount of energy gained or lost by

the soil in this process should theoretically be subtracted or added to Rn when estimating

evapotranspiration.

Units

The standard unit used in this handbook to express energy received on a unit surface per unit

time is megajoules per square metre per day (MJ m-2 day-1). In meteorological bulletins other

units might be used or radiation might even be expressed in units no longer accepted as

standard S.I. units, such as calories cm-2 day-1.

Meteorological data

44

In the FAO Penman-Monteith equation (Equation 6), radiation expressed in MJ m-2 day-1 is

converted (Box 8) to equivalent evaporation in mm day-1 by using a conversion factor equal to

the inverse of the latent heat heat of vaporization (1/λ = 0.408):

[

]

[

]

121 daymMJRadiationx408.0daymmnevaporatioequivalent −−− = (20)

BOX 8

Conversion from energy values to equivalent evaporation

The conversion from energy values to depths of water or vice versa is given by:

[]

w

surface/energyRadiation

waterofdepthRadiation ρλ

=

where λ latent heat of vaporization [MJ kg-1],

ρw density of water, i.e., 1 000 kg m-3,

[depth of water] is expressed in m,

[energy/surface] is expressed in MJ m-2.

By using a single value of 2.45 MJ kg-1 for λ (see section on atmospheric parameters and Annex 3)

and multiplying the above equation by 1 000 to obtain mm:

ú

û

ù

ê

ë

é

=

ú

û

ù

ê

ë

é

≈

ú

û

ù

ê

ë

é−−

−−

−12

12

1daymMJRadiationx408.0

45.2

daymMJRadiation

daymmRadiation

FIGURE 15

Various components of radiation

clouds

Earth

at

mos

ph

ere

R

s

R

s

R

a

R

l,down

l,up

R

ns

R

nl

α

Crop evapotranspiration

45

Common units used to express energy received on a unit surface per unit time, and

conversion factors are summarized in Table 3.

TABLE 3

Conversion factors for radiation

multiplier to obtain

energy received

on a unit surface per unit time

equivalent

evaporation

MJ

m-2 day-1

J

cm-2 day-1

cal

cm-2 day-1

W m-2 mm day-1

1 MJ m-2 day-1 1 100 23.9 11.6 0.408

1 cal cm-2 day-1 4.1868 10-2 4.1868 1 0.485 0.0171

1 W m-2 0.0864 8.64 2.06 1 0.035

1 mm day-1 2.45 245 58.5 28.4 1

Measurement

Solar radiation can be measured with pyranometers, radiometers or solarimeters. The

instruments contain a sensor installed on a horizontal surface that measures the intensity of the

total solar radiation, i.e., both direct and diffuse radiation from cloudy conditions. The sensor is

often protected and kept in a dry atmosphere by a glass dome that should be regularly wiped

clean.

Net longwave and net shortwave radiation can be measured by recording the difference

in output between sensors facing upward and downward. In a net radiometer, the glass domes

are replaced by polyethylene domes that have a transmission range for both shortwave and

longwave radiation.

Where pyranometers are not available, solar radiation is usually estimated from the

duration of bright sunshine. The actual duration of sunshine, n, is measured with a Campbell-

Stokes sunshine recorder. This instrument records periods of bright sunshine by using a glass

globe that acts as a lens. The sun rays are concentrated at a focal point that burns a hole in a

specially treated card mounted concentrically with the sphere. The movement of the sun

changes the focal point throughout the day and a trace is drawn on the card. If the sun is

obscured, the trace is interrupted. The hours of bright sunshine are indicated by the lengths of

the line segments.

The quantity of heat conducted into the soil, G, can be measured with systems of soil

heat flux plates and thermocouples or thermisters.

Calculation procedures

Extraterrestrial radiation for daily periods (Ra )

The extraterrestrial radiation, Ra, for each day of the year and for different latitudes can be

estimated from the solar constant, the solar declination and the time of the year by:

Meteorological data

46

[]

)sin()cos()cos()sin()sin(dG

)60(24

Rssrsca ωδϕ+δϕω

π

= (21)

where Ra extraterrestrial radiation [MJ m-2 day-1],

Gsc solar constant = 0.0820 MJ m-2 min-1,

dr inverse relative distance Earth-Sun (Equation 23),

ωs sunset hour angle (Equation 25 or 26) [rad],

ϕ latitude [rad] (Equation 22),

δ solar declination (Equation 24) [rad].

Ra is expressed in the above equation in MJ m-2 day-1. The corresponding equivalent

evaporation in mm day-1 is obtained by multiplying Ra by 0.408 (Equation 20). The latitude, ϕ,

expressed in radians is positive for the northern hemisphere and negative for the southern

hemisphere (Example 7). The conversion from decimal degrees to radians is given by:

[] []

degreesdecimalRadians 180

π

= (22)

EXAMPLE 7

Conversion of latitude in degrees and minutes to radians

Express the latitudes of Bangkok (Thailand) at 13°44'N and Rio de Janeiro (Brazil) at 22°54’S in

radians.

Latitude Bangkok

(northern hemisphere)

Rio de Janeiro

(southern hemisphere)

degrees & minutes

decimal degrees

radians

13°44'N

13 + 44/60 = 13.73

(π/180) 13.73 = + 0.240

22°54'S

(-22) + (-54/60) = - 22.90

(π/180) (-22.90) = - 0.400

The latitudes of Bangkok and Rio de Janeiro are respectively + 0.240 and - 0.400 radians.

The inverse relative distance Earth-Sun, dr, and the solar declination, δ, are given by:

÷

ø

ö

ç

è

æπ

+= J

365

2

cos033.01d r (23)

÷

ø

ö

ç

è

æ−

π

=δ 39.1J

365

2

sin409.0 (24)

where J is the number of the day in the year between 1 (1 January) and 365 or 366 (31

December). Values for J for all days of the year and an equation for estimating J are given in

Annex 2 (Table 2.5).

The sunset hour angle, ωs, is given by:

[]

)(tan)(tanarccos

sδϕ−=ω (25)

Crop evapotranspiration

47

As the arccos function is not available in all computer languages, the sunset hour angle

can also be computed using the arctan function:

ú

û

ù

ê

ë

éδϕ−

−

π

=ω 5.0

sX

)tan()tan(

arctan

2 (26)

where

[][]

22 )(tan)(tan1X δϕ−= (27)

and X if X=≤0 00001 0.

Values for Ra for different latitudes are given in Annex 2 (Table 2.6). These values

represent Ra on the 15th day of each month. These values deviate from values that are averaged

over each day of the month by less than 1% for all latitudes during non-frozen periods and are

included for simplicity of calculation. These values deviate slightly from the values in the

Smithsonian Tables. For the winter months in latitudes greater than 55° (N or S), the equations

for Ra have limited validity. Reference should be made to the Smithsonian Tables to assess

possible deviations.

EXAMPLE 8

Determination of extraterrestrial radiation

Determine the extraterrestrial radiation (Ra ) for 3 September at 20°S.

From Eq. 22

From Table 2.5:

20°S or ϕ = (π/180) (-20) =

(the value is negative for the southern hemisphere)

The number of day in the year, J =

- 0.35

246

rad

days

From Eq. 23 dr = 1 + 0.033 cos(2π(246)/365) = 0.985 rad

From Eq. 24 δ = 0.409 sin(2π(246)/365-1.39) = 0.120 rad

From Eq. 25:

Then:

and:

ωs = arccos[-tan(-0.35)tan(0.120)] =

sin(ϕ)sin(δ) =

cos(ϕ)cos(δ) =

1.527

-0.041

0.933

rad

-

From Eq. 21 Ra = 24(60)/π (0.0820)(0.985)[1.527

(-0.041)+0.933sin(1.527)] =

32.2

MJ m-2 d-1

From Eq. 20 expressed as equivalent evaporation = 0.408 (32.2)= 13.1 mm/day

The extraterrestrial radiation is 32.2 MJ m-2 day-1.

Extraterrestrial radiation for hourly or shorter periods (Ra )

For hourly or shorter periods the solar time angle at the beginning and end of the period should

be considered when calculating Ra:

[]

))sin()sin(()cos()cos()sin()sin()(dG

)60(12

R1212rsca ω−ωδϕ+δϕω−ω

π

= (28)

where Ra extraterrestrial radiation in the hour (or shorter) period [MJ m-2 hour-1],

Gsc solar constant = 0.0820 MJ m-2 min-1,

dr inverse relative distance Earth-Sun (Equation 23),

δ solar declination [rad] (Equation 24),

ϕ latitude [rad] (Equation 22),

ω1 solar time angle at beginning of period [rad] (Equation 29),

ω2 solar time angle at end of period [rad] (Equation 30).

Meteorological data

48

The solar time angles at the beginning and end of the period are given by:

24

t1

1

π

−ω=ω (29)

24

t1

2

π

+ω=ω (30)

where ω solar time angle at midpoint of hourly or shorter period [rad],

tl length of the calculation period [hour]: i.e., 1 for hourly period or 0.5 for a

30-minute period.

The solar time angle at midpoint of the period is:

[]

12)S)LL(06667.0t(

12 cmz −+−+

π

=ω (31)

where t standard clock time at the midpoint of the period [hour]. For example for a

period between 14.00 and 15.00 hours, t = 14.5,

Lz longitude of the centre of the local time zone [degrees west of Greenwich].

For example, Lz = 75, 90, 105 and 120° for the Eastern, Central, Rocky

Mountain and Pacific time zones (United States) and Lz = 0° for Greenwich,

330° for Cairo (Egypt), and 255° for Bangkok (Thailand),

Lm longitude of the measurement site [degrees west of Greenwich],

Sc seasonal correction for solar time [hour].

Of course, ω < -ωs or ω > ωs from Equation 31 indicates that the sun is below the

horizon so that, by definition, Ra is zero.

The seasonal correction for solar time is:

)bsin(025.0)bcos(1255.0)b2sin(1645.0Sc−−= (32)

bJ

=−281

364

π

()

(33)

where J is the number of the day in the year.

Daylight hours (N)

The daylight hours, N, are given by:

s

24

Nω

π

= (34)

where ωs is the sunset hour angle in radians given by Equation 25 or 26. Mean values for N

(15th day of each month) for different latitudes are given in Annex 2, Table 2.7.

Crop evapotranspiration

49

EXAMPLE 9

Determination of daylight hours

Determine the daylight hours (N) for 3 September at 20°S.

From Example 8: ωs = arccos[-tan(-0.35)tan(0.120)] = 1.527 rad

From Eq. 34: N = 24/π (1.527) = 11.7 hour

The number of daylight hours is 11.7 hours.

BOX 9

Calculation sheet for extraterrestrial radiation (Ra) and daylight hours (N)

Latitude Degrees and minutes are

+ positive for northern hemisphere

- negative for southern hemisphere

Degrees

°

---------------------------------------->

°

Minutes

'

-−-------------- / 60 -------------->

°

Decimal degrees = Sum(degrees+minutes/60)

°

ϕ = π/180 * [decimal degrees] Eq. 22

rad

Day of the year

Day

Month

J Table 2.5 (Annex 2)

d

r = 1+0.033 cos(2πJ/365) Eq. 23

δ = 0.409 sin(2πJ/365 - 1.39) Eq. 24

rad

sin(ϕ)sin(δ)

cos(ϕ)cos(δ)

ωs = arccos[-tan(ϕ)tan(δ)] Eq. 25

rad

(24 (60)/π) Gsc

37.59

MJ m-2day-1

Extraterrestrial radiation: Ra

[]

)(sin)(cos)(cos)(sin)(sindG

)60(24

Rssrsca ωδϕ+δϕω

π

=

Eq.21

MJ m-2day-1

Daylight hours: N

s

24

Nω

π

= Eq. 34

hour/day

Meteorological data

50

Solar radiation (Rs )

If the solar radiation, Rs, is not measured, it can be calculated with the Angstrom formula,

which relates solar radiation to extraterrestrial radiation and relative sunshine duration:

where Rs solar or shortwave radiation [MJ m-2 day-1],

n actual duration of sunshine [hour],

N maximum possible duration of sunshine or daylight hours [hour],

n/N relative sunshine duration [-],

R

a extraterrestrial radiation [MJ m-2 day-1],

a

s regression constant, expressing the fraction of extraterrestrial radiation

reaching the earth on overcast days (n = 0),

a

s+bs fraction of extraterrestrial radiation reaching the earth on clear days (n = N).

R

s is expressed in the above equation in MJ m-2 day-1. The corresponding equivalent

evaporation in mm day-1 is obtained by multiplying Rs by 0.408 (Equation 20). Depending on

atmospheric conditions (humidity, dust) and solar declination (latitude and month), the

Angstrom values as and bs will vary. Where no actual solar radiation data are available and no

calibration has been carried out for improved as and bs parameters, the values as = 0.25 and bs

= 0.50 are recommended.

The extraterrestrial radiation, Ra, and the daylight hours or maximum possible duration

of sunshine, N, are given by Equations 21 and 34. Values for Ra and N for different latitudes

are also listed in Annex 2 (Tables 2.6 and 2.7). The actual duration of sunshine, n, is recorded

with a Campbell Stokes sunshine recorder.

EXAMPLE 10

Determination of solar radiation from measured duration of sunshine

In Rio de Janeiro (Brazil) at a latitude of 22°54’S, 220 hours of sunshine were recorded in May.

Determine the solar radiation.

From Eq. 22:

From Table 2.5:

From Eq. 21 or Table 2.6:

latitude = 22°54’S = 22.90°S

or π/180 (-22.90) =

for 15 May, the day in the year (J) =

Ra =

-0.40

135

25.1

rad

--

MJ m-2 day-1

From Eq. 34 or Table 2.7 N =

n = 220 hours / 31 days =

10.9

7.1

hours day-1

From Eq. 35:

Rs = [0.25 + 0.50 (7.1/10.9)] Ra =

= 0.58 Ra = 0.58 (25.1) =

14.5

MJ m-2 day-1

From Eq. 20:

expressed as equivalent evaporation

= 0.408 (14.5) =

5.9

mm/day

The estimated solar radiation is 14.5 MJ m-2 day-1.

asss R

N

n

baR ÷

ø

ö

ç

è

æ+= (35)

Crop evapotranspiration

51

Clear-sky solar radiation (Rso )

The calculation of the clear-sky radiation, Rso, when n = N, is required for computing net

longwave radiation.

• For near sea level or when calibrated values for as and bs are available:

where Rso clear-sky solar radiation [MJ m-2 day-1],

a

s+bs fraction of extraterrestrial radiation reaching the earth on clear-sky days (n = N).

• When calibrated values for as and bs are not available:

where z station elevation above sea level [m].

Other more complex estimates for Rso, which include turbidity and water vapour effects,

are discussed in Annex 3 (Equations 3.14 to 20).

Net solar or net shortwave radiation (Rns )

The net shortwave radiation resulting from the balance between incoming and reflected solar

radiation is given by:

sns R)1(R α−= (38)

where Rns net solar or shortwave radiation [MJ m-2 day-1],

α albedo or canopy reflection coefficient, which is 0.23 for the hypothetical

grass reference crop [dimensionless],

Rs the incoming solar radiation [MJ m-2 day-1].

R

ns is expressed in the above equation in MJ m-2 day-1.

Net longwave radiation (Rnl )

The rate of longwave energy emission is proportional to the absolute temperature of the surface

raised to the fourth power. This relation is expressed quantitatively by the Stefan-Boltzmann

law. The net energy flux leaving the earth's surface is, however, less than that emitted and given

by the Stefan-Boltzmann law due to the absorption and downward radiation from the sky.

Water vapour, clouds, carbon dioxide and dust are absorbers and emitters of longwave

radiation. Their concentrations should be known when assessing the net outgoing flux. As

humidity and cloudiness play an important role, the Stefan-Boltzmann law is corrected by these

two factors when estimating the net outgoing flux of longwave radiation. It is thereby assumed

that the concentrations of the other absorbers are constant:

assso R)ba(R += (36)

(

)

a

5

so Rz10275.0R −

+= (37)

Meteorological data

52

where Rnl net outgoing longwave radiation [MJ m-2 day-1],

σ Stefan-Boltzmann constant [ 4.903 10-9 MJ K-4 m-2 day-1],

Tmax,K maximum absolute temperature during the 24-hour period [K = °C + 273.16],

Tmin,K minimum absolute temperature during the 24-hour period [K = °C + 273.16],

ea actual vapour pressure [kPa],

Rs/Rso relative shortwave radiation (limited to ≤ 1.0),

Rs measured or calculated (Equation 35) solar radiation [MJ m-2 day-1],

Rso calculated (Equation 36 or 37) clear-sky radiation [MJ m-2 day-1].

An average of the maximum air temperature to the fourth power and the minimum air

temperature to the fourth power is commonly used in the Stefan-Boltzmann equation for 24-

hour time steps. The term (0.34-0.14√ea) expresses the correction for air humidity, and will be

smaller if the humidity increases. The effect of cloudiness is expressed by (1.35 Rs/Rso - 0.35).

The term becomes smaller if the cloudiness increases and hence Rs decreases. The smaller the

correction terms, the smaller the net outgoing flux of longwave radiation. Note that the Rs/Rso

term in Equation 39 must be limited so that Rs/Rso ≤ 1.0.

Where measurements of incoming and outgoing short and longwave radiation during

bright sunny and overcast hours are available, calibration of the coefficients in Equation 39 can

be carried out.

Annex 2 (Table 2.8) lists values for σTK4 for different air temperatures.

EXAMPLE 11

Determination of net longwave radiation

In Rio de Janeiro (Brazil) at a latitude of 22°54’S (= -22.70°), 220 hours of bright sunshine, a mean

monthly daily maximum and minimum air temperature of 25.1 and 19.1°C and a vapour pressure of 2.1

kPa were recorded in May. Determine the net longwave radiation.

From Example 10: Rs = 14.5 MJ m-2 day-1

From Eq. 36: Rso = 0.75 Ra = 0.75 . 25.1 = 18.8 MJ m-2 day-1

From Table 2.8 or for:

Then:

and:

σ =

Tmax = 25.1°C =

σTmax,K4 =

4.903 10-9

298.3

38.8

MJ K-4 m-2 day-1

K

MJ m-2 day-1

and:

Tmin = 19.1°C =

σ Tmin,K4 = 35.8 MJ m-2 day-1

292.3

35.8

K

MJ m-2 day-1

and:

ea =

0.34 - 0.14 √ea =

2.1

0.14

kPa

-

and:

-

Rs/Rso = (14.5)/(18.8)

1.35(0.77)-0.35 =

0.77

0.69

-

From Eq. 39: Rnl = [(38.7 + 35.7)/2] (0.14) (0.69) = 3.5 MJ m-2 day-1

From Eq. 20: expressed as equivalent evaporation =

0.408 (3.5) =

1.4

mm/day

The net longwave radiation is 3.5 MJ m-2 day-1.

()

÷

ø

ö

ç

è

æ−−

ú

û

ù

ê

ë

é+

σ= 35.0

R

35.1e14.034.0

2

TT

R

so

s

a

4

K,min

4

K,max

nl (39)

Crop evapotranspiration

53

Net radiation (Rn )

The net radiation (Rn) is the difference between the incoming net shortwave radiation (Rns)

and the outgoing net longwave radiation (Rnl):

nlnsn RRR −= (40)

EXAMPLE 12

Determination of net radiation

Determine the net radiation in Rio de Janeiro in May with the data from previous examples.

From Example 10: Rs = 14.5 MJ m-2 day-1

From Eq. 39: Rns = (1 - 0.23) Rs = 11.1 MJ m-2 day-1

From Example 11: Rnl = 3.5 MJ m-2 day-1

From Eq. 40: Rn = 11.1 - 3.5 = 7.6 MJ m-2 day-1

From Eq. 20: expressed as equivalent evaporation =

0.408 (7.7) =

3.1 mm/day

The net radiation is 7.6 MJ m-2 day-1.

BOX 10

Calculation sheet for net radiation (Rn)

Latitude °

Day R

a (Box 9 or Table 2.6) MJ m-2 d-1

Month N (Box 9 or Table 2.7) hours

n hours (in absence of Rs) n/N

Net solar radiation: Rns

If n is measured instead of Rs:

R

s = (0.25+0.50 n/N) Ra Eq. 35

MJ m-2 d-1

R

so = [0.75 + 2 (Altitude)/ 100 000] Ra Eq. 37 MJ m-2 d-1

R

s /Rso (≤ 1.0)

R

ns = 0.77 Rs Eq. 38 MJ m-2 d-1

Net longwave radiation: Rnl

with σ = 4.903 10-9 MJ K-4 m-2 day-1

and TK = T[°C] + 273.16

Tmax °C Tmax,K = Tmax + 273.16 K

Tmin °C Tmin,K = Tmin + 273.16 K

σTmax,K4 (Table 2.8) MJ m-2 d-1

σTmin,K4 (Table 2.8) MJ m-2 d-1

(σTmax,K4 + σTmin,K4)/2 MJ m-2 d-1

ea kPa (0.34-0.14√ea)

Rs/Rso (1.35 Rs/Rso-0.35)

Rnl = (σTmax,K4 + σTmin,K4)/2 (0.34-0.14√ea)(1.35Rs/Rso-0.35) Eq. 39 MJ m-2 d-1

Net radiation: Rn

MJ m-2 d-1 R

n = Rns - Rnl Eq. 40

Meteorological data

54

Soil heat flux (G)

Complex models are available to describe soil heat flux. Because soil heat flux is small

compared to Rn, particularly when the surface is covered by vegetation and calculation time

steps are 24 hours or longer, a simple calculation procedure is presented here for long time

steps, based on the idea that the soil temperature follows air temperature:

where G soil heat flux [MJ m-2 day-1],

c

s soil heat capacity [MJ m-3 °C-1],

T

i air temperature at time i [°C],

T

i-1 air temperature at time i-1 [°C],

∆t length of time interval [day],

∆z effective soil depth [m].

As the soil temperature lags air temperature, the average temperature for a period should

be considered when assessing the daily soil heat flux, i.e., ∆t should exceed one day. The depth

of penetration of the temperature wave is determined by the length of the time interval. The

effective soil depth, ∆z, is only 0.10-0.20 m for a time interval of one or a few days but might

be 2 m or more for monthly periods. The soil heat capacity is related to its mineral composition

and water content.

• For day and ten-day periods:

As the magnitude of the day or ten-day soil heat flux beneath the grass reference surface is

relatively small, it may be ignored and thus:

• For monthly periods:

When assuming a constant soil heat capacity of 2.1 MJ m-3 °C-1 and an appropriate soil depth,

Equation 41 can be used to derive G for monthly periods:

or, if Tmonth,i+1 is unknown:

where Tmonth,i mean air temperature of month i [°C],

Tmonth,i-1 mean air temperature of previous month [°C],

Tmonth,i+1 mean air temperature of next month [°C].

• For hourly or shorter periods:

For hourly (or shorter) calculations, G beneath a dense cover of grass does not correlate well

with air temperature. Hourly G can be approximated during daylight periods as:

z

t

TT

cG 1ii

s∆

∆

+

=− (41)

0G day ≈ (42)

)TT(07.0G 1i,month1i,monthi,month −+ −= (43)

)TT(14.0G 1i,monthi,monthi,month −

−= (44)

Crop evapotranspiration

55

and during nighttime periods as:

Where the soil is warming, the soil heat flux G is positive. The amount of energy

required for this process is subtracted from Rn when estimating evapotranspiration.

EXAMPLE 13

Determination of soil heat flux for monthly periods

Determine the soil heat flux in April in Algiers (Algeria) when the soil is warming. The mean monthly

temperatures of March, April and May are 14.1, 16.1, and 18.8°C respectively.

From Eq. 43

for the month of April:

Gmonth = 0.07 (18.8 - 14.1) =

0.33

MJ m-2 day-1

From Eq. 20

expressed as equivalent evaporation = 0.408(0.33) =

0.13

mm/day

The soil heat flux is 0.33 MJ m-2 day-1.

WIND SPEED

Measurement

Wind is characterized by its direction and velocity. Wind direction refers to the direction from

which the wind is blowing. For the computation of evapotranspiration, wind speed is the

relevant variable. As wind speed at a given location varies with time, it is necessary to express

it as an average over a given time interval. Wind speed is given in metres per second (m s-1) or

kilometres per day (km day-1).

Wind speed is measured with anemometers. The anemometers commonly used in

weather stations are composed of cups or propellers which are turned by the force of the wind.

By counting the number of revolutions over a given time period, the average wind speed over

the measuring period is computed.

Wind profile relationship

Wind speeds measured at different heights above the soil surface are different. Surface friction

tends to slow down wind passing over it. Wind speed is slowest at the surface and increases

with height. For this reason anemometers are placed at a chosen standard height, i.e., 10 m in

meteorology and 2 or 3 m in agrometeorology. For the calculation of evapotranspiration, wind

speed measured at 2 m above the surface is required. To adjust wind speed data obtained from

instruments placed at elevations other than the standard height of 2 m, a logarithmic wind speed

profile may be used for measurements above a short grassed surface:

nhr R1.0G = (45)

nhr R5.0G = (46)

Meteorological data

56

where u2 wind speed at 2 m above ground surface [m s-1],

uz measured wind speed at z m above ground surface [m s-1],

z height of measurement above ground surface [m].

The corresponding multipliers or conversion factors are given in Annex 2 (Table 2.9)

and are plotted in Figure 16.

EXAMPLE 14

Adjusting wind speed data to standard height

Determine the wind speed at the standard height of 2 m, from a measured wind speed of 3.2 m/s at

10 m above the soil surface.

For:

And:

Then:

uz =

z =

Conversion factor = 4.87 / ln(67.8 (10) - 5.42) =

3.2

10

0.75

m/s

m

-

From Eq. 47: u2 = 3.2 (0.75) = 2.4 m/s

The wind speed at 2 m above the soil surface is 2.4 m/s.

)42.5z8.67(ln

87.4

uu z2 −

= (47)

FIGURE 16

Conversion factor to convert wind speed measured at a certain height above ground level to

wind speed at the standard height (2 m)

0

1

2

3

4

5

6

7

8

9

10

0.7 0.8 0.9 1 1.1 1.2 1.3

conversion factor

measured height above ground surface (m)

Crop evapotranspiration

57

CLIMATIC DATA ACQUISITION

Weather stations

Meteorological data are recorded at various types of weather stations. Agrometeorological

stations are sited in cropped areas where instruments are exposed to atmospheric conditions

similar to those for the surrounding crops. In these stations, air temperature and humidity, wind

speed and sunshine duration are typically measured at 2 m above an extensive surface of grass

or short crop. Where needed and feasible, the cover of the station is irrigated. Guidelines for the

establishment and maintenance of agrometeorological stations are given in the FAO Irrigation

and Drainage Paper No. 27. This handbook also describes the different types of instruments,

their installation and reliability.

Data collected at stations other than agrometeorological stations require a careful

analysis of their validity before their use. For example, in aeronautic stations, data relevant for

aviation are measured. As airports are often situated near urban conditions, temperatures may

be higher than those found in rural agricultural areas. Wind speed is commonly measured at 10

m height above the ground surface.

The country’s national meteorological service should be contacted for information on the

climatic data collected at various types of weather stations in the country. National services

commonly publish meteorological bulletins listing processed climatic data from the various

stations.

The annexes list procedures for the statistical analysis, assessment, correction and

completion of partial or missing weather data:

Annex 4: Statistical analysis of weather data sets;

Annex 5: Measuring and assessing integrity of weather data;

Annex 6: Correction of weather data observed at non-reference sites for computing ETo.

Agroclimatic monthly databases

Starting in 1984, FAO has published mean monthly agroclimatic data from 2 300 stations in the

FAO Plant Production and Protection Series. Several volumes exist:

No. 22: Volume 1: data for Africa, countries north of the equator (1984),

Volume 2: data for Africa, countries south of the equator (1984);

No. 24: Agroclimatic data for Latin America and the Caribbean (1985);

No. 25: Volume 1: Agroclimatic data for Asia (A-J) (1987),

Volume 2: Agroclimatic data for Asia (K-Z) (1987).

CLIMWAT for CROPWAT (FAO Irrigation and Drainage Paper No. 46) contains

monthly data from 3 262 climatic stations contained on five separate diskettes. The stations are

grouped by country and by continent. Monthly averages of maximum and minimum

temperatures, mean relative humidity, wind speed, sunshine hours, radiation data as well as

rainfall and ETo calculated with the FAO Penman-Monteith method are listed on the diskettes

for mean long-term conditions.

Meteorological data

58

FAOCLIM provides a user friendly interface on compact disc to the agroclimatic

database of the Agrometeorology Group in FAO. The data presented are an extension of the

previously published FAO Plant Production and Protection Series and the number of stations

has been increased from 2 300 to about 19 000, with an improved world wide coverage.

However, values for all principal weather parameters are not available for all stations. Many

contain air temperature and precipitation only.

These databases can be consulted in order to verify the consistency of the actual database

or to estimate missing climatic parameters. However, they should only be used for preliminary

studies as they contain mean monthly data only. FAOCLIM provides monthly time series for

only a few stations. The information in these databases should never replace actual data.

Other electronic databases for portions of the globe have been published by the

International Water Management Institute (IWMI). These databases include daily and monthly

air temperature, precipitation and ETo predicted using the Hargreaves ETo equation that is

based on differences between daily maximum and minimum air temperature.

ESTIMATING MISSING CLIMATIC DATA

The assessment of the reference evapotranspiration ETo with the Penman-Monteith method is

developed in Chapter 4. The calculation requires mean daily, ten-day or monthly maximum and

minimum air temperature (Tmax and Tmin), actual vapour pressure (ea), net radiation (Rn) and

wind speed measured at 2 m (u2). If some of the required weather data are missing or cannot be

calculated, it is strongly recommended that the user estimate the missing climatic data with one

of the following procedures and use the FAO Penman-Monteith method for the calculation of

ETo. The use of an alternative ETo calculation procedure, requiring only limited

meteorological parameters, is less recommended. Procedures to estimate missing humidity,

radiation and wind speed data are given in this section.

Estimating missing humidity data

Where humidity data are lacking or are of questionable quality, an estimate of actual vapour

pressure, ea, can be obtained by assuming that dewpoint temperature (Tdew) is near the daily

minimum temperature (Tmin). This statement implicitly assumes that at sunrise, when the air

temperature is close to Tmin, that the air is nearly saturated with water vapour and the relative

humidity is nearly 100%. If Tmin is used to represent Tdew then:

ú

û

ù

ê

ë

é

+

== 3.237T

T27.17

exp611.0)T(ee

min

o

a (48)

The relationship Tdew ≈ Tmin holds for locations where the cover crop of the station is

well watered. However, particularly for arid regions, the air might not be saturated when its

temperature is at its minimum. Hence, Tmin might be greater than Tdew and a further

calibration may be required to estimate dewpoint temperatures. In these situations, “Tmin” in

the above equation may be better approximated by subtracting 2-3°C from Tmin. Appropriate

correction procedures are given in Annex 6. In humid and subhumid climates, Tmin and Tdew

measured in early morning may be less than Tdew measured during the daytime because of

condensation of dew during the night. After sunrise, evaporation of the dew will once again

Crop evapotranspiration

59

humidify the air and will increase the value measured for Tdew during the daytime. This

phenomenon is demonstrated in Figure 5.4 of Annex 5. However, it is standard practice in 24-

hour calculations of ETo to use Tdew measured or calculated durin early morning.

The estimate for ea from Tmin should be checked. When the prediction by Equation 48 is

validated for a region, it can be used for daily estimates of ea.

Estimating missing radiation data

Net radiation measuring devices, requiring professional control, have rarely been installed in

agrometeorological stations. In the absence of a direct measurement, longwave and net

radiation can be derived from more commonly observed weather parameters, i.e., solar

radiation or sunshine hours, air temperature and vapour pressure. Where solar radiation is not

measured, it can perhaps be estimated from measured hours of bright sunshine. However,

where daily sunshine hours (n) are not available, solar radiation data cannot be computed with

the calculation procedures previously presented. This section presents various methods to

estimate solar radiation data with an alternative methodology.

Solar Radiation data from a nearby weather station

This method relies on the fact that for the same month and often for the same day, the variables

affecting incoming solar radiation, Rs, and sunshine duration, n, are similar throughout a given

region. This implies that: (i) the size of the region is small; (ii) the air masses governing rainfall

and cloudiness are nearly identical within parts of the region; and (iii) the physiography of the

region is almost homogenous. Differences in relief should be negligible as they strongly

influence the movement of air masses. Under such conditions, radiation data observed at nearby

stations can be used.

Caution should be used when applying this method to mountainous and coastal areas

where differences in exposure and altitude could be important or where rainfall is variable due

to convective conditions. Moreover, data from a station located nearby but situated on the other

side of a mountain may not be transferable as conditions governing radiation are different. The

user should observe climatic conditions in both locations and obtain information from local

persons concerning general differences in cloud cover and type.

Where the north-south distance to a weather station within the same homogeneous region

exceeds 50 km so that the value for Ra changes, the Rs measurement should be adjusted using

the ratio of the solar to extraterrestrial radiation, Rs/Ra:

a

reg,a

reg,s

sR

R

R= (49)

where Rs,reg solar radiation at the regional location [MJ m-2 day-1],

Ra,reg extraterrestrial radiation at the regional location [MJ m-2 day-1].

Once the solar radiation has been derived from the radiation data of a nearby station, the

net longwave radiation (Equation 39) and the net radiation (Equation 40) can be calculated.

The estimation method of Equation 49 is recommended for monthly calculations of ETo.

If using the method for daily estimates of ETo, a more careful analysis of weather data in the

importing and exporting meteorological stations has to be performed to verify whether both

Meteorological data

60

stations are in the same homogeneous climatic region and are close enough to experience

similar conditions within the same day. The analysis should include the comparison of daily

weather data from both stations, particularly the maximum and minimum air temperature and

humidity. In fact, similar cloudiness and sunshine durations are related to similarities in

temperature and humidity trends.

Generally, daily calculations of ETo with estimated radiation data are justified when

utilized as a sum or an average over a several-day period. This is the case for the computation

of the mean evapotranspiration demand between successive irrigations or when planning

irrigation schedules. Under these conditions, the relative error for one day often

counterbalances the error for another day of the averaging period. Daily estimates should not be

utilized as true daily estimates but only in averages over the period under consideration.

Solar Radiation data derived from air temperature differences

The difference between the maximum and minimum air temperature is related to the degree of

cloud cover in a location. Clear-sky conditions result in high temperatures during the day

(Tmax) because the atmosphere is transparent to the incoming solar radiation and in low

temperatures during the night (Tmin) because less outgoing longwave radiation is absorbed by

the atmosphere. On the other hand, in overcast conditions, Tmax is relatively smaller because a

significant part of the incoming solar radiation never reaches the earth's surface and is absorbed

and reflected by the clouds. Similarly, Tmin will be relatively higher as the cloud cover acts as

a blanket and decreases the net outgoing longwave radiation. Therefore, the difference between

the maximum and minimum air temperature (Tmax - Tmin) can be used as an indicator of the

fraction of extraterrestrial radiation that reaches the earth's surface. This principle has been

utilized by Hargreaves and Samani to develop estimates of ETo using only air temperature data.

The Hargreaves’ radiation formula, adjusted and validated at several weather stations in a

variety of climate conditions, becomes:

a

R)

min

T

max

(T

Rs

k

s

R−= (50)

where Ra extraterrestrial radiation [MJ m-2 d-1],

Tmax maximum air temperature [°C],

Tmin minimum air temperature [°C],

kRs adjustment coefficient (0.16 .. 0.19) [°C-0.5].

The square root of the temperature difference is closely related to the existing daily solar

radiation in a given location. The adjustment coefficient kRs is empirical and differs for

‘interior’ or ‘coastal’ regions:

• for ‘interior’ locations, where land mass dominates and air masses are not strongly

influenced by a large water body, kRs ≅ 0.16;

• for ‘coastal’ locations, situated on or adjacent to the coast of a large land mass and where

air masses are influenced by a nearby water body, kRs ≅ 0.19.

The relationship between Rs/Ra and the temperature difference is plotted in Figure 17 for

interior and coastal locations. The fraction of extraterrestrial radiation that reaches the earth's

surface, Rs/Ra, ranges from about 0.25 on a day with dense cloud cover to about 0.75 on a

Crop evapotranspiration

61

cloudless day with clear sky. Rs predicted by Equation 50 should be limited to ≤ R

so from

Equation 36 or 37.

The temperature difference method is recommended for locations where it is not appropriate

to import radiation data from a regional station, either because homogeneous climate conditions

do not occur, or because data for the region are lacking. For island conditions, the methodology

of Equation 50 is not appropriate due to moderating effects of the surrounding water body.

Caution is required when daily computations of ETo are needed. The advice given for

Equation 49 fully applies. It is recommended that daily estimates of ETo that are based on

estimated Rs be summed or averaged over a several-day period, such as a week, decade or

month to reduce prediction error.

EXAMPLE 15

Determination of solar radiation from temperature data

Determine the solar radiation from the temperature data of July in Lyon (France) at a latitude of

45°43'N and at 200 m above sea level. In July, the mean monthly maximum and minimum air

temperatures are 26.6 and 14.8°C respectively.

From Table 2.5:

From Eq. 21 or

Annex 2 Table 2.6):

latitude = 45°43’ = +45.72° decimal degrees =

The day of the year for 15 July is

Ra =

0.80

196

40.6

radian

-

MJ m-2 day-1

From Eq. 50

(same latitude):

Rs = 0.16 [√(26.6-14.8)] Ra = 0.55 (40.6) = 22.3 MJ m-2 day-1

From Eq. 20

(same latitude):

equivalent evaporation = 0.408 (22.3) = 9.1 mm/day

In July, the estimated solar radiation, Rs, is 22.3 MJ m-2 day-1

FIGURE 17

Relationship between the fraction of extraterrestrial radiation that reaches the earth's surface,

Rs/Ra, and the air temperature difference Tmax - Tmin for interior (kRs = 0.16) and coastal

(kRs = 0.19) regions

0 2 4 6 8 10 12 14 16 18 20 22

temperature difference (Tmax - Tmin ) °C

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

R

s

/ R

a

interior

coastal

Meteorological data

62

EXAMPLE 16

Determination of net radiation in the absence of radiation data

Calculate the net radiation for Bangkok (13°44’N) by using Tmax and Tmin. The station is located at the

coast at 2 m above sea level. In April, the monthly average of the daily maximum temperature, daily

minimum temperature and daily vapour pressure are 34.8°C, 25.6°C and 2.85 kPa respectively.

For Latitude 13°44’N = +13.73° decimal degrees = -0.24 radian

and for 15 April, J =105:

From Eq. 21 or

Table 2.6:

Ra =

(in coastal location) kRs =

(Tmax - Tmin) = (34.8 - 25.6) =

38.1

0.19

9.2°C

MJ m-2 day-1

°C

From Eq. 50: Rs = 0.19 √(9.2) Ra 21.9 MJ m-2 day-1

From Eq. 36: Rso = 0.75 Ra 28.5 MJ m-2 day-1

From Eq. 38: Rns = 0.77 Rs 16.9 MJ m-2 day-1

σ =

Tmax =

σTmax,K4 =

4.903 10-9

34.8

44.1

MJ K-4 m-2 day-1

°C

MJ m-2 day-1

T

min

σTmin,K4 =

(σTmax,K4 + σTmin,K4)/2 = (44.1 + 39.1)/2 =

25.6

39.1

41.6

°C

MJ m-2 day-1

For: ea = 2.85 kPa

(0.34-0.14√ea) =

2.85

0.10

kPa

-

For:

Then:

Rs/Rso =

(1.35(Rs/Rso)-0.35) =

0.77

0.69

-

From Eq. 39: Rnl = 41.6 (0.10) 0.69 = 3.0 MJ m-2 day-1

From Eq. 40: Rn = (16.9 - 2.9) = 13.9 MJ m-2 day-1

From Eq. 20:

equivalent evaporation = 0.408 (13.9) =

5.7

mm/day

The estimated net radiation is 13.9 MJ m-2 day-1.

Empirical methodology for island locations

For island locations, where the land mass has a width perpendicular to the coastline of 20 km or

less, the air masses influencing the atmospheric conditions are dominated by the adjacent water

body in all directions. The temperature method is not appropriate for this situation. Where

radiation data from another location on the island are not available, a first estimate of the

monthly solar average can be obtained from the empirical relation:

where Rs solar radiation [MJ m-2 day-1],

Ra extraterrestrial radiation [MJ m-2 day-1],

b empirical constant, equal to 4 MJ m-2 day-1.

This relationship is only applicable for low altitudes (from 0 to 100 m). The empirical

constant represents the fact that in island locations some clouds are usually present, thus

making the mean solar radiation 4 MJ m-2 day-1 below the nearly clear sky envelope (0.7 Ra).

Local adjustment of the empirical constant may improve the estimation.

bR7.0R as −= (51)

Crop evapotranspiration

63

The method is only appropriate for monthly calculations. The constant relation between Rs

and Ra does not yield accurate daily estimates.

Missing wind speed data

Wind speed data from a nearby weather station

Importing wind speed data from a nearby station, as for radiation data, relies on the fact that the

air flow above a ‘homogeneous’ region may have relatively large variations through the course

of a day but small variations when referring to longer periods or the total for the day. Data from

a nearby station may be imported where air masses are of the same origin or where the same

fronts govern air flows in the region and where the relief is similar.

When importing wind speed data from another station, the regional climate, trends in

variation of other meteorological parameters and relief should be compared. Strong winds are

often associated with low relative humidity and light winds are common with high relative

humidity. Thus, trends in variation of daily maximum and minimum relative humidities should

be similar in both locations. In mountainous areas, data should not necessarily be imported

from the nearest station but from nearby stations with similar elevation and exposure to the

dominant winds. The paired stations may even vary from one season to another, depending on

the dominant winds.

Imported wind speed data can be used when making monthly estimates of

evapotranspiration. Daily calculations are justified when utilized as a sum or average over a

several-day period, such as a week or decade.

Empirical estimates of monthly wind speed

As the variation in wind speed average over monthly periods is relatively small and fluctuates

around average values, monthly values of wind speed may be estimated. The ‘average’ wind

speed estimates may be selected from information available for the regional climate, but should

take seasonal changes into account. General values are suggested in Table 4.

TABLE 4

General classes of monthly wind speed data

Description

mean monthly wind speed at 2 m

light wind

light to moderate wind

moderate to strong wind

strong wind

... ≤ 1.0 m/s

1 – 3 m/s

3 – 5 m/s

... ≥ 5.0 m/s

Where no wind data are available within the region, a value of 2 m/s can be used as a

temporary estimate. This value is the average over 2 000 weather stations around the globe.

In general, wind speed at 2 m, u2, should be limited to about u2 ≥ 0.5 m/s when used in

the ETo equation (Equation 6). This is necessary to account for the effects of boundary layer

instability and bouyancy of air in promoting exchange of vapour at the surface when air is calm.

This effect occurs when the wind speed is small and buoyancy of warm air induces air

exchange at the surface. Limiting u2 ≥ 0.5 m/s in the ETo equation improves the estimation

accuracy under the conditions of very low wind speed.

Meteorological data

64

Minimum data requirements

This section has shown how solar radiation, vapour pressure and wind data can be estimated

when missing. Many of the suggested procedures rely upon maximum and minimum air

temperature measurements. Unfortunately, there is no dependable way to estimate air

temperature when it is missing. Therefore it is suggested that maximum and minimum daily air

temperature data are the minimum data requirements necessary to apply the FAO Penman-

Monteith method.

An alternative equation for ETo when weather data are missing

When solar radiation data, relative humidity data and/or wind speed data are missing, they

should be estimated using the procedures presented in this section. As an alternative, ETo can

be estimated using the Hargreaves ETo equation where:

a

5.0

minmaxmeano R)TT()8.17T(0023.0ET −+= (52)

where all parameters have been previously defined. Units for both ETo and Ra in Equation 52

are mm day-1. Equation 52 should be verified in each new region by comparing with estimates

by the FAO Penman-Monteith equation (Equation 6) at weather stations where solar radiation,

air temperature, humidity, and wind speed are measured. If necessary, Equation 52 can be

calibrated on a monthly or annual basis by determining empirical coefficients where ETo = a +

b ETo Eq.52, where the “Eq. 52” subscript refers to ETo predicted using Equation 52. The

coefficients a and b can be determined by regression analyses or by visual fitting. In general,

estimating solar radiation, vapor pressure and wind speed as described in Equations 48 to 51

and Table 4 and then utilizing these estimates in Equation 6 (the FAO Penman-Monteith

equation) will provide somewhat more accurate estimates as compared to estimating ETo

directly using Equation 52. This is due to the ability of the estimation equations to incorporate

general climatic characteristics such as high or low wind speed or high or low relative humidity

into the ETo estimate made using Equation 6.

Equation 52 has a tendency to underpredict under high wind conditions (u2 > 3 m/s) and

to overpredict under conditions of high relative humidity.

Crop evapotranspiration

65

Chapter 4

Determination of ETo

This chapter demonstrates how the crop reference evapotranspiration (ETo) is determined

either from meteorological data or from pan evaporation.

The FAO Penman-Monteith method is maintained as the sole standard method for the

computation of ETo from meteorological data. The method itself is introduced in Chapter 2,

and the computation of all data required for the calculation of ETo is discussed in Chapter 3.

This chapter presents guidelines to calculate ETo with different time steps, ranging from hours

to months, and with missing climatic data. The ETo calculation can be done by hand with the

help of a calculation sheet, or by means of a computer.

ETo can also be estimated from the evaporation loss from a water surface. The procedure

to obtain ETo from pan evaporation and the coefficients for different types of pans are

presented in this chapter.

PENMAN-MONTEITH EQUATION

From the original Penman-Monteith equation and the equations of the aerodynamic and canopy

resistance, the FAO Penman-Monteith equation has been derived in Chapter 2:

where ETo reference evapotranspiration [mm day-1],

Rn net radiation at the crop surface [MJ m-2 day-1],

G soil heat flux density [MJ m-2 day-1],

T air temperature at 2 m height [°C],

u2 wind speed at 2 m height [m s-1],

es saturation vapour pressure [kPa],

ea actual vapour pressure [kPa],

es-ea saturation vapour pressure deficit [kPa],

∆ slope vapour pressure curve [kPa °C-1],

γ psychrometric constant [kPa °C-1].

The FAO Penman-Monteith equation determines the evapotranspiration from the

hypothetical grass reference surface and provides a standard to which evapotranspiration in

)u34.01(

)ee(u

273T

900

)GR(408.0

2

as2n

o

ET +γ+∆

−

+

γ+−∆

= (6)

Determination of ETo

66

different periods of the year or in other regions can be compared and to which the

evapotranspiration from other crops can be related.

Calculation procedure

Calculation sheet

ETo can be estimated by means of the calculation sheet presented in Box 11. The calculation

sheet refers to tables in Annex II for the determination of some of the climatic parameters. The

calculation procedure consists of the following steps:

1. Derivation of some climatic parameters from the daily maximum (Tmax) and minimum

(Tmin) air temperature, altitude (z) and mean wind speed (u2).

2. Calculation of the vapour pressure deficit (es-ea). The saturation vapour pressure (es) is

derived from Tmax and Tmin, while the actual vapour pressure (ea) can be derived

from the dewpoint temperature (Tdew), from maximum (RHmax) and minimum

(RHmin) relative humidity, from the maximum (RHmax), or from mean relative

humidity (RHmean).

3. Determination of the net radiation (Rn) as the difference between the net shortwave

radiation (Rns) and the net longwave radiation (Rnl). In the calculation sheet, the effect

of soil heat flux (G) is ignored for daily calculations as the magnitude of the flux in this

case is relatively small. The net radiation, expressed in MJ m-2 day-1, is converted to

mm/day (equivalent evaporation) in the FAO Penman-Monteith equation by using

0.408 as the conversion factor within the equation.

4. ETo is obtained by combining the results of the previous steps.

Examples 17 and 20 present typical examples using the calculation sheet.

Computerized calculations

Calculations of the reference crop evapotranspiration ETo are often computerized. The

calculation procedures of all data required for the calculation of ETo by means of the FAO

Penman-Monteith equation are presented in Chapter 3. Typical sequences in which the

calculations can be executed are given in the calculation sheets. The procedures presented in

Boxes 7 (vapour pressure deficit), 9 (extraterrestrial radiation and daylight hours), 10 (net

radiation) and 11 (ETo) can be used when developing a spreadsheet or computer program to

calculate ETo.

Many software packages already use the FAO Penman-Monteith equation to assess the

reference evapotranspiration. As an example, the output of CROPWAT, the FAO software for

irrigation scheduling, is presented in Figure 18.

ETo calculated with different time steps

The selection of the time step with which ETo is calculated depends on the purpose of the

calculation, the accuracy required and the time step of the climatic data available.

Crop evapotranspiration

67

BOX 11

Calculation sheet for ETo (FAO Penman-Monteith) using meteorological tables of Annex 2

Parameters

Tmax

°C

Tmin

°C

Tmean = (Tmax+Tmin)/2

°C

Tmean

°C

∆ (Table 2.4 of Annex 2)

kPa/°C

Altitude

m

γ (Table 2.2 of Annex 2)

kPa/°C

u2

m/s

(1 + 0.34 u2)

∆ / [ ∆ + γ (1 + 0.34 u2)]

γ / [ ∆ + γ (1 + 0.34 u2)]

[ 900 / (Tmean + 273) ] u2

Vapour pressure deficit

Tmax

°C

e°(Tmax) (Table 2.3)

kPa

Tmin

°C

e°(Tmin) (Table 2.3)

kPa

Saturation vapour pressure es = [(e°(Tmax) + e°(Tmin)]/2

kPa

ea derived from dewpoint temperature:

Tdew

°C

ea = e°(Tdew) (Table 2.3)

kPa

OR ea derived from maximum and minimum relative humidity:

RHmax

%

e°(Tmin) RHmax/100

kPa

RHmin

%

e°(Tmax) RHmin/100

kPa

ea: (average)

kPa

OR ea derived from maximum relative humidity: (recommended if there are errors in RHmin)

RHmax

%

ea = e°(Tmin) RHmax/100

kPa

OR ea derived from mean relative humidity: (less recommended due to non-linearities)

RHmean

%

ea = es RHmean/100

kPa

Vapour pressure deficit (es - ea) kPa

Determination of ETo

68

Radiation

Latitude °

Day Ra (Table 2.6) MJ m-2 d-1

Month N (Table 2.7) hours

n hours n/N

If no Rs data available: Rs = (0.25 + 0.50 n/N) Ra MJ m-2 d-1

Rso = [0.75 + 2 (Altitude)/ 100 000] Ra MJ m-2 d-1

Rs / Rso

Rns = 0.77 Rs MJ m-2 d-1

Tmax σ Tmax,K4 (Table 2.8) MJ m-2 d-1

Tmin σ Tmin,K4 (Table 2.8) MJ m-2 d-1

(σTmax,K4 + σTmin,K4)/2 MJ m-2 d-1

ea kPa (0.34 - 0.14 √ea)

Rs/Rso (1.35 Rs/Rso - 0.35)

Rnl = (σTmax,K4 + σTmin,K4)/2 (0.34 - 0.14 √ea) (1.35 Rs/Rso - 0.35) MJ m-2 d-1

Rn = Rns - Rnl MJ m-2 d-1

Tmonth °C Gday (assume) 0 MJ m-2 d-1

Tmonth-1 °C Gmonth = 0.14 (Tmonth - Tmonth-1) MJ m-2 d-1

Rn – G MJ m-2 d-1

0.408 (Rn - G) mm/day

Grass reference evapotranspiration

[]

)GR(408.0

)u34.01( n

2

−

ú

û

ù

ê

ë

é

+γ+∆

∆

mm/day

[]

)ee(u

273T

900

)u34.01( as2

2

−

ú

û

ù

ê

ë

é

+

ú

û

ù

ê

ë

é

+γ+∆

γ

mm/day

)u34.01(

)ee(u

273T

900

)GR(408.0

ET

2

as2n

o+γ+∆

−

+

γ+−∆

=

mm/day

Crop evapotranspiration

69

Ten-day or monthly time step

Notwithstanding the non-linearity in the Penman-Monteith equation and some weather

parameter methods, mean ten-day or monthly weather data can be used to compute the mean

ten-day or monthly values for the reference evapotranspiration. The value of the reference

evapotranspiration calculated with mean monthly weather data is indeed very similar to the

average of the daily ETo values calculated with daily average weather data for that month.

The meteorological data consist of:

• Air temperature: ten-day or monthly average daily maximum (Tmax) and average daily

minimum temperature (Tmin).

• Air humidity: ten-day or monthly average of the daily actual vapour pressure (ea) derived

from psychrometric, dewpoint or relative humidity data.

• Wind speed: ten-day or monthly average of daily wind speed data measured at 2 m

height (u2).

• Radiation: ten-day or monthly average of daily net radiation (Rn) computed from the

mean ten-day or monthly measured shortwave radiation or from actual duration of daily

sunshine hours (n). The extraterrestrial radiation (Ra) and daylight hours (N) for a

specific day of the month can be computed using Equations 21 and 34 or can be selected

from Tables 2.5 and 2.6 in Annex 2.

When the soil is warming (spring) or cooling (autumn), the soil heat flux (G) for monthly

periods may become significant relative to the mean monthly Rn. In these cases G cannot be

ignored and its value should be determined from the mean monthly air temperatures of the

previous and next month. Chapter 3 outlines the calculation procedure (Equations 43 and 44).

FIGURE 18

ETo computed by CROPWAT

+-----------------------------------------------------------------------------+

¦ MONTHLY REFERENCE EVAPOTRANSPIRATION PENMAN MONTEITH ¦

¦-----------------------------------------------------------------------------¦

¦ Meteostation: CABINDA Country: Angola ¦

¦ Altitude: 20 m. Coordinates: -5.33 South 12.11 East¦

¦-----------------------------------------------------------------------------¦

¦ Month MinTemp MaxTemp Humid. Wind Sunshine Radiation ETo-PenMon¦

¦ °C °C % km/day Hours MJ/m²/day mm/day ¦

¦-----------------------------------------------------------------------------¦

¦ January 22.8 29.6 81 78 4.0 15.7 3.4 ¦

¦ February 22.7 30.3 82 69 4.6 16.9 3.7 ¦

¦ March 23.0 30.6 80 78 5.1 17.4 3.8 ¦

¦ April 23.0 30.2 82 69 5.0 16.4 3.5 ¦

¦ May 22.0 28.6 84 69 3.8 13.5 2.9 ¦

¦ June 19.2 26.5 81 69 3.3 12.2 2.6 ¦

¦ July 17.6 25.1 78 78 3.2 12.3 2.6 ¦

¦ August 18.6 25.3 78 78 2.6 12.4 2.6 ¦

¦ September 20.5 26.5 78 104 2.0 12.4 2.8 ¦

¦ October 22.5 28.0 79 130 2.2 12.9 3.1 ¦

¦ November 23.0 28.7 80 104 3.2 14.4 3.3 ¦

¦ December 23.0 29.1 82 95 3.8 15.2 3.4 ¦

¦-----------------------------------------------------------------------------¦

¦ Year 21.5 28.2 80 85 3.6 14.3 3.1 ¦

+-----------------------------------------------------------------------------+

CROPWAT 7.0 Climate file: C:\PROF-P~1\CROPWAT\CROPWAT\CLI\CABINDA.PEN 03/07/98

Determination of ETo

70

EXAMPLE 17

Determination of ETo with mean monthly data

Given the monthly average climatic data of April of Bangkok (Thailand) located at 13 °44’N and at an

elevation of 2 m:

- Monthly average daily maximum temperature (Tmax) = 34.8 °C

- Monthly average daily minimum temperature (Tmin) = 25.6 °C

- Monthly average daily vapour pressure (ea) = 2.85 kPa

Measured at 2 m Monthly average daily wind speed (u2) = 2 m/s

- Monthly average sunshine duration (n) = 8.5 hours/day

For April Mean monthly average temperature (Tmonth,i) = 30.2 °C

For March Mean monthly average temperature (Tmonth,i-1) = 29.2 °C

Determination according to outline of Box 11 (calculation sheet ETo)

Parameters

- Tmean = [(Tmax = 34.8) + (Tmin = 25.6)]/2 = 30.2 °C

From Table 2.4 or

Eq. 13:

∆ = 0.246 kPa/°C

From Table 2.1

and Table 2.2 or

Eq. 7 and Eq. 8:

Altitude =

P =

γ =

2

101.3

0.0674

m

kPa

kPa/°C

-

(1 + 0.34 u2) =

1.68

-

∆/[∆+γ(1+0.34u2)] =0.246/[(0.246 + 0.0674 (1.68)] =

0.685

-

γ/[∆+γ(1+0.34u2)] = 0.0667/[0.246 + 0.0674 (1.68)] =

0.188

-

900/(Tmean+273) u2 =

5.94

-

Vapour pressure deficit

From Table 2.3 or

Eq. 11:

Tmax =

e°(Tmax) =

34.8

5.56

°C

kPa

From Table 2.3 or

Eq. 11:

Tmin =

e°(Tmin) =

25.6

3.28

°C

kPa

-

Given

es = (5.56 + 3.28)/2 =

ea =

4.42

2.85

kPa

- Vapour pressure deficit (es-ea) = (4.42 - 2.85) = 1.57 kPa

Crop evapotranspiration

71

Radiation (for month = April)

From Table 2.6 or

2.5 or Eq. 21:

J = (for 15 April)

Latitude = 13°44’N = (13 + 44/60)=

Ra =

105

13.73

38.06

-

°N

MJ m-2 day-1

N (Table 2.7 or

Eq. 34):

Daylength N =

12.31 hours

- n/N = (8.5/12.31) = 0.69 -

- Rs = [0.25 + 0.50 (0.69)] 38.06 = 22.65 MJ m-2 day-1

- Rso = (0.75 + 2 (2)/100 000) 38.06 = 28.54 MJ m-2 day-1

- Rs/Rso = (22.65/28.54) = 0.79 -

- Rns = 0.77 (22.65) = 17.44 MJ m-2 day-1

From Table 2.8: Tmax =

σTmax,K4 =

34.8

44.10

°C

MJ m-2 day-1

From Table 2.8: Tmin =

σTmin,K4 =

25.6

39.06

°C

MJ m-2 day-1

- (σTmax,K4 + σTmin,K4)/2 = (44.10 + 39.06)/2 = 41.58 MJ m-2 day-1

For:

Then:

ea =

(0.34-0.14√ea) =

2.85

0.10

kPa

-

For:

Then:

Rs/Rso =

(1.35 Rs/Rso - 0.35) =

0.79

0.72

-

Rnl = 41.58 (0.10) 0.72 =

Rn = (17.44 - 3.11) =

G = 0.14 (30.2 - 29.2) =

(Rn - G) = (14.33 - 0.14) =

0.408 (Rn - G) =

3.11

14.33

0.14

14.19

5.79

MJ m-2 day-1

mm/day

Grass reference evapotranspiration

-

0.408 (Rn-G) ∆/[∆+γ(1+0.34u2)] =

(5.79) 0.685 =

900u2/(T+273) (es-ea) γ/[∆+γ(1+0.34u2)] =

5.94(1.57)0.188 =

ETo = (3.97 + 1.75) =

3.97

1.75

5.72

mm/day

The grass reference evapotranspiration is 5.7 mm/day.

Determination of ETo

72

Daily time step

Calculation of ETo with the Penman-Monteith equation on 24-hour time scales will generally

provide accurate results. The required meteorological data consist of:

• Air temperature: maximum (Tmax) and minimum (Tmin) daily air temperatures.

