Page 1

HYDROLOGICAL PROCESSES

Hydrol. Process. 20, 2393–2413 (2006)

Published online 20 March 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/hyp.6051

Wadi flow in the Arabian Gulf states

M. Nouh*

College of Engineering, University of Sharjah, PO Box 27272, Sharjah, United Arab Emirates

Abstract:

Real data on wadi flood flows from Saudi Arabia, Yemen, Oman, Kuwait, UAE, Bahrain and Qatar were used to

develop methodologies for the prediction of annual maximum flows and average monthly flows in the Arabian Gulf

states. For the prediction of annual maximum floods, three methods have been investigated. In the first method, regional

curves were developed and used together with the mean annual flood flow, estimated from the characteristics of the

drainage basin, to estimate flood flows at a location in the basin. The second method fits data to various probability

distribution functions, with a developed methodology introduced to account for floods generated by more than one

system of climate, and the best fitted function was used for flood estimates. In the third method, only floods over a

threshold, which depends on characteristics of the drainage basin, were considered and modelled. For the prediction of

average monthly flows, stochastic simulation approaches of flood frequency analysis were used. Each of the prediction

methods was verified by being applied in 40 different drainage basins. Based on the results obtained, recommendations

were made on the best method to be applied (at present) by design engineers in the Arabian Gulf states. Copyright

2006 John Wiley & Sons, Ltd.

KEY WORDS

surface water; arid areas; flood prediction; frequency analysis

INTRODUCTION

The Arabian Peninsula is a vast terrain characterized by its aridity, scarcity of fresh water resources and

an ever-increasing demand on fresh water supplies. It is probably the largest surface (about 3 ð 106km2)

on Earth not traversed by a perennial stream. Despite these facts, the surface of the peninsula, except those

areas covered by moving sand dunes, like the An-Nafud Desert and Ar Rub Al-Khali, is intersected by a

large number of ephemeral streams or wadis. A high density of wadis is found in the western part of Saudi

Arabia, parallel to the Red Sea coast and increasing in a southerly direction. The highest density occurs in

the southern part of the Tihama Plain and continues in the southern part of Yemen parallel to the Gulf of

Aden. The network of wadis is somewhat less dense in the coastal plains of Oman and decreases further in

the UAE. The density of wadis appears to be strongly related to the long-term average annual rainfall depth,

as shown in Figure 1.

A wadi is a stream that runs full for only a short time, mostly during and after a rainstorm. Not every

rainstorm, however, necessarily produces surface runoff. It is seldom that wadi flow at a certain section can

be described as perennial. If so, the flow is then extremely variable from one year to another. The recorded

annual flow in Wadi Dayqah at Mazara, Oman, for example, ranges from 3 ð 106m3to 192 ð 106m3(over

26 years of record) after excluding the exceptionally high flow of the year 1982 (Kaul, 1995).

The countries comprising the Arabian Peninsula, as well as the surrounding countries, are crowded with

wadis, though to varying extents. It is important that wadi flow should be viewed as a precious surface

water resource in these arid countries. Efficient use of wadi flow requires certain conservation measures to be

undertaken in order to reduce the losses by drainage into the sea or under ground. Proper management and

*Correspondence to: M. Nouh, College of Engineering, University of Sharjah, PO Box 27272, Sharjah, United Arab Emirates.

E-mail: m nouh@sharjah.ac.ae

Received 14 June 2004

Accepted 12 May 2005

Copyright 2006 John Wiley & Sons, Ltd.

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M. NOUH

Figure 1. Map of the Arabian Peninsula showing the isohyetal lines of annual rainfall and some of the principal wadis (Korzon, 1974)

utilization of wadi flow depend essentially on the availability of the data, as well as of their analysis. In this

study, a trial is made to develop a reliable methodology for surface water flow analysis and prediction in the

Arabian Gulf states.

The climate of the Arabian Gulf states is classified as arid to semi-arid. It is affected, mainly, by two

contrasting and conflicting systems of climate: a cyclonic system, which moves in from the north during

winter, and the monsoon from the southwest in summer. Variations in the duration and relative strength of

these two, successive, seasonal systems over the study area account for the immense uncertainty of its climate.

The range of daily temperature variation is 32Ð7°C, and the average annual relative humidity is 62%.

Altitude is the major factor controlling the quantity and pattern of rainfall, with the amount of rain increasing

progressively with elevation. The distribution of monthly rainfall is such that the main peak period of rain

in the year (summer rain) starts in late April and May, in parallel with the prevalence of low pressure, and

lasts until high pressure moves in in late autumn. However, the absolute maximum and minimum monthly

rainfall usually occur in this high peak period. Summer monsoon rain, which is derived primarily from tropical

storms and convectional thunderstorms, seems to represent some 20 to 30% of the total annual in the coastal

zone. Over the high lands (altitude more than 1000 m) it represents only 80% of the total. In contrast, winter

rain, which is of cyclonic frontal origin, represents the bulk of precipitation on the coastal zone, decreasing

gradually to the east from 80% on the coast to 20% in the highlands of the study area.

Field experience indicates that storm rainfall has to exceed a total of about 20 mm before there is any

substantial surface runoff. Annual runoff coefficients fluctuate between 0Ð133 and 0Ð185, resulting in an

overall average of 0Ð158. Runoff is characterized by spates that last, on average, 12 h from start to finish.

Copyright 2006 John Wiley & Sons, Ltd.

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These flows are typical torrential catchments, namely having a steep rise and rapid recession. The behaviour

of flow discharge is parallel to that of the rise–recession pattern. Available data reveal that it takes about

25 min, on average, to reach the target of 100 m3s?1.

In general, the soil characteristics indicate that it belongs to the light-coloured soil group (sierozems) of the

semi-arid and arid zones of the world. Generally, the soils are of varied texture, but silty or sandy profiles are

the order throughout the watersheds. Soil moisture loss is greatly aggravated under the prevailing conditions

of high temperature, low humidity, intense isolation and radiation, and strong dry surface wind movement,

with evaporation from surface soil tending to be high.

Vegetation of the highlands is classified as savanna–wood and formation, and that of the coastal plain as

formation with tropical bias. However, a large part of the vegetation is composed of mesophytic herbs that

only grow following the rain. It was observed that vegetation on the higher lands is often more uniform and

denser than in lower, depressed areas of wadis and closed basins.

OVERVIEW OF SURFACE WATER FLOWS IN SELECTED COUNTRIES

In the Arabian Gulf states the largest amount of available surface flow data exists in Saudi Arabia, Oman and

Yemen. The following is a brief review of these data.

Surface water flows in Saudi Arabia

Almost 60% of the volume of wadi flow in Saudi Arabia runs off the southwest part, from the terrain

situated between the Red Sea coast and the adjacent mountains, occupying only 10% of the total surface area

of the kingdom. Additionally, about 40% of the flow develops in the southern part of Tihama Plain at the

extreme south of the western coast, which represents no more than 2% of the surface area of the kingdom.

It has been reported (ACSAD/AFESD/KFAED, 1986a) that the surface water flows in the Red Sea zone can

be estimated at about 39Ð8 m3s?1, of which 27 m3s?1occurs south of Jeddah, 8Ð2 m3s?1north of Jeddah

and the remaining 4Ð6 m3s?1around Jeddah itself.

