Evaluation of a Parametric Approach for Estimating Potential Evapotranspiration Across Different Climates

Abstract

Potential evapotranspiration (PET) is key input in water resources, agricultural and environmental modelling. For many decades, numerous approaches have been proposed for the consistent estimation of PET at several time scales of interest. The most recognized is the Penman-Monteith formula, which is yet difficult to apply in data-scarce areas, since it requires simultaneous observations of four meteorological variables (temperature, sunshine duration, humidity, wind velocity). For this reason, parsimonious models with minimum input data requirements are strongly preferred. Typically, these have been developed and tested for specific hydroclimatic conditions, but when they are applied in different regimes they provide much less reliable (and in some cases misleading) estimates. Therefore, it is essential to develop generic methods that remain parsimonious, in terms of input data and parameterization, yet they also allow for some kind of local adjustment of their parameters, through calibration. In this study we present a recent parametric formula, based on a simplified formulation of the original Penman-Monteith expression, which only requires mean daily or monthly temperature data. The method is evaluated using meteorological records from different areas worldwide, at both the daily and monthly time scales. The outcomes of this extended analysis are very encouraging, as indicated by the substantially high validation scores of the proposed approach across all examined data sets. In general, the parametric model outperforms well-established methods of the everyday practice, since it ensures optimal approximation ofpotential evapotranspiration.

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Agriculture and Agricultural Science Procedia 4 ( 2015 ) 2 – 9
2210-7843 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of Data Research and Consulting
doi: 10.1016/j.aaspro.2015.03.002
Available online at www.sciencedirect.com
ScienceDirect
IRLA2014. The Effects of Irrigation and Drainage on Rural and Urban Landscapes, Patras, Greece
Evaluation of a Parametric Approach for Estimating Potential
Evapotranspiration across Different Climates
Aristoteles Tegos
a
*, Andreas Efstratiadisa, Nikolaos Malamosb, Nikolaos Mamassisa,
Demetris Koutsoyiannisa
aDepartment of Water Resources, School of Civil Engineering, National Technical University of Athens, Heroon Polytechneiou 5, GR-157 80
Zographou, Greece
bDepartment of Agricultural Technology, Technological Educational Institute of Western Greece, Amaliada, Greece
Abstract
Potential evapotranspiration (PET) is key input in water resources, agricultural and environmental modelling. For many decades,
numerous approaches have been proposed for the consistent estimation of PET at several time scales of interest. The most
recognized is the Penman-Monteith formula, which is yet difficult to apply in data-scarce areas, since it requires simultaneous
observations of four meteorological variables (temperature, sunshine duration, humidity, wind velocity). For this reason,
parsimonious models with minimum input data requirements are strongly preferred. Typically, these have been developed and
tested for specific hydroclimatic conditions, but when they are applied in different regimes they provide much less reliable (and
in some cases misleading) estimates. Therefore, it is essential to develop generic methods that remain parsimonious, in terms of
input data and parameterization, yet they also allow for some kind of local adjustment of their parameters, through calibration. In
this study we present a recent parametric formula, based on a simplified formulation of the original Penman-Monteith expression,
which only requires mean daily or monthly temperature data. The method is evaluated using meteorological records from
different areas worldwide, at both the daily and monthly time scales. The outcomes of this extended analysis are very
encouraging, as indicated by the substantially high validation scores of the proposed approach across all examined data sets. In
general, the parametric model outperforms well-established methods of the everyday practice, since it ensures optimal
approximation ofpotential evapotranspiration.
© 2015 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of Technological Educational Institute of Epirus, Hydroconcept R&D (www.hydroconcept.gr)
Keywords: Potential evapotranspiration; Penman-Monteith formula; parametric model; calibration
* Corresponding author. Tel.: +30 2121029102 E-mail address: tegosaris@yahoo.gr
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of Data Research and Consulting