• Air humidity: mean daily actual vapour pressure (ea) derived from psychrometric,

dewpoint temperature or relative humidity data.

• Wind speed: daily average for 24 hours of wind speed measured at 2 m height (u2).

• Radiation: net radiation (Rn) measured or computed from solar and longwave radiation

or from the actual duration of sunshine (n). The extraterrestrial radiation (Ra) and

daylight hours (N) for a specific day of the month should be computed using Equations

21 and 34. As the magnitude of daily soil heat flux (G) beneath the reference grass

surface is relatively small, it may be ignored for 24-hour time steps.

EXAMPLE 18

Determination of ETo with daily data

Given the meteorological data as measured on 6 July in Uccle (Brussels, Belgium) located at 50°48’N

and at 100 m above sea level:

-

Maximum air temperature (Tmax) =

Minimum air temperature (Tmin) =

Maximum relative humidity (RHmax) =

Minimum relative humidity (RHmin) =

Wind speed measured at 10 m height =

Actual hours of sunshine (n) =

21.5

12.3

84

63

10

9.25

°C

%

km/h

hours

Conversion of wind speed

At 10 m height

From Eq. 47, with

z = 10 m:

Wind speed = 10 km/h or uz =

At standard height, u2 = 0.748 (2.78) =

2.78

2.078

m/s

Parameters

From Eq. 7, for:

altitude =

P =

100

100.1

m

kPa

-

Tmean = (21.5 + 12.3)/2 =

16.9

°C

From Eq. 13, for:

Tmean =

∆ =

16.9

0.122

°C

kPa/°C

From Eq. 8, for:

P =

γ =

100.1

0.0666

kPa

kPa/°C

-

(1 + 0.34 u2) =

1.71

-

∆/[∆+γ(1+0.34u2)] =

0.122/[(0.122 + 0.0666 (1.71)] =

0.518

-

γ/[∆+γ(1+0.34u2)] =

0.0666/[0.122 + 0.0666 (1.71)] =

0.282

-

900/(Tmean+273) u2 =

6.450

-

Crop evapotranspiration

73

Vapour pressure deficit

From Eq. 11, for:

Then:

Tmax =

e°(Tmax) =

21.5

2.564

°C

kPa

From Eq. 11, for:

Then:

Tmin =

e°(Tmin) =

12.3

1.431

°C

kPa

-

es = (2.564 + 1.431) =

1.997

kPa

Given relative

humidity data

RHmax =

RHmin =

84

63

%

From Eq. 17: ea = [1.431 (0.84) + 2.564 (0.63)]/2 = 1.409 kPa

-

Vapour pressure deficit (es-ea) = (1.997 - 1.409) =

0.589

kPa

Radiation

From Table 2.5: Month 7, Day = 6

J =

187

-

From Eq. 21: Latitude = 50°48’N =

J =

Ra =

50.80

187

41.09

°N

-

MJ m-2 day-1

From Eq. 34:

Latitude = 50°48’N =

J =

N = 16.1

n/N = 9.25/16.3 =

50.80

187

16.1

0.57

°N

-

hours

-

From Eq. 35

From Eq. 37

-

From Eq. 38

Rs = [0.25 + 0.50 (0.57)] 41.09

Rso = (0.75 + 2 (100)/100 000) 41.09 =

Rs/Rso =

Rns = 0.77 (22.07) =

22.07

30.90

0.71

17.00

MJ m-2 day-1

-

MJ m-2 day-1

For:

Then:

For:

Then:

-

Tmax =

Tmax,K = 21.5 + 273.16 =

σTmax,K4 =

Tmin =

Tmin,K = 12.3 + 273.16 =

σTmin,K4 =

(σTmax,K4 + σTmin,K4)/2 = (36.96 + 32.56)/2 =

21.5

294.7

36.96

12.3

285.5

32.56

34.76

°C

K

MJ m-2 day-1

°C

K

MJ m-2 day-1

-

(0.34-0.14√ea) =

(1.35(Rs/Rso)-0.35) =

0.17

0.61

-

From Eq. 39

From Eq. 40

From Eq. 42

-

Rnl = 34.76 (0.17) 0.61 =

Rn = (17.00 - 3.71) =

G =

(Rn - G) = (13.28 - 0) =

0.408 (Rn-G) =

3.71

13.28

0

13.28

5.42

MJ m-2 day-1

mm/day

Grass reference evapotranspiration

-

0.408 (Rn-G) ∆/[∆+γ(1+0.34u2)] =

900/(T+273) u2 (es-ea) γ/[∆+γ(1+0.34u2)] =

ETo (Eq. 6) = 2.81 + 1.07 = 3.88 ≈

2.81

1.07

3.9

mm/day

The grass reference evapotranspiration is 3.9 mm/day.

Determination of ETo

74

Hourly time step

In areas where substantial changes in wind speed, dewpoint or cloudiness occur during the day,

calculation of the ETo equation using hourly time steps is generally better than using 24-hour

calculation time steps. Such weather changes can cause 24-hour means to misrepresent

evaporative power of the environment during parts of the day and may introduce error into the

calculations. However, under most conditions, application of the FAO Penman-Monteith

equation with 24-hour data produces accurate results.

With the advent of electronic, automated weather stations, weather data are increasingly

reported for hourly or shorter periods. Therefore, in situations where calculations are

computerized, the FAO Penman-Monteith equation can be applied on an hourly basis with good

results. When applying the FAO Penman-Monteith equation on an hourly or shorter time scale,

the equation and some of the procedures for calculating meteorological data should be adjusted

for the smaller time step. The FAO Penman-Monteith equation for hourly time steps is:

)u34.01(

)e)T(e(u

273T

37

)GR(408.0

2

ahr

o

2

hr

n

o

ET +γ+∆

−

+

γ+−∆

= (53)

where ETo reference evapotranspiration [mm hour-1],

Rn net radiation at the grass surface [MJ m-2 hour-1] (Equation 40),

G soil heat flux density [MJ m-2 hour-1] (Equations 45 and 46),

Thr mean hourly air temperature [°C],

∆ saturation slope vapour pressure curve at Thr [kPa °C-1] (Equation 13),

γ psychrometric constant [kPa °C-1] (Equation 8),

e°(Thr) saturation vapour pressure at air temperature Thr [kPa] (Equation 11),

ea average hourly actual vapour pressure [kPa] (Equation 54),

u2 average hourly wind speed [m s-1].

Given relative humidity measurements, the actual vapour pressure is determined as:

where ea average hourly actual vapour pressure [kPa],

e°(Thr) saturation vapour pressure at air temperature Thr [kPa] (Equation 11),

RHhr average hourly relative humidity [%].

The net radiation is the difference between the net shortwave radiation (Rns) and the net

longwave radiation (Rnl) at the hourly time steps. Consequently:

• If Rns and Rnl need to be calculated, the extraterrestrial radiation value (Ra) for the

hourly period (Equation 28) should be used.

• In the computation of Rnl by means of Equation 39, (σTmax,K4 + σTmin,K4)/2 is

replaced by σThr,K4 and the Stefan-Boltzman constant becomes:

σ = (4.903/24) 10-9 = 2.043 10-10 MJ m-2 hour-1.

100

RH

)T(ee hr

hr

o

a= (54)

Crop evapotranspiration

75

Since the ratio Rs/Rso is used to represent cloud cover, when calculating Rnl for hourly

periods during the nighttime, the ratio Rs/Rso can be set equal to the Rs/Rso calculated for a

time period occurring 2-3 hours before sunset, before the sun angle becomes small. This will

generally serve as a good approximation of cloudiness occurring during the subsequent

nighttime. The hourly period that is 2 to 3 hours before sunset can be identified during

computation of Ra as the period where ω, calculated from Equation 31, is within the range (ωs -

0.79) ≤ ω ≤ (ωs - 0.52), where ωs is calculated using Equation 25. As a more approximate

alternative, one can assume Rs/Rso = 0.4 to 0.6 during nighttime periods in humid and

subhumid climates and Rs/Rso = 0.7 to 0.8 in arid and semiarid climates. A value of Rs/Rso =

0.3 presumes total cloud cover.

Soil heat flux is important for hourly calculations. Equations 45 and 46 can be used to

derive G for the hourly periods.

The required meteorological data consist of:

• Air temperature: mean hourly temperature (Thr).

• Air humidity: average hourly relative humidity (RHhr).

• Wind speed: average hourly wind speed data measured at 2 m height (u2).

• Radiation: total hourly solar (Rs) or net radiation (Rn).

Because of the need for standardization, the constants in Equation 53 presume a constant

surface resistance (rs) of 70 s/m during all periods. This constant resistance may cause some

underprediction of hourly ETo during some daytime periods when actual rs may be somewhat

lower. The constant resistance may cause some overprediction of hourly ETo during evening

periods when actual rs may be somewhat higher. However, when the calculations of hourly

ETo from Equation 53 are summed over 24 hour periods to produce an equivalent 24-hour

ETo, the hourly differences tend to compensate one another and the results are generally

equivalent to calculations of ETo made on a 24-hour time step. Precise estimates of ETo for

specific hourly periods may require the use of aerodynamic stability functions and functions

for modifying the value for rs based on levels of radiation, humidity and temperature.

Application of these functions are not normally required when hourly calculations are to be

summed to 24-hour totals. Therefore, these functions are not described here.

EXAMPLE 19

Determination of ETo with hourly data

Given mean average hourly data between 02.00 and 03.00 hours and 14.00 and 15.00 hours on 1

October in N’Diaye (Senegal) at 16°13'N and 16°15'W and 8 m above sea level. In the absence of

calibrated coefficients, indicative values for as and bs (Eq. 35 Angstrom formula) and for the coefficients

of the net longwave radiation (Eq. 39) are used.

Measured climatic data 02.00-03.00 h 14.00-15.00 h Units

Thr: mean hourly temperature =

RHhr: mean hourly relative humidity =

u2: mean hourly wind speed =

Rs: total solar radiation =

28

90

1.9

-

38

52

3.3

2.450

°C

%

m/s

MJ m-2 hour-1

Parameters

From Eq. 13

From Eq. 8

∆ =

γ =

0.220

0.0673

0.358

0.0673

kPa °C-1

Vapour pressure deficit

From Eq. 11

From Eq. 54

-

eº(T) =

ea =

es - ea =

3.780

3.402

0.378

6.625

3.445

3.180

kPa

Determination of ETo

76

Extraterrestrial radiation 02.00-03.00 h 14.00-15.00 h Units

From Table 2.5

for 1 October:

From Eq. 22:

From Eq. 23:

From Eq. 24:

From Eq. 33:

From Eq. 32:

-

J = 274

ϕ = π/180 (16.22) = 0.2830

dr = 1.0001

δ = - 0.0753

b = 3.3315

Sc = 0.1889

Lz = 15

Lm = 16.25

-

rad

-

rad

-

hour

degrees

-

From Eq. 31:

-

From Eq. 29:

From Eq. 30:

From Eq. 28:

t =

ω =

tl =

ω1 =

ω2 =

Ra =

2.5

-2.46

1

-

0

14.5

0.682

1

0.5512

0.8130

3.543

hour

rad

hour

rad

MJ m-2 hour-1

Radiation

Given

From Eq. 37:

From Eq. 38:

-

From Eq. 39:

From Eq. 40:

From Eq. 46, 45:

-

Rs =

Rso =

Rns =

σ TK4 =

(0.34-0.14 √ea) =

Rs/Rso =

(1.35 Rs/Rso - 0.35) =

Rnl =

Rn =

G =

(Rn - G) =

0.408(Rn - G) =

0

1.681

0.082

0.8 (assumed)

0.730

0.100

-0.100

-0.050

-0.020

2.450

2.658

1.887

1.915

0.080

0.922

0.894

0.137

1.749

0.175

1.574

0.642

MJ m-2 hour-1

-

MJ m-2 hour-1

mm/hour

Grass reference evapotranspiration

-

From Eq. 53:

0.408(Rn-G)

∆/[∆+γ(1+0.34u2)] =

37/(T+273) u2 (es-ea)

γ/[∆+γ( 1 +0.34u2)=

ETo =

-0.01

0.01

0.00

0.46

0.17

0.63

mm/hour

The grass reference evapotranspiration is 0.00 mm/hour between 02.00 and 03.00 hours and

0.63 mm/hour between 14.00 and 15.00 hours.

CALCULATION PROCEDURES WITH MISSING DATA

The meteorological data, required to estimate ETo by means of the FAO Penman-Monteith

equation, consist of air temperature, air humidity, wind speed and radiation. Where some of

these data are missing or cannot be calculated, it is strongly recommended that the user estimate

the missing climatic data with one of the procedures presented in Chapter 3 and that the FAO

Penman-Monteith method be used for the calculation of ETo. The use of an alternative ETo

calculation procedure, requiring only limited meteorological parameters, is less recommended.

Example 20 illustrates the estimation of monthly ETo with the FAO Penman-Monteith

for a data set containing only maximum and minimum air temperature. The procedures given in

Chapter 3 to estimate missing humidity, radiation and wind speed data should be validated by

comparing ETo calculated with full and with limited data sets for weather stations in the region

with complete data sets.

Crop evapotranspiration

77

EXAMPLE 20

Determination of ETo with missing data

Given the monthly average daily maximum and average daily minimum air temperature of July from a

station near Lyon, France (45°43' N, altitude 200 m). No other climatic data were recorded.

-

Monthly average daily maximum temperature (Tmax) =

Monthly average daily minimum temperature (Tmin) =

26.6

14.8

°C

Determination according to Box 11 (calculation sheet ETo)

Estimation of wind speed:

2 m/s is used as a temporary estimate. Due to the relatively small crop height of 0.12 m of the

reference crop and the appearance of u2 in both the nominator and denominator of the FAO Penman-

Monteith equation, ETo is not highly sensitive to normal ranges of wind speed.

Parameters:

- Tmean = (26.6 + 14.8)/2 = 20.7 °C

From Table 2.4 or

Eq. 13:

Tmean =

∆ =

20.7

0.150

°C

kPa/°C

From Table 2.2 or

Eq. 8:

Altitude =

γ =

200

0.066

m

kPa/°C

- (1 + 0.34 u2) = (1 + 0.34 (2)) = 1.68 -

- ∆/[∆+γ(1+0.34u2)] =0.150/[(0.150 +

0.066 (1.68)] =

0.576

-

- γ/[∆+γ(1+0.34u2)] = 0.0658/[0.150 +

0.066 (1.68)] =

0.252

-

- 900/(Tmean+273) u2 = 6.13 -

Estimation of humidity data:

Assume (Eq. 48):

Consequently

(Eq. 14 or Table

2.3) for:

Then

Tdew ≈ Tmin =

Tdew =

ea =

14.8

1.68

°C

kPa

From Table 2.3 or

Eq. 11, for:

Then:

Tmax =

e°(Tmax) =

26.6

3.48

°C

kPa

From Table 2.3 or

Eq. 11, for: Then:

Tmin =

e°(Tmin) =

14.8

1.68

°C

kPa

-

es = (3.48 + 1.68)/2 =

(es-ea) = (2.58 - 1.68) =

2.58

0.90

kPa

This corresponds with:

-

RHmax = 100 ea/e°(Tmin) =

RHmin = 100 ea/e°(Tmax) = 100 (1.68/3.48) =

RHmean = (RHmax + RHmin)/2 =

100

48

74

%

Estimation of radiation data:

Rs can be derived from the difference between Tmax and Tmin:

Determination of ETo

78

From Eq. 50

-

Rs = 0.16 √(26.6-14.8) Ra

Rs = 0.55 Ra

-

MJ m-2 day-1

Table 2.6 or Eq.

21, for:

Then:

For Day 15, Month = July, J =

Latitude = 45°43’N =

Ra =

196

45.72

40.55

-

°N

MJ m-2 day-1

-

Rs = 0.55 Ra = 0.55 (40.55) =

Rso = (0.75 + 2 (200)/100 000) 40.55 =

Rs/Rso =

Rns = 0.77 (22.29) =

22.29

30.58

0.73

17.16

MJ m-2 day-1

-

MJ m-2 day-1

Table 2.8, for:

Then:

Table 2.8, for:

Then:

-

Tmax = 26.6°C

Tmax,K = 26.6 + 273.16 =

σTmax,K4

Tmin = 14.8°C

Tmin,K = 14.8 + 273.16

σTmin,K4

(σTmax,K4 + σTmin,K4)/2 = (39.59 + 33.71)/2 =

26.6

299.76

39.59

14.8

287.96

33.71

36.65

°C

K

MJ m-2 day-1

°C

K

MJ m-2 day-1

For:

Then:

ea =

(0.34-0.14√ea) =

1.68

0.16

kPa

-

For:

Then:

-

Rs/Rso =

(1.35 Rs/Rso - 0.35) =

Rnl = 36.65 (0.16) 0.63 =

Rn = (17.16 - 3.68) =

0.73

0.63

3.68

13.48

-

MJ m-2 day-1

Assume:

-

G =

(Rn - G) = (13.48 - 0) =

0.408 (Rn - G) =

0

13.48

5.50

MJ m-2 day-1

mm/day

Grass reference evapotranspiration:

-

0.408 (Rn-G) ∆/[∆+γ(1+0.34u2)] =

900/(T+273) u2 (es-ea) γ/[∆+γ(1+0.34u2)] =

ETo = (3.17 + 1.39) =

3.17

1.39

4.56

mm/day

The estimated grass reference evapotranspiration is 4.6 mm/day. If instead of 2 m/s, the wind speed is

estimated as 1 or 3 m/s, ETo would have been 7% lower (4.2 mm/day) or 6% higher (4.8 mm/day)

respectively. In comparison, the Hargreaves equation (Equation 52) predicts ETo = 5.0 mm/day

PAN EVAPORATION METHOD

Pan evaporation

The evaporation rate from pans filled with water is easily obtained. In the absence of rain, the

amount of water evaporated during a period (mm/day) corresponds with the decrease in water

depth in that period. Pans provide a measurement of the integrated effect of radiation, wind,

temperature and humidity on the evaporation from an open water surface. Although the pan

responds in a similar fashion to the same climatic factors affecting crop transpiration, several

factors produce significant differences in loss of water from a water surface and from a cropped

surface. Reflection of solar radiation from water in the shallow pan might be different from the

assumed 23% for the grass reference surface. Storage of heat within the pan can be appreciable

and may cause significant evaporation during the night while most crops transpire only during

the daytime. There are also differences in turbulence, temperature and humidity of the air

immediately above the respective surfaces. Heat transfer through the sides of the pan occurs

and affects the energy balance.

Crop evapotranspiration

79

Notwithstanding the difference between pan-evaporation and the evapotranspiration of

cropped surfaces, the use of pans to predict ETo for periods of 10 days or longer may be

warranted. The pan evaporation is related to the reference evapotranspiration by an empirically

derived pan coefficient:

where ETo reference evapotranspiration [mm/day],

Kp pan coefficient [-],

Epan pan evaporation [mm/day].

Pan coefficient (Kp)

Pan types and environment

Different types of pans exist. Descriptions of Class A and Colorado sunken pans are given in

Boxes 12 and 13. As the colour, size, and position of the pan have a significant influence on the

measured results, the pan coefficients are pan specific.

In selecting the appropriate pan coefficient, not only the pan type, but also the ground

cover in the station, its surroundings as well as the general wind and humidity conditions,

should be checked. The siting of the pan and the pan environment also influence the results.

This is particularly so where the pan is placed in fallow rather than cropped fields. Two cases

are commonly considered: Case A where the pan is sited on a short green (grass) cover and

surrounded by fallow soil; and Case B where the pan is sited on fallow soil and surrounded by a

green crop (Figure 19).

panpo EKET = (55)

FIGURE 19

Two cases of evaporation pan siting and their environment

wind wind

Case A Case B

dry

surface

dry

surface

green

crop

green

crop

pan pan

fetch fetch

50 m or more 50 m or more

Determination of ETo

80

Pan coefficients

Depending on the type of pan and the size and state of the upwind buffer zone (fetch), pan

coefficients will differ. The larger the upwind buffer zone, the more the air moving over the pan

will be in equilibrium with the buffer zone. At equilibrium with a large fetch, the air contains

more water vapour and less heat in Case A than in Case B. Pan coefficients for the Class A pan

and for the Colorado sunken pan for different ground cover, fetch and climatic conditions are

presented in Tables 5 and 6. Regression equations derived from the tables are presented in

Table 7. Where measured data from other types of sunken pans are available, such data should

first be related to Colorado sunken pan data or to the FAO Penman-Monteith equation to

develop Kp. Ratios between evaporation from sunken pans and from the Colorado sunken pan

for different climatic conditions and pan environment are given in Table 8.

Where data are missing, wind speed can be estimated by taking a global value of u2 = 2

m s-1 or as indicated in Table 4 (page 63). RHmean can be approximated from air temperature

as RHmean = 50 eo(Tmin)/eo(Tmax) + 50.

Adjustments

Under some conditions not accounted for in the tables, the presented Kp coefficients may need

some adjustment. This is the case in areas with no agricultural development, or where the pans

are enclosed by tall crops. Not maintaining the standard colour of the pan or installing screens

can affect the pan readings and will require some adjustment of the pan coefficient.

In areas with no agricultural development and extensive areas of bare soils (large fetch,

Case B), as found under desert or semi-desert conditions, the listed values for Kp given for arid,

windy areas may need to be reduced by up to 20%; for areas with moderate levels of wind,

temperature and relative humidity, the listed values may need to be reduced by 5-10%; no or

little reduction in Kp is needed in humid, cool conditions.

Where pans are placed in a small enclosure but surrounded by tall crops, for example 2.5

m high maize, the listed pan coefficients will need to be increased by up to 30% for dry windy

climates whereas only a 5-10% increase is required for calm, humid conditions.

Painting the pans may affect the pan evaporation. The pan coefficients presented apply to

galvanized pans annually painted with aluminium and to stainless steel pans. Little difference in

Epan will occur where the inside and outside surfaces of the pan are painted white. An increase

in Epan of up to 10% may occur when they are painted black. The material from which the pan

is made may account for variations of only a few percent.

The level at which the water is maintained in the pan is important; resulting errors may

be up to 15% when water levels in the Class A pan fall 10 cm below the accepted standard of

between 5 and 7.5 cm below the rim. Screens mounted over pans will reduce Epan by up to

10%. In an attempt to avoid pans being used by birds for drinking, pans filled to the rim with

water can be placed near the Class A pan; birds may prefer to use the fully filled pan. The

evaporation pan should be placed in a large, secure, wire enclosure to prevent animals from

entering and drinking. The turbidity of the water in the pan usually does not affect Epan by

more than 5%. The overall variation in Epan is not constant with time because of ageing,

deterioration and repainting.

Crop evapotranspiration

81

TABLE 5

Pan coefficients (Kp) for Class A pan for different pan siting and environment and different levels

of mean relative humidity and wind speed (FAO Irrigation and Drainage Paper No. 24)

Class A pan Case A: Pan placed in short green cropped

area

Case B: Pan placed in dry fallow area

RH mean

(%) à

low

< 40

medium

40 -70

high

> 70

low

< 40

medium

40 -70

high

> 70

Wind speed

(m s-1)

Windward

side

distance of

green crop

(m)

Windward

side

distance of

dry fallow

(m)

Light 1 .55 .65 .75 1 .7 .8 .85

< 2 10 .65 .75 .85 10 .6 .7 .8

100 .7 .8 .85 100 .55 .65 .75

1 000 .75 .85 .85 1 000 .5 .6 .7

Moderate 1 .5 .6 .65 1 .65 .75 .8

2-5 10 .6 .7 .75 10 .55 .65 .7

100 .65 .75 .8 100 .5 .6 .65

1 000 .7 .8 .8 1 000 .45 .55 .6

Strong 1 .45 .5 .6 1 .6 .65 .7

5-8 10 .55 .6 .65 10 .5 .55 .65

100 .6 .65 .7 100 .45 .5 .6

1 000 .65 .7 .75 1 000 .4 .45 .55

Very strong 1 .4 .45 .5 1 .5 .6 .65

> 8 10 .45 .55 .6 10 .45 .5 .55

100 .5 .6 .65 100 .4 .45 .5

1 000 .55 .6 .65 1 000 .35 .4 .45

TABLE 6

Pan coefficients (Kp) for Colorado sunken pan for different pan siting and environment and

different levels of mean relative humidity and wind speed (FAO Irrigation and Drainage Paper

No. 24)

Sunken

Colorado

Case A: Pan placed in short green

cropped area

Case B: Pan placed in dry fallow area

(1)

RH mean

(%) à

low

< 40

medium

40 -70

high

> 70

low

< 40

medium

40 -70

high

> 70

Wind speed

(m s-1)

Windward

side

distance of

green crop

(m)

Windward

side

distance of

dry fallow

(m)

Light 1 .75 .75 .8 1 1.1 1.1 1.1

< 2 10 1.0 1.0 1.0 10 .85 .85 .85

≥ 100 1.1 1.1 1.1 100 .75 .75 .8

1 000 .7 .7 .75

Moderate 1 .65 .7 .7 1 .95 .95 .95

2-5 10 .85 .85 .9 10 .75 .75 .75

≥ 100 .95 .95 .95 100 .65 .65 .7

1 000 .6 .6 .65

Strong 1 .55 .6 .65 1 .8 .8 .8

5-8 10 .75 .75 .75 10 .65 .65 .65

≥ 100 .8 .8 .8 100 .55 .6 .65

1 000 .5 .55 .6

Very strong 1 .5 .55 .6 1 .7 .75 .75

> 8 10 .65 .7 .7 10 .55 .6 .65

≥ 100 .7 .75 .75 100 .5 .55 .6

1 000 .45 .5 .55

(1) For extensive areas of bare-fallow soils and no agricultural development, reduce Kpan by 20% under

hot, windy conditions; by 5-10% for moderate wind, temperature and humidity conditions.

Determination of ETo

82

TABLE 7

Pan coefficients (Kp): regression equations derived from Tables 5 and 6

Class A pan with

green fetch

)

mean

ln(RH

2

[ln(FET)]0.000631

)

mean

ln(RH0.1434ln(FET)0.0422

2

u0.02860.108

p

K

−

++−=

Class A pan with

dry fetch

)u4.86ln()][ln(00063.0

)ln()u4.86ln(0106.0)u4.86ln(u00289.0

)ln(u00327.0FETu00000959.0

RHu000162.0RH00341.061.0K

2

222

22

mean2meanp

FET

+

−−

+−

−+=

Colorado sunken

pan with green

fetch

mean2mean

22

2

2p

RH)FETln()u4.86ln(000053.0)RHln(

)u4.86ln(013.0)u4.86ln()]FET[ln(0019.0

)]u4.86[ln(0157.0)FETln(119.087.0K

−

+−

−+=

Colorado sunken

pan with dry fetch

)RHln()]FET[ln(0015.0

)FETln(u0031.0)FETln(0964.0

)RHln()u(000903.0u080.0145.1K

mean

2

mean

2

22p

+

+−

+−=

Coefficients and

parameters

Kp pan coefficient [ ]

u2 average daily wind speed at 2 m height (m s-1)

RHmean average daily relative humidity [%] = (RHmax + RHmin)/2

FET fetch, or distance of the identified surface type (grass or

short green agricultural crop for case A, dry crop or bare

soil for case B upwind of the evaporation pan)

Range

for

variables

1 m ≤ FET ≤ 1 000 m (these limits must be observed)

30% ≤ RHmean ≤ 84%

1 m s-1 ≤ u2 ≤ 8 m s-1

Recommendations

The above considerations and adjustments indicate that the use of tables or the corresponding

equations may not be sufficient to consider all local environmental factors influencing Kp and

that local adjustment may be required. To do so, an appropriate calibration of Epan against

ETo computed with the Penman-Monteith method is recommended.

It is recommended that the pan should be installed inside a short green cropped area with

a size of a square of at least 15 by 15 m. The pan should not be installed in the centre but at a

distance of at least 10 m from the green crop edge in the general upwind direction.

Where observations of wind speed and relative humidity, required for the computation of

Kp, are not available at the site, estimates of the weather variables from a nearby station have to

be utilized. It is then recommended that these variables be averaged for the computation period

and that Epan be averaged for the same period.

Equation 1 in Table 7 yields Kp = 0.83 for data in Example 21 as shown in Example 22.

Crop evapotranspiration

83

TABLE 8

Ratios between the evaporation from sunken pans and a Colorado sunken pan for different

climatic conditions and environments (FAO Irrigation and Drainage Paper No. 24)

Ratio Epan mentioned and Epan Colorado

Climate Humid-temperate climate Arid to semi-arid

(dry season)

Ground cover surrounding pan (50 m or

more)

Short green

cover

Dry fallow

Short green

cover

Dry fallow

Pan area

(m2)

GGI 20 diameter 5 m, depth 2 m

(former Soviet Union)

19.6

1.0

1.1

1.05

1.25

Sunken pan diameter 12 ft

(3.66 m) depth 3.3 ft (Israel)

10.5

BPI diameter 6 ft (1.83 m), depth

2 ft (0.61 m) (USA)

2.6

Kenya pan diameter 4 ft

(1.22 m) depth 14 in (0.356 m)

1.2

Australian pan diameter 3 ft

(0.91 m) depth 3 ft (0.91 m)

0.7

1.0

Symmons pan 6 ft2 (0.56 m2)

depth 2 ft (0.61 m)

0.6

Aslyng pan 0.33 m2, depth 1 m

(Denmark)

0.3

1.0

GGI 3000 diameter 0.618 cm,

depth 60-80 cm (former Soviet

Union)

0.3

Sunken pan diameter 50 cm,

depth 25 cm (Netherlands)

0.2

1.0

0.95

1.0

0.95

EXAMPLE 21

Determination of ETo from pan evaporation using tables

Given the daily evaporation data for the first week of July for a Class A pan installed in a green area

surrounded by short irrigated field crops: 8.2, 7.5, 7.6, 6.8, 7.6, 8.9 and 8.5 mm/day. In that period the

mean wind speed is 1.9 m/s and the daily mean relative humidity is 73%. Determine the 7-day average

reference evapotranspiration.

Pan is installed on a green surface: Case A

Pan is surrounded by

irrigated crops:

Wind speed is light:

Relative humidity is high:

fetchmax =

u <

RHmean >

1 000

2

70

m

m/s

%

From Table 5 (for above

conditions):

Kp =

0.85

-

From Eq. 55:

Epan = (8.2+7.5+7.6+6.8+7.6+8.9+8.5)/7 =

ETo = 0.85 (7.9) =

7.9

6.7

mm/day

The 7-day average of the crop reference evapotranspiration is 6.7 mm/day

Determination of ETo

84

BOX 12

Description of Class A pan

The Class A Evaporation pan is circular, 120.7 cm in diameter and 25 cm deep. It is made of

galvanized iron (22 gauge) or Monel metal (0.8 mm). The pan is mounted on a wooden open frame

platform which is 15 cm above ground level. The soil is built up to within 5 cm of the bottom of the pan.

The pan must be level. It is filled with water to 5 cm below the rim, and the water level should not be

allowed to drop to more than 7.5 cm below the rim. The water should be regularly renewed, at least

weekly, to eliminate extreme turbidity. The pan, if galvanized, is painted annually with aluminium paint.

Screens over the pan are not a standard requirement and should preferably not be used. Pans should

be protected by fences to keep animals from drinking.

The site should preferably be under grass, 20 by 20 m, open on all sides to permit free circulation of

the air. It is preferable that stations be located in the centre or on the leeward side of large cropped

fields.

Pan readings are taken daily in the early morning at the same time that precipitation is measured.

Measurements are made in a stilling well that is situated in the pan near one edge. The stilling well is a

metal cylinder of about 10 cm in diameter and some 20 cm deep with a small hole at the bottom.

120.7 cm

15 cm

25 cm

stilling well

water level

5 - 7.5 cm from rim

Crop evapotranspiration

85

BOX 13

Description of Colorado sunken pan

The Colorado sunken pan is 92 cm (3 ft) square and 46 cm (18 in) deep, made of 3 mm thick iron,

placed in the ground with the rim 5 cm (2 in) above the soil level. Also, the dimensions 1 m square and

0.5 m deep are frequently used. The pan is painted with black tar paint. The water level is maintained

at or slightly below ground level, i.e., 5-7.5 cm below the rim.

Measurements are taken similarly to those for the Class A pan. Siting and environment requirements

are also similar to those for the Class A pan.

Sunken Colorado pans are sometimes preferred in crop water requirements studies, as these pans

give a better direct estimation of the reference evapotranspiration than does the Class A pan. The

disadvantage is that maintenance is more difficult and leaks are not visible.

5 cm

46 cm

92 cm

water level

5 - 7.5 cm from rim

stilling well

Determination of ETo

86

EXAMPLE 22

Determination of ETo from pan evaporation using equations

Given the 7-day average evaporation measurement from Example 21, estimate the ETo for the two types of pans and two types of fetch conditions represented by equations in Table 7. Assume

that fetch is 1000 m in both fetch cases (green and dry).

Fetch =

u2 =

RHmean =

1000

1.9

73

m

m/s

%

Class A pan

with green fetch )RHln()]FET[ln(000631.0

)RHln(1434.0)FETln(0422.0u0286.0108.0K

mean

2

mean2p

−

++−=

Epan =

ETo =

)73ln()]1000[ln(000631.0

)73ln(1434.0)1000ln(0422.0)9.1(0286.0108.0K

2

p

−

++−=

ETo = 0.83 (7.9)

0.83

7.9

6.6

-

mm/day

Class A pan

with dry fetch

)u4.86ln()][ln(00063.0

)ln()u4.86ln(0106.0)u4.86ln(u00289.0

)ln(u00327.0FETu00000959.0

RHu000162.0RH00341.061.0K

2

222

22

mean2meanp

FET

+

−−

+−

−+=

Epan =

ETo =

))9.1(4.86ln()]1000[ln(00063.0

)1000ln())9.1(4.86ln(0106.0))9.1(4.86ln()9.1(00289.0

)1000ln()9.1(00327.0)1000()9.1(00000959.0

)73()9.1(000162.0)73(00341.061.0K

2

p

+

−−

+−

−+=

ETo = 0.61 (7.9)

0.61

7.9

4.8

-

mm/day

Colorado

sunken pan with

green fetch

mean2mean

22

2

2p

RH)FETln()u4.86ln(000053.0)RHln(

)u4.86ln(013.0)u4.86ln()]FET[ln(0019.0

)]u4.86[ln(0157.0)FETln(119.087.0K

−

+−

−+=

Epan =

ETo =

)73()1000ln())9.1(4.86ln(000053.0)73ln(

))9.1(4.86ln(013.0))9.1(4.86ln()]1000[ln(0019.0

))]9.1(4.86[ln(0157.0)100ln(119.087.0K

2

p

−

+−

−+=

ETo = 0.97 (7.9)

0.97

7.9

7.7

-

mm/day

Colorado

sunken pan with

dry fetch

)RHln()]FET[ln(0015.0

)FETln(u0031.0)FETln(0964.0

)RHln()u(000903.0u080.0145.1K

mean

2

mean

2

22p

+

+−

+−=

Epan =

ETo =

)73ln()]1000[ln(0015.0

)1000ln()9.1(0031.0)1000ln(0964.0

)73ln()9.1(000903.0)9.1(080.0145.1K

2

p

+

+−

+−=

ETo = 0.69 (7.9)

0.69

7.9

5.4

-

mm/day

The 7-day average of the crop reference evapotranspiration for the four pan/fetch conditions is 6.6, 4.8, 7.7, and 5.4 mm/day

Crop evapotranspiration

87

Part B

Crop evapotranspiration under

standard conditions

This part examines crop evapotranspiration under standard conditions (ETc). This is the

evapotranspiration from disease-free, well-fertilized crops, grown in large fields, under

optimum soil water conditions and achieving full production under the given climatic

conditions.

The effects of various weather conditions on evapotranspiration are incorporated into ETo (Part

A). The effects of characteristics that distinguish the cropped surface from the reference surface

are integrated into the crop coefficient. By multiplying ETo by the crop coefficient, ETc is

determined.

Typical crop coefficients, calculation procedures for adjusting the crop coefficients and for

calculating ETc are presented in this part. Two calculation approaches are outlined: the single

and the dual crop coefficient approach. In the single crop coefficient approach, the difference in

evapotranspiration between the cropped and reference grass is combined into one single

coefficient. In the dual crop coefficient approach, the crop coefficient is split into two factors

describing separately the differences in evaporation and transpiration between the crop and

reference surface.

As discussed in Chapter 5 and summarized in Table 10, the single crop coefficient approach

is used for most applications related to irrigation planning, design, and management. The dual

crop coefficient approach is relevant in calculations where detailed estimates of soil water

evaporation are required, such as in real time irrigation scheduling applications, water quality

modelling, and in research.

88

Crop evapotranspiration

89

Chapter 5

Introduction to crop

evapotranspiration (ETc)

This chapter outlines the crop coefficient approach for calculating the crop evapotranspiration

under standard conditions (ETc). The standard conditions refer to crops grown in large fields

under excellent agronomic and soil water conditions. The crop evapotranspiration differs

distinctly from the reference evapotranspiration (ETo) as the ground cover, canopy properties

and aerodynamic resistance of the crop are different from grass. The effects of characteristics

that distinguish field crops from grass are integrated into the crop coefficient (Kc). In the crop

coefficient approach, crop evapotranspiration is calculated by multiplying ETo by Kc.

Differences in evaporation and transpiration between field crops and the reference grass

surface can be integrated in a single crop coefficient (Kc) or separated into two coefficients: a

basal crop (Kcb) and a soil evaporation coefficient (Ke), i.e., Kc = Kcb + Ke. The approach to

follow should be selected as a function of the purpose of the calculation, the accuracy required

and the data available.

CALCULATION PROCEDURES

Direct calculation

The evapotranspiration rate from a cropped surface can be directly measured by the mass

transfer or the energy balance method. It can also be derived from studies of the soil water

balance determined from cropped fields or from lysimeters.

Crop evapotranspiration can also be derived from meteorological and crop data by means

of the Penman-Monteith equation (Eq. 3). By adjusting the albedo and the aerodynamic and

canopy surface resistances to the growing characteristics of the specific crop, the

evapotranspiration rate can be directly estimated. The albedo and resistances are, however,

difficult to estimate accurately as they may vary continually during the growing season as

climatic conditions change, as the crop develops, and with wetness of the soil surface. The

canopy resistance will further be influenced by the soil water availability, and it increases

strongly if the crop is subjected to water stress.

As there is still a considerable lack of consolidated information on the aerodynamic and

canopy resistances for the various cropped surfaces, the FAO Penman-Monteith method is used

in this handbook only for estimating ETo, the evapotranspiration from a well-watered

hypothetical grass surface having fixed crop height, albedo and surface resistance.

Introduction to crop evapotranspiration (ETc)

90

Crop coefficient approach

In the crop coefficient approach the crop evapotranspiration, ETc, is calculated by multiplying

the reference crop evapotranspiration, ETo, by a crop coefficient, Kc:

occ ETKET = (56)

where ETc crop evapotranspiration [mm d-1],

Kc crop coefficient [dimensionless],

ETo reference crop evapotranspiration [mm d-1].

Most of the effects of the various weather conditions are incorporated into the ETo

estimate. Therefore, as ETo represents an index of climatic demand, Kc varies predominately

with the specific crop characteristics and only to a limited extent with climate. This enables the

transfer of standard values for Kc between locations and between climates. This has been a

primary reason for the global acceptance and usefulness of the crop coefficient approach and

the Kc factors developed in past studies.

The reference ETo is defined and calculated using the FAO Penman-Monteith equation

(Chapter 4). The crop coefficient, Kc, is basically the ratio of the crop ETc to the reference

ETo, and it represents an integration of the effects of four primary characteristics that

distinguish the crop from reference grass. These characteristics are:

• Crop height. The crop height influences the aerodynamic resistance term, ra, of the FAO

Penman-Monteith equation and the turbulent transfer of vapour from the crop into the

atmosphere. The ra term appears twice in the full form of the FAO Penman-Monteith

equation.

• Albedo (reflectance) of the crop-soil surface. The albedo is affected by the fraction of

ground covered by vegetation and by the soil surface wetness. The albedo of the crop-soil

surface influences the net radiation of the surface, Rn, which is the primary source of the

energy exchange for the evaporation process.

• Canopy resistance. The resistance of the crop to vapour transfer is affected by leaf area

(number of stomata), leaf age and condition, and the degree of stomatal control. The

canopy resistance influences the surface resistance, rs.

• Evaporation from soil, especially exposed soil.

The soil surface wetness and the fraction of ground covered by vegetation influence the

surface resistance, rs. Following soil wetting, the vapour transfer rate from the soil is high,

especially for crops having incomplete ground cover. The combined surface resistance of the

canopy and of the soil determines the (bulk) surface resistance, rs. The surface resistance term

in the Penman-Monteith equation represents the resistance to vapour flow from within plant

leaves and from beneath the soil surface.

The Kc in Equation 56 predicts ETc under standard conditions. This represents the upper

envelope of crop evapotranspiration and represents conditions where no limitations are placed

on crop growth or evapotranspiration due to water shortage, crop density, or disease, weed,

insect or salinity pressures. The ETc predicted by Kc is adjusted if necessary to non-standard

Crop evapotranspiration

91

conditions, ETc adj, where any environmental condition or characteristic is known to have an

impact on or to limit ETc. Factors for correcting ETc to ETc adj are described in Part C.

FACTORS DETERMINING THE CROP COEFFICIENT

The crop coefficient integrates the effect of characteristics that distinguish a typical field crop

from the grass reference, which has a constant appearance and a complete ground cover.

Consequently, different crops will have different Kc coefficients. The changing characteristics

of the crop over the growing season also affect the Kc coefficient. Finally, as evaporation is an

integrated part of crop evapotranspiration, conditions affecting soil evaporation will also have

an effect on Kc.

Crop type

Due to differences in albedo, crop height, aerodynamic properties, and leaf and stomata

properties, the evapotranspiration from full grown, well-watered crops differs from ETo.

The close spacings of plants and taller canopy height and roughness of many full grown

agricultural crops cause these crops to have Kc factors that are larger than 1. The Kc factor is

often 5-10% higher than the reference (where Kc = 1.0), and even 15-20% greater for some tall

crops such as maize, sorghum or sugar cane (Figure 20). Typical values for the crop coefficient

for full grown crops (Kc mid) are listed in Table 12.

Crops such as pineapples, that close their stomata during the day, have very small crop

coefficients. In most species, however, the stomata open as irradiance increases. In addition to

the stomatal response to environment, the position and number of the stomata and the resistance

of the cuticula to vapour transfer determine the water loss from the crop. Species with stomata

on only the lower side of the leaf and/or large leaf resistances will have relatively smaller Kc

values. This is the case for citrus and most deciduous fruit trees. Transpiration control and

spacing of the trees, providing only 70% ground cover for mature trees, may cause the Kc of

those trees, if cultivated without a ground cover crop, to be smaller than one (Figure 20).

Climate

The Kc values of Table 12 are typical values expected for average Kc under a standard climatic

condition, which is defined as a sub-humid climate with average daytime minimum relative

humidity (RHmin) ≈ 45% and having calm to moderate wind speeds averaging 2 m/s.

Variations in wind alter the aerodynamic resistance of the crops and hence their crop

coefficients, especially for those crops that are substantially taller than the hypothetical grass

reference. The effect of the difference in aerodynamic properties between the grass reference

surface and agricultural crops is not only crop specific. It also varies with the climatic

conditions and crop height. Because aerodynamic properties are greater for many agricultural

crops as compared to the grass reference, the ratio of ETc to ETo (i.e., Kc) for many crops

increases as wind speed increases and as relative humidity decreases. More arid climates and

conditions of greater wind speed will have higher values for Kc. More humid climates and

conditions of lower wind speed will have lower values for Kc.

Introduction to crop evapotranspiration (ETc)

92

FIGURE 20

Typical Kc for different types of full grown crops

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

reference

Kc

Pineapple

Citrus

Peaches Cotton

Small

Vegetables

Maize

Sugar

cane

Large

Vegetables

Apples

Cherries

grass

at mid-season

FIGURE 21

Extreme ranges expected in Kc for full grown crops as climate and weather change

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

Kc at mid-season

cabbage

extremely

arid & windy

very humid

& calm

cotton

maize

sugar cane

tall

crops

grass

reference

short

beans

onion

Crop evapotranspiration

93

The relative impact of the climate on Kc for full grown crops is illustrated in Figure 21.

The upper bounds represent extremely arid and windy conditions, while the lower bounds are

valid under very humid and calm weather conditions. The ranges expected in Kc as climate and

weather conditions change are quite small for short crops but are large for tall crops. Guidelines

for the adjustment of Kc to the climatic conditions as a function of crop height are given in

Chapter 6.

Under humid and calm wind conditions, Kc becomes less dependent on the differences

between the aerodynamic components of ETc and ETo and the Kc values for ‘full-cover’

agricultural crops do not exceed 1.0 by more than about 0.05. This is because full-cover

agricultural crops and the reference crop of clipped grass both provide for nearly maximum

absorption of shortwave radiation, which is the primary energy source for evaporation under

humid and calm conditions. Generally, the albedos, α, are similar over a wide range of full-

cover agricultural crops, including the reference crop. Because the vapour pressure deficit (es -

ea) is small under humid conditions, differences in ET caused by differences in aerodynamic

resistance, ra, between the agricultural crop and the reference crop are also small, especially

with low to moderate wind speed.

Under arid conditions, the effect of differences in ra between the agricultural crop and

the grass reference crop on ETc become more pronounced because the (es - ea) term may be

relatively large. The larger magnitudes of (es - ea) amplify differences in the aerodynamic term

in the numerator of the Penman-Monteith equation (Equation 3) for both the crop and the

reference crop. Hence, Kc will be larger under arid conditions when the agricultural crop has a

leaf area and roughness height that are greater than that of the grass reference.

Because the 1/ra term in the numerator of the Penman-Monteith equation (Equation 3) is

multiplied by the vapour pressure deficit (es - ea), the ET from tall crops increases

proportionately more relative to ETo than does ET from short crops when relative humidity is

low. The Kc for tall crops, such as those 2-3 m in height, can be as much as 30% higher in a

windy, arid climate as compared with a calm, humid climate. The increase in Kc is due to the

influence of the larger aerodynamic roughness of the tall crop relative to grass on the transport

of water vapour from the surface.

Soil evaporation

Differences in soil evaporation and crop transpiration between field crops and the reference

surface are integrated within the crop coefficient. The Kc coefficient for full-cover crops

primarily reflects differences in transpiration as the contribution of soil evaporation is relatively

small. After rainfall or irrigation, the effect of evaporation is predominant when the crop is

small and scarcely shades the ground. For such low-cover conditions, the Kc coefficient is

determined largely by the frequency with which the soil surface is wetted. Where the soil is wet

for most of the time from irrigation or rain, the evaporation from the soil surface will be

considerable and Kc may exceed 1. On the other hand, where the soil surface is dry,

evaporation is restricted and Kc will be small and might even drop to as low as 0.1 (Figure 22).

Differences in soil evaporation between the field crop and the reference surface can be

forecast more precisely by using a dual crop coefficient.

Introduction to crop evapotranspiration (ETc)

94

FIGURE 22

The effect of evaporation on Kc. The horizontal line represents Kcwhen the soil surface is kept

continuously wet. The curved line corresponds to Kcwhen the soil surface is kept dry but the

crop receives sufficient water to sustain full transpiration

0

0.2

0.4

0.6

0.8

1.0

1.2

seedling full cover

dry

so

il

evaporation

transpiration

Kcwet soil surface

sur

face

cb

(K )

e

(K )

FIGURE 23

Crop growth stages for different types of crops

initial late

season

Crop type

- rangeland

- deciduous

trees & shrubs

- evergreen

- grass reference

growing season

mid-season

Hypothetical

Annuals

Perennials

crop

development

Crop evapotranspiration

95

Crop growth stages

As the crop develops, the ground cover, crop height and the leaf area change. Due to

differences in evapotranspiration during the various growth stages, the Kc for a given crop will

vary over the growing period. The growing period can be divided into four distinct growth

stages: initial, crop development, mid-season and late season. Figure 23 illustrates the general

sequence and proportion of these stages for different types of crops.

Initial stage

The initial stage runs from planting date to approximately 10% ground cover. The length of the

initial period is highly dependent on the crop, the crop variety, the planting date and the

climate. The end of the initial period is determined as the time when approximately 10% of the

ground surface is covered by green vegetation. For perennial crops, the planting date is replaced

by the ‘greenup’ date, i.e., the time when the initiation of new leaves occurs.

During the initial period, the leaf area is small, and evapotranspiration is predominately

in the form of soil evaporation. Therefore, the Kc during the initial period (Kc ini) is large

when the soil is wet from irrigation and rainfall and is low when the soil surface is dry. The

time for the soil surface to dry is determined by the time interval between wetting events, the

evaporation power of the atmosphere (ETo) and the importance of the wetting event. General

estimates for Kc ini as a function of the frequency of wetting and ETo are given in Table 9. The

data assume a medium textured soil. The procedure for estimating Kc ini is presented in

Chapter 6.

TABLE 9

Approximate values for Kc ini for medium wetting events (10-40 mm) and a medium textured soil

wetting interval

evaporating power of the atmosphere (ETo)

low

1 - 3

mm/day

moderate

3 - 5 mm/day

high

5 - 7 mm/day

very high

> 7

mm/day

less than weekly

weekly

longer than once per

week

1.2 - 0.8

0.8

0.7 - 0.4

1.1 - 0.6

0.6

0.4 - 0.2*

1.0 - 0.4

0.4

0.3 - 0.2*

0.9 - 0.3

0.3

0.2* - 0.1*

Values derived from Figures 29 and 30

(*) Note that irrigation intervals may be too large to sustain full transpiration for some young annual crops.

Crop development stage

The crop development stage runs from 10% ground cover to effective full cover. Effective full

cover for many crops occurs at the initiation of flowering. For row crops where rows commonly

interlock leaves such as beans, sugar beets, potatoes and corn, effective cover can be defined as

the time when some leaves of plants in adjacent rows begin to intermingle so that soil shading

becomes nearly complete, or when plants reach nearly full size if no intermingling occurs. For

some crops, especially those taller than 0.5 m, the average fraction of the ground surface

covered by vegetation (fc) at the start of effective full cover is about 0.7-0.8. Fractions of sunlit

Introduction to crop evapotranspiration (ETc)

96

and shaded soil and leaves do not change significantly with further growth of the crop beyond

fc ≈ 0.7 to 0.8. It is understood that the crop or plant can continue to grow in both height and

leaf area after the time of effective full cover. Because it is difficult to visually determine when

densely sown vegetation such as winter and spring cereals and some grasses reach effective full

cover, the more easily detectable stage of heading (flowering) is generally used for these types

of crops.

For dense grasses, effective full cover may occur at about 0.10-0.15 m height. For thin

stands of grass (dry rangeland), grass height may approach 0.3-0.5 m before effective full cover

is reached. Densely planted forages such as alfalfa and clover reach effective full cover at about

0.3-0.4 m.

Another way to estimate the occurrence of effective full cover is when the leaf area index

(LAI) reaches three. LAI is defined as the average total area of leaves (one side) per unit area of

ground surface.

As the crop develops and shades more and more of the ground, evaporation becomes

more restricted and transpiration gradually becomes the major process. During the crop

development stage, the Kc value corresponds to amounts of ground cover and plant

development. Typically, if the soil surface is dry, Kc = 0.5 corresponds to about 25-40% of the

ground surface covered by vegetation due to the effects of shading and due to microscale

transport of sensible heat from the soil into the vegetation. A Kc = 0.7 often corresponds to

about 40-60% ground cover. These values will vary, depending on the crop, frequency of

wetting and whether the crop uses more water than the reference crop at full ground cover (e.g.,

depending on its canopy architecture and crop height relative to clipped grass).

Mid-season stage

The mid-season stage runs from effective full cover to the start of maturity. The start of

maturity is often indicated by the beginning of the ageing, yellowing or senescence of leaves,

leaf drop, or the browning of fruit to the degree that the crop evapotranspiration is reduced

relative to the reference ETo. The mid-season stage is the longest stage for perennials and for

many annuals, but it may be relatively short for vegetable crops that are harvested fresh for

their green vegetation.

At the mid-season stage the Kc reaches its maximum value. The value for Kc (Kc mid) is

relatively constant for most growing and cultural conditions. Deviation of the Kc mid from the

reference value '1' is primarily due to differences in crop height and resistance between the

grass reference surface and the agricultural crop and weather conditions.

Late season stage

The late season stage runs from the start of maturity to harvest or full senescence. The

calculation for Kc and ETc is presumed to end when the crop is harvested, dries out naturally,

reaches full senescence, or experiences leaf drop.

For some perennial vegetation in frost free climates, crops may grow year round so that

the date of termination may be taken as the same as the date of ‘planting’.

Crop evapotranspiration

97

The Kc value at the end of the late season stage (Kc end) reflects crop and water

management practices. The Kc end value is high if the crop is frequently irrigated until

harvested fresh. If the crop is allowed to senesce and to dry out in the field before harvest, the

Kc end value will be small. Senescence is usually associated with less efficient stomatal

conductance of leaf surfaces due to the effects of ageing, thereby causing a reduction in Kc.

Figure 24 illustrates the variation in Kc for different crops as influenced by weather

factors and crop development.

CROP EVAPOTRANSPIRATION (ETC)

Crop evapotranspiration is calculated by multiplying ETo by Kc, a coefficient expressing the

difference in evapotranspiration between the cropped and reference grass surface. The

difference can be combined into one single coefficient, or it can be split into two factors

describing separately the differences in evaporation and transpiration between both surfaces.

The selection of the approach depends on the purpose of the calculation, the accuracy required,

the climatic data available and the time step with which the calculations are executed. Table 10

presents the general selection criteria.

FIGURE 24

Typical ranges expected in Kc for the four growth stages

1.0 frequent

in-

frequent

sugar cane

cotton

maize

cabbage, onions

apples

harvested

fresh

dried

wetting

events

0.8

0.6

0.4

0.2

1.2

Kc

(short)

(long)

mid-season late season

initial crop

ment

develop-

60 .. %

.. 25

ground cover

. 40 .

c

soil ground cover crop type

evapo-

ration

plant

development (wind speed)

(humidity)

main factors affecting K

crop type

harvesting date

in the 4 growth stages

Introduction to crop evapotranspiration (ETc)

98

TABLE 10

General selection criteria for the single and dual crop coefficient approaches

Single crop coefficient

Kc

Dual crop coefficient

Kcb + Ke

Purpose of

calculation

- irrigation planning and design

- irrigation management

- basic irrigation schedules

- real time irrigation scheduling for

non-frequent water applications

(surface and sprinkler irrigation)

- research

- real time irrigation scheduling

- irrigation scheduling for high

frequency water application

(microirrigation and automated

sprinkler irrigation)

- supplemental irrigation

- detailed soil and hydrologic

water balance studies

Time step

daily, 10-day, monthly

(data and calculation)

daily

(data and calculation)

Solution

method

graphical

pocket calculator

computer

Single and dual crop coefficient approaches

Single crop coefficient approach (Kc )

In the single crop coefficient approach, the effect of crop transpiration and soil evaporation are

combined into a single Kc coefficient. The coefficient integrates differences in the soil

evaporation and crop transpiration rate between the crop and the grass reference surface. As

soil evaporation may fluctuate daily as a result of rainfall or irrigation, the single crop

coefficient expresses only the time-averaged (multi-day) effects of crop evapotranspiration.

As the single Kc coefficient averages soil evaporation and transpiration, the approach is

used to compute ETc for weekly or longer time periods, although calculations may proceed on a

daily time step. The time-averaged single Kc is used for planning studies and irrigation system

design where the averaged effects of soil wetting are acceptable and relevant. This is the case

for surface irrigation and set sprinkler systems where the time interval between successive

irrigation is of several days, often ten days or more. For typical irrigation management, the

time-averaged single Kc is valid.

Dual crop coefficient approach (Kcb + Ke )

In the dual crop coefficient approach, the effects of crop transpiration and soil evaporation are

determined separately. Two coefficients are used: the basal crop coefficient (Kcb) to describe

plant transpiration, and the soil water evaporation coefficient (Ke) to describe evaporation from

the soil surface. The single Kc coefficient is replaced by:

ecbc KKK += (57)

where Kcb basal crop coefficient,

Ke soil water evaporation coefficient.

Crop evapotranspiration

99

The basal crop coefficient, Kcb, is defined as the ratio of ETc to ETo when the soil

surface layer is dry but where the average soil water content of the root zone is adequate to

sustain full plant transpiration. The Kcb represents the baseline potential Kc in the absence of

the additional effects of soil wetting by irrigation or precipitation. The soil evaporation

coefficient, Ke, describes the evaporation component from the soil surface. If the soil is wet

following rain or irrigation, Ke may be large. However, the sum of K

cb and Ke can never

exceed a maximum value, Kc max, determined by the energy available for evapotranspiration at

the soil surface. As the soil surface becomes drier, Ke becomes smaller and falls to zero when

no water is left for evaporation. The estimation of Ke requires a daily water balance

computation for the calculation of the soil water content remaining in the upper topsoil.

The dual coefficient approach requires more numerical calculations than the procedure

using the single time-averaged Kc coefficient. The dual procedure is best for real time irrigation

scheduling, for soil water balance computations, and for research studies where effects of day-

to-day variations in soil surface wetness and the resulting impacts on daily ETc, the soil water

profile, and deep percolation fluxes are important. This is the case for high frequency irrigation

with microirrigation systems or lateral move systems such as centre pivots and linear move

systems.

Crop coefficient curve

After the selection of the calculation approach, the determination of the lengths for the crop

growth stages and the corresponding crop coefficients, a crop coefficient curve can be

constructed. The curve represents the changes in the crop coefficient over the length of the

growing season. The shape of the curve represents the changes in the vegetation and ground

cover during plant development and maturation that affect the ratio of ETc to ETo. From the

curve, the Kc factor and hence ETc can be derived for any period within the growing season.

Single crop coefficient

The generalized crop coefficient curve is shown in Figure 25. Shortly after the planting of

annuals or shortly after the initiation of new leaves for perennials, the value for Kc is small,

often less than 0.4. The Kc begins to increase from the initial Kc value, Kc ini, at the beginning

of rapid plant development and reaches a maximum value, Kc mid, at the time of maximum or

near maximum plant development. During the late season period, as leaves begin to age and

senesce due to natural or cultural practices, the Kc begins to decrease until it reaches a lower

value at the end of the growing period equal to Kc end.

Dual crop coefficient

The single 'time-averaged' Kc curve in Figure 25 incorporates averaged wetting effects into the

Kc factor. The value for Kc mid is relatively constant for most growing and cultural conditions.

However, the values for Kc ini and Kc end can vary considerably on a daily basis, depending

on the frequency of wetting by irrigation and rainfall. The dual crop coefficient approach

calculates the actual increases in Kc for each day as a function of plant development and the

wetness of the soil surface.

