Maximum Monthly Rainfall Analysis Using L-Moments for an Arid Region in Isfahan Province, Iran

Transcription

1 494 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 46 Maximum Monthly Rainfall Analysis Using L-Moments for an Arid Region in Isfahan Province, Iran S. SAEID ESLAMIAN* Department of Civil and Environmental Engineering, Princeton University, Princeton, New Jersey HUSSEIN FEIZI Isfahan University of Technology, Isfahan, Iran (Manuscript received 26 September 2004, in final form 11 July 2006) ABSTRACT Developing methods that can give a suitable prediction of hydrologic events is always interesting for both hydrologists and ians, because of its importance in designing hydraulic structures and water resource management. Because of the computer revolution in al computation and lack of robustness in at-site frequency analysis, since early 1990 the application of regional frequency analysis based on L-moments has been considered more for flood analysis. In this study, the above-mentioned method has been used for the selection of parent distributions to fit maximum monthly rainfall data of 18 sites in the Zayandehrood basin, Iran, and as a consequence the generalized extreme-value and Pearson type-iii distributions have been selected and model parameters have been estimated. The obtained extreme rainfall values can be used for meteorological drought management in the arid zone. 1. Introduction An essential step in the design of any containment structure and in water resource management is estimating the probabilities of occurrence for the events that the structure is designed to alleviate or that will play an important role in our decisions. The al method for making such probability predictions has been revolutionized by computers, and it seems that the current method based on regional analysis is much better than the conventional at-site procedure. Many problems exist with the at-site procedure, and the cumulative effect of these problems is that this approach can no longer be condoned for real applications. These problems include the following: violation of the assumption of independence between the event values; the quantile function depends upon the method of fit, * Current affiliation: Department of Water Engineering, Ishafan University of Technology, Isfahan, Iran. Corresponding author address: S. Saeid Eslamian, Associate Professor, Department of Water Engineering, Isfahan University of Technology, Isfahan, Iran. saeid@cc.iut.ac.ir and its arbitrary rules; problems with small samples; and, last, the lack of power inherent in goodness-of-fit tests (Wallis 1988). Another problem with the at-site procedures that is often unclear, in particular at sites with short records, is that some of the sample s are bounded by infection of sample sizes (Dalen 1987). It is the result of such estimator limitations as skew and kurtosis. In fact, the purpose of regional analysis based on L-moments is to improve the results and predictions of frequency analysis by improving the goodness-of-fit tests and the estimates of the distribution parameters used in single-site analysis. This is in contrast to a traditional at-site frequency analysis using the method of moments to find the data distribution and to estimate parameters. This study applies a regional frequency analysis approach based on L-moments for the first time on Iranian rainfall data. In a regional frequency analysis, regional information is used to increase the reliability of rainfall intensity duration frequency estimates at any particular site (Hosking and Wallis 1991). One limitation is that the procedure assumes that sites from a homogeneous region have the same frequency distribution apart from a site-specific scaling factor. The L-moments are defined as expectations of cer- DOI: /JAM American Meteorological Society

2 APRIL 2007 E S L A M I A N A N D F E I Z I 495 No. Site name TABLE 1. Main characteristics of rainfall-gauging sites under investigation and data series analyzed. Alt (m) Lon (E) Lat (N) Record length (yr) Data collection period First L-moment (mm) L-C ( ) L-C s ( 3 ) L-C k ( 4 ) Fifth L-moment ratio ( 5 ) 1 Isfahan Pol-Kalleh Pole-Mazraeh Garmaseh Khajoo Ziar Damaneh Eskandari Paiabe-Sad Pole-Zamankhan Tiran Khomeini-Shahr Meimeh Shahrokh Koohpaieh Firoozabad Maqsoodbeig Mahiar Regional average of L-moments ratio tain linear combinations of order s (Hosking 1990). They are analogous to conventional moments with measures of location (mean), scale (standard deviation), and shape (skewness and kurtosis). Application of these moments in hydrology and frequency analysis of hydrologic events began in the late 1970s (Greenwood and et al. 1979). Because L-moments are linear combinations of ranked observations and do not involve squaring or cubing the observations, as is done for the conventional method of moments estimators, they are generally more robust and less sensitive to outliers. Zayandehrood basin is one of the large catchments in the central region of Iran, with the Qom desert in the north, the Zagros Mountain Range in the west and the Iran central desert in the east and south. The Zagros Mountain Range prevents humid air masses from moving eastward to reach this region; therefore, this basin receives no precipitation for at least 9 months each year. Although it is known as an arid basin, the existence of the Zayandehrood River and coastal farmlands in this basin cause this arid region to be one of the important agricultural regions of central Iran. Because rainfall occurrence and its quantity during specific durations, such as daily, monthly, and seasonal depths, play an important role in water resources planning and crop water management, estimation of these values has been of particular interest for engineers in this region. They have faced the major problem of small data samples derived from short record lengths of hydrologic events, such as rainfall, resulting from a lack of historical data and old gauging stations. The procedure of this study is based on the use of L-moments in a regional frequency analysis for estimation of maximum monthly rainfalls in Zayandehrood basin. The data used have been collected from 18 raingauging stations, the records of which are published by the Iranian National Meteorology Department. Altitude and latitude have been assumed as initial s of hydrologic homogeneity, and station selection criteria were based on these characteristics. All sites used in this procedure are located between and N latitudes, and 1530 and 2300 m altitudes (MSL). Records used for the analysis have ended the same year, and there are no gaps in the records (Table 1). All of the data were tested using the method of Wald and Wolfowitz (1943) for possible correlation. There was not any at-site and regional correlations between data. 2. Method a. L-moments: Definition The L-moments are the summary s for probability distributions and data samples and are analogous to ordinary moments (Hosking 1990). They provide measures of location, dispersion, skewness, kurtosis, and other aspects of the shape of probability distributions or data samples. Using the uniform distribution function as its foundation and based on shifted Legendre polynomials, each al L-moment is