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Aristoteles Tegos et al. / Agriculture and Agricultural Science Procedia 4 ( 2015 ) 2 – 9
1. Introduction
The accurate estimation of evapotranspiration has a great importance in hydrological modeling, irrigation
planning and water resources management. Several related studies have been performed during past decades and the
attempts of estimating water requirements for irrigation purposes; go back to 1890 in U.S.A (Jensen and Haise,
1963).
More than 50 important evapotranspiration models can be found in literature (Lu et al., 2005, McMahon et al.
2013), which can be grouped into seven categories: (i) empirical, (ii) water budget (iii) energy budget, (iv) mass
transfer, (v) combination, (vi) radiation and (vii) measurement (Xu and Singh, 2000).
The variety of models and frameworks is related to the complexity of the natural phenomenon and depends on
the wide range of input climate data and local climate conditions.
The Penman-Monteith formulation (Monteith, 1981) for computing potential ET proposed from FAO as
standardized method (Allen et al., 1998) That method had numerous successful applications in the fields of
hydrology and agrometeorology and in a variety of hydroclimatic regimes (Wang and Georgakakos, 2007). Basic
disadvantage of Penman–Monteith model is the simultaneous requirement of several meteorological data as
temperature, wind speed, relative humidity and sunshine measures.
The interdependence of these meteorological parameters and their variability in space and time, lead in
difficulties to formulate an equation that can be used to estimate ET from various crops under different climate
conditions (Temesgen B. et al., 2005). Notably, the difficulties due the sparse hydrometeorological networks in
several regions like Africa and the instability in the records of radiation and relative humidity (Samani, 2000)
reveals the demand of new simplifies models.
Therefore parsimonious model developed and implemented worldwide, such as radiation-based or temperature-
based models (Valiantzas, 2013). From numerous publications (Tabari, 2010; Samaras et al., 2014) demonstrated
that radiation-based methods are powerful models for the ET estimation.
In this study a new radiation based model is proposed, which include a new strategy in the estimation of potential
evapotranspiration (PET).
2. Overview of PET models
2.1. Penman- Monteith model
The classic model of the Penman-Monteith (1963) equation to estimate potential evaporation or
evapotranspiration is represented from the form:
PET = , γ΄ = γ (1 + rs/ra)"(1)
where PET is potential evaporation or evapotranspiration (mm/d), Rn is net radiation at the surface, Δ is the slope of
the saturation vapor pressure curve,γis psychometric coefficient while rs and ra are the surface and aerodynamic
resistance factors.
The FAO Penman–Monteith method was developed 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.
2.2. Radiation- Based Methods
Jensen and Haise (1963)evaluated 3000 observations of PET as determined by soil sampling procedures over a
35-year period, and developed the following relation. This equation has only known the average daily temperature
and extraterrestrial radiation and calculated easily form:

4 Aristoteles Tegos et al. / Agriculture and Agricultural Science Procedia 4 ( 2015 ) 2 – 9
nt
40
aaTR
PET ?
(2)
One decade later Mcguiness and Bordne (1972) using measured values of lysimeter suggested a slight modification
of Jensen’s formulation with the expression:
nt
68
)5( -
?aa TR
PET
(3)
Another widely used approach is the Hargreaves model (Hargeanes and Samani, 1982) that estimates the
reference evapotranspiration at the monthly and daily scale by:
PET = 0.0023 (Tα + 17.8) (Tmax – Tmin)0.5 (4)
The method has received considerable attention because it can produce very acceptable results under diverse
climates using only temperature measurements. According to several researchers (Samani, 2000;Xu and Singh
2002) the method tends fails in extreme humidity and wind conditions.
A recent research (Oudinet al., 2005) evaluated a number of evapotranspiration methods, on the basis of
precipitation and streamflow data from a large sample of catchments in U.S., France and Australia. After extended
analysis with the use of four hydrological models, the researchers proposed a modification of Jensen and McGuiness
model:
nt
100
)5( -
?a
T
a
R
PET
(5)
In the four radiation-based formulas PET(mm/d) is the potential evapotranspiration, Ra (kJ m-2d-1) is the
extraterrestrial shortwave radiation, Ta (°C) is the air temperature, λ latent heat of vaporization (kj/kgr) and p is the
water density (kgL-1).
3. Implementation of the parametric approach
3.1. The PET parametric model
Koutsoyiannis and Xanthopoulos (1999), Tegos et al. (2013), Tegos et al. (2015) examined the structure and the
sensitivity of input data in Penmann-Monteith model. They concluded that there are “one to one” relationship
between potential evapotranspiration, extraterrestrial radiation and temperature. In the parametric simplification of
the Penman-Monteith formula, the numerator is approximated by a linear function of extraterrestrial solar radiation,
Ra, while the denominator is approximated by a linear descending function of temperature.
The generalized mathematic equation of the parametric model is:
a
cT
baS
PET /
/
?1
0
(6)
where PET (mm) is the potential evapotranspiration, S0 (kJ m-2) is the extraterrestrial shortwave radiation, Ta (°C)
is the air temperature, and c (°C-1), a (kgkJ-1) and b (kg m-2) are parameters.