As the single Kc coefficient includes averaged effects of evaporation from the soil, the

basal crop coefficient, Kcb d escribing only plant transpiration, lies below the Kc value (Figure

Introduction to crop evapotranspiration (ETc)

100

FIGURE 25

Generalized crop coefficient curve for the single crop coefficient approach

Kc

K

c ini

c mid

c end

initial crop development mid-season late season

0

0.2

0.4

0.6

0.8

1

1.2

1.4

time (days)

FIGURE 26

Crop coefficient curves showing the basal Kcb (thick line), soil evaporation Ke (thin line) and

the corresponding single Kc = Kcb + Ke curve (dashed line)

Kc

initial crop development mid-season late season

0

0.2

0.4

0.6

0.8

1

1.2

1.4

time (days)

K

cb

K

Ke

= K + Ke

c

Crop evapotranspiration

101

26). The largest difference between Kc and Kcb is found in the initial growth stage where

evapotranspiration is predominantly in the form of soil evaporation and crop transpiration is

still small. Because crop canopies are near or at full ground cover during the mid-season stage,

soil evaporation beneath the canopy has less effect on crop evapotranspiration and the value for

Kcb in the mid-season stage will be nearly the same as Kc. Depending on the ground cover, the

basal crop coefficient during the mid-season may be only 0.05-0.10 lower than the Kc value.

Depending on the frequency with which the crop is irrigated during the late season stage, Kcb

will be similar to (if infrequently irrigated) or less than the Kc value.

Figure 26 presents typical shapes for the Kcb, Ke and single Kc curves. The Kcb curve in

the figure represents the minimum Kc for conditions of adequate soil water and dry soil surface.

The Ke ‘spikes’ in the figure represent increased evaporation when precipitation or irrigation

has wetted the soil surface and has temporarily increased total ETc. These wet soil evaporation

spikes decrease as the soil surface layer dries. The spikes generally reach a maximum value of

1.0-1.2, depending on the climate, the magnitude of the wetting event and the portion of soil

surface wetted.

Summed together, the values for Kcb and for Ke represent the single crop coefficient,

Kc. The total Kc curve, shown as the dashed line in Figure 26, illustrates the effect of averaging

Kcb + Ke over time and is displayed as a ‘smoothed’ curve. It is this smoothed curve that is

represented by the single Kc calculation procedure. The Kc curve lies above the Kcb curve,

with potentially large differences during the initial and development stages, depending on the

frequency of soil wetting.

FLOW CHART OF THE CALCULATIONS

The calculation procedures required for the crop coefficient approaches are developed in the

following chapters. In Chapter 6, a single time-averaged crop coefficient is used to calculate

ETc. The approach using two coefficients to describe the effects of crop and soil separately is

outlined in Chapter 7. Figure 27 presents the general calculation procedures.

Introduction to crop evapotranspiration (ETc)

102

FIGURE 27

General procedure for calculating ETc

Calculate Reference ET

KK + K

ET = K ET

Construct Kc curve

K = K + K

o

ccb e

c ini c mid c end

Select values for

K, K & K

cini

c mid c end

A

djust

K

local climatic conditions

to

Determine daily K values

for surface evaporation

cb ini cb mid cb end

Select values for

K, K & K

cb mid cb end

A

djust

K

& K to

local climatic conditions

cb

Construct K curve

e

ccb e

cco

& K

A

djust K to reflect

wetting frequency

of soil surface

Single crop coefficient

Chapter 6

Dual crop coefficient

Chapter 7

Select stage lengths (Table 11)

Verify and supplement locally

from Table 12 from Table 17

Crop evapotranspiration

103

Chapter 6

ETc – single crop coefficient (Kc)

This chapter deals with the calculation of crop evapotranspiration (ETc) under standard

conditions. No limitations are placed on crop growth or evapotranspiration from soil water and

salinity stress, crop density, pests and diseases, weed infestation or low fertility. ETc is

determined by the crop coefficient approach whereby the effect of the various weather

conditions are incorporated into ETo and the crop characteristics into the Kc coefficient:

The effect of both crop transpiration and soil evaporation are integrated into a single crop

coefficient. The Kc coefficient incorporates crop characteristics and averaged effects of

evaporation from the soil. For normal irrigation planning and management purposes, for the

development of basic irrigation schedules, and for most hydrologic water balance studies,

average crop coefficients are relevant and more convenient than the Kc computed on a daily

time step using a separate crop and soil coefficient (Chapter 7). Only when values for Kc are

needed on a daily basis for specific fields of crops and for specific years, must a separate

transpiration and evaporation coefficient (Kcb + Ke) be considered.

The calculation procedure for crop evapotranspiration, ETc, consists of:

1. identifying the crop growth stages, determining their lengths, and selecting the

corresponding Kc coefficients;

2. adjusting the selected Kc coefficients for frequency of wetting or climatic conditions

during the stage;

3. constructing the crop coefficient curve (allowing one to determine Kc values for any

period during the growing period); and

4. calculating ETc as the product of ETo and Kc.

LENGTH OF GROWTH STAGES

FAO Irrigation and Drainage Paper No. 24 provides general lengths for the four distinct growth

stages and the total growing period for various types of climates and locations. This

information has been supplemented from other sources and is summarized in Table 11.

In some situations, the time of emergence of vegetation and the time of effective full

cover can be predicted using cumulative degree-based regression equations or by more

sophisticated plant growth models. These types of models should be verified or validated for

the local area or for a specific crop variety using local observations.

occ ETKET = (58)

ETc -–single crop coefficient (Kc)

104

TABLE 11

Lengths of crop development stages* for various planting periods and climatic regions (days)

Crop Init.

(Lini)

Dev.

(Ldev)

Mid

(Lmid)

Late

(Llate)

Total Plant Date Region

a. Small Vegetables

Broccoli 35 45 40 15 135 Sept Calif. Desert, USA

Cabbage 40 60 50 15 165 Sept Calif. Desert, USA

Carrots 20

30

40

50

50/30

60

90

20

30

100

150

200

Oct/Jan

Feb/Mar

Oct

Arid climate

Mediterranean

Calif. Desert, USA

Cauliflower 35 50 40 15 140 Sept Calif. Desert, USA

Celery 25

25

30

40

55

95

45

105

20

15

20

180

125

210

Oct

April

Jan

(Semi)Arid

Mediterranean

(Semi)Arid

Crucifers1 20

25

30

35

20

25

90

10

40

80

95

195

April

February

Oct/Nov

Mediterranean

Lettuce 20

30

25

35

30

40

35

50

15

25

30

45

10

75

105

100

140

April

Nov/Jan

Oct/Nov

Feb

Mediterranean

Arid Region

Mediterranean

Onion (dry) 15

20

25

35

70

110

40

45

150

210

April

Oct; Jan.

Mediterranean

Arid Region; Calif.

Onion (green) 25

20

30

45

55

10

20

55

5

10

40

70

95

180

April/May

October

March

Mediterranean

Arid Region

Calif., USA

Onion (seed) 20 45 165 45 275 Sept Calif. Desert, USA

Spinach 20

20

30

15/25

40

5

10

60/70

100

Apr; Sep/Oct

November

Mediterranean

Arid Region

Radish 5

10

15

5

35

40

Mar/Apr

Winter

Medit.; Europe

Arid Region

b. Vegetables – Solanum Family (Solanaceae)

Egg plant 30

30

40

45

40

20

25

130\14

0

October

May/June

Arid Region

Mediterranean

Sweet peppers

(bell)

25/30

30

35

40

110

20

30

125

210

April/June

October

Europe and Medit.

Arid Region

Tomato 30

35

25

35

30

40

45

40

50

60

70

45

25

30

135

155

180

145

January

Apr/May

Jan

Oct/Nov

April/May

Arid Region

Calif., USA

Calif. Desert, USA

Arid Region

Mediterranean

c. Vegetables - Cucumber Family (Cucurbitaceae)

Cantaloupe 30

10

45

60

35

25

10

25

120

Jan

Aug

Calif., USA

Cucumber 20

25

30

35

40

50

15

20

105

130

June/Aug

Nov; Feb

Arid Region

Pumpkin,

Winter squash

20

25

30

35

30

35

20

25

100

120

Mar, Aug

June

Mediterranean

Europe

Squash,

Zucchini

25

20

35

30

25

15

100

90

Apr; Dec.

May/June

Medit.; Arid Reg.

Medit.; Europe

continued…

* Lengths of crop development stages provided in this table are indicative of general conditions, but

may vary substantially from region to region, with climate and cropping conditions, and with crop

variety. The user is strongly encouraged to obtain appropriate local information.

1 Crucifers include cabbage, cauliflower, broccoli, and Brussel sprouts. The wide range in lengths

of seasons is due to varietal and species differences.

Crop evapotranspiration

105

Table 11 continued

Crop Init.

(Lini)

Dev.

(Ldev)

Mid

(Lmid)

Late

(Llate)

Total Plant Date Region

Sweet melons 25

30

15

30

35

30

40

45

40

50

65

20

30

15

20

120

140

135

160

May

March

Aug

Dec/Jan

Mediterranean

Calif., USA

Calif. Desert, USA

Arid Region

Water melons 20

10

30

20

30

20

30

110

80

April

Mat/Aug

Italy

Near East (desert)

d. Roots and Tubers

Beets, table 15

25

30

20

25

10

70

90

Apr/May

Feb/Mar

Mediterranean

Mediterranean & Arid

Cassava: year 1

year 2

20

150

40

90

110

60

210

360

Rainy

season

Tropical regions

Potato 25

25

30

45

30

35

30

35

30/45

45

50

70

50

30

20

25

115/130

130

145

165

140

Jan/Nov

May

April

Apr/May

Dec

(Semi)Arid Climate

Continental Climate

Europe

Idaho, USA

Calif. Desert, USA

Sweet potato 20

15

30

60

50

40

30

150

125

April

Rainy

seas.

Mediterranean

Tropical regions

Sugarbeet 30

25

50

25

45

35

45

30

65

40

35

75

60

90

100

50

80

70

15

10

65

40

50

30

40

180

155

255

180

160

230

205

March

June

Sept

April

May

November

Calif., USA

Calif. Desert, USA

Idaho, USA

Mediterranean

Arid Regions

e. Legumes (Leguminosae)

Beans (green) 20

15

30

25

30

25

10

90

75

Feb/Mar

Aug/Sep

Calif., Mediterranean

Calif., Egypt, Lebanon

Beans (dry) 20

15

25

30

25

40

35

30

20

110

95

100

May/June

June

Continental Climates

Pakistan, Calif.

Idaho, USA

Faba bean,

broad bean

- dry

- green

15

20

90

25

30

45

35

40

15

60

0

90

100

235

175

May

Mar/Apr

Nov

Europe

Mediterranean

Europe

Green gram,

cowpeas

20 30 30 20 110 March Mediterranean

Groundnut 25

35

45

35

25

35

25

130

140

Dry season

May

May/June

West Africa

High Latitudes

Mediterranean

Lentil 20

25

30

35

60

70

40

150

170

April

Oct/Nov

Europe

Arid Region

Peas 15

20

35

25

30

25

35

30

15

20

90

100

110

May

Mar/Apr

April

Europe

Mediterranean

Idaho, USA

Soybeans 15

20

15

30/35

25

40

60

75

15

25

30

85

140

150

Dec

May

June

Tropics

Central USA

Japan

continued…

ETc -–single crop coefficient (Kc)

106

Table 11 continued.

Crop Init.

(Lini)

Dev.

(Ldev)

Mid

(Lmid)

Late

(Llate)

Total Plant Date Region

f. Perennial Vegetables (with winter dormancy and initially bare or mulched soil)

Artichoke 40

20

40

25

250

30

360

325

Apr (1st yr)

May (2nd

yr)

California

(cut in May)

Asparagus 50

90

30

100

200

50

45

230

365

Feb

Warm Winter

Mediterranean

g. Fibre Crops

Cotton 30

45

30

50

90

50

60

45

60

55

45

55

45

195

225

195

180

Mar-May

Mar

Sept

April

Egypt; Pakistan; Calif.

Calif. Desert, USA

Yemen

Texas

Flax 25

30

35

40

50

100

40

50

150

220

April

October

Europe

Arizona

h. Oil Crops

Castor beans 25

20

40

65

50

25

180

135

March

Nov.

(Semi)Arid Climates

Indonesia

Safflower 20

25

35

55

45

55

60

25

30

40

125

145

190

April

Mar

Oct/Nov

California, USA

High Latitudes

Arid Region

Sesame 20 30 40 20 100 June China

Sunflower 25 35 45 25 130 April/May Medit.; California

i. Cereals

Barley/Oats/

Wheat

15

20

15

40

20

25

30

60

50

60

65

40

60

30

40

20

40

30

120

135

150

130

200

160

November

March/Apr

July

Apr

Nov

Dec

Central India

35-45 oL

East Africa

Calif. Desert, USA

Winter Wheat 202

30

160

602

140

75

70

40

75

30

25

180

240

335

December

November

October

Calif., USA

Mediterranean

Idaho, USA

Grains (small) 20

25

30

35

60

65

40

150

165

April

Oct/Nov

Mediterranean

Pakistan; Arid Reg.

Maize (grain) 30

25

20

30

50

40

35

40

60

45

40

50

40

30

50

180

140

125

150

170

April

Dec/Jan

June

October

April

East Africa (alt.)

Arid Climate

Nigeria (humid)

India (dry, cool)

Spain (spr, sum.); Calif.

Idaho, USA

Maize (sweet) 20

20

30

20

25

30

40

30

25

50/30

30

70

10

103

10

80

90

110

140

March

May/June

Oct/Dec

April

Jan

Philippines

Mediterranean

Arid Climate

Idaho, USA

Calif. Desert, USA

Millet 15

20

25

30

40

55

25

35

105

140

June

April

Pakistan

Central USA

continued…

2 These periods for winter wheat will lengthen in frozen climates according to days having zero

growth potential and wheat dormancy . Under general conditions and in the absence of local

data, fall planting of winter wheat can be presumed to occur in northern temperate climates when

the 10-day running average of mean daily air temperature decreases to 17o C or December 1,

whichever comes first. Planting of spring wheat can be presumed to occur when the 10-day

running average of mean daily air temperature increases to 5o C. Spring planting of maize-grain

can be presumed to occur when the 10-day running average of mean daily air temperature

increases to 13o C.

3 The late season for sweet maize will be about 35 days if the grain is allowed to mature and dry.

Crop evapotranspiration

107

Table 11 continued

Crop Init.

(Lini)

Dev.

(Ldev)

Mid

(Lmid)

Late

(Llate)

Total Plant Date Region

Sorghum 20

20

35

40

45

30

130

140

May/June

Mar/April

USA, Pakis., Med.

Arid Region

Rice 30

30

60

80

30

40

150

180

Dec; May

May

Tropics; Mediterranean

Tropics

j. Forages

Alfalfa, total

season4

10 30 var. var. var. last -4oC in spring until

first -4oC in fall

Alfalfa4

1st cutting cycle

10

20

30

20

25

10

60

75

Jan

Apr

(last -4o C)

Calif., USA.

Idaho, USA.

Alfalfa4, other

cutting cycles

5

10

20

10

5

10

30

45

Mar

Jun

Calif., USA.

Idaho, USA.

Bermuda for

seed

10 25 35 35 105 March Calif. Desert, USA

Bermuda for hay

(several cuttings)

10 15 75 35 135 --- Calif. Desert, USA

Grass Pasture4 10 20 -- -- -- 7 days before last -4oC in

spring until 7 days after

first -4oC in fall

Sudan,

1st cutting cycle

25 25 15 10 75 Apr Calif. Desert, USA

Sudan, other

cutting cycles

3 15 12 7 37 June Calif. Desert, USA

k. Sugar Cane

Sugarcane, virgin 35

50

75

60

70

105

190

220

330

120

140

210

405

480

720

Low Latitudes

Tropics

Hawaii, USA

Sugarcane,

ratoon

25

30

35

70

50

105

135

180

210

50

60

70

280

320

420

Low Latitudes

Tropics

Hawaii, USA

l. Tropical Fruits and Trees

Banana, 1st yr 120 90 120 60 390 Mar Mediterranean

Banana, 2nd yr 120 60 180 5 365 Feb Mediterranean

Pineapple 60 120 600 10 790 Hawaii, USA

m. Grapes and Berries

Grapes 20

20

30

40

50

60

120

75

90

40

60

20

80

240

205

180

210

April

Mar

May

April

Low Latitudes

Calif., USA

High Latitudes

Mid Latitudes (wine)

Hops 25 40 80 10 155 April Idaho, USA

n. Fruit Trees

Citrus 60 90 120 95 365 Jan Mediterranean

Deciduous

Orchard

20

30

70

50

90

120

130

30

60

30

210

270

240

March

High Latitudes

Low Latitudes

Calif., USA

continued…

4 In climates having killing frosts, growing seasons can be estimated for alfalfa and grass as:

alfalfa: last -4oC in spring until first -4oC in fall (Everson, D.O., M. Faubion and D.E. Amos

1978. "Freezing temperatures and growing seasons in Idaho." Univ. Idaho Agric. Exp. station

bulletin 494. 18 p.)

grass: 7 days before last -4oC in spring and 7 days after last -4oC in fall (Kruse E.G. and Haise,

H.R. 1974. "Water use by native grasses in high altitude Colorado meadows." USDA Agric. Res.

Service, Western Region report ARS-W-6-1974. 60 pages)

ETc -–single crop coefficient (Kc)

108

Table 11 continued

Crop Init.

(Lini)

Dev.

(Ldev)

Mid

(Lmid)

Late

(Llate)

Total Plant Date Region

Olives 30 90 60 90 2705 March Mediterranean

Pistachios 20 60 30 40 150 Feb Mediterranean

Walnuts 20 10 130 30 190 April Utah, USA

o. Wetlands - Temperate Climate

Wetlands

(Cattails,

Bulrush)

10

180

30

60

80

90

20

35

140

365

May

November

Utah, USA; killing frost

Florida, USA

Wetlands (short

veg.)

180 60 90 35 365 November frost-free climate

5 Olive trees gain new leaves in March. See footnote 24 of Table 12 for additional information,

where the Kc continues outside of the “growing period”.

Primary source: FAO Irrigation and Drainage Paper 24 (Doorenbos and Pruitt, 1977), Table 22.

The lengths of the initial and development periods may be relatively short for deciduous

trees and shrubs that can develop new leaves in the spring at relatively fast rates (Figure 23).

The rate at which vegetation cover develops and the time at which it attains effective full

cover are affected by weather conditions in general and by mean daily air temperature in

particular. Therefore, the length of time between planting and effective full cover will vary with

climate, latitude, elevation and planting date. It will also vary with cultivar (crop variety).

Generally, once the effective full cover for a plant canopy has been reached, the rate of further

phenological development (flowering, seed development, ripening, and senescence) is more

dependent on plant genotype and less dependent on weather. As an example, Figure 28 presents

the variation in length of the growing period for one cultivar of rice for one region and for

various planting dates.

The end of the mid-season and beginning of the late season is usually marked by

senescence of leaves, often beginning with the lower leaves of plants. The length of the late

season period may be relatively short (less than 10 days) for vegetation killed by frost (for

example, maize at high elevations in latitudes > 40°N) or for agricultural crops that are

harvested fresh (for example, table beets and small vegetables).

High temperatures may accelerate the ripening and senescence of crops. Long duration of

high air temperature (> 35°C) can cause some crops such as turf grass to go into dormancy. If

severely high air temperatures are coupled with moisture stress, the dormancy of grass can be

permanent for the remainder of the growing season. Moisture stress or other environmental

stresses will usually accelerate the rate of crop maturation and can shorten the mid and late

season growing periods.

The values in Table 11 are useful only as a general guide and for comparison

purposes. The listed lengths of growth stages are average lengths for the regions and

periods specified and are intended to serve only as examples. Local observations of the

specific plant stage development should be used, wherever possible, to incorporate effects

of plant variety, climate and cultural practices. Local information can be obtained by

interviewing farmers, ranchers, agricultural extension agents and local researchers, by

conducting local surveys, or by remote sensing. When determining stage dates from local

observations, the guidelines and visual descriptions may be helpful.

Crop evapotranspiration

109

CROP COEFFICIENTS

Changes in vegetation and ground cover mean that the crop coefficient Kc varies during the

growing period. The trends in Kc during the growing period are represented in the crop

coefficient curve. Only three values for Kc are required to describe and construct the crop

coefficient curve: those during the initial stage (Kc ini), the mid-season stage (Kc mid) and at

the end of the late season stage (Kc end).

Tabulated Kc values

Table 12 lists typical values for Kc ini, Kc mid, and Kc end for various agricultural crops. The

coefficients presented are organized by group type (i.e., small vegetables, legumes, cereals,

etc.) to assist in locating the crop in the table and to aid in comparing crops within the same

group. There is usually close similarity in the coefficients among the members of the same crop

group, as the plant height, leaf area, ground coverage and water management are normally

similar.

The coefficients in Table 12 integrate the effects of both transpiration and evaporation

over time. The effects of the integration over time represent an average wetting frequency for a

'standard' crop under typical growing conditions in an irrigated setting. The values for Kc

during the initial and crop development stages are subject to the effects of large variations in

wetting frequencies and therefore refinements to the value used for Kc ini should always be

made. For frequent wettings such as with high frequency sprinkler irrigation or rainfall, the

values for Kc ini may increase substantially.

FIGURE 28

V

ariation in the length of the growing period of rice (cultivar: Jaya) sown during various

months of the year at different locations along the Senegal River (Africa)

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

136 days

134 days

110 days

120 days

Delta

Valley 132 days

112 days

114 days

ETc -–single crop coefficient (Kc)

110

TABLE 12

Single (time-averaged) crop coefficients, Kc, and mean maximum plant heights for non stressed,

well-managed crops in subhumid climates (RHmin ≈

≈≈

≈ 45%, u2 ≈

≈≈

≈ 2 m/s) for use with the FAO

Penman-Monteith ETo.

Crop

Kc ini1

Kc mid

Kc end

Maximum

Crop Height

(h)

(m)

a. Small Vegetables 0.7 1.05 0.95

Broccoli 1.05 0.95 0.3

Brussel Sprouts 1.05 0.95 0.4

Cabbage 1.05 0.95 0.4

Carrots 1.05 0.95 0.3

Cauliflower 1.05 0.95 0.4

Celery 1.05 1.00 0.6

Garlic 1.00 0.70 0.3

Lettuce 1.00 0.95 0.3

Onions - dry 1.05 0.75 0.4

- green 1.00 1.00 0.3

- seed 1.05 0.80 0.5

Spinach 1.00 0.95 0.3

Radish 0.90 0.85 0.3

b. Vegetables – Solanum Family (Solanaceae) 0.6 1.15 0.80

Egg Plant 1.05 0.90 0.8

Sweet Peppers (bell) 1.052 0.90 0.7

Tomato 1.152 0.70-0.90 0.6

c. Vegetables – Cucumber Family (Cucurbitaceae) 0.5 1.00 0.80

Cantaloupe 0.5 0.85 0.60 0.3

Cucumber – Fresh Market 0.6 1.002 0.75 0.3

– Machine harvest 0.5 1.00 0.90 0.3

Pumpkin, Winter Squash 1.00 0.80 0.4

Squash, Zucchini 0.95 0.75 0.3

Sweet Melons 1.05 0.75 0.4

Watermelon 0.4 1.00 0.75 0.4

d. Roots and Tubers 0.5 1.10 0.95

Beets, table 1.05 0.95 0.4

Cassava – year 1 0.3 0.803 0.30 1.0

– year 2 0.3 1.10 0.50 1.5

Parsnip 0.5 1.05 0.95 0.4

Potato 1.15 0.754 0.6

Sweet Potato 1.15 0.65 0.4

Turnip (and Rutabaga) 1.10 0.95 0.6

Sugar Beet 0.35 1.20 0.705 0.5

continued…

1 These are general values for Kc ini under typical irrigation management and soil wetting. For frequent

wettings such as with high frequency sprinkle irrigation or daily rainfall, these values may increase

substantially and may approach 1.0 to 1.2. Kc ini is a function of wetting interval and potential

evaporation rate during the initial and development periods and is more accurately estimated using

Figures 29 and 30, or Equation 7-3 in Annex 7, or using the dual Kcb ini + Ke.

2 Beans, Peas, Legumes, Tomatoes, Peppers and Cucumbers are sometimes grown on stalks reaching

1.5 to 2 meters in height. In such cases, increased Kc values need to be taken. For green beans,

peppers and cucumbers, 1.15 can be taken, and for tomatoes, dry beans and peas, 1.20. Under these

conditions h should be increased also.

3 The midseason values for cassava assume non-stressed conditions during or following the rainy

season. The Kc end values account for dormancy during the dry season.

4 The Kc end value for potatoes is about 0.40 for long season potatoes with vine kill.

5 This Kc end value is for no irrigation during the last month of the growing season. The Kc end value

for sugar beets is higher, up to 1.0, when irrigation or significant rain occurs during the last month.

Crop evapotranspiration

111

Table 12 continued

Crop

Kc ini1

Kc mid

Kc end

Maximum

Crop Height

(h)

(m)

e. Legumes (Leguminosae) 0.4 1.15 0.55

Beans, green 0.5 1.052 0.90 0.4

Beans, dry and Pulses 0.4 1.152 0.35 0.4

Chick pea 1.00 0.35 0.4

Fababean (broad bean) – Fresh 0.5 1.152 1.10 0.8

– Dry/Seed 0.5 1.152 0.30 0.8

Grabanzo 0.4 1.15 0.35 0.8

Green Gram and Cowpeas 1.05 0.60-0.356 0.4

Groundnut (Peanut) 1.15 0.60 0.4

Lentil 1.10 0.30 0.5

Peas – Fresh 0.5 1.152 1.10 0.5

– Dry/Seed 1.15 0.30 0.5

Soybeans 1.15 0.50 0.5-1.0

f. Perennial Vegetables (with winter dormancy and

initially bare or mulched soil)

0.5 1.00 0.80

Artichokes 0.5 1.00 0.95 0.7

Asparagus 0.5 0.957 0.30 0.2-0.8

Mint 0.60 1.15 1.10 0.6-0.8

Strawberries 0.40 0.85 0.75 0.2

g. Fibre Crops 0.35

Cotton 1.15-1.20 0.70-0.50 1.2-1.5

Flax 1.10 0.25 1.2

Sisal8 0.4-0.7 0.4-0.7 1.5

h. Oil Crops 0.35 1.15 0.35

Castorbean (Ricinus) 1.15 0.55 0.3

Rapeseed, Canola 1.0-1.159 0.35 0.6

Safflower 1.0-1.159 0.25 0.8

Sesame 1.10 0.25 1.0

Sunflower 1.0-1.159 0.35 2.0

i. Cereals 0.3 1.15 0.4

Barley 1.15 0.25 1

Oats 1.15 0.25 1

Spring Wheat 1.15 0.25-0.410 1

Winter Wheat - with frozen soils

- with non-frozen soils

0.4

0.7

1.15

0.25-0.410

0.25-0.410 1

Maize, Field (grain) (field corn) 1.20 0.60,0.3511 2

Maize, Sweet (sweet corn) 1.15 1.0512 1.5

Millet 1.00 0.30 1.5

Sorghum – grain 1.00-1.10 0.55 1-2

– sweet 1.20 1.05 2-4

Rice 1.05 1.20 0.90-0.60 1

continued…

6 The first Kc end is for harvested fresh. The second value is for harvested dry.

7 The Kc for asparagus usually remains at Kc ini during harvest of the spears, due to sparse ground

cover. The Kc mid value is for following regrowth of plant vegetation following termination of harvest

of spears.

8 K

c for sisal depends on the planting density and water management (e.g., intentional moisture stress).

9 The lower values are for rainfed crops having less dense plant populations.

10 The higher value is for hand-harvested crops.

11 The first Kc end value is for harvest at high grain moisture. The second Kc end value is for harvest

after complete field drying of the grain (to about 18% moisture, wet mass basis).

12 If harvested fresh for human consumption. Use Kc end for field maize if the sweet maize is allowed to

mature and dry in the field.

ETc -–single crop coefficient (Kc)

112

Table 12 continued

Crop

Kc ini1

Kc mid

Kc end

Maximum

Crop Height

(h)

(m)

j. Forages

Alfalfa Hay – averaged cutting effects 0.40 0.9513 0.90 0.7

– individual cutting periods 0.4014 1.2014 1.1514 0.7

– for seed 0.40 0.50 0.50 0.7

Bermuda hay – averaged cutting effects 0.55 1.0013 0.85 0.35

– Spring crop for seed 0.35 0.90 0.65 0.4

Clover hay, Berseem – averaged cutting effects 0.40 0.9013 0.85 0.6

– individual cutting periods 0.4014 1.1514 1.1014 0.6

Rye Grass hay – averaged cutting effects 0.95 1.05 1.00 0.3

Sudan Grass hay (annual) – averaged cutting effects 0.50 0.9014 0.85 1.2

– individual cutting periods 0.5014 1.1514 1.1014 1.2

Grazing Pasture - Rotated Grazing 0.40 0.85-1.05 0.85 0.15-0.30

- Extensive Grazing 0.30 0.75 0.75 0.10

Turf grass - cool season15 0.90 0.95 0.95 0.10

- warm season15 0.80 0.85 0.85 0.10

k. Sugar Cane 0.40 1.25 0.75 3

l. Tropical Fruits and Trees

Banana – 1st year 0.50 1.10 1.00 3

– 2nd year 1.00 1.20 1.10 4

Cacao 1.00 1.05 1.05 3

Coffee – bare ground cover 0.90 0.95 0.95 2-3

– with weeds 1.05 1.10 1.10 2-3

Date Palms 0.90 0.95 0.95 8

Palm Trees 0.95 1.00 1.00 8

Pineapple16 – bare soil 0.50 0.30 0.30 0.6-1.2

– with grass cover 0.50 0.50 0.50 0.6-1.2

Rubber Trees 0.95 1.00 1.00 10

Tea – non-shaded 0.95 1.00 1.00 1.5

– shaded17 1.10 1.15 1.15 2

m. Grapes and Berries

Berries (bushes) 0.30 1.05 0.50 1.5

Grapes – Table or Raisin 0.30 0.85 0.45 2

– Wine 0.30 0.70 0.45 1.5-2

Hops 0.3 1.05 0.85 5

continued…

13 This Kc mid coefficient for hay crops is an overall average Kc mid coefficient that averages Kc for both

before and following cuttings. It is applied to the period following the first development period until

the beginning of the last late season period of the growing season.

14 These Kc coefficients for hay crops represent immediately following cutting; at full cover; and

immediately before cutting, respectively. The growing season is described as a series of individual

cutting periods (Figure 35).

15 Cool season grass varieties include dense stands of bluegrass, ryegrass, and fescue. Warm season

varieties include bermuda grass and St. Augustine grass. The 0.95 values for cool season grass

represent a 0.06 to 0.08 m mowing height under general turf conditions. Where careful water

management is practiced and rapid growth is not required, Kc's for turf can be reduced by 0.10.

16 The pineapple plant has very low transpiration because it closes its stomates during the day and opens

them during the night. Therefore, the majority of ETc from pineapple is evaporation from the soil. The

Kc mid < Kc ini since Kc mid occurs during full ground cover so that soil evaporation is less. Values

given assume that 50% of the ground surface is covered by black plastic mulch and that irrigation is

by sprinkler. For drip irrigation beneath the plastic mulch, Kc's given can be reduced by 0.10.

17 Includes the water requirements of the shade trees.

Crop evapotranspiration

113

Table 12 continued

Crop

Kc ini1

Kc mid

Kc end

Maximum

Crop Height

(h)

(m)

n. Fruit Trees

Almonds, no ground cover 0.40 0.90 0.6518 5

Apples, Cherries, Pears19

- no ground cover, killing frost 0.45 0.95 0.7018 4

- no ground cover, no frosts 0.60 0.95 0.7518 4

- active ground cover, killing frost 0.50 1.20 0.9518 4

- active ground cover, no frosts 0.80 1.20 0.8518 4

Apricots, Peaches, Stone Fruit19, 20

- no ground cover, killing frost 0.45 0.90 0.6518 3

- no ground cover, no frosts 0.55 0.90 0.6518 3

- active ground cover, killing frost 0.50 1.15 0.9018 3

- active ground cover, no frosts 0.80 1.15 0.8518 3

Avocado, no ground cover 0.60 0.85 0.75 3

Citrus, no ground cover21

- 70% canopy 0.70 0.65 0.70 4

- 50% canopy 0.65 0.60 0.65 3

- 20% canopy 0.50 0.45 0.55 2

Citrus, with active ground cover or weeds22

- 70% canopy 0.75 0.70 0.75 4

- 50% canopy 0.80 0.80 0.80 3

- 20% canopy 0.85 0.85 0.85 2

Conifer Trees23 1.00 1.00 1.00 10

Kiwi 0.40 1.05 1.05 3

Olives (40 to 60% ground coverage by canopy)24 0.65 0.70 0.70 3-5

Pistachios, no ground cover 0.40 1.10 0.45 3-5

Walnut Orchard19 0.50 1.10 0.6518 4-5

continued…

18 These Kc end values represent Kc prior to leaf drop. After leaf drop, Kc end ≈ 0.20 for bare, dry soil

or dead ground cover and Kc end ≈ 0.50 to 0.80 for actively growing ground cover (consult Chapter

11).

19 Refer to Eq. 94, 97 or 98 and footnotes 21 and 22 for estimating Kc for immature stands.

20 Stone fruit category applies to peaches, apricots, pears, plums and pecans.

21 These Kc values can be calculated from Eq. 98 for Kc min = 0.15 and Kc full = 0.75, 0.70 and 0.75

for the initial, mid season and end of season periods, and fc eff = fc where fc = fraction of ground

covered by tree canopy (e.g., the sun is presumed to be directly overhead). The values listed

correspond with those in Doorenbos and Pruitt (1977) and with more recent measurements. The

midseason value is lower than initial and ending values due to the effects of stomatal closure during

periods of peak ET. For humid and subhumid climates where there is less stomatal control by citrus,

values for Kc ini, Kc mid, and Kc end can be increased by 0.1 - 0.2, following Rogers et al. (1983).

22 These Kc values can be calculated as Kc = fc Kc ngc + (1 - fc) Kc cover where Kc ngc is the Kc of

citrus with no active ground cover (calculated as in footnote 21), Kc cover is the Kc for the active

ground cover (0.95), and fc is defined in footnote 21. The values listed correspond with those in

Doorenbos and Pruitt (1977) and with more recent measurements. Alternatively, Kc for citrus with

active ground cover can be estimated directly from Eq. 98 by setting Kc min = Kc cover. For humid

and subhumid climates where there is less stomatal control by citrus, values for Kc ini, Kc mid, and Kc

end can be increased by 0.1 - 0.2, following Rogers et al. (1983).

For non-active or only moderately active ground cover (active indicates green and growing ground

cover with LAI > about 2 to 3) , Kc should be weighted between Kc for no ground cover and Kc for

active ground cover, with the weighting based on the "greenness" and approximate leaf area of the

ground cover.

23 Confers exhibit substantial stomatal control due to reduced aerodynamic resistance. The Kc can easily

reduce below the values presented, which represent well-watered conditions for large forests.

ETc -–single crop coefficient (Kc)

114

Table 12 continued

Crop

Kc ini1

Kc mid

Kc end

Maximum

Crop Height

(h)

(m)

o. Wetlands – temperate climate

Cattails, Bulrushes, killing frost 0.30 1.20 0.30 2

Cattails, Bulrushes, no frost 0.60 1.20 0.60 2

Short Veg., no frost 1.05 1.10 1.10 0.3

Reed Swamp, standing water 1.00 1.20 1.00 1-3

Reed Swamp, moist soil 0.90 1.20 0.70 1-3

p. Special

Open Water, < 2 m depth

or in subhumid climates or tropics

1.05 1.05

Open Water, > 5 m depth, clear of turbidity,

temperate climate

0.6525 1.2525

24 These coefficients represent about 40 to 60% ground cover. Refer to Eq. 98 and footnotes 21 and

22 for estimating Kc for immature stands. In Spain, Pastor and Orgaz (1994) have found the following

monthly Kc‘s for olive orchards having 60% ground cover: 0.50, 0.50, 0.65, 0.60, 0.55, 0.50, 0.45,

0.45, 0.55, 0.60, 0.65, 0.50 for months January through December. These coefficients can be

invoked by using Kc ini = 0.65, Kc mid = 0.45, and Kc end = 0.65, with stage lengths = 30, 90,

60 and 90 days, respectively for initial, development, midseason and late season periods, and using Kc

during the winter (“off season”) in December to February = 0.50.

25 These Kc's are for deep water in temperate latitudes where large temperature changes in the water

body occur during the year, and initial and peak period evaporation is low as radiation energy is

absorbed into the deep water body. During fall and winter periods (Kc end), heat is released from the

water body that increases the evaporation above that for grass. Therefore, Kc mid corresponds to the

period when the water body is gaining thermal energy and Kc end when releasing thermal energy.

These Kc's should be used with caution.

Primary sources: Kc ini: Doorenbos and Kassam (1979)

Kc mid and Kc end: Doorenbos and Pruitt (1977); Pruitt (1986); Wright (1981,

1982), Snyder et al., (1989)

The values for Kc mid and Kc end in Table 12 represent those for a sub-humid climate

with an average daytime minimum relative humidity (RHmin) of about 45% and with calm to

moderate wind speeds averaging 2 m/s. For more humid or arid conditions, or for more or less

windy conditions, the Kc coefficients for the mid-season and end of late season stage should be

modified as described in this chapter.

The values for Kc in Table 12 are values for non-stressed crops cultivated under

excellent agronomic and water management conditions and achieving maximum crop yield

(standard conditions). Where stand density, height or leaf area are less than that attained under

such conditions, the value for Kc mid and, for most crops, for Kc end will need to be modified

(Part C, Chapters 8, 9 and 10).

Crop coefficient for the initial stage (Kc ini)

Calculation procedure

The values for Kc ini in Table 12 are only approximations and should only be used for

estimating ETc during preliminary or planning studies. For several group types only one value

Crop evapotranspiration

115

for Kc ini is listed and it is considered to be representative of the whole group for a typical

irrigation water management. More accurate estimates of Kc ini can be obtained by

considering:

Time interval between wetting events

Evapotranspiration during the initial stage for annual crops is predominately in the form of

evaporation. Therefore, accurate estimates for Kc ini should consider the frequency with which

the soil surface is wetted during the initial period. Where the soil is frequently wet from

irrigation or rain, the evaporation from the soil surface can be considerable and Kc ini will be

large. On the other hand, where the soil surface is dry, evaporation is restricted and the Kc ini

will be small (Table 9).

Evaporation power of the atmosphere

The value of Kc ini is affected by the evaporating power of the atmosphere, i.e., ETo. The

higher the evaporation power of the atmosphere, the quicker the soil will dry between water

applications and the smaller the time-averaged Kc will be for any particular period.

Magnitude of the wetting event

As the amount of water available in the topsoil for evaporation and hence the time for the soil

surface to dry is a function of the magnitude of the wetting event, Kc ini will be smaller for

light wetting events than for large wettings.

Depending on the time interval between wetting events, the magnitude of the wetting

event, and the evaporation power of the atmosphere, Kc ini can vary between 0.1 and 1.15. A

numerical procedure to compute Kc ini is provided in Annex 7.

Time interval between wetting events

In general, the mean time interval between wetting events is estimated by counting all rainfall

and irrigation events occurring during the initial period that are greater than a few millimetres.

Wetting events occurring on adjacent days can be counted as one event. The mean wetting

interval is estimated by dividing the length of the initial period by the number of events.

Where only monthly rainfall values are available without any information on the number

of rainy days, the number of events within the month can be estimated by dividing the monthly

rainfall depth by the depth of a typical rain event. The typical depth, if it exists, can vary widely

from climate to climate, region to region and from season to season. Table 13 presents some

information on the range of rainfall depths. After deciding what rainfall is typical for the region

and time of the year, the number of rainy days and the mean wetting interval can be estimated.

TABLE 13

Classification of rainfall depths

rain event depth

Very light (drizzle)

Light (light showers)

Medium (showers)

Heavy (rainstorms)

≤

≤≤

≤ 3 mm

5 mm

≥

≥≥

≥ 10 mm

≥

≥≥

≥ 40 mm

Where rainfall is insufficient, irrigation is needed to keep the crop well watered. Even

where irrigation is not yet developed, the mean interval between the future irrigations should be

ETc -–single crop coefficient (Kc)

116

estimated to obtain the required frequency of wetting necessary to keep the crop stress free. The

interval might be as small as a few days for small vegetables, but up to a week or longer for

cereals depending on the climatic conditions. Where no estimate of the interval can be made,

the user may refer to the values for Kc ini of Table 12.

EXAMPLE 23

Estimation of interval between wetting events

Estimate, from mean monthly rainfall data, the interval between rains during the rainy season for a

station in a temperate climate (Paris, France: 50 mm/month), dry climate (Gafsa, Tunisia: 20

mm/month) and tropical climate (Calcutta, India: 300 mm/month).

Station monthly rain

(mm/month)

typical rainfall

(mm)

number of rainy

days

interval between

rains

Paris

Gafsa

Calcutta

50

20

300

3

5

20

17

4

15

~ 2 days

weekly

~ 2 days

Determination of Kc ini

The crop coefficient for the initial growth stage can be derived from Figures 29 and 30 which

provide estimates for Kc ini as a function of the average interval between wetting events, the

evaporation power ETo, and the importance of the wetting event.

Light wetting events (infiltration depths of 10 mm or less): rainfall and high frequency

irrigation systems

Figure 29 is used for all soil types when wetting events are light. When wetting during the

initial period is only by precipitation, one will usually use Figure 29 to determine Kc ini. The

graph can also be used when irrigation is by high frequency systems such as microirrigation and

centre pivot and light applications of about 10 mm or less per wetting event are applied.

EXAMPLE 24

Graphical determination of Kc ini

A silt loam soil receives irrigation every two days during the initial growth stage via a centre pivot

irrigation system. The average depth applied by the centre pivot system is about 12 mm per event and

the average ETo during the initial stage is 4 mm/day. Estimate the crop evapotranspiration during that

stage.

From Fig. 29 using the 2-day

interval curve:

Kc ini =

0.85

-

- ETc = Kc ETo = 0.85 (4.0) = 3.4 mm/day

The average crop evapotranspiration during the initial growth stage is 3.4 mm/day

Heavy wetting events (infiltration depths of 40 mm or more): surface and sprinkler irrigation

Figure 30 is used for heavy wetting events when infiltration depths are greater than 40 mm,

such as for when wetting is primarily by periodic irrigation such as by sprinkler or surface

irrigation. Following a wetting event, the amount of water available in the topsoil for

evaporation is considerable, and the time for the soil surface to dry might be significantly

increased. Consequently, the average Kc factor is larger than for light wetting events. As the

time for the soil surface to dry is, apart from the evaporation power and the frequency of

wetting, also determined by the water storage capacity of the topsoil, a distinction is made

between soil types.

Crop evapotranspiration

117

Figure 30a is used for coarse textured soils and Figure 30b is used for fine and medium

textured soils. Coarse textured soils include sands and loamy sand textured soils. Medium

textured soils include sandy loam, loam, silt loam and silt textured soils. Fine textured soils

include silty clay loam, silty clay and clay textured soils.

Average wetting events (infiltration depths between 10 and 40 mm):

Where average infiltration depths are between 10 and 40 mm, the value for Kc ini can be

estimated from Figures 29 and 30:

[]

)29.Fig(inic)30.Fig(inic)29.Fig(inicinic KK

)1040(

)10I(

KK −

−

+= (59)

where Kc ini (Fig.29) value for Kc ini from Figure 29,

Kc ini (Fig.30) value for Kc ini from Figure 30,

I average infiltration depth [mm].

The values 10 and 40 in Equation 59 are the average depths of infiltration (millimetres)

upon which Figures 29 and 30 are based.

FIGURE 29

Average Kc ini as related to the level of EToand the interval between irrigations and/o

r

significant rain during the initial growth stage for all soil types when wetting events are light to

medium (3-10 mm per event)

0123456789101112

0.0

0.2

0.4

0.6

0.8

1.0

1.2

K

c ini

ET mm/day

o

1 day

2 day

4 day

7 day

10 day

20 day

small infiltration depths

~ 10 mm

low moderate high very high

ETc -–single crop coefficient (Kc)

118

FIGURE 30

Average Kc ini as related to the level of EToand the interval between irrigations greater than o

r

equal to 40 mm per wetting event, during the initial growth stage for a) coarse textured soils; b)

medium and fine textured soils

0123456789101112

0.0

0.2

0.4

0.6

0.8

1.0

1.2

K

c ini

ET mm/day

o

1 day

2 day

4 day

7 day

10 day

20 day

large infiltration depths

> 40 mm

low moderate high very high

a. coarse textures

0123456789101112

0.0

0.2

0.4

0.6

0.8

1.0

1.2

K

c ini

ET mm/day

o

1 day

2 day

4 day

7 day

10 day

20 day

large infiltration depths

> 40 mm

low moderate high very high

b. fine and medium textures

Crop evapotranspiration

119

EXAMPLE 25

Interpolation between light and heavy wetting events

Small vegetables cultivated in a dry area on a coarse textured soil receive 20 mm of water twice a

week by means of a sprinkler irrigation system. The average ETo during the initial stage is 5 mm/day.

Estimate the crop evapotranspiration during that stage.

For:

7/2 =

ETo =

and a coarse textured soil

3.5

5

day interval

mm/day

From Fig. 29:

Kc ini(Fig. 29) ≈

0.55 -

From Fig. 30.a:

Kc ini(Fig. 30a) ≈ 0.7 -

For: I = 20 mm

From Eq. 59:

Kc ini = 0.55 + [(20-10)/(40-10)] (0.7-0.55)

= 0.55+0.33(0.15)=

0.60

-

From Eq. 58: ETc = 0.60 (5) = 3.0 mm/day

The average crop evapotranspiration during the initial growth stage for the small vegetables is

3.0 mm/day.

Adjustment for partial wetting by irrigation

Many types of irrigation systems wet only a fraction of the soil surface. For example, for a

trickle irrigation system, the fraction of the surface wetted, fw, may be only 0.4. For furrow

irrigation systems, the fraction of the surface wetted may range from 0.3 to 0.8. Common values

for the fraction of the soil surface wetted by irrigation or precipitation are given in Table 20.

When only a fraction of the soil surface is wetted, the value for Kc ini obtained from Table 12

or from Figures 29 or 30 should be multiplied by the fraction of the surface wetted to adjust for

the partial wetting:

Fig),(Tabinicwinic KfK = (60)

where fw the fraction of surfaced wetted by irrigation or rain [0 - 1],

Kc ini (Tab,Fig) the value for Kc ini from Table 12 or Figure 29 or 30.

In addition, in selecting which figure to use (i.e., Figure 29 or 30), the average infiltrated

depth, expressed in millimetres over the entire field surface, should be divided by fw to

represent the true infiltrated depth of water for the part of the surface that is wetted (Figure 31):

w

wf

I

I= (61)

where Iw irrigation depth for the part of the surface that is wetted [mm],

fw fraction of surface wetted by irrigation,

I the irrigation depth for the field [mm].

When irrigation of part of the soil surface and precipitation over the entire soil surface

both occur during the initial period, fw should represent the average of fw for each type of

wetting, weighted according to the total infiltration depth received by each type.

ETc -–single crop coefficient (Kc)

120

EXAMPLE 26

Determination of Kc ini for partial wetting of the soil surface

Determine the evapotranspiration of the crop in Example 24 if it had been irrigated using a trickle

system every two days (with 12 mm each application expressed as an equivalent depth over the field

area), and where the average fraction of surface wet was 0.4, and where little or no precipitation

occurred during the initial period.

The average depth of infiltration per event in the wetted fraction of the surface:

From Eq. 61:

Iw = I / fw = 12 mm / 0.4 =

30 mm

Therefore, one can interpolate between Fig. 29 representing light wetting events (~10 mm per event)

and Fig. 30.b representing medium textured soil and large wetting events (~40 mm per event).

For:

and:

Fig. 29 produces:

Fig. 30.b produces

From Eq. 59:

ETo = 4 mm/day

a 2 day wetting interval:

Kc ini = 0.85

Kc ini = 1.15

Kc ini = 0.85 + [(30 - 10)/(40 - 10)] (1.15 - 0.85) =

4

-

0.85

1.15

1.05

mm/day

-

Because the fraction of soil surface wetted by the trickle system is 0.4, the actual Kc ini for the trickle

irrigation is calculated as:

From Eq. 60:

-

Kc ini = fw Kc ini Fig = 0.4 (1.05) =

This value (0.42) represents the Kc ini as applied over

the entire field area.

ETc = Kc ini ETo = 0.42 (4) =

0.42

1.7

-

mm/day

The average crop evapotranspiration during the initial growth stage for this trickle irrigated crop is

1.7 mm/day.

FIGURE 31

Partial wetting by irrigation

dry

Soil surface

wet

dry

fw

1

I = average infiltration depth over entire field surface

I = I/f

ww

Crop evapotranspiration

121

Kc ini for trees and shrubs

Kc ini for trees and shrubs should reflect the ground condition prior to leaf emergence or

initiation in case of deciduous trees or shrubs, and the ground condition during the dormancy or

low active period for evergreen trees and shrubs. The Kc ini depends upon the amount of grass

or weed cover, frequency of soil wetting, tree density and mulch density. For a deciduous

orchard in frost-free climates, the Kc ini can be as high as 0.8 or 0.9, where grass ground cover

exists, and as low as 0.3 or 0.4 when the soil surface is kept bare and wetting is infrequent. The

Kc ini for an evergreen orchard (having no concerted leaf drop) with a dormant period has less

variation from Kc mid, as exemplified for citrus in Table 12, footnotes 21 and 22. For 50%

canopy or less, the Kc ini also reflects ground cover conditions (bare soil, mulch or active grass

or weed cover).

Kc ini for paddy rice

For rice growing in paddy fields with a water depth of 0.10-0.20 m, the ETc during the initial

stage mainly consists of evaporation from the standing water. The Kc ini in Table 12 is 1.05 for

a sub-humid climate with calm to moderate wind speeds. The Kc ini should be adjusted for the

local climate as indicated in Table 14.

TABLE 14

Kc ini for rice for various climatic conditions

Humidity Wind speed

light moderate strong

arid - semi-arid

sub-humid - humid

very humid

1.10

1.05

1.00

1.15

1.10

1.05

1.20

1.15

1.10

Crop coefficient for the mid-season stage (Kc mid)

Illustration of the climatic effect

Typical values for the crop coefficient for the mid-season growth stage, Kc mid, are listed in

Table 12 for various agricultural crops.

As discussed in Chapter 5, the effect of the difference in aerodynamic properties between

the grass reference surface and agricultural crops is not only crop specific but also varies with

the climatic conditions and crop height (Figure 21). More arid climates and conditions of

greater wind speed will have higher values for Kc mid. More humid climates and conditions of

lower wind speed will have lower values for Kc mid.

The relative impact of climate on Kc mid is illustrated in Figure 32 where the

adjustments to the values from Table 12 are shown for various types of climates, mean daily

wind speeds and various crop heights. As an example, expected variations for Kc mid for

tomatoes in response to regional climatic conditions are presented in Box 14.

Determination of Kc mid

For specific adjustment in climates where RHmin differs from 45% or where u2 is larger or

smaller than 2.0 m/s, the Kc mid values from Table 12 are adjusted as:

[]

3.0

min2)Tab(midcmidc 3

h

)45RH(004.0)2u(04.0KK ÷

ø

ö

ç

è

æ

−−−+= (62)

ETc -–single crop coefficient (Kc)

122

FIGURE 32

Adjustment (additive) to the Kc mid values from Table 12 for different crop heights and mean

daily wind speeds (u2) for different humidity conditions

0.1

0.2

0.3

0.5

0.7

1.0

2.0

3.0

5.0

7.0

10.0

RHmin

= 25%

RHmean = 50%

- 0.3 - 0.2 - 0.1 0 0.1 0.2 0 .3

adjustment K

cmid

crop height (m)

0 m/s

4 m/s

6 m/s

u

2

= 2 m/s

Arid

Semi-arid

- 0.3 - 0.2 - 0.1 0 0.1 0.2 0.3

adjustment Kc mid

0.1

0.2

0.3

0.5

0.7

1.0

2.0

3.0

5.0

7.0

10.0

crop height (m)

RHmin = 45%

RHmean = 70%

Sub-

humid

0 m/s

4 m/s

6 m/s

u

2

= 2 m/s

0.1

0.2

0.3

0.5

0.7

1.0

2.0

3.0

5.0

7.0

10.0

crop height (m)

- 0.3 - 0.2 - 0.1 0 0.1 0.2 0.3

adjustment K

c

mid

RHmin = 80 %

RHmean = 90%

Very-

humid

0 m/s

4 m/s

6 m/s

u = 2 m/s

2

Crop evapotranspiration

123

where Kc mid (Tab) value for Kc mid taken from Table 12,

u2 mean value for daily wind speed at 2 m height over grass during the mid-

season growth stage [m s-1], for 1 m s-1 ≤ u2 ≤ 6 m s-1,

RHmin mean value for daily minimum relative humidity during the mid-season

growth stage [%], for 20% ≤ RHmin ≤ 80%,

h mean plant height during the mid-season stage [m] for 0.1 m < h < 10 m.

The Kc mid values determined with equations 62 and 65 are average adjustments for the

midseason and late season periods. The values for parameters u2 and RHmin should be

accordingly taken as averages for these periods (see example, Annex 8). The limits expressed

for parameters u2, RHmin and h should be observed.

BOX 14

Demonstration of effect of climate on Kc mid for wheat crop grown under field conditions

From Table 12 for wheat: Kc mid = 1.15 and h = 1.0 m

For semi-arid to arid conditions

- for strong wind (4 m/s) Kc mid = 1.15 + 0.10 = 1.25

- for moderate wind (2 m/s) Kc mid = 1.15 + 0.05 = 1.20

- for calm wind (1 m/s) Kc mid = 1.15 + 0.00 = 1.17

For sub-humid conditions

- for strong wind (4 m/s) Kc mid = 1.15 + 0.05 = 1.20

- for moderate wind (2 m/s) Kc mid = 1.15 + 0.00 = 1.15

- for calm wind (1 m/s) Kc mid = 1.15 - 0.05 = 1.12

For humid and very humid conditions

- for strong wind (4 m/s) Kc mid = 1.15 - 0.05 = 1.10

- for moderate wind (2 m/s) Kc mid = 1.15 - 0.10 = 1.05

- for calm wind (1 m/s) Kc mid = 1.15 - 0.15 = 1.02

Depending on the aridity of the climate and the wind conditions, the crop coefficient for wheat during

the mid-season stage ranges from 1.02 (humid and calm wind) to 1.25 (arid and strong wind).

Where the user does not have access to a calculator with an exponential function, the

solution of the (h/3)0.3 expression can be approximated as [(h/3)0.5]0.5 where the square root

key is used.

RHmin is used rather than RHmean because it is easier to approximate RHmin from

Tmax where relative humidity data are unavailable. Moreover, under the common condition

where Tmin approaches Tdew (i.e., RHmax ≈ 100%), the vapour pressure deficit (es - ea), with

es from Equation 12 and ea from Equation 17, becomes [(100-RHmin)/200] e°(Tmax), where

e°(Tmax) is saturation vapour pressure at maximum daily air temperature. This indicates that

RHmin better reflects the impact of vapour pressure deficit on Kc than does RHmean.

RHmin is calculated on a daily or average monthly basis as:

100

)T(e

RH

max

o

dew

o

min = (63)

where Tdew is mean dewpoint temperature and Tmax is mean daily maximum air temperature

during the mid-season growth stage. Where dewpoint temperature or other hygrometric data are

ETc -–single crop coefficient (Kc)

124

not available or are of questionable quality, RHmin can be estimated by substituting mean daily

minimum air temperature, Tmin, for Tdew1. Then:

100

)T(e

RH

max

o

min

o

min = (64)

The values for u2 and RHmin need only be approximate for the mid-season growth stage.

This is because Equation 62 is not strongly sensitive to these values, changing 0.04 per 1 m/s

change in u2 and per 10% change in RHmin for a 3 m tall crop. Measurements, calculation and

estimation of missing wind and humidity data are provided in Chapter 3. Wind speed measured

at other than 2 m height should be adjusted to reflect values for wind speed at 2 m over grass

using Equation 47. Where no data on u2 or RHmin are available, the general classification for

wind speed and humidity data given in Tables 15 and 16 can be used.

TABLE 15

Empirical estimates of monthly wind speed data

description mean monthly wind speed at 2 m

light wind

light to moderate wind

moderate to strong wind

strong wind

...≤ 1.0 m/s

2.0 m/s

4.0 m/s

... ≥ 5.0 m/s

general global conditions 2 m/s

TABLE 16

Typical values for RHmin compared with RHmean for general climatic classifications

Climatic classification RHmin

(%)

RHmean

(%)

Arid

Semi-arid

Sub-humid

Humid

Very humid

20

30

45

70

80

45

55

70

85

90

Equation 62 is valid for mean plant heights up to 10 m. For plant heights smaller than 0.1

m, vegetation will behave aerodynamically similar to grass reference and eq. 62 should not be

applied. Example values for h are listed in Table 12 for various crops. However, the mean plant

height will greatly vary with crop variety and with cultural practices. Therefore, wherever

possible, h should be obtained from general field observations. However, the presence of the

0.3 exponent in Equation 62 makes these equations relatively insensitive to small errors in the

value used for h. Generally, a single value for h is used to represent the mid-season period.

Adjustment for frequency of wetting

Kc mid is less affected by wetting frequency than is Kc ini, as vegetation during this stage is

generally near full ground cover so that the effects of surface evaporation on Kc are smaller.

For frequent irrigation of crops (more frequently than every 3 days) and where the Kc mid of

Table 12 is less than 1.0, the value can be replaced by approximately 1.1-1.3 to account for the

combined effects of continuously wet soil, evaporation due to interception (sprinkler irrigation)

and roughness of the vegetation, especially where the irrigation system moistens an important

fraction of the soil surface (fw > 0.3).

1 In the case of arid and semi-arid climates, Tmin in equation (64) should be adjusted as indicated in

Annex 6 (equation 6-6) by subtracting 2°C from the average value of Tmin to better approximate Tdew.

Crop evapotranspiration

125

EXAMPLE 27

Determination of Kc mid

Calculate Kc mid for maize crops near Taipei, Taiwan and near Mocha, Yemen. The average mean

daily wind speed (u2) during the mid-season stage at Taipei is about 1.3 m/s and the minimum relative

humidity (RHmin) during this stage averages 75%. The average u2 during the mid-season near Mocha

is 4.6 m/s and the RHmin during this stage averages 44%.

From Table 12, the value for Kc mid is 1.20 for maize. The value for h from Table 12 is 2 m. Using

Eq. 62

For Taipei (humid climate):

[]

07.1

3

2

)4575(004.0)23.1(04.020.1K

3.0

midc =

÷

ø

ö

ç

è

æ

−−−+=

For Mocha (arid climate):

[]

30.1

3

2

)4544(004.0)26.4(04.020.1K

3.0

midc =

÷

ø

ö

ç

è

æ

−−−+=

The average crop coefficient predicted during the mid-season stage is 1.07 for Taipei and 1.30 for

Mocha.

Crop coefficient for the end of the late season stage (Kc end)

Typical values for the crop coefficient at the end of the late season growth stage, Kc end, are

listed in Table 12 for various agricultural crops. The values given for Kc end reflect crop and

water management practices particular to those crops. If the crop is irrigated frequently until

harvested fresh, the topsoil remains wet and the Kc end value will be relatively high. On the

other hand, crops that are allowed to senesce and dry out in the field before harvest receive less

frequent irrigation or no irrigation at all during the late season stage. Consequently, both the

soil surface and vegetation are dry and the value for Kc end will be relatively small (Figure 33).