The wadi flows in the interior drainage system east of the Red Sea mountains are less in quantity compared

with the wadi runoff west of these mountains. The largest wadi runoff in this system develops in the west

and southwest, where precipitation is relatively heavy. The largest volumes of runoff in the interior drainage

system occur in the catchment areas of Wadi Al-Dawasir and Wadi Najran. The flow of the former, almost

10Ð5 m3s?1, on average, depends on the runoff of the Wadis Turabah, Ranya, Bishah and Tathlieth. The

second largest yield, 4Ð28 m3s?1, is produced by the Asir-Najran group of wadi subcatchments, of which

Wadi Najran alone produces 3Ð17 m3s?1on average. Table I gives estimates of the wadi runoff for different

regions of Saudi Arabia (ACSAD/AFESD/KFAED, 1986b).

Table I. Estimated runoff flows of main wadis in Saudi Arabia (ACSAD/AFESD/KFAED, 1986a)

Region Surface area

(km2)

Main wadisEstimated runoff

(106m3year?1)

Red Sea coastal belt

Asir-Wadi Dawasir

Asir-Wadi Najran

Wadi Birk Nisah

North Tuwayq

Taif, Fadat Al-Misbah

Wadi Ar-Rimah Al-Batin

Al-Nafud

Al-Jawf

Total

241600

180000

38400

162300

152800

43200

174400

161000

192000

1345700

Jizan, Damad, Baysh, Hali, Yiba, Qanonah, Al-Laith

Turabah, Ranya, Bishah, Tathlieth, Dawasir

Najran

Nawtah, Nisah, Hanifa

Sudair, Meshgar, Namil

Jadwal

Wajj, Liyyah, Aqiq, Awali

—

As-Sirhan, Rabigh

—

1265

330

135

100

95

55

25

20

35

2060

Copyright 2006 John Wiley & Sons, Ltd.

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Table II. Estimated runoff flows of main wadis in Oman (ACSAD/AFESD/KFAED, 1986b)

Region Main wadis Surface runoff

(106m3year?1)

Musandam

Al-Batinah, east and centre

— 23Ð40

180Ð02

168Ð10

143Ð20

95Ð90

121Ð70

90Ð10

67Ð90

27Ð30

917Ð80

Alansab, Rasyl, Smael, Taw, Ma’awel, Beni Kharouss, Fara’Beni

Ghafer, Hager, Mabbara

Helw, Qarr, Hatta, Feid, Rajemi/Zaben Fazz, Beni Omar, Souk,

Al-Gazy, Yanbo, Gelati/A’hen, Saken, Sarami, Hawasna, Shafan

Al-Ula, Sifam, Al-Ghoul, Bahla, Tenouf, Nazwi, Moayden, Yahle,

Helfen

Andam, Qanet, Upper Sound, Upper Ethli, Fath, Mangrid, Aghda,

Al-Zaher, Ebra, Tima, Selim, Al-Qabil and Beni Khaled

Al-Batinah, north

Oman, interior

Eastern region

Az’zahera (north interior basins)

Quryat and Sur

Salalah and southern wadis

Dhofar and northern wadis

Total

—

—

Dorbat, Arzat, Sandah, Dikan

Eidyem, Andour, Hallouf, Gurrah

—

Surface water flows in Oman

The available data on the wadi flows in Oman suggest that the average runoff per square kilometre per

year is about 4Ð3 l. The corresponding figure for Saudi Arabia is barely 0Ð95 l, or just 22% of the yield of the

Omani wadis. This can be attributed partly to the difference in the average rainfall between the two countries:

71 mm for Oman and 59 mm for Saudi Arabia. More important in this respect is that the topography, land

slope and other conditions in Oman are more favourable for producing runoff, compared with the topography

and other factors affecting wadi runoff in Saudi Arabia. The surface runoff in selected wadis is estimated by

traditional methods and presented in Table II. It can be seen that the estimates in the table are considerably

larger than the corresponding values supported by real measurement (i.e. 4Ð3 l km?2), which supports the idea

of developing a reasonably accurate methodology for surface water prediction in the investigated wadis of

the Arabian Gulf states.

Surface water flows in Yemen

Wadi catchments in Yemen are classified into four major basins. These are: the Red Sea basin, consisting

of eight major wadis and a number of minor wadis; the Gulf of Aden basin; consisting of seven major wadis

and a number of minor wadis; the Arabian Sea basin, consisting of 13 major wadis and a large number of

minor wadis; and the Ar Rub Al-Khali basin, of which Wadi Najran is probably the most important one.

A limited number of wadis in Yemen have already been gauged for some years. Some of the available

measurements, together with estimates flows of these wadis, are listed in Table III. From this table it is clear

that some of the computed flows are not entirely in good agreement with the measured flows. Considering the

available data measured from different sources, the wadi flows in Yemen total about 1Ð247 ð 103m3year?1.

This figure, when divided by the surface area of Yemen, i.e. 277195 km2, gives about 4Ð5 l km?2, which is

some 5% higher than the corresponding figure for Oman and more than 500% more than for Saudi Arabia.

In connection with the comparison between these three countries, one should not forget that Yemen is the

richest of all three in terms of rainfall. The overall areal average rainfall is about 140 mm year?1.

ESTIMATION OF WADI FLOWS IN THE ARABIAN GULF STATES

Estimation of surface water is needed for proper operation and management of water resources systems.

Methods available for such estimation may be classified into three major groups: empirical, analytical and

Copyright 2006 John Wiley & Sons, Ltd.

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WADI FLOW

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Table III. Measured and estimated runoff in Yemena

Basin and wadiCatchment area (km2) Annual runoff (m3)

Measured Estimated

(a) (c)

(a) (b)(c) (d) (a) (b)

Red Sea

Harad

Mawr

Surdud

Siham

Rima

Zabid

Rasyan

Mawza

Minor wadis

Gulf of Aden

Tuban

Sukaybiya

Bana

Hassan

Ahwar

Maifa’ah

Hajr

Minor wadis

Arabian Sea

Al-Jawf

Adhana

Hareeb

Behan

Markha

Amd/Doan

Al-Ain

Sarr

Idim

Masila

Jiza

Minor wadis

Rub Al-Khali

Najran

Other wadis

Total

—

1700

1900

2700

4050

2750

4450

1990

1600

5850

—

5060

1400

6200

3000

6410

4300

9900

10400

—

12000

8300

2500

3000

4300

6550

1500

2540

5485

24000

15000

27460

—

4400

86500

277195

—

—

—

2450

3200

2540

4910

1750

1300

5000

—

7150

—

7200

3300

6350

—

—

—

—

—

—

—

3300

—

—

—

—

—

—

—

—

—

—

—

—

—

—

162

69

—

99

125

12

—

—

—

109

—

170

—

71

—

—

—

—

87

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

120

67

—

51

97

19

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

125

—

162

32

69

44

40

—

—

—

—

—

48

—

20

10

—

—

216

61

—

106

164

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

——

—

117

121

130

103

164

54

38

120

—

—

—

—

41

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

35

207

65

89

83

135

51

29

97

—

129

19

126

—

71

24

54

47

—

116

82

14

21

24

29

6

63

41

—

—

—

—

—

—

—

21

79

95

151

—

30

171

2145

a(a) Ministry of Oil and Minerals Resources of Yemen and TNO Institute of Applied Geoscience (1995); (b) Yemen (Arab) Republic and

DHV Consulting Engineers (1989); (c) Al-Garoo (1987); (d) ACSAD/AFESD/KFAED (1986c).

experimental. The experimental group includes statistical and stochastic simulation methods. The annual

maximum (referred to as ANNMAX) model and the peak over a threshold (referred to as POT) model

are among the statistical methods, whereas the time series illustrated by the shot noise and the integrated

autoregressive moving average (ARIMA) models is among the stochastic simulation methods. The time series

models approximate all stages of the natural process, both high and low flows, whereas the other models

(i.e. ANNMAX and POT) represent only the flood peak aspects of the parent process. On the other hand,

the time series models consider the flow at a time t as the sum of three components (i.e. trend, seasonal and

stochastic), whereas the peak flows are considered by the ANNMAX and the POT models to be random.

