Academic Journal of Science CD-ROM. ISSN: 165-68 :: 1(1):41 55 (01) RAINFALL INTENSITY-DURATION-FREQUENCY RELATIONSHIP FOR SOME REGIONS IN SAUDI ARABIA Ibrahim H. Elsebaie King Saud University, Kingdom of Saudi Arabia Intensity-Duration-Frequency (IDF) relationship of rainfall amount is one of the most commonly used tools in water resources engineering for planning designing, and operation of water resources projects, or for various engineering projects against floods. The aim of this study is to develop (IDF) equation for the rainfall for two regions in the Kingdom of Saudi Arabia. These relationships are useful in the design of urban drainage works, culverts and other hydraulic structures. Two commonly frequency analysis techniques were used to develop the relationship between the rainfall intensity, storm duration, and return periods from rainfall data for two regions in Saudi Arabia. These techniques are Gumbel and the Log Pearson Type III distribution. Rainfall data was obtained for different durations. A set of curves was plotted using the intensity-duration-frequency data to be the basis of the analysis. An equation for calculating rainfall intensity for every region was derived based on the results obtained from the IDF data. The results obtained using Gumbel distribution are slightly higher than the results obtained using the Log Pearson III distribution. In general, the results obtained using the two approaches are very close and have the same trend. Rainfall intensities were drawn from these two methods and the results showed good consistency with what has been done before in some parts of the study area. The chisquared goodness-of-fit tests were used to choose the best statistical distribution among them. The results show that the precipitation data for the regions under study follow fairly well the gamma distribution. Further studies are recommended whenever there will be more data to verify the results obtained or updating those IDF curves and the equation parameters obtained from the study. Keywords: IDF curve, Rainfall intensity, Rainfall duration, Rainfall frequency relationships. INTRODUCTION Rainfall Intensity-Duration-Frequency curves (IDF curves) are graphical representations of the amount of water that falls within a given period of time [1]. Rainfall Intensity-Duration-Frequency is used to aid the engineer when creating the design. The establishment of such relationships was done as early as in 193 (see [] and [3]). Since then, many sets of relationships have been constructed for several parts of the globe. However, such map with rainfall intensity contours has not been constructed in many developing countries [4]. Koutsoyiannis [4-5] cited that the IDF relationship is a mathematical relationship between the rainfall intensity i, the duration d, and the return period T (or, equivalently, the annual frequency of exceedance, typically referred to as frequency only). Indeed the IDF-curves allow for the estimation of the return period of an observed rainfall event or conversely of the rainfall amount corresponding to a given return period for different aggregation times. A set of Intensity-Duration-Frequency (IDF) curves constitutes a relation between the intensity (more precisely, the mean intensity) of precipitation (measured in mm/h), the duration or the aggregation time of the rainfall (in min) and the return period of the event. The return period of an event indicates how 41
4 Ibrahim H. Elsebaie rare/how frequent is this event and is defined by the inverse of the annual exceedance probability (see references [6] and [7]) In Kentucky, rainfall IDF curves are used in conjunction with the Rational Method to calculate runoff from a particular watershed. The information from the curves is then used in hydraulic design to size culverts and pipes [1]. Further studies by [8] performed Rainfall analysis and regionalization computing intensity-duration-frequency curves for different regions. There is a high need for IDF-curves in the tropical region of KSA but unfortunately the adequate long-term data sets are frequently not available. The purpose of this study is mainly to produce IDFcurves for precipitation for different regions in Saudi Arabia. In recent studies, various authors are tempting to relate the IDF-relationship to the synoptic meteorological conditions in the area of the stations (see references [3], [6]). Reference [9] derived rainfall depth-duration-frequency relationships (DDF) for Saudi Arabia through analysis of available rainfall intensity data. Reference [] mentioned that individual station rainfall estimates can be used with confidence for shorter return periods. Reference [] found that the regional rainfall estimates can be used with confidence for longer return period up to 100 years. Reference [9] added that Saudi Arabia could be divided into different rainfall zones. These zones are: I South-western region, III Northern region, IV Central and Eastern region, V Southern region, as shown in Figure (1). Reference [9] recommended that rainfall