Rainfall distribution function for Libya and rainfall prediction

Hydroîogical Sciences—Journal—des Sciences Hydrologiques, 44(5) October 1999 £65

Rainfall distribution function for Libya and rainfall prediction

ZEKAI §EN & ALI GEATH ELJADID Meteorology Department, Istanbul Technical University, Maslak 80626, Istanbul, Turkey e-mail: [email protected]

Abstract Monthly rainfall amounts are distributed according to different frequency distribution functions in different parts of the world. However, in extremely arid regions gamma probability distribution functions are most often found to fit the existing data well. Libyan monthly rainfall distributions are found to abide by gamma probability distribution function which is confirmed on the basis of chi-square tests. Almost all the rainfall sequences recorded for at least the last 20 years in Libya are investigated statistically and gamma distribution parameters are calculated at existing stations. The shape and scale parameters are then regionalized and hence it becomes possible to find the parameter values at any desired location within the study area and then to generate synthetic sequences according to the gamma distribution. Predictions of 10, 25, 50 and 100 mm rainfall amounts are achieved by this probability function.

Distribution statistique et prédétermination des pluies mensuelles en Libye Résumé Les hauteurs de pluies mensuelles des différentes parties du monde sont distribuées selon différentes fonctions de probabilité. Dans les régions du monde particulièrement arides cependant, c'est le plus souvent une fonction Gamma qui rend compte des distributions de probabilité observées. Nous avons pu montrer que les totaux pluviométriques mensuels de la Libye obéissaient à une loi Gamma, ce qui a été confirmé sur la base de tests du Chi-2. Pratiquement toutes les séries pluviométriques enregistrées au moins au cours des vingt dernières années dans les stations de la Libye ont fait l'objet d'une analyse statistique et les paramètres correspondants de la loi Gamma ont été estimés. Les paramètres de forme et d'échelle ont ensuite été régionalisés et il était donc possible de les estimer en n'importe quel point de la région étudiée puis de générer des séries synthétiques distribuées selon une loi Gamma. Les probabilités d'occurrence des totaux pluviométriques de 10, 25, 50 et 100 mm ont ainsi pu être prédéterminés sur la base de cette loi de probabilité.

INTRODUCTION

Any meteorological variable has a random behaviour in addition to deterministic trend and seasonal fluctuations. The latter two components can be identified and extended to future time periods by classical mathematical methods but the first, uncertain, component does not lend itself well to extension. Identification requires some uncertainty procedures based on the statistical, probability, stochastic, fuzzy or chaotic concepts (Thorn, 1957; Vines & Tomlinson, 1980; Tabony, 1981). The first two concepts are employed in this paper for the identification of statistical methods and for future probabilistic predictions. The main reason for such a choice is the independent evolution of, especially, the rainfall occurrences in Libya. A modern statistical approach is the basis of climatological analysis with the objective of climatological

Open for discussion until 1 April 2000

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666 Zekai §en & Ali Geath Eljadid

prediction. In particular, in arid regions, the most dominant meteorological variable is rainfall which reflects wet and dry period characteristics. Rainfall is quite sporadic and it is necessary to save each drop of rain for water supply purposes, without wastage.

The random behaviour of rainfall patterns in arid or semiarid regions such as Libya makes their prediction rather difficult. However, probability and statistical methodologies provide the necessary ways for such a goal. It is, therefore, a prerequisite to derive empirically the frequency distributions of rainfall records at each available meteorological station. Furthermore, a theoretically convenient probability distribution function (PDF) is fitted to empirical histograms in order to obtain the population distribution of the rainfall amounts. Similar studies have been considered by many researchers in the literature, such as Gregory (1957), Colling (1961), Chow (1964), Jackson (1969), Taylor & Lawes (1971), Canterford & Pierrehumert (1977), Fiddes (1977), Martin (1988), Abouammoh (1991) and Wallis & Hosking (1993). The main purpose of this paper is to identify the most suitable probability distribution function for the monthly rainfall records all over the study area in Libya. Therefore, maps of "shape" and "scale" parameters have been prepared in order to determine the regional rainfall characteristics at points of no station.