Copyright 2006 John Wiley & Sons, Ltd.

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Table IV. Characteristics of data used for study

Country Number of

wadis

Length of

record (years)

Range of

altitude (m)

Saudi Arabia

Oman

Yemen

Kuwait

Bahrain

Qatar

UAE

34

26

12

2

4

19–54

15–51

17–62

16–64

12–42

11–51

18–58

620–2150

540–2630

874–2960

156–280

142–352

432–786

552–940

4

11

Depending on the characteristics of data at a given location, the flood flow estimated by one of these models

may differ significantly from that by any of the other models (Nouh, 1987a; Adamoski, 2000; Smithers and

Schulze, 2001).

Previous experience (Nouh, 1987b) has favoured the use of the experimental methods over the empirical

and analytical methods of estimation, as they provide results of reasonable accuracy in humid sites of sufficient

flood records (Nouh, 1982). Unfortunately, there is a lack of information in the literature concerning the use

of the experimental methods in extremely arid sites of limited flood records and where the dry period between

two successive flood events at a site is long. The present study is a trial to fill this gap in the knowledge.

The emphasis is to develop a reliable experimental methodology for the surface water flow prediction in the

Arabian Gulf states. Special consideration is given to the prediction of the annual maximum and monthly

average wadi flow at a location in the Arabian Gulf states. To reach the stated objective, wadi flow data from

Saudi Arabia, Oman, Yemen, UAE, Qatar, Kuwait, Bahrain and Qatar were collected and used. A general

description of these data is shown in Table IV.

Prediction of annual maximum floods

Several probability distribution functions were proposed to fit annual maximum flood distributions. These

functions are the extreme-value type 1 distribution (EV1), the two-parameter and three-parameter log–normal

distribution (LN2 and LN3 respectively), the Pearson type 3 distribution (P), the Gamma distribution (G), and

the log-Pearson type 3 distribution (LP). Descriptions of these distributions can be found elsewhere (Kendall

and Stuart, 1961). In this study, and owing to the property of rainfall in the Arabian Gulf states being affected

by two different types of climate (namely the cyclonic system during winter seasons and the monsoon during

summers), a mixture of two extreme-value type 1 distributions (MEV) is proposed to best fit the annual

maximum floods in wadis in the Arabian Gulf states. The mixture of distributions (MEV) can be expressed

as

f?x? D pf1?x? C ?1 ? p?f2?x?

in which p is the proportion units that have variate values that come from an extreme-value type 1 distribution

f1?x?, and the remainder (1 ? p) have variate values that come from a different extreme-value type 1

distribution f2?x?.

In the above distribution functions, which represent a class of the ANNMAX models, a major assumption

is that some set of data is a random sample from a single unknown probability distribution. Thus, series

of annual maximum floods should be checked against this assumption before any trial to fit probability

distribution functions to them. Randomness cannot be proved, but it can be disproved by the presence of

some feature of a nonrandom nature, such as a trend or serial persistence. In this study, distribution-free

methods (Kendall and Stewart, 1961) were applied to the wadi flood records in the Arabian Gulf states to test

these conditions. The Spearman rank order serial correlation coefficient and the arithmetic serial correlation

tests were used to check the persistence of the records. The rank order correlation coefficient of trend and

?1?

Copyright 2006 John Wiley & Sons, Ltd.

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the Mann–Whitney tests were used to check the progressive change in the mean value of a flood series with

time. In addition, the Wald–Wolfowitz runs test was performed to check whether there was any change with

time of a distribution’s measures (i.e. mean, variance, etc.). Moreover, the randomness of the flood series was

tested by performing the number of turning points test. Details of these tests are reported elsewhere (Kendall

and Stewart, 1961). The above tests were performed on 60 cases selected randomly from the available data

and the results at different levels of significance are given in Table V.

The values in Table V indicate that the records are not free of persistence and trend. However, the occurrence

of only five cases, out of 60, at the 10% level of significance may lead to the conclusion that the records, if

they are viewed as a whole, are almost free of persistence. On the other hand, the seven rejections, out of the

60 cases considered for the trend test at the 10% significance level, refers to the strong evidence for having

a trend in the records. The results of the Mann–Whitney and Wald–Wolfowitz tests and the turning points

test indicate that, if the records are viewed as a whole, the measures (i.e. mean, variance, etc.) of the annual

maximum flood distribution do not change with time and that the records can be considered random.

Based on the above findings, the above-mentioned probability distribution functions were fitted to the annual

maximum flood records of each of the selected wadis (see Table IV). The parameters of these distributions were

estimated by the maximum likelihood method. A goodness of fit was performed by applying the chi-square

test, the Kolmogorov–Smirnov test, the Cramer–Von Mises test, and the Anderson–Darling test. Details

of the distributions, the method of parameter estimation, and the goodness-of-fit tests are given elsewhere

(Kendall and Stewart, 1961). To fit the mixture of distributions (MEV) in Equation (1), the annual maximum

flood series of each station was split into two series (the first series contained summer floods and the second

series contained winter floods of the station) and each series was fitted to the EV1 (Gumbel) distribution. A

computer program was developed to change the proportion value of the two distributions automatically in

order to minimize the sampling standard error of the series. The results of the goodness-of-fit tests are shown

in Table VI.

Inspection of Table VI reveals that the MEV is the best-fitted probability distribution function. Such a

superior fit may be explained by the fact that the two successive systems of climate in the Arabian Gulf states

region (i.e. cyclonic in winter and monsoon in summer) result in that the effective rainfall components belong

Table V. Number of cases in which the null hypothesis is rejected at stated level (out of 60 cases)

Test signific-

ance level (%)

Persistence Trend: rank order

correlation test

Progressive changes in measures

Rank order

serial

correlation

test

Arithmetic

serial

correlation

test

Mann–

Whitney

test (U test)

Wald–

Wolfowitz

runs test

Randomness

turning

points test

1

5

0

2

5

1

3

6

2

5

7

2

4

9

1

2

5

1

2

4 10

Table VI. Percentage of times that distributions for annual maximum discharges were totally accepted at a stated

level of significance

Test significance

level (%)

EV1LN2LN3PG LP MEV

1074

79

84

61

63

68

67

69

74

72

78

85

65

71

74

73

78

83

84

87

95

5

1

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M. NOUH

to different populations. This means that the population units of the fitted mixture of Gumbel distributions

have variate values that come from a Gumbel distribution influenced mainly by the cyclonic system of climate

while the remainder of the variate values come from another Gumbel distribution influenced mainly by the

monsoon system of climate.

