The hydrology of areas of low precipitation - L'hydrologie des régions à faibles précipitations (Proceedings of the Canberra Symposium, December 1979; Actes du Colloque de Canberra, décembre 1979): IAHS-AISH Publ. no. 128.

Regional analysis of peak discharges in the Negev

OFRA COHEN and ARIE BEN-ZVI Jerusalem, Israel

Abstract. A statistical model has been formulated to describe the magnitude and frequency distribution of the complete series of peak discharges in the Negev. The model is composed of two functions: the log Pearson type HI and the Poisson distributions. The parameters of the model are the mean, the standard deviation and the coefficient of skewness of the logarithms of the peak discharges, and the mean number of flow events per year. The parameters have been fitted to describe the statistics of the peak discharges at 17 stations located in the arid Negev of Israel. Regional comparison of the parameters reveals a good correlation of the mean logarithm with the basin area, a low variability of the standard deviation, a poor regionalization of the skewness coefficient, and a fair estimation of the Poisson parameter with respect to geographical location and mean annual rainfall.

Analyse régionale des débits maximum dans le Neguev Résumé. Un modèle statistique été mis au point pour présenter les valeurs et la distribution fréquentielles des séries complètes de débits de pointe relevés dans le Neguev. Le modèle est composé de deux fonctions: la loi de log Pearson III et la loi de Poisson. Les paramètres du modèle sont la moyenne, l'écart-type et le coefficient d'asymétrie des valeurs logarithmiques des débits maximaux et la moyenne du nombre d'événements par an. Les paramètres représentent de façon statistique les débits de pointe pour 17 stations hydrométriques du Neguev. Une comparaison régionale des paramètres révèle une bonne corrélation de la moyenne des logarithmes avec la sur­face du bassin versant, une faible variance de Fécart-type, une mauvaise régionalisation du coeffi­cient d'asymétrie et une assez bonne estimation du paramètre de Poisson en tenant compte des données géographiques et de la moyenne annuelle des précipitations.

THE NEGEV

The Negev is the arid southern portion of Israel. It extends from the line from Gaza— Be'er Sheva—Sodom to the line Eilat—Rafiah, and from the Arava Valley to the Mediterranean Sea (Fig. 1). The shape of the Negev is a triangle and its area is about 10000 km2. The central portion of the Negev is mountainous, the northwestern portion is a flat wide valley, and the eastern edge is a narrow rift valley. The Negev drains towards the Mediterranean, the Dead Sea and the Red Sea.

The mean annual precipitation in the Negev is from 300 mm/year at Nirim in the northwestern comer to 25 mm/year at Eilat in the southern corner. About 90 per cent of the Negev area lies south of the 200 mm/year isohyetal line, and about 60 per cent south and east of the 100 mm/year line. The 100 mm/year isohyetal line runs near the major watershed line which divides the Mediterranean on the western side, and the Dead Sea and the Red Sea on the eastern side.

The soil types found in the western drainage differ from those found in the eastern drainage (Dan et ai, 1975). The soils of the western drainage are generally loessial and brown soils, lithosols and serozems, or sand dunes, sandy rogosols and arid brown soils. The soils of the eastern drainage are generally red and coarse sediment alluvium, or bare rocks and desert lithosols. A significant deviation from this soil type division is noted for a 230 km2 basin which by morphological evidence has been captured to the eastern drainage (Nir, 1978).

The rivers in the central mountainous portion are mostly wide dry channels. They obtain a narrow canyon shape where they cross very steep mountain slopes and faults.

23

Ofra Cohen and Arie Ben-Zvi

FIGURE 1. Hydrometric stations in the Negev.

Regional analysis of peak discharges in the Negev 25 Gaps are found where the rivers cross mountain ridges. The channel beds are composed of sandy or coarser deposits into which large volumes of water can infiltrate. As a result much of the water and a considerable number of the flow events which occur in the rivers do not reach the sea. The waters that infiltrate flow in the beds and evaporate, feed vegetation or enrich groundwater resources.