MONTHLY RAINFALL DATA

The study area lies approximately within the latitude 34°~33°N and longitude 10°-23°E. This area stretches about 950 km away from the Mediterranean coast towards the south, and about 1450 km from west to east along the Mediterranean Sea. It is noted that there is a fair number of meteorological stations in the study area, but the periods of rainfall records are frequently different. However, most of the meteorological stations are concentrated on the northwestern part of the study area. In this study, monthly rather than daily rainfall data are used since most daily rainfall records have zeros in arid regions.

There are 37 meteorological stations within the study area from about 45 stations established all over the country. However, eight of these have been omitted because their records include a lot of missing data during many months in one year due to shortages in technical and human capabilities and therefore they are considered as inappropriate for further evaluation. The remaining 29 stations, which were considered for calculation, analysis and prediction purposes, are listed in Table 1. The locations of these meteorological stations are given in Fig. 1.

The rainfall records have at least 23 years duration, which is enough to assess the temporal and spatial characteristics. They are distributed over an area covering about 73% from the Libyan coastline where the climatic conditions are much more favourable than in the centre and southern parts of the country.

Prior to processing the rainfall data, it is necessary to assess the reliability and quality of the available observations and to correct unreliable data by filling the few missing data. It was observed during the study that inter-station correlation of monthly rainfall values was very low, as was expected. Consequently, the nearest three stations were selected in the filling procedure of the data. Missing data for many stations were filled from observations taken from at least three of the nearest stations which are possibly evenly spaced around the station by using the normal-ratio method. The rain-

Rainfall distribution function for Libya and rainfall prediction 667

Table 1 Meteorological station information.

Number

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

Station name

Zwara Beniwalid Azizya Hun Tripoli Jaghbub Al-Khums Awbari Misrata Braq Ghudamis Derj Sinawn Mizda Tobruq Marada

Jado Zentan Sarir Garyan Gnat Abo-Gren Abnjem Sirt Ajdabiya Benghazi Al-Marj Derna Slouq

Longitude (E)

12°06' 14°02' 12°44' 16°02'

13°11' 24°58'

14°16' 12°40' 15°06' 14°25' 09°30' 10°26' 10°36' 12°59' 24°01' 19°20'

i2°or 12°02' 21°30' 13°01' 09°58' 09°58' 15°02' 16°35' 20°14' 21°04' 20°54'

22°39' 20°15'

Latitude (N)

32°56' 32°48'

31°10' 29°01' 32°54'

29°45' 32°39' 26°35' 32°23' 27°15' 30°08' 29°58' 31°02'

31°26' 31°58' 29° 10' 31°57' 29°50' 25°50' 32°10' 25°02' 25°02' 30°40' 31°12' 30°48' 32°07' 32°30' 32°46' 31°39'

Altitude (m a.s.l.)

2 42 15 86 25

108 69 83

105 108 345 266 253 797 621 259 266 506 691 662 176 49

455 13 7

130 272

17

56

Record period (months)

157 250 287 636 647 267 443 443 323 159 93

335 70 70 93 59

443 431

71

96 96

287

70 159 120 575

120 120 575

fall amounts at the index station were weighted by the ratio of normal annual precipitation values (Summer, 1988). Estimates for long intervals such as month or year are more reliable than those for short intervals. Kohler (1949) and Berndtsson & Niemczynowicz (1986) noted that a double mass curve may be derived to test short and long periods of rainfall data for homogeneity over time. The mass curve for monthly and annual rainfall will normally approach a straight line, assuming that there is no consistent trend in month-to-month or year-to-year precipitation totals. Over a long data period, therefore, the normal expectation is that, whilst there may be irregularities in the curve, the overall trend will be clear and consistent. If, however, consistent changes in the rainfall regime are present, then a clear inflection will appear on the mass curve near the point where the change occurs (Cornish, 1977). In this study, the mass curves were examined in order to know whether the available data are homogenous or not. Mass curve results show that the rainfall data of about 18 stations followed a straight line, which implies that the data are homogeneous. However, data at 11 stations consist of a series of straight lines, and they were corrected accordingly.


0 100 200 303 400 500 km

SCALE Fig. 1 Representative meteorological stations.