The least accurate results of the MEV (ANNMAX) model were found in wadis in Kuwait, Bahrain, UAE,

Qatar and the centre of Saudi Arabia, where the available flood data are limited. In these wadis, floods were

generated suddenly (flash floods) during only the winter season and after receiving certain rain amounts. For

these wadis, it was decided to use a statistical model (POT) that acknowledges the variation between years

in the number of flood peaks exceeding a specific threshold, but the variation between seasons within the

year is ignored. The number of peaks exceeding the threshold each year is assumed to be a random variable

having a Poisson distribution with parameter ? and the distribution of peak magnitudes is exponential with

parameters q0and ˇ. If the threshold q0is specified, then the number of peaks over the threshold M and

their magnitudes q1,q2,...,qMcan be abstracted from the record whose length is N years. The maximum

likelihood estimate of parameters (Kendall and Stewart, 1961) gives

? D M/Nˇ D q ? qq D

M

?

iD1

qi/M?2?

The Y-year flood can then be written (Kendall and Stewart, 1961) as

Q?T? D q0C ˇ ln ? C ˇ ln T?3?

and the variance of Q?T? is

var[Q?T?] D ?ˇ2/N?[1 C ?ln T C ln ??2/?]

?4?

The distribution of the annual maxima can then be written as

p?Qm? q? D exp[??e??q?q0?ˇ]

?5?

where Qmis the annual maximum flood.

The flood record of the stations was used in the above model. For each individual station record the threshold

value q0was selected in such a way that the abstracted peak flood magnitudes were a reasonable fit to the

exponential distribution. Figure 2 shows the abstracted flood peak magnitudes of some selected individual

stations fitted to the exponential distributions. The solid fitted line in the figure shows the abstracted small-

magnitude flood peaks of all stations fitted to the exponential distribution. Table VII shows the characteristics

of the best-fitted exponential distribution in the different regions.

Inspection of Figure 2 indicates that the exponential distribution in the POT model can be used for flood

prediction in the regions investigated. Table VII shows that a large number of peaks, a large threshold value,

a small Poisson parameter and a large scale parameter of the exponential distribution are characteristics of

the floods recorded at high altitude.

Correlation analyses were performed to examine the statistical independence of the successive peaks in the

records. The results indicated that the successive peaks at sites of high altitude are correlated more than those

at low-altitude sites. A correlation coefficient in the order of 0Ð10 was obtained at low-altitude sites, whereas

at high-altitude sites the correlation reached a value of 0Ð24 (at the south of Oman sites). In addition to the

correlation coefficients, scatter diagrams of peaks q1against q2that occur within different time intervals were

prepared for 60 individual stations. The diagrams indicated that the successive peak magnitudes in the model

are independent. Based on the above results, the POT model (Equations (3)–(5)) is considered further for

flood prediction in wadis in the Arabian Gulf states.

To compare the performance of the POT model (Equations (3)–(5)) with that of the ANNMAX model in

estimating Q?T?, the sampling variance estimated by the two models was evaluated and related to the Gumbel

Copyright 2006 John Wiley & Sons, Ltd.

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Figure 2. Fitting abstracted flood peaks at some stations to an exponential distribution

Table VII. Characteristics of best-fitted distributions

CountryRegion Altitude (m) Peaks exceeding

the threshold

?q0

(m3s?1)

ˇ

Saudi ArabiaSouthwest

Northeast

Centre

South

North

Centre

Red Sea

Gulf of Aden

Rub Al-Khali

—

—

—

—

1700–2150

620–870

740–918

2110–2630

540–1760

870–2100

2650–2960

1490–2100

874–910

552–940

142–352

156–280

432–786

5

3

2

6

4

3

7

6

2

2

2

2

2

1Ð76

3Ð15

2Ð15

1Ð81

1Ð95

2Ð74

1Ð84

2Ð15

2Ð35

3Ð34

4Ð20

4Ð24

4Ð29

85

62

39

105

85

65

92

54

36

29

26

21

29

62

59

49

71

66

59

76

69

67

36

29

28

31

Oman

Yemen

UAE

Bahrain

Kuwait

Qatar

reduced variate. In the ANNMAX model, a Gumbel distribution was utilized, which gives the sampling

variance estimated as

var[Q?T?ANNMAX] D ?ˇ2/N??1Ð11 C 0Ð52y C 0Ð61y2??6?

Copyright 2006 John Wiley & Sons, Ltd.

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M. NOUH

where y is the Gumbel reduced variate of return period T, given as

y D ?lnfln[T/?T ? 1?]g

?7?

The sampling variances estimated by the POT model (Equation (4)) and by the ANNMAX model

(Equation (6)) were evaluated at 60 stations, selected by random draw. The estimated sampling variances

were averaged for a specified return period, and this process of computations was repeated for different return

periods T. A relationship was established to relate the average sampling variances, divided by ˇ2/N, estimated

by the models, to the Gumbel reduced variate. This relationship is shown in Figure 3. The figure indicates

that the sampling variances estimated by the POT model are smaller than those estimated by the ANNMAX

model for a return period of less than 10 years. For longer return periods, the sampling variances by the

ANNMAX model are smaller than those estimated by the POT model.

To avoid errors from misuse of the exponential assumption in the POT model, the peak discharges over

the threshold in each of individual station records were plotted against the exponential reduced variate. The

fit obtained was reasonable for all low-altitude sites. A typical example of such a fit is shown in Figure 4.

Therefore, the POT model may be used in flood estimation at low-altitudes sites in the Arabian Peninsula.

Figure 3. Average sampling variances/ˇ2/N estimated by the ANNMAX and POT models

Figure 4. Peak discharges over the threshold in Wadi Awali (Saudi Arabia) against an exponential reduced variate (R2D 0Ð8692)

Copyright 2006 John Wiley & Sons, Ltd.

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WADI FLOW

2403

Regional prediction of annual maximum floods

The previously developed ANNMAX and POT models successfully estimate the annual maximum floods

at a station with historical flood records. To predict the annual maximum flood flow at any location in the

Arabian Gulf states, regional curves are developed and explained in the following.

The intended regional curve is a frequency distribution of Q?T?/Qav, where Qavis the mean annual flood

flow (defined as the arithmetic mean of annual maximum instantaneous flow series). It associates a return

period T with Q?T?/Qav, and this relation is assumed to be valid for all drainage basins in one region. Where

no flood flow records exist at a site in the region, this curve may be used together with an estimate of the

mean annual flood flow to predict Q?T? at the site. The mean annual flood flow may be estimated using a

relationship between floods and drainage basin characteristics obtained from maps. A previous investigation

undertaken by Cole (1966) related mean annual flood to drainage basin area. A study by Nash and Shaw

(1966) took into account an index of climate or slope, and Boulton (1965) listed 13 indices of drainage basin

morphometry that may be included in the relationship. However, the inclusion of a large number of variables

in the relationship is not possible from a practical point of view. Gray (1964) summarized the successful

variables: size and shape of drainage basin; density and distribution of watercourses; general land slope; slope

of channels; and storage. The results of reported multivariate analyses (Nouh and El-Laithy, 1988) showed

that four groups of variables (namely size, slope, geology and drainage density) are sufficient to explain about

92% of the variance of such a type of relationship. A further study by Nouh (1990) proposed a simplified

procedure to estimate floods from arid catchment morphology.

In this study, a further reduction in the significant number of variables in the relationship between floods

and drainage basin characteristics is proposed in the study zone of the Arabian Gulf states. A previous

investigation by Nouh (1989) indicated that the slope, the geology and the density of a drainage basin in the

Arabian Gulf states have a strong relation with the elevation of the drainage basin; that is, the slope is steeper,

the permeability is lower, the rainfall amount is larger and more intense, the runoff coefficient is higher, and

the flood flows are larger in drainage basins of higher elevation than in those of lower elevation. Therefore,

with regard to this relationship in the context of the Arabian Gulf states region, it is proposed to replace the

drainage basin’s variables of slope, geology and density with only one variable, namely elevation.