3 496 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 46 computed linearly (hence the L) giving a more robust estimate for a given amount of data than other methods. For the random variables X 1,...,X n of sample size n drawn from the distribution of a random variable X with the mean m and variance of s 2, let X 1:1 X 1:n be the order s such that the L-moments of X are defined by r 1 r 1 r r 1 k 0 1 k E X k r k:r, r 1,2,..., Hosking and Wallis (1991) provide two s for a test of homogeneity. The first is a measure of dissimilarity. This can determine the sites that are disharmonious from a group of other sites. Estimates for this measure determine the status of distance of a site from the center of a group. If the vector u i [t (i) t (i) 3 t (i) 4 ] T includes values of sample L-moments ratios t, t 3, and t 4 for site (i) (the superscript T denotes the transposition of a vector or matrix), the average of the group for sites can be written as where r is the rth L-moment of a distribution and E[X i:r ] is the expected value of the ith smallest observation in a sample of size r. The first four L-moments of a random variable X can be written as 1 u 1 u i, i 1 and the matrix of covariance of sample is S 1 1 u i u u i u T ; i E X, E X 2:2 X 1:2, E X 3:3 2X 2:3 X 1:3, and E X 4:4 3X 3:4 3X 2:4 X 1:4. Hosking (1990) demonstrated the utility of estimators based on the L-moment ratios in hydrological extreme analysis. The second moment is often scaled by the mean so that a coefficient of variability is determined: L-C 2 1, where 1 is the measure of location. Similar to the definitions and the meaning of the ratios between ordinary moments, the coefficients of L-kurtosis and L-skewness are defined as r r 2, r 3, where 3 is the measure of skewness (L-C s ) and 4 is the measure of kurtosis (L-C k ). Unlike standard moments, 3 and 4 are constrained to be between 1 and 1 and 4 is constrained by 3 to be no lower than Because precipitation is nonnegative, is also constrained to the range from 0 to 1. b. L-moment: Applications 1) HOMOGENEITY TEST In a regional frequency analysis, it is necessary that all of the sites located in one region have data with the same probability distribution. For this reason, the homogeneity of the region should be tested before performing the regional analysis then the measure of dissimilarity can be written as D i 1 3 u i u T S 1 u i u. 10 If D i 3, the site i is disharmonious. The second is a measure of heterogeneity. This estimates the degree of heterogeneity in a group of sites and evaluates whether these sites are reasonably homogeneous. In fact, this compares the variations of L-moments of sites for a group of these sites with the expected values of variations for a homogeneous region. Three measures of variability, V 1, V 2 and V 3, are available. The weighted standard deviation of t, based on L-C (t) is V 1 i 1 N i t i t 2 i 1 N i, 11 where is number of sites, N i is length of records at each site, and t is average of t (i) values computed by t i 1 N i t i i 1 N i. 12 The weighted average distance between the site and weighted average of the group based on L-C and L-C s is computed by V 2 i 1 N i t i t 2 t i 3 t i 1 N i. 13 Weighted average distance between the site and weighted average of the group in a diagram of t 3 versus t 4, based on L-skewness and L-kurtosis, can be written as