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Aristoteles Tegos et al. / Agriculture and Agricultural Science Procedia 4 ( 2015 ) 2 – 9
The mode parameters have physical interpretation of model parameters while:
‚ The dimensionless term “a / λρ” represents the average percentage of the energy provided by the sun (in terms of
Ra) and, after reaching the Earth’s terrain, is transformed to latent heat, thus driving the evapotranspiration
process.
‚ Parameter b lumps the missing information associated with aerodynamic processes, driven by the wind and the
vapour deficit in the atmosphere.
‚ The term “1 – c Ta” approximates “1 + γ΄/Δ”which is function of surface and aerodynamic resistance and Δ is the
slope vapour pressure curve, which is function of Ta.
3.2. Study areas and processes
We used monthly meteorological data from 37 stations distributed over Greece, run by the National
Meteorological Service of Greece, from 39 stations of CIMIS hydrometeological network in California, 10 from
Germany and finally 4 from Spain.
The organization of the time series and the calculation of potential evapotranspiration with different methods
(Penman-Monteith, Parametric, Hargreaves) were carried out using the Hydrognomon software. Finally, the other
expression (Jensen, Mcguiness and Oudin) modeled through appropriate spreadsheets.
Every time series was split to two control periods (calibration and validation), where in the first developed the
parametric model and in the second tested its predictive ability. At each station, the three parameters of parametric
model were calibrated against the reference potential evapotranspiration timeseries. This procedure was
automatically employed via a least square optimization technique, embedded in the evapotranspiration module of
Hydrognomon. The optimized values of a, b and c were next embedded to the parametric model.
3.3. Evaluation of the new parametric formula in Greece
The distribution of the coefficient of efficiency (CE),introduced by Nash and Sutcliffe (1970), is presented in the
Table 1. The results for the parametric model are satisfactory while CE values are greater than 95% at all locations
(90% for validation). The globally used radiation-based approaches by McGuiness et al. (1972) and Oudin et al.
(2005) present moderate results.
In order to provide a further parsimonious parametric formulas alternative parameterizations were also examined
through optimization techniques, i.e. (a) by omitting parameter b, and (b) by omitting b and substituting c by its
average value over Greece; in formulation (a) the reduction of CE was negligible.
Table 1 : Distribution of CE values of radiation-based approaches in Greece
CE
Parametric
Mcguiness
Oudin
Cal
Val
Cal
Val
Cal
Val
95-100
37
30
0
2
5
2
90-95
0
6
8
9
5
9
70-90
0
1
12
19
12
15
50-70
0
0
15
6
12
7
<50
0
0
2
1
3
4
3.4. Evaluation of the new parametric formula in California and in Europe
The distribution of the CE for the CIMIS stations is presented in Table 2 and for European stations in Table 3.
The results for both period and in different climatic regimes are satisfactory for the parametric model, while the
average CE in calibration are 94.80% (CIMIS), 96.52% (European) and in validation period are 94.34% (CIMIS)
and 90.06% (European). Similar satisfactory results shown Hargreaves model especially in CIMIS network (average