Based on this concept, the size and elevation of the drainage basin are considered as independent variables

in a regression model proposed to estimate mean annual flood discharges. The proposed model for drainage

basins of altitude >8 m is

QavD ?0AREA?1ELEV?2ε

where Qav?m3s?1? is the mean annual flood discharge, AREA (km2) is the size of the drainage basin, ELEV

(m) is the mean elevation of the drainage basin in above sea level (i.e. altitude), ?0, ?1and ?2are regression

parameters; and ε is the error term (residual) or unobservable random variables that measures the difference

between log Qavand the value estimated by the equation. It is assumed that log Qavis normally distributed

with mean zero and unknown variance.

The Arabian Gulf states zone is split into four regions according to similarity in hydrologic characteristics.

Region 1 includes the southwest of Saudi Arabia and Yemen. Regions 2 and 3 include Oman and the centre

of Saudi Arabia respectively. Region 4 includes UAE, Qatar and Bahrain. The station records of each region

were split into two sections: the first section (containing about 70% of the station records, and referred to

as the calibration records) was used for calibrating the regional method; the second section (containing the

remaining records, and referred to as the verification records) was used for verification of the method.

A multiple regression analysis of the proposed model (Equation (8)) was performed using the calibration

records. The parameters estimated by means of a forward stepwise algorithm for region 1 (26 stations in

the southwest of Saudi Arabia and Yemen) were ?0D 0Ð322, ?1D 0Ð559, and ?2D 0Ð436. The correlation

coefficient between the observed and predicted log Qavvalues was 0Ð803. An analysis-of-variance table based

on the multiple linear regression is shown in Table VIII. From this table, it can be seen that the calculated F

value was 48Ð6406. If the percentage points of the F(2, 23) distribution are looked up, it can be seen that the

?8?

Copyright 2006 John Wiley & Sons, Ltd.

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Table VIII. Analysis-of-variance table from multiple linear regression of 26 stations

Source of variationSum of

squares

Degrees of

freedom

Mean

square

Calculated F

Regression

Error (residual)

Total, corrected for mean

2Ð5378

0Ð6000

—

21Ð2689

0Ð0261

—

48Ð6406

—

—

23

25

95% point F?2, 23, 0Ð95? D 3Ð42. As the calculated F value exceeds the critical F value in the table (i.e.

F D 48Ð6406 > 3Ð42), the hypothesis H0: ?1D 0 and ?2D 0 can be rejected, running a less than 5% risk of

being wrong.

In addition, the intercorrelation coefficient of two independent variables, log(AREA) and log(ELEV), was

0Ð242, meaning that these two independent variables are not strongly correlated. The 95% confidence interval

of ?1D 0Ð559 was (0Ð109, 1Ð009), and that of ?2D 0Ð436 was (0Ð018, 0Ð854). Since both confidence intervals

do not cover zero, it may be said, with a risk of less than 5% of being wrong, that both ?1and ?2are not

zero. In other words, both independent variables are important.

With only one variable in the model, log(AREA), the simple correlation coefficient was 0Ð741. In order to

measure the contribution of the second variable, log(ELEV), the partial correlation was estimated and found

to be 0Ð68. In other words, 46% (square of 0Ð68) of the unexpected variance can be explained by adding the

second variable, log(ELEV). Based on the above, it can be concluded that both independent variables in the

model (Equation (8)) are important and cannot be neglected.

To study the effect of drainage basin size and elevation on the relationship developed (Equation (8)), two

investigations were carried out. In the first investigation, the calibration station records were split into two

sections of large- and small-sized drainage basins, whereupon three tests were conducted with the division

size at 1000, 2000 and 3500 km2. In the second investigation, the calibration station records were split into

two sections of high- and low-altitude drainage basins, whereupon three tests were also conducted with the

division altitude at 500, 1000 and 1200 m. The relationship developed (Equation (8)) was then applied in the

drainage basins of each section, and measures of the average size of residuals were calculated. In this regard,

the standard error of estimate and the combined standard error were computed. The standard error of estimate

of log Qav, i.e. s, is calculated by the following formula:

???

and the combined standard error of log Qav, i.e. sc, is calculated by

s D

ε2?

/?n ? m ? 1?

?0Ð5

?9?

scD [ss/?n ? 2m ? 2?]0Ð5

?10?

where ss is the sum of the squares of the residuals over both sections (less and more than the division size,

or lower and higher than the division altitude of the drainage basin in the first or in the second investigation

respectively); n is the number of stations in each test and m in the number of independent variables in the

relationship (i.e. two).

The results of the tests performed are given in Tables IX and X. The values of the standard errors in

Table IX indicate that larger sized drainage basins are fitted better by the derived relationship, whereas those

in Table X indicate that higher altitude drainage basins are fitted better by the relationship. However, the effect

of drainage basin size on predicted mean annual flood flows is larger than that of drainage basin altitude.

Because the number of available gauged runoff stations is small and the period of each station record is short,

only the effect of drainage basin size on predicted mean annual flood flows was considered. The calibration

records of 26 stations were divided into three areal size classes, and a regression analysis of the proposed

Copyright 2006 John Wiley & Sons, Ltd.

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WADI FLOW

2405

Table IX. Effect of drainage basin size on predicted mean annual flood flowsa

Less than dividing sizeDividing size of

drainage basin (km2)

Greater than dividing sizeCombined standard

error (m3s?1)

No. of station

records

Standard error of

estimate (m3s?1)

No. of station

records

Standard error of

estimate (m3s?1)

10

14

18

6Ð149

5Ð081

3Ð938

1000

2000

3500

16

12

8

2Ð964

2Ð075

1Ð255

3Ð866

3Ð862

3Ð864

3Ð864

aAll drainage basins: 26.

Table X. Effect of drainage basin altitude on predicted mean annual flood flowsa

Lower than dividing altitudeDividing altitude of

drainage basin (m)

Higher than dividing altitudeCombined standard

error (m3s?1)

No. of station

records

Standard error of

estimate (m3s?1)

No. of station

records

Standard error of

estimate (m3s?1)

94Ð319

4Ð106

3Ð911

500

1000

1200

17

10

8

3Ð637

3Ð605

3Ð440

3Ð863

3Ð864

3Ð863

3Ð864

16

18

aAll drainage basins: 26.

Table XI. Summary of regression analyses performed

Regression Areal size class

range of drainage

basin (km2)

No. of

station

records

Correlation

coefficient

Inter-

correlation

coefficient

?0a

?1a

?2a

Original

regression

1st areal size

class regression

2nd areal size

class regression

3rd areal size

class regression

All calibration records260Ð812

0Ð897

0Ð881

0Ð827

0Ð242

0Ð102

0Ð273

0Ð319

0Ð322

0Ð278

0Ð310

0Ð346

0Ð559

0Ð492

0Ð621

0Ð705

0Ð436

0Ð408

0Ð45

0Ð500

(0Ð139, 0Ð505) (0Ð109, 1Ð009) (0Ð018, 0Ð854)

(0Ð248, 0Ð308) (0Ð061, 0Ð923)

(0Ð216, 0Ð404) (0Ð321,0Ð921) (0Ð240, 0Ð660)

(0Ð201, 0Ð491) (0Ð355, 1Ð055) (0Ð260, 0Ð740)

<1000 10

(0Ð016, 0Ð80)

3500 ½ 1000

½3500

8

8

aThe 95% confidence level is in parentheses.

model (Equation (8)) for each areal size class of records was made. The results of the regression analyses are

summarized in Table XI. It can be seen that

1. The accuracy of the predicted log Qav, measured by the correlation coefficient between the measured and

predicted log Qavvalues, is improved for the divided areal size classes of drainage basin.