Where the local water management and harvest timing practices are known to deviate

from the typical values presented in Table 12, then the user should make some adjustments to

the values for Kc end. Some guidance on adjustment of Kc values for wetting frequency is

provided in Chapter 7. For premature harvest, the user can construct a Kc curve using the

Kc end value provided in Table 12 and a late season length typical of a normal harvest date; but

can then terminate the application of the constructed curve early, corresponding to the time of

the early harvest.

The Kc end values in Table 12 are typical values expected for average Kc end under the

standard climatic conditions. More arid climates and conditions of greater wind speed will have

higher values for Kc end. More humid climates and conditions of lower wind speed will have

lower values for Kc end. For specific adjustment in climates where RHmin differs from 45% or

where u2 is larger or smaller than 2.0 m/s, Equation 65 can be used:

[]

3.0

min2)Tab(endcendc 3

h

)45RH(004.0)2u(04.0KK ÷

ø

ö

ç

è

æ

−−−+= (65)

ETc -–single crop coefficient (Kc)

126

FIGURE 33

Ranges expected for Kc end

time

K

c

coefficient

c mid

K

cabbage

onion

cotton

sugar cane

maize (sillage)

green

beans

maize (moist grain)

dry beans

maize

(dry grain)

harvested

fresh

dried

c end

K

FIGURE 34

Crop coefficient curve

Kc

K

time of season (days)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

K

initial

crop

development

mid-

season

late

season

c ini

c mid

c end

Crop evapotranspiration

127

where Kc end (Tab) value for Kc end taken from Table 12,

u2 mean value for daily wind speed at 2 m height over grass during

the late season growth stage [m s-1], for 1 m s-1 ≤ u2 ≤ 6 m s-1,

RHmin mean value for daily minimum relative humidity during the late

season stage [%],for 20% ≤ RHmin ≤ 80%,

h mean plant height during the late season stage [m], for 0.1 m ≤ h

≤ 10 m.

Equation 65 is only applied when the tabulated values for Kc end exceed 0.45. The

equation reduces the Kc end with increasing RHmin. This reduction in Kc end is characteristic

of crops that are harvested ‘green’ or before becoming completely dead and dry (i.e., Kc end ≥

0.45).

No adjustment is made when Kc end (Table) < 0.45 (i.e., Kc end = Kc end (Tab)). When

crops are allowed to senesce and dry in the field (as evidenced by Kc end < 0.45), u2 and

RHmin have less effect on Kc end and no adjustment is necessary. In fact, Kc end may decrease

with decreasing RHmin for crops that are ripe and dry at the time of harvest, as lower relative

humidity accelerates the drying process.

CONSTRUCTION OF THE Kc CURVE

Annual crops

Only three point values for Kc are required to describe and to construct the Kc curve. The curve

such as that shown in Figure 34 is constructed using the following three steps:

1. Divide the growing period into four general growth stages that describe crop phenology

or development (initial, crop development, mid-season, and late season stage), determine

the lengths of the growth stages, and identify the three Kc values that correspond to Kc

ini, Kc mid and Kc end from Table 12.

2. Adjust the Kc values to the frequency of wetting and/or climatic conditions of the growth

stages as outlined in the previous section.

3. Construct a curve by connecting straight line segments through each of the four growth

stages. Horizontal lines are drawn through Kc ini in the initial stage and through Kc mid

in the mid-season stage. Diagonal lines are drawn from Kc ini to Kc mid within the

course of the crop development stage and from Kc mid to Kc end within the course of the

late season stage.

Kc curves for forage crops

Many crops grown for forage or hay are harvested several times during the growing season.

Each harvest essentially terminates a ‘sub’ growing season and associated Kc curve and

initiates a new ‘sub’ growing season and associated Kc curve. The resulting Kc curve for the

entire growing season is the aggregation of a series of Kc curves associated with each sub-

cycle. Figure 35 presents a Kc curve for the entire growing season constructed for alfalfa grown

for hay in southern Idaho.

ETc -–single crop coefficient (Kc)

128

In the southern Idaho climate, greenup (leaf initiation) begins in the spring on about day

90 of the year. The crop is usually harvested (cut) for hay three or four times during the

growing season. Therefore, Figure 35 shows four Kc sub-cycles or cutting cycles: sub-cycle 1

follows greenup in the spring and the three additional Kc sub-cycles follow cuttings. Cuttings

create a ground surface with less than 10% vegetation cover. Cutting cycle 1 is longer in

duration than cycles 2, 3 and 4 due to lower air and soil temperatures during this period that

reduce crop growth rates. The lengths for cutting cycle 1 were taken from the first entry for

alfalfa (“1st cutting cycle”) in Table 11 for Idaho, the United States (10/30/25/10). The lengths

for cutting cycles 2, 3 and 4 were taken from the entry for alfalfa in Table 11 for “individual

cutting periods” for Idaho, the United States (5/20/10/10). These lengths were based on

observations. In the southern Idaho climate, frosts terminate the growing season sometime in

the fall, usually around day 280-290 of the year (early to mid-October).

The magnitudes of the Kc values during the mid-season periods of each cutting cycle

shown in Figure 35 vary from cycle to cycle due to the effects of adjusting the values for

Kc mid and Kc end for each cutting cycle period using Equations 62 and 65. In applying these

two adjustment equations, the u2 and RHmin values were averages for the mid-season and late

season stages within each cutting cycle. Basal Kcb curves similar to Figure 35 can be

constructed for forage or hay crops, following procedures presented in Chapter 7.

Kc mid when effects of individual cutting periods are averaged

Under some conditions, the user may wish to average the effects of cuttings for a forage crop

over the course of the growing season. When cutting effects are averaged, then only a single

value for Kc mid and a only single Kc curve need to be employed for the whole growing

FIGURE 35

Constructed curve for Kc for alfalfa hay in southern Idaho, the United States using values from

Tables 11 and 12 and adjusted using Equations 62 and 65

(data from Wright, 1990)

75 100 125 150 175 200 225 250 275 300 325

day of the year

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

crop coefficient

ini

dev mid

late

second

cutting

third

cutting frost

first

cycle

Crop evapotranspiration

129

season. When this is the case, a “normal” Kc curve is constructed as in Figure 25, where only

one midseason period is shown for the forage crop. The Kc mid for this total midseason period

must average the effects of occasional cuttings or harvesting. The value that is used for Kc mid

is therefore an average of the Kc curve for the time period starting at the first attainment of full

cover and ending at the beginning of the final late season period near dormancy or frost. The

value used for Kc mid under these averaged conditions may be only about 80% of the Kc value

that represents full ground cover. These averaged, full-season Kc mid values are listed in Table

12. For example, for alfalfa hay, the averaged, full-season Kc mid is 1.05, whereas, the Kc mid

for an individual cutting period is 1.20.

Fruit trees

Values for the crop coefficient during the mid-season and end of late season stages are given in

Table 12. As mentioned before, the Kc values listed are typical values for standard climatic

conditions and need to be adjusted by using Equations 62 and 65 where RHmin or u2 differ. As

the mid and late season stages of deciduous trees are quite long, the specific adjustment of Kc

to RHmin and u2 should take into account the varying climatic conditions throughout the

season. Therefore, several adjustments of Kc are often required if the mid and late seasons

cover several climatic seasons, e.g., spring, summer and autumn or wet and dry seasons. The

Kc ini and Kc end for evergreen non dormant trees and shrubs are often not different, where

climatic conditions do not vary much, as happens in tropical climates. Under these conditions,

seasonal adjustments for climate may therefore not be required since variations in ETc depend

mostly on variations in ETo.

CALCULATING ETC

From the crop coefficient curve the Kc value for any period during the growing period can be

graphically or numerically determined. Once the Kc values have been derived, the crop

evapotranspiration, ETc, can be calculated by multiplying the Kc values by the corresponding

ETo values.

Graphical determination of Kc

Weekly, ten-day or monthly values for Kc are necessary when ETc calculations are made on

weekly, ten-day or monthly time steps. A general procedure is to construct the Kc curve,

overlay the curve with the lengths of the weeks, decade or months, and to derive graphically

from the curve the Kc value for the period under consideration (Figure 36). Assuming that all

decades have a duration of 10 days facilitates the derivation of Kc and introduces little error

into the calculation of ETc.

The constructed Kc curve in Box 15 was used to construct the curve in Figure 36. This

curve has been overlaid with the lengths of the decades. Kc values of 0.15, 1.19 and 0.35 and

the actual lengths for growth stages equal to 25, 25, 30 and 20 days were used. The crop was

planted at the beginning of the last decade of May and was harvested 100 days later at the end

of August.

For all decades the Kc values can be derived directly from the curve. The value at the

middle of the decade is considered to be the average Kc of that 10 day period. Only the second

decade of June, where the Kc value changes abruptly, requires some calculation.

ETc -–single crop coefficient (Kc)

130

BOX 15

Case study of a dry bean crop at Kimberly, Idaho, the United States (single crop coefficient)

An example application for using the Kc procedure under average soil wetness conditions is presented

for a dry bean crop planted on 23 May 1974 at Kimberly, Idaho, the United States (latitude = 42.4°N).

The initial, development, mid-season and late season stage lengths are taken from Table 11 for a

continental climate as 20, 30, 40 and 20 days (the stage lengths listed for southern Idaho were not

used in this example in order to demonstrate the only approximate accuracy of values provided in

Table 11 when values for the specific location are not available). Initial values for Kc ini, Kc mid and Kc

end are selected from Table 12 as 0.4, 1.15, and 0.35.

The mean RHmin and u2 during both the mid-season and late season growth stages were 30% and

2.2 m/s. The maximum height suggested in Table 12 for dry beans is 0.4 m. Therefore, Kc mid is

adjusted using Eq. 62 as:

[]

19.1

3

4.0

)4530(004.0)22.2(04.015.1K

3.0

midc =

÷

ø

ö

ç

è

æ

−−−+=

As Kc end = 0.35 is less than 0.45, no adjustment is made to Kc end. The value for Kc mid is not

significantly different from that in Table 12 as u2 ≈ 2 m/s, RHmin is just 15% lower than the 45%

represented in Table 12, and the height of the beans is relatively short. The initial Kc curve for dry

beans in Idaho can be drawn, for initial, planning purposes, as shown in the graph (dotted line), where

Kc ini, Kc mid and Kc end are 0.4, 1.19, and 0.35 and the four lengths of growth stages are 20, 30, 40

and 20 days. Note that the Kc ini = 0.4 taken from Table 12 serves only as an initial, approximate

estimate for Kc ini.

140 160 180 200 220 240

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0

40

80

120

160

200

day of the Year

Measured

crop coefficient

I

P

I

P

precipitation and irrigation (mm)

Constructed Kc curves using values from Tables 11 and 12 directly (dotted line) and

modified using Kc ini from Fig. 29 and Lini = 25, Ldev = 25, Lmid = 30, and Llate = 20 days

(heavy line) for dry beans at Kimberly, Idaho. Also shown are daily measured Kc (lysimeter

data from Wright, 1990).

Crop evapotranspiration

131

Kc ini can be more accurately estimated using the approach described in this chapter. ETo during

the initial period at Kimberly (late May - early June, 1974) averaged 5.3 mm/day, and the wetting

interval during this period was approximately 14 days (2 rainfall events occurred averaging 5 mm

per event). Therefore, as the wetting events were light (< 10 mm each), Fig. 29 is used. The soil

texture at Kimberly, Idaho is silt loam. From Fig. 29, Kc ini for the 14 day wetting interval and ETo =

5.3 mm/day is about 0.15. This value is substantially less than the general 0.4 value suggested by

Table 12, and emphasizes the need to utilize local, actual precipitation and irrigation data when

determining Kc ini.

Comparison of constructed curves with measurements

Because the ETc data for the dry bean crop at Kimberly, Idaho were measured using a precision

lysimeter system during 1974 by Wright (1990), the actual Kc measurements can be compared

with the constructed Kc curves, where actual Kc was calculated by dividing lysimeter

measurements of ETc by daily ETo estimated using the FAO Penman-Monteith equation.

As illustrated in the graph, the mid-season length as taken from Table 11 for the general,

continental climate overestimated the true mid-season length for dry beans in southern Idaho,

which averaged only about 30 days rather than 40 days as suggested by Table 11. This illustrates

the importance of using the local observation of 30 days for mid-season period length rather than

the general value from Table 11.

The final, best estimate for the Kc curve for the dry bean crop in southern Idaho is plotted (lower

curve in graph) using Kc values of 0.15, 1.19, and 0.35 and the actual observed lengths of growth

stages equal to 25, 25, 30 and 20 days. Note the impact that the error in estimating mid-season

length has on the area under the Kc curve. This supports the need to obtain local observations of

growth stage dates and lengths.

The value calculated for Kc mid (1.19) appears to have underestimated the measured value for Kc

during portions of the mid-season period at Kimberly. Some of this effect was due to effects of

increased soil water evaporation following four irrigations during the 1974 mid-season which

increased the effective Kc. This is illustrated in Box 16, where the basal Kcb + Ke approach is

introduced and demonstrated for this same example.

The 0.15 value calculated for Kc ini using Fig. 29 agrees closely with measured Kc during the initial

period. Measured Kc during the development period exceeded the final Kc curve during days on or

following wetting events. The day to day variation in the lysimeter measured Kc is normal and is

caused by day to day variations in weather, in wind direction, by errors in prediction of Rn and ETo,

and by some random errors in the lysimeter measurements and weather measurements.

ETc -–single crop coefficient (Kc)

132

During the first five days of that decade, Kc = 0.15, while during the second part of the

decade Kc varies from 0.15 to 0.36 at the end of day 10. The Kc for that decade is

consequently: 5/10 (0.15) + 5/10 (0.15+0.36)/2 = 0.20.

Numerical determination of Kc

The Kc coefficient for any period of the growing season can be derived by considering that

during the initial and mid-season stages Kc is constant and equal to the Kc value of the growth

stage under consideration. During the crop development and late season stage, Kc varies

linearly between the Kc at the end of the previous stage (Kc prev) and the Kc at the beginning

of the next stage (Kc next), which is Kc end in the case of the late season stage:

()

prevcnextc

stage

L

)L(i

KK −

ú

û

ù

ê

ë

éΣ−

+= (66)

where i day number within the growing season [1 .. length of the growing

season],

Kc i crop coefficient on day i,

Lstage length of the stage under consideration [days],

Σ(Lprev) sum of the lengths of all previous stages [days].

Equation 66 applies to all four stages.

FIGURE 36

Kc curve and ten-day values for Kc and ETcderived from the graph for the dry bean crop

example (Box 15)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

crop coefficient

123123123 12 3

May June July August

mm/day

Kc

ETo

ETc

0.15

1.19

0.35

0.15 0.15 0.20 0.57 0.98 1.19 1.19 1.19 0.98 0.56

3.0 3.4 4.0 4.2 4.5 5.1 5.6 6.0 5.5 5.2

0.5 0.5 0.8 2.4 4.4 6.1 6.7 7.1 5.4 2.9

Crop evapotranspiration

133

EXAMPLE 28

Numerical determination of Kc

Determine Kc at day 20, 40, 70 and 95 for the dry bean crop (Figure 36).

Crop growth stage

Length (days)

Kc

initial

crop development

mid-season

late season

25

30

20

Kc ini = 0.15

0.15 ... 1.19

Kc mid = 1.19

1.19 .. Kc end = 0.35

At i = 20:

At i = 40

For:

and:

From Eq. 66:

initial stage, Kc = Kc ini =

Crop development stage,

Σ(Lprev) = Lini =

Lstage = Ldev =

Kc = 0.15+[(40-25)/25](1.19-0.15)=

0.15

25

0.77

-

days

-

At i = 70:

At i = 95

For:

and:

From Eq. 66:

mid-season stage, Kc = Kc mid =

late season stage,

Σ(Lprev) = Lini +Ldev + Lmid = (25+25+30)=

Lstage = Llate =

Kc = 1.19+[(95-80)/20](0.35-1.19)=

1.19

80

20

0.56

-

days

-

The crop coefficients at day 20, 40, 70 and 95 for the dry bean crop are 0.15, 0.77, 1.19 and 0.56

respectively.

ALFALFA-BASED CROP COEFFICIENTS

As two reference crop definitions (grass and alfalfa) are in use in various parts of the world,

two families of Kc curves for agricultural crops have been developed. These are the alfalfa-

based Kc curves by Wright (1981; 1982) and grass-based curves by Pruitt (Doorenbos and

Pruitt 1977; Jensen et al. 1990) and those reported in this paper. The user must exercise caution

to avoid mixing grass-based Kc values with alfalfa reference ET and vice versa. Usually, a Kc

based on the alfalfa reference can be ‘converted’ for use with a grass reference by multiplying

by a factor ranging from about 1.0 to 1.3, depending on the climate (1.05 for humid, calm

conditions, and 1.2 for semi-arid, moderately windy conditions, and 1.35 for arid, windy

conditions):

)alfalfa(cratio)grass(c KKK = (67)

where Kc (grass) grass-based Kc (this handbook),

Kc (alfalfa) alfalfa-based Kc,

Kratio conversion factor (1.0 ... 1.3).

A reference conversion ratio can be established for any climate by using the Kc mid =

1.20 listed for alfalfa in Table 12 and then adjusting this Kc mid for the climate using Equation

62. For example, at Kimberly, Idaho, the United States, where RHmin = 30% and u2 = 2.2 m/s

are average values during the summer months, a reference conversion ratio between alfalfa and

grass references using Equation 62 is approximately:

ETc -–single crop coefficient (Kc)

134

[]

24.1

3

5.0

)4530(004.0)22.2(04.02.1K

3.0

ratio =

÷

ø

ö

ç

è

æ

−−−+= (68)

where h = 0.5 m is the standard height for the alfalfa reference.

TRANSFERABILITY OF PREVIOUS KC VALUES

The values for Kc mid and Kc end listed in Table 12 are for a large part based on the original

values presented in FAO Irrigation and Drainage Papers No. 24 and 33 (FAO-24 and FAO-33),

with some adjustment and revisions to reflect recent findings. Similarly adjustments in Kc mid

to compensate for differences in aerodynamic roughness and leaf area, as introduced in

Equation 62 are derived from the Kc values given for different wind and RHmin conditions in

the concerned Kc table in FAO-24, with some upward adjustment to better reflect increased

ETcrop values under high wind and low RHmin when applied with the FAO Penman-Monteith

equation.

The Kc’s from FAO-24 were based primarily on a living grass reference crop. The FAO

Penman-Monteith equation presented in this publication similarly represents the same

standardized grass reference. For that reason Kc values are in general not very different

between these publications except under high wind and low RHmin.

The No. 24 modified Penman was found frequently to overestimate ETo even up to 25

% under high wind and low evaporative conditions and required often substantial local

calibration (see chapter 2). Kc values derived from crop water use studies which used the FAO-

24 Penman equation to compute grass reference crop evapotranspiration, can therefore not be

used and need to be adjusted using ETo values estimated from the FAO Penman-Monteith

equation. Similarly crop water requirement estimates based on the FAO-24 Modified Penman

equation will need to be reassessed in view of the found differences between the FAO-24

Penman and the FAO Penman-Monteith reference equations.

Crop evapotranspiration

135

Chapter 7

ETc – dual crop coefficient (Kc = Kcb + Ke)

Like Chapter 6, this chapter also deals with the calculation of crop evapotranspiration (ETc)

under standard conditions where no limitations are placed on crop growth or

evapotranspiration. This chapter presents the procedure for predicting the effects of specific

wetting events on the value for the crop coefficient Kc. The solution consists of splitting Kc

into two separate coefficients, one for crop transpiration, i.e., the basal crop coefficient (Kcb),

and one for soil evaporation (Ke):

oecbc ET)K(KET += (69)

The dual crop coefficient approach is more complicated and more computationally

intensive than the single crop coefficient approach (Kc) of Chapter 6. The procedure is

conducted on a daily basis and is intended for applications using computers. It is recommended

that the approach be followed when improved estimates for Kc are needed, for example to

schedule irrigations for individual fields on a daily basis.

The calculation procedure for crop evapotranspiration, ETc, consists of:

1. identifying the lengths of crop growth stages, and selecting the corresponding Kcb

coefficients;

2. adjusting the selected Kcb coefficients for climatic conditions during the stage;

3. constructing the basal crop coefficient curve (allowing one to determine Kcb values for

any period during the growing period);

4. determining daily Ke values for surface evaporation; and

5. calculating ETc as the product of ETo and (Kcb + Ke).

TRANSPIRATION COMPONENT (Kcb ETo)

Basal crop coefficient (Kcb)

The basal crop coefficient (Kcb) is defined as the ratio of the crop evapotranspiration over the

reference evapotranspiration (ETc/ETo) when the soil surface is dry but transpiration is

occurring at a potential rate, i.e., water is not limiting transpiration (Figure 22). Therefore, 'Kcb

ETo' represents primarily the transpiration component of ETc. The Kcb ETo does include a

residual diffusive evaporation component supplied by soil water below the dry surface and by

soil water from beneath dense vegetation.

ETc -–dual crop coefficient (Kc = Kcb + Ke)

136

As the Kc values of Chapter 6 include averaged effects of evaporation from the soil

surface, the Kcb values lie below the Kc values as illustrated in Figure 26 and a separate table

for Kcb is required. Recommended values for Kcb are listed in Table 17 for the same crops

listed in Table 12. As with Table 12, the values for Kcb in the table represent Kcb for a sub-

humid climate and with moderate wind speed. For specific adjustment in climates where

RHmin differs from 45% or where the wind speed is larger or smaller than 2 m/s, the Kcb mid

and Kcb end values larger than 0.45 must be adjusted using the following equation:

[]

3.0

min2)Tab(cbcb 3

h

)45RH(004.0)2u(04.0KK ÷

ø

ö

ç

è

æ

−−−+= (70)

where Kcb (Tab) the value for Kcb mid or Kcb end ( if ≥ 0.45) taken from Table 17,

u

2 the mean value for daily wind speed at 2 m height over grass during the

mid or late season growth stage [m s-1] for 1 m s-1 ≤ u2 ≤ 6 m s-1,

RHmin the mean value for daily minimum relative humidity during the mid- or

late season growth stage [%] for 20% ≤ RHmin ≤ 80%,

h the mean plant height during the mid or late season stage [m] (from

Table 12) for 20% ≤ RHmin ≤ 80%.

For a full discussion on the impact of the climatic correction, and the numerical

determination of Kcb mid and Kcb end, the user is referred to the discussions on Kc mid and

Kc end in Chapter 6.

Table 18 summarizes the general guidelines that were used in deriving Kcb values from

the Kc values listed in Table 17. Where local research results are available, values for Kcb from

Table 17 can be modified to reflect effects of local conditions, cultural practices or crop

varieties on Kcb. However, local values for Kcb should not be expected to deviate by more than

0.2 from the values in Table 17. A greater deviation should signal the need to investigate or

evaluate the local research technique, equipment and cultural practices. Where local Kcb values

are used, no adjustment for climate using Equation 70 is necessary.

EXAMPLE 29

Selection and adjustment of basal crop coefficients, Kcb

Select Kcb ini, Kcb mid and Kcb end for the dry bean crop of Box 15.

Kcb ini, Kcb mid and Kcb end can be selected directly from Table 17 for dry beans as 0.15, 1.10 and

0.25. When adjusted for climate using Eq. 70:

Kcb ini = 0.15

Kcb mid = 1.10 + (0.04 (2.2-2) - 0.004 (30-45) )(0.4/3)0.3 = 1.14

Kcb end = 0.25 (as Kcb < 0.45)

Height for beans was taken from Table 12 as 0.4 m.

The corresponding Kcb curve is shown in Figure 37.

Crop evapotranspiration

137

TABLE 17

Basal crop coefficients, Kc, for non stressed, well-managed crops in subhumid climates (RHmin ≈

≈≈

≈

45%, u2 ≈

≈≈

≈ 2 m/s) for use with the FAO Penman-Monteith ETo.

Crop

Kcb ini1

Kcb mid1

Kcb end1

a. Small Vegetables 0.15 0.95 0.85

Broccoli 0.95 0.85

Brussel Sprouts 0.95 0.85

Cabbage 0.95 0.85

Carrots 0.95 0.85

Cauliflower 0.95 0.85

Celery 0.95 0.90

Garlic 0.90 0.60

Lettuce 0.90 0.90

Onions - dry 0.95 0.65

- green 0.90 0.90

- seed 1.05 0.70

Spinach 0.90 0.85

Radishes 0.85 0.75

b. Vegetables – Solanum Family (Solanaceae) 0.15 1.10 0.70

EggPlant 1.00 0.80

Sweet Peppers (bell) 1.002 0.80

Tomato 1.102 0.60-0.80

c. Vegetables – Cucumber Family (Cucurbitaceae) 0.15 0.95 0.70

Cantaloupe 0.75 0.50

Cucumber – Fresh Market 0.952 0.70

– Machine harvest 0.95 0.80

Pumpkin, Winter Squash 0.95 0.70

Squash, Zucchini 0.90 0.70

Sweet Melons 1.00 0.70

Watermelon 0.95 0.70

d. Roots and Tubers 0.15 1.00 0.85

Beets, table 0.95 0.85

Cassava – year 1 0.703 0.20

– year 2 1.00 0.45

Parsnip 0.95 0.85

Potato 1.10 0.654

Sweet Potato 1.10 0.55

Turnip (and Rutabaga) 1.00 0.85

Sugar Beet 1.15 0.505

continued…

1 These are values for Kcb representing conditions having a dry soil surface. These values are intended

for use with the dual Kcb ini + Ke approach, only. Values for maximum crop height, h, are given in

Table 12 for adjusting Kcb for climate.

2 Beans, Peas, Legumes, Tomatoes, Peppers and Cucumbers are sometimes grown on stalks reaching

1.5 to 2 meters in height. In such cases, increased Kcb values need to be taken. For green beans,

peppers and cucumbers, 1.10 can be taken, and for tomatoes, dry beans and peas, 1.15. Under these

conditions h should be increased also.

3 The misdseason values for cassava assume nonstressed conditions during or following the rainy

season. The Kcb end values account for domancy during the dry season.

4 The Kcb end value for potatoes is about 0.35 for long season potatoes with vine kill.

5 This Kcb end value is for no irrigation during the last month of the growing season. The Kcb end value

for sugar beets is higher, up to 0.9, when irrigation or significant rain occurs during the last month of

the growing season.

ETc -–dual crop coefficient (Kc = Kcb + Ke)

138

Table 17 continued

Crop Kcb ini1 Kcb mid1 K

cb end1

e. Lugumes (Leguminosae) 0.15 1.10 0.50

Beans, green 1.002 0.80

Beans, dry and Pulses 1.102 0.25

Chick pea 0.95 0.25

Fababean (broad bean) – Fresh 1.102 1.05

– Dry/Seed 1.102 0.20

Grabanzo 1.05 0.25

Green Gram and Cowpeas 1.00 0.55-0.256

Groundnut (Peanut) 1.10 0.50

Lentil 1.05 0.20

Peas – Fresh 1.102 1.05

– Dry/Seed 1.10 0.20

Soybeans 1.10 0.30

f. Perennial Vegetables (with winter dormancy and initially

bare or mulched soil)

Artichokes 0.15 0.95 0.90

Asparagus 0.15 0.907 0.20

Mint 0.40 1.10 1.05

Strawberries 0.30 0.80 0.70

g. Fibre Crops 0.15

Cotton 1.10-1.15 0.50-0.40

Flax 1.05 0.20

Sisal8 0.4-0.7 0.4-0.7

h. Oil Crops 0.15 1.10 0.25

Castorbean (Ricinus) 1.10 0.45

Rapeseed, Canola 0.95-1.109 0.25

Safflower 0.95-1.109 0.20

Sesame 1.05 0.20

Sunflower 0.95-1.109 0.25

i. Cereals 0.15 1.10 0.25

Barley 1.10 0.15

Oats 1.10 0.15

Spring Wheat 1.10 0.15-0.310

Winter Wheat 0.15-

0.511

1.10 0.15-0.310

Maize - Field (grain) (field corn) 0.15 1.15 0.50,0.1512

- Sweet (sweet corn) 1.10 1.0013

Millet 0.95 0.20

Sorghum – grain 0.95-1.05 0.35

– sweet 1.15 1.00

Rice 1.00 1.15 0.70-0.45

continued…

6 The first Kcb end is for harvested fresh. The second value is for harvested dry.

7 The Kcb for asparagus usually remains at Kcb ini during harvest of the spears, due to sparse ground cover.

The Kcb mid value is for following regrowth of vegetation following termination of harvest of spears.

8 K

cb for sisal depends on the planting density and water management (e.g., intentional moisture stress).

9 The lower values are for rainfed crops having less dense plant populations.

10 The higher value is for hand-harvested crops.

11 The two Kcb ini values for winter wheat are for less than 10% ground cover and for during the dormant,

winter period, if the vegetation fully covers the ground, but conditions are nonfrozen.

12 The first Kcb end value is for harvest at high grain moisture. The second Kcb end value is for harvest after

complete field drying of the grain (to about 18% moisture, wet mass basis).

13 If harvested fresh for human consumption. Use Kcb end for field maize if the sweet maize is allowed to

mature and dry in the field.

Crop evapotranspiration

139

Table 17 continued.

Crop

Kcb ini1

Kcb mid1

Kcb end1

j. Forages

Alfalfa Hay – individual cutting periods 0.3014 1.1514 1.1014

– for seed 0.30 0.45 0.45

Bermuda hay – averaged cutting effects 0.50 0.9515 0.80

– Spring crop for seed 0.15 0.85 0.60

Clover hay, Berseem – individual cutting periods 0.3014 1.1014 1.0514

Rye Grass hay – averaged cutting effects 0.85 1.0015 0.95

Sudan Grass hay (annual) – individual cutting periods 0.3014 1.1014 1.0514

Grazing Pasture - Rotated Grazing 0.30 0.80-1.00 0.80

- Extensive Grazing 0.30 0.70 0.70

Turf grass - cool season16 0.85 0.90 0.90

- warm season16 0.75 0.80 0.80

k. Sugar cane 0.15 1.20 0.70

l. Tropical Fruits and Trees

Banana – 1st year 0.15 1.05 0.90

– 2nd year 0.60 1.10 1.05

Cacao 0.90 1.00 1.00

Coffee – bare ground cover 0.80 0.90 0.90

– with weeds 1.00 1.05 1.05

Date Palms 0.80 0.85 0.85

Palm Trees 0.85 0.90 0.90

Pineapple17 (multiyear crop) – bare soil 0.15 0.25 0.25

– with grass cover 0.30 0.45 0.45

Rubber Trees 0.85 0.90 0.90

Tea – nonshaded 0.90 0.95 0.90

– shaded18 1.00 1.10 1.05

m. Grapes and Berries

Berries (bushes) 0.20 1.00 0.40

Grapes – Table or Raisin 0.15 0.80 0.40

– Wine 0.15 0.65 0.40

Hops 0.15 1.00 0.80

continued…

14 These Kcb coefficients for hay crops represent immediately following cutting; at full cover; and

immediately before cutting, respectively. The growing season is described as a series of individual

cutting periods.

15 This Kcb mid coefficient for bermuda and ryegrass hay crops is an overall average Kcb mid coefficient

that averages Kcb for both before and following cuttings. It is applied to the period following the first

development period until the beginning of the last late season period of the growing season.

16 Cool season grass varieties include dense stands of bluegrass, ryegrass, and fescue. Warm season

varieties include bermuda grass and St. Augustine grass. The 0.90 values for cool season grass

represent a 0.06 to 0.08 m mowing height under general turf conditions. Where careful water

management is practiced and rapid growth is not required, Kcb's for turf can be reduced by 0.10.

17 The pineapple plant has very low transpiration because it closes its stomates during the day and opens

them during the night. Therefore, the majority of ETc from pineapple is evaporation from the soil.

18 Includes the water requirements of the shade trees.

ETc -–dual crop coefficient (Kc = Kcb + Ke)

140

Table 17 continued.

Crop Kcb ini1 Kcb mid1 K

cb end1

n. Fruit Trees

Almonds, no ground cover 0.20 0.85 0.6019

Apples, Cherries, Pears20

- no ground cover, killing frost 0.35 0.90 0.6519

- no ground cover, no frosts 0.50 0.90 0.7019

- active ground cover, killing frost 0.45 1.15 0.9019

- active ground cover, no frosts 0.75 1.15 0.8019

Apricots, Peaches, Stone Fruit20,21

- no ground cover, killing frost 0.35 0.85 0.6019

- no ground cover, no frosts 0.45 0.85 0.6019

- active ground cover, killing frost 0.45 1.10 0.8519

- active ground cover, no frosts 0.75 1.10 0.8019

Avocado, no ground cover 0.50 0.80 0.70

Citrus, no ground cover22

70% canopy 0.65 0.60 0.65

50% canopy 0.60 0.55 0.60

20% canopy 0.45 0.40 0.50

Citrus, with active ground cover or weeds23

70% canopy 0.75 0.70 0.75

50% canopy 0.75 0.75 0.75

20% canopy 0.80 0.80 0.85

Conifer Trees24 0.95 0.95 0.95

Kiwi 0.20 1.00 1.00

Olives (40 to 60% ground coverage by canopy)25 0.55 0.65 0.65

Pistachios, no ground cover 0.20 1.05 0.40

Walnut Orchard20 0.40 1.05 0.6019

19 These Kcb end values represent Kcb prior to leaf drop. After leaf drop, Kcb end ≈ 0.15 for bare, dry

soil or dead ground cover and Kcb end ≈ 0.45 to 0.75 for actively growing ground cover (consult

Chapter 11).

20 Refer to Eq. 94, 97 or 98 and footnotes 22 and 23 for estimating Kcb for immature stands.

21 Stone fruit category applies to peaches, apricots, pears, plums and pecans.

22 These Kcb values can be calculated from Eq. 98 for Kc min = 0.15 and Kcb full = 0.70, 0.65 and

0.70 for the initial, mid season and end of season periods, and fc eff = fc where fc = fraction of

ground covered by tree canopy (e.g., the sun is presumed to be directly overhead). The midseason

value is lower than initial and ending values due to the effects of stomatal closure during periods of

peak ET. For humid and subhumid climates where there is less stomatal control by citrus, values for

Kcb ini, Kcb mid, and Kcb end can be increased by 0.1 - 0.2, following Rogers et al. (1983).

23 These Kcb values can be calculated as Kcb = fc Kcb ngc + (1 - fc) Kcb cover where Kcb ngc is the

Kcb of citrus with no active ground cover (calculated as in footnote 22), Kcb cover is the Kcb for the

active ground cover (0.90), and fc is defined in footnote 22. Alternatively, Kcb for citrus with active

ground cover can be estimated directly from Eq. 98 by setting Kc min = Kcb cover. For humid and

subhumid climates where there is less stomatal control by citrus, values for Kcb ini, Kcb mid, and Kcb

end can be increased by 0.1 - 0.2, following Rogers et al. (1983).

For non-active or only moderately active ground cover (active indicates green and growing ground

cover with LAI > about 2 to 3) , Kcb should be weighted between Kcb for no ground cover and Kcb

for active ground cover, with the weighting based on the "greeness" and approximate leaf area of the

ground cover.

24 Conifers exhibit substantial stomatal control due to reduced aerodynamic resistance. The Kcb can

easily reduce below the values presented, which represent well-watered conditions for large forests.

25 These coefficients represent about 40 to 60% ground cover. Refer to Eq. 98, example 43, and

footnotes 22 and 23 for estimating Kcb for immature stands.

Primary sources: Kcb ini: Doorenbos and Kassam (1979); Kcb mid and Kcb end: Doorenbos and Pruitt

(1977); Pruitt (1986); Wright (1981, 1982), Snyder et al. (1989)

Crop evapotranspiration

141

TABLE 18

General guidelines to derive Kcb from the Kc values listed in Table 12

Growth

stage

Ground condition, irrigation and cultural

practices

Kcb further

adjustment

Initial

Annual crop - (nearly) bare soil surface

Perennial crop - (nearly) bare soil surface

Grasses, brush and trees - killing frost

Perennial crop - some ground cover or leaf cover

- infrequently irrigated (olives, palm trees, fruit

trees, ...)

- frequently irrigated (garden-type vegetables, ...)

0.15

0.15 - 0.20

0.30 - 0.40

Kc ini(Tab.12) - 0.1

Kc ini(Tab.12) - 0.2

-

Mid-

season

Ground cover more than 80%

Ground cover less than 80% (vegetables)

Kc mid(Tab.12) - 0.05

Kc mid(Tab.12) - 0.10

Climate (Eq.

70)

Climate (Eq.

70)

At end of

season

infrequently irrigated or wetted during late season

frequently irrigated or wetted during late season

~ Kc end - 0.05

Kc end - 0.1

-

Climate (Eq.

70)

Climate: adjustment for climate using Eq. 70 where Kcb > 0.45

Determination of daily Kcb values

As outlined in Chapter 6, only three point values are required to describe and to construct the

crop coefficient curve. After dividing the growing period into the four general growth stages

and selecting and adjusting the Kcb values corresponding to the initial (Kcb ini), mid-season

(Kcb mid) and end of the late season stages (Kcb end), the crop coefficient curve can be drawn

(Figure 37) and the Kcb coefficients can be derived (Example 30).

EXAMPLE 30

Determination of daily values for Kcb

Calculate the basal crop coefficient for the dry beans (Example 29, Figure 37) at the middle of each of

the four growth stages.

Initial stage (Lini = 25 days), at day 12 of the growing period:

K

cb = Kcb ini = 0.15

Crop development stage (Ldev = 25 days), at day (25+25/2=) 37 of the growing period,

using Eq. 66:

K

cb = 0.15 + [(37 - 25)/25] (1.14 - 0.15) = 0.63

Mid-season stage (Lmid = 30 days), at day (25+25+30/2=) 65 of the growing period:

K

cb = Kcb mid = 1.14

Late season stage (Llate = 20 days), at day (25+25+30+20/2=) 90 of the growing period, Eq. 66:

K

cb = 1.14 + [(90-(25+25+30))/20] (0.25-1.14) = 0.70

The basal crop coefficients, Kcb, at days 12, 37, 65 and 90 of the growing period are 0.15, 0.63, 1.14

and 0.70 respectively.

ETc -–dual crop coefficient (Kc = Kcb + Ke)

142

EVAPORATION COMPONENT (Ke ETo)

The soil evaporation coefficient, Ke, describes the evaporation component of ETc. Where the

topsoil is wet, following rain or irrigation, Ke is maximal. Where the soil surface is dry, Ke is

small and even zero when no water remains near the soil surface for evaporation.

Calculation procedure

Where the soil is wet, evaporation from the soil occurs at the maximum rate. However, the crop

coefficient (Kc = Kcb + Ke) can never exceed a maximum value, Kc max. This value is

determined by the energy available for evapotranspiration at the soil surface (Kcb + Ke ≤ Kc

max), or Ke ≤ (Kc max - Kcb).

When the topsoil dries out, less water is available for evaporation and a reduction in

evaporation begins to occur in proportion to the amount of water remaining in the surface soil

layer, or:

maxcewcbmaxcre Kf)KK(KK ≤−= (71)

where Ke soil evaporation coefficient,

Kcb basal crop coefficient,

Kc max maximum value of Kc following rain or irrigation,

Kr dimensionless evaporation reduction coefficient dependent on the

cumulative depth of water depleted (evaporated) from the topsoil,

few fraction of the soil that is both exposed and wetted, i.e., the fraction of

soil surface from which most evaporation occurs.

FIGURE 37

Constructed basal crop coefficient (Kcb) curve for a dry bean crop (Example 29) using growth

stage lengths of 25, 25, 30 and 20 days

0.00

0.20

0.40

0.60

0.80

1.00

1.20

K

initial crop

development

mid

season

late

season

Kcb

cb ini

Kcb mid

cb end

1 25 50 80 100

days

Crop evapotranspiration

143

In computer programming terminology, Equation 71 is expressed as Ke = min(Kr (Kc

max - Kcb), few Kc max).

Following rain or irrigation Kr is 1, and evaporation is only determined by the energy

available for evaporation. As the soil surface dries, Kr becomes less than one and evaporation is

reduced. Kr becomes zero when no water is left for evaporation in the upper soil layer.

Evaporation occurs predominantly from the exposed soil fraction. Hence, evaporation is

restricted at any moment by the energy available at the exposed soil fraction, i.e., Ke cannot

exceed few Kc max, where few is the fraction of soil from which most evaporation occurs, i.e.,

the fraction of the soil not covered by vegetation and that is wetted by irrigation or

precipitation.

The calculation procedure consists in determining:

• the upper limit Kc max;

• the soil evaporation reduction coefficient Kr; and

• the exposed and wetted soil fraction few.

The estimation of Kr requires a daily water balance computation for the surface soil layer.

Upper limit Kc max

Kc max represents an upper limit on the evaporation and transpiration from any cropped

surface and is imposed to reflect the natural constraints placed on available energy represented

by the energy balance difference Rn - G - H (Equation 1). Kc max ranges from about 1.05 to

1.30 when using the grass reference ETo:

[]

{}

÷

ø

ö

ç

è

æ

+

ï

þ

ï

ý

ü

ï

î

ï

í

ì

÷

ø

ö

ç

è

æ

−−−+= 05.0K,

3

h

)45RH(004.0)2u(04.02.1maxK cb

3.0

min2maxc (72)

where h mean maximum plant height during the period of calculation (initial,

development, mid-season, or late-season) [m],

Kcb basal crop coefficient,

max ( ) maximum value of the parameters in braces { } that are separated by

the comma.

Equation 72 ensures that Kc max is always greater or equal to the sum Kcb + 0.05. This

requirement suggests that wet soil will always increase the value for Kcb by 0.05 following

complete wetting of the soil surface, even during periods of full ground cover. A value of 1.2

instead of 1 is used for Kc max in Equation 72 because of the effect of increased aerodynamic

roughness of surrounding crops during development, mid-season and late season growth stages

which can increase the turbulent transfer of vapour from the exposed soil surface. The “1.2”

coefficient also reflects the impact of the reduced albedo of wet soil and the contribution of

heat stored in dry soil prior to the wetting event. All of these factors can contribute to increased

evaporation relative to the reference.

ETc -–dual crop coefficient (Kc = Kcb + Ke)

144

The “1.2” coefficient in Equation 72 represents effects of wetting intervals that are

greater than 3 or 4 days. If irrigation or precipitation events are more frequent, for example

daily or each two days, then the soil has less opportunity to absorb heat between wettings, and

the “1.2” coefficient in Equation 72 can be reduced to about 1.1. The time step to compute Kc

max may vary from daily to monthly.

Soil evaporation reduction coefficient (Kr)

Soil evaporation from the exposed soil can be assumed to take place in two stages: an energy

limiting stage, and a falling rate stage. When the soil surface is wet, Kr is 1. When the water

content in the upper soil becomes limiting, Kr decreases and becomes zero when the total

amount of water that can be evaporated from the topsoil is depleted.

Maximum amount of water that can be evaporated

In the simple evaporation procedure, it is assumed that the water content of the evaporating

layer of the soil is at field capacity, θFC, shortly following a major wetting event and that the

soil can dry to a soil water content level that is halfway between oven dry (no water left) and

wilting point, θWP. The amount of water that can be depleted by evaporation during a complete

drying cycle can hence be estimated as:

eWPFC Z)5.0(1000TEW θ−θ= (73)

where TEW total evaporable water = maximum depth of water that can be

evaporated from the soil when the topsoil has been initially completely

wetted [mm],

θFC soil water content at field capacity [m3 m-3],

θWP soil water content at wilting point [m3 m-3],

Ze depth of the surface soil layer that is subject to drying by way of

evaporation [0.10-0.15 m].

Where unknown, a value for Ze, the effective depth of the soil evaporation layer, of 0.10-

0.15 m is recommended. Typical values for θFC, θWP and TEW are given in Table 19.

TABLE 19

Typical soil water characteristics for different soil types

Soil type Soil water characteristics Evaporation parameters

θ

θθ

θFC θ

θθ

θWP (θ

θθ

θFC-θ

θθ

θWP) Amount of water that can be

depleted by evaporation

(USA Soil

Texture

Classification) stage 1

REW

stages 1 and 2

TEW*

(Ze = 0.10 m)

m3/m3 m

3/m3 m

3/m3 mm mm

Sand

Loamy sand

Sandy loam

Loam

Silt loam

Silt

Silt clay loam

Silty clay

Clay

0.07 - 0.17

0.11 - 0.19

0.18 - 0.28

0.20 - 0.30

0.22 - 0.36

0.28 - 0.36

0.30 - 0.37

0.30 - 0.42

0.32 - 0.40

0.02 - 0.07

0.03 - 0.10

0.06 - 0.16

0.07 - 0.17

0.09 - 0.21

0.12 - 0.22

0.17 - 0.24

0.17 - 0.29

0.20 - 0.24

0.05 - 0.11

0.06 - 0.12

0.11 - 0.15

0.13 - 0.18

0.13 - 0.19

0.16 - 0.20

0.13 - 0.18

0.13 - 0.19

0.12 - 0.20

2 - 7

4 - 8

6 - 10

8 - 10

8 - 11

8 - 12

6 - 12

9 - 14

15 - 20

16 - 22

18 - 25

22 - 26

22 - 27

22 - 28

22 - 29

*TEW = (θFC - 0.5 θWP) Ze

Crop evapotranspiration

145

Stage 1: energy limiting stage

At the start of a drying cycle, following heavy rain or irrigation, the soil water content in the

topsoil is at field capacity and the amount of water depleted by evaporation, De, is zero. During

stage 1 of the drying process, the soil surface remains wet and it is assumed that evaporation

from soil exposed to the atmosphere will occur at the maximum rate limited only by energy

availability at the soil surface. This stage holds until the cumulative depth of evaporation, De, is

such that the hydraulic properties of the upper soil become limiting and water cannot be

transported to the soil surface at a rate that can supply the potential demand. During stage 1

drying, Kr = 1.

The cumulative depth of evaporation, De, at the end of stage 1 drying is REW (Readily

evaporable water, which is the maximum depth of water that can be evaporated from the topsoil

layer without restriction during stage1). The depth normally ranges from 5 to 12 mm and is

generally highest for medium and fine textured soils. Typical values for REW are given in

Table 19.

Stage 2: falling rate stage

The second stage (where the evaporation rate is reducing) is termed the ‘falling rate stage’

evaporation and starts when De exceeds REW. At this point, the soil surface is visibly dry, and

the evaporation from the exposed soil decreases in proportion to the amount of water remaining

in the surface soil layer:

FIGURE 38

Soil evaporation reduction coefficient, Kr

energy

limiting

stage

falling

rate

stage

θFC WP WP

0.5

Kr

(depletion)

0TEWREW

REW

TEW

θ: soil water content (m3 /m3 )

D cumulative depth of evaporation

e

0.00

0.20

0.40

0.60

0.80

1.00

θθ

ETc -–dual crop coefficient (Kc = Kcb + Ke)

146

REWTEW

1-i,e

DTEW

r

K

−

= for De,i-1 > REW (74)

where Kr dimensionless evaporation reduction coefficient dependent on the soil

water depletion (cumulative depth of evaporation) from the topsoil

layer (Kr = 1 when De,i-1 ≤ REW),

De,i-1 cumulative depth of evaporation (depletion) from the soil surface layer

at the end of day i-1 (the previous day) [mm],

TEW maximum cumulative depth of evaporation (depletion) from the soil

surface layer when Kr = 0 (TEW = total evaporable water) [mm],

REW cumulative depth of evaporation (depletion) at the end of stage 1 (REW

= readily evaporable water) [mm].

EXAMPLE 31

Determination of the evapotranspiration from a bare soil

Determine the evapotranspiration from a bare loamy soil surface (Kcb ≈ 0.15) for ten successive days

following a heavy rain. The reference evapotranspiration during the drying period is ETo = 4.5 mm/day,

and the climate is subhumid with light wind.

From Table 19 For Loam: TEW ≈ 20 mm and REW ≈ 9 mm

For rain on bare soil few = 1

From Eq. 72 Kc max = 1.20

(1)

Day

(2)

De start

mm

(3)

Stage

(4)

Kr

(5)

Ke

(6)

Ke ETo

mm/day

(7)

De end

mm

(8)

ETc

mm/day

1

2

3

4

5

6

7

8

9

10

0.00

4.73

9.45

13.98

16.57

18.04

18.88

19.36

19.64

19.79

1

2

1

(20-9.45)/(20-9)=0.96

(20-13.98)/(20-9)=0.55

(20-16.57)/(20-9)=0.31

(20-18.04)/(20-9)=0.18

(20-18.88/(20-9)=0.10

(20-19.36)/(20-9)=0.06

(20-19.64)/(20-9)=0.03

(20-19.79)/(20-9)=0.02

1.05

1.01

0.57

0.33

0.19

0.11

0.06

0.03

0.02

4.73

4.53

2.59

1.47

0.84

0.48

0.27

0.16

0.09

4.73

9.45

13.98

16.57

18.04

18.88

19.36

19.64

19.79

19.88

5.4

5.2

3.3

2.1

1.5

1.2

0.9

0.8

(1) Day number.

(2) Depletion at beginning of the day (= depletion at end of previous day).

(3) Soil evaporation stage (stage 2 starts if De > REW = 9 mm).

(4) Kr (Kr = 1 for stage 1. Use Eq. 74 for stage 2).

(5) From Eq. 21: Ke = Kr (Kc max - Kcb) = Kr (1.20-0.15) = 1.05 Kr ≤ 1.20.

(6) Evaporation component: Ke ETo = Ke (4.5 mm/day).

(7) Depletion at end of day = (2) - (6).

(8) ETc = (Kcb + Ke) ETo = (0.15 + Ke) ETo = (0.15+Ke) 4.5 mm/day,

where Kcb ETo = (0.15 ETo) ≈ 0.7 mm/day is basal, “diffusive” evaporation from the soil,

possibly from beneath the Ze depth (∼ 0.10 to 0.15 m). Since the soil in this situation is bare,

one could set the Kcb equal to zero so that maximum Ke becomes Ke = Kc max = 1.20. Then

all of the evaporation would be deducted from the surface soil layer.

The example demonstrates that the estimation of Kr requires a daily water balance calculation. This is

further developed in the section on the daily calculation of Ke.

Crop evapotranspiration

147

Exposed and wetted soil fraction (few)

few: calculation procedure

In crops with incomplete ground cover, evaporation from the soil often does not occur

uniformly over the entire surface, but is greater between plants where exposure to sunlight

occurs and where more air ventilation is able to transport vapour from the soil surface to above

the canopy. This is especially true where only part of the soil surface is wetted by irrigation.

It is recognized that both the location and the fraction of the soil surface exposed to

sunlight change to some degree with the time of day and depending on row orientation. The

procedure presented here predicts a general averaged fraction of the soil surface from which the

majority of evaporation occurs. Diffusive evaporation from the soil beneath the crop canopy is

assumed to be largely included in the basal Kcb coefficient.

Where the complete soil surface is wetted, as by precipitation or sprinkler, then the

fraction of soil surface from which most evaporation occurs, few, is essentially defined as (1-

fc), where fc is the average fraction of soil surface covered by vegetation and (1-fc) is the

approximate fraction of soil surface that is exposed. However, for irrigation systems where only

a fraction of the ground surface is wetted, few must be limited to fw, the fraction of the soil

surface wetted by irrigation (Figure 39). Therefore, few is calculated as:

()

wcew f,f1minf −= (75)

where 1-fc average exposed soil fraction not covered (or shaded) by vegetation [0.01 - 1],

f

w average fraction of soil surface wetted by irrigation or precipitation [0.01 - 1].

The ‘min()’ function selects the lowest value of the ‘1-fc’ and ‘fw’ values. Figure 39

illustrates the relation of few to (1-fc) and fw.

The limitation imposed by Equation 75 assumes that the fraction of soil wetted by

irrigation occurs within the fraction of soil exposed to sunlight and ventilation. This is

generally the case, except perhaps with drip irrigation (Figure 39).

In the case of drip irrigation, where the majority of soil wetted by irrigation may be

beneath the canopy and may therefore be shaded, more complex models of the soil surface and

wetting patterns may be required to accurately estimate total evaporation from the soil. In this

case, the value for fw may need to be reduced to about one-half to one-third of that given in

Table 20 to account for the effects of shading of emitters by the plant canopy on the

evaporation rate from wetted soil (Example 34). A general approach could be to multiply fw by

[1-(2/3)fc] for drip irrigation.

fw: fraction of soil surface wetted by irrigation or precipitation

Table 20 presents typical values for fw. Where a mixture of irrigation and precipitation occur

within the same drying period or on the same day, the value for fw should be based on a

weighted average of the fw for precipitation (fw = 1) and the fw for the irrigation system. The

weighting should be approximately proportional to the infiltration depths from each water

source.

ETc -–dual crop coefficient (Kc = Kcb + Ke)

148

FIGURE 39

Determination of variable few (cross-hatched areas) as a function of the fraction of ground

surface coverage (fc) and the fraction of the surface wetted (fw)

initial

crop development

mid and late

season

few = (1 - f

c)

Rain

Basin

Border

Sprinkler

Irrigation

1 - f f

cc

f = 1

wf = 1

w

1 - f f

cc

few = f wfew = (1 - f

c)

few = 1.0 ... 0.3 f

w

Furrow

Irrigation

Drip

Irrigation

1 - f

c1 - f

c

1 - fc1 - fcfc

fc

fcfc

fwfw

fw

Crop evapotranspiration

149

TABLE 20

Common values of fraction fw of soil surface wetted by irrigation or precipitation

Wetting event fw

Precipitation

Sprinkler irrigation

Basin irrigation

Border irrigation

Furrow irrigation (every furrow), narrow bed

Furrow irrigation (every furrow), wide bed

Furrow irrigation (alternated furrows)

Trickle irrigation

1.0

0.6 ... 1.0

0.4 ... 0.6

0.3 ... 0.5

0.3 ... 0.4

Alternatively, on each day of the application, the following rules can be applied to

determine fw for that and subsequent days in a more simplified manner:

• Surface is wetted by irrigation and rain: fw is the fw for the irrigation system;

• Surface is wetted by irrigation: fw is the fw for the irrigation system;

• Surface is wetted by significant rain (i.e., > 3 to 4 mm) with no irrigation: fw = 1;

• Where there is neither irrigation nor significant precipitation: fw is the fw of the previous

day.

1-fc: exposed soil fraction

The fraction of the soil surface that is covered by vegetation is termed fc. Therefore, (1-fc)

represents the fraction of the soil that is exposed to sunlight and air ventilation and which

serves as the site for the majority of evaporation from wet soil. The value for fc is limited to

< 0.99. The user should assume appropriate values for the various growth stages. Typical

values for fc and (1-fc) are given in Table 21.

TABLE 21

Common values of fractions covered by vegetation (fc) and exposed to sunlight (1-fc)

Crop growth stage fc 1-fc

Initial stage

Crop development stage

Mid-season stage

Late season stage

0.0 - 0.1

0.1 - 0.8

0.8 - 1.0

0.8 - 0.2

1.0 - 0.9

0.9 - 0.2

0.2 - 0.0

0.2 - 0.8

Where fc is not measured, fc can be estimated using the relationship:

)h5.01(

mincmaxc

mincbc

cKK

KK

f

+

÷

ø

ö

ç

è

æ

−

= (76)

where fc the effective fraction of soil surface covered by vegetation [0-0.99],

Kcb the value for the basal crop coefficient for the particular day or period,

Kc min the minimum Kc for dry bare soil with no ground cover [≈ 0.15 - 0.20],

Kc max the maximum Kc immediately following wetting (Equation 72),

h mean plant height [m] .

ETc -–dual crop coefficient (Kc = Kcb + Ke)

150

This equation should be used with caution and validated from field observations. Kc min

is the minimum crop coefficient for dry bare soil when transpiration and evaporation from the

soil are near baseline (diffusive) levels. Kc min ≈ 0.15 - 0.20 is recommended. The value of Kc

min is an integral part of all Kcb coefficients. Kc min ordinarily has the same value as the Kcb

ini used for annual crops under nearly bare soil conditions (0.15 - 0.20).

Equation 76 assumes that the value for Kcb is largely affected by the fraction of soil

surface covered by vegetation. This is a good assumption for most vegetation and conditions.

The ‘1+0.5h’ exponent in the equation represents the effect of plant height on shading the soil

surface and in increasing the value for Kcb given a specific value for fc. The user should limit

the difference Kcb - Kc min to ≥ 0.01 for numerical stability. The value for fc will change daily

as Kcb changes. Therefore, the above equation is applied daily.

Application of Equation 76 predicts that fc decreases during the late season period in

proportion to Kcb, even though the ground may remain covered with senescing vegetation. This

prediction helps to account for the local transport of sensible heat from senescing leaves to the

soil surface below.

EXAMPLE 32

Calculation of the crop coefficient (Kcb + Ke) under sprinkler irrigation

A field of cotton has just been sprinkler irrigated. The Kcb for the specific day (during the development

period) has been computed using Table 17 and Eq. 70 and then interpolated from the Kcb curve as

0.9. The ETo = 7 mm/day, u2 = 3 m/s and RHmin = 20%. Estimate the crop coefficient (Kcb + Ke).

Assuming h = 1 m, from Eq. 72, Kc max for this arid climate is:

[]

{}

30.105.09.0,

3

1

)4520(004.0)23(04.02.1maxK

3.0

maxc =

÷

ø

ö

ç

è

æ

+

ï

þ

ï

ý

ü

ï

î

ï

í

ì

÷

ø

ö

ç

è

æ

−−−+=

From Eq. 76, where Kc min = 0.15:

fc = [(Kcb - Kc min)/(Kc max - Kc min)](1 + 0.5h)

= [(0.9 - 0.15)/(1.3 - 0.15)](1 + 0.5(1)) = 0.53.

As the field was sprinkler irrigated, fw = 1.0 and from Eq. 75:

few = min(1 - fc, fw)

= min(1 - 0.53, 1.0) = 0.47.

Assuming that the irrigation was sufficient to fill the evaporating layer to field capacity,

so that Kr = 1, evaporation would be in stage 1.

From Eq. 71: Ke = 1.00 (1.30-0.90) = 0.40

The value is compared against the upper limit few Kc max to ensure that it is less than the upper limit:

few Kc max = 0.47 (1.30) = 0.61, which is greater than the value for Ke. Therefore, the value

for Ke can be used with no limitation.

The total Kc for the field, assuming no moisture stress due to a dry soil profile, is

Kc = Kcb + Ke

= 0.9 + 0.40 = 1.30.

This value is large because of the very wet soil surface, the relatively tall rough crop as compared to

the grass reference, and the arid climate (u2 = 3 m/s and RHmin = 20%). In this situation, Kc happens

to equal Kc max, as the field has just been wetted by sprinkler irrigation.

Crop evapotranspiration

151

EXAMPLE 33

Calculation of the crop coefficient (Kcb + Ke) under furrow irrigation

The cotton field in the previous example (Ex. 32) has been irrigated by furrow irrigation of alternate

rows rather than by sprinkler, and the fraction of the field surface wetted by the irrigation is 0.3.

The few in this case is calculated from Eq. 75 as:

few = min(1-fc, fw) = min(1 - 0.53, 0.3) = 0.3.

Assuming that the irrigation was sufficient to fill the few portion of the evaporating layer to field

capacity, so that Kr = 1, evaporation would be in stage 1.

From Eq. 71: Ke = 1.00 (1.30 - 0.9) = 0.40

The value is compared to the upper limit few Kc max which is 0.30 (1.30) = 0.39. Because 0.40 > 0.39,

Ke from the few surface area is constrained to 0.39.