THE DATA

The Israel Hydrological Service has been operating a hydrometric network in the Negev for almost 30 years. All the stations in this network are equipped with water level recorders, some of them are equipped also with series of special staff gauges for record­ing the peak stages (by which the slope of the water level is computed), and in some stations an artificial control is constructed. Discharge measurements are performed only at two of the stations, while proper formulae are employed for the computations of the flow at the other stations. Records of at least five years are available for 25 stations, but complete statistical analysis has been prepared for only 17 stations. The computation of the discharges in the other stations is not yet completed.

A list of the 17 stations and their period of operation is given in Table 1. The water level records of the stations, and the notes of the operators have been thoroughly scrutinized and all the flow events have been listed. Yet, in some stations, due to technical shortcomings, a few of the small events might have been missed. Details of the. events prior to September 1971 and the techniques employed for the computations of the peak discharges have been published by Ben-Zvi and Cohen (1975). The lists for the present work have been extended till August 1977, and a number of the records have been revised.

The present analysis is performed on the series of individual flow events. The distinction between separate and connected events is done by the length of the time between the events. When the flow ceases for more than 24 h the newly commencing flow is considered as being another flow event. If the cessation of flow is for less than 24 h the newly commencing flow is considered as being a continuation of the previous flow. An exact time measurement for employing this criterion was required for less than 1 per cent of the events.

MAGNITUDE DISTRIBUTION OF PEAK DISCHARGE

The peaks of all the flow events are analysed in this work. The reasons for the use of the complete series rather than of the annual series, are the relatively long time inter­vals between events, the dryness of the air which encourages evaporation, the rare occurrences of rainfall events which might wet the soil between flow events, the relative shortness of the series, the high variability of the peaks, and the gain in information. We do not think that any significant internal dependence exists in the analysed series, and therefore consider the flow events in a station as being independent of one another.

For every series of peaks the mean, the standard deviation and the coefficient of skewness of the discharges and of their logarithms have been computed. Lines and curves for several probability functions have been plotted on proper papers. These functions are the normal, lognormal, Pearson III, log Pearson III by the technique recommended by the US Water Resources Council (1967), and the log Pearson III by the technique proposed by Bobée (1975). It has been found that the log Pearson HI by the technique recommended by the US Water Resources Council (1967) best fits the data, and consequently it has been selected to represent them. Examples for this fit are shown in Figs. 2 and 3. Thereafter the fit of the function to the data of several stations was improved by recomputing the statistical moments without considering the very small peak discharges. The effect is similar to that shown by Bowers et al. (1971)

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. PERIOD

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of peak discharges in the Negev 27

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FIGURE 3. Peak discharges at Be'er Sheva River (from 1949/1950 to 1964/1965 and from 1972/1973 to 1976/1977).

28 Ofra Cohen and Arie Ben-Zvi in their Fig. 11. These very small flow events are still considered in the other statistical aspects of the work.

The statistical parameters of the series are given in Table 2. These parameters are as follows: TV is the number of events used for the computation of the statistical moments, X is the mean of the common logarithms of the peak discharges, a is the standard deviation of these logarithms, g is the coefficient of skewness of the logarithms, Qm

is the maximal (minimal) peak discharge as defined by Gilroy (1972) due to the nega­tive (positive) skewness of the records.

logQm=X~2a/g (1)

TABLE 2. Statistical data

6 K / S ]

No. Station NX a g 16% 50% 84% g,

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10. 11. 12. 13. 14. 15. 16. 17.

Besor - Daiqa Be'er Sheva (Zarnuq) Be'er Sheva at Be'er Sheva

and Be'er Sheva at JHatserim Besor - Zeelim Besor — Reim Gerar - Reim Hemar Zin - Elion Zin - Mapal Mamshit Zin - Ein Aqrabim Hiyyon Paran - Halamish Paran - Zavar Habaqbuq Karkom Arod Neqarot

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0.505 0.622

0.654 0.69 0.45 0.62 0.659 0.598 0.76 0.59 0.43 0.74 0.854 0.8 0.64 0.9 0.82

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N_ = number of events used for the analysis. X = mean of the discharge logarithms. a = standard deviation of the discharge logarithms. g = skew coefficient. Q = peak discharge of flow events (16,50, 84, Exceedance probability percentages). Sm = extreme discharge.