2. The 95% confidence intervals of the regression parameters for each areal size class of records do not cover

zero, meaning, as has been mentioned earlier, that both independent variables are important and cannot be

neglected.

Copyright 2006 John Wiley & Sons, Ltd.

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M. NOUH

3. The values of the regression parameters increase with increasing drainage basin size, meaning that the

independent variables are more important in predicting log Qav in larger sized drainage basins than in

smaller sized ones.

Based on the above results, the proposed model (Equation (8)) with parameters appropriate to a drainage

size (Table XI) was considered reasonable for predicting the mean annual flood discharge of the drainage

basin and, therefore, was used to predict Q?T? through regional curves.

In addition, the calibration records were assembled in the form Q?T?/Qavtogether with the plotting positions

appropriate to the Gumbel distribution expressed as reduced variate values y?T?. Details of the procedure are

reported by Nouh (1989). The Gumbel reduced variate was selected on the basis of the results reported before

in this research, which showed that the MEV distributions (Equation (1)) fit reasonably well the annual

maximum flood flow series at a site in the Arabian Gulf states. The plotted points were smoothed by one

region curve. A typical example for region 1 (southwest of Saudi Arabia and Yemen) is shown in Figure 5.

A trial was instigated to study the effect of various drainage basin characteristics on the characteristics of

the region curves. The region curve was expressed in the usual form of the general extreme value variate as

Q?T?/QavD ? C ˛[?1 ? e?ky?/k]

?11?

where y is the MEV reduced variate for a return period of T years, ? is the value of Q?T?/Qavwhen y D 0,

˛ is the gradient of the curve when y D 0, and k is a curvature parameter.

The parameters of Equation (11) were estimated for individual station records differing in their drainage

basin size and elevation. The procedure adopted for estimating the parameters of Equation (11) for an

individual station record is exactly the same as that adopted in region curve development. For each station

record, the values of Q?T?/Qavwere plotted against y?T?, and the plotted points were then smoothed by a

Figure 5. Regional curves for southwest of Saudi Arabia and Yemen

Copyright 2006 John Wiley & Sons, Ltd.

Hydrol. Process. 20, 2393–2413 (2006)

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WADI FLOW

2407

hand-drawn curve. This curve was regarded as the definitive estimate of the population curve to begin with,

and by means of a series of trials a curve of the form of Equation (11) was chosen. The procedure adopted

was to set ? to equal the ordinate of the hand-drawn curve at y D 0, ˛ to equal the slope of the hand-drawn

curve at y D 0, and then to sketch in the curves corresponding to a few trial values of k. The curve that

provided the best description of the hand-drawn curve was chosen.

The values obtained indicate that the region curve characteristics (reflected by values of ?, ˛ and k) are

more sensitive to a variation in drainage basin altitude than to a variation in drainage basin size. Thus, it

was decided to divide the calibration records into two groups and to derive a regional curve for each group

of records. The first group contained the records of low-altitude drainage basins (<1000 m), and the second

group contained those of high-altitude drainage basins (½1000 m). Because the length of record is relatively

short, quantile estimates of return periods up to 500 years were obtained at each station. Typical regional

curves for region 1 (southwest of Saudi Arabia and Yemen) are shown in Figure 5. The characteristics of the

derived regional curves in the Arabian Gulf states are given in Table XII.

The regional curves developed were verified by comparing the predicted Q?T? with that observed at each

of the verification stations. The results of such a comparison are presented later in this paper. The procedure

adopted for the prediction is summarized as follows:

1. At each station, the size and altitude of the drainage basin were determined by means of a 1:25 000

topographic survey map.

2. The mean annual flood flow was estimated using Equation (8) with parameters appropriate to the drainage

basin.

3. For a specific return period, the region curve (Equation (11) and Table XII) appropriate to the drainage

basin altitude was used to predict Q?T?/Qav.

4. The predicted Q?T?/Qavand the estimated mean annual flood flow Qav(step 2) were used to predict Q?T?

at the station.

Accuracy of the prediction of annual maximum floods

Each of the methods of Q?T? prediction considered was verified through its application in the verification

station records. Only the first 30 years of each station’s record was retained for the assessment of flood

predictions, for return periods of 2, 3, 5, 10 and 20 years, by comparison with the corresponding flood flow

events in the station record. A measure of the prediction accuracy is given by the prediction error PE, defined

as

PE DjQ0?T? ? Qc?T?j

Q0?T?

?12?

Table XII. Characteristics of the regional curves in the Arabian Gulf states

Region

?˛k

Southwest of Saudi Arabia and Yemen

Altitude <1000 m

Altitude ½1000 m

Oman

Altitude <1000 m

Altitude ½1000 m

Centre of Saudi Arabia

Altitude <1000 m

Altitude ½1000 m

UAE, Kuwait, Qatar and Bahrain

Altitude <1000 m

Altitude ½1000 m

0Ð37

0Ð29

0Ð41

0Ð39

0Ð32

0Ð46

0Ð23

0Ð21

0Ð29

0Ð17

0Ð11

0Ð19

0Ð26

0Ð22

0Ð36

0Ð33

0Ð23

0Ð39

0Ð26

0Ð19

0Ð29

0Ð12

0Ð07

0Ð15

?0Ð49

?0Ð26

?0Ð59

?0Ð32

?0Ð21

?0Ð43

?0Ð22

?0Ð15

?0Ð27

?0Ð09

?0Ð03

?0Ð16

Copyright 2006 John Wiley & Sons, Ltd.

Hydrol. Process. 20, 2393–2413 (2006)

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2408

M. NOUH

where Q0?T? and Qc?T? are respectively the observed and calculated (predicted) discharges for a return period

T years.

The results of the comparison are given in Table XIII. It appears that the region curve method, followed by

the POT model, is the most appropriate one for the Arabian Gulf states. It can also be seen that the accuracy of

the prediction decreases with an increase in return period. In addition, the accuracy of the prediction methods

considered is higher in higher altitude drainage basins than in low-altitude drainage basins.

Mean monthly flows

Monthly flows are characterized by a wide variation over a short period of time. Thus, averaging the flow

over a period such as a week or a month does not make much sense. However, long monthly records might

help establish a distribution pattern of the average annual flow over the months of the year. This might be

useful when the monthly demands and supplies are considered.