The total Kc for the furrow irrigated field, assuming no moisture stress due to dry soil, is

Kc = Kcb + Ke = 0.9 + 0.39 = 1.29. This value is essentially the same as for the previous example (Ex.

32) because the procedure assumes that the soil between alternate rows is the portion that is wetted

by the irrigation, so that the majority of the field surface has either vegetation cover or wet soil.

EXAMPLE 34

Calculation of the crop coefficient (Kcb + Ke) under drip irrigation

The cotton field in the previous example (Ex. 32) has been irrigated by drip irrigation, where the

emitters are placed beneath the cotton canopy. The fraction of the field surface wetted by the irrigation

is 0.3.

The few in this case is calculated from Eq. 75 as few = min(1-fc, fw). Because the emitters are beneath

the canopy so that less energy is available for evaporation, the value for fw is reduced by multiplying by

1 - (2/3)fc, so that:

few = min[(1-fc),(1 - 0.67 fc) fw)] = min[(1-0.53), (1 - 0.67(0.53))(0.3)] = 0.19

Assuming that the irrigation was sufficient to fill the fw portion of the evaporating layer to field capacity,

so that Kr = 1, evaporation would be in stage 1.

From Eq. 71: Ke = 1.00 (1.30-0.90) = 0.40.

The value is compared to the upper limit few Kc max = 0.19 (1.30) = 0.25. Because 0.25 < 0.40, Ke

from the fw fraction of the surface area is constrained by the available energy. Therefore Ke = 0.25.

The total Kc for the drip irrigated field, assuming no moisture stress due to dry soil, is:

Kc = Kcb + Ke = 0.9 + 0.25 = 1.15. This Kc value is less than that for sprinkler and furrow irrigation

(Examples 32 and 33).

Daily calculation of Ke

Daily water balance

The estimation of Ke in the calculation procedure requires a daily water balance computation

for the surface soil layer for the calculation of the cumulative evaporation or depletion from the

wet condition. The daily soil water balance equation for the exposed and wetted soil fraction

few is (Figure 40):

ETc -–dual crop coefficient (Kc = Kcb + Ke)

152

i e,i,ew

ew

i

w

i

ii1-i,ei,e DPT

f

E

+

f

I

)RO(PDD ++−−−= (77)

where De,i-1 cumulative depth of evaporation following complete wetting from the

exposed and wetted fraction of the topsoil at the end of day i-1 [mm],

De,i cumulative depth of evaporation (depletion) following complete wetting

at the end of day i [mm],

Pi precipitation on day i [mm],

ROi precipitation runoff from the soil surface on day i [mm],

Ii irrigation depth on day i that infiltrates the soil [mm],

Ei evaporation on day i (i.e., Ei = Ke ETo) [mm],

Tew,i depth of transpiration from the exposed and wetted fraction of the soil

surface layer on day i [mm],

DPe, i deep percolation loss from the topsoil layer on day i if soil water content

exceeds field capacity [mm],

fw fraction of soil surface wetted by irrigation [0.01 - 1],

few exposed and wetted soil fraction [0.01 - 1].

FIGURE 40

Water balance of the topsoil layer

f < 1

Irrigation

Rain

few

T = K ET

E = K ET

cb

eo

o

w

Ze

fw=1

De,I-1 - (Pi - Roi) - Ii/fw + Ei/few + Tew,i + DPi = De,i

Crop evapotranspiration

153

Limits on De, i

By assuming that the topsoil is at field capacity following heavy rain or irrigation, the minimum

value for the depletion De,i is zero. As the soil surface dries, De,i increases and in absence of

any wetting event will steadily reach its maximum value TEW (Equation 73). At that moment

no water is left for evaporation in the upper soil layer, Kr becomes zero, and the value for De,i

remains at TEW until the topsoil is wetted once again. The limits imposed on De,i are

consequently:

TEWD0 i,e ≤≤ (78)

Initial depletion

To initiate the water balance for the evaporating layer, the user can assume that the topsoil is

near field capacity following a heavy rain or irrigation, i.e., De,i-1 = 0. Where a long period of

time has elapsed since the last wetting, the user can assume that all evaporable water has been

depleted from the evaporation layer at the beginning of calculations, i.e., De, i-1 = TEW =

1 000 (θFC - 0.5 θWP) Ze.

Precipitation and runoff

Pi is equivalent to daily precipitation. Daily precipitation in amounts less than about 0.2 ETo is

normally entirely evaporated and can usually be ignored in the Ke and water balance

calculations. The amount of rainfall lost by runoff depends on: the intensity of rainfall; the

slope of land; the soil type, its hydraulic conditions and antecedent moisture content; and the

land use and cover. For general situations, ROi can be assumed to be zero or can be accounted

for by considering only a certain percentage of Pi. This is especially true for the water balance

of the topsoil layer, since almost all precipitation events that would have intensities or depths

large enough to cause runoff would probably replenish the water content of the topsoil layer to

field capacity. Therefore, the impact of the runoff component can be ignored. Light

precipitation events will generally have little or no runoff.

Irrigation

Ii is generally expressed as a depth of water that is equivalent to the mean infiltrated irrigation

depth distributed over the entire field. Therefore, the value Ii/fw is used to describe the actual

concentration of the irrigation volume over the fraction of the soil that is wetted (Figure 31).

Evaporation

Evaporation beneath the vegetation canopy is assumed to be included in Kcb and is therefore

not explicitly quantified. The computed evaporation is fully concentrated in the exposed,

wetted topsoil. The evaporation Ei is given by Ke ETo. The Ei/few provides for the actual

concentration of the evaporation over the fraction of the soil that is both exposed and wetted.

Transpiration

Except for shallow rooted crops (i.e., where the depth of the maximum rooting zone is < 0.5 to

0.6 m), the amount of transpiration from the evaporating soil layer is small and can be ignored

(i.e., Tew = 0). In addition, for row crops, most of the water extracted by the roots may be

extracted from beneath the vegetation canopy. Therefore, Tew from the few fraction of soil

surface can be assumed to be zero in these cases.

ETc -–dual crop coefficient (Kc = Kcb + Ke)

154

EXAMPLE 35

Estimation of crop evapotranspiration with the dual crop coefficient approach

Estimate the crop evapotranspiration, ETc, for ten successive days. It is assumed that:

- the soil is a sandy loam soil, characterized by θFC = 0.23 m3 m-3 and θWP = 0.10 m3 m-3,

- the depth of the surface soil layer that is subject to drying by way of evaporation, Ze, is 0.1 m,

- during the period, the height of the vegetation h = 0.30 m, the average wind speed u2 = 1.6 m s-1,

and RHmin = 35%,

- the Kcb on day 1 is 0.30 and increases to 0.40 by day 10,

- the exposed soil fraction, (1-fc), decreases from 0.92 on day 1 to 0.86 on day 10,

- all evaporable water has been depleted from the evaporation layer at the beginning of calculations

(De, i-1 = TEW),

- irrigation occurs at the beginning of day 1 (I = 40 mm), and the fraction of soil surface wetted b

y

irrigation, fw = 0.8,

- a rain of 6 mm occurred at the beginning of day 6.

From Tab. 19 REW ≈ 8 mm

From Eq. 73 TEW = 1 000 (0.23 - 0.5(0.10)) 0.1 = 18 mm

From Eq. 72 Kc max = 1.2 + [0.04(1.6-2) - 0.004(35-45)] (0.3/3)0.3 = 1.21

All evaporable water has been depleted at the beginning of calculations, De, i-1 = TEW = 18 mm

K

e

cb

Irrigation Rain

K =

c

cb

e

K + K

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

012345678910

day

crop coefficient K

c

= K

cb

+ K

e

Crop evapotranspiration

155

(1)

Day

(2)

ETo

(3)

P-RO

(4)

I/fw

(5)

1-fc

(6)

fw

(7)

few

(8)

Kcb

(9)

De, i

start

mm/d mm mm mm

start

1

2

3

4

5

6

7

8

9

10

-

4.5

5.0

3.9

4.2

4.8

2.7

5.8

5.1

4.7

5.2

-

0

6

0

-

50

0

-

0.92

0.91

0.90

0.89

0.88

0.87

0.86

-

0.8

1

-

0.80

0.89

0.88

0.87

0.86

-

0.30

0.31

0.32

0.33

0.34

0.36

0.37

0.38

0.39

0.40

-

0

5

11

14

16

11

13

16

17

18

(10)

Kr

(11)

Ke

(12)

E/few

(13)

DPe

(14)

De,i

end

(15)

E

(16)

Kc

(17)

ETc

mm mm mm mm/d mm/d

-

1.00

0.70

0.40

0.20

0.75

0.53

0.20

0.09

0.05

-

0.91

0.90

0.62

0.35

0.18

0.64

0.45

0.17

0.08

0.04

-

5.1

5.6

3.0

1.8

1.1

2.0

3.0

1.0

0.4

0.2

-

32

0

18

5

11

14

16

17

13

16

17

18

-

4.1

4.5

2.8

1.5

0.8

1.7

2.6

0.9

0.4

0.2

-

1.21

1.04

0.70

0.52

1.00

0.82

0.55

0.47

0.44

-

5.5

6.1

4.0

2.9

2.5

2.7

4.7

2.8

2.2

2.3

(1) Day number.

(2) ETo is given. Note that ETo would be forecast values in real time irrigation scheduling but are known

values after the occurrence of the day, during an update of the calculations.

(3) (P-RO) are known values after the occurrence of the day, during an update of the calculations.

(4) Net irrigation depth for the part of the soil surface wetted by irrigation.

(5) (1-fc) is given (interpolated between 0.92 m on day 1 and 0.86 m on day 10).

(6) If significant rain: fw,i = 1.0 (Tab. 20)

If irrigation: fw,i = 0.8 (given),

otherwise: fw,i = fw,i-1.

(7) Eq. 75. Fraction of soil surface from which most evaporation occurs.

(8) Kcb is given (interpolated between 0.30 on day 1 and 0.40 on day 10).

(9) De,i start (depletion at start of day)

If precipitation and irrigation occur early in the day then the status of depletion from the soil surface layer

(at the start of the day) should be updated:

= Max(De,i-1 - In,i/fwi - (P-RO)i , or 0). where De,i-1 is taken from column 14 of previous

day.

If precipitation and irrigation occur late in the day, then column 6 should be set equal to De,i-1 (column 14

of previous day).

(10) If De,i ≤ REW Kr = 1 If De,i > REW Kr = Eq. 74.

(11) Eq. 71 where Ke = Kr (Kc max - Kcb) ≤ few Kc max. (e.g., Ke = min(Kr (Kc max - Kcb), few Kc max).

(12) Evaporation from the wetted and exposed fraction of the soil surface = (Ke ETo)/few.

(13) Eq. 79 where DPe ≥ 0. (This is deep percolation from the evaporating layer).

(14) De,i (depletion at end of day) is from Eq. 77 where De,i-1 is value in column 14 of previous day.

(15) Mean evaporation expressed as distributed over the entire field surface = Ke ETo.

(16) Kc = Kcb + Ke.

(17) Eq. 69.

ETc -–dual crop coefficient (Kc = Kcb + Ke)

156

The daily water balance calculation for the surface layer, even for shallow rooted crops,

is not usually sensitive to Tew, as Tew is a minor part of the flux from the Ze depth for the first

3-5 days following a wetting event. Tew can, therefore, generally be ignored. The effects of the

reduction of the water content of the evaporating soil layer due to Tew can be accounted for

ulteriorly when it is assumed that Tew = 0 by decreasing the value for Ze, for example from

0.15 to 0.12 m or from 0.10 to 0.08 m.

Deep percolation

Following heavy rain or irrigation, the soil water content in the topsoil (Ze layer) might exceed

field capacity. However, in this simple procedure it is assumed that the soil water content is at

θFC nearly immediately following a complete wetting event, so that the depletion De,i in

Equation 77 is zero. Following heavy rain or irrigation, downward drainage (percolation) of

water from the topsoil layer is calculated as:

0D

f

I

)RO(PDP 1-i,e

w

i

iii e, ≥−+−= (79)

As long as the soil water content in the evaporation layer is below field capacity (i.e., De, i >

0), the soil will not drain and DPe, i = 0.

Order of calculation

In making calculations for the Kcb + Ke procedure, for example when using a spreadsheet, the

calculations should proceed in the following order: Kcb, h, Kc max, fc, fw, few, Kr, Ke, E,

DPe, De, I, Kc, and ETc.

CALCULATING ETC

The calculation procedure lends itself to application by computer, either in the form of

electronic spreadsheets (Example 35) or in the form of structured programming languages. The

calculation procedure consists in determining:

a. Reference evaporation, ETo:

Estimate ETo: the procedure is given in Chapter 4.

b. Growth stages:

Determine the locally adjusted lengths of the four growth stages (for general information

consult Table 11):

- Initial growth stage: Lini,

- Crop development stage: Ldev,

- Mid-season stage: Lmid,

- Late season stage: Llate.

Crop evapotranspiration

157

c. Basal crop coefficient, Kcb:

Calculate basal crop coefficients for each day of the growing period:

- select Kcb ini, Kcb mid and Kcb end from Table 17;

- adjust Kcb mid and Kcb end to the local climatic conditions (Equation 70);

- determine the daily Kcb values

• initial growth stage: Kcb = Kcb ini,

• crop development stage: from Kcb ini to Kcb mid (Equation 66),

• mid-season stage: Kcb = Kcb mid,

• late season stage: from Kcb mid to Kcb end (Equation 66).

d. Evaporation coefficient, Ke:

Calculate the maximum value of Kc, i.e., the upper limit Kc max (Equation 72), and

Determine for each day of the growing period:

- the fraction of soil covered by vegetation, fc (Table 21 or Equation 76),

- the fraction of soil surface wetted by irrigation or precipitation, fw (Table 20),

- the fraction of soil surface from which most evaporation occurs, few (Equation 75),

- the cumulative depletion from the evaporating soil layer, De,

determined by means of a daily soil water balance of the topsoil (Equation 77),

- the corresponding evaporation reduction coefficient, Kr (Equation 74), and

- the soil evaporation coefficient, Ke (Equation 71).

e. Crop evapotranspiration, ETc:

Calculate ETc = (Kcb + Ke) ETo (Equation 69).

ETc -–dual crop coefficient (Kc = Kcb + Ke)

158

BOX 16

Case study of dry bean crop at Kimberly, Idaho, the United States (dual crop coefficient)

Results from applying the Kcb + Ke procedure for a snap bean crop harvested as dry seed are shown

in the figure below. This example uses the same data set that was used in the case study of Box 15.

The measured ETc data were measured using a precision lysimeter system at Kimberly, Idaho. Values

for Kcb ini, Kcb mid, and Kcb end were calculated in Example 29 as 0.15, 1.14, and 0.25. The lengths

of growth stages were 25, 25, 30, and 20 days. The Kcb values are plotted in Fig. 37. The value for Kc

max from Eq. 72 for the mid-season period averaged 1.24, based on u2 = 2.2 m/s and RHmin = 30%

for Kimberly. The soil at Kimberly was a silt loam texture. Assuming that the depth of the evaporation

soil layer, Ze, was 0.1 m, values for TEW = 22 mm and REW = 9 mm, based on Eq. 73 and using soil

data from Table 19.

The occurrence and magnitudes of individual wetting events are shown in the figure below. Nearly all

wetting events were from irrigation. Because the irrigation was by furrow irrigation of alternate rows, the

value for fw was set equal to 0.5. Irrigation events occurred at about midday or during early afternoon.

The agreement between the estimated values for daily Kcb+Ke (thin continuous line) and actual 24-

hour measurements (symbols) is relatively good. Measured and predicted Kcb+Ke was higher following

wetting by rainfall or irrigation, as expected. The two wet soil evaporation ‘spikes’ occurring during the

late initial period and early development period (between days 160 and 180) were less than Kc max,

because this evaporation was from wetting by furrow irrigation where fw = 0.5. The value for few was

constrained to fw by Eq. 75 during these two events, because during this period, fw < 1-fc. Therefore,

less than all of the ‘potential energy’ was converted into evaporation due to the limitation on maximum

evaporation per unit surface area that was imposed by Eq. 71.

140 160 180 200 220 240

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0

40

80

120

160

200

day of the Year

I

II

I

PP

precipitation and irrigation (mm)

crop coefficient

Measured

Measured (symbols) and predicted (thin line) daily coefficients (Kcb+Ke) and the basal crop

curve (thick line) for a dry bean crop at Kimberly, Idaho. P in the figure denotes a precipitation

event and I denotes an irrigation (data from Wright, 1990).

Crop evapotranspiration

159

Part C

Crop evapotranspiration under

non-standard conditions

In well-managed fields, the standard conditions are generally the actual field conditions. The

ETc calculated by means of the procedures described in Part B is the crop evapotranspiration

under the standard field conditions.

Where the conditions encountered in the field differ from the standard conditions, a

correction on ETc is required. Low soil fertility, salt toxicity, soil waterlogging, pests, diseases

and the presence of hard or impenetrable soil horizons in the root zone may result in scanty

plant growth and lower evapotranspiration. Soil water shortage and soil salinity may reduce soil

water uptake and limit crop evapotranspiration. The evapotranspiration from small isolated

stands of plants or from fields where two different crops are grown together or where mulches

are used to reduce evaporation may also deviate from the crop evapotranspiration under

standard conditions.

This part discusses the effect on ET of management and environmental conditions that

deviate from the standard conditions. The environmental effects are described by introducing

stress coefficients and by adjusting Kc to the field conditions.

160

Crop evapotranspiration

161

Chapter 8

ETc under soil water stress conditions

Forces acting on the soil water decrease its potential energy and make it less available for plant

root extraction. When the soil is wet, the water has a high potential energy, is relatively free to

move and is easily taken up by the plant roots. In dry soils, the water has a low potential energy

and is strongly bound by capillary and absorptive forces to the soil matrix, and is less easily

extracted by the crop.

When the potential energy of the soil water drops below a threshold value, the crop is

said to be water stressed. The effects of soil water stress are described by multiplying the basal

crop coefficient by the water stress coefficient, Ks:

oecbsdjac ET)KK(KET += (80)

For soil water limiting conditions, Ks < 1. Where there is no soil water stress, Ks = 1.

Ks describes the effect of water stress on crop transpiration. Where the single crop

coefficient is used, the effect of water stress is incorporated into Kc as:

ocsdjac ETKKET = (81)

Because the water stress coefficient impacts only crop transpiration, rather than

evaporation from soil, the application of Ks using Equation 80 is generally more valid than is

application using Equation 81. However, in situations where evaporation from soil is not a

large component of ETc, use of Equation 81 will provide reasonable results.

SOIL WATER AVAILABILITY

Total available water (TAW)

Soil water availability refers to the capacity of a soil to retain water available to plants. After

heavy rainfall or irrigation, the soil will drain until field capacity is reached. Field capacity is

the amount of water that a well-drained soil should hold against gravitational forces, or the

amount of water remaining when downward drainage has markedly decreased. In the absence of

water supply, the water content in the root zone decreases as a result of water uptake by the

crop. As water uptake progresses, the remaining water is held to the soil particles with greater

force, lowering its potential energy and making it more difficult for the plant to extract it.

Eventually, a point is reached where the crop can no longer extract the remaining water. The

water uptake becomes zero when wilting point is reached. Wilting point is the water content at

which plants will permanently wilt.

ETc under soil water stress conditions

162

As the water content above field capacity cannot be held against the forces of gravity and

will drain and as the water content below wilting point cannot be extracted by plant roots, the

total available water in the root zone is the difference between the water content at field

capacity and wilting point:

rWPFC Z)(1000TAW θ−θ= (82)

where TAW the total available soil water in the root zone [mm],

θFC the water content at field capacity [m3 m-3],

θWP the water content at wilting point [m3 m-3],

Zr the rooting depth [m].

TAW is the amount of water that a crop can extract from its root zone, and its magnitude

depends on the type of soil and the rooting depth. Typical ranges for field capacity and wilting

point are listed in Table 19 for various soil texture classes. Ranges of the maximum effective

rooting depth for various crops are given in Table 22.

Readily available water (RAW)

Although water is theoretically available until wilting point, crop water uptake is reduced well

before wilting point is reached. Where the soil is sufficiently wet, the soil supplies water fast

enough to meet the atmospheric demand of the crop, and water uptake equals ETc. As the soil

water content decreases, water becomes more strongly bound to the soil matrix and is more

difficult to extract. When the soil water content drops below a threshold value, soil water can

no longer be transported quickly enough towards the roots to respond to the transpiration

demand and the crop begins to experience stress. The fraction of TAW that a crop can extract

from the root zone without suffering water stress is the readily available soil water:

TAWpRAW = (83)

where RAW the readily available soil water in the root zone [mm],

p average fraction of Total Available Soil Water (TAW) that can be

depleted from the root zone before moisture stress (reduction in ET)

occurs [0 - 1].

Values for p are listed in Table 22. The factor p differs from one crop to another. The

factor p normally varies from 0.30 for shallow rooted plants at high rates of ETc (> 8 mm d-1)

to 0.70 for deep rooted plants at low rates of ETc (< 3 mm d-1). A value of 0.50 for p is

commonly used for many crops.

The fraction p is a function of the evaporation power of the atmosphere. At low rates of

ETc, the p values listed in Table 22 are higher than at high rates of ETc. For hot dry weather

conditions, where ETc is high, p is 10-25% less than the values presented in Table 22, and the

soil is still relatively wet when the stress starts to occur. When the crop evapotranspiration is

low, p will be up to 20% more than the listed values. Often, a constant value is used for p for a

specific growing period, rather than varying the value each day. A numerical approximation for

adjusting p for ETc rate is p = pTable 22 + 0.04 (5 - ETc) where the adjusted p is limited to

0.1 ≤ p ≤ 0.8 and ETc is in mm/day. The influence of the numerical adjustment is shown in

Figure 41.

Crop evapotranspiration

163

TABLE 22

Ranges of maximum effective rooting depth (Zr), and soil water depletion fraction for no stress (p),

for common crops

Crop

Maximum Root

Depth1

(m)

Depletion Fraction2

(for ET _ 5 mm/day)

p

a. Small Vegetables

Broccoli 0.4-0.6 0.45

Brussel Sprouts 0.4-0.6 0.45

Cabbage 0.5-0.8 0.45

Carrots 0.5-1.0 0.35

Cauliflower 0.4-0.7 0.45

Celery 0.3-0.5 0.20

Garlic 0.3-0.5 0.30

Lettuce 0.3-0.5 0.30

Onions - dry 0.3-0.6 0.30

- green 0.3-0.6 0.30

- seed 0.3-0.6 0.35

Spinach 0.3-0.5 0.20

Radishes 0.3-0.5 0.30

b. Vegetables – Solanum Family (Solanaceae)

Egg Plant 0.7-1.2 0.45

Sweet Peppers (bell) 0.5-1.0 0.30

Tomato 0.7-1.5 0.40

c. Vegetables – Cucumber Family (Cucurbitaceae)

Cantaloupe 0.9-1.5 0.45

Cucumber – Fresh Market 0.7-1.2 0.50

– Machine harvest 0.7-1.2 0.50

Pumpkin, Winter Squash 1.0-1.5 0.35

Squash, Zucchini 0.6-1.0 0.50

Sweet Melons 0.8-1.5 0.40

Watermelon 0.8-1.5 0.40

d. Roots and Tubers

Beets, table 0.6-1.0 0.50

Cassava – year 1 0.5-0.8 0.35

– year 2 0.7-1.0 0.40

Parsnip 0.5-1.0 0.40

Potato 0.4-0.6 0.35

Sweet Potato 1.0-1.5 0.65

Turnip (and Rutabaga) 0.5-1.0 0.50

Sugar Beet 0.7-1.2 0.553

continued…

1 The larger values for Zr are for soils having no significant layering or other characteristics that can

restrict rooting depth. The smaller values for Zr may be used for irrigation scheduling and the

larger values for modeling soil water stress or for rainfed conditions.

2 The values for p apply for ETc ≈ 5 mm/day. The value for p can be adjusted for different ETc

according to

p = p

table 22 + 0.04 (5 – ETc)

where p is expressed as a fraction and ETc as mm/day.

3 Sugar beets often experience late afternoon wilting in arid climates even at p < 0.55, with

usually only minor impact on sugar yield.

ETc under soil water stress conditions

164

Table 22 continued

Crop

Maximum Root

Depth1

(m)

Depletion Fraction2

(for ET _ 5 mm/day)

p

e. Lugumes (Leguminosae)

Beans, green 0.5-0.7 0.45

Beans, dry and Pulses 0.6-0.9 0.45

Beans, lima, large vines 0.8-1.2 0.45

Chick pea 0.6-1.0 0.50

Fababean (broad bean) – Fresh 0.5-0.7 0.45

– Dry/Seed 0.5-0.7 0.45

Grabanzo 0.6-1.0 0.45

Green Gram and Cowpeas 0.6-1.0 0.45

Groundnut (Peanut) 0.5-1.0 0.50

Lentil 0.6-0.8 0.50

Peas – Fresh 0.6-1.0 0.35

– Dry/Seed 0.6-1.0 0.40

Soybeans 0.6-1.3 0.50

f. Perennial Vegetables (with winter dormancy and

initially bare or mulched soil)

Artichokes 0.6-0.9 0.45

Asparagus 1.2-1.8 0.45

Mint 0.4-0.8 0.40

Strawberries 0.2-0.3 0.20

g. Fibre Crops

Cotton 1.0-1.7 0.65

Flax 1.0-1.5 0.50

Sisal 0.5-1.0 0.80

h. Oil Crops

Castorbean (Ricinus) 1.0-2.0 0.50

Rapeseed, Canola 1.0-1.5 0.60

Safflower 1.0-2.0 0.60

Sesame 1.0-1.5 0.60

Sunflower 0.8-1.5 0.45

i. Cereals

Barley 1.0-1.5 0.55

Oats 1.0-1.5 0.55

Spring Wheat 1.0-1.5 0.55

Winter Wheat 1.5-1.8 0.55

Maize, Field (grain) (field corn) 1.0-1.7 0.55

Maize, Sweet (sweet corn) 0.8-1.2 0.50

Millet 1.0-2.0 0.55

Sorghum – grain 1.0-2.0 0.55

– sweet 1.0-2.0 0.50

Rice 0.5-1.0 0.204

continued…

4 The value for p for rice is 0.20 of saturation.

Crop evapotranspiration

165

Table 22 continued

Crop

Maximum Root

Depth1

(m)

Depletion Fraction2

(for ET _ 5 mm/day)

p

j. Forages

Alfalfa – for hay 1.0-2.0 0.55

– for seed 1.0-3.0 0.60

Bermuda – for hay 1.0-1.5 0.55

– Spring crop for seed 1.0-1.5 0.60

Clover hay, Berseem 0.6-0.9 0.50

Rye Grass hay 0.6-1.0 0.60

Sudan Grass hay (annual) 1.0-1.5 0.55

Grazing Pasture - Rotated Grazing 0.5-1.5 0.60

- Extensive Grazing 0.5-1.5 0.60

Turf grass - cool season5 0.5-1.0 0.40

- warm season5 0.5-1.0 0.50

k. Sugar Cane 1.2-2.0 0.65

l. Tropical Fruits and Trees

Banana – 1st year 0.5-0.9 0.35

– 2nd year 0.5-0.9 0.35

Cacao 0.7-1.0 0.30

Coffee 0.9-1.5 0.40

Date Palms 1.5-2.5 0.50

Palm Trees 0.7-1.1 0.65

Pineapple 0.3-0.6 0.50

Rubber Trees 1.0-1.5 0.40

Tea – non-shaded 0.9-1.5 0.40

– shaded 0.9-1.5 0.45

m. Grapes and Berries

Berries (bushes) 0.6-1.2 0.50

Grapes – Table or Raisin 1.0-2.0 0.35

– Wine 1.0-2.0 0.45

Hops 1.0-1.2 0.50

n. Fruit Trees

Almonds 1.0-2.0 0.40

Apples, Cherries, Pears 1.0-2.0 0.50

Apricots, Peaches, Stone Fruit 1.0-2.0 0.50

Avocado 0.5-1.0 0.70

Citrus

- 70% canopy 1.2-1.5 0.50

- 50% canopy 1.1-1.5 0.50

- 20% canopy 0.8-1.1 0.50

Conifer Trees 1.0-1.5 0.70

Kiwi 0.7-1.3 0.35

Olives (40 to 60% ground coverage by canopy) 1.2-1.7 0.65

Pistachios 1.0-1.5 0.40

Walnut Orchard 1.7-2.4 0.50

5 Cool season grass varieties include bluegrass, ryegrass and fescue. Warm season varieties include

bermuda grass, buffalo grass and St. Augustine grass. Grasses are variable in rooting depth.

Some root below 1.2 m while others have shallow rooting depths. The deeper rooting depths for

grasses represent conditions where careful water management is practiced with higher depletion

between irrigations to encourage the deeper root exploration.

ETc under soil water stress conditions

166

EXAMPLE 36

Determination of readily available soil water for various crops and soil types

Estimate RAW for a full-grown onion, tomato and maize crop. Assume that the crops are cultivated on

loamy sand, silt and silty clay soils.

From Table 22 Onion Zr ≈ 0.4 m, p = 0.30

Tomato Zr ≈ 0.8 m, p = 0.40

Maize Zr ≈ 1.2 m, p = 0.55

From Table 19 Loamy sand θFC ≈ 0.15 m3 m-3, θWP ≈ 0.06 m3 m-3

1 000 (θFC - θWP) = 90 mm(water)/m(soil depth)

Silt θFC ≈ 0.32 m3 m-3, θWP ≈ 0.15 m3 m-3

1 000 (θFC - θWP) = 170 mm(water)/m(soil depth)

Silty clay θFC ≈ 0.35 m3 m-3, θWP ≈ 0.23 m3 m-3

1 000 (θFC - θWP) = 120 mm(water)/m(soil depth)

Loamy sand Silt Silty clay

TAW

(Eq. 82)

mm

RAW

(Eq. 83)

mm

TAW

(Eq. 82)

mm

RAW

(Eq. 83)

mm

TAW

(Eq. 82)

mm

RAW

(Eq. 83)

mm

Onion

Tomato

Maize

36

72

108

11

29

59

68

136

204

20

54

112

48

96

144

14

38

79

FIGURE 41

Depletion factor for different levels of crop evapotranspiration

01234567891011

crop evapotranspiration (mm/day)

0.0

0.2

0.4

0.6

0.8

1.0

depletion factor

maximum

minimum

0.5

0.4

0.3

0.2

p(Table 22) = 0.6

Crop evapotranspiration

167

To express the tolerance of crops to water stress as a function of the fraction (p) of TAW

is not wholly correct. The rate of root water uptake is in fact influenced more directly by the

potential energy level of the soil water (soil matric potential and the associated hydraulic

conductivity) than by water content. As a certain soil matric potential corresponds in different

soil types with different soil water contents, the value for p is also a function of the soil type.

Generally, it can be stated that for fine textured soils (clay) the p values listed in Table 22 can

be reduced by 5-10%, while for more coarse textured soils (sand), they can be increased by 5-

10%.

RAW is similar to the term Management Allowed Depletion (MAD) introduced by

Merriam. However, values for MAD are influenced by management and economic factors in

addition to the physical factors influencing p. Generally, MAD < RAW where there is risk

aversion or uncertainty, and MAD > RAW where plant moisture stress is an intentional part of

soil water management.

WATER STRESS COEFFICIENT (KS)

The effects of soil water stress on crop ET are described by reducing the value for the crop

coefficient. This is accomplished by multiplying the crop coefficient by the water stress

coefficient, Ks (Equations 80 and 81).

Water content in the root zone can also be expressed by root zone depletion, Dr, i.e.,

water shortage relative to field capacity. At field capacity, the root zone depletion is zero (Dr =

0). When soil water is extracted by evapotranspiration, the depletion increases and stress will

be induced when Dr becomes equal to RAW. After the root zone depletion exceeds RAW (the

water content drops below the threshold θt), the root zone depletion is high enough to limit

evapotranspiration to less than potential values and the crop evapotranspiration begins to

decrease in proportion to the amount of water remaining in the root zone (Figure 42).

FIGURE 42

Water stress coefficient, Ks

0.00

0.20

0.40

0.60

0.80

1.00

: soil water content

FC tWP

Ks

0RAW TAW

RAW

TAW

ETc under soil water stress conditions

168

EXAMPLE 37

Effect of water stress on crop evapotranspiration

Estimate the effect of water stress on the evapotranspiration of a full grown tomato crop (Zr = 0.8 m

and p = 0.40) cultivated on a silty soil (θFC = 0.32 and θWP = 0.12 m3 m-3) for the coming 10 days

when the initial root zone depletion is 55 mm and neither rain nor irrigations are either forecasted or

planned. The expected ETo for the coming decade is 5 mm/day and Kc = 1.2.

12345678910

day

0

1

2

3

4

5

6

7

crop ET (mm/day)

ETc

ET

c adj

water stress

From Eq. 82 TAW = 1 000 (0.32-0.12) 0.8 = 160 mm

From Eq. 83 RAW = 0.40 (160) = 64 mm

(1)

Day

(2)

ETo

(3)

Kc

(4)

ETc

(5)

Dr,i

start

(6)

Ks

(7)

ETc adj

(8)

Dr,i

end

mm/day mm/day mm mm/day mm

start

1

2

3

4

5

6

7

8

9

10

-

5.0

-

1.2

-

6.0

-

55.0

61.0

67.0

72.8

78.3

83.4

88.2

92.6

96.9

100.8

-

1.00

0.97

0.91

0.85

0.80

0.75

0.70

0.66

0.62

-

6.0

5.8

5.4

5.1

4.8

4.5

4.2

3.9

3.7

55.0

61.0

67.0

72.8

78.3

83.4

88.2

92.6

96.9

100.8

104.5

(1) Day number.

(2) Reference crop evapotranspiration.

(3) Crop coefficient.

(4) Eq. 58, crop ET if no water stress.

(5) Root zone depletion at the beginning of the day (column 8 of previous day).

(6) Eq. 84 where Ks = 1 if Dr,i < RAW.

(7) Eq. 81, crop ET under soil water stress conditions.

(8) Depletion at end of day .

The example demonstrates that the estimate of Ks requires a daily water balance calculation. This is

developed further in the next section.

Crop evapotranspiration

169

For Dr > RAW, Ks is given by:

TAW)p(1

DTAW

=

RAWTAW

DTAW

=K rr

s−

−

− (84)

where Ks is a dimensionless transpiration reduction factor dependent on

available soil water [0 - 1],

Dr root zone depletion [mm],

TAW total available soil water in the root zone [mm],

p fraction of TAW that a crop can extract from the root zone without

suffering water stress [-].

After the computation of Ks, the adjusted evapotranspiration ETc adj is computed by

means of Equation 80 or 81, depending on the coefficients used to describe crop

evapotranspiration. When the root zone depletion is smaller than RAW, Ks = 1.

SOIL WATER BALANCE

The estimation of Ks requires a daily water balance computation for the root zone.

Schematically (Figure 43), the root zone can be presented by means of a container in which the

water content may fluctuate. To express the water content as root zone depletion is useful. It

makes the adding and subtracting of losses and gains straightforward as the various parameters

FIGURE 43

Water balance of the root zone

depletion

RAW

TAW

saturation

field capacity

threshold

wilting point

Evapotranspiration

Irrigation

Rain

Runoff

Capillary Deep

Percolation

Rise

ETc under soil water stress conditions

170

of the soil water budget are usually expressed in terms of water depth. Rainfall, irrigation and

capillary rise of groundwater towards the root zone add water to the root zone and decrease the

root zone depletion. Soil evaporation, crop transpiration and percolation losses remove water

from the root zone and increase the depletion. The daily water balance, expressed in terms of

depletion at the end of the day is:

ii,ciii1-i,ri,r DP+ETCRIRO)(PDD +−−−−= (85)

where Dr,i root zone depletion at the end of day i [mm],

Dr,i-1 water content in the root zone at the end of the previous day, i-1 [mm],

Pi precipitation on day i [mm],

ROi runoff from the soil surface on day i [mm],

Ii net irrigation depth on day i that infiltrates the soil [mm],

CRi capillary rise from the groundwater table on day i [mm],

ETc,i crop evapotranspiration on day i [mm],

DPi water loss out of the root zone by deep percolation on day i [mm].

Limits on Dr, i

In Figure 43 it is assumed that water can be stored in the root zone until field capacity is

reached. Although following heavy rain or irrigation the water content might temporally exceed

field capacity, the total amount of water above field capacity is assumed to be lost the same day

by deep percolation, following any ET for that day. By assuming that the root zone is at field

capacity following heavy rain or irrigation, the minimum value for the depletion Dr,i is zero. As

a result of percolation and evapotranspiration, the water content in the root zone will gradually

decrease and the root zone depletion will increase. In the absence of any wetting event, the

water content will steadily reach its minimum value θWP. At that moment no water is left for

evapotranspiration in the root zone, Ks becomes zero, and the root zone depletion has reached

its maximum value TAW. The limits imposed on Dr,i are consequently:

TAWD0 i,r ≤≤ (86)

Initial depletion

To initiate the water balance for the root zone, the initial depletion Dr,i-1 should be estimated.

The initial depletion can be derived from measured soil water content by:

r1-iFC1-i,r Z)(1000D θ−θ= (87)

where θi-1 is the average soil water content for the effective root zone. Following heavy rain or

irrigation, the user can assume that the root zone is near field capacity, i.e., Dr,i-1 ≈ 0.

Precipitation (P), runoff (RO) and irrigation (I)

Pi is equivalent to daily precipitation. Daily precipitation in amounts less than about 0.2 ETo is

normally entirely evaporated and can usually be ignored in the water balance calculations

especially when the single crop coefficient approach is being used. Ii is equivalent to the mean

Crop evapotranspiration

171

infiltrated irrigation depth expressed for the entire field surface. Runoff from the surface during

precipitation can be predicted using standard procedures from hydrological texts.

Capillary rise (CR)

The amount of water transported upwards by capillary rise from the water table to the root zone

depends on the soil type, the depth of the water table and the wetness of the root zone. CR can

normally be assumed to be zero when the water table is more than about 1 m below the bottom

of the root zone. Some information on CR was presented in FAO Irrigation and Drainage Paper

No. 24. CR will be a topic in a future FAO publication.

Evapotranspiration (ETc)

Where the soil water depletion is smaller than RAW, the crop evapotranspiration equals ETc =

Kc ETo. As soon as Dr,i exceeds RAW, the crop evapotranspiration is reduced and ETc can be

computed from Equation 80 or 81.

Deep percolation (DP)

Following heavy rain or irrigation, the soil water content in the root zone might exceed field

capacity. In this simple procedure it is assumed that the soil water content is at θFC within the

same day of the wetting event, so that the depletion Dr, i in Equation 85 becomes zero.

Therefore, following heavy rain or irrigation

0DETI)RO(PPD 1-i,ri,ciiii ≥−−+−= (88)

As long as the soil water content in the root zone is below field capacity (i.e., Dr, i > 0),

the soil will not drain and DPi = 0.

The DPi term in Equations 85 and 88 is not to be confused with the DPe,i term used in

Equations 77 and 79 for the evaporation layer. Both terms can be calculated at the same time,

but are independent of one another.

FORECASTING OR ALLOCATING IRRIGATIONS

Irrigation is required when rainfall is insufficient to compensate for the water lost by

evapotranspiration. The primary objective of irrigation is to apply water at the right period and

in the right amount. By calculating the soil water balance of the root zone on a daily basis

(Equation 85), the timing and the depth of future irrigations can be planned. To avoid crop

water stress, irrigations should be applied before or at the moment when the readily available

soil water is depleted (Dr,i ≤ RAW). To avoid deep percolation losses that may leach relevant

nutrients out of the root zone, the net irrigation depth should be smaller than or equal to the root

zone depletion (Ii ≤ Dr,i).

Example 38 illustrates the application of a water balance of the root zone to predict

irrigation dates to avoid water stress. The example utilizes various calculations for Ke from

Example 35. A complete “spreadsheet” that includes all necessary calculations for predicting

both irrigation schedules and to predict Kc = Kcb + Ke for daily timesteps is presented in

Annex 8.

ETc under soil water stress conditions

172

EXAMPLE 38

Irrigation scheduling to avoid crop water stress

Plan the irrigation applications for Example 35. It is assumed that:

- irrigations are to be applied when RAW is depleted,

- the depletion factor (p) is 0.6,

- all irrigations and precipitations occur early in the day,

- the depth of the root zone (Zr) on day 1 is 0.3 m and increases to 0.35 m by day 10,

- the root zone depletion at the beginning of day 1 (Dr,i-1) is RAW.

From Eq. 82 TAW = 1 000 (0.23 - 0.10) Zr,i = 130 Zr,i [mm]

From Eq. 83 RAW = 0.6 TAW = 78 Zr,i [mm]

On day 1, when Zr = 0.3 m: Dr,i-1 = RAW = 78 (0.3) = 23 mm

0246810

day

11

135 7 9

40

30

20

10

0

-10

-20

root zone depletion (mm)

I = 40 I = RAWP = 6

RAW

DP

FC

Dr

Crop evapotranspiration

173

(1)

Day

(2)

ETo

(3)

Zr

(4)

RAW

(5)

Dr,i

start

(6)

P-RO

(7)

I

(8)

Ks

(9)

Kcb

(10)

Ke

(11)

Kc

(12)

ETc

(13)

DP

(14)

Dr,i

end

mm/d

m

mm

start

1

2

3

4

5

6

7

8

9

10

-

4.5

5.0

3.9

4.2

4.8

2.7

5.8

5.1

4.7

5.2

-

0.30

0.31

0.32

0.33

0.34

0.35

-

23

24

25

26

27

-

0

5

12

16

18

15

18

22

25

0

-

0

6

0

-

40

0

27

-

1

-

0.30

0.31

0.32

0.33

0.34

0.36

0.37

0.38

0.39

0.40

-

0.91

0.90

0.72

0.37

0.18

0.64

0.45

0.17

0.08

0.81

-

1.21

1.04

0.70

0.52

1.00

0.82

0.55

0.47

1.21

-

5.5

6.1

4.0

2.9

2.5

2.7

4.7

2.8

2.2

6.3

-

17

0

23

5

12

16

18

21

18

22

25

27

6

(1) Day number.

(2) From Example 35.

(3) Zr is given (interpolated between 0.3 m on day 1 and 0.35 m on day 10).

(4) Eq. 83.

(5) Dr,i start (root zone depletion at the beginning of the day)

If precipitation and irrigation occur early in the day then

D

r,i start = Max(Dr,i-1 end - I - (P-RO), or 0)

If precipitation and irrigation occur late in the day, then

D

r,i start = Dr,i-1end

where Dr,i-1end is taken from column 14 of previous day

Since the depth of the root zone increases each day, the water content of the subsoil (θsub)

has to be considered to update Dr,i

D

r,i = Dr,i-1 + 1 000 (θFC - θsub,i-1) ∆Zr,I

In the example it is assumed that θsub is at field capacity (due to prior overirrigation and

excessive rainfall on previous days). Therefore, a combination of the equations for Dr,i can be

utilized.

(6) From Example 35.

(7) Irrigation is required when Dr,i ≥ RAW.

On day 1, the irrigation depth (infiltrating the soil) is given (from Example 35: I = 40 mm)

On day 10, another irrigation is required. An irrigation with a net depth of 27 mm

refills the root zone and avoids water loss by deep percolation (DP = 0 mm).

(8) Eq. 84, where Ks = 1 for Dr,i ≤ RAW.

(9) From Example 35.

(10) Day 1 to 9: From Example 35.

Day 10: Following the extra irrigation early in the day, the topsoil will be wet and

K

r is 1 or from Eq. 71: Ke = (1.21 - 0.40) = 0.81.

(11) Kc = Ks Kcb + Ke.

(12) Eq. 80.

(13) Eq. 88, where Dr,i-1 is taken from column 14 of previous day.

(14) Dr,i (root zone depletion at end of one day) = the starting Dr,i at the beginning of the next day

(see footnote 5). From Eq. 85, where Dr,i-1 is taken from column 14 of previous day.

ETc under soil water stress conditions

174

EFFECTS OF SOIL SALINITY

Salts in the soil water solution can reduce evapotranspiration by making soil water less

"available" for plant root extraction. Salts have an affinity for water and hence additional force

is required for the crop to extract water from a saline soil. The presence of salts in the soil

water solution reduces the total potential energy of the soil water solution. In addition, some

salts cause toxic effects in plants and can reduce plant metabolism and growth. A function is

presented here that predicts the reduction in evapotranspiration caused by salinity of soil water.

The function is derived by combining yield-salinity equations from the FAO Irrigation and

Drainage Paper No29 with yield-ET equations from FAO Irrigation and Drainage Paper No33.

The resulting equation provides a first approximation of the reduction in evapotranspiration

expected under various salinity conditions.

There is evidence that crop yield and transpiration are not as sensitive to low osmotic

potential as they are to low matric potential. Under saline conditions, many plants are able to

partially compensate for low osmotic potential of the soil water by building up higher internal

solute contents. This is done by absorbing ions from the soil solution and by synthesizing

organic osmolytes. Both of these reactions reduce the impact of osmotic potential on water

availability. However, synthesis of organic osmolytes does require expenditure of metabolic

energy. Therefore plant growth is often reduced under saline conditions. The reduced plant

growth impacts transpiration by reducing ground cover and is sometimes additionally due to

partial stomatal closure.

Other impacts of salts in the soil include direct sodium and chloride toxicities and

induced nutrient deficiencies. These deficiencies reduce plant growth by reducing the rate of

leaf elongation, the enlargement, and the division of cells in leaves. The modality depends on

the method of irrigation. With sprinkler irrigation, adsorption of sodium and chloride through

the leaf can result in toxic conditions for all crop species. With surface or trickle irrigaiton,

direct toxic conditions generally occur only in vine and tree crops; however, high levels of

sodium can induce calcium deficieincies for all crop species.

Since salt concentration changes as the soil water content changes, soil salinity is

normally measured and expressed on the basis of the electrical conductivity of the saturation

extract of the soil (ECe). The ECe is defined as the electrical conductivity of the soil water

solution after the addition of a sufficient quantity of distilled water to bring the soil water

content to saturation. ECe is typically expressed in deciSiemens per meter (dS m-1). Under

optimum management conditions, crop yields remain at potential levels until a specific,

threshold electrical conductivity of the saturation soil water extract (ECe threshold) is reached.

If the average ECe of the root zone increases above this critical threshold value, the yield is

presumed to begin to decrease linearly in proportion to the increase in salinity. The rate of

decrease in yield with increase in salinity is usually expressed as a slope, b, having units of %

reduction in yield per dS/m increase in ECe.

All plants do not respond to salinity in a similar manner; some crops can produce

acceptable yields at much higher soil salinity levels than others. This is because some crops are

better able to make the needed osmotic adjustments that enable them to extract more water from

a saline soil, or they may be more tolerant of some of the toxic effects of salinity. Salt tolerance

for many agricultural crops are provided in the FAO Irrigation and Drainage Papers No. 33 and

48. The ECe,threshold and slope b from these sources are listed in Table 23.

Crop evapotranspiration

175

As can be observed from the data in Table 23, there is an 8 to 10-fold range in salt

tolerance of agricultural crops. The effect of soil salinity on yield and crop evapotranspiration

is hence crop specific.

The ECe, threshold and b parameters in Table 23 were determined primarily in research

experiments using nearly steady-state irrigation where soil water contents were maintained at

levels close to field capacity. However, under most types of irrigation scheduling for sprinkler

and surface irrigation, the soil water content is typically depleted to well below field capacity,

so that the EC of the soil water solution, ECsw, increases prior to irrigation, even though the

EC of the saturation extract does not change. The increased salt concentration in the soil water

solution reduces the osmotic potential of the soil water solution (it becomes more negative), so

that the plant must expend more metabolic energy and may exert more mechanical force to

absorb water. In addition, metabolic and toxic effects of salts on plants may become more

pronounced as the soil dries and concentrations increase. However, the variation in soil water

content during an irrigation interval has not been found to strongly influence crop

evapotranspiration. This is because of the rise of soil water content to levels that are above that

experienced under steady state irrigation early in a long irrigation interval. There is a similar,

counteractive decrease in soil water content later in a long irrigation interval. In addition, the

distribution of salts in the root zone under low frequency irrigation can reduce salinity impacts

during the first portion of the irrigation interval. Also, under high frequency irrigation of the

soil surface, soil evaporation losses are higher. Consequently, given the same application

depth, the leaching fraction is reduced. For these reasons, the length of irrigation interval and

the change in EC of soil water during the interval have usually not been found to be factors in

the reduction of ET, given that the same depths of water are infiltrated into the root zone over

time.

In some cases, increased evaporation under high frequency irrigation of the soil surface

can counteract reductions in Kc caused by high ECe of the root zone. Under these conditions,

the total Kc and ETc are not very different from the non-saline, standard conditions under less

frequent irrigation, even though crop yields and crop transpiration are reduced. Because of this,

under saline conditions, the Ks reducing factor should only be applied with the dual Kc

approach.

In review articles on impacts of salinity on crop production, Letey et al. (1985) and

Shalhevet (1994) concluded that effects of soil salinity and water stress are generally additive

in their impacts on crop evapotranspiration. Therefore, the same yield-ET functions may hold

for both water shortage induced stress and for salinity induced stress.

YIELD-SALINITY RELATIONSHIP

A widely practiced approach for predicting the reduction in crop yield due to salinity has been

described in the FAO Irrigation and Drainage Paper No29. The approach presumes that, under

optimum management conditions, crop yields remain at potential levels until a specific,

threshold electrical conductivity of the soil water solution is reached. When salinity increases

beyond this threshold, crop yields are presumed to decrease linearly in proportion to the

increase in salinity. The soil water salinity is expressed as the electrical conductivity of the

saturation extract, ECe. In equation form, the procedure followed in FAO Irrigation and

Drainage Paper No29 is:

ETc under soil water stress conditions

176

()

100

b

ECEC1

Y

thresholdee

m

a−−= (89)

for conditions where ECe > ECe threshold where:

Y

a actual crop yield

Y

m maximum expected crop yield when ECe < ECe threshold

ECe mean electrical conductivity of the saturation extract for the root zone

[dS m-1]

ECe threshold electrical conductivity of the saturation extract at the threshold of ECe

when crop yield first reduces below Ym [dS m-1]

b reduction in yield per increase in ECe [%/(dS m-1)]

Values for ECe threshold and b have been provided in the FAO Irrigation and Drainage

Paper No29 and 48 and are listed in Table 23 for many agricultural crops.

Salinity-yield data from the FAO Irrigation and Drainage papers Nos. 29 and 48 were

mostly from studies where soil water content was held at about –3 m potential (-30 kPa) or

higher at the 0.3 to 0.6 m depth, depending on the crop. However, these papers indicate that for

most crops, the data are transferable to typical field situations where the readily available soil

water (RAW) is depleted between irrigations.

YIELD-MOISTURE STRESS RELATIONSHIP

A simple, linear crop-water production function was introduced in the FAO Irrigation and

Drainage Paper No33 to predict the reduction in crop yield when crop stress was caused by a

shortage of soil water:

÷

ø

ö

ç

è

æ

−=

÷

ø

ö

ç

è

æ−

c

adjc

y

mET

ET

1K

Y

Ya

1 (90)

where: Ky a yield response factor [-]

ETc adj adjusted (actual) crop evapotranspiration [mm d-1]

ETc crop evapotranspiration for standard conditions (no water stress) [mm d-1]

K

y is a factor that describes the reduction in relative yield according to the reduction in

ETc caused by soil water shortage. In FAO No33, Ky values are crop specific and may vary

over the growing season. In general, the decrease in yield due to water deficit during the

vegetative and ripening period is relatively small, while during the flowering and yield

formation periods it will be large. Values for Ky for individual growth periods and for the

complete growing season have been included in the FAO Irrigation and Drainage Paper No33.

Seasonal values for Ky are summarized in Table 24.

COMBINED SALINITY-ET REDUCTION RELATIONSHIP

No water stress (Dr < RAW)

When salinity stress occurs without water stress, Equations 89 and 90 can be combined and

solved for an equivalent Ks, where Ks = ETc adj/ETc:

Crop evapotranspiration

177

()

thresholdee

y

sECEC

100K

b

1K −−= (91)

for conditions when ECe > ECe threshold and soil water depletion is less than the readily

available soil water depth (Dr < RAW). Dr and RAW are defined in the previous section.

With water stress (Dr > RAW)

When soil water stress occurs in addition to salinity stress, Equation 84 in Chapter 8 and

Equations 89 and 90 are combined to yield:

()

÷

ø

ö

ç

è

æ

−

÷

ø

ö

ç

è

æ

−−= RAWTAW

DTAW

ECEC

100K

b

1K r

thresholdee

y

s (92)

for conditions when ECe > ECe threshold and Dr > RAW. Figure 44 shows the impact of

salinity reduction on Ks as salinity increases. Note that the approach presumes that RAW (and

p) do not change with increasing salinity. This may or may not be a good assumption for some

crops.

Limitations

Because the impact of salinity on plant growth and yield and on crop evapotranspiration is a

time-integrated process, generally only the seasonal value for Ky is used to predict the

reduction in evapotranspiration. There are Ky values in FAO Irrigation and Drainage paper

No33 for only about 23 crops. The seasonal values for Ky from paper No33 are summarized in

Table 24. For many crops, the seasonal Ky is nearly 1. For crops where Ky is unknown, the

user may use Ky = 1 in Equations 91 and 92 or may select the Ky for a crop type that has

similar behaviour.

Equations 91 and 92 are suggested as only approximate estimates of salinity impacts on

ET, and represent general effects of salinity on evapotranspiration as occurring over an

extended period of time (as measured in weeks or months). These equations are not expected

to be accurate for predicting ETc for specific days. Nor do they include other complicating

effects such as specific ion toxicity. Application of equations 91 and 92 presumes that the ECe

represents the average ECe for the root zone.

The equations presented may not be valid at high salinitiy, where the linear relationships

between ECe, crop yield and Ks may not hold. The use of Equations 91 and 92 should usually

be restricted to ECe < ECthreshold + 50/b. In addition, the equations predict Ya = 0 before Ks

= 0 when Ky > 1 and vice versa.

As indicated earlier, reduction in ETc in the presence of soil salinity is often partially

caused by reduced plant size and fraction of ground cover. These effects are largely included in

the coefficient values in Table 23. Therefore, where plant growth is affected by salinity and

Equations 91 and 92 are applied, no other reductions in Kc are required, for example using LAI

or fraction of ground cover, as described in Chapter 9.

ETc under soil water stress conditions

178

TABLE 23

Salt tolerance of common agricultural crops expressed as electrical conductivity of the soil saturation

extract at the threshold when crop yield first reduces below the full yield potential (ECe, threshold) and

as the slope (b) of reduction in crop yield with increasing salinity beyond ECe, threshold.

Crop1 ECe, threshold2

(dS m-1)3

b4

(% / dS m-1)

Rating5

a. Small vegetables

Broccoli 2.8 9.2 MS

Brussels sprouts 1.8 9.7 MS

Cabbage 1.0–1.8 9.8-14.0 MS

Carrots 1.0 14.0 S

Cauliflower 1.8 6.2 MS

Celery 1.8-2.5 6.2-13.0 MS

Lettuce 1.3-1.7 12.0 MS

Onions 1.2 16.0 S

Spinach 2.0-3.2 7.7-16.0 MS

Radishes 1.2-2.0 7.6-13.0 MS

b. Vegetables - Solanum Family (Solanaceae)

Egg Plant - - MS

Peppers 1.5-1.7 12.0-14.0 MS

Tomato 0.9-2.5 9.0 MS

c. Vegetables Cucumber Family (Cucurbitaceae)

Cucumber 1.1-2.5 7.0-13.0 MS

Melons - - MS

Pumpkin, winter squash 1.2 13.0 MS

Squash, Zucchini 4.7 10.0 MT

Squash (scallop) 3.2 16.0 MS

Watermelon - - MS

d. Roots and Tubers

Beets, red 4.0 9.0 MT

Parsnip - - S

Potato 1.7 12.0 MS

Sweet potato 1.5-2.5 10.0 MS

Turnip 0.9 9.0 MS

Sugar beet 7.0 5.9 T

e. Legumes (Leguminosae)

Beans 1.0 19.0 S

Broadbean (faba bean) 1.5-1.6 9.6 MS

Cowpea 4.9 12.0 MT

Groundnut (Peanut) 3.2 29.0 MS

Peas 1.5 14.0 S

Soybeans 5.0 20.0 MT

1 The data serve only as a guideline - Tolerance vary depending upon climate, soil conditions and

cultural practices. Crops are often less tolerant during germination and seedling stage.

2 ECe,

threshold means average root zone salinity at which yield starts to decline

3 Root zone salinity is measured by electrical conductivity of the saturation extract of the soil,

reported in deciSiemens per metre (dS m-1) at 25°C

4 b is the percentage reduction in crop yield per 1 dS/m increase in ECe beyond ECe threshold

5 Ratings are: T = Tolerant, MT = Moderately Tolerant, MS = Moderately Sensitive and S =

Sensitive

Crop evapotranspiration

179

Table 23 Continued

Crop ECe, threshold

(dS m-1)

b

(% / dS m-1)

Rating

f. Perennial Vegetables (with winter dormancy and

initially bare or mulched soil)

Artichokes - - MT

Asparagus 4.1 2.0 T

Mint - - -

Strawberries 1.0-1.5 11.0-33.0 S

g. Fibre crops

Cotton 7.7 5.2 T

Flax 1.7 12.0 MS

h. Oil crops

Casterbean - - MS

Safflower - - MT

Sunflower - - MS

i. Cereals

Barley 8.0 5.0 T

Oats - - MT

Maize 1.7 12.0 MS

Maize, sweet (sweet corn) 1.7 12.0 MS

Millet - - MS

Sorghum 6.8 16.0 MT

Rice6 3.0 12.0 S

Wheat (Triticum aestivum) 6.0 7.1 MT

Wheat, semidwarf (T. aestivum) 8.6 3.0 T

Wheat, durum (Triticum turgidum) 5.7-5.9 3.8-5.5 T

j. Forages

Alfalfa 2.0 7.3 MS

Barley (forage) 6.0 7.1 MT

Bermuda 6.9 6.4 T

Clover, Berseem 1.5 5.7 MS

Clover (alsike, ladino, red, strawberry) 1.5 12.0 MS

Cowpea (forage) 2.5 11.0 MS

Fescue 3.9 5.3-6.2 MT

Foxtail 1.5 9.6 MS

Hardinggrass 4.6 7.6 MT

Lovegrass 2.0 8.4 MS

Maize (forage) 1.8 7.4 MS

Orchardgrass 1.5 6.2 MS

Rye-grass (perennial) 5.6 7.6 MT

Sesbania 2.3 7.0 MS

Sphaerophysa 2.2 7.0 MS

Sudangrass 2.8 4.3 MT

Trefoil, narrowleaf birdsfoot 5.0 10.0 MT

Trefoil, big 2.3 19.0 MS

Vetch, common 3.0 11.0 MS

Wheatgrass, tall 7.5 4.2 T

Wheatgrass, fairway crested 7.5 6.9 T

Wheatgrass, standard crested 3.5 4.0 MT

Wildrye, beardless 2.7 6.0 MT

k. Sugar cane 1.7 5.9 MS

6 Because paddy rice is grown under flooded conditions, values refer to the electrical conductivity of

the soil water while the plants are submerged

ETc under soil water stress conditions

180

Table 23 Continued

Crop ECe, threshold

(dS m-1)

b

(% / dS m-1)

Rating

l. Tropical Fruits and Trees

Banana - - MS

Coffee - - -

Date Palms 4.0 3.6 T

Palm trees - - T

Pineapple (multi-year crop) - - MT

Tea - - -

m. Grapes and berries

Blackberry 1.5 22.0 S

Boysenberry 1.5 22.0 S

Grapes 1.5 9.6 MS

Hops - - -

n. Fruit trees

Almonds 1.5 19.0 S

Avocado - - S

Citrus (Grapefruit) 1.8 16.0 S

Citrus (Orange) 1.7 16.0 S

Citrus (Lemon) - - S

Citrus (Lime) - - S

Citrus (Pummelo) - - S

Citrus (Tangerine) - - S

Conifer trees - - MS/MT

Deciduous orchard

- Apples - - S

- Peaches 1.7 21.0 S

- Cherries - - S

- Pear - - S

- Apricot 1.6 24.0 S

- Plum, prune 1.5 18.0 S

- Pomegranate - - MT

Olives - - MT

Primary sources :

Ayers and Westcot, 1985. FAO Irrigation and Drainage Paper N° 29. Water quality for agriculture ;

Rhoades, Kandiah and Mashali, 1992. FAO Irrigation and Drainage Paper N° 48. The use of saline

waters for crop productions.