Extrapolation of the discharges by the log Pearson III function does not exceed (fall below) this value. Fifteen of the Qms are large (small) enough not to cause any prob­lem, whereas the Karkom River maximal discharge is 900 m3/s which seems too low, and the Besor River minimal discharge at Reim is 1.5 m3/s which seems too high.

FREQUENCY OF EVENTS

Analysis of the flow events reveals that no flow was recorded in the month of August and only one event was recorded by one station in July. Consequently the division of the records into water years was made from 1 August to 31 July to give the least prob­ability of inter-effect between flows registered in different water years. This division differs from the regular division of the Israel Hydrological Service (October—September) but it is the same as that of the Israel Meteorological Service, and therefore here we define our water year as a meteorological year.

Regional analysis of peak discharges in the Negev 29 Analysis of the frequency of occurrence of the flow events reveals that the number

of events in a meteorological year is a small random number which is closely described by the Poisson probability function (Ben-Zvi and Ben-Zvi, 1971). The parameter of this function («) for every station is given in Table 1. No relationship has been found between the highest peak discharge in a year and the number of events in this year.

The frequency distribution of the occurrence of the events provides the connection between the magnitude distribution of the peak discharges and the recurrence interval of rare events. Since the events at a station are independent of one another the product of the recurrence interval and the mean number of events per season is the inverse of the event probability. For example, in a station where the mean number of events per season is four, the peak discharge having a recurrence interval of 100 years is the dis­charge whose exceedance probability is 1:400 or 0.25 per cent.

REGIONAL PARAMETERS

In the foregoing analysis we have developed a model for the description of peak dis­charges in a station. The model is composed of two functions and four parameters. The functions are the log Pearson III and the Poisson. The parameters are the mean, the standard deviation and the skewness coefficient of the logarithms of the peak dis­charges and the mean number of flow events per year. Application of this model to ungauged sites requires a technique for the estimation of the parameters. A logical way to develop this technique is by means of a regional analysis of the parameters derived for the gauged sites.

In the analysis we haye found that the mean of the peak discharge logarithms, X, is related to the basin area A. The relationship has been transformed to the geometrical mean of the peak discharges, Qg, which is

2 g = i o r (2)

FIGURE 4. Mean peak logarithm versus area.

30 Ofra Cohen arid Arie Ben-Zvi The relationship is shown in Fig. 4. For this relationship a regression has been formu­lated

Qg = 0.56AOA1 (3)

with a correlation coefficient of 0.72. The standard deviation of the peak discharge logarithms is between 0.4 and 0.9

logarithmic units. Six of the 17 values lie between 0.6 and 0.7 units, and the other values are evenly distributed out of these bounds. We have not found any relationship between the standard deviation of the peak discharge logarithms and the basin charac­teristics. Consequently in estimating for ungaugéd sites we assume either a value similar to one found for a nearby gauged site or an average value of 0.65 units.

The skewness of the peak discharge logarithms is sensitive to sampling error and to the length of the records (Matalas et al, 1975 ; Wallis et al, 1977). The standard error of its estimate, Se, from a sample of N data is defined by

Sg = [6N(N - 1)/(N -2)(N+ 1) (N + 3) ] 1 / 2 (4)

In Fig. 5 we present the skewness coefficient against the number of flow events. The skewness coefficients range from —0.62 to 0.65. The number of events for a station are from 14 to 99. From equation (4) the standard error of estimate of the skewness in a gauged site is from 0.60 to 0.24. The stations for which the skewness coefficient is smaller than —0.2 have 14 to 35 events and their Sg is from 0.6 to 0.40. The stations for which the skewness coefficient is larger than 0.2 have 30 to 45 events and their Sg is from 0.42 to 0.35. Is general, the large absolute values of skewness are involved with large errors of determination.