Table XIII. Prediction error PE (%) for different return periods

RegionPrediction method PE (%) Mean

PE (%)

2 years 3 years5 years10 years20 years

Southwest of Saudi Arabia and YemenRegion curve

Altitude <1000 m

Altitude ½1000 m

ANNMAX model

Altitude <1000 m

Altitude ½1000 m

POT model

Altitude <1000 m

Altitude ½1000 m

Region curve

Altitude <1000 m

Altitude ½1000 m

ANNMAX model

Altitude <1000 m

Altitude ½1000 m

POT model

Altitude <1000 m

Altitude ½1000 m

Region curve

Altitude <1000 m

Altitude ½1000 m

ANNMAX model

Altitude <1000 m

Altitude ½1000 m

POT model

Altitude <1000 m

Altitude ½1000 m

Region curve

Altitude <1000 m

Altitude ½1000 m

ANNMAX model

Altitude <1000 m

Altitude ½1000 m

POT model

Altitude <1000 m

Altitude ½1000 m

9Ð8

11Ð1

5Ð3

11Ð3

13Ð3

7Ð3

10Ð2

11Ð9

6Ð9

7Ð9

9Ð6

5Ð4

10Ð7

13Ð2

7Ð5

8Ð9

12Ð4

5Ð2

11Ð3

11Ð3

N/A

15Ð8

15Ð8

N/A

13Ð7

13Ð7

N/A

11Ð9

11Ð9

N/A

15Ð9

15Ð9

N/A

14Ð3

14Ð3

N/A

13Ð6

14Ð1

7Ð2

16Ð3

18Ð2

9Ð4

14Ð7

15Ð5

10Ð4

9Ð6

11Ð8

6Ð9

13Ð5

15Ð6

9Ð2

12Ð2

15Ð2

7Ð9

13Ð7

13Ð7

17Ð3

17Ð3

14Ð2

14Ð2

14Ð1

14Ð1

18Ð4

18Ð4

16Ð7

16Ð7

14Ð7

15Ð3

9Ð2

18Ð3

19Ð8

11Ð2

15Ð2

17Ð8

13Ð2

12Ð6

13Ð9

10Ð8

15Ð7

19Ð0

11Ð4

13Ð4

18Ð4

9Ð8

16Ð2

16Ð2

22Ð5

22Ð5

18Ð9

18Ð9

17Ð8

17Ð8

24Ð3

24Ð3

19Ð2

19Ð2

15Ð2

15Ð9

10Ð4

23Ð6

24Ð9

14Ð3

17Ð6

19Ð7

14Ð7

14Ð7

16Ð3

12Ð8

19Ð3

22Ð4

15Ð3

15Ð8

19Ð7

13Ð3

19Ð8

19Ð8

29Ð4

29Ð4

22Ð4

22Ð4

21Ð3

21Ð3

31Ð2

31Ð2

26Ð2

26Ð2

16Ð7

17Ð3

13Ð2

27Ð5

29Ð5

16Ð5

21Ð6

25Ð4

16Ð3

17Ð1

19Ð2

14Ð6

28Ð2

31Ð4

19Ð7

21Ð2

29Ð4

15Ð9

21Ð8

21Ð8

33Ð2

33Ð2

28Ð4

28Ð4

23Ð6

23Ð6

33Ð2

33Ð2

29Ð9

29Ð9

14Ð00

14Ð74

9Ð06

19Ð40

21Ð14

11Ð74

15Ð86

18Ð06

12Ð30

12Ð38

14Ð16

10Ð10

17Ð48

20Ð32

12Ð62

14Ð30

19Ð02

10Ð42

16Ð56

16Ð56

23Ð64

23Ð64

19Ð52

19Ð52

17Ð74

17Ð74

24Ð60

24Ð60

21Ð26

21Ð26

Oman

Centre of Saudi Arabia

UAE, Kuwait, Qatar and Bahrain

Copyright 2006 John Wiley & Sons, Ltd.

Hydrol. Process. 20, 2393–2413 (2006)

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WADI FLOW

2409

Very few trials have been made to estimate the monthly flows in the Arabian Gulf states. In a study on

water resources assessment of Wadi Surdud in Yemen (Sana’a University, 1995), a multiple regression model

has been suggested for estimating the monthly flow. In this model the runoff at month t, Qt, is regressed on

rainfall P at the same month t, 1 month earlier t ? 1, 2 months earlier t ? 2, etc. The model can be written

as

QtD a C b0?Pt? C? C b1?Pt?1? C? C b2?Pt?2? C? C ÐÐÐ C bi?Pt?i? C??13?

where a and C are constants, and b0,b1,...,biare regression coefficients. The study claims that the correlation

coefficient between the observed and predicted flows is close to 0Ð80 when C is assumed as 25 mm and

i D 5 months.

In opinion of this author, the model (Equation (13)) can be much improved by adjusting the values obtained

from Equation (13) to become Qt?adjusted?, where

Qt?adjusted?D 0Ð25 C 0Ð50QtC 0Ð12Q2

t

?14?

The difference in results obtained from both Equations (13) and (14) can be seen in Figure 6.

In this study a trial was undertaken to predict the mean monthly flows by using the stochastic simulation

method. The objective of the stochastic method is to describe natural hydrologic processes mathematically in

such a way that the new samples generated cannot be distinguished statistically from the historical sample.

A detailed treatment of this subject is given by Box and Jenkins (1971).

The ARIMA models (Box and Jenkins, 1971), which have been used extensively for modelling river flow

sequences, were applied to estimate the average monthly flow time series in selected wadis in the Arabian

Gulf states. To apply the models to an individual station record, the within-year cyclicity must be removed to

obtain a stationary time series. Further, the data of the time series analysed should be independent according to

Box and Jenkins (1971), have a constant variance and it is advantageous that they be normally distributed. To

obtain a series that is approximately normally distributed, the mean monthly flows Qp,twere log-transformed

as

qp,tD logQp,t

?15?

in which t is the sequence of months (i.e. t D 1,2,..., 12) and p is the sequence of years in the series.

Figure 6. Predicted and observed monthly flows of some wadis in Yemen

Copyright 2006 John Wiley & Sons, Ltd.

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2410

M. NOUH

Computed general means mtand standard deviations stof the log-transformed series exhibit annual cyclicity

in all individual station records considered. The process qp,tmay be represented by

qp,tD ?tC ?tzp,t

?16?

in which ?t and ?t are respectively the periodic mean and standard deviation of the process and zp,t is a

dependent stochastic component. Using sample estimate mtand stinstead of ?tand ?t, the series is standardized

by a nonparametric approach as

zp,tD ?qp,t? mt?/st

The stochastic process zp,t, obtained by Equation (17) having mean equal to zero and standard deviation

equal to unity, shows a time dependence. This is illustrated typically by the Jizan station (in the southwest of

Saudi Arabia) series autocorrelogram in Figure 7. Thus, a time-varying ARIMA model was used to represent

the time dependency of the zp,tprocess adequately.

The autocorrelation and the partial autocorrelation functions are used in the identification of the time series

model. These were calculated for 40 drainage basins in the Arabian Gulf states for the lag-1 differenced series

and for the standardized series. The technique used is that of Box and Jenkins (1971). The Anderson (1942)

and the Box and Pierce (1970) Portmanteau lack-of-fit were used to investigate the hypothesis of white noise

suggested by the earlier work of Roesner and Yevjevich (1966). The tests for the white noise hypothesis

were applied first to the lag-1 differenced series ARIMA(0, 1, 0) model and it failed at the 5% level. All the

autocorrelation functions were observed to have a cutoff after lag-1 and all the partial autocorrelations tail

off. This behaviour suggests an MA(1) model. Therefore, the ARIMA(0, 1, 1) model was selected for further

consideration.