Application

Under steady state conditions, the value for ECe can be predicted as a function of EC of the

irrigation water (ECiw) and the leaching fraction, using a standard leaching formula. For

example, the FAO-29 leaching formula LR = ECiw / (5 ECe – ECiw) predicts the leaching

requirement when approximately a 40-30-20-10 percent water extraction pattern occurs from

the upper to lower quarters of the root zone prior to irrigation. ECiw is the electrical

conductivity of the irrigation water. From this equation, ECe is estimated as:

5

EC

LF

LF1

EC iw

e

+

= (93)

Crop evapotranspiration

181

where LF, the actual leaching fraction, is used in place of LR, the leaching requirement.

Equation 93 predicts ECe = 1.5 ECiw under conditions where a 15-20 percent leaching

fraction is employed. Other leaching fraction equations can be used in place of the FAO-29

equation to fit local characteristics. Equation 93 is only true if the irrigation water quality and

the leaching fraction are constant over the growing season. Time is required to attain a salt

equilibrium in the soil. If there are important winter rains of high quality water and often

excellent leaching, the salt balance in the soil will be quite different at the beginning of the

season and with a lower average ECe of the root zone than would be predicted from Equation

93. An appropriate local calibration of Equation 93 is desirable under these particular

conditions.

TABLE 24

Seasonal yield response functions from FAO Irrigation and Drainage Paper No. 33.

Crop K

y

Crop K

y

Alfalfa 1.1 Potato 1.1

Banana 1.2-1.35 Safflower 0.8

Beans 1.15 Sorghum 0.9

Cabbage 0.95 Soybean 0.85

Citrus 1.1-1.3 Spring Wheat 1.15

Cotton 0.85 Sugarbeet 1.0

Grape 0.85 Sugarcane 1.2

Groundnet 0.70 Sunflower 0.95

Maize 1.25 Tomato 1.05

Onion 1.1 Watermelon 1.1

Peas 1,15 Winter wheat 1.05

Pepper 1.1

FIGURE 44

The effect of soil salinity on the water stress coefficient Ks

0.00

0.20

0.40

0.60

0.80

1.00

: soil water content

FC tWP

D : de

p

letion from root zone

(

mm

)

r

Ks

0RAW TAW

low

high

with soil salinity

no soil salinity

ETc under soil water stress conditions

182

EXAMPLE 39

Effect of soil salinity on crop evapotranspiration

A field of beans is cultivated on a silt loam soil and is irrigated during the midseason period using water

having salinity ECiw = 1 dS m-1. A 15 percent leaching fraction is employed. The ECe,threshold and

slope from Table 23 are 1.0 dS m-1 and 19 %/(dS m-1) respectively. The seasonal Ky from FAO

Irrigation and Drainage Paper No 33 and Table 24 for beans is Ky = 1.15. Compare the effect on crop

evapotranspiration for various levels of soil water depletion in the root zone under saline and nonsaline

conditions. The TAW and RAW for the bean crop are 110 and 44 mm (for p = 0.4).

Since the leaching fraction is 0.15, ECe is estimated from Equation 93 as ECe = 1.5 ECw = 1.5 (1) =

1.5 dS m-1. The Ks in the presence of salinity stress and absence of moisture stress is:

()

92.00.15.1

)100(15.1

19

1ECEC

100K

b

1K thresholdee

y

s=

÷

ø

ö

ç

è

æ−−=

÷

ø

ö

ç

è

æ

−−=

The Ks in the presence of moisture stress, but in the absence of salinity stress is:

÷

ø

ö

ç

è

æ−

=

÷

ø

ö

ç

è

æ

−

=

÷

ø

ö

ç

è

æ

−

=66

D110

44100

D110

RAWTAW

DTAW

Krrr

s

The Ks in the presence of both moisture stress and salinity stress is:

()

÷

ø

ö

ç

è

æ−

=

÷

ø

ö

ç

è

æ

−

÷

ø

ö

ç

è

æ

−−= 66

D110

92.0

RAWTAW

DTAW

ECEC

100K

b

1K rr

thresholdee

y

s

The effect on crop evapotranspiration for various soil water depletions in the root zone (Dr) are:

Dr

(mm)

Ks

no soil salinity

Ks

with soil salinity

(ECe = 1.5 dS m-1)

(Eq. 92)

Additional reduction

in potential ETc due

to salinity

0 1.00 no reduction in ETc 0.92 8% reduction in ETc 8%

35 1.00

no reduction in ETc 0.92 8% reduction in ETc 8%

40 1.00

no reduction in ETc 0.92 8% reduction 8%

44 1.00

no reduction in ETc 0.92 8% reduction 8%

50 0.91 9% reduction 0.83 17% reduction 8%

60 0.76 24% reduction 0.69 31% reduction 7%

70 0.61 39% reduction 0.56 44% reduction 5%

80 0.45 55% reduction 0.42 58% reduction 3%

90 0.30 70% reduction 0.28 72% reduction 2%

100 0.15 85% reduction 0.14 86% reduction 1%

110 0.00

ETc = 0 0.00 ETc = 0 --

Crop evapotranspiration

183

Chapter 9

ETc for natural, non-typical and

non-pristine vegetation

Non-typical refers to types or arrangements of agricultural crops that are not listed or described

in Tables 12 and 17. Non-pristine vegetation is defined, in the usage here, as vegetation having

less than perfect growing conditions or stand characteristics (i.e., relatively poorer conditions of

density, height, leaf area, fertility, or vitality) as compared to ‘pristine’ conditions.

The approach whereby a crop is characterized by a crop coefficient, Kc, and the crop

evapotranspiration is given by the product of Kc and the reference evapotranspiration ETo,

provides a simple and convenient way of also characterizing the evapotranspiration from

natural vegetation and for non-typical cultivation practices. This chapter presents procedures

for estimating Kc values for natural vegetation and for agricultural vegetation for which Kc

values are not available.

CALCULATION APPROACH

As described in Figure 27, the first step in the KcETo approach is the estimation of lengths of

growth stages. This also applies to natural and other vegetation. The next step is the

development of crop coefficient curves that represent the ratios of ETc to ETo during the

various growth stages of the vegetation.

Initial growth stage

The procedure to estimate crop coefficients for the initial growth stage for natural, non-typical

and non-pristine vegetation is identical to that described in Chapter 6 (single crop coefficient

Kc ini) or Chapter 7 (dual crop coefficient, Kcb ini + Ke). The crop coefficient in this stage is

primarily determined by the frequency with which the soil is wetted.

Mid and late season stages

The Kc during the mid-season period (Kc mid and Kcb mid) and to a lesser extent the Kc

during the late season period differ from that described in previous chapters. As the ground

cover for natural and non-pristine vegetation is often reduced, the Kc is affected to a large

extent by the frequency of precipitation and/or irrigation and by the amount of leaf area and

ground cover.

Dual crop coefficient approach

The determination of Kc for natural, non-typical or non-pristine vegetation should ordinarily

follow the approach described in Chapter 7 whereby separate transpiration (Kcb) and

evaporation (Ke) coefficients are used. The effects of evaporation from the soil surface can be

directly estimated as such.

ETc for natural, non-typical and non-pristine vegetation

184

Two procedures that can be used to adjust the basal crop coefficient (Kcb mid adj) for

sparse vegetation are presented in this section. In these approaches, Kcb mid adj is estimated

either from LAI (Equation 97) or from effective ground cover (Equation 98). After the

determination of Kcb mid adj, the soil evaporation coefficient, Ke, should be determined to

obtain the crop coefficient for the mid-season stage: Kc mid adj = Kcb mid adj + Ke.

Procedures for calculating Ke are presented in Chapter 7.

Even where the estimated Kcb mid adj is small, the total Kc adj (= Kcb adj + Ke)

following precipitation may sometimes be as high or higher than the Kc for pristine vegetation

due to surface evaporation from among sparse vegetation.

Single crop coefficient approach

When the single crop coefficient Kc of Chapter 6 is used, the average effects of soil wetting are

incorporated into a general mean Kc. Some guidelines for the estimation of Kc adj are given in

the following sections. The single crop coefficient can also be derived from the adjusted Kcb

by considering the frequency of soil wetting, i.e., during the midseason period, Kc adj = Kcb

adj + 0.05 for infrequent wetting and Kcb adj + 0.10 for wettings of up to once a week. For

more frequent wettings, the dual crop coefficient approach should be used.

Alternatively, Equations 97 and 98 can be used to determine Kc instead of Kcb. Then,

Kc min in Equations 97 and 98 can be set equal to Kc ini, where Kc ini is estimated from

Figure 29 or 30. The use of Kc ini incorporates effects of soil evaporation and therefore serves

as a lower limit on the estimate for Kc mid.

Water stress conditions

Where rainfall or irrigation is low, water stress might be induced and the evapotranspiration

will drop below the standard crop evapotranspiration, ETc. The reduction in the value for Kc

under conditions of low soil water availability is determined using the stress coefficient Ks as

described in Chapter 8.

MID-SEASON STAGE - ADJUSTMENT FOR SPARSE VEGETATION

Adjustment from simple field observations

As a rough approximation for Kc during the mid-season stage for crops that usually nearly

completely shade the soil under pristine conditions, but where cover is reduced due to disease,

stress, pests, or planting density, the values for Kc mid and Kcb mid can be reduced by a factor

depending on the actual vegetation development:

cmcdjac AKK −= (94)

where Kc the Kc from Table 12 (Kc mid) or 17 (Kcb mid) after adjusting it for

climate (Equation 62 or 70),

Kc adj the adjusted Kc (Kc mid adj or Kcb mid adj).

The Kc adjustment using Equation 94 does not apply when crops are frequently wetted

and increased soil evaporation compensates for the reduced ground cover. Under these

conditions Equation 94 should be applied only to Kcb.

Crop evapotranspiration

185

The adjustment coefficient, Acm, is estimated from:

5.0

dense

cm LAI

LAI

1A ú

û

ù

ê

ë

é

−= (95)

where LAI is the actual leaf area index (Box 17) and LAIdense is the leaf area index expected

for the same crop under normal, standard crop management practices. The values for LAI in the

above equation can be replaced by values for the ground cover fraction (fc):

5.0

densec

c

cm f

f

1A ú

ú

û

ù

ê

ë

é

−= (96)

EXAMPLE 40

First approximation of the crop coefficient for the mid-season stage for sparse vegetation

A tomato crop was grown at Davis, California, United States in 1980 and only developed 50%

ground cover during the midseason period (Pruitt et al., 1984). The height of the tomato crop was

0.75 m. The typical percentage of ground cover for tomatoes at effective full cover at Davis is 85 to

90% and corresponds to the Kcb mid listed in Table 17 for tomatoes. The mean values for wind

speed and minimum relative humidity during the midseason period were u2 = 1.1 m/s and RHmin =

30%. The latitude at Davis is 38.5o N and the midpoint of the midseason occurs on July 20. What

is an adjusted Kcb mid for tomatoes that reflects the 50% ground cover condition?

From Tables 12 and 17, Kc mid = 1.2 and Kcb mid = 1.15.

Following adjustments for climate (Eq. 62 and Eq. 70) where u2 = 1.1 m/s, RHmin = 30% and mean

crop height = 0.75 m,

K

c = Kc,Table + [0.04(1.1-2)-0.004(30-45)](0.75/3)0.30 = Kc,Table + 0.02

yields, Kc mid = 1.22 and Kcb mid = 1.17.

The ground cover fraction implied in the tabulated values for tomatoes grown under pristine conditions

is about 85% (fc dense = 0.85). For a sparse tomato crop where fc = 0.50,

From Eq. 96 Acm = 1 - (0.50/0.85)0.5 = 0.23

The Kcb mid adj and Kc mid adj for 50% ground cover is

(from Eq. 94) Kcb mid adj = 1.17 - 0.23 = 0.94

K

c mid adj = 1.22 - 0.23 = 0.99

Compare the results with Example 42 where a more precise derivation of Kcb mid adj is made.

As a first estimate, the crop coefficient is expected to be 20% lower than the value under pristine

conditions.

Estimation of Kcb mid from Leaf Area Index (LAI)

Natural vegetation typically has less leaf area or fraction of ground cover than does agricultural

vegetation that has been developed for full ground cover and for soil water conditions favouring

vigorous growth. This is especially true in semi-arid and arid climates. The value for Kcb mid

for natural or non-pristine vegetation should be reduced when plant density and/or leaf area are

lower than for full cover conditions (generally defined as when LAI ≥ 3). Where LAI can be

ETc for natural, non-typical and non-pristine vegetation

186

measured or approximated, a peak Kcb mid for natural, non-typical or non-pristine agricultural

vegetation can be approximated similar to a procedure used by Ritchie as:

(

)

()

]LAI7.0[exp1KKKK mincfullcbmincmidcb −−−+= (97)

where Kcb mid estimated basal Kcb during the mid-season when plant density

and/or leaf area are lower than for full cover conditions,

Kcb full estimated basal Kcb during the mid-season (at peak plant size

or height) for vegetation having full ground cover or LAI > 3

(Equations 99 and 100),

Kc min the minimum Kc for bare soil (Kc min ≈ 0.15 - 0.20),

LAI actual leaf area index, defined as the area of leaves per area of

underlying ground surface averaged over a large area. Only one

side of leaves is counted [m2 m-2].

Equation 97 is recommended for annual types of vegetation that are either natural or are

in a non-pristine state due to sparse density or effects of some type of environmental stress on

growth.

The relationship expressed in Equation 97 produces results similar to those suggested by

Ritchie (1974). For LAI > 3, Kcb mid ≈ Kcb full. The LAI used in Equation 97 should be the

‘green’ LAI representing only healthy leaves that are active in vapour transfer.

BOX 17

Measuring and estimating LAI

LAI can be measured directly by harvesting all green healthy leaves from vegetation over a measured

or prescribed area, for example, 1 m2 or 10 m2, and then measuring and summing the areas of

individual leaves using photometric methods or by measuring areas of several representative leaves,

averaging, and then multiplying by the total number of leaves counted.

In the absence of measurements for LAI, LAI can be estimated for sparse, annual vegetation as:

a

dense

dense Population

Population

LAILAI ú

û

ù

ê

ë

é

=

where LAIdense LAI for the particular plant species under normal, ‘dense’ or pristine

growing conditions. LAIdense can be obtained from various

physiological sources and textbooks on crops and vegetation.

Population number of plants per unit area of soil surface under the actual

growing conditions [No. m-2].

Populationdense number of plants per unit area of soil surface under the ‘dense’

or pristine growing conditions [No. m-2].

a a = 0.5 when population is formed from vigorous growing plants; a

= 1 when plants are less vigorous.

The 0.5 exponent in the equation simulates the tendency for vegetation to compensate for reduced

stand density by increasing the size and total leaf areas for individual plants. Therefore, LAI does not

fall in direct proportion to plant population. Under conditions where the plant size does not increase

with reduced stand density, the ’a’ exponent in the equation should be set to 1 (a = 1). These latter

conditions may occur where soil fertility is poor or where soil salinity, soil water stress, or waterlogging

inhibit both growth and stand density, so that the growth of individual plants is retarded.

Crop evapotranspiration

187

Estimation of Kcb mid from effective ground cover (fc eff)

Where only estimates of the fraction of soil surface effectively covered by vegetation are

available, the following approximation for Kcb mid adj can be used:

()()

÷

ø

ö

ç

è

æ

÷

ø

ö

ç

è

æ

−+=

÷

ø

ö

ç

è

æ

+h1

1

effccmincfullcbmincmidcb f,f2,1minKKKK (98)

where Kcb mid estimated basal Kcb during the mid-season when plant density and/or

leaf area are lower than for full cover conditions,

Kcb full estimated basal Kcb during the mid-season (at peak plant size or height)

for vegetation having full ground cover or LAI > 3 (see Equations 99 and

100),

Kc min the minimum Kc for bare soil (in the presence of vegetation) (Kc min ≈

0.15 - 0.20),

fc observed fraction of soil surface that is covered by vegetation as

observed from nadir (overhead) [0.01 - 1],

fc eff the effective fraction of soil surface covered or shaded by vegetation

[0.01-1] (see Box 18),

h the plant height [m].

Stomatal conductance and water transport within plants may limit ET under conditions of

sparse, tall vegetation. Under these conditions, Kcb mid is limited by the “2fc” term in

Equation 98. Equation 98 applies well to trees and shrubs.

BOX 18

Measuring and estimating fc eff

fc eff should normally represent the fraction of the soil surface that is shaded by vegetation. This value

is generally larger than fc, the actual fraction of the soil surface that is covered by vegetation as

observed from directly overhead. The amount of shading represents the amount of solar radiation

intercepted by plants for potential conversion into evapotranspiration. The total fraction of shading is a

function of the sun angle and the horizontal size and shape of individual plants (or rows) relative to

their height.

fc eff for ‘rectangular’ shaped canopies such as most agricultural plant rows can be approximated as:

1

)tan(

HWR

1ff ceffc ≤

ú

û

ù

ê

ë

é

η

+=

w

here fc observed fraction of soil surface that is covered by vegetation as observed from nadir

(overhead) [0.01 - 1],

HWR height to width ratio of individual plants or groups of plants when viewed from the east

or from the west [ ],

tan(η) tangent of the mean angle of the sun, η, above the horizon during the period of

maximum evapotranspiration (generally between 11.00 and 15.00 hours) [ ]. For most

applications, η can be computed at solar noon (12.00 hours).

ETc for natural, non-typical and non-pristine vegetation

188

HWR is computed as:

()

Width

(cos

canopy

h

HWR

)Γ

=

where hcanopy mean vertical height of the canopy area of the plant [m],

Width mean width of the canopy of a plant or group of plants (e.g., row) [m]

Γ angle of plant row from east-west direction [rad] (for east-west rows, Γ= 0;

for north-south rows, Γ = π/2)

S

N N

N

Width

S

h

canopy

For north-south rows, the HWR would be zero, as cos(π/2) = 0. This implies that rows of plants that run

from north to south would have fc eff ≈ fc as all soil is exposed to the sun at various times of the day

and as the shaded area is the same as the fraction of vegetation cover at midday.

For trees or vegetation that do not have canopies that extend to the ground, hcanopy does not

include the lower trunk length, but only the active canopy. Therefore, in these situations, hcanopy < h

where h is mean plant height.

For round or spherical shaped canopies such as trees, fc eff can be estimated as:

1

)sin(

f

fc

effc ≤

η

=

where sin(η) is the sine of the mean angle of the sun, η, above the horizon during the period of

maximum evapotranspiration (generally between 11.00 and 15.00) [ ]

Mean angle of the sun above the horizon during the period of maximum evapotranspiration (η

ηη

η)

The sine of η can be calculated for any specific time of day as:

)cos()cos()cos()sin()sin()sin( ωδϕ+δϕ=η

where ϕ latitude [rad] (negative for southern latitudes)

δ solar declination [rad] (Eq. 24)

ω solar time angle [rad] (Eq. 31)

Generally, fc eff can be calculated at solar noon (12.00), so that ω = 0 and the above equation reduces

to:

)cos()cos()sin()sin()sin( δϕ+δϕ=η

The value for η can be obtained by taking the arcsine of the above equation.

Crop evapotranspiration

189

Estimation of Kcb full

Agricultural crops:

Non-pristine agricultural crops represent crops that have not developed to their potential due to

environmental stresses caused by soil water shortage, fertility, disease, grazing or insect

damage or due to low plant density. The value for Kcb full in Equations 97 and 98 can be taken

as the Kcb mid value listed for any “full-cover” crop (fc eff ~ 1) in Table 17, after adjusting it

for climate (Equation 70):

[]

3.0

min2Table,midcbfullcb 3

h

)45RH(004.0)2u(04.0KK ÷

ø

ö

ç

è

æ

−−−+= (99)

where u2 mean value for wind speed at 2 m height during the mid-season [m s-1],

RHmin mean value for minimum daily relative humidity during the mid-season

[%],

h mean maximum plant height [m].

Natural vegetation and crops not listed in Table 17:

For natural vegetation, nonfull-cover crops, or for crops not listed in Table 17, Kcb full can be

approximated as a function of climate and mean plant height for areas of vegetation that are

greater than a few hectares:

[]

3.0

min2h,cbfullcb 3

h

)45RH(004.0)2u(04.0KK ÷

ø

ö

ç

è

æ

−−−+= (100)

where Kcb,h K

cb mid for full cover vegetation (LAI > 3) under sub-humid and calm

wind conditions (RHmin = 45% and u2 = 2 m s-1), (Equation 101),

u2 mean value for wind speed at 2 m height during the mid-season [m s-1],

RHmin mean value for minimum daily relative humidity during the mid-season

[%],

h mean maximum plant height [m].

The value for Kcb,h is estimated as:

m2hforh1.00.1K h,cb ≤+= (101)

where Kcb,h is limited to ≤ 1.20 when h > 2 m. The value of 1.2 represents a general upper

limit on Kcb mid for tall vegetation having full ground cover and LAI > 3 under the sub-humid

and calm wind conditions. This limit of 1.2 is adjusted for other climatic conditions in Equation

100. Equations 100 and 101 produce a general approximation for the increase in Kcb full with

plant height and climate. The form of these equations adheres to trends represented in Equation

70.

For small, isolated stand sizes, Kcb full may need to be increased beyond the value given

by Equation 99 or 100, as discussed in Chapter 10.

ETc for natural, non-typical and non-pristine vegetation

190

Conclusion

Equations 97 and 98 can be used to estimate or to reduce Kcb for non-pristine agricultural

vegetation. The exponents in Equations 97 and 98 reflect the effects of microscale advection

(transfer) of sensible heat from dry soil surfaces between plants toward plant leaves, thereby

increasing ET per unit leaf area, and the effects of increased aerodynamic roughness as the

value for LAI decreases. Equation 98 suggests that as h increases, total leaf area and effective

roughness of vegetation increase, thereby increasing the crop coefficient. In addition, as h

increases, more opportunity for microadvection of heat from soil to canopy occurs and

turbulent exchange within the canopy increases for the same amount of ground coverage. All of

these factors affect the relative magnitude of Kcb mid.

Equations 97 and 98 should be used with caution as they provide only an estimate of the

maximum Kcb expected during peak plant growth for vegetation with healthy transpiring leaves

and a dry soil surface. Where stomatal control is greater than for typical agricultural vegetation,

then the Kcb should be further reduced using the recommendations set out in the next section

(Equation 102).

EXAMPLE 41

Estimation of mid-season crop coefficient

Estimate Kcb mid and Kc mid for rectangular shaped 2 m tall vegetation that is as tall as it is wide,

where 30% of the ground is covered by vegetation (fc = 0.3) on 19 July (day 200 of the year) and at

latitude 40°N. RHmin = 55% and u2 = 1.5 m/s during the mid-season period.

On day J = 200 at latitude ϕ = 40 (π/180) = 0.70 radians (40°N), from Eq. 24, the solar declination δ =

0.36 radians.

At solar noon (ω = 0):

sin(η) = sin(ϕ)sin(δ) + cos(ϕ)cos(δ) = 0.94

The value for η by taking the arcsine of above value is 1.24 radians and tan(η) = 2.8. If fc = 0.3 and the

HWR for the vegetation is 1, then fc eff from Box 18 is: 0.3(1+1/2.8) = 0.41.

From Eq. 101 Kcb,h = 1.0 + 0.1(2) = 1.20 (≤ 1.20, so OK)

From Eq. 100 Kcb full = 1.20 + (0.04(1.5-2) - 0.004(55-45)) (2/3)0.3 = 1.15

Therefore, Kcb mid estimated by Eq. 98 for Kc min = 0.15 is

Kcb mid = Kc min + (Kcb full - Kc min) min[1, 2fc, (fc eff)(1/(1+h))]

= 0.15+(1.15-0.15) min[1, 2(0.3), (0.4)(1/(1+2))] =0.75

This value does not need any further adjustment for climate.

Kc mid (where Kc mid includes average wetting effects) can be derived from Kcb mid using the

guidelines presented in the calculation procedures at the beginning of this chapter

Kc mid = Kcb mid + (0.05 ... 0.10) = 0.80 .. 0.85

depending on the frequency of soil wetting.

The estimated crop coefficients for the mid-season stage are Kcb mid = 0.75 and Kc mid = 0.80 to

0.85.

Crop evapotranspiration

191

EXAMPLE 42

Estimation of mid-season crop coefficient for reduced ground cover

A more precise estimate of Kcb mid for the tomato crop of Example 40 that only developed 50%

ground cover at Davis, California, the United States can be calculated if one knows that the tomato

crop was planted in 1.52 m rows running east-west, that the crop reached a plant height of 0.75 m and

that the height to width ratio of the tomato crop can be estimated as about 1.0 for the east-west rows.

The latitude is 38.5°N and the midpoint of the mid-season occurs on July 20.

What is the adjusted Kcb mid for tomatoes that reflects the 50% ground cover condition?

On day J = 201 (20 July) at latitude ϕ = 38.5 (π/180) = 0.67 radians (38.5°N), from Eq. 24 the solar

declination δ = 0.36 radians. At solar noon (ω = 0):

sin(η) = sin(ϕ)sin(δ) + cos(ϕ)cos(δ) = 0.95

The value for η by taking the arcsine of the above value is 1.26 radians. Therefore, for the observed

HWR = 1 and fc = 0.5, the effective soil cover for the east-west rows can be calculated as (Box 18):

fc eff = 0.5 [1 + 1/tan(1.26)] = 0.66

The Kcb mid in Table 17 representing a full cover tomato crop is 1.15 and the average h for fully

developed tomatoes (this variety) is about 0.75 also. Following adjustment for climate (using Eq. 99)

Kcb full = 1.15 + [0.04(1.1-2) - 0.004(30-45)] (0.75/3)0.30 = 1.17

From Eq. 98 and using Kc min = 0.15, the adjusted Kcb mid for 50% ground cover would be:

Kcb mid adj = 0.15 + (1.17 - 0.15) min(1, 2(0.5), 0.661/(1+0.76)) = 0.95

The results Kcb mid adj = 0.95 for 50% ground cover are similar to the first estimate calculated in

Example 40 and compare with the measured Kcb mid ∼ 0.90 to1.00 as determined by precision

lysimeter by Pruitt et al. (1984).

MID-SEASON STAGE - ADJUSTMENT FOR STOMATAL CONTROL

The value for Kcb full in Equations 97 and 98 may need to be reduced for vegetation that has a

high degree of stomatal control. For vegetation such as some types of desert vegetation or trees

with leaf resistance significantly greater than that of most agricultural crops where rl is

commonly about 100 s m-1, the Kcb mid estimated using Equations 97 and 98 can be modified

by multiplying by a resistance correction factor, Fr. The resistance correction factor is

developed based on the FAO Penman-Monteith equation:

÷

ø

ö

ç

è

æ+γ+∆

+

γ

+∆

≈

100

r

u34.01

)u34.01(

F

l

2

r (102)

where rl mean leaf resistance for the vegetation in question [s m-1].

The mean leaf resistance rl is 100 s m-1 for the grass ETo reference and for many

agricultural crops. Values for rl for many agricultural and non-agricultural plants can be found

in Körner et al. (1978) and Allen et al. (1996). Equation 102 reflects the fixed aerodynamic

roughness of grass rather than the roughness of the specific vegetation, since the adjusted Kc is

multiplied by the grass ETo and the Kc already reflects the effects of the aerodynamic

roughness for the specific vegetation.

ETc for natural, non-typical and non-pristine vegetation

192

EXAMPLE 43

Estimation of Kcb mid from ground cover with reduction for stomatal control

A grove of olive trees has a tree spacing of 10 m. The horizontal diameter of the trees as viewed from

overhead is 5 m. The tree height is 5 m. The lower 1.5 m of the trees have no foliage. The ground

cover between the trees is bare. The mean u2 during the mid-season growth stage is 2 m/s and the

mean RHmin = 25%. The midpoint of the mid-season growth stage is on 29 June (i.e., day 180 of the

year). The latitude of the location is 30°N.

Estimate Kcb mid using Eq. 98 for the 10x10 m and for a 5x10 m spacing.

On day J = 180 (29 June) at latitude ϕ = 30 (π/180) = 0.52 radians (30°N) and from Eq. 24 the solar

declination δ = 0.405 radians. At solar noon (ω = 0):

sin(

η) = sin(ϕ)sin(δ) + cos(ϕ)cos(δ) = 0.99

As olive trees have somewhat round shapes, the effective fraction of ground cover (Box 18) can be

estimated as fc eff = fc/(sin(η)).

f

c = area of canopy/area tree spacing = (π(5)2/4)/(10)(10) = 0.196

f

c eff = 0.196/0.99 = 0.20

From Eq. 101: Kcb,h = 1.0 +0.1(5) > 1.2 or Kcb,h = 1.2

From Eq. 100: Kcb full = 1.2 + [0-0.004(25-45)] (5/3)0.3 = 1.29

From Eq. 98 and using Kc min = 0.15:

Kcb mid = 0.15 + (1.29 - 0.15) min(1, 2(0.196), (0.20)1/(1+5)) = 0.60

Körner et al. (1979) indicate that olives (Olea europaea) have rl of about 420 s/m. Therefore, assuming

that average Tmean = 25°C and that the elevation of the grove is 0 m (sea level), so that ∆ = 0.189 kPa

(Eq. 13) and γ = 0.0676 kPa (Eq. 8), Fr is estimated from Eq. 102 as:

67.0

100

420

)2(34.010676.0189.0

))2(34.01(0676.0189.0

Fr=

÷

ø

ö

ç

è

æ÷

ø

ö

ç

è

æ

++

≈

The Kcb mid adjusted for increased stomatal control using Fr is then

K

cb mid adj = Fr Kcb mid = 0.67 (0.60) = 0.40

The value Kcb mid adj = 0.40 estimated for fc = 0.20 is less than the value for Kcb mid in Table 17 for

olives for fc = 0.40 to 0.67, due to the differences in fc. The value from Table 17 is 0.70, which after

adjustment for climate using Eq. 70 equals 0.79.

If the olives had been planted on a 5x10 m spacing, as is common in California, the United States, and

which is reflected in the Kcb values for olives in Table 17, then fc = 0.39, fc eff = 0.40, and Kcb mid

from Eq. 98 is Kcb mid = 1.04, so that the estimated Kcb mid adjusted for stomatal control using Fr =

0.67 is Kcb mid adj = 0.67(1.04) = 0.70. This value compares with the value of 0.79 obtained from

Table 17 for mature trees, after adjustment for climate.

The basal crop coefficient, Kcb mid, taking the low density, climatic condition and stomatal control into

account is 0.40. It increases to 0.70 for the 5x10 m spacing.

Crop evapotranspiration

193

The equation would underestimate Fr (overestimate the reduction in Kcb) if used with

the actual roughness of the vegetation when rl > 100 s m-1 because of the lack in Equation 102

of feedback effects that reduced ETc has on temperature and vapour pressure deficit profiles

over the crop. These parameters generally increase with decreasing ETc and therefore dampen

the reduction in ETc.

LATE SEASON STAGE

During the late season stage, the Kcb begins to decrease until it reaches Kcb end at the end of

the growing period. Values for Kcb end can be scaled from Kcb mid in proportion to the health

and leaf condition of the vegetation at termination of the growing season and according to the

length of the late season period (i.e., whether leaves senesce slowly or are killed by frost).

Values for Kc end can be similarly scaled from Kc mid; however, the reduction in Kc end will

be affected by the frequency of wetting by irrigation or precipitation and Kc end may be

proportionally less.

If estimated from Equations 97 and 98, Kcb end should be reduced if it is to represent Kc

values for plants with stomatal control that is greater than that for agricultural vegetation

(where rl ≈ 100 s m-1) or to reflect effects of ageing and senescence on stomatal control. In

these situations, the estimated Kcb end values should be multiplied by the Fr from Equation

102. Alternatively, they can be reduced by about 10% for each doubling of rl above 100 s m-1

when mean daily air temperature (Tmean) is about 30o C and by about 20% for each doubling

of rl above 100 s m-1 when Tmean is about 15o C.

Alternatively, the value for Kcb end can be reduced relative to the calculated value for

Kcb mid in proportion to the fraction of green healthy leaves remaining at the end of the late

season stage relative to that during the mid-season. This can often be based on a visual survey

of the field and may therefore be a subjective observation.

The fc parameter and h are probably the simplest indices to estimate in the field. Again,

Equations 97 and 98 should be used only as general or preliminary estimates of Kcb end.

ESTIMATING ETc adj USING CROP YIELDS

A simple, linear crop-water production function was introduced in the FAO Irrigation and

Drainage Paper No. 33 to predict the reduction in crop yield when crop stress is caused by a

shortage of soil water. This function was presented earlier as Equation 90:

÷

ø

ö

ç

è

æ−=

÷

ø

ö

ç

è

æ−

c

adjc

y

m

a

ET

1K

Y

1 (90)

where Ya = actual yield of the crop [kg ha-1]

Y

m = maximum (expected) yield in absence of environmental or water

stresses

K

y = yield response factor [ ]

ETc = potential (expected) crop evapotranspiration in the absence of

environmental or water stresses (Kc ETo)

ETc adj = actual (adjusted) crop evapotranspiration as a result of environmental

or water stresses

ETc for natural, non-typical and non-pristine vegetation

194

Values for Ky have been reported in Paper No. 33 for a wide range of crops for

predicting the effect of water stress and associated reduction in ETc adj on crop yield. Factors

are presented there for predicting yield reductions for when stress occurs in only one crop

growth stage, or when stress is distributed throughout the growing period. Seasonal yield

response functions are summarized in Table 24.

Many environmental stresses such as water shortage, salinity, low fertility and disease

impact yield by reducing the amount of ETc adj relative to the potential amount ETc. The same

can be true for when yields are reduced due to the use of low densities for plant populations.

Therefore, for very general estimates of ETc adj, one can invert Equation 90 and solve for the

stress factor, Ks:

÷

ø

ö

ç

è

æ−−=

m

a

y

sY

Y

1

K

1

1K (103)

where Ks is multiplied by Kcb or by Kc in equations 80 or 81 to predict the ETc adj in the

presense of the water or other environmental stresses or for low plant populations or virility.

The ETc adj predicted using Ks from equation 103 provides only a very general and

approximate estimate of monthly or even seasonal evapotranspiration. Equation 103 works best

for forage or other indiscriminate crops where the value for Ky is relatively constant during the

season.

Equation 103 is generally only valid for use in predicting actual crop evapotranspiration

for use in regional water balance studies, for studies of ground-water depletion and recharge, or

for estimating historical water use. The procedure is not valid for predicting ETc for daily or

weekly time periods due to the very general nature of the Ky coefficient and the seasonal time

scale of the crop yield. The procedures presented previously for adjusting ETc using a daily

soil water balance, salinity functions, or reductions in Kc based on leaf area or fraction of

ground cover are recommended over the use of equation 103.

EXAMPLE 44

Approximate estimation of Ks from crop yield data

An irrigation scheme (project) cultivates dry, edible beans. There is known to be a shortage of

irrigation water and a corresponding reduction in crop yield. The reported yield for the scheme

averages 1100 kg/ha. The potential yield for the region and variety of beans, in the absence of water

or environmental stresses and with good soil fertility is 1800 kg/ha.

From FAO Irrigation and Drainage Paper No. 33 or Table 24, the Ky for dry beans, assuming that

stresses are distributed uniformly through the growing season, is 1.15. Therefore, from Equation 103,

the estimated Ks to apply with Equation 80 for the growing season is:

66.0

1800

1100

1

15.1

1

1Ks=

÷

ø

ö

ç

è

æ−−=

Therefore, the ETc adj for the season is predicted to be only 0.66 of maximum ETc under pristine

growing conditions.

The estimated seasonal ETc adj is predicted to be ETc adj = 0.66 ETc where ETc is predicted as

ETc = Kc ETo.

Crop evapotranspiration

195

Chapter 10

ETc under various management practices

This chapter discusses various types of factors that may cause the values for Kc and ETc to

deviate from the standard values described in the Chapters 6 and 7. These factors refer to the

effects of surface mulches, intercropping, small areas of vegetation and specific cultivation

practices.

This chapter is intended to serve as a resource for situations where cultivation practices

are known to deviate from those assumed in Tables 12 and 17, but where estimates of Kc and

ETc are still necessary. This chapter is by no means exhaustive. The intent is to demonstrate

some of the procedures that can be used to make adjustments to Kc to account for deviations

from standard conditions.

EFFECTS OF SURFACE MULCHES

Mulches are frequently used in vegetable production to reduce evaporation losses from the soil

surface, to accelerate crop development in cool climates by increasing soil temperature, to

reduce erosion, or to assist in weed control. Mulches may be composed of organic plant

materials or they may be synthetic mulches consisting of plastic sheets.

Plastic mulches

Plastic mulches generally consist of thin sheets of polyethylene or a similar material placed

over the ground surface, especially along the plant rows. Holes are cut into the plastic film at

plant spacings to allow the plant vegetation to emerge. Plastic mulches can be transparent,

white or black. Colour influences albedo mainly during the early stages of the crop. However,

as the intention is to use a simple procedure for adjusting Kc for mulched crops, no distinction

is made between the different types of plastic mulches.

Plastic mulches substantially reduce the evaporation of water from the soil surface,

especially under trickle irrigation systems. Associated with the reduction in evaporation is a

general increase in transpiration from vegetation caused by the transfer of both sensible and

radiative heat from the surface of the plastic cover to adjacent vegetation. Even though the

transpiration rates under mulch may increase by an average of 10-30% over the season as

compared to using no mulch, the Kc values decrease by an average of 10-30% due to the 50-

80% reduction in soil evaporation. A summary of observed reductions in Kc, in evaporation,

and increases in transpiration over growing seasons is given in Table 25 for five horticultural

crops. Generally, crop growth rates and vegetable yields are increased by the use of plastic

mulches.

ETc under various management practices

196

TABLE 25

Approximate reductions in Kc and surface evaporation and increases in transpiration for various

horticultural crops under complete plastic mulch as compared with no mulch using trickle

irrigation

Crop Reduction1

in Kc

(%)

Reduction1 in

evaporation

(%)

Increase1 in

transpiration

(%)

Source

Squash 5-15 40-70 10-30 Safadi (1991)

Cucumber 15-20 40-60 15-30 Safadi (1991)

Cantaloupe 5-10 80 35 Battikhi and Hill (1988)

Watermelon 25-30 90 -10 Battikhi and Hill (1986),

Ghawi and Battikhi (1986)

Tomato 35 not reported not reported Haddadin and Ghawi (1983)

Average 10-30 50-80 10-30

1 Relative to using no mulch

Single crop coefficient, Kc

To consider the effects of plastic mulch on ETc, the values for Kc mid and Kc end for the

horticultural crops listed in Table 12 can be reduced by 10-30%, depending on the frequency of

irrigation (use the higher value for frequent trickle irrigation). The value for Kc ini under mulch

is often as low as 0.10. When the plastic mulch does not entirely cover the soil wetted by the

drip emitters, or where substantial rainfall occurs, then the reduction in Kc mid or Kc end will be

less, in proportion to the fraction of wet surface covered by the mulch.

Dual crop coefficient, Kcb + Ke

When estimating basal Kcb for mulched crops, less adjustment is normally needed to the Kcb

curve, being of the order of a 5-15% reduction in Kcb, as it is generally understood that the

‘basal’ evaporation of water from the soil surface is less with a plastic mulch, though the

transpiration is increased. The effect on Kcb could be greater in some situations and with some

types of low density crops. Local calibration of Kcb (and Kc) for use with mulch culture is

encouraged.

When calculating the soil evaporation coefficient Ke with plastic mulch, the fw should

represent the relative equivalent fraction of the ground surface that can contribute to

evaporation through the vent holes in the plastic cover and to the fraction of surface that is

wetted, but is not covered by the mulch. The effective area of vent holes is normally two to four

times the physical area of the vents (or even higher) to account for vapour transfer from under

the sheet.

Organic mulches

Organic mulches are often used with orchard production and with row crops under reduced

tillage operations. Organic mulches may consist of unincorporated plant residues or foreign

material imported to the field such as straw. The depth of the organic mulch and the fraction of

the soil surface covered can vary widely. These two parameters will affect the amount of

reduction in evaporation from the soil surface.

Crop evapotranspiration

197

EXAMPLE 45

Effects of surface mulch

A plastic mulch is placed over cucumbers under drip irrigation. The mulch is clear plastic covering the

entire field surface, with small openings at each plant. Adjust both the mean and basal Kc values for

this crop to reflect the presence of the mulch.

From Table 12, Kc ini, Kc mid and Kc end for fresh market cucumbers have values equal to 0.4, 1.0 and

0.75.

As the plastic mulch is continuous with only small vents at each plant, the Kc ini is assumed to be only

0.10 (this value should be adjusted upward if precipitation occurs).

The Kc mid and Kc end values are estimated as:

Kc mid = 0.85 (1.0) = 0.85

Kc end = 0.85 (0.75) = 0.64

where the 0.85 multipliers are derived from Table 25 and reflect an assumed 15% reduction in ETc due

to the mulch, assuming an approximately weekly irrigation frequency.

From Table 17, the values for Kcb ini, Kcb mid, and Kcb end are 0.15, 0.95 and 0.7 for this same cucumber

crop. The Kcb ini is assumed to be similar to the Kc ini for mulched cover and is therefore set equal to

0.10. The Kcb mid and Kcb end values are estimated to be reduced by 10% so that:

Kcb mid = 0.9 (0.95) = 0.86

Kcb end = 0.9 (0.7) = 0.63

These basal values are similar to the adjusted values for Kc. This is expected as evaporation from the

mulch covered surface can be ignored. Additional adjustment to these Kc values to account for climate

is necessary using Eq. 62 and 70.

The values for mean Kc and Kcb are similar with values of 0.10 for the initial stage, 0.85 for the mid-

season stage and 0.64 at the end of the late season stage.

Single crop coefficient, Kc

A general rule when applying Kc from Table 12 is to reduce the amount of soil water

evaporation by about 5% for each 10% of soil surface that is effectively covered by an organic

mulch.

For example, if 50% of the soil surface were covered by an organic crop residue mulch,

then the soil evaporation would be reduced by about 25%.

• In the case of Kc ini, which represents mostly evaporation from soil, one would reduce Kc

ini by about 25% in this situation.

• In the cases of Kc mid and Kc end, one would reduce these values by 25% of the difference

between (Kc mid - Kcb mid) and (Kc end - Kcb end) from Tables 12 and 17. Generally, the

differences between values in Tables 12 and 17 are only 5-10% so that the adjustment to

Kc mid and Kc end to account for an organic mulch may not be very large.

ETc under various management practices

198

Dual crop coefficient, Kcb + Ke

When applying the approach with a separate water balance of the surface soil layer, the

magnitude of the evaporation component (Ke ETo) should be reduced by about 5% for each 10%

of soil surface covered by the organic mulch. Kcb is not changed.

These recommendations are only approximate and attempt to account for the effects of

partial reflection of solar radiation from residue, microadvection of heat from residue into the

soil, lateral movement of soil water from below residue to exposed soil, and the insulating

effect of the organic cover. As these parameters can vary widely, local observations and

measurements are required if precise estimates are required.

INTERCROPPING

Intercropping refers to the situation where two different crops are grown together within one

field. For the estimation of the crop coefficient, a distinction is made between (Figure 45):

• Contiguous vegetation, where the canopies of the two crops intermingle at some height

(e.g., corn and beans intercropping);

• Overlapping crops, where the canopy of one crop is well above that of the other so that

the canopies cannot be considered to be contiguous (e.g., date trees overlapping

pomegranate trees at an oasis); and

• Border crops, where tall crops such as windbreaks border fields of shorter crops, or high

trees border a field crop.

There is an upper limit to the energy available to evaporate water. This is represented by

Kc max (Equation 72 of Chapter 7) for all crops in cultivated fields larger than 3-5 ha:

FIGURE 45

Different situations of intercropping

contiguous overlapping border

crop

Crop evapotranspiration

199

[]

{}

÷

ø

ö

ç

è

æ

+

ï

þ

ï

ý

ü

ï

î

ï

í

ì

÷

ø

ö

ç

è

æ

−−−+= 05.0K,

3

h

)45RH(004.0)2u(04.02.1maxK cb

3.0

min2maxc (72)

where h is the height for the taller crop. Under all conditions when combining crop coefficients

for multiple crops, Kc should be constrained by this upper bound (Kc ≤ Kc max).

Contiguous vegetation

Where the taller crop has canopy foliage that extends down to the same elevation as that of the

top of the shorter crop, the vegetation canopy can be considered to be contiguous. For example,

in Africa and South America, maize and beans are frequently intercropped as contiguous

vegetation, with one row of maize planted per one or more rows of beans. Another example is

the cultivation of five to seven rows of wheat intercropped with three rows of maize in many

areas of China.

Similar ground cover

Where the leaf area or fraction of ground covered by the vegetation (fc) is similar for each crop,

the Kc in Tables 12 and 17 for the taller crop (if this Kc is higher) can be taken to represent the

entire field. The taller crop will act in some sense as a clothesline so that Kc (and ETc) for the

taller crop per unit of ground area is increased over that given in Table 12 or 17. However, the

Kc (and ETc) for the shorter crop will be reduced due to the windbreak effect by the taller crop.

As a result, the Kc for the field as a whole may be similar to the weighted average of the Kc

values for the two crops from Tables 12 and 17, or, the total Kc may more closely follow the Kc

predicted for a field sown entirely to the taller crop (Kc field ≈ Kc taller crop). Yields for the shorter

crop may be reduced relative to those for single cultivar production due to the effects of

shading by the taller crop and the competition for soil water.

Different ground cover

Where the fractions of ground covered by each crop are different, the Kc for an intercropped

field can be estimated by weighting the Kc values for the individual crops according to the

fraction of area covered by each crop and by the height of the crop:

2211

c222c111

fieldc hf+hf

Khf+Khf

K= (104)

where f1 and f2 are the fractions of the field surface planted to crops 1 and 2, h1 and h2 are the

heights of crops 1 and 2, and Kc1 and Kc2 are the Kc values for crops 1 and 2.

Overlapping vegetation

Where intercropping entails overlapping of spacings, the canopy of one crop is well above the

other. This is the case, for example in southern California, where citrus trees are planted in date

palm groves. Where a normal dense spacing is used for both the dates and for the citrus trees,

the Kc may increase as the density of the combined vegetation increases, proportional to the

increase in LAI (Example 47), with maximum Kc constrained by either Kc max (Equation 72) or

by Kcb full (Equations 99 and 100) unless the total field area is small so that there is an additional

clothesline or oasis effect as discussed in the next section.

ETc under various management practices

200

EXAMPLE 46

Intercropped maize and beans

Determine the representative Kc mid for a situation where a single 1 m wide row of maize is grown for

each 2 m of squash, where RHmin ≈ 45% and u2 ≈ 2 m/s.

From Table 12, the Kc mid and h for maize is 1.20 and 2 m and the Kc mid and h for squash is 0.95 and

0.3 m. No correction is needed for climate. The representative Kc mid can be obtained by weighting the

individual Kc mid values according to the fraction of the field surface allocated to each crop (f1 ≈ 0.3 for

maize and f2 ≈ 0.7 for squash) and according to the heights of the crops as (Eq. 104):

14.1

)3.0(70.0+)2(30.0

)95.(0)3.0(70.0+)20.1()2(30.0

Kmidc ==

Values can be obtained for daily Kc in a similar manner by constructing individual Kc curves and then

weighting interpolated values from the individual Kc curves for any specific day using Eq. 104.

The crop coefficient for the mid-season and entire field is 1.14.

Border crops

Where tall crops such as windbreaks or date palms border fields of shorter crops, the upper

storey of the taller crop can intercept extra sensible heat energy from the air stream. Under

these conditions, the Kc is weighted according to the areas for each crop. However, prior to the

weighting, the Kc for the border crop, if taller than the field (interior) crop, should be adjusted

for potential clothesline impact (next section).

SMALL AREAS OF VEGETATION

Natural vegetation and some subsistence agriculture frequently occurs in small groups or stands

of plants. The value for Kc for these small stands depends on the type and condition of other

vegetation surrounding the small stand.

Areas surrounded by vegetation having similar roughness and moisture conditions

In the majority of cases for natural vegetation or for non-pristine agricultural vegetation, the

value for Kc must adhere to upper limits for Kc of approximately 1.20-1.40, when the areal

expanse of the vegetation is larger than about 2 000 m2. This is required as ET from large areas

of vegetation is governed by one-dimensional energy exchange principles and by the principle

of conservation of energy. ET from small stands (< 2 000 m2) will adhere to these same

principles and limits only where the vegetation height, leaf area, and soil water availability are

similar to that of the surrounding vegetation.

Crop evapotranspiration

201

EXAMPLE 47

Overlapping vegetation

A 20 ha date palm grove in Palm Desert, California, the United States has a tree spacing of 6 m.

Interplanted among the rows of palms are small orange trees (50% canopy) on a 6 m spacing. The

palm and citrus trees are 3 m from one another in the rows. Height of the palms is 10 m and height of

the citrus is 3 m. The canopy foliage of the palms is well above that of the citrus so that the canopies

cannot be considered contiguous. Mean average RHmin during the mid-season is 20% and u2 = 2 m/s.

The Kc mid from Table 12 for dates is 0.95 and when adjusted for humidity and wind using Eq. 62 is Kc

mid = 1.09. The Kc mid from Table 12 for citrus having 50% canopy with no ground cover is 0.60 and

when adjusted for humidity and wind using Eq. 62 is Kc mid = 0.70.

The interplanting of citrus among the date palms has increased the total leaf area of the orchard so

that ETc for the combined planting (palms and citrus together) will be greater than for either planting

alone. The estimated combined Kc mid will be estimated as a function of the increase in total LAI. First

the LAI values of the individual plantings are estimated by inverting Eq. 97 to solve for LAI:

ú

û

ù

ê

ë

é

−

−−=

mincfullcb

minccb

KK

1ln4.1LAI

where Kc min is the minimum basal Kc for bare soil (≈ 0.15 to 0.20) and Kcb full is the maximum mid-

season Kc expected for the crop if there were complete ground cover, calculated using Eq. 99. Based

on Eq. 99, with h = 10 m for the date palms and h = 3 m for the citrus, the Kcb full values for the two

crops, assuming complete ground cover for each, are Kcb full = 1.34 for palms and Kcb full = 1.30 for

citrus (using RHmin = 20% and u2 = 2 m/s). These estimates ignore effects of any unique stomatal

control. Therefore, using the above equation, the effective LAI values of the date palms and citrus are

estimated to be approximately:

LAIpalms = - 1.4 ln[1-(1.09-0.15)/(1.34-0.15)] = 2.2

LAIcitrus = - 1.4 ln[1-(0.70-0.15)/(1.30-0.15)] = 0.9

Therefore, the effective LAI for the date palm-citrus combination is estimated to be approximately

LAIcombined = LAIpalms + LAIcitrus = 2.2 + 0.9 = 3.1.

The increase in Kc mid for the date palm orchard resulting from the increase in LAI from the interplanting

of citrus is estimated using a ratio of the LAI-based function introduced in Eq. 97. This results in the

relationship:

()

ú

û

ù

ê

ë

é

−−

=

cropsingle

combined

cropsingle,midcmid c LAI0.7exp1

LAI0.7exp1

KK

where LAIcombined is the LAI for the two intercropped plantings combined and LAIsingle crop is the LAI for

the taller, single crop. Kc mid single crop is the mid-season Kc for the taller, single crop (in this case the date

palms). In this application, the above equation is solved as:

()

23.1

)2.2(0.7exp1

)1.3(0.7exp1

09.1K mid c =

ú

û

ù

ê

ë

é

−−

=

Therefore, the Kc mid estimated for the complex of date palms and citrus together is 1.23. This value is

compared with the maximum expected Kc based on energy limitations, represented by Kc max of Eq. 72

which in this case for h=10 m is Kc max = 1.34. Because Kc mid < Kc max (i.e., 1.23 < 1.34), the Kc mid =

1.23 is accepted as the approximate estimate of the Kc mid for the intercropped field.

ETc under various management practices

202

Clothesline and oasis effects

Under the clothesline effect, where vegetation height is greater than that of the surroundings

(different roughness conditions), or under the oasis effect, where vegetation has higher soil

water availability than the surroundings (different moisture conditions), the peak Kc values may

exceed the 1.20-1.40 limit. The user should exercise caution when extrapolating ET

measurements taken from these sorts of vegetation stands or plots to larger stands or regions as

an overestimation of regional ET may occur.

Small expanses of tall vegetation that are surrounded by shorter cover can exhibit a

clothesline effect. This occurs where turbulent transport of sensible heat into the canopy and

transport of vapour away from the canopy is increased by the ‘broadsiding’ of wind

horizontally into the taller vegetation. In addition, the internal boundary layer above the

vegetation may not be in equilibrium with the new surface. Therefore, ET from the isolated

expanses, on a per unit area basis, may be significantly greater than the corresponding ETo

computed for the grass reference. Examples of the clothesline or oasis effects would be ET

from a single row of trees surrounded by short vegetation or surrounded by a dry non-cropped

field, or ET from a narrow strip of cattails (a hydrophytic vegetation) along a stream channel.

Kc values up to and exceeding two have been recorded for such situations.

Where ET estimates are needed for such small, isolated expanses of vegetation

surrounded by shorter cover (clothesline effect) or dry land (oasis effect), then the Kc may

exceed the grass reference by 100% or more. Estimates of Kc for the expanses of vegetation

should contain u2, RHmin and h parameters to adjust Kc values, and parameters expressing the

aridity of the surrounding area, the general width of the vegetation stand and the ability of the

wind to penetrate into the vegetation. The equation should also consider the LAI of the

vegetation to account for the ability of the vegetation to conduct and transpire the amount of

water demanded by the clothesline/climatic condition. An upper limit of 2.5 is usually placed

on Kc to represent an upper limit on the stomatal capacity of the vegetation to supply water

vapour to the air stream under the clothesline or oasis conditions. For vegetation with a great

leaf resistance, such as for some types of desert vegetation or trees, the upper limit should be

multiplied by a resistance correction factor, Fr, calculated in Chapter 9 using Equation 102.

Figure 46 presents example curves of Kc for small areas of vegetation versus vegetation

stand width, for conditions where u2 = 2 m/s, RHmin = 30%, vegetation height = 2 m, and LAI =

3. The upper curve represents conditions where the specific vegetation is surrounded by dead

vegetation, dry bare soil, or even gravel or asphalt. In this situation, large amounts of sensible

heat are generated from the surrounding area due to the lack of ET. Some of this sensible heat

is advected into the vegetation downwind. The lower curve represents conditions where the

vegetation is surrounded by well-watered grass. In this situation, there is much less sensible

heat available from the surrounding area to increase ET (and Kc) of vegetation downwind. The

influence of the aridity of the surroundings on the Kc for a small expanse is apparent. The two

curves shown will change with changes in u2, RHmin, h, and LAI. The user is cautioned that

Figure 46 provides only general estimates of Kc under clothesline and oasis conditions. These

estimates should be verified where possible using valid local measurements.

ET estimates from large expanses of vegetation or from small expanses of vegetation that

are surrounded by mixtures of other vegetation having similar roughness and moisture

conditions should almost always be less than or equal to 1.4 ETo, even under arid conditions.

Crop evapotranspiration

203

For tall wind breaks, such as single rows of trees, an approximate estimate for Kc is:

÷

ø

ö

ç

è

æ

+= 5.2,

Width

canopy

h

r

F

2.1min

c

K (105)

where Fr stomatal resistance correction factor (Equation 102)

h

canopy mean vertical height of canopy area [m]

Width width (horizontal thickness) of the windbreak [m]

The Kc = 2.5 limit imposed in Equation 105 represents an approximate upper limit on

ETc of trees per unit ground area. However, this value has large uncertainty. Because

availability of soil water may limit evapotranspiration from wind breaks, a soil water balance

and calculation of the Ks stress factor should be conducted.

MANAGEMENT INDUCED ENVIRONMENTAL STRESS

Many agricultural crops are intentionally water stressed during specific crop growth periods to

encourage particular crop characteristics. The water stress is initiated by withholding or by

reducing irrigations. In situations where this type of cultural management is practised, the Kc

should be reduced to account for the reduction in evapotranspiration.

Environmental stress from soil water shortage, low soil fertility, or soil salinity can cause

some types of plants to accelerate their reproductive cycle. In these situations, the length of the

growing season may be shortened, particularly the mid-season period. Stress during the

FIGURE 46

Kc curves for small areas of vegetation under the oasis effect as a function of the width of the

expanse of vegetation for conditions where RHmin = 30%, u2= 2 m/s, vegetation height (h) = 2 m

and LAI = 3

K

c

dry surroundings

surrounded by well-watered grass

upper limit on K c

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 50 100 150 200 250 300

width of ve

g

etation stand

(

m

)

ETc under various management practices

204

development period may increase the length of that period. Therefore, the length of the mid-

season, Lmid, and perhaps the lengths of the development and late seasons may need to be

adjusted for environmentally stressed or damaged vegetation. Local research and observation is

critical to identify the magnitudes and extent of these adjustments. Some examples of

modifications to Kc and to lengths of growing periods are described below.

Alfalfa seed

Some forage crops such as alfalfa that are grown for seed production are intentionally water

stressed to reduce the amount of vegetation and to encourage increased production of flowers

and seed. In areas subject to freezing winters, the reduction in Kc for deep rooted crops such as

alfalfa depends upon the amount of water made available from precipitation during the dormant

(winter) season and upon the amount of rainfall and limited irrigation during the growing

season. Therefore, the effects of the intentional stress on the values for Kc should be modelled

using the basal crop procedure presented in Chapter 7 and the Ks coefficient and water balance

procedure presented in Chapter 8.

Cotton

In cotton production, soil water stress may be initiated during the development period to delay

flower development and to encourage boll development. This practice retards the growth rate of

the cotton plant and delays the date of full cover. For cotton, the attainment of full cover and

the beginning of the mid-season generally occurs when the LAI reaches approximately three.

When soil water stress and growth retardation is practised, full cover may occur after the

beginning of flowering. The effect of stress during the development period on ETc can be

incorporated into the Kc curve by extending the length of the development period into the mid-

season period. The length of the total season generally remains the same.

Sugar beets

Sugar beets are frequently managed to initiate mild soil water stress during the late season

period to dehydrate roots and concentrate sugars. A terminal irrigation may be needed just prior

to harvest to assist in root extraction. When this type of water stress is practised, the value for

Kc end is reduced from 1.0 to 0.6 (Table 12, Footnote 5).