4 APRIL 2007 E S L A M I A N A N D F E I Z I 497 TABLE 2. Dissimilarity measures for 18 Zayandehrood sites with monthly rainfall data. No. Site name Record length (yr) D(i) 1 Isfahan Pol-Kalleh Pole-Mazraeh Garmaseh Khajoo Ziar Damaneh Eskandari Paiabe-Sad Pole-Zamankhan Tiran Khomeini-Shahr Meimeh Shahrokh Koohpaieh Firoozabad Maqsoodbeig Mahiar V 3 i 1 N i t 3 i t 3 2 t 4 i t i 1 N i. 14 By simulation of a large number of sites by fitting a four-parameter kappa distribution (Hosking and Wallis 1988), we can compute the heterogeneity measure by H i V i, 15 where and are the mean and the standard deviation of simulated data, respectively. The four-parameter kappa distribution has several useful features. As a generalization of the generalized logistic, generalized extreme-value, and generalized Pareto distributions, it is a candidate for being fitted to data when these three-parameter distributions give an inadequate fit, or when the experimentalist does not want to be committed to the use of a particular threeparameter distribution. Hosking and Wallis (1993) used the four-parameter kappa distribution to generate artificial data for assessing the goodness of fit for different distributions. Hosking and Wallis (1991) suggested that TABLE 3. Dissimilarity measures for six Zayandehrood sites with monthly rainfall data. No. Site name Record length (yr) D(i) 1 Pole-Mazraeh Ziar Eskandari Khomeini-Shahr Meimeh Firoozabad TABLE 4. Results of testing hypothesis for definition of homogeneous regions (18 sites). No. of simulations No. of stations in region H 1 a region is reasonably homogeneous if H i 1 and that a region is fairly homogeneous if 1 H i 2. If H i 2, then the region is absolutely heterogeneous. 2) GOODNESS-OF-FIT TEST Hosking and Wallis (1991) presented a measure of goodness of fit based on sample average regional kurtosis t r. This is more applicable for threeparameter distributions, because all of the threeparameter distributions fitted to data have the same t 3 in L-C s versus the L-C k diagram. Quality of fitness can be adjusted by the measure of difference between t r and the value of DIST 4 upon the fitted distribution. The Z DIST that is a measure of goodness of fit can be written as Z DIST t 4 4 DIST 4, 16 where 4 is the standard deviation of t 4. The value of 4 can be computed by simulation after fitting a kappa distribution to the data (Hosking and Wallis 1988). If Z DIST 1.64, we can say the fitted distribution is reasonable. 3) PARAMETER ESTIMATION H 2 H A common problem in s and its application in hydrology, such as frequency analysis, is the estimation, from a random sample of size n, of a probability distribution whose specification involves a finite number p of the unknown parameters. Analogous to the ordinary method of moments, the method of L-moments obtains parameters by equating the first p sample L-moments to the corresponding population quantiles. The exact distribution of parameter estimators obtained by this method is difficult to derive in general. Asymptotic distributions can be found by treating the estimators as a function of sample L-moments and applying Taylor series methods. TABLE 5. Results of testing hypothesis for definition of homogeneous regions (six sites). No. of simulations No. of stations in region H 1 H 2 H

5 498 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 46 TABLE 7. Result of goodness-of-fit test for distributions fitted (six sites). Generalized Pareto GEV Generalized logistic Results and discussion FIG. 1. L-moments ratio diagram: L-C vs L-C s. Hosking et al. (1985) and Hosking and Wallis (1987) found that asymptotic approximations are usually reliable for samples of 50 or more. Therefore, for sample sizes less than 50 we must use the regional analysis by the method of L-moments. For estimating parameters from a given distribution in the regional method, first the shape parameter of the distribution should be estimated from regional weighted L-moments. In the regional estimation of parameters, all of the sites that are in one region should have the same shape parameter value. After the estimation of the shape parameter, the other parameters of the distribution will be estimated for each site. As will be shown, because of the goodness-of-fit test, the generalized extreme-value (GEV) distribution should fit the data of the region. The probability density function of this distribution is f x 1 1 k k u 1 k 1 Maximum monthly rainfall records (the wettest month in each year), with more than 12 data points, related to 18 gauging stations in Zayandehrood basin, have been analyzed in the context of a regional analysis. Table 1 reports the values of some geophysical parameters and record length for these sites, as well as the L-moments and L-moment ratios for all stations. Table 2 shows the results of applying the L-moment test for the measure of dissimilarity. It seems that our grouping does not look particularly intuitive, because there are no sites with D(i) greater than 3 (the maximum value of this measure in this group is 2.49). Because of these results, we can say that this region does not have any dissimilar stations. However, as shown in Table 2, there are some sites with the values greater than D(i), such as Isfahan, that may affect the other results of the procedure if the homogeneity assumption was not correct. For this reason, we have taken a random sample from this region and have calculated D(i) and other s necessary for the procedure. Table 3 shows the D(i) values for this sample region. As shown in this table, the values of D(i) for the sample region sites are closer to D(i) for the same sites in the region with 18 sites. To test hypothesis for the definition of homogeneous regions, Monte Carlo simulation was performed with the four-parameter kappa parent distribution, with re- exp 1 k[ x / ] 1 k. The parameter estimators of this distribution are 17 z 2 3 t 3 log2 log3, kˆ z z 2, ˆ l 2 kˆ 1 2 kˆ 1 kˆ, and 20 û l 1 ˆ 1 kˆ 1 kˆ. 21 TABLE 6. Result of goodness-of-fit test for distributions fitted (18 sites). Generalized Pareto GEV Generalized logistic FIG. 2. L-moments ratio diagram: L-C s vs L-C k.