The white noise hypothesis was then applied to the standardized series. It was accepted at the 5% level

by Anderson’s test and at the 10% level by the Portmanteau lack-of-fit test in 12 out of 60 cases. The white

noise model was thus found inadequate to explain the stationary component of the monthly runoff time series

for the geographical region considered. Under the hypothesis that the lag-k autocorrelation ?kD 0 for k > 0

and that the time series is a white noise, ?1was found significant at the 5% level in 51 out of 60 cases by

Anderson’s test. The partial autocorrelation ?kk has a cutoff after lag-1 in all series. Thus, ?1and ?11are

small but significant. Because the autocorrelation and the partial autocorrelation functions of the ARIMA(1,

0, 1) model decay from the first lag on, this model of the standardized monthly flow series was retained for

further consideration.

?17?

Figure 7. Correlograms of stochastic components of mean monthly flows of Jizan (southwest of Saudi Arabia): (1) for the dependent zp,t

series; (2) for the independent ap,tseries of the fitted ARIMA(1, 0, 1) model

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The initial estimates of the parameters were obtained from the equations governing their autocovariance

structure. Refined estimates of the parameters that maximize the log-likelihood function were then obtained

by minimizing the sum of squares of the residuals. These methods of estimation of the model parameters have

been described elsewhere (Salas et al., 1980). Because of the presence of the MA component in the ARIMA

model, the white noise is a nonlinear function of the model parameters. Thus, once the approximate least-

squares estimates of the parameters are obtained from the sum of squares of the residuals, these estimates are

used as initial values in a nonlinear iterative estimation of the parameters that minimizes the sum of squares

of the residuals (Box and Jenkins, 1971). For the ARIMA(1, 1, 1) model, the moving average parameter

O? > 0Ð99 in all 60 cases suggests that differencing was not necessary, as with ? D 1Ð00 the ARIMA(1, 1, 1)

model reduces to an AR(1) model. However, the sole purpose of experimenting with the lag-1 differencing

was the removal of the periodic component of the time series. Thus, the ARIMA(1, 1, 1) model is not a

suitable model for the monthly flow series in the Arabian Gulf states.

The Portmanteau lack-of-fit test of Box and Pierce (1970) with 25 lags and the cumulative periodogram test

were applied to the ARIMA(1, 1, 1) and ARIMA(1, 0, 1) models. The results are summarized on Table XIV.

These tests show that standardization of the time series effectively removes the periodic component of the

monthly flow series and that the ARIMA(1, 0, 1) model fitted to the log-transformed standardized monthly

flow series is an adequate model of the monthly flow series in the Arabian Gulf states. Table XV gives the

average estimates of ?1, ? and ? for drainage basins in different regions.

The ARIMA(1, 0, 1) model is defined as

zp,tD ?1,tzp,t?1? ?1,tap,t?1C ap,t

?18?

where ?1and ?1,tare seasonal first-order autoregressive and moving-average parameters respectively and ap,t

is an independent identically distributed stochastic component (random variable, white noise).

By removing the dependence in the zp,t process, the resulting stochastic process ap,t, according to the

hypothesis (Yevjevich, 1976a), is a second-order stationary independent process. One of the possible tests of

goodness of fit of the model is, therefore, to compute the ap,tcomponent from the historic sample and test it

for independence. From Equation (18), ap,tis obtained as

ap,tD zp,t? ?1,tzp,t?1C ?1,tap,t?1

?19?

Table XIV. Lack-of-fit tests

TestARIMA(1, 1, 1) ARIMA(1, 0, 1)

No. passingNo. failing No. passingNo. failing

Portmanteau lack-of-fit test

Cumulative periodogram test

12

16

48

44

54

51

6

9

Table XV. Average parameter estimates of the ARIMA(1, 0, 1) model

Region Drainage basins

???

Southwest of Saudi Arabia and YemenAll drainage basins

Altitude <1000 m

Altitude ½1000 m

All drainage basins

Altitude <1000 m

Altitude ½1000 m

0Ð32

0Ð23

0Ð39

0Ð19

0Ð11

0Ð27

0Ð21

0Ð15

0Ð24

0Ð07

0Ð02

0Ð15

0Ð15

0Ð09

0Ð11

0Ð21

0Ð16

0Ð32

Oman

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M. NOUH

Figure 7 shows the correlogram of the ap,tseries and the 95% tolerance limits using the individual station

record of Jizan, Saudi Arabia. The ap,t stochastic process passes the test of independence. Similar results

were obtained for the other station records considered.

The independent stochastic process ap,twas further investigated to find its probability distribution function.

For this purpose the two-parameter normal and the three-parameter log–normal distribution functions were

fitted to the ap,tempirical distribution and both failed the chi-square test of goodness of fit. It was decided

to use the empirical ap,tfrequency distribution in the generation of new samples.

The generation of new samples of the mean monthly flows follows the inverse procedure of the historic

series decomposition outlined above. The simulation starts with generating the white noise series ap,t. The

nonparametric method of the white noise generation (Yevjevich, 1976b) was used in the present analysis.

The generated noise is substituted into Equation (18) to compute the dependent stochastic component series,

which, in turn, is used in Equation (17) to compute the qp,tseries. An inverse logarithmic transformation, i.e.

exponentiation of the qp,tvalues, is then performed to obtain the synthetic monthly flow series.

The above procedure was used to generate 50 series of mean monthly flows, each 100 years long. Such a

process of computation was made for each of the 40 stations considered. The approach used in this generation

of new samples is based on the concept of preserving the most reliable parameters of the historic sample,

namely the general means, the general standard deviations, and the autoregressive coefficients. Statistical tests

using Z and F statistics were performed to determine whether the differences of the historic and generated

means and standard deviations respectively are significant at the 95% confidence level. A review of these

tests is given elsewhere (Kendall and Stewart, 1971). The results indicate that the historic means were not

preserved for the months of January and February, and the historic standard deviations were not preserved,

in general, for July and August. Generally, the ARIMA(1, 0, 1) model can be utilized for flood estimates in

the southwest of Saudi Arabia, Yemen and Oman.

CONCLUSIONS AND RECOMMENDATIONS

According to the findings of this study, it can be concluded that wadi flood flows in Arabian Gulf states, and

probably in similar arid regions, constitute a reasonable source of fresh water. Measurements of wadi flows,

however, are too limited to allow water resources engineers to come up with design figures. This conclusion

is supported by the fact that existing records show wide variations, temporally and spatially, for each of the

gauged wadis. Estimation of wadi flood flows using the existing approaches and formulas is not difficult, yet

the results obtained should be used with caution. These formulas contain parameters that cannot be estimated

accurately unless the available data are of satisfactory quality and quantity. Based on the quality and quantity

of data available at present, it can be concluded that:

1. The regional curves derived, followed by the POT and the ANNMAX models, provided predictions of

reasonable accuracy for the maximum annual flood flows for return periods up to 20 years.

2. The ARIMA(1, 0, 1) model can reasonably simulate the monthly flow sequences in the Arabian Gulf states.

Thus, it should be used for such estimates.

3. The frequency of annual maximum floods at sites affected by two different systems of climate can be best

described by a mixture of two extreme-value type 1 distributions.

ACKNOWLEDGEMENTS

This work is part of a comprehensive regional study to investigate fresh water sources in the Arab world.

Funds received from King Abdulaziz City for Science and Technology (Saudi Arabia) and UNESCO and

assistants from the Arab League contributed to achieve the reported results. The help received from the

Deputy Minister of Agriculture and Water in Saudi Arabia, from the Deputy Minister of Water Resources

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in Oman, and from the UNESCO water science section and wadi floods working group are recognized and

appreciated.

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