estimates from individual stations and regional analysis may be modified in future when new rainfall data become available. Reference [9] performed study IDF Curves and derived an equation to calculate the rainfall intensity for the AlRiyadh region. Reference [9] found that the rainfall intensity obtained using Gumbel method is higher than the rainfall intensities calculated by Log Pearson type III. Reference [11] performed a study to estimate the rainfall depth duration frequency relationships for Qasim region in the kingdom at various return periods, using two continuous probability distributions, the extreme value type I distribution (Gumbel) and the log Pearson type III distribution. Reference [11] found that among the two distributions used in the study, the log pearson type III distribution method gave some larger rainfall estimates with small standard errors. Reference [1] conducted a study for predicting short-duration, high-intensity rainfall in Saudi Arabia. Reference [1] found that the results that the short duration/high intensity rainfall was far from the universal relationship suggested by other researchers and concluded that a relation for each region has to be obtained to act as a useful tools in estimating rainfall intensities for different durations and return periods ranges between to 100 years. Further studies by [13] and [14] performed Rainfall Frequency Distribution and analysis for Riyadh, Shaqra and Al-Zilfi Areas in Saudi Arabia Data collection Data gathering was perhaps, the most difficult part of the project. Different climatological stations were available from the Ministry of Water and Electricity (KSA) but only few of these stations have a good record (1967-001) with few missing data and the other stations have very few records of the data which are not presentable at all to be considered in that study. So, it was decided to use the stations which have a good record and the time intervals (10, 0, 30, 60, 10, 180 minutes, etc ). These stations represent two regions of the kingdom (region I and IV). So, it was decided to develop IDF relationships for some stations in zone (I) and for zone (IV), these zones including the stations are shown in Figure (1). Once the data was collected, it was exported and analyzed by the computer to determine maximum or peak rainfall for each year of data for the available stations. The data was then imported into an excel spreadsheet for further process.
Rainfall intensity-duration-frequency relationship for some regions in saudi arabia 43 Figure 1. Rainfall Zones in Saudi Arabia (Reference [9]). DEVELOPMENT OF RAINFALL INTENSITY DURATION FREQUENCY CURVES For many hydrologic analyses, planning or design problems, reliable rainfall intensity estimates are necessary. Rainfall intensity duration frequency relationship comprises the estimates of rainfall intensities of different durations and recurrence intervals. There are commonly used theoretical distribution function were applied in different regions all over the world (e.g. GEV, Gumbel, Pearson type III distributions). Gumbel distribution methodology was applied on different region all over the world by [1]. Also that technique was applied in Jordan by [15]. Two commonly frequency analysis techniques were used to develop the relationship between the rainfall intensity, storm duration, and return periods from rainfall data for the regions under study. These techniques are: Gumbel distribution and the Log Pearson Type III (LPIII) distribution. Gumbel Theory of Distribution Gumbel distribution methodology was selected to perform the flood probability analysis. The Gumbel theory of distribution is relatively simple and uses only extreme events (maximum values or peak rainfalls). The Gumbel method calculates the, 5, 10, 5, 50 and 100-year return intervals for each duration period and require several calculations. Two of the commonly used distributions for rainfall analysis are the Gumbel s extreme value Distribution and the Log Pearson Type III distribution. Either may be used as a formula or as a graphical approach (see references [1] and []). Frequency precipitation P T (in mm) for each duration with specified return period T (in year) is given by the following equation. Where K is Gumbel frequency factor given by:
44 Ibrahim H. Elsebaie K = 6 0.577 π T + ln ln T 1 where P ave is the average of the Maximum precipitation corresponding to a specific duration. In utilizing Gumbel s distribution, the arithmetic average in Eq. (1) is used: P = 1 n ave P i n i= 1 where p i is the individual extreme value of rainfall and n is the number of events or years of record. The standard deviation is calculated by Eq. (4) computed using the following relation: n 1 S = ( P i P ave ) n 1 i= 1 Where S is the standard deviation of P data. The frequency factor (K) (given in Table: Appendix), which is a function of the return period and sample size, when multiplied by the standard deviation gives the departure of a desired return period rainfall from the average. Then the rainfall intensity, I (in