6 Aristoteles Tegos et al. / Agriculture and Agricultural Science Procedia 4 ( 2015 ) 2 – 9
CE 94.39% for the calibration period and 91.80% for the validation period) where the model has been developed,
while in European stations the indexes are lower (91.80% in validation period and 87.53% in calibration period).
Mcguiness model gives lower results than parametric and Hargreaves with 87.14% in calibration period and 87.76%
in the validation period.
Oudin’s model which is a modern improved version of radiation-based methods presents moderate results in
CIMIS network (52.18% calibration and 46.82% validation period) but quite better results in European stations
(89.37 % calibration and 82.82% validation period).
By combining the results with the previous study (Tegos et al., 2013) the model’s performance is more
acceptable in humid than in arid climatic regimes. Finally, Jensen-Haise model totally failed to produce physical
results.
Table 2: Distribution of CE values of radiation-based approaches in CIMIS network
CE
Parametric
Hargreaves
Jensen-Haise
Mcguiness
Oudin
Cal
Val
Cal
Val
Cal
Val
Cal
Val
Cal
Val
95-100
26
26
26
23
0
7
16
15
0
0
90-95
11
5
10
7
0
2
6
7
0
0
80-90
2
8
3
9
1
2
10
10
1
0
70-80
0
0
0
0
6
3
3
3
3
5
60-70
0
0
0
0
1
6
2
3
7
4
50-60
0
0
0
0
3
4
1
1
12
6
0-50
0
0
0
0
16
9
1
0
16
24
<0
0
0
0
0
12
6
0
0
0
0
Table3 : Distribution of CE values of radiation-based approaches in European stations
CE
Parametric
Hargreaves
Jensen-Haise
Mcguiness
Oudin
Cal
Val
Cal
Val
Cal
Val
Cal
Val
Cal
Val
95-100
10
9
6
0
0
0
0
0
9
1
90-95
4
4
4
6
0
0
0
0
2
8
80-90
0
0
3
7
0
0
0
0
0
2
70-80
0
0
1
1
0
0
7
1
1
1
60-70
0
0
0
0
0
0
3
1
1
1
50-60
0
0
0
0
0
0
3
1
1
0
0-50
0
1
0
0
5
1
2
9
0
1
<0
0
0
0
0
9
13
1
2
0
0
4. Spatial variability of the parameters
4.1. Mapping of the parameters over Greece
Assuming the simplified parameterization, in which b is omitted, we re-calibrated the local values of a and c, and
mapped them over Greece, using typical interpolation tools (e.g. Inverse Distance Weighting, Kriging etc.).

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Aristoteles Tegos et al. / Agriculture and Agricultural Science Procedia 4 ( 2015 ) 2 – 9
As shown in Figure 1 parameter a exhibits a systematic geographical pattern, since it increases from SE to NW
Greece, following the increase of sunshine duration and wind velocity as moving from the continental to insular
Greece, while parameter c is site-specific.
Fig. 1. Geographical distribution of parameters a and c over Greece
4.2. Spatial interpolation of models parameters over California
We implemented three well-known interpolation methods, i.e. Inverse Distance Weighting (I.D.W.), Kriging,
Natural Neighbours (NaN) and a recently proposed Bilinear Surface Smoothing (Malamos and Koutsoyiannis, in
press) in California territory.
After extended analysis with these alternatives interpolation techniques in a validation set of 11 stations, the
Inverse Weighting Distance, i.e. the simplest of interpolation methods, provides the more accurate point estimations
of model parameters.
The mapping of the three parameters over California through the I.D.W. approach is illustrated in Figure 2.
Generally, we detect that the parameters a, c increase from North to SE and the opposite occurs for the parameter b.
Par a
Par c

8 Aristoteles Tegos et al. / Agriculture and Agricultural Science Procedia 4 ( 2015 ) 2 – 9
Fig. 2. Geographical distribution of the parameters over California
5. Conclusions
The proposed parametric model can be considered as simplification of the Penman-Monteith formula, in an
attempt to compromise parsimony, in terms of model structure and data requirements, and physical consistency. The
parameters a, b and c have some physical background, since they substitute, to some extent, the three missing
meteorological variables.
The model ensures excellent predictive capacity (in terms of reproducing monthly PET estimations through the
Penman-Monteith) in all examined locations in Greece and California, as well as in Germany and Spain (full results
shown in Tegos et al. 2015). Additionally even simpler parameterizations in Greece through optimization (i.e. the
formulation with two parameters, a and c ) provide similarly good results.
The appropriateness of the method is further revealed through extensive comparisons with other radiation-based
approaches, most of which exhibit poor performance.
The comparisons across different climates reveal the great advantage of parametric approaches against radiation-
based ones, since calibration allows the coefficients that are involved in the mathematical formulas to be fitted to
local climatic conditions.

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Aristoteles Tegos et al. / Agriculture and Agricultural Science Procedia 4 ( 2015 ) 2 – 9
Reliable estimations of PET, both at point basis as well as over extended areas of interest (i.e. river basins), can
be obtained by interpolating the known (i.e., locally optimized) parameter values and next employing the parametric
formula.
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