FREQUENCY DISTRIBUTION

Important characteristics of a large data set can be readily assessed by grouping them into different classes and then determining the number of observations that fall into each of the classes. Such an arrangement in tabular form is called a frequency distribution, which is a basic tool in any climatological analysis. It is possible to find from such distributions relevant probabilities that provide a basis for climatological predictions. At present there are several distributions for estimating precipitation amounts, but the merits of their applicability to different types of data and for different purposes have not been established. The distributions adopted vary between countries and even within the same country in different regions. Nevertheless, since Fuller (1914), much work has been done on selecting a relatively suitable distribution. As a consequence, new distribution methods for choosing between them, as well as methods of parameter estimation, have been developed. So far, there are distributions for all practical possibilities and at least one distribution can fit the observations quite well or better than others.


Several fundamental frequency distributions are available, including the normal, gamma, extreme value, binomial, Poisson, and negative binomial distributions (Walpole, 1982). It is possible to observe quickly from a bar chart that, if most of the observations are concentrated around the zero value, then gamma distribution, or its special case exponential distribution, is suitable. The gamma function is applicable to zero bounded continuous variables. It has been widely used in hydrology. Rainfall probabilities for durations of days, weeks, months and years have been estimated by gamma distribution (Barger & Thom, 1949; Moolley & Crutcher, 1968). Annual runoff has been described by gamma distribution (Young, 1970).

The frequency diagrams of the selected meteorological station rainfall records of Libya have been identified. According to available data, the relative frequency histograms have been drawn as shown in Fig. 2 for the same representative stations. It is clear that the most frequently occurring rainfall amounts have small values. All the

Fig. 2 Some histograms from the study area.


histograms appear as skewed to the right or positively skewed, since they have long right tails with almost zero frequencies at the higher data values. As a result, the large values in the right tail are not off-set by correspondingly low values in the left tail and, consequently, the arithmetic mean value will be greater than the median. Hence, the difference between the mean and median divided by the standard deviation, a, can be used as a definition of the numerical measure of skewness. In the light of this statement, the Pearsonian coefficient of skewness, y, is given by:

3(mean- median) y = standard deviation

(1)

According to this formula, for a perfectly symmetrical distribution, the mean and median are identical and the value of y is zero. For the Libyan data, the skewness related parameters and the measured values of skewness coefficient are listed in Table 2 for various meteorological stations. When the distribution is skewed to the right, as in the Libyan data set, the mean is greater than the median and the skewness will be positive. In general, the values of y will fall between -3 and +3 ( Walpole, 1982).

Table 2 Statistical parameters.

Number

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

Station name

Zwara Beniwalid Azizya Hun Tripoli Jaghbub Al-Khums Awbari Misrata Braq Ghudamis Derj Sinawn Mizda Tobruq Marada Jado Zentan Sarir Gary an Gnat Abo-Gren Abnjem Sirt Ajdabiya Benghazi Al-Marj Derna Slouq

Arithmetic mean (mm)

17.65 14.71 21.88 17.52 27.60 22.42 16.16 19.46 19.83 12.00 2.98 2.07 3.17 5.50

13.30 9.00

17.17 16.83 9.26

22.56 16.19 4.80 3.48

16.80 12.12 23.10 23.80 25.92 13.93

Median (mm)

6 1 6 3 8 5 0 4 4 0 0 0 0 0 2 0 2 2 0 2 3 0 0 3 1 6 0 3 1

Standard deviation (mm)

31.5 23.0 38.4 33.0 41.9 35.4 30.5 31.7 33.7 23.2

9.7 6.6 8.7

13.2 23.5 20.0 31.9 27.0 21.6 37.8 26.9 13.8 8.8

31.8 27.1 39.6 46.5 50.6 24.0

Skewness

1.10 1.78 1.24 1.32 1.40 1.47 1.60 1.46 1.41 1.55 0.92 0.94 1.09 1.25 1.44 1.35 1.61 1.65 1.28 1.63 1.47 1.04 1.18 1.30 1.23 1.29 1.53 1.35 1.62


There are many benefits from a relative frequency distribution function, which are also referred to as the probability distribution functions. A first glance at the histograms in Fig. 2 indicates the following distinctive characteristics: (a) All the empirical frequency functions have their maximum frequencies at the

lowest monthly rainfall amounts; i.e. very low intensity rainfalls occur rather frequently all over Libya.