mm/hr) for return period T is obtained from: P I t = T T d Where Td is durations in hrs (10/60, 0/60, 30/60 & so on) The frequency of the rainfall is usually defined by reference to the annual maximum series, which comprises the largest values observed in each year. An alternative data format for rainfall frequency studies is that based on the peak-over-threshold concept, which consist of all large precipitation amounts above certain thresholds selected for different durations. Due to its simpler structure, the annualmaximum-series-based method is more popular in practice (see reference [18]). From the raw data, the Maximum precipitation (P) and the statistical variables (average and standard deviation) for each duration (10, 0, 30, 60, 10, 180, 360, 70, 1440 minutes) were computed. The computed frequency precipitation (P T ) values and intensities (I T ) for different durations and six return periods were computed following the methodology before. 1/ Log Pearson Type III (LPIII) The Pearson Type III probability model is used to calculate the rainfall intensity at different rainfall durations and return periods to form the historical IDF curves for each station. It is commonly used in Vietnam. LP III distribution involves logarithms of the measured values. The mean and the standard deviation are determined using the logarithmically transformed data, using this frequency distribution functions, the maximum rainfall intensity for considered durations and, 5, 10, 0, 50 and 100 years return periods, have been determined.
Rainfall intensity-duration-frequency relationship for some regions in saudi arabia 45 In same manner as Gumbel method, the frequency precipitation is obtained using Log Pearson III method but using the logarithm of variables in the relation. The simplified expression for this distribution is given as: * * * P = P K S T ave + T n * 1 * Pave = P n i = 1 n 1 / * 1 * * S = ( P P ave ) 1 n i = 1 Where P* T, P* ave and S* as defined before, and K T is Pearson frequency factor which depends on return period (T) and skew coefficient (C s ). The skewness coefficient, Cs, is required to compute the frequency factor for this distribution. The skewness coefficient is computed by Eq. (11) (see references [] and [16]). C s ni * * 3 n ( P P ) i i ave = * ( n 1)( n )( S ) 3 K T values can be obtained from Tables in many Hydrology references. By knowing the skewness coefficient and the recurrence interval, the frequency factor, K T for the LPIII distribution can be extracted. The antilog of the solution in Equation (7) will provide the estimated extreme value for the given return period. Derivation of IDF equation The IDF formulas are the empirical equations representing a relationship among maximum rainfall intensity (as dependant variable) and other parameters of interest such as rainfall duration and frequency (as independent variables). There are several commonly used functions found in the literature of hydrology applications (see references ([], [16] and [7]). To derive an equation for calculating rainfall intensity (I) for the region (I) and the Central & Eastern province (Zone IV), there are some required steps for establishing an equation suit the calculation of rainfall intensity (I) for a certain recurrence interval and specific rainfall period which
46 Ibrahim H. Elsebaie depends mainly on the results obtained from the (IDF curves) and the corresponding logarithmic conversion, where it is possible to convert the equation into a linear equation, and thus to calculate all the parameters related to the equation. The following steps are followed to derive an equation and its parameters for every region. 1 convert the original equation d By applying the logarithmic function to get I = CT T m r e m K = CT r Calculate the natural logarithm for (K) value found from Gumbel method or form Log Pearson Type III method as well as the natural logarithmic for rainfall period T d 3 Plot the values of (log I ) on the y axis and the value of (Log T d ) on the X- axis for all the recurrence intervals for the two methods. 4 From the graphs (or mathematically) we find the value of (e) for all recurrence intervals where e represents the slope of the straight line. Then we find out the average e value called e ave by using the following equation: e e ave = n where n represents recurrence intervals (year ) value noted as Tr 5 From the graph we find (log K) value for each recurrence intervals where (log K) represents the Y intercept values as per Gumbel method or LP III method. Then we convert equation (13) into a linear equation by applying the natural logarithm to become: 6 Plot the values of (log K ) on the y- axis and the values of the (log Tr) on the x-axis to find out the values of parameters c&m as per Gumbel method or Log Pearson III where (m) represents the slope of the straight line and ( C ) represents the ( anti log ) for the y intercept. Finally, by substituting the parameters (c, e and m) which in original equation is a conclusive equation for the calculation of rainfall for every region.