The skewness coefficient is poorly correlated (r = 0.53) with the number of flow events. The dependence described by Klemes (1976) would result in an increasing function for the absolute value of the skewness with respect to the number of events. Such a relationship is not found in our results.

Since we have not found any good relationship between the skewness coefficient and the basin characteristics, and following the tendency of approaching a value of

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Regional analysis of peak discharges in the Negev 31 0.2 with the increase in the number of events, we estimate for an ungauged site either a regional value of 0.2 or a value similar to that of a nearby station.

The fourth parameter is the mean number of events per year. Its range in the Negev is between 1 and 6.2, and it is roughly related to the mean annual rainfall. In basins where the mean rainfall is from 25 to 125 mm/year there are from 1 to 3 events/year. In basins where the mean rainfall is from 150 to 250 mm/year there are from 3 to 5 flow events per year. The average number for the first group is about two events/year and for the second group about 4.2 events/year.

We have not found relationships between the mean number of events and the basin morphometric variables, although there is an indication of a decrease in the mean number of events with an increase in the drainage density. We explain this by the con­tradicting effects of the basin area and the channel length which are interrelated between themselves.

In the selection of regional values we relate a basin to a group according to the mean annual rainfall and then assume either the average value of the group or a value similar to that of a nearby gauged site.

CONCLUSIONS

(1) The magnitude and the frequency of peak discharges from an arid basin can be described by a statistical model.

(2) The magnitude distribution of the peaks of all the flow events follows the log Pearson III function.

(3) The frequency of the occurrence of flow events follows the Poisson function. (4) The model, which is composed of these functions, contains four parameters.

Estimation of the parameters for ungauged sites can be made from a regional analysis. Two parameters, the mean and the standard deviation of the peak discharge logarithms, are estimated well. The third parameter, the skewness of the logarithms, is estimated poorly. The fourth parameter, the mean number of events per year, is estimated fairly.

Acknowledgement. The article summarizes a chapter of the authors' regular duties with the Israel Hydrological Service. This chapter was supported also by the Dead Sea Works Ltd.

REFERENCES

Ben-Zvi, A. and Cohen, O. (1975) Peak discharges in the Negev. Report no. 4/1975, Israel Hydro-logical Service, Jerusalem, Israel.

Ben-Zvi, M. and Ben-Zvi, A. (1971) The probability distribution of flow events in the Negev./. Hydrol. 14, no. 4, 348-353 .

Bobée, B. (1975) The log Pearson type III distribution and its application in Hydrology. Wat. Resour. Res. 11, no. 5 ,681 -689 .

Bowers, C. E., Patost, A. F . and Larson, S. P. (1971) Computer program for statistical analysis of annual flood data by the log Pearson type III method. Bull. 39, Water Resources Research Center, University of Minnesota, Minneapolis, Minn., USA.

Dan, Y., Raz, Z., Yaalon, D. H. and Koyumjisky, H. (1975) Soil map of Israel 1:500 000: Survey of Israel, Tel Aviv, Israel.

Gilroy, E. J. (1972) The upper bound of a log Pearson type III random variable with negatively skewed logarithms. US Geol. Surv. Prof. Paper 800-B, B273-B275, Washington, DC, USA.

Klemes, V. (1976) Comment on 'Regional skew in search of a parent' by N. C. Matalas, J. R. Slack and J. R. Wallis. Wat. Resour. Res. 12, no. 6, 1325-1326.

Matalas, N. C , Slack,,!. R. and Wallis, J. R. (1975) Regional skew in search of a parent. Wat. Resour. Res. I I , no. 6, 816-825 .

Nir, D. (1978) Israel geomorphological map 1: 500 000: Survey of Israel, Tel Aviv, Israel. US Water Resources Council (1967) A uniform technique for determining flood flow frequencies.

Bull. 15, Water Resources Council, Washington, DC, USA. Wallis, J. R., Matalas, N. C. and Slack, J. R. (1977) Apparent regional skew. Wat. Resour. Res.

13, no. 1 ,159-182.