(b) Very intensive rainfall occurrences are rather rare but once they happen the amount of rainfall is very high.

(c) At some stations (see Tripoli, Misrata and Jado), the transition from very low to high intensity rainfall amounts is smooth in a continuously decreasing form.

(d) At some of the stations (e.g. Zwara and Ghudamis), there are moderately high frequency occurrences between small and large intensities.

(e) In some of the frequency distributions there are continuous rainfall amounts without "zero" frequency over the range of the data. If all the rainfall amounts have continuous non-zero frequencies, it corresponds to a rather humid location.

(f) The distribution of frequencies within each class interval of the data has different variability ranges.

(g) It is possible to make interpretations as to the change of frequencies with the data value. Hence, one can decide about the most frequently occurring data values.

(h) The frequency of extreme value occurrences is easily observed. (i) It is possible to identify the skewness of the data towards lower or higher values.

Although the histograms (Fig. 2) give the impression that they are of exponential type, the statistical tests on the basis of the chi-square criteria have shown that they all belong to a theoretical gamma distribution family. The most convenient theoretical gamma functions are also shown as continuous curves in Fig. 2, especially the rainfall frequency distributions at Tripoli, Azizya, Misrata, Jado, Ghudamis, Mizda, Zentan, Abo-Gren, Benghazi, Al-Marj, Derna and Slouq, which comply almost perfectly with the gamma distribution function.

The selection of a suitable distribution is done in most cases by using statistical tests or by comparing the observed data with theoretical distribution on the basis of analytical and graphical methods. The data series may be tested for basic assumptions using the convenient tests available in the literature. Whatever the goodness of a test may be, it can only indicate the occurrence and not the cause of the disturbance. The latter is a matter of interpretation. In most cases, the test together with a meteorological interpretation of its results presents a satisfactory solution.

GOODNESS-OF-FIT TEST

The choice of suitable distribution is often dictated by convenience or procedural policy, primarily owing to the lack of a basis for selecting the distribution of best fit. Frequently, various distributions are fitted to the data and the one with the best fit is selected. Thus, the decision is made on an empirical basis by using subjective visual techniques or more objective statistical tests. According to the available monthly rainfall data in the present analysis, the most suitable procedure is based on fitting gamma distributions. The chi-square test has been used to examine the type of the


distribution which is suitable for such analysis. The results show that the Libyan precipitation data follow fairly well the gamma distribution (see Table 3).

The "shape" and "scale" parameters are estimated by using the maximum likelihood solutions. It is acknowledged that there are other ways to estimate these two parameters, such as method of moments (Gabriel et al., 1993). This may not be the most efficient way to obtain such estimates, but efficiency is not an important consideration for the range of shape values (3-4) and sample sizes (around 60 or more) considered here. In general, the goodness-of-fit test between observed, 0„ and expected, Eh frequencies is based on the calculation of chi-square, yj, as follows:

x'-ttofï- (2) where k is the number of classes in each histogram and the sampling distribution of %2

is approximated very closely by the chi-square distribution. If the observed frequencies are close to the corresponding expected frequencies, the x2 value is small, indicating a

Table 3 Chi-square and the critical values.

Number

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

Station name

Zwara Beniwalid Azizya Hun Tripoli Jaghbub Al-Khums Awbari Misrata Braq Ghudamis Derj Sinawn Mizda Tobruq Marada Jado Zentan Sarir Garyan Gnat Abo-Gren Abnjem Sirt Ajdabiya Benghazi Al-Marj Derna Slouq