Rainfall intensity-duration-frequency relationship for some regions in saudi arabia 47 GOODNESS OF FIT TEST The aim of the test is to decide how good a fit is between the frequency of occurrence observed in a sample and the expected frequencies obtained from the hypothesized distribution. A goodness-of-fit test between observed and expected frequencies is based on the chi-square quantity, which is expressed as, k ( ) i χ = O i E i / E i = 1 Where χ is a random variable whose sampling distribution is approximated very closely by the Chisquare distribution. The symbols O i and E i represent the observed and expected frequencies, respectively, for the i-th class interval in the histogram. The symbol k represents the number of class intervals. If the observed frequencies are close to the corresponding expected frequencies, the χ value will be small, indicating a good fit; otherwise, it is a poor fit. A good fit leads to the acceptance of (null hypothesis), whereas a poor fit leads to its rejection. The critical region will, therefore, fall in the right tail of the chi-square distribution. For a level of significance equal to α, the critical value χ is found from readily available Chi-square tables and χ a > χ constitutes the critical region (see reference [9]) a RESULTS AND ANALYSIS The purpose of this study was to develop Rainfall Intensity-Duration-Frequency curves (IDF) for some areas in the kingdom. When constructing an engineering project that must consider storm runoff, the IDF curves are used as an aid when designing drainage structures. The curves allow the engineer to design safe and economical flood control measures. Rainfall estimates in mm and their intensities (mm/hr) for various return period and different durations were analyzed using the two techniques (Gumbel and LP III). It was noticed that the rainfall estimates are increasing with increasing the return period (T) and the results obtained from the two methods have good consistency. Figures (-9) show results of the IDF curves obtained by Gumbel and LP III methods for Region I, which include Najran, Sabia, Madhah and Bisha stations. It was shown that there were small differences between the results obtained from the two methods, where Gumbel method gives slightly higher results than the results obtained by LP III for Najran station while slightly higher results by LP III than the results obtained by Gumbel for Sabia station. In the case of Midhah and Bisha stations, it was noticed small differences also between the results obtained by the two methods where Gumbel gives little higher results than LPIII for the majority of the rainfall durations starting from 30 min to 70 min while little higher results obtained by LPIII than Gubmel for only 10 min rainfall duration. Figures (10-11) show results of the IDF curves obtained by Gumbel and LP III (zone IV). Also, Gumbel method gives higher results than the results obtained by LP III when they applied for zone IV. Also the trend of the results is the same and they close each other. Table (5) shows the parameters values obtained by analyzing the IDF data using the two methods used in the deriving formulas for the two regions. Also, goodness-of-fit tests were used to choose the best statistical distribution among those techniques. Results of the chi-square goodness of fit test on annual series of rainfall are shown in Table 6. As it is seen most of the data fit the distributions at the level of significance of α=0.05, which yields χ cal < 3.84. Only few data of the stations in the region for specific rainfall durations do not give good fit using the LP III distribution. Also, few data of two regions for different rainfall durations do not give good fit, even at the level of significance of α = 0.01.