Coffee

Coffee plants are often intentionally water stressed to reduce vegetation growth and to

encourage development of berries. Under these conditions, Kc values from Table 12 should be

reduced. In addition, coffee fields may be bordered by trees that serve as windbreaks. The

effect of windbreaks is to reduce the Kc of the coffee plants due to a reduction in wind and solar

radiation over the plants. The reduction in Kc could be significant where windbreaks are tall

and frequent. However, the Kc for the entire field area, including the windbreaks, may be

increased by the presence of the trees, relative to the values for Kc for coffee shown in

Table 12, due to increased total leaf area of the coffee-tree combination and the increased

aerodynamic roughness.

Tea

Initiation and development of new leaves on tea plants often occurs following the start of the

rainy season. During the dry season, initiation of new leaves is slow or non-existent. The

transpiration from older leaves is lower than for new leaves due to effects of leaf age on

Crop evapotranspiration

205

stomatal conductance. Therefore, the Kc, when leaves have aged (more than 2-3 months old),

will be perhaps 10-20% lower than shown in Tables 12 and 17. Similar to coffee, tea fields may

be bordered by trees that serve as windbreaks. The effect of windbreaks is to reduce the Kc of

the tea plants, but to potentially increase the Kc for the entire plantation, as described for

coffee.

Olives

Growers may increase spacings of olive trees under rainfed conditions in areas with less

rainfall. This is done to increase the ground area per tree that contributes infiltrated rain to

transpiration of the tree. For example, in Tunisia, the spacing of olive trees changes from the

north to the south, in proportion to annual rainfall. The tree spacing influences the Kc for the

crop (Example 43).

ETc under various management practices

206

Crop evapotranspiration

207

Chapter 11

ETc during non-growing periods

This chapter describes procedures for predicting ETc during non-growing periods. Non-

growing periods are defined as periods during which no agricultural crop has been planted. In

temperate climates, non-growing periods may include periods of frost and continuously frozen

conditions.

TYPES OF SURFACE CONDITIONS

The type and condition of the ground surface during non-growing periods dictates the range for

ETc. Where the surface is bare soil, then the Kc will be quite similar to the Kc ini predicted in

Chapter 6. Where the surface is covered by nearly dead vegetation or some type of organic

mulch or crop residue, then the Kc will be similar to that for agriculture that uses a surface

mulch. Where the surface is covered by weed growth or growth of ‘volunteer’ plants, then the

Kc will vary according to the leaf area or fraction of ground covered by the vegetation and by

the availability of soil water. Where the surface is snow covered or frozen, then the Kc is

difficult to predict and a constant value for ETc may have to be assumed.

Bare soil

Single crop coefficient

Where the ground is left mostly bare following harvest, then the Kc following harvest will be

strongly influenced by the frequency and amount of precipitation. Kc for bare soil can be

calculated as Kc = Kc ini where Kc ini is calculated using the procedure of Chapter 6.

Dual crop coefficient

Where a daily soil water balance can be applied, the user may elect to apply the dual Kc

approach of Chapter 7. In this situation, the topsoil layer may dry to very low water contents

during periods having no precipitation. Therefore, the values for Kcb and for Kc min in

Equations 71 and 76 should be set equal to zero. This provides for the opportunity to predict

ETc = 0 during long periods having no rainfall. This is necessary to preserve the water balance

of the evaporation layer and of the root zone in total. The daily water balance calculation, given

Kcb = 0, will provide the most accurate estimates of ETc during the non-growing periods.

Surface covered with dead vegetation

Single crop coefficient

Where the ground surface has a plant residue or other dead organic mulch cover, or where part

of the unharvested crop remains suspended above the surface in a dead or senesced condition,

then the surface will respond similarly to a surface covered by mulch. In this case, Kc can be

ETc during non-growing periods

208

set equal to Kc ini as predicted from figures 29 and 30, but the value for Kc ini can be reduced

by about 5% for each 10% of soil surface that is effectively covered by an organic mulch.

Dual crop coefficient

Evaporation from dead, wet vegetation can be substantial for a few days following a

precipitation event. Therefore, in the dual Kc approach, the value for fc should be set equal to

zero to reflect the lack of green cover and fw should be set equal to 1.0 to reflect the wetting of

both soil and mulch cover by precipitation.

The dead mulch or vegetation will dry more quickly than would the underlying soil if it

were exposed. In addition, the soil will be protected somewhat from evaporation by the dead

mulch or vegetation cover. Therefore, total evaporation losses will be less than the TEW

predicted from Equation 73. This can be accounted for by reducing the value for TEW by 5%

for each 10% of soil surface that is effectively covered by an organic mulch. The value for

REW should be limited to less than or equal to that for TEW.

Surface covered with live vegetation

During frost-free periods following harvest, weeds may begin to germinate and grow. This

vegetation is supplied with water from storage in the soil profile and from any rainfall. In

addition, crop seed lost during harvest may germinate following rainfall events and add to the

ground cover. The amount of ground surface covered by vegetation will depend on the severity

of weed infestation; the density of the volunteer crop; the frequency and extent of soil tillage;

the availability of soil water or rain, and any damage by frost.

The value for Kcb during the non-growing period can be predicted over time according to

the amount of vegetation covering the surface. This can be done through estimates of LAI using

Equation 97 or estimates of the fraction of ground cover, fc, using Equation 98.

Single crop coefficient

In the single crop coefficient approach, the value for Kcb determined using procedures in

Chapter 9 can be converted into an equivalent Kc by adding 0.05 to 0.15 according to the

frequency of rainfall and surface wetting.

It is important that the Kc for vegetation during the non-growing period be limited

according to the amount of soil water available to supply evapotranspiration. Otherwise, the

law of conservation of mass will be violated. Under all conditions, the integration of Kc ETo

over the course of the non-growing period cannot exceed the sum of the precipitation occurring

during the period plus any residual soil water in the root zone following harvest that can be

depleted by the subsequent vegetation. The root zone in this case is the root zone for the weed

or volunteer crops. A daily soil water balance may provide for the best estimate of soil water

induced stress and associated reduction in Kc and ETc.

Dual crop coefficient

Under the dual crop coefficient approach, Kcb can be predicted according to the amount of

surface that is covered by vegetation using Equation 97 or 98. Then, a full daily soil water

Crop evapotranspiration

209

balance of the topsoil together with a full daily soil water balance of the root zone can be

employed as described in Chapter 7. The soil water balances will automatically adhere to the

law of conservation of mass, so that total ETc from the weed or volunteer vegetation will not be

overestimated. Again, because the topsoil layer may dry to below wilting point under

conditions of sparse rainfall, the values for Kcb and Kc min used in Equations 71 and 76 should

be set equal to zero. In this manner, the daily soil water balance with dual Kc calculations can

progress throughout the non-growing period with good results.

Frozen or snow covered surfaces

Where the ground surface is snow covered or frozen, any vegetation will be largely non-

responsive and non-contributing to ETc, and the amount of ETc will be closely related to the

availability of free water at the surface and to the albedo of the surface.

The albedo of snow covered surfaces can range from 0.40 for old, dirty snow cover to

0.90 for fresh, dry snow. Therefore, the ETc for snow cover will be less than ETo for grass, as

25-85% less shortwave energy is available. In addition, some energy must be used to melt the

snow before evaporation.

The use of ETo under such conditions is of limited value, as the assumption of conditions

sustaining a green grass cover is violated. It is even possible to obtain negative values for ETo

on some winter days where the longwave radiation from the surface is large and the vapour

pressure deficit is small. It is under these conditions that net condensation of water from the

atmosphere is possible. This would be similar to negative evaporation.

Given the limited value of ETo (or even ETp) under snow covered or frozen conditions, a

single, average value may be best used to predict ETc. Wright (1993) found that ETc averaged

1 mm/day over winter periods at Kimberly, Idaho, the United States, that were six months long

(1 October to 30 March). The latitude of Kimberly is 42°N and the elevation is about 1 200 m.

Over the six-year study period, the ground was 50% covered by snow for 25% of the time from

1 October to 30 March. The ground, when exposed, was frozen about 50% of the time. The Kc

averaged 0.25 during periods when the soil was not frozen but where frosts were occurring

(October and early November). When the ground had 50% snow cover or greater, the ETc

averaged only 0.4 mm/day. Wright found that over the six-month non-growing period, total

cumulative ETc exceeded precipitation by about 50 mm.

Figure 47 shows the mean measurements of ETc during the 1985-1991 study period. The

measurements have high correspondence to the total shortwave radiation energy available on a

clear day, Rso, estimated as 0.75 Ra. There is some lag between ETc and Rso and Rs caused by

cooler temperatures in January – March as compared to the October – December period. The

ETc/Rso ratio averaged only 0.17 over the six-month period, and averaged 0.11 from 1 Dec. –

10 Mar. The ETc/Rs ratio averaged 0.23 over the six-month period, and averaged 0.15 from 1

Dec. – 10 Mar.

A similar study conducted in Logan, Utah, the United States (latitude 41.6°N, elevation

1 350 m) over an eight-year period showed that ETc varied widely with soil surface wetness

and air temperature during the winter months. The ‘average’ Kc from November to March was

0.5 for days having no snow cover. For days with snow cover, ‘ETc’ ranged from 0 to 1.5

mm/day. Similarly, Kc is about 0.4 for winter wheat during frozen periods in the region of

northern China (latitude near 39°N.

ETc during non-growing periods

210

Single Crop Coefficient

The above procedure can provide estimates for the single Kc during non-growing season

periods having snow cover or freezing conditions. However, the actual value for Kc is known to

vary widely and will be less when water is less available at the soil surface.

Dual Crop Coefficient

A daily soil water balance using the dual crop coefficient approach is necessary to accurately

predict ETc under freezing and snow cover conditions. In the dual crop coefficient method, a

daily water balance is conducted for the topsoil and the estimate for Kc can be reduced

according to available water. However, in addition to the limited validity of the concept of ETo

under frozen or snow covered conditions, the evaporation coefficient, Ke, may be reduced

when the ground surface is frozen, as the water in a frozen state is less available.

Other, more complex models for predicting ETc under non-growing season conditions,

snow cover, and freezing, are available in the literature and should be consulted and perhaps

applied when precise estimates for ETc are needed. Some of these are listed in section K of the

References.

FIGURE 47

Mean evapotranspiration measured during non-growing, winter periods at Kimberly, Idaho,

United States by Wright (1993)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0

2

4

6

8

10

ET , mm/day and ratios

Solar radiation (R ), mm/day

c

so

ETc

ET /R

cso

Rso

280 300 320 340 360 20 40 60 80

day of the year

Rs

ET /R

cs

Crop evapotranspiration

211

Annex 1

Units and symbols

PREFIXES

Units can be used as such or in multiples:

Tera (T) and which is 1012 Centi (c) 10

-2

Giga (G) 109 Milli (m) 10-3

Mega (M) 106 Micro (µ) 10-6

Kilo (k) 103 Nano (n) 10-9

Hecto (h) 102 Pico (p) 10-12

Deca (da) 101 Femto (f) 10

-15

Deci (d) 10-1 Atto (a) 10

-18

TEMPERATURE

Standard unit: degree Celsius (°C)

_________________________________________________________

degree Fahrenheit (°F) (°C) = (°F-32) 5/9

Kelvin (K) 1 K = (°C) + 273.16

PRESSURE

(air pressure, vapour pressure)

Standard unit: kilopascal (kPa)

_______________________________________________________________

millibar (mbar) 1 mbar = 0.1 kPa

bar 1 bar = 100 kPa

centimetre of water (cm) 1 cm of water = 0.09807 kPa

millimetre of mercury (mmHg) 1 mmHg = 0.1333 kPa

atmospheres (atm) 1 atm = 101.325 kPa

pound per square inch (psi) 1 psi = 6.896 kPa

Annex I: Units and symbols

212

WIND SPEED

Standard unit: metre per second (m s-1)

________________________________________________________________

kilometre per day (km day-1) 1 km day-1 = 0.01157 m s-1

nautical mile/hour (knot) 1 knot = 0.5144 m s-1

foot per second (ft s-1) 1 ft/s = 0.3048 m s-1

RADIATION

Standard unit: megajoule per square metre and per day (MJ m-2 day-1)

or as equivalent evaporation in mm per day (mm day-1)

____________________________________________________________________

equivalent evaporation (mm/day) 1 mm day-1 = 2.45 MJ m-2 day-1

joule per cm2 per day (J cm-2 day-1) 1 J cm-2 day-1 = 0.01 MJ m-2 day-1

calorie per cm2 per day (cal cm-2 day-1) 1 cal = 4.1868 J = 4.1868 10-6 MJ

1 cal cm-2 day-1 = 4.1868 10-2 MJ m-2 day-1

watt per m2 (W m-2) 1 W = 1 J s-1

1 W m-2 = 0.0864 MJ m-2 day-1

EVAPOTRANSPIRATION

Standard unit: millimetre per day (mm day-1)

__________________________________________________________________

m3 per hectare per day (m3 ha-1 day-1) 1 m3 ha-1 day-1 = 0.1 mm day-1

litre per second per hectare (l s-1 ha-1) 1 l s-1 ha-1 = 8.640 mm day-1

equivalent radiation in megajoules per

square metre per day (MJ m-2 day-1) 1 MJ m-2 day-1 = 0.408 mm day-1

Crop evapotranspiration

213

Annex 2

Meteorological tables

2.1 Atmospheric pressure (P) for different elevations above sea level (z)

2.2 Psychrometric constant (γ) for different elevations above sea level (z)

2.3 Saturation vapour pressure (eo(T)) for different temperatures (T)

2.4 Slope of vapour pressure curve (∆) for different temperatures (T)

2.5 Number of the day in the year (J)

2.6 Daily extraterrestrial radiation (Ra) for different latitudes

2.7 Mean daylight hours (N) for different latitudes

2.8 σTK4 (Stefan-Boltzmann law) at different temperatures (T)

2.9 Conversion factors to convert wind speed measured at given height to wind speed

measured at standard height of 2 m above ground surface

TABLE 2.1

Atmospheric pressure (P) for different altitudes (z)

÷

ø

ö

ç

è

æ−

=293

z0065.0293

3.101P

26.5

(Eq. 7)

Z

(m)

P

(kPa)

z

(m)

P

(kPa)

z

(m)

P

(kPa)

z

(m)

P

(kPa)

0

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

850

900

950

1000

101.3

100.7

100.1

99.5

99.0

98.4

97.8

97.2

96.7

96.1

95.5

95.0

94.4

93.8

93.3

92.7

92.2

91.6

91.1

90.6

90.0

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1500

1550

1600

1650

1700

1750

1800

1850

1900

1950

2000

90.0

89.5

89.0

88.4

87.9

87.4

86.8

86.3

85.8

85.3

84.8

84.3

83.8

83.3

82.8

82.3

81.8

81.3

80.8

80.3

79.8

2000

2050

2100

2150

2200

2250

2300

2350

2400

2450

2500

2550

2600

2650

2700

2750

2800

2850

2900

2950

3000

79.8

79.3

78.8

78.3

77.9

77.4

76.9

76.4

76.0

75.5

75.0

74.6

74.1

73.7

73.2

72.7

72.3

71.8

71.4

71.0

70.5

3000

3050

3100

3150

3200

3250

3300

3350

3400

3450

3500

3550

3600

3650

3700

3750

3800

3850

3900

3950

4000

70.5

70.1

69.6

69.2

68.8

68.3

67.9

67.5

67.1

66.6

66.2

65.8

65.4

65.0

64.6

64.1

63.7

63.3

62.9

62.5

62.1

Annex 2: Meteorological tables

214

TABLE 2.2

Psychrometric constant (γ

γγ

γ) for different altitudes (z)

P10x665.0

Pc 3

p−

=

λε

=γ (Eq. 8)

Z

(m)

γ

γγ

γ

kPa/°C

z

(m)

γ

γγ

γ

kPa/°C

z

(m)

γ

γγ

γ

kPa/°C

z

(m)

γ

γγ

γ

kPa/°C

0

100

200

300

400

500

600

700

800

900

1000

0.067

0.066

0.065

0.064

0.063

0.062

0.061

0.060

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

0.060

0.059

0.058

0.057

0.056

0.055

0.054

0.053

2000

2100

2200

2300

2400

2500

2600

2700

2800

2900

3000

0.053

0.052

0.051

0.050

0.049

0.048

0.047

3000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

0.047

0.046

0.045

0.044

0.043

0.042

0.041

Based on λ= 2.45 MJ kg-1 at 20°C.

Crop evapotranspiration

215

TABLE 2.3

Saturation vapour pressure (eo(T)) for different temperatures (T)

ú

û

ù

ê

ë

é

+

=3.237T

T27.17

exp6108.0)T(eo (Eq. 11)

T

°C

es

kPa

T

°C

e°(T)

kPa

T

°C

e°(T)

kPa

T

°C

es

kPa

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

0.657

0.681

0.706

0.731

0.758

0.785

0.813

0.842

0.872

0.903

0.935

0.968

1.002

1.037

1.073

1.110

1.148

1.187

1.228

1.270

1.313

1.357

1.403

1.449

13.0

13.5

14.0

14.5

15.0

15.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

23.5

24.0

24.5

1.498

1.547

1.599

1.651

1.705

1.761

1.818

1.877

1.938

2.000

2.064

2.130

2.197

2.267

2.338

2.412

2.487

2.564

2.644

2.726

2.809

2.896

2.984

3.075

25.0

25.5

26.0

26.5

27.0

27.5

28.0

28.5

29.0

29.5

30.0

30.5

31.0

31.5

32.0

32.5

33.0

33.5

34.0

34.5

35.0

35.5

36.0

36.5

3.168

3.263

3.361

3.462

3.565

3.671

3.780

3.891

4.006

4.123

4.243

4.366

4.493

4.622

4.755

4.891

5.030

5.173

5.319

5.469

5.623

5.780

5.941

6.106

37.0

37.5

38.0

38.5

39.0

39.5

40.0

40.5

41.0

41.5

42.0

42.5

43.0

43.5

44.0

44.5

45.0

45.5

46.0

46.5

47.0

47.5

48.0

48.5

6.275

6.448

6.625

6.806

6.991

7.181

7.376

7.574

7.778

7.986

8.199

8.417

8.640

8.867

9.101

9.339

9.582

9.832

10.086

10.347

10.613

10.885

11.163

11.447

Annex 2: Meteorological tables

216

TABLE 2.4

Slope of vapour pressure curve (∆

∆∆

∆) for different temperatures (T)

2

)37.32+(T

237.3+T

T17.27

exp0.61084098 ú

û

ù

ê

ë

é

÷

ø

ö

ç

è

æ

=∆ (Eq. 13)

T

°C

∆

∆∆

∆

kPa/°C

T

°C

∆

∆∆

∆

kPa/°C

T

°C

∆

∆∆

∆

kPa/°C

T

°C

∆

∆∆

∆

kPa/°C

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

0.047

0.049

0.050

0.052

0.054

0.055

0.057

0.059

0.061

0.063

0.065

0.067

0.069

0.071

0.073

0.075

0.078

0.080

0.082

0.085

0.087

0.090

0.092

0.095

13.0

13.5

14.0

14.5

15.0

15.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

23.5

24.0

24.5

0.098

0.101

0.104

0.107

0.110

0.113

0.116

0.119

0.123

0.126

0.130

0.133

0.137

0.141

0.145

0.149

0.153

0.157

0.161

0.165

0.170

0.174

0.179

0.184

25.0

25.5

26.0

26.5

27.0

27.5

28.0

28.5

29.0

29.5

30.0

30.5

31.0

31.5

32.0

32.5

33.0

33.5

34.0

34.5

35.0

35.5

36.0

36.5

0.189

0.194

0.199

0.204

0.209

0.215

0.220

0.226

0.231

0.237

0.243

0.249

0.256

0.262

0.269

0.275

0.282

0.289

0.296

0.303

0.311

0.318

0.326

0.334

37.0

37.5

38.0

38.5

39.0

39.5

40.0

40.5

41.0

41.5

42.0

42.5

43.0

43.5

44.0

44.5

45.0

45.5

46.0

46.5

47.0

47.5

48.0

48.5

0.342

0.350

0.358

0.367

0.375

0.384

0.393

0.402

0.412

0.421

0.431

0.441

0.451

0.461

0.471

0.482

0.493

0.504

0.515

0.526

0.538

0.550

0.562

0.574

Crop evapotranspiration

217

TABLE 2.5

Number of the day in the year (J)

Day

January

February

March*

April*

May*

June*

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

(60)

-

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

-

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

-

ΤΑΒΛΕ 2.

ΤΑΒΛΕ 2.ΤΑΒΛΕ 2.

ΤΑΒΛΕ 2.





add 1 if leap year

J can be determined for each day (D) of month (M) by

J = INTEGER(275 M/9 – 30 + D) – 2

IF (M < 3) THEN J = J + 2

also, IF (leap year and (M > 2)) THEN J = J + 1

For ten-day calculations, compute J for day D = 5, 15 and 25

For monthly calculations, J at the middle of the month is approximately given by

J = INTEGER(30.4 M – 15)

Annex 2: Meteorological tables

218

TABLE 2.5 (continued)

Number of the day in the year (J)

Day

July*

August*

September*

October*

November*

December*

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

-

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

-

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

* add 1 if leap year

Crop evapotranspiration

219

Annex 2: Meteorological tables

220

Crop evapotranspiration

221

TABLE 2.8

σ

σσ

σTK4 (Stefan-Boltzmann law) at different temperatures (T)

With σ = 4.903 10-9 MJ K-4 m-2 day-1

and TK = T[°C] + 273.16

T

(°C)

σTK4

(MJ m-2 d-1)

T

(°C)

σTK4

(MJ m-2 d-1)

T

(°C)

σTK4

(MJ m-2 d-1)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

14.5

15.0

15.5

16.0

16.5

27.70

27.90

28.11

28.31

28.52

28.72

28.93

29.14

29.35

29.56

29.78

29.99

30.21

30.42

30.64

30.86

31.08

31.30

31.52

31.74

31.97

32.19

32.42

32.65

32.88

33.11

33.34

33.57

33.81

34.04

34.28

34.52

17.0

17.5

18.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

23.5

24.0

24.5

25.0

25.5

26.0

26.5

27.0

27.5

28.0

28.5

29.0

29.5

30.0

30.5

31.0

31.5

32.0

32.5

34.75

34.99

35.24

35.48

35.72

35.97

36.21

36.46

36.71

36.96

37.21

37.47

37.72

37.98

38.23

38.49

38.75

39.01

39.27

39.53

39.80

40.06

40.33

40.60

40.87

41.14

41.41

41.69

41.96

42.24

42.52

42.80

33.0

33.5

34.0

34.5

35.0

35.5

36.0

36.5

37.0

37.5

38.0

38.5

39.0

39.5

40.0

40.5

41.0

41.5

42.0

42.5

43.0

43.5

44.0

44.5

45.0

45.5

46.0

46.5

47.0

47.5

48.0

48.5

43.08

43.36

43.64

43.93

44.21

44.50

44.79

45.08

45.37

45.67

45.96

46.26

46.56

46.85

47.15

47.46

47.76

48.06

48.37

48.68

48.99

49.30

49.61

49.92

50.24

50.56

50.87

51.19

51.51

51.84

52.16

52.49

Annex 2: Meteorological tables

222

TABLE 2.9

Conversion factors to convert wind speed measured at given height (over grass) to wind speed

measured at standard height of 2 m above ground surface

)42.5z8.67ln(

87.4

factorconversion −

= (Eq. 47)

z

height

(m)

con-

version

factor

z

height

(m)

con-

version

factor

z

height

(m)

con-

version

factor

z

height

(m)

con-

version

factor

-

1.0

1.2

1.4

1.6

1.8

2.0

-

1.178

1.125

1.084

1.051

1.023

1.000

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

0.980

0.963

0.947

0.933

0.921

0.910

0.899

0.889

0.881

0.872

4.2

4.4

4.6

4.8

5.0

5.2

5.4

5.6

5.8

6.0

0.865

0.857

0.851

0.844

0.838

0.833

0.827

0.822

0.817

0.812

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

0.812

0.802

0.792

0.783

0.775

0.767

0.760

0.754

0.748

0.742

Crop evapotranspiration

223

Annex 3

Background on physical parameters used in

evapotranspiration computations

Latent Heat of Vaporization (λ

λλ

λ)1

T ) 10 x(2.361 - 2.501 = 3−

λ (3-1)

where: λ latent heat of vaporization [MJ kg-1]

T air temperature [°C]

The value of the latent heat varies only slightly over normal temperature ranges. A single

value may be taken (for T = 20 °C): λ = 2.45 MJ kg-1.

Atmospheric Pressure (P)2

÷

ø

ö

ç

è

æαα

T

)z-(z- T R

g

P = P

Ko

o1Ko 1

o (3-2)

where: P atmospheric pressure at elevation z [kPa]

P

o atmospheric pressure at sea level = 101.3 [kPa]

z elevation [m]

zo elevation at reference level [m]

g gravitational acceleration = 9.807 [m s-2]

R specific gas constant = 287 [J kg-1 K-1]

al constant lapse rate moist air = 0.0065 [K m-1]

TKo reference temperature [K] at elevation zo given by

T + 273.16 = TKo (3-3)

where: T mean air temperature for the time period of calculation [oC]

1 Reference: Harrison (1963)

2 Reference: Burman et al. (1987)

Annex 3: Background on physical parameters used in evapotranspiration computations

224

When assuming Po = 101.3 [kPa] at zo = 0, and TKo = 293 [K] for T = 20 [°C],

equation (3-3) becomes:

÷

ø

ö

ç

è

æ

293

z0.0065-293 5.26

101.3 = P (3-4)

Atmospheric Density (ρ

ρρ

ρ)3

KvKv T

P

3.486 =

R T

P 1000

= ρ (3-5)

where: ρ atmospheric density [kg m-3]

R specific gas constant = 287 [J kg-1 K-1]

T

Kv virtual temperature [K]

÷

ø

ö

ç

è

æ

P

e

0.378-1

1-

T

=

Ta

KKv (3-6)

where: TK absolute temperature [K] : TK = 273.16 + T [oC]

e

a actual vapour pressure [kPa]

For average conditions (ea in the range 1 - 5[kPa] and P between 80 - 100 [kPa]),

equation (3-6) may be substituted by:

273)1.01(T+T Kv ≈ (3-7)

T is set equal to mean daily temperature for 24-hour calculation time steps.

Saturation Vapour Pressure (es)4

÷

ø

ö

ç

è

æ

237.3T+

T 17.27

exp 0.611 = (T)eo (3-8)

where: eo(T) saturation vapour pressure function [kPa]

T air temperature [°C]

3 Reference: Smith et al. (1991)

4 Reference: Tetens (1930)

Crop evapotranspiration

225

Slope Vapour Pressure Curve (∆

∆∆

∆)5

)2

237.3(T+

237.2T+

T17.27

exp 2504

=

)2

237.3(T+

)(4098

=

÷

ø

ö

ç

è

æ

∆°Te (3-9)

where: ∆ slope vapour pressure curve [kPa°C-1]

T air temperature [°C]

e°(T) saturation vapour pressure at temperature T [kPa]

In 24-hour calculations, ∆ is calculated using mean daily air temperature. In hourly

calculations T refers to the hourly mean, Thr.

Psychrometric Constant (γ

γγ

γ)6

λλε

γP

0.00163 =

10-3

x

P

cp

= (3-10)

where: γ psychrometric constant [kPa °C-1]

c

p specific heat of moist air = 1.013 [kJ kg-1 °C-1]

P atmospheric pressure [kPa]: equations 2 or 4

ε ratio molecular weight of water vapour/dry air = 0.622

λ latent heat of vaporization [MJ kg-1]

Dew Point Temperature (Tdew)7

When it is not observed, Tdew can be computed from ea by:

()

a

dew eln78.16

eln3.23791.116

T−

+

= (3-11)

where: Tdew dew point temperature [oC]

e

a actual vapour pressure [kPa]

For the case of measurements with the Assmann psychrometer, Tdew can be calculated

from

()

wet

8/1

wet

o

a

wetdew T1.0112

Te

e

T9.0112T +−

÷

ø

ö

ç

è

æ

+= (3-12)

5 References: Tetens (1930), Murray (1967)

6 Reference: Brunt (1952)

7 Reference: Bosen (1958); Jensen et al. (1990)

Annex 3: Background on physical parameters used in evapotranspiration computations

226

Short Wave Radiation on a Clear-Sky Day (Rso)8

The calculation of Rso is required for computing net long wave radiation and for checking

calibration of pyranometers and integrity of Rso data. Q good approximatino for Rso for

daily and hourly periods is:

(

)

a

5

so Rz10x275.0R −

+= (3-13)

where: z station elevation [m]

R

a extraterrestrial radiation [MJ m-2 d-1]

Equation (3-13) is valid for station elevations less than 6000 m having low air turbidity.

The equation was developed by linearizing Beer’s radiation extinction law as a function of

station elevation and assuming that the average angle of the sun above the horizon is about

50o.

For areas of high turbidity caused by pollution or airborne dust or for regions where

the sun angle is significantly less than 50o so that the path length of radiation through the

atmosphere is increased, an adaption of Beer’s law can be employed where P is used to

represent atmospheric mass:

÷

ø

ö

ç

è

æ

φ

−

=sinK

P0018.0

expRR

t

aso (3-14)

where: Kt turbidity coefficient [], 0 < Kt ≤ 1.0 where Kt = 1.0 for clean air and

Kt = 1.0 for extremely trubid, dusty or polluted air.

P atmospheric pressure [kPa]

φ angle of the sun above the horizon [rad]

Ra extraterrestrial radiation [MJ m-2 d-1]

For hourly or shorter periods φ is calculated as:

ω

δ

ϕ

+

δ

ϕ

=

φ

coscoscossinsinsin (3-15)

where: ϕ latitude [rad]

δ solar declination [rad] (Equation 24 in Chapter 3)

ω solar time angle at midpoint of hourly or shorter period [rad] (Equation

(31) in chapter 3)

For 24-hour periods, the mean daily sun angle, weighted according to Ra, can be

approximated as:

8 Reference: Allen (1996)

Crop evapotranspiration

227

ú

û

ù

ê

ë

éϕ−

÷

ø

ö

ç

è

æ−

π

ϕ+=φ 2

24 42.039.1J

365

2

sin3.085.0sinsin (3-16)

where: φ24 average φ during the daylight period, weighted according to Ra [rad]

ϕ latitude [rad]

J day in the year []

The φ24 variable is used in Equation (3-14) or (3-18) to represent the average sun angle

during daylight hours and has been weighted to represent integrated 24-hour transmission

effects on 24-hour Rso by the atmosphere. φ24 in Equation (3-16) should be limited to ≥ 0.

In some situations, the estimation for Rso can be improved by modifying Equation (3-

14) to consider the effects of water vapour on short wave absorption, so that:

aDBso R)KK(R += (3-17)

where: KB the clearness index for direct beam radiation []

KD the corresponding index for diffuse beam radiation []

Ra extraterrestrial radiation [MJ m-2 d-1]

ú

û

ù

ê

ë

é

÷

ø

ö

ç

è

æ

φ

−

φ

−

=

25.0

t

Bsin

W

091.0

sinK

P00146.0

exp98.0K (3-18)

where: Kt turbidity coefficient [], 0 < Kt ≤ 1.0 where Kt = 1.0 for clean air and

Kt = 1.0 for extremely trubid, dusty or polluted air.

P atmospheric pressure [kPa]

φ angle of the sun above the horizon [rad]

W precipitable water in the atmosphere [mm]

1.2Pe14.0W a+= (3-19)

where: W precipitable water in the atmosphere [mm]

e

a actual vapour pressure [kPa]

P atmospheric pressure [kPa]

The diffuse radiation index is estimated from KB:

15.0KforK82.018.0K

15.0KforK33.035.0K

BBD

<+=

≥−= (3-20)

As with Equation (3-14), the φ24 variable from Equation (16) is used for φ in Equation

(3-18) for 24-hour estimates of Rso.

Annex 3: Background on physical parameters used in evapotranspiration computations

228

Ordinarily, Rso computed using Equations (3-13), (3-14) or (3-16) should plot as an

upper envelope of measured Rs and is useful for checking calibration of instruments. This is

illustrated in Annex 5.

Crop evapotranspiration

229

Annex 4

Statistical analysis of weather data sets1

COMPLETING A DATA SET

Quite often data sets containing a weather variable Yi observed at a given station are

incomplete due to short interruptions in observations. Interruptions can be due to a large

number of causes, the most frequent being the breakage or malfunction of instruments during

a certain time period. When data are missing, it may be appropriate to complete these data

sets from observations Xi from another nearby and reliable station. However, to use portions

of data set Xi to replace data set Yi, both data sets Xi and Yi must be homogeneous. In other

words, they need to represent the same conditions. The procedure for completing data sets is

applied after the test for homogeneity and any needed correction for nonhomogeneity has

been performed. The substitution procedure proposed herein consists of using an appropriate

regression analysis.

The procedure for substituting nearby data into an incomplete data set can be

summarized as follows:

1. Select a nearby weather station for which the data set length covers all periods for

which data are missing.

2. Characterize the data sets from the nearby station, Xi, and of the station having missing

data, Yi, by computing the mean xand the standard deviation sx for the data set Xi:

x = x / n

i

i = 1

n

å (4-1)

()

sx = x - x / n - 1

i

2

i = 1

n

å

æ

è

ç

ö

ø

÷

12/

(4-2)

and the mean

y

and standard deviation sy for data set Yi:

y = y / n

i

i = 1

n

å (4-3)

1 With contributions from J.L. Teixeira, Instituto Superior de Agronomia, Lisbon, Portugal.

Annex 4: Statistical analysis of weather data sets

230

()

sy = y - y / n - 1

i

2

i = 1

n

å

æ

è

ç

ö

ø

÷

12/

(4-4)

for the periods when the data in both data sets are present, where xi and yi are

individual observations from data sets Xi and Yi, and n is the number of observations

in each set.

3. Perform a regression of y on x for the periods when the data in both data sets are

present:

∃

y

i = a + b x

i (4-5)

with

()()

()

b =

cov

s =

x - x y - y

x - x

xy

x

2

ii

i = 1

n

i

i = 1

n

å

å2

(4-6)

a =

y

- b x (4-7)

where a and b are emprical regression constants, and covxy is the covariance between

Xi and Yi. Plot all points xi and yi and the regression line for the range of observed

values. If deviations from the regression line increase as y increases then substitution is

not recommended because this indicates that the two sites have a different behaviour

relative to the particular weather variable, and they may not be homogeneous. Another

nearby station should be selected.

4. Compute the correlation coefficient r:

()()

r =

cov

s s =

x - x y - y

xy

ii

i = 1

n

ii

i = 1

n

i = 1

n

å

åå

æ

è

ç

ö

ø

÷

22

12/ (4-8)

Both a high r2 (r2 ≥ 0.7) and a value for b that is within the range (0.7 ≤ b ≤ 1.3)

indicate good conditions and perhaps sufficient homogeneity for replacing missing data

in the incomplete data series. These parameters r2 and b can be used as criteria for

selecting the best nearby station.

5. Compute the data for the missing periods k = n + 1, n + 2..., m using the regression

equation caracterized by the parameters a and b (equations 4-6 and 4-7), thus

k

xb + a =

ˆk

y (4-9)

Crop evapotranspiration

231

6. The complete data set with dimension m will now be

()

Y

Yj

j

= y j = i = 1,...,n

= y = k = n + 1, n + 2,..., m

i

k

∃ (4-10)

Note that estimates Yj =

y

k

∃

taken from the regression equations are useful for predicting

evapotranspiration. However, they cannot be treated as random variables(2).

ANALYSIS OF THE HOMOGENEITY OF DATA SERIES

Weather data collected at a given weather station during a period of several years may be not

homogeneous, i.e., the data set representing a particular weather variable may present a

sudden change in its mean and variance in relation to the original values. This phenomenon

may occur due to several causes, some of which are related to changes in instrumentation and

observation practices, and others which relate to modification of the environmental conditions

of the site, such as rapid urbanization or, on the contrary, perhaps development of irrigation

in the area.

Changes relative to data collection may be caused by:

• change in type of sensor or instrument;

• change in the observer and or change in the timing of observations;

• “sleeping” data collector;

• deterioration of sensors, such as with some types of pyranometers and RH sensors, or

malfunctionning of mechanical parts, such as with a tipping bucket rain gauge, or by an

intermittently broken or shorted wire;

• aging of bearings on anemometers;

• use of incorrect calibration coefficients;

• variation in power supply or electronic behaviour of instruments;

• growth of trees or planting of tall crops or construction of buildings or fences near a

raingauge, anemometer, or evaporation pan;

• change in the location of the weather station, or in the types of shelters for housing

temperature and humidity sensors;

• change in the watering, type or maintenance of vegetation in the vicinity of the weather

station;

• significant change in the watering or type of vegetation of the region surrounding the

weather station.

These changes cause observations made prior to the change to belong to a statistically

different population than data collected after the change. It is therefore necessary to apply

appropriate techniques to evaluate whether a given data set can be considered to be

2

To create random values,, one can add to

∃

yk (equation 4-9) the residuals εk synthetically generated

from a population N (0, sy,x). The residuals are created using tables of random numbers. In that

case the estimates Yj can be treated as random variables.

Annex 4: Statistical analysis of weather data sets

232

homogeneous and, if not, to introduce the appropriate corrections. To do so requires the

identification of which sub-data series is to be corrected. To do this requires local information.

Procedures indicated herein are simple but are well proven in practice. They rely upon

the statistical comparison of two data sets, one considered homogeneous and constituted by

the observations Xi, the other being the one under analysis and consitutued by the

observations Yi of the same weather variable (Tmax, Tmin, u2, RHmax,..., etc). Both sets

Xi and Yi should be collected at two stations that are in the same climatic region, i. e., Xi

and Yi should present the same trends in time despite the space variability when short time

scales (daily, weekly or decadaily) are utilized.

The reference observations Xi are selected from a weather station for which the data set

can be considered to be homogeneous.(3) The Xi data set should have the same time length of

observations as the set of observations Yi.

Method of Cumulative Residuals

When relating two weather data sets from two weather stations, where the first is considered

to be homogeneous, the data set of the second station can be considered to be homogeneous if

the cumulative residuals of the second data set from a regression line based on the first data

set are not biased. The bias hypothesis can be tested for a given probability p. This is done

by verifying whether the residuals can be contained within an elipsis that has axis α and axis

β. The magnitudes of α and β depend on the size of the data set, on the standard deviation of

the sample being tested and on the probability p used to test the hypothesis(4).

The procedure for analysing the homogeneity of a weather data set Yi collected in a given

weather station environment can be summarized as follows:

1. Select a reference weather station inside the same climatic region that is known to have

an homogeneous data set Xi of the same weather variable. As an alternative, construct

a “regional” homogeneous data set by averaging the observations at several weather

stations in the same region.

2. Organize both data sets xi and yi in chronological order i = 1, 2,..., n, where the

starting time and time increment are identical for both data sets.

3. For both data sets, compute the mean and standard deviation (equations 1 to 4) for the

homogeneous variable (xi) and for the variable to be tested (yi).

3 When, for a given climatic region, there is no information concerning the homogeneity of data, then

the average of observations of the same variable from all stations (excluding the one in the analysis),

Xi = å Xj, i / m, can be used to constitute the homogeneous data set.

4 This test utilizes results from residuals from the linear regression of Y on X. The residuals should

follow a normal distribution with mean zero and standard deviation sy,x, i.e. the error εi ∈ N

(0,sy,x). The residuals from the regression should be considered to be independent random

variables (i.e., they should exhibit homoscedaticity).

Crop evapotranspiration

233

4. Calculate the regression line between the two variables yi and xi and the associated

correlation coefficient (equations 4-5 to 4-8). The regression equation among the full

sets is expressed as

∃

y

i =

a

+ b x

f

i (i = 1, 2, ..., n) (4-11)

where the subscript f refers to the full set. Whenever possible, plot xi, yi and the

regression line to visually verify whether the homoscedaticity hypothesis(5) can be

accepted (see Figure 4.1)6

5. Compute the residuals of the observed yi values to the regression line (equation 4-5),

the standard deviation sy,x of the residuals and the corresponding cumulative residual

Ei:

5 The homoscedaticity hyphotesis is accepted when the residuals εi of the dependent variable to the

regression line (equation 4-5) can be considered to be independent random variables. This can be

visually assessed when the deviations of yi to the regression estimates

∃

yi are within the same range

for all xi, i.e., when these deviations are not increasing (or decreasing) with increasing values of xi.

6 Data in this example were provided by J.L. Teixeira (personal communication, 1995).

FIGURE 4.1

Regression between two sets of weather data, with the X data set being homogeneous. The

example shows that the homoscedaticity condition was satisfied.

0

200

400

600

800

1000

1200

0 200 400 600 800 1000 1200

.

Station Y

P

(mm)

Station X, Pi (mm)

Annex 4: Statistical analysis of weather data sets

234

εi =

y

-

y

ii

∃

(4-12)

()

sy,

/

xy 2

= s 1 - r

12

(4-13)

Ei = +

ij

j = 1

i - 1

εε

å

()

j i = ,... - 11 (4-14)

6. Select a probability p for accepting the hypothesis of homogeneity. The value

p = 80% is commonly utilized. Then compute the elipsis equation having axes

α

= n/2 (4-15)

()

β = n

n - 1

z s

py, x

12/ (4-16)

where:

n size of the sample under analysis

zp standard normal variate for the probability p (usually p = 80% for non

excedancy): Table 4.1

sy, x standard deviation of the residuals of y (equation 4-13)

The parametric equation of the elipsisis is then

()

X

Y

= cos

= sin

αθ

βθ

(4-17)

with θ [rad] varying from 0 to 2 π.

Note: given the symmetry of

the normal distribution, the

values for p < 50%

correspond to (100 - p) but

with the opposite sign. Ex: p

= 20% is associated with z

= -z80 = -0.84

It can therefore be concluded, at the level of probability p, that there is no bias in the

distribution of residuals, i.e., the data set yi is considered to be homogeneous, when

the computed values for Ei fall inside the elipsis (equation 4-17).

7. Plot the cumulative residuals Ei against time using the time scale (interval) of the

variable under analysis (Figure 4-2).

8. Draw the elipsis on the same plot and verify whether the Ei all lie inside the elipsis. If

they do, then the hypothesis of homogeneity is accepted at the p level of confidence

(Figure 4.4).

TABLE 4.1

Value of the standard normal variate zp for selected

probabilities P of non-excedance

p (%) zp p (%) zp

50

60

70

75

0.00

0.25

0.52

067

80

85

90

95

0.84

1.04

1.28

164

Crop evapotranspiration

235

9. If the hypothesis of homogeneity cannot be accepted (this is the case in Figure 4.2),

then one can select the break point where it appears that Ei ceases to increase (or to

decrease) and begins to decrease (or to increase), for example at I = 16 in Figure 4.2.

This break point is termed k = i.

10. The data set is now divided into two subsets, the first from 1 to k, the second from

k + 1 to n. Then, new regression equations are computed between Y and X for both

subsets. If we presume that the second subset is homogeneous but that the first is not,

then we have

∃

y

i =

a

+b x

nh nh i (i = 1, 2, ...,k) (4-18)

and

∃

y

i =

a

+ b x

hhi

(i = k + 1, k + 2, ...,n) (4-19)

where the subscripts h and nh identify the regression coefficients of the homogeneous

and the non homogeneous subsets, respectively (see Figure 4-3).

11. Compute the differences between the two regression lines

()( )

∆ y = a + b x - a + b x

ihhinhnhi

∃ (4-20)

for the non homogeneous set (i = 1, 2, ..., k)

FIGURE 4.2

Plot of cumulative residuals against time and associated elipsis for the probability p = 80%, with

results indicating that data set Y is not homogeneous (relative to data set X).

.

-1200

-1000

-800

-600

-400

-200

0

200

400

600

800

03691215182124273033363942

i

Ei

(mm)

Annex 4: Statistical analysis of weather data sets

236

FIGURE 4.3

The regression lines for the two subsets obtained from the data sets of Figures 4.1 and 4.2.

Selection was made after definition of the break point in Figure 4.2.

0

200

400

600

800

1000

1200

0 200 400 600 800 1000 1200

.

Values of Pi to be cor rected

True vales of Pi

Equation 16

Equation 17

Station Y

Pi

(mm)

Station X, Pi (mm)

18

19

FIGURE 4.4

Plot of cumulative residuals against time and the associated elipsis for p=80% after correction of

variable y.

.

-800

-600

-400

-200

0

200

400

600

800

03691215182124273033363942

i

Ei

(mm)

Crop evapotranspiration

237

12. Correct the non homogeneous subset portion of data set Y

y

ci,

∃

=

y

+

y

ii

∆ (i = 1, 2, ...,k) (4-21)

where the subscript c identifies the corrected values. Thus, the corrected, homogeneous

full set for weather variable Y is composed by

Yfori

for i

i = y = 1, 2,..., k

Y = y = k + 1, k + 2,..., n

c,i

ii

(4-22)

A similar procedure would be utilized if it was presumed that the second sub-set

requires correction, rather than the first sub-set.

Note that the variables Yi are still considered to be random variables despite that the

mean and the variance have been modified due to the correction introduced. To confirm the

results of the correction of data set Y for homogeneity, the homogeneity test methodology can

be applied again to the corrected variable Y to provide evidence of homogeneity in the graph

of residuals. This has been done in Figure 4.4.

In this example, it was presumed that the latter sub-set (k to I) was the correct

(representative) data set, or the data set displaying the desired attributes. It was therefore

presumed that prior to time k, the readings were biased by instrument calibration, different

location of the station or the instrument within the station, change in type or manufacturer of

the instrument, or change in general environment of the station. It appears in Figure 3 that

the data prior to i = k were biased downward by approximately 100 mm of annual

precipitation.

Double-Mass Technique

The double-mass technique is also useful for assessing homogeneity in a weather parameter.

As with the method of cumulative residuals discussed in the last section, the double-mass

technique requires data sets from two weather stations, where Xi (i = 1, 2,...,n) is a

chronologic data set for a given weather variable observed for a certain time length at a

“reference” station, and which is considered to be homogeneous, and where Yi is a data set

of the same variable, with the same time length, observed at another station and for which

homogeneity needs to be analysed.

In the double-mass technique, starting with the first observed pair of values X1 and Y1,

cumulative data sets are created by progressively summing values of Xi and Yi to verify

whether the long term trends in variation of Xi and Yi are the same. Thus the following

cumulative variables are obtained

xi = X + X

ij

j = 1

i - 1

å (4-23)

and

Annex 4: Statistical analysis of weather data sets

238

yi = Y + Y

ij

j = 1

i - 1

å (4-24)

with i = 1,..., n and j = 1,..., i - 1.

These variables xi and yi are still considered to be random variables and are

characterized by the mean and the standard deviation (equations 4-1 to 4-4). The yi and xi

variables can be related through linear regression (equations 4-5 to 4-8). However, the double

mass technique is typically applied as a graphical procedure.

The graphical application of the double-mass analysis is done by plotting all coordinate

points xi and yi. The plot is then visually analysed to determine whether successive points of

xi and yi follow an unique straight line, indicating the homogeneity of the data set Yi relative

to data set Xi. If there appears to be a break (or more than one break) in the the plot of yi to

xi, then there is a visual indication that the data series Yi (or perhaps Xi) is not homogeneous

(Figure 4.5). The break at coordinates xk and yk can be used to separate two subsets (i = 1,

2, ..., k) and (k + 1, k + 2,..., n). One of the subsets is to be corrected. The appropriate

one to correct needs to be identified by consulting the records of the weather station, when

available.

FIGURE 4.5

Double mass analysis applied to two series of precipitation when data from station Y are not

homogeneous

.

0

4000

8000

12000

16000

20000

24000

28000

32000

0 4000 8000 12000 16000 20000 24000 28000 32000

Station Y

Pi

(mm)

Station X,

Σ

ΣPi (mm)

Crop evapotranspiration

239

Often, visual interpretation of the double-mass balance is difficult. Thus the following

numerical regression procedure is recommended:

1. Compute the regression line through the origin for the full set of data xi and yi

∃

y

i = b x

i (i = 1, 2, ..., n) (4-25)

with

()()()

b = x - x y - y / x - x

ii i

åå 2

2 Compute the residuals to the regression line

εi = y - b x

ii

(4-26)

3. Analyse the distribution of residuals. If the residuals plot as independent, random

variables, then the set can be considered to be homogeneous. However, if the

distribution of residuals is biased over i = k, then the homogeneity hypothesis is

rejected. The bias can be visually assessed by plotting (εi, i). The example in Figure

4.6 shows that residuals follow a trend of decreasing εi until i = k (= 16). Following

that, the trend is to increase. This plot demonstrates a bias indicating that the data set Y

is not homogeneous.

FIGURE 4.6

Residuals of double mass to the straight line (equation 26) indicating the non homogeneity of the

residuals of the series of precipitation of station Y.

.

-600

-400

-200

0

200

400

600

800

1 4 7 101316192225283134374043

i

(mm)

Residuals

εi

Annex 4: Statistical analysis of weather data sets

240

4. The break point at i = k defines two subsets (i = 1, 2, ..., k) and

(i = k + 1, k + 2, ..., n). Using local information on data collection, the user must

decide which subset requires correction.

5. When the first subset is homogeneous the following correction procedure can be

applied:

a) compute the two regression lines, the first through the origin

∃

y

i = b x

hi

(i = 1, 2, ..., k) (4-27)

and

∃

,

y

nh i = a + b x

nh nh i (i = k + 1, k + 2, ..., n) (4-28)

where subscripts h and nh identify respectively the homogenous and non homogeneous

subsets.

b) compute the differences between both regression lines for i = k + 1, k + 2, ..., n

()

∆ y = b x - a + b x

ihinhnhi

∃ (4-29)

6. When the second subset is homogeneous:

a) compute the regression line for the homogeneous subset (i = k +1, k + 2, ..., n)

after correcting the coordinates (xi, yi) using the coordinates of the break point (xk,

yk), i.e. moving the origin of coordinates from (0, 0) to (xk, yk). This regression is

therefore

()

yi - y = b x - x

khi k

(4-30)

thus

()

∃

yi = y - b x + b x

k

h

k

hi

(i = k +1, k + 2, ..., n.) (4-31)

b) compute the regression line for the non homogeneous subset forced to the origin

∃

y

i = b x

nh i (i = 1, 2, ..., k) (4-32)

c) compute the differences between the regression lines (4-31) and (4-32)

()

[]

∆ y = y - b x + b x - b x

ikhkhinhi

∃ (4-33)

7. For both cases, correct the variables yi corresponding to the non homogeneous subset

as

y

ci,

∃

=

y

+

y

ii

∆ (4-34)

Crop evapotranspiration

241

with ∆

y

i

∃

given by equations (4-29) or (4-33).

8. Compute the corrected estimates of the weather variables Yi by solving equation (4-24)

for Yi.

Figure 4.7 illustrates the double mass after correction of subset Y in Figure 4.3, where

the cumulative sums now follow a straight line.

Figure 4.8 is a plot of the corresponding residuals, which now follow a normal

distribution. Similar verification can be easily made by the user. This procedure can be easily

applied using a spreadsheet computation and graphical packages that are currently available.

FIGURE 4.7

Double mass after correction of data set Y (case of Figure 4.3)

.

0

4000

8000

12000

16000

20000

24000

28000

32000

0 4000 8000 12000 16000 20000 24000 28000 32000

Station Y

Pi

(mm)

Station X,

Σ

ΣPi (mm)

Annex 4: Statistical analysis of weather data sets

242

SELECTED BIBLIOGRAPHY ON STATISTICAL ANALYSIS

Dubreuil, P. 1974. Initiation à l’analyse hydrologique. Masson & Cie. et ORSTOM, Paris.

Haan, C. T. 1977. Statistical Methods in Hydrology. The Iowa State University Press, Ames.

Kite, G. W. 1988. Frequency and Risk Analyses in Hydrology. Water Resources

Publications, Littleton, CO, 257 pp.

Natural Environment Research Council (NERC) 1975. Flood Studies Report, Vol I -

Hydrology Studies. Natural Environmental Research Council, London, 550 pp.

FIGURE 4.8

Residuals of the double mass after correction of data set Y (compare to Figure 4.4)

.

-100

-80

-60

-40

-20

0

20

40

60

1 4 7 101316192225283134374043

i

(mm)

Residuals

εi

Crop evapotranspiration

243

NOTATION IN STATISTICAL ANALYSIS

a regression coefficient

b regression coefficient

covxy covariance of variables x and y

Ei cumulative residuals

i number of order of variable xi in the sample

j, k number of a variable in a subset

n size of the sample

p probability

p (x) probability distribution density function

r correlation coefficient

r2 coefficient of determination

sx estimate of the standard deviation of the variable x

sx

2 estimate of the variance of the variable x

sy estimate of the standard deviation of the variable y

sy

2 estimate of the variance of the variable y

sy,x standard deviation of the residuals of y estimated from the regression

X random variable

Xi value of a variable in a data set

xi random variable

∃

xp estimated value for the variable x with probability of non excedance p

x estimate of the mean, or mean of a sample of the random variable xi

Y transformed variable from X

Yi value of a variable in a data set

yi random variable

∃

y

i value of yi estimated from the regression

y

estimate of the mean, or mean of a sample of the random variable yi

Z standard normal variable

zp value of the standard normal variable for the probability p

εi residuals of y estimated from the regression

µ mean of a population

σ standard deviation of a population

Annex 4: Statistical analysis of weather data sets

244

Crop evapotranspiration

245

Annex 5

Measuring and assessing integrity of

weather data

Estimates of reference evapotranspiration (ETo) are no better than weather data upon which

they are based. Assessments of weather data integrity and quality need to be conducted before

data are utilized in ETo equations. When necessary and when possible, corrections to the data

should be made to account for poor sensor calibration. Some of these corrections are

described in section 1 of Annex 4.

A good cautionary statement in data analysis and application is that “no data are better

than bad data.” This statement applies primarily to measurements of evapotranspiration that

are used to develop or to calibrate reference ET equations or to determine crop coefficients.

However, it also applies to weather data. When one has no data, one can look to regional

weather or ET data summaries for information that might be useful to represent conditions

within the local area. In the case of ET data, one might go to a publication such as this one

to make reasonably accurate estimates of ETo and ETc. However, in the case of “bad” data,

meaning biased, or faulty, or nonrepresentative data collected locally, one is “stuck” with

weather data and associated predictions of ETo and ETc, or with local measurements of ETc

that can be biased, faulty, or nonrepresentative. The result is application of

evapotranspiration data or evapotranspiration calculations to irrigation water management, to

water resources operations, or to irrigation and water resources systems design that can

actually cause more economic and hydrologic problems than if only reasonable estimates or

even “textbook” values for ETc had been used instead. Humanity can be worse off because

of faulty data as compared to no data.

Some years ago, when computer modeling was in its infancy, a common cautionary

advice was to “do not trust any model until it has been validated using independent data.”

Today, with some of the more common mathematical models becoming proven and

trustworthy, the corollary of this expression is commonly advocated, where “one should not

trust any data until they are validated using a model!” Certainly, some place in between these

two cautionary advocations is appropriate. Often a valid model can be valuable for evaluating

data to identify errors, outliers and biases. And of course, valid data are required for

selecting or calibrating a particular model.

This Annex presents guidelines to be used to calculate both extreme ranges for weather

data measurements and also means to assess integrity of data that fall between the extremes.

A review on instrumentation for agricultural weather stations is given first.

Annex 5: Measuring and assessing integrity of weather data

246

INSTRUMENTATION FOR MEASURING WEATHER VARIABLES1

Data Acquisition and Instrumentation

Radiation

Solar radiation is commonly measured with pyranometers. Pyranometers measure the

shortwave incoming radiation in a solid angle in the shape of a hemisphere oriented upwards.

Currently, in the most common “glassed dome” pyranometers, a thermopile is used within

the instrument as the sensor, where thermal gradients are measured across hot and cold areas

(black and white). The radiation intensity is proportional to the temperature differences

between the two sensing areas. Accuracy depends upon the sensitivity of the material used in

the sensors, the response time and the distortion characteristics of the material constituting the

dome covering the sensors. A second type of pyranometer that is less expensive and that is

gaining acceptance is the silicon diode instrument where electric current is generated by a

photo sensitive diode in proportion to solar intensity. Ordinarily, silicon diode pyranometers

are not fully sensitive to the full spectrum of visible light, so that the calibration of the

instrument is only valid for upward solar measurements.

When a pyranometer is oriented downwards it measures the reflected shortwave

radiation, and is thus called an albedometer. When two pyranometers are associated, one

oriented upwards and the other downwards, the net short wave radiation is measured. The

instrument is then called a net pyranometer. A point of caution is that any instrument used

as an albedometer or net pyranameter must have full sensitivity to all spectra of visible light.

This is important since the composition of reflected light from vegetation is highly biased

toward green. Therefore, most albedometers must be of the glass domed thermopile type and

not the photo diode type.

Net radiation is measured by pyradiometers (or net radiometers), which sense both

short and long wave radiation. They have two bodies, one oriented upwards and the other

downwards, both covering a solid angle in the shape of a hemisphere. The sensors are made

from several thermocouples sensing heat generated by radiation from all wavelengths, and are

protected by domes made in general of polyethylene treated in a specific manner. The black

bodies can loose their sensing capabilities with time, so that these instruments require regular

and frequent calibrations. Other net radiometers are comprised of ventilated differential

thermopiles, but they are very seldom utilized. All radiometers refered to above transform the

radiation energy into thermal energy, a portion of which is transformed into an electric

voltage gradient that provides appropriate conditions for continuous recording using

dataloggers.

Sunshine duration is most commonly recorded with the Campbell-Stokes heliograph. A

glass globe focuses the radiation beam to a special recording paper and a trace is burned on

the paper as the sun is moving. No records occur when no bright sunshine is sensed.

Measurements are reliable when the recording paper is placed in the right position according

to the relative position of the sun. Care is required to avoid accumulation of rain water on the

paper. The heliograph has to be oriented South in the northern hemisphere and North in the

1 Details on weather station instrumentation can be found in FAO 27 (Doorenbos, 1976), in the WMO

Guide to Agrometeorological Practices (WMO, 1981, 1983), or in meteorology handbooks

(Seemann et al., 1979; Rosenberg et al., 1983; Kessler et al., 1990).

Crop evapotranspiration

247

southern hemisphere. In China, another type of heliograph is used. The solar beam penetrates

through an orifice and traces a recording paper treated with a sensible chemical substance.

Electronic records of sunshine duration are obtained through the photo-electric or the rotating

optical fibre sunshine recorders.

Windspeed

Windspeed is measured using anemometers, always placed at an height not less than 2 m

above the ground, and often at 5 m, following recommendations by WMO. Most common are

the three-cup anemometers. Also common are propeller anemometers. Measurements by

both types are reliable provided that maintenance ensures appropriate functioning of the

mechanical parts. Older designs of anemometers utilize mechanical counters as the output

device. Modern anemometers may be equipped with generators giving a voltage signal that is

proportional to the windspeed. Other anemometers may be equipped with small magnetic reed

switches or with opto-electronic couplers that generate electric impulses in proportion to the

windspeed. The electronic devices are utilized in automatic weather stations. Accuracy of

windspeed measurements depends on the upwind fetch as much as on instrumentation. A

large upwind fetch that is free of buildings and trees is definitely required for representative

measurements.