Chi-square statistic

7.417 4.467 2.028

12.783 29.363 2.230

14.996 23.197 27.175 9.244

14.057 237.257 27.624 18.287 3.522

15.947 56.120 29.865 22.473 5.437 1.012

171.463 28.893 7.785

18.691 7.418

10.868 21.189

196.183

Chi-square goodness-of-fit test

2.806 3.280 1.924 6.500

29.079 2.505 1.573

22.150 27.838

1.865 13.988

234.546 23.400 11.713 1.999 6.876

54.473 29.673 6.248 2.370 1.420

160.198 28.788 7.832

19.355 0.417 6.442

17.265 146.450

Degree of freedom

3 2 2 7

19 2 8

14 17 4 8

19 18 11 2

10 17 19 14 3 2

13 19 3

11 4 5

13 16

Critical value 0.05

7.815 5.991 5.991

14.067 30.143 5.991

15.507 9.487

27.587 9.488

15.234 30.143 28.869 19.675 5.991

18.307 27.587 30.143 23.685 7.814 5.991

22.361 30.143 7.814

19.670 9.487

11.070 22.362 26.296


good fit. Otherwise, %2 is large and the fit is then poor. The chi-square test which is calculated here, can be used on the one hand to examine the combinations or contingency of the observed and theoretical frequencies, and on the other hand, to make the decision about the type of distribution which the available data set follows. According to the results that are given in Table 3, it is clear that all the stations follow a gamma distribution. This is because all the values of chi-square for whole stations are acceptable, i.e. the critical %2 values are greater than empirical values.

GAMMA DISTRIBUTION AND PROBABILITY PREDICTIONS

The application of the gamma function was suggested by Barger & Thom (1949), and since then has been adapted to precipitation data sets by various researchers. The gamma function was also employed for Indian rainfall calculations by Mooley & Crutcher (1968), and Wihl & Nobilis (1975) calculated 2.5 and 97.5 percentiles in Austrian rainfall distribution based on the gamma distribution. Furthermore, Mooley (1973) found the gamma distribution as the most suitable model for fitting monthly rainfall amounts in Asia. In the same sense, the results which were obtained in this study show that the gamma distribution is the most suitable model for monthly rainfall amounts in Libya.

The gamma distribution has been found to give good fit to precipitation climato-logical series (WMO, 1966). It is one of the most useful continuous distributions available for many natural events, especially those occurring in arid regions. The gamma distribution is defined by two parameters, a and p, where a is the shape parameter and p the scale parameter. Numerical values of these parameters for stations analysed are given in Table 4.

A wide variety of shapes ranging from reverse J-shaped for P < 1 to single peaked, with the peak (mode) at x = (p ~~ l)/a for P > 1, can be produced by the gamma density function. It is defined by its frequency, or probability density function, as:

PY(X) = —— a pX p~V a X x,a, b>0 (3) x T(p)

where T(fi) is called the gamma function with the properties

r(P) = ( p - l ) for p = 1,2,3,,., (4)

T(P)= jVMe"'d* f o r p > 0 (5) 0

and

r( l ) = T(2)= l , T(l/2) = V ^ (6)

The cumulative gamma distribution function, p(A), is given as:

JaV^'d? p(X) = -2 (7) F T(p)

674 Zekai Çen & Ali Geath Eljadid

which can be evaluated using a table of the incomplete gamma function given in several handbooks (e.g. Pearson, 1957).

The moments in this instance give poor estimates of the parameters, and thus the maximum likelihood fit is recommended. However, sufficient estimates are available and these are closely approximated by the following formula (WMO, 1966):

1 f r ^ T ^ P = 1 it 4 A

1 + J1+-V ' 3J 4A

where A is given by:

(8)

_ 5>*, A = lnX--& (9)

n The parameter a is then estimated by:

a = L (10) X

Table 4 Numerical values of gamma distribution parameters.

Number Station name a P

1 Zwara 0.041 0.728 2 Beniwalid 0.100 1.475 3 Azizya 0.039 0.860 4 Hun 0.058 1.022 5 Tripoli 0.040 1.121 6 Jaghbub 0.045 1.014 7 Al-Khums 0.059 0.963 8 Awbari 0.044 0.857 9 Misrata 0.046 0.930 10 Braq 0.120 1.568 11 Ghudamis 0.150 0.465 12 Derj 0.076 0.158 13 Sinawn 0.173 0.551 14 Mizda 0.088 0.488 15 Tobruq 0.055 0.733 16 Marada 0.071 0.643 17 Jado 0.024 0.441 18 Zentan 0.041 0.690 19 Sarir 0.011 1.302 20 Garyan 0.057 1.302 21 Gnat 0.048 0.793 22 Abo-Gren 0.044 0.213 23 Abnjem 0.137 0.476 24 Sirt 0.089 1.497 25 Ajdabiya 0.088 1.085 26 Benghazi 0.077 1.793 27 Al-Marj 0.056 1.333 28 Derna 0.186 4.841 29 Slouq 0.074 1.032