48 Ibrahim H. Elsebaie Table 5. The parameters values used in the deriving formulas. Table 6. Results of Chi-square goodness of fit test on annual maximum of rainfall. Duration in minutes Region/Station Distribution 10 0 30 60 10 180 360 7 0 1440 Gumbel 0.575 0.175 0.36 0 0.373 0.489 0.9 9 0.19 0 0. 8 0.60 Najran LP III 0.300 1.08 0* 4.66 7 5.161 0.5 0.54 0 0.4 1.4 38 7.60* Gumbel 0.139 0.144 LP III 0.6 0.473 0.577 0.63 1 0.83 5 0. 87 0.59 14.44 * 0.57 1.8 14.9 31.9 0.70.74 1.6 - Gumbel 3. 1.0 LP III 0.69 7.7 7.7 1.5 5 1.4 1.4 1.15 1.7 1. 4. * 7.8* 4.59 1.13 3 6.5 - - Gumbel 1.40 0.364 0.33 0.64 0.3 0.38 0.37 LP III 10.80 * 0.63.61 10.5 * 4.49 4.44 0.67 0.6 11 -.9 8 -
Rainfall intensity-duration-frequency relationship for some regions in saudi arabia 49 Central & Gumbel 0. 0.151 0.43 1 0.535 0.379 0.56 0.77 4 0.5 5 0.446 Eastern region (Region IV) LP III.081 3.40 0.56 6.310 1.04 1* 1.75 1 6.86 1 8.0 48 * 1.96 Figure : IDF Curves by Gumbel Method (Najran station).
50 Ibrahim H. Elsebaie
Rainfall intensity-duration-frequency relationship for some regions in saudi arabia 51
5 Ibrahim H. Elsebaie CONCLUSIONS This research presents some insight into the way in which the rainfall is produced in Saudi Arabia. Since the Kingdom is large and has different climatic conditions from region to region, a relation for each region has to be obtained to act as a useful tool in estimating rainfall intensities for different durations and return periods ranges between to 100 years. This study has been conducted to the formulation and construction of IDF curves using data from recording station by using two distribution methods (Gumbel and LP III Distribution). Gumbel method gave some larger rainfall intensities estimates compared to LP III Distribution. IDF equations for the two regions (region I and region IV) were obtained for duration varying from 10 to 1440 min and return period from to 100 years. The results obtained using Gumbel distribution are slightly higher than the results obtained using the LP III distribution for the majority of the rainfall durations when they were applied on the study regions with the kngdom. In general, the results obtained using the two approaches are almost close at all return periods except in few rainfall durations and also have the same trend. Rainfall intensities are drawn from these two methods and the results obtained from that work showed good consistency with what has been done before in some parts of the study area. It may be concluded that the difference observed between the results of this study and the results have done
Rainfall intensity-duration-frequency relationship for some regions in saudi arabia 53 before by [9] may be considered accepted and in good agreement. The difference between the results of this study and the study done by [9] can be attributed to the record lengths used for this study and the studies before. The chi-square test 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. The results of the chi-square test of goodness of fit showed that in all durations the null hypothesis that the extreme rainfall series have the Gumbel distribution is acceptable at the 5% level of significance. On the other hand, few cases in which the fitting was not good for the LP III distribution. Inspite of the chi-square values are appreciably below the critical region using Gumbel distribution and few values are higher than the critical region using LP III distribution, it is difficult to say that one distribution is superior to the other. Further studies are still recommended whenever there will be more data to verify the results obtained or updating those IDF curves and the equation parameters obtained from that study. NOTATIONS AND ABBREVIATIONS