Temperature

The most commonly utilized sensor for measuring temperature are still the mercury

thermometers. Maximum and minimum thermometers use mercury and alcohol. Bimetallic

thermographs are the most common mechanical temperature recorders. They are easy to read

and maintain. However, mechanical thermographs do require verification and adjustment of

the position of the pen recorder. These instruments are installed in shelters that are naturally

ventilated.

Modern temperature sensors have been developed, namely the thermistor and the

thermocouple. These provide very accurate analogue measurements and are normally utilized

in automatic weather stations. Thermistors provide independent measurements of air or soil

temperature, whereas thermocouples require an additional base temperature reading, normally

provided by a thermistor. To maintain the accuracy and representativeness of these

instruments, they are installed in special radiation shields (shelters) having natural ventilation.

Occasionally the shields or shelters are artificially aspirated to reduce biases caused by heat

loading from the sun.

Humidity

Dew point temperature is often measured with a mirrorlike metallic surface that is

artificially cooled. When dew forms on the surface, its temperature is sensed as Tdew. Other

dew sensor systems use chemical or electric properties of certain materials that are altered

when absorbing water vapour. Instruments for measuring dew point temperature require

careful operation and maintenance and are seldom available in weather stations. The

accuracy of estimation of the actual vapour pressure from Tdew is generally very high.

Relative humidity is measured using hygrometers. Most frequently used in

mechanically-based field stations are the hair hygrometers, normally operated as mechanical

hygrographs. Measurements loose accuracy with dust and ageing of the hairs. Modern

hygrometers use a film from a dielectric polymer that changes its dielectric constant with

Annex 5: Measuring and assessing integrity of weather data

248

changes in surface moisture, thus inducing a variation of the capacity of a condensator using

that dielectric. These instruments are normally called dielectric polymer capacitive

hygrometers. Accuracy can be higher than for hair hygrometers. These electronic devices are

utilized in most modern automatic weather stations.

The dry and wet bulb temperatures are measured using psychrometers. Most

common are those using two mercury thermometers, one of them having the bulb covered

with a wick saturated with distilled water, and which measures a temperature lowered due to

the evaporative cooling. When they are naturally ventilated inside a shelter, problems can

arise if air flow is not sufficient to maintain an appropriate evaporation rate and associated

cooling. The Assmann psychrometer has a forced ventilation of the wet bulb and dry bulb

thermometers.

The dry and wet bulb temperature can be measured by thermocouples or by thermistors, the

so called thermocouple psychrometers and thermosound psychrometers. These psychrometers

are used in automatic weather stations and, when properly maintained and operated, provide

very accurate measurements.

ASSESSING INTEGRITY OF WEATHER DATA2

Solar Radiation using Clear Sky Comparisons

Pyranometer operation and calibration accuracy can be evaluated for a particular weather

location by plotting hourly or daily average readings of solar radiation (Rs) against computed

short wave radiation that is expected to occur under clear sky conditions (Rso). Rso can be

computed for any day or hour as

aTso R K = R (5-1)

where Ra is extraterrestrial radiation3 and KT is a “clearness” or transmission index.

R

so computed with equation (5-1) should plot as an upper envelope of measured Rs and

is useful to check the calibration of pyranometers. Equations (3-13), (3-14), or (3-17) to (3-

20) of Annex 3 should be used for predicting KT for low air turbidity. Equations (3-14) or

(3-17) to (3-20) of Annex 3 are appropriate for areas with high turbidity caused by pollution

or airborne dust or for regions where the sun angle is significantly less than 50º.

The example in Figure 5.1 shows one application concerning 24-hour calculations for

Logan, Utah, where the highest observed values for Rs correspond to the envelope of

calculated Rso, thus showing appropriate calibration of the pyranometer being utilized. In

Figure 5.2, the half-hour observations of Rs for Logan are compared with the computed Rso

envelopes. This figure shows good agreement between observed and computed values.

However, as shown for day 7, Rs may sometimes exceed the predicted Rso when there is

reflection of radiation from nearby clouds during periods when no clouds shade the

pyranometer.

2 These guidelines are based on an article by Allen (1996).

3 For Ra daily computations see Chapter 3, Equations (21) - (24) and for hourly computations see

Equations (28) - (33). For KT see Rso equations (3-13) - (3-20) of Annex 3.

Crop evapotranspiration

249

When the Rs observations on obviously clear days fall significantly above or below the

computed Rso curves, then corrective action may be warranted. The correction may be in the

form of applying a correction multiplier to the observed data, so that (Rs)cor = a Rs, where a

is a derived correction factor. Or, an additive correction may be warranted, where (Rs)cor

= Rs + c. Or, correction may be made by a combination of multiplicative and additive

factors. Obviously, the corrections based on the computed Rso curves presume that the curve

is accurate. The accuracy of the Rso envelope may need to be confirmed in a region by using

accurate radiation measurements obtained from a calibration-grade pyranometer that has a

calibration coefficient that is traceable to the international standard. The calibration

pyranometer should be used only for short term periods of 10 – 14 days, and then should be

stored in darkness to extend its life and to preserve the calibrated accuracy. Care should be

exercised in selection of the turbidity coefficient in Equation (3-14) and (3-18) of Annex 3.

Unfortunately, little information is available on this topic.

Net radiation

Equations for estimating hourly and 24-hour average rates of net radiation (Rn) using Rs

measurements are generally accurate under most conditions. Therefore, measured Rn data

should always be plotted against Rn that has been estimated using equations4 that are based on

4 See equations (38) through (40) in Chapter 3.

FIGURE 5.1

24-hour average Rs and estimated Rso envelopes at Logan, UT during 1992 showing an

appropriate calibration of the pyranometer utilized

-2

Logan, UT

1992

0 30 60 90 120 150 180 210 240 270 300 330 360

0

100

200

300

400

500

Day of the Year

Ave. 24-hour Solar Radiation, W m

Eq. 3-13 Eq. 3-14 Eq. 3-17 to 3-20

Annex 5: Measuring and assessing integrity of weather data

250

measured Rs, air temperature and vapour pressure. The value for albedo (α) used in the Rn

estimating equation should represent conditions of the surface beneath the radiometer.

If measured values for Rn chronically deviate from estimated values by more than 3-

5%, then the calibration or operation of the Rn device (radiometer) should be scrutinized.

This type of comparison can readily identify days or periods during which the radiometer

device has malfunctioned due to effects of dust, bird droppings, moisture condensation inside

the plastic domes, a lack of levelness of the intrument, or a lack of green vegetation beneath

the sensor. Of course, the Rs measurement used in the Rn equations should also be

scrutinized as discussed in the previous section.

The user of net radiometer data must be aware that net radiometers manufactured by

different companies may not yield the same measurements of radiation even when placed over

the same surface. These differences are due to differences in sensitivities of various

radiometers to long wave and short wave radiation and variations among methods for

calibrating sensors during manufacturing.

The type, density and height of vegetation beneath the net radiometer and relative soil

moisture content should be monitored and reported with the data. Care should be exercised

when positioning the radiometer to avoid shading the sensed vegetation with other instruments

or structures and to insure that the radiometer is not shaded by other instruments or structures

at any time of the day or year.

FIGURE 5.2

30-minute average Rs and estimated Rso envelopes at Logan, UT during July 7 and July 25,

1992

-2

Logan, UT

July, 1992

July 7 July 25

Reflection by

nearby clouds

0 600 1200 1800 0 600 1200 1800 2400

0

200

400

600

800

1,000

1,200

1,400

Hour of the D ay

Solar R adiation, W m

Eq. 3.13 Eq. 3.14 Eq.3.17-20

Crop evapotranspiration

251

Figure 5.3 shows measured and estimated Rn for cattail vegetation near Logan, UT

during 1993. The measurement and calculation time step was 20 minutes. The agreement

between measurements and equation estimates was fairly good. Perfect agreement between

the Rn measurements and Rn equations should not be expected, due to limitations of

assumptions used in the equations (e.g., the value for albedo, means for estimating the net

long wave radiation component, etc.).

Soil Heat Flux

A relationship proposed by Choudhury (1989) for predicting soil heat flux density (G) under

daylight conditions5 is:

G = 0.4 exp(-0.5 LAI) Rn (5-2)

where LAI is the leaf area index, exp() is the natural number raised to the exponent, and G

has the same units as Rn.

5 This equation predicts G = 0.1 Rn for LAI = 2.8, which is typical for clipped grass (equation (45)

in Chapter 3). Soil heat flux under forage grass during nighttime hours was found to be about 0.5

Rn. Pruitt (1995, personal communication) observed G = 0.3 Rn during nighttime hours under

clipped grass at Davis, CA.

FIGURE 5.3

Measured and estimated Rn during 20 minute periods over mature cattail vegetation near

Logan, UT during August, 1993 (from Allen et al., 1994)

(200) 0 200 400 600 800 1,00

0

(200)

0

200

400

600

800

1,000

Calculated R , W m-2

n

Estimated R , W m

-2

n

Pelican Pond, Logan, Utah

August, 1993

Annex 5: Measuring and assessing integrity of weather data

252

Equation (5-2) can be used to check the functioning and relative accuracy of soil heat

flux plates after correcting measurements for temperature change of soil above the plates. The

relationship of Equation (5-2) does not hold for 24-hour data, as a positive 24-hour soil heat

flux estimate would always result. The user must be aware that Eq. (5-2) is only approximate

and does not consider effects of plant spacing, sun angle, soil colour, soil moisture, or soil

texture, nor the sensible heat balance at the surface on the ratio of G to Rn. Generally, more

than one soil heat flux plate is used due to spatial variation in soil, soil water content, and

vegetation.

Windspeed

Accuracy of wind measurements is difficult to assess unless duplicate instruments are used.

One should always scan wind records for the presence of consistently low wind recordings.

For electronic instruments, these recordings may represent a numerical “offset” in the

anemometer calibration equation. The presence of these constant and consistent offsets in the

data set indicates either the presence of exceptionally calm conditions (wind speeds less than

about 0.5 m s-1 during the entire sampling period (which is rare)) or a malfunctioning of the

wind speed sensor due to electrical shorting or perhaps due to fatigue of bearings. These

problems may not be noticed by the station operator.

When possible, a second anemometer6 of the same design, but with fresh bearings,

should be placed at the weather location for a three or four day period at least once each year,

and recordings compared with the permanent instrument. Variations between recordings can

signal a need to replace bearings, switches, or other parts.

Relative Humidity and Vapour Pressure

Vapour pressure of air is difficult to measure accurately. Some older electronic humidity

sensors were commonly plagued by hysteresis, nonlinearity and calibration errors. Some of

these errors are inherent in the sensor design and still plague some modern sensors. Other

errors result from dust, moisture, insects, pollution, and age.

Replication of RH Instruments

It is strongly recommended that duplicate RH and air temperature sensors be permanently

employed in electronic weather stations, at least for some period each year. When duplicate

RH and air temperature sensors yield similar measurments, then it is likely that both sensors

are functioning properly, provided proper calibration equations have been used. However,

even though duplicate sensors are in agreement does not mean that the readings are free from

calibration errors and biases due to nonlinearity, etc..

Trends in Computed Dew Point Temperature with Time

When air humidity is measured using RH sensors, the actual vapour pressure of the air (ea) is

calculated as:

6 If a second data logger is used to record the temporary anemometer, one should be careful to

synchronize data logger clocks. Also, one should be careful that anemometers do not interfere with

one another’s wind stream.

Crop evapotranspiration

253

()

T eº

100

RH

=

a

e (5-3)

where eº (T) is the saturation vapour pressure at air temperature T and RH is in %. RH and

T must be taken for the same time period, preferably for ≤ 1 hour .

Hourly (or shorter) measurements of RH, dew point temperature (Tdew) or vapour

pressure (ea) can be preliminarily assessed by plotting hourly measurements of computed

Tdew or ea with time. Relative humidity will vary significantly with time of day, and

inversely with air temperature as shown in Figure 12 of Chapter 3. However, both Tdew and

ea, either measured directly, or computed using RH and T measurements, should remain

somewhat constant throughout a 24-hour period when the air mass is stable and advection of

dry air from outside the area does not occur. During these stable periods, one should expect

some rise in Tdew and ea during daytime periods, when ET fluxes humidify the equilibrium

boundary layer. However, this increase is usually less than about 10 to 20%. Variation in

Tdew increases significantly when a weather front passes overhead. Since ea is calculated as

the product of RH and saturation vapour pressure at air temperature, any error in the RH

calibration tends to cause false variation in Tdew and ea with changing air temperature.

Figure 5.4 shows Tdew computed from measurements of RH and air temperature at a

weather station in the center of a wetland near Logan, UT (20-minute data). Tdew generally

varied from hour to hour due to air mass instability and increased during most days of this

period as evaporation from the local wetland vegetation added humidity to the air. The data

sequence shows some periods of relatively constant measurement (calculation) of Tdew

throughout a 24-hour period (for example day of year 199), even though air temperature

varied substantially. This is a good indication that the RH sensor was probably functioning

correctly and that the instrument calibrations were probably valid.

Figure 5.4 also shows, for the same weather station, a comparison between RH

measured using two different and independent relative humidity sensors. The two sensors,

one a “chilled-mirror” device that measures Tdew directly, and the other, a device that

measures RH directly, agreed very well with each other during the 8 days shown. The value

of having “redundancy” in instrumentation is demonstrated in this example, where the two

different devices measuring the same parameter (in this case RH) leave no question

concerning the validity and accuracy of the RH measurements, due to the close agreement.

The use of only a single instrument would leave some question as to accuracy.

One can notice in Figure 5.4 that the RH approached 100% on day 200, which is

expected for a well-watered setting. The difference between minimum daily temperature and

Tdew was generally 1 to 2 oC for many of the days. This is expected in dry, advective

environments, as discussed in Chapter 3 and Annex 6.

Observations During Periods of Dew and Rainfall

In many climates, especially those where nightime dew occurs, Td during early morning

hours before sunrise should coincide closely with recorded Tmin and RH should approach

100%. For automatic recording weather stations where recording rain gauges are used, one

should expect RH recordings during periods of rain or light drizzle to exceed 95%. Relative

humidity recordings that exceed 100% by more than 3-5% during early morning hours or

during precipitation events indicate a need for recalibration and numerical adjustment of

collected data.

Annex 5: Measuring and assessing integrity of weather data

254

Maximum Daily Relative Humidity

When humidity data are measured in a reference setting, early morning RH will often

approach 100%, even in semiarid areas if measurements are taken inside an irrigated region.

Values of maximum relative humidity (RHmax) that consistently fall below 80% to 90%

when in an irrigated or well-watered setting may indicate problems in RH sensor calibration

or functioning or may indicate aridity of the measurement site and deviation from reference

conditions.

Figure 5.5 shows daily measurements of RHmax from an electronic agricultural

weather station located near North Baltimore, Ohio over a five year period. One would

expect RHmax to approach 100% in this subhumid setting. However, one can see clear

evidence in Figure 5.5 that the RH sensor was undermeasuring RHmax during several years,

with decreasing trends in RHmax visible during these years. This indicates that the RH sensor

was functioning electronically, except during the first half of 1988. However the calibration

of the sensor element had seriously decayed and was not valid for 1988, 1990 and 1992.

Sensor elements were typically replaced in September of each year. RH data for 1990 and

1992 could potentially be corrected by multiplying the RH measurements by a correction

factor or by adding an offset.

FIGURE 5.4

Tdew and RH from measurements near Logan, Utah, the United States during 1995 (20-minute

data)

193 194 195 196 197 198 199 200

0

10

20

30

40

50

60

70

80

0

20

40

60

80

100

Day of the year

Chilled-mirror RH "Vaisala" RH

Air and dewpoint temperature, C

o

Relative humidity, %

dewpoint temperature

air temperature

relative humidity

Relatively

Uniform Tdew

RH approaching 100%

Crop evapotranspiration

255

The type of plotting and screening demonstrated in Figures 5.4 and 5.5 shows the

simple types of integrity assessments that can be utilized in near-real-time or with historical

data. These types of assessments can be applied to all weather data used in evapotranspiration

estimation and should be adopted by operators of agricultural weather networks.

FIGURE 5.5

Daily values of measured RHmax at North Baltimore, Ohio (1988-92) showing inappropriate

calibration of the sensor for 1988, 1990 and 1992

50 100 150 200 250 300

0

20

40

60

80

100

120

Day of the year

Ma ximum Daily Relative Humidity, %

1989, 1991

1990, 1992

1988

Elements replaced

Annex 5: Measuring and assessing integrity of weather data

256

Crop evapotranspiration

257

Annex 6

Correction of weather data observed in non-

reference weather sites to compute ETo

The concept on which the FAO Penman – Monteith method for computing ETo is based

requires that weather data be measured in environmental conditions that correspond to the

definition of reference evapotranspiration. In other words, the weather data are to be

measured above an extensive grass crop that is actively evapotranspiring, or in an

environment with healthy vegetation not short of water1. Under these reference conditions,

the energy available at the surface (Rn - G) is partitioned between sensible and latent heat (H

and λE, respectively) in such a way that, in general, the ratio β = H/λEref ≤ 0.5. The

subscript ref indicates reference conditions.

Environmental conditions of arid lands that surround a non reference (arid) weather site

do not allow for the reference rate of evapotranspiration to be attained. This is generally

caused by lack of well-watered conditions. Thus, λEn/ref < λEref (subscript n/ref for non

reference conditions). If the available energy (Rn - G) is the same, then the partitioning

among sensible and latent heat changes, with Hn/ref > Href and, often, βn/ref > 0.5.

Consequently, since air temperature increases with increasing H, the air temperatures

measured at non reference sites are higher than those that would have been measured if

reference conditions had existed, i.e. Tn/ref > Tref. On the contrary, humidity measured at a

non reference site is lower than that which would have occurred under reference conditions,

thus ea n/ref < ea ref and VPDn/ref > VPDref.

When computing ETo using standard estimates for Rn - G, ra and rs, ETo will be

overestimated when calculated using Tn/ref and VPDn/ref. A correction is therefore required

to bring temperature and humidity data closer to the reference conditions.

In an environment having healthy vegetation and adequate soil moisture (reference

conditions), minimum air temperature Tmin usually approaches dew point temperature, Tdew,

(see Figure 6.2 for Kimberly, Idaho, the United States)2. This especially occurs if the wind

dies down by early morning and when soil moisture is high (illustrated through the ratio

precipitation/ETo, in Figure 6.1). Air temperatures decrease during night time due to surface

cooling caused by long-wave emission and evaporation when VPD is positive. When near

surface air temperature T approaches Tdew, T is prevented from decreasing below Tdew by

condensation of vapour from the air and the correspondent heating effect of released latent

heat. Thus, for reference conditions the relationship (Tmin)ref = (Tdew)ref is generally valid.

1 More detailed discussions are given in Allen (1996) and Allen et al (1996).

2 However, air temperature may not decrease to the dew point when large amounts of warm and dry

air are transported to the surface by wind.

Annex 6: Correction of weather data observed in non-reference weather sites to compute ETo

258

FIGURE 6.1

Comparison of differences between the monthly values of minimum and dew point temperature

(Tmin - Tdew) corresponding to monthly ratios of precipitation/ETo Sudan, Africa and the United

States

00.511.522.53

(5)

0

5

10

15

20

25

Precipitation / ETo

T - T , C

o

dew

min

Mean M onthly D ata

Sudan, Africa

63 stations

00.511.522.53

(5)

0

5

10

15

20

25

P re cip ita tio n / E To

T - T , C

o

dew

min

Mean Monthly Data

United States

26 Stations

Crop evapotranspiration

259

For non reference sites, soil moisture and/or vegetation limitations make λETn/ref < λ

ETref or ETn/ref < ETo. Thus Tmin may remain above Tdew. One cause of this phenomenon

is the large "reservoir" of sensible heat created during daytime in the atmosphere (Hn/ref >

Href, as suggested before), which is transferred towards the surface during the night,

reducing the effect of cooling by long wave radiation. Another cause is the lack of soil

moisture for evaporative cooling during night time.

This phenomenon can be observed in Figure 6.1, where monthly means for Tmin - Tdew

are plotted for weather stations operated by national governments of two countries, Sudan and

the United States. The data are plotted against the monthly ratios of precipitation to reference

ETo. The P/ETo ratios indicate the availability of adequate soil water to support reference

(well-watered) conditions in the absense of irrigation. As illustrated by the data, Tmin

approaches Tdew for nearly all stations when the ratio P/ETo approaches and exceeds 1.

When P/ETo < 1, then the aridity of the station causes Tmin to substantially exceed measured

Tdew. The exception is for those weather stations that have P/ETo < 1, but are irrigated or

have adequate soil water reserves from a prior month. The similarity between data of Sudan

and the United States indicates that this is a general phenomenon.

An additional comparison is given in Figure 2, where Tmin - Tdew are compared for

two semiarid locations in Idaho, the United States that are separated by 200 km. One

location, Kimberly, is a reference site in the middle of a large irrigated area. The other,

Boise, is a non reference site, located at an airport and surrounded by a mixture of irrigated

and non irrigated rangeland. It can be seen that Tmin approaches Tdew frequently for the

irrigated site at Kimberly, with only small differences occuring during months where a dry

climate occurs (low precipitation/ETo ratio). On the contrary, Tmin was as much as 10ºC

higher than Tdew for the nonreference Boise station. From this graphical comparison, one can

conclude that data for the nonreference Boise site require appropriate correction before being

utilized to compute ETo using the FAO - PM method. This is necessary to avoid

overestimation of ETo due to overestimation of air temperature and VPD.

Adjustment of Tmax, Tmin and Tdew

The empirical method described herein intends to correct the observed temperatures, Tmax

and Tmin in proportion to the difference (Tmin - Tdew), which works as an indicator of the

overestimation of (Tn/ref - Tref). Since Tdew defines the actual vapour pressure (ea = eº

(Tdew)), correcting Tdew also provides an adjustment for VPD.

The methodology proposed is the following:

1. Compare Tmin - Tdew (Tdew measured or computed from ea using equations (11) or

(12) in Annex 3) from a non reference site with those from a reference site using a

graphical procedure such as in Figure 6.2 and using monthly ratios of Precipitation/ETo

as the abcissa. Daily or monthly data are utilized to compute Tmin - Tdew.

2. When differences Tmin - Tdew for the non reference site are systematically higher than

about 2ºC relative to the reference site, then compute the average differences

∆T = T - T

min dew (6-1)

for the months which require correction (in general this will occur when the monthly

ratio Precipitation/ETo does not exceed 0.5).

Annex 6: Correction of weather data observed in non-reference weather sites to compute ETo

260

Or, if comparing Tmin - Tdew from the nonreference sets to Tmin - Tdew from the

reference site, then calculate ∆T as:

(

)

(

)

ref

dewmin

ref/n

dewmin T - T T - T = T −∆ (6-2)

3. Correct temperatures for each month (or day) by:

()()

÷

ø

ö

ç

è

æ∆

2

K -T

- T = T o

obs

max

cor

max

(6-3)

() ()

÷

ø

ö

ç

è

æ∆

2

K -T

- T = T o

obs

min

cor

min

(6-4)

for ∆T > Ko, where subscripts cor and obs refer to corrected and observed values,

respectively. Ko is a “conservative” factor equal to 2ºC when the nonreference station

is not compared to a reference station (∆T is from Equation (1)). Ko = 0 when ∆T is

from Equation (6-2).

4. Correct Tdew for the same months or days as:

FIGURE 6.2

Comparison of differences between the monthly values of minimum and dew point temperature

(Tmin - Tdew) corresponding to monthly ratios of precipitation/ETofor a reference site (Kimberly,

Idaho, the United States) and for a non reference site (Boise, Idaho, the United States) in the

same region

0 0.2 0.4 0.6 0.8 1 1.2

(5)

0

5

10

15

20

25

Precipitation / ETo

T - T , C

o

dew

min

Boise, Idaho, 1961 - 1990

Mean Monthy

Kimberly, Idaho, 1965 - 1986

Monthy Means

Crop evapotranspiration

261

()()

÷

ø

ö

ç

è

æ∆

2

K-T

+ T = T o

obs

dew

cor

dew

(6-5)

where Ko has the same value as for Equations (6-3) and (6-4), and utilizing either the

observed or the calculated values for Tdew (equations (3-11) or (3-12) in Annex 3).

The user should always insure that (Tmin)cor ≥ (Tdew)cor.

5. Compute ETo with the corrected values for Tmax, Tmin and Tdew.

Adjustment of Tdew only

When RH, ea, or Tdew data are not dependable or where “correction” of Tmax and Tmin as is

done in the previous section is undesireable, a second means for “correcting” the weather

data set for station aridity is possible. This second method is to merely set

omindew KTT −= (6-6)

in the calculation for ETo where Ko = 0ºC for humid and subhumid climates and Ko = 2ºC

for arid and semiarid climates. The result of this procedure is to increase Tdew to reflect the

higher humidity anticipated under reference conditions. It is noted that in a nonreference

setting, the measured Tmin may be too high, as compared to Tmin expected for a reference

setting, so that Equation (6-6) may result in values for Tdew that are overestimated, even for a

reference condition. However, since the computation of vapor pressure deficit, VPD, in the

ETo equation, where VPD = 0.5(eo(Tmax) + eo(Tmin)) - eo(Tdew), utilizes values for air

temperature and dew point temperature that may both be too high, the upward bias in all

temperature parameters will tend to cancel, thereby presenting a VPD that is representative of

a reference condition.

Index for station aridity

For non reference sites, when humidity data are available, one can compute an aridity bias

index Abi (for monthly time scales)

()

A

ET

bi

obs

oTdew

=

ET

- 1

o

=T

min

(6-7)

between the ETo computed from the observed (non-corrected) data (subscript obs) and, for

the same period, using Tmin as an estimate of Tdew. If there is not a large difference between

Tmin and Tdew, then Abi~0. When ∆T = Tmin - Tdew is large (i.e., for a nonreference

condition), then the aridity bias index Abi becomes > 0.

The user should compare aridity bias indices for the dry and humid months and decide

whether higher values for Abi result from aridity or from other causes. A correction may be

required when Abi are consistently greater than 0.05. The correction of temperature and

humidity data can be performed as indicated in the previous sections.

It is important for the user to realize that these corrections are to improve the calculations of

ETo only, since ETo is defined for a well-watered environment. For hydrology studies where

actual ET is required, then no adjustment should be made to air temperature and dew point

temperature, since the ETo n/ref characterises the natural evaporation demands of the climate.

Annex 6: Correction of weather data observed in non-reference weather sites to compute ETo

262

Any corrected Tmax, Tmin, Tdew data should not be reintroduced into the original historical

data series. Also, the user should note that all of the correction procedures presented here

are only approximate attempts to bring the ETo calculations closer to the “real” ETo that

reflects a well-watered environment. Any errors or uncertainties introduced by these

adjustments at a specific site will remain largely unknown. Therefore the user is encouraged

to use caution.

Crop evapotranspiration

263

Annex 7

Background and computations for Kc for

the initial stage for annual crops

ET during the initial stage for annual crops is predominately in the form of evaporation.

Therefore, accurate estimates for Kc ini must consider the frequency that the soil surface is

wetted during the initial period. The initial period was defined in Chapter 6 for annual crops

as the period between the planting date and the date of approximately 10% ground cover.

Chapter 6 presents background and figures for predicting Kc ini as a function of

reference evapotranspiration (ETo), soil texture, and frequency and depth of wetting.

Additional background and equations are given in Chapter 7. This annex provides further

background on development of the Kc ini curves that are presented in Figures 29 and 30 of

Chapter 6. Equations are presented here that can be used in place of Figures 29 and 30 when

computers are used.

INTRODUCTION

Evaporation from bare soil (Es) can be characterized as occurring in two distinct stages. The

stage 1 is termed the "energy limited" stage. During this stage, moisture is transported to the

soil surface at a rate sufficient to supply the potential rate of evaporation (Eso), which, in

turn, is governed by energy availability at the soil surface. In this procedure, Eso is estimated

from

oso ET15.1E = (7-1)

where Eso is the potential rate of evaporation [mm d-1] and ETo is the the mean ETo during

the initial period [mm d-1]. The value 1.15 represents increased evaporation potential due to

low albedo of wet soil and the possibility of heat stored in the surface layer during previous

dry periods.

Stage 2 is termed the "soil limited" stage, where hydraulic transport of subsurface

water to the soil surface is unable to supply water at the potential evaporation rate. During

stage 2, the soil surface appears partially dry and a portion of the evaporation occurs from

below the soil surface. The energy required for subsurface evaporation is supplied by

transport of heat from the soil surface into the soil profile. The evaporation rate during stage

2 drying decreases as soil water content decreases as shown in Figure 7.1 (see also Figure 38

of Chapter 7). The evaporation rate can therefore be expressed as being proportional to the

water remaining in the evaporation layer relative to the maximum depth of water that can be

evaporated from the same soil layer during stage 2 drying.

Annex 7: Background and computations for Kc for the initial stage for annual crops

264

The maximum total depth of water that can be evaporated from the surface soil layer is

termed “total evaporable water” or TEW. Equation 73 of Chapter 7 is used to predict TEW.

In turn, the maximum total depth of water that can be evaporated during stage 1 is termed

“readily evaporable water” or REW. Table 19 of Chapter 7 includes recommended values

for REW.

If the evaporation rate during stage 2 drying is assumed to be linearly proportional to

the equivalent depth of water remaining in the evaporation layer, as shown in Figure 7.1,

then the average soil water evaporation rate during stage 2 can be estimated, similar to

Equation 74 of Chapter 7:

ú

û

ù

ê

ë

é

−

=REWTEW

DTEW

EE e

sos (7-2)

for when De > REW, where Es is the actual evaporation rate [mm d-1] at any particular time

when the depletion from the soil surface layer equals De. De is the depletion from the surface

layer [mm] and REW is the readily evaporable water in the surface layer [mm]. The length of

time required to complete stage 1 drying (t1) is equal to t1 = REW/Eso.

GENERAL EQUATION FOR Kc ini

Equation 7-2 can be integrated over the range REW to TEW, resulting in the basic equation

for Kc ini during stage 2:

FIGURE 7.1

Two stage model for soil evaporation during the initial period

E / ET

s

Depth of Soil Water Evaporated, D , mm

e

Stage 1

Drying

Stage 2

Drying

REW TEW

De

o

1.15

0

Crop evapotranspiration

265

()

ow

so1w

inic ETt

TEW

REWTEW

REW

1E)tt(

expREWTEWTEW

K

÷

ø

ö

ç

è

æ

÷

ø

ö

ç

è

æ

−

+−−

−−

= (7-3)

for tw > t1, where Kc ini = Es/ETo, tw is the mean interval between wetting events [days]

and t1 is the time when stage 1 drying is completed (t1 = REW / Eso) [days]. The “exp”

parameter represents the exponential of the value contained within the parenthesis following

the parameter. The Kc ini calculated from Equation 7-3 is limited to Kc ini ≤ 1.15.

When tw < t1, i.e. the entire process resides within stage 1, so that:

()

1w

o

so

inic ttfor15.1

ET

E

K<== (7-4)

Where furrow or trickle irrigation is practiced, and only a portion of the soil surface is

wetted, the value calculated for Kc ini in Equations 7-3 and 7-4 should be reduced in

proportion to the average fraction of surface wetted, fw [0,1]. Indicative values for fw are

given in Table 20 of Chapter 7. Equation 60 of Chapter 6 is used to make the adjustment:

)1f(inicwinic w

KfK =

=

where fw is the fraction of surfaced wetted by irrigation or rain [0 - 1], and Kc ini (fw=1) is the

value for Kc ini for fw = 1 from Equation 7-3 or 7-4.

Accordingly, the value for the infiltration depth from irrigation (Iw) should be adjusted using

Equation 61 of Chapter 6:

w

wf

I

I=

where Iw is the depth of irrigation water that is infiltrated over the part of the surface that is

wetted [mm] and I is the depth of water infiltrated from irrigation, expressed as one-

dimensional depth over the entire surface area [mm].

TOTAL EVAPORABLE WATER

The value for TEW is the maximum depth of water that can be evaporated from the soil

following wetting. The value for TEW is governed by the depth of the soil profile

contributing to soil water evaporation and by the soil water holding properties within the

evaporating layer. In addition, the value for TEW is affected by the unsaturated hydraulic

Annex 7: Background and computations for Kc for the initial stage for annual crops

266

conductivity, by the presence of a hydraulically limiting layer beneath the evaporating layer,

and by the conduction of sensible heat into the soil to supply energy for subsurface

evaporation. An approximation for the maximum value of TEW for initial periods having ETo

≥ 5 mm d-1is:

()

eWPFC Z5.01000TEW θ−θ= (7-5)

where TEW has units of mm, θFC is soil water content at field capacity [m3 m-3], θWP is soil

water content at wilting point [m3 m-3], and Ze is the depth of the soil surface soil layer that

is subject to drying by way of evaporation [0.10 to 0.15 m]. If unknown, a value of Ze =

0.15 m is recommended. Typical values for θFC and θWP are given in Table 19 of Chapter 7.

During winter and other cool season months, less radiation energy is available to

penetrate the soil surface and to evaporate water from within a drying soil, and TEW may be

less. Therefore, when ETo < 5 mm d-1, TEW for use in Equation 7-3 is estimated as:

()

5

ET

Z5.01000TEW o

eWPFC θ−θ= (7-6)

where ETo is reference ET in mm/day. Equation 7-6 is intended to correct TEW for use

during the initial stage with mostly bare soil. It is not intended for use with the dual Kc

procedure of Chapter 7. REW is limited so that REW ≤ TEW.

NUMBER OF WETTING EVENTS AND AVERAGE DEPTH

Estimating the number of wetting events and the corresponding time between wetting events

during the initial period is described in Chapter 6. The number of wetting events (both from

precipitation and irrigation) occurring during the initial period is determined by considering

that two wetting events occurring on adjoining days can be counted as one event, and

individual wetting events of less than 0.2 ETo can be ignored.

The average time between wetting events during the initial period (tw) is approximated

as:

tL

n

wini

w

=+05. (7-7)

where tw is in days, Lini is the length of the initial period [days], and nw is the number of

wetting events during the initial period.

The average depth of water added to the evaporating layer at each wetting event is

determined by dividing the sum of the precipitation and irrigation infiltration occurring during

all wetting events by the number of events, thus:

(

)

w

wn

mean n

IP

Påå

+

= (7-8)

Crop evapotranspiration

267

where Pmean is the average depth of infiltrated water per wetting event [mm], Pn is the depth

of infiltrated precipitation occurring during the initial period, and Iw is the infiltrated

irrigation depth for the part of the surface that is wetted [mm] (Equation 61). Each individual

value of Pn and Iw must be limited in Equation 7-8 so that Pn ≤ TEW and Iw ≤ TEW where

TEW is from Equation 7-5 or 7-6.

LIMITATIONS ON TEW AND REW

In the case of wetting depths (Pmean) that are smaller than the TEW, the evaporation process,

including stage 1 drying, may terminate sooner than expected. The actual values for TEW

and REW must be corrected according to Pmean. Therefore, TEW and REW are calculated

according to the average total water available during each drying cycle:

÷

ø

ö

ç

è

æ+=

w

ini

meancor n

W

P,TEWminTEW (7-9)

and

ú

û

ù

ê

ë

é

÷

ø

ö

ç

è

æ+

=1,

TEW

n

W

P

minREWREW w

ini

mean

cor (7-10)

where “min ()” is a function to select the minimum value of those in braces that are separated

by the comma, and where TEW is from Equation 7-5 or 7-6. Wini is the equivalent depth of

water [mm] in the evaporation layer (of thickness Ze ) at the time of planting (beginning of

the initial period). Wini has a maximum value of TEW when the initial soil water content of

the evaporation layer is at field capacity. Values for TEWcor and REWcor from Equations 7-

9 and 7-10 are used in place of TEW and REW in Equation 7-3.

EQUATIONS FOR FIGURES 29 AND 30 OF CHAPTER 6

Figures 29 and 30 of Chapter 6 can be reproduced numerically by applying Equation 7-3

under the following conditions. For all applications:

t1 = REWcor / Eso and Eso = 1.15 ETo (Equation 1).

If t1 < tw then Kc ini = 1.15 (Equation 4), and Equation 7-3 is not applied.

Otherwise, apply Equation 3 using the following parameters (TEWcor and REWcor are

used in place of TEW and REW in Equation 3):

For Figure 29 (all soil textures having light infiltration depths (< 10 mm)):

TEWcor = 10 mm

REWcor = min( max(2.5, 6 / (ETo)0.5), 7 )

Annex 7: Background and computations for Kc for the initial stage for annual crops

268

For Figure 30a (coarse soil textures having large infiltration depths (≥

≥≥

≥40 mm)):

TEWcor = min(15, 7 (ET)0.5)

REWcor = min(6, TEWcor - 0.01)

For Figure 30b (medium and fine soil textures having large infiltration depths (≥

≥≥

≥ 40

mm)):

TEWcor = min(28, 13 (ET)0.5)

REWcor = min(9, TEWcor - 0.01)

The max() and min() functions indicate the selection of the maximum or minimum

value of the parameters that are separated by the comma. Most programming languages and

spreadsheet programs include these functions.

The numerical application of Equation 7-3 using the parameters and constraints listed here

will fully reproduce Figures 29 and 30a and b, with the exception that calculations made in the

vicinity of ETo = 5 mm d-1 may deviate from the curves in Figures 30a and b, since curves in

the vicinity of ETo = 5 mm d-1 were smoothed before plotting. The smoothing caused small,

insignificant differences between the figures and the numerical procedure. The parameters

listed above are reduced from equations 5 through 10 and using typical values for θFC and θWP.

In situations where wetting events are not equally spaced during the initial period, the

dual Kc approach of Chapter 7, along with a daily soil water balance, can provide for more

accurate results.

EXAMPLE 7-1

Application of Equation 7-3 to Example 25

As in Example 25 in Section B, small vegetables are cultivated in a dry area on a coarse textured soil

and receive 20 mm of water twice a week by means of a sprinkler irrigation system. The average ETo

during the initial stage is 5 mm/day. Estimate the crop evapotranspiration during that stage.

For:

tw = 7/2 =

ETo =

Eso = 1.15 ETo = 1.15 (5) =

3.5

5

5.75

day interval

mm/day

For Fig. 29:

TEWcor =

REWcor = min(max(2.5, 6/(50.5), 7) =

t1 = REW/Eso = 2.7/5.75 =

since tw > t1, use Eq. 7-3: Kc ini(Fig. 29) = (10 – (10-

2.7)

exp[-(3.5-0.47)(5.75)(1+2.7/(10-2.7))/10])/(3.5(5)) =

10

2.7

0.47

0.57

mm

days

-

For Fig. 30.a:

TEWcor = min(15, 7(50.5)) =

REWcor = min(6, 15-0.001) =

t1 = REW/Eso = 6/5.75 =

since tw > t1, use Eq. 7-3: Kc ini(Fig. 30a) = (15 – (15-

6)

exp[-(3.5-1.04)(5.75)(1+6/(15-6))/15])/(3.5(5)) =

15

6

1.04

0.75

mm

days

-

For: I = 20 mm

From Eq. 59:

Kc ini = 0.57 + [(20-10)/(40-10)] (0.75-0.57)

= 0.57+0.33(0.12)=

0.63

-

From Eq. 58: ETc = 0.63 (5) = 3.2 mm/day

The average crop evapotranspiration during the initial growth stage for the small vegetables is

3.2 mm/day. The values in this example agree relatively closely with those obtained from Example 25.

Crop evapotranspiration

269

Annex 8

Calculation example for applying the dual

Kc procedure in irrigation scheduling

This annex illustrates in more detail the application of the various equations for calculating

Kcb, Ke and ETc using the dual Kc approach of Chapter 7. The example is in the form of a

computer spreadsheet and is applied to the dry, edible bean crop that was used in example

boxes 15 and 16 of Chapters 6 and 7. The spreadsheet is shown in Figure 8.1, where the

irrigation schedule is determined using the daily soil-water balance procedure described in

Chapter 8. The timing of irrigations is based on the management allowed depletion (MAD) of

the available water that can be stored in the root zone. The irrigation schedule and the

corresponding estimated wet soil evaporation are different from the actual values shown in Box

16 of Chapter 7, since Box 16 represents the actual irrigation schedule used at Kimberly during

1974. The actual schedule deviated somewhat from the theoretical schedule of Figure 8.1.

The spreadsheet formulas used for calculations and the references to equations in the text

are indicated in Box 8.1. The variable names used for parameters follow the same convention

used in Chapters 1 to 9. The variable names are defined in the List of principal symbols and

acronymns in the introduction to this paper. A few exceptions are defined in Table 8.1.

The spreadsheet in Figure 8.1 includes columns for variables Tmax, u2, and Tdew. The

Tmax and Tdew columns are used to calculate daily RHmin. The u2 and RHmin columns are

used to adjust Kcb mid and Kcb end using Equation 70 of Chapter 7 and to calculate Kc max

using Equation 72 on a daily basis. The data in the first 7 rows of Figure 8.1 that appear within

boxes represent the specific crop and soils information that is entered by the user for a

particular crop and soil combination. All other information (outside of boxes) is calculated

automatically by the spreadsheet program. The columns having double underlined headings

represent the data that are input by the user into the spreadsheet.

The calculations in Figure 8.1 can be used to verify other computer programs or

spreadsheet calculations for Ke, Kc and ETc. Small differences may result, depending on the

assumptions of timing of irrigations. The spreadsheet of Figure 8.1 presumes that all irrigation

and precipitation events occur early in the morning. The scheduling and magnitudes of

irrigations are based on the soil water depletion at the end of the previous day. The spreadsheet

also presumes that all drainage from the root zone due to excess precipitation occurs on the day

of the precipitation event. It is assumed that runoff from precipitation is zero. If necessary,

procedures for predicting precipitation runoff can be entered into the spreadsheet using

procedures described in most standard hydrology textbooks. It is assumed that the net depth of

irrigation that is retained in the crop root zone is exactly equal to the depletion depth of the

previous day. This assumption presumes perfect knowledge of soil water depletion by the

irrigator or that all irrigations are adequate or excessive. This assumption may not hold for

some irrigation conditions and can be changed by the user as needed.

Annex 8:Calculation example for applying the dual Kc procedure in irrigation scheduling

270

Spreadsheet formulas used to create the spreadsheet of Figure 8.1 are listed in Box 8.1 for the

Microsoft Excel language (versions 5 and higher). Formulae for other types of spreadsheets

would be similar. Formulae for the Corel Quattro-Pro language (versions 5 and higher) can be

downloaded from the FAO internet site.

Crop evapotranspiration

271

Annex 8:Calculation example for applying the dual Kc procedure in irrigation scheduling

272

Crop evapotranspiration

273

Annex 8:Calculation example for applying the dual Kc procedure in irrigation scheduling

274

BOX 8.1. Spreadsheet formulas and corresponding equations for Excel spreadsheet programs.

Formulas for Rows 1 to 15 of Figure 8.1 (for MicroSoft Excel, versions 5/95 and

later)

Equation

in text or Underlined numeric values are input by the user

footnote Cell Text, value, or formula

A1: Example Spreadsheet for Calculating ETc = (Kcb + Ke)ETo and an

Irrigation Schedule

P2: Computed Dates for Stages:

A3: Crop:

B3: Dry, Edible Beans

F3: Table 11:

I3: Table 12:

J3: Following Adjustment:

P3: JPlant

Table 2.5 Q3: =TRUNC(275*C5/9-30+C6)+IF(C5>2,-2,0)+ IF(MOD(C14,4)=0,+1,0)

V3: fw (irrig.):

X3: 0.5

AE3: Rootmin

AF3: 0.2

AG3: m

AH3: MAD during Initial Stage

AK3: 70

AL3: %

E4: Lini

F4: 25

H4: Kcb ini

I4: 0.15

J4: =I4

L4: Kcmin

M4: =J4

P4: JDev

Q4: =Q3+F4

V4: REW:

X4: 8

Y4: mm

AE4: Rootmax

AF4: 0.8

AG4: m

AH4: MAD after Initial Stage

AK4: 45

AL4: %

A5: Planting:

B5: Month

C5: 5

E5: Ldev

F5: 25

H5: Kcb mid

I5: 1.1

Eq. 70 J5: =I5+(0.04*($K$8-2)-0.004*($K$9-45))*($M$5/3)^0.3

L5: Max.Ht.:

M5: 0.4

N5: m

P5: JMid

Q5: =Q4+F5

V5: TEW:

X5: 22

Y5: mm

Crop evapotranspiration

275

BOX 8.1, continued.

AE5: Avail.Water

AF5: 160

AG5: mm/m

B6: Day

C6: 22

E6: Lmid

F6: 30

H6: Kcb end

I6: 0.25

Eq. 70 J6: =IF(I6<0.45,I6,I6+(0.04*($K$8-2)-0.004*($K$9-45))*($M$5/3)^0.3)

P6: JLate

Q6: =Q5+F6

V6: initial De:

X6: 18

Y6: mm

E7: Llate

F7: 20

P7: JHarv

Q7: =Q6+F7

V7: Initial fw:

X7: 1

H8: Midseas. Av. Wind Speed:

(1) K8: =(VLOOKUP(Q6,D14:AP183,38)-

VLOOKUP(Q5,D14:AP183,38))/(Q6-Q5)

L8: m/s

M8: <----Computed automatically from Lookup on column AO

AH8: (Irrigation that is needed is presumed applied at beginning of

next day)

H9: Midseas. Av. RHmin:

(1) K9: =(VLOOKUP(Q6,D14:AP183,39)-

VLOOKUP(Q5,D14:AP183,39))/(Q6-Q5)

L9: %

M9: <----Computed automatically from Lookup on column AP

First row of formulas (row 14)

Note: some formulas in row 14 (first day) vary from those in rows 15 onward. See row 15 for

example calculations for all subsequent days.

A14: 5

B14: 15

C14: 74

Table 2.5 D14: =TRUNC(275*A14/9-30+B14)+IF(A14>2,-2,0)+IF(MOD(C14,4)=0,+1,0)

E14: 10

F14: 5.7655

G14: 0

H14: 3.4

Eq. 14 I14: =0.6108*EXP((17.27*G14)/(G14+237.3))

Eq. 11 J14: =0.6108*EXP((17.27*E14)/(E14+237.3))

Eq. 63 K14: =I14/J14*100

L14: 0

Eq. 66(2) O14: =IF(D14<$Q$4,$J$4,IF(D14<$Q$5,$J$4+(D14-$Q$4)/

$F$5*($J$5- $J$4),IF(D14<$Q$6,$J$5,IF(D14<$Q$7,

$J$5+(D14-$Q$6)/$F$7*($J$6-$J$5),$J$4))))

(3) P14: =MAX(O14/$J$5*$M$5,P13)

Eq. 72 Q14: =MAX(1.2+(0.04*(F14*0.9-2)-0.004*(K14-45))*(P14/3)^0.3, O14+0.05)

(4) R14: 0

Eq. 76 S14: =MAX(((O14-M$4)/(Q14-M$4))^(1+0.5*P14),0.01)

(5) T14: =IF(R14>0,X$3,IF(L14>0,1,X7))

Annex 8:Calculation example for applying the dual Kc procedure in irrigation scheduling

276

BOX 8.1, continued.

Eq. 75 U14: =MIN(1-S14,T14)

(6) V14: =X6

Eq. 74 W14: =MAX(IF(V14<X$4,1,(X$5-V14)/(X$5-X$4)),0)

Eq. 71 X14: =MIN(+W14*(Q14-O14),U14*Q14)

Y14: =X14*H14

Eq. 79 Z14: =MAX(L14+R14,0)

Eq. 77(6) AA14: =V14-L14-R14+Y14/U14+Z14

(7) AB14: =O14+X14

Eq. 69(7) AC14: =AB14*H14

Eq. 8.1(8) AE14: =MAX((O14-$J$4)/($J$5-$J$4)*($AF$4-$AF$3)+$AF$3,AE13)

Eq. 82 AF14: =MAX(IF(D14<Q$4,AK$3,AK$4)/100*AE14*$AF$5,AF13)

Eq. 85(9) AG14: =$X$6-L14+AC14

(10) AH14: =IF(D14>=Q$3,IF(D14<(Q$6+Q$7)/2,IF(AG14>AF14, AG14,0), 0), 0)

Eq. 88 AI14: =MAX(+L14-AC14-$X$6,0)

Eq. 84(11) AJ14: =IF(AG14>AF14,(AE14*AF$5-AG14)/(AE14*AF$5-AF14),1)

Eq. 80 AK14: =X14+O14*AJ14

Eq. 85(9) AL14: =+$X$6-L14+AK14*H14+AI14

(12) AO14: =F14

(12) AP14: =K14

Second row of formulas

All rows below row 15 are similar.

A15: 5

B15: 16

C15: 74

Table 2.5 D15: =TRUNC(275*A15/9-30+B15)+IF(A15>2,-2,0)+IF(MOD(C15,4)= 0,+1,0)

E15: 13.3

F15: 2.2175

G15: -5

H15: 4.1

Eq. 14 I15: =0.6108*EXP((17.27*G15)/(G15+237.3))

Eq. 11 J15: =0.6108*EXP((17.27*E15)/(E15+237.3))

Eq. 63 K15: =I15/J15*100

L14: 0

Eq. 66 (2) O15: =IF(D15<$Q$4,$J$4,IF(D15<$Q$5,$J$4+(D15-$Q$4)/$F$5*

($J$5-$J$4),IF(D15<$Q$6,$J$5,IF(D15<$Q$7,$J$5+

(D15-$Q$6)/$F$7*($J$6-$J$5),$J$4))))

(3) P15: =MAX(O15/$J$5*$M$5,P14)

Eq. 72 Q15: =MAX(1.2+(0.04*(F15*0.9-2)-0.004*(K15-45))*(P15/3)^0.3, O15+0.05)

(4) R15: =IF(AH14>0,AH14/$X$3,0)

Eq. 76 S15: =MAX(((O15-M$4)/(Q15-M$4))^(1+0.5*P15),0.01)

(5) T15: =IF(R15>0,X$3,IF(L15>0,1,T14))

Eq. 75 U15: =MIN(1-S15,T15)

(6) V15: =MAX(AA14-L15-R15,0)

Eq. 74 W15: =MAX(IF(V15<X$4,1,(X$5-V15)/(X$5-X$4)),0)

Eq. 71 X15: =MIN(+W15*(Q15-O15),U15*Q15)

Y15: =X15*H15

Eq. 79 Z15: =MAX(L15+R15-AA14,0)

Eq. 77(6) AA15: =AA14-L15-R15+Y15/U15+Z15

(7) AB15: =15+X15

Eq. 69(7) AC15: =AB15*H15

Eq. 8.1(8) AE15: =MAX((O15-$J$4)/($J$5-$J$4)*($AF$4-$AF$3)+$AF$3,AE14)

Eq. 82 AF15: =MAX(IF(D15<Q$4,AK$3,AK$4)/100*AE15*$AF$5,AF14)

Eq. 85(9) AG15: =AK14 -L15-AH14+AC15

(10) AH15: =IF(D15>=Q$3,IF(D15<(Q$6+Q$7)/2,IF(AG15>AF15, AG15,0), 0),0)

Eq. 88 AI15: =MAX(+L15+AH14-AC15-AK14,0)

Eq. 84(11) AJ14: =IF(AG15>AF15,(AE15*AF$5-AG15)/(AE15*AF$5-AF15),1)

Eq. 80 AK14: =X15+O15*AJ15

Eq. 85(9) AL15: =+AL14-L15-AH14+AK15*H15+AI15

(12) AO15: =AO14+F15

(12) AP15: =AP14+K15

Crop evapotranspiration

277

BOX 8.1, continued.

Footnotes:

(1) Cells K8 and K9 use the vertical lookup function to automatically calculate the average wind

speed and average daily minimum relative humidity during the midseason period. The lookup

function uses cumulative totals of wind speed and RHmin that are calculated in columns AO and

AP.

(2) The formula to calculate Kcb for each day uses a series of imbedded IF statements to determine

which growing period the day is in. Linear interpolation is applied when the day is within the

development and late season growing periods.

(3) The crop height on any day is calculated as proportional to the value of Kcb on that day to the

Kcb mid value, multiplied by the maximum crop height that has been entered by the user in cell

M5. The value for crop height is not allowed to decrease with time. Hence, the MAX() function is

employed, comparing with the value of the previous day.

(4) The value for irrigation depth (divided by fwto express the depth over the wetted fraction of the

soil, only) is presumed to occur early in the day. This value is based on a decision made at the

end of the previous day (column AH), based on whether or not the ending soil water depletion on

the previous day has exceeded the readily available water (RAW). The irrigation depth on the

first day is presumed to be zero.

(5) The value for fw is determined according to the last occurrence of precipitation or irrigation, as

described in Chapter 7.

(6) The depletion of the evaporation layer (top soil layer) at the beginning of the day is presumed to

equal the depletion at the end of the previous day less any precipitation or irrigation, which is

assumed to occur very early in the day. The value for De,i is limited to ≥ 0.

The depletion of the evaporation layer at the end of the day is calculated according to Eq. 77,

with root extraction of plant transpiration from the evaporation layer assumed to equal zero.

(7) The value for Kc is calculated as Kc = Kcb + Ke and the value for ETc is calculated as Kc x ETo.

(8) The depth of the effective root zone on any day is calculated as being proportional to the ratio o

f

the value of the Kcb on that day (above the value of Kc min) to the Kcb mid – Kc min, as

described in Eq. 1 of this annex. The rooting depth is not allowed to decrease with time.

Therefore, the MAX() function is utilised, where the value for the previous day is compared.

(9) The “first” estimate for ending depletion of the root zone (Dr,,i) is estimated using Eq. 85, with

drainage assumed to be zero and with ETc for nonstressed conditions. The value for Dr, i is

then recalculated in Column AK, after any drainage loss is estimated and after any reduction in

ETc, to account for low soil water content. The value for Dr, i in column AK represents depletion

of the root zone at the end of the day.

(10) The net depth of irrigation needed is based on the value of soil water depletion at the end of the

day. It is assumed that irrigation will be applied at the beginning of the following day. The

formula in column AH checks to insure that the specific day is within the growing period. The

formula assumes that no irrigation will be desired during the last one-half of the late-season

period. This assumption may need to be modified for some other crops. The value fo

r

management allowed depletion is allowed to have a different (normally higher) value during the

initial period as compared to during the rest of the growing season.

(11) The stress coefficient Ks represents the Ksunder the current conditions of soil water. The value

for Ks is reduced below 1.0 using Eq. 84 if the depletion of the root zone (following any irrigation

or precipitation earlier in the day) is greater than the readily available water (RAW). It is

presumed that the stress point, p, is the same as the value entered for MAD. This presumption

can be modified as needed.

(12) Columns AO and AP contain cumulative sums of daily wind speed and daily minimum relative

humidity. These columns are used to calculate mean values for u2 and RHmin during the

midseason period (footnote 1).

Annex 8:Calculation example for applying the dual Kc procedure in irrigation scheduling

278

TABLE 8.1

List of variable names in the spreadsheet that are not included in the List of principal symbols and

acronyms in the introduction section of this paper.

Avail. Water water available to plant (field capacity – wilting point) [mm/m]

JPlant number of day of the year at time of planting [-]

JDev number of day of the year at beginning of development period [-]

JMid number of day of the year at beginning of midseason period [-]

JLate number of day of the year at beginning of late season period [-]

JHarv number of day of the year at time of harvest or death [-]

Max. Ht. mean height of vegetation during the midseason period [m]

MAD during initial stage management allowed depletion fraction during the initial growing period [-]

MAD after initial stage management allowed depletion fraction following the initial growing period

(during all other periods) [-]

Rootmin average depth of “effective” root zone during the initial period (also

described as Zr min) [m]

Rootmax maximum depth of “effective” root zone (also described as Zr max) [m]

The daily values calculated for Kcb and Kc are illustrated in Figure 8.2. The daily soil

water depletion at the end of each day calculated in the spreadsheet example is graphed in

Figure 8.3. Figure 8.3 illustrates the effect of an increasing root zone on the allowable

depletion. The allowable depletion is the same as the readily available water (RAW) when it

is assumed that MAD = p, the evapotranspiration depletion factor. The depth of the

effective root zone is calculated on each day as:

FIGURE 8.2

Daily values for Kcb from the calculation example of Figure 8.1

140 160 180 200 220 240

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0

40

80

120

160

200

Day of the year

K

c

precipitation and irrigation, mm

Dry, Edible Beans Kimberly, Idaho, 1974

I

II II

PP

I

Kc

Kcb

Crop evapotranspiration

279

()

mid

inicbmidcb

inicbicb

minrmaxrminrir JJfor

KK

ZZZZ <

−

−+= (8-1)

and

midmaxrir JJforZZ ≥= (8-2)

where

Z

r i effective depth of the root zone on day i [m]

Z

r min initial effective depth of the root zone (at the beginning of the initial

period (planting))

Z

r max maximum effective depth of the root zone during the midseason

period (from Table 22 of Chapter 8)

J Day of year [1 to 366]

Z

r min is the same as variable Rootmin that is used in Figure 8.1 and Zr max is the

same as Rootmax. Equations 8-1 and 8-2 presume that the development of the root zone

increases in proportion to the increase in Kcb. This implies that the maximum effective root

depth is reached by the beginning of the midseason. Other approaches to estimate Zr can be

used, including interpolations based on time of season, for example:

()

startmax

start

minrmaxrminrir JJ

JJ

ZZZZ −

−

−+= for Jstart ≤ J ≤ Jmax (8-3)

and Zr i = Zr min when J < Jstart, and Zr i = Zr max when J > Jmax, where:

J

start Day of year at beginning of the increase in Zr i beyond Zr min

J

max Day of year at the attainment of maximum rooting depth

Z

r min for annual crops should represent the depth of seed placement plus an additional

depth of soil that may contribute water to the seed as it extends its initial roots downward

following germination. For many annual crops, Zr min can be be estimated as 0.15 to 0.20

m.

The value used for MAD is given a separate and larger value during the initial period to

account for the ability of roots for some crops to extract water at relatively dry water contents

during germination and during the initial period with little impact by stress. In this example, it

is assumed that p =MAD.

The irrigation period for the bean crop is presumed to begin at planting and to

terminate half-way through the late season period. Therefore, the last irrigation date is on day

225. The bean crop exhibited only a small amount of stress following day 225, since the Kc

was declining. The stress coefficient (Ks) is calculated in column AJ of the spreadsheet.

The fact that irrigations are not applied in the spreadsheet until the the soil water

depletion at the end of the previous day is greater than or equal to RAW occasionally causes a

small amount of stress on the day prior to irrigation (see Ks in column AJ). The impact of Ks

on Kc adj was small before planting and near the end of the growing season because Kcb is

small relative to the potential value for Ke during these periods.

Annex 8:Calculation example for applying the dual Kc procedure in irrigation scheduling

280

This particular example is intended only to demonstrate how to apply the soil

evaporation equations during scheduling of irrigations. The procedure used to determine the

irrigation schedule and the assumptions used may not always be appropriate. The reader

should modify the irrigation scheduling procedure to fit the conditions of the local area.

FIGURE 8.3

Soil water depletion at the end of each day calculated in Figure 8.1

140 160 180 200 220 240

0.0

20.0

40.0

60.0

80.0

0

100

200

300

400

500

600

700

800

Day of the year

RAW and Depletion, mm

precipitation and irrigation, mm

Dry, Edible Beans

Kimberly, Idaho, 1974

IIIIIII

PP

II

RAW

depletion

Crop evapotranspiration

281

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