As summarized by Stevens & Smulders (1979), the values of a and P can be obtained from the method of moments, the method of energy pattern factor, the maximum likelihood method, the Weibull probability paper method, or by the use of percentile estimators. Stevens & Smulders (1979) also calculated the values of both parameters using all five of the above methods and obtained almost the same values without any significant differences.

APPLICATION

All 29 meteorological stations over Libya were selected to develop the future probability predictions of rainfall amounts. For prediction purposes, two maps, one for the shape and the other for scale parameters, were developed as shown in Figs 3 and 4. These are sufficient to make predictions for the precipitation over the study area, especially at sites where the stations are not available (i.e. given the latitude and longitude, it is possible to obtain the values of a and (3 from these maps).

10° 1? 14' 2CP 22° 24° 0.6

0.5

04

0.3

0.2

0.1

00

. 3 Shape parameter distribution (a)

10° 12° 14° 16° 18° 20° 22° 24°

30°-;

Fig. 4 Scale parameter distribution (p).

^


In order to estimate the probability of rainfall exceeding 10, 25, 50 and 100 mm over the study area, it is necessary to consider the standard forms of the prediction. Hence, prediction of rainfall exceedence of 10 mm is given explicitly: for 10.0 mm rainfall, the probability statement is Prob(yield > 10.0 mm) = 1 - P A { 1 0 . 0 mm), which can be expressed in terms of gamma distribution function as:

Pjr(10)= japZp" ,e_aArdA r (11)

The use of this equation together with the incomplete gamma function tables enables calculations to be made for rainfall exceedence predictions as presented in Table 5.

Interpolations to points where there are no meteorological stations can be made from the incomplete gamma tables by using x2 and v values which are listed in Table 5 on the basis of the scale and shape parameter values given in Table 4. The probabilities

Table 5 Values of %2 and v for the study area at different rainfall amounts.

Number

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

Station name

Zwara Beniwalid Azizya Hun Tripoli Jaghbub Al-Khums Awbari Misrata Braq Ghudamis Derj Sinawn Mizda Tobruq Marada Jado Zentan Sarir Gary an Gnat Abo-Gren Abnjem Sirt Ajdabiya Benghazi Al-Marj Derna Slouq

%2 values:

10mm 0.8 2.0 0.8 1.2 0.8 0.9 1.2 0.9 0.9 2.4 3.0 1.5 3.5 1.8 1.1 1.4 0.5 0.8 0.2 1.2 1.0 0.9 2.7 1.8 1.8 1.6 1.1 3.7 1.5

25 mm 2.0 5.0 2.0 3.0 2.0 2.6 3.0 2.0 2.5 6.0 7.6 4.0 8.6 4.6 3.0 3.6 1.0 2.0 0.5 3.0 3.0 2.0 7.0 4.6 4.6 4.0 3.0 9.6 3.6

50 mm

4.0 10.0 4.0 6.0 4.0 5.0 6.0 4.0 5.0

12.0 15.0 8.0

17.0 9.0 6.0 7.0 2.0 4.0 1.0 6.0 5.0 4.0

13.0 9.0 9.0 8.0 6.0

19.0 7.0

100 mm 8.0

20.0 8.0

12.0 8.0 9.0

12.0 9.0 9.0

24.0 30.0 15.0 35.0 18.0 11.0 14.0 5.0 8.0 2.0

12.0 10.0 9.0

27.0 18.0 18.0 16.0 11.0 37.0 15.0

V

1 3 2 2 2 2 2 2 2 3 1 1 1 1 1 1 1 1 3 3 2 1 1 3 2 4 3

10 3


of 10, 25, 50 and 100 mm rainfall over the northern part of Libya have been calculated similarly and the results are listed in Table 6.