54 Ibrahim H. Elsebaie ACKNOWLEDGEMENTS The writer expresses his sincere gratitude to the Research Center of the Faculty of Engineering, King Saud University, for supporting this work. The writer is thankful to the Ministry of Water and Electricity, KSA for providing the data needed for the research. REFERENCES B.S. Dupont and D.L. Allen, Revision of the Rainfall Intensity Duration Curves for the Commonwealth of Kentucky, Kentucky Transportation Center, College of Engineering, University of Kentucky, (U.S.A), March 000. V. T. Chow, Handbook of Applied Hydrology, McGraw-Hill Book, 1988. B.S. Dupont and D.L. Allen, Establishment of Intensity-Duration-Frequency Curves for Precipitation in the Monsoon Area of Vietnam, Kentucky Tranportation Center, College of Engineer, University of Kentucky in corporation with U.S. Department of Transportation, 006. D. Koutsoyiannis, D. Kozonis, and A. Manetas, A mathematical Framework for Studying Rainfall Intensity- Duration-Frequency Relationships, J. Hydrol., 06: 118-135, 1998. D. Koutsoyiannis, On the appropriateness of the Gumbel distribution for modelling extreme rainfall, Prceedings of the ESF LESC Exploratory Workshop, Hydrological Risk: recent advances in peak river flow modelling, prediction and real-time forecasting, Assessment of the impacts of land-use and climate changes, European Science Foundation, National Research Council of Italy, University of Bologna, Bologna, October 003. B. Mohymont1, G.R. Demar ee1, and D.N. Faka, Establishment of IDF-curves for precipitation in the tropical area of Central Africa comparison of techniques and results, Department of Meteorological Research and Development, Royal Meteorological Institute of Belgium, Ringlaan 3, B 1180 Brussels, Belgium, 004. L.M. Nhat; Y. Tachikawa; and K. Takara, Establishment of Intensity-Duration-Frequency Curves for Precipitation in the Monsoon Area of Vietnam, Annuals of Disas. Prev. Res. Inst., Kyoto University, No. 49 B, 006. V. Ilona and F. Francés, Rainfall analysis and regionalization computing intensity-duration-frequency curves, Universidad Politecnica de Valencia Departamento de Ingenieria Hidraulica Medio Ambiente APDO. 01 46071 Valencia Spain, 00. A.A. Al-Shaikh, Rainfall frequency studies for Saudi Arabia, M.S. Thesis, Civil Engineering Department; king Saud University, Riaydh (K.S.A), 1985. K.N. Khamees, IDF Curves, B.S. Project, Civil Engineering Department; king Saud University, Riaydh (K.S.A), CE 4/5-I-06-01, 004. A.A. Al-Dokhayel, Regional rainfall frequency analysis for Qasim, B.S. Project, Civil Engineering Department; king Saud University, Riaydh (K.S.A), April, 1986. H.A. Al-Khalaf, Predicting short-duration, high-intensity rainfall in Saudi Arabia, M.S. Thesis, Faculty of the college of Graduate Studies; king Fahad University of Petroleum & Minerals, Dahran, (K.S.A), 1997. A.E. Al Sobayel, Rainfall Frequency Distribution for Riyadh, B.S. Project, Civil Engineering Department; king Saud University, Riaydh (K.S.A), Jan. 1983. H.S. Al-Salem, Rainfall Frequency Distribution in Shaqra and Al-Zilfi Areas, B.S. Project, Civil Engineering Department; king Saud University, Riaydh (K.S.A), Jan. 1985.
Rainfall intensity-duration-frequency relationship for some regions in saudi arabia 55 N.A. Hadadin, Rainfall Intensity Duration Frequency Relationship in the Mujib basin in Jordan, Journal of Applied Science 8(10): 1777-1784, 005. C.B. Burke and T.T. Burke, Storm Drainage Manual, Indiana LTAP, February 008. M. Samaw and, N. Sabbagh, Application of Methods for Analysis of Rainfall Intensity in Areas of Israeli, Jordanian, and Palestinian Interest, Jordanian Meteorological Department, Ministry of Water and Irrigation, Amman, Jordan, 004. M. Borga; C. Vezzani; and G. D. Fontana, A Regional Rainfall Depth Duration Frequency Equations for an Alpine Region, Department of Land and AgroForest Environments, University of Padova, Legnaro 3500, Italy, Natural Hazards 36: 1 35, 005.