According to the general pattern of the actual rainfall data set for Libya, it is clear that Libya receives less amounts of rainfall within its peak periods. However, any future prediction based on the actual data set should follow the same pattern. It is clear from Table 6 that there is an inverse relationship between the probability of occurrence and the assumed rainfall amounts over all regions in the study area, i.e. the probabilities of occurrence decrease as the rainfall amounts increase. This is tantamount to saying that lower amounts of rainfall occur relatively more frequently than large amounts. In addition, the results reflect some spatial distribution of the rainfall over the coastal and non-coastal stations. It can be seen that coastal stations, especially in the western and eastern zones, such as at Zwara, Tripoli, Al-Khums, Benghazi, Derna, Al-Marj and Slouq, have greater rainfall amounts than other, non-coastal, stations and, in general, rainfall amounts decrease southwards (Fig. 5). Sixteen meteorological stations have more than 50% probability of occurrence of 10 mm rainfall amounts (Table 7). Further, rainfall with values of 25 and 50 mm may occur, especially over the coastal stations, but the percentages of 100 mm rainfall occurrences are quite insignificant within that region.

Table 6 Probabilities of rainfall amounts based on gamma function.

Number 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

Station name Zwara Beniwalid Azizya Hun Tripoli Jaghbub Al-Khums Awbari Misrata Braq Ghudamis Derj Sinawn Mizda Tobruq Marada Jado Zentan Sarir Garyan Gnat Abo-Gren Abnjem Sirt Ajdabiya Benghazi Al-Marj Derna Slouq

(10 mm) 0.371 0.572 0.670 0.549 0.670 0.638 0.549 0.638 0.638 0.494 0.083 0.221 0.058 0.192 0.294 0.237 0.479 0.371 0.905 0.753 0.506 0.343 0.094 0.615 0.407 0.809 0.777 0.956 0.682

(25 mm) 0.157 0.919 0.368 0.223 0.368 0.272 0.223 0.368 0.272 0.112 0.006 0.045 0.003 0.032 0.083 0.058 0.317 0.157 0.919 0.392 0.272 0.157 0.008 0.203 0.100 0.406 0.392 0.476 0.308

(50 mm) 0.045 0.019 0.135 0.050 0.135 0.082 0.050 0.135 0.082 0.007 0.000 0.005 0.000 0.003 0.014 0.008 0.157 0.045 0.801 0.112 0.082 0.045 0.000 0.029 0.011 0.096 0.112 0.040 0.072

(100 mm)

0.005 0.000 0.018 0.002 0.018 0.011 0.002 0.011 0.011 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.025 0.005 0.572 0.007 0.007 0.003 0.000 0.000 0.000 0.003 0.012 0.000 0.036

678 Zekai Sen & Ali Geath Eljadid

M l min—-=̂ 1

^ VM


Table 7 Stations with 50% or more rainfall occurrences at 10 mm.

No. Station name Rainfall occurrence (%) 1 Zwara 57.2 2 Al-Zawiya 67.0 3 Azizya 54.8 4 Tripoli 67.0 5 Al-Qrabolli 63.7 6 Al-Khums 54.8 7 Zlitan 63.7 8 Misrata 63.7 9 Yefrn 90.0 10 Garyan 75.3 11 Trhona 60.0 12 Sirt 61.4 13 Benghazi 80.0 14 Al-Marj 77.7 15 Derna 95.5 16 Slouq 68.2

CONCLUSION

Consideration of monthly rainfall amounts at 29 meteorological stations over more than 20 years indicates that their station-based frequency distributions accord with a gamma probability distribution function. Hence, it becomes possible to generalize that monthly rainfall amounts in Libya have gamma distribution, which enables one to construct regional maps for the study area of the two gamma parameters, shape and scale. From these two maps, the parameters can be found for any desired station provided that its position within the study area is known. It is then possible to make rainfall probability occurrence predictions of any desired amount, or alternatively, to predict the amount of rainfall at any point for any given level of probability. The necessary regional map predictions for 10, 25, 50 and 100 mm rainfall amounts were made for the study area.

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Received 26 August 1997; accepted 11 January 1999

Rainfall distribution function for Libya and rainfall prediction

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