6 APRIL 2007 E S L A M I A N A N D F E I Z I 499 TABLE 8. Estimated parameters for GEV distribution using at-site and regional procedures (18 sites). At site Regional Site name u k U K Isfahan Pol-Kalleh Pole-Mazraeh Garmaseh Khajoo Ziar Damaneh Eskandari Paiabe-Sad Pole-Zamankhan Tiran Khomeini-Shahr Meimeh Shahrokh Koohpaieh Firoozabad Maqsoodbeig Mahiar TABLE 9. Estimated parameters for GEV distribution using regional procedure (six sites). Site name U K Pole-Mazraeh Ziar Eskandari Khomeini-Shahr Meimeh Firoozabad sults shown in Table 3 for the 18-site region and in Table 4 for the sample region. Using a four-parameter kappa parent distribution and deriving H i for moments, we obtained values of H i 1, indicating that all values of H i are less than 1; on the other hand, this region is a homogeneous region (Table 5). The negative value indicates that the data have dispersion that is less than the amount we expect for a homogeneous region (Rao and Hamed 1999). The important point in using this homogeneity test is that the primary measure for this test is the value of H 1. This is because, for small regions, H 2 and H 3 values may result in incorrect and virtual status of homogeneity. Furthermore, the L-moments ratio diagram (L-C vs L-C s ) confirms this homogeneity (Fig. 1). However, it seems that the Mahiar station has some differences with the other sites of the group because it is shown as an outlier event in this figure. For making a decision about the primary distributions of this region, the goodness-of-fit test is applied as mentioned above and then the GEV distribution is selected through the candidate distributions, based on their Z DIST 1.64 (Table 6). For the sample region, it also shows that the GEV distribution is a good choice for this region, although the generalized logistic distribution has the same conditions (Table 7 and Fig. 2). The final step for a frequency analysis of hydrologic events is the parameter estimation of the selected distributions and, ultimately, the estimation of their quantiles. For this reason, we used both the at-site and regional procedures. Tables 8 and 9 report the results of estimating the parameters of GEV for the region and TABLE 10. Estimated quantiles for GEV distribution using at-site and regional procedures (18 sites). Here, T is return period. At site T (yr) Regional T (yr) Site name Isfahan Pol-Kalleh Pole-Mazraeh Garmaseh Khajoo Ziar Damaneh Eskandari Paiabe-Sad Pol-Zamankhan Tiran Khomeini-Shahr Meimeh Shahrokh Koohpaieh Firoozabad Maqsoodbeig Mahiar

7 500 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 46 TABLE 11. Estimated quantiles for GEV distribution using regional procedures (six sites). Here, T is return period. T (yr) Site name Pole-Mazraeh Ziar Eskandari Khomeini-Shahr Meimeh Firoozabad FIG. 4. Quantile plot for site Eskandari. sample region. As shown in these tables, the major differences in these results are seen in the shape parameters. Conversely, the estimated parameters of location and scale from the regional procedure do not have the numerous differences with the estimated value from at-site procedure using GEV distribution. Comparing parameters estimated from the region and the sample region indicates that the values have no major differences and are very close to each other for the location, scale, and shape parameters. Sticking points about the predicted quantile from comparing at-site and regional procedures, we can say that the quantiles estimated from the at-site procedure for small return periods are similar to values predicted from the regional method, particularly for sites with record lengths of more than 30 yr. Through decreasing the record length for some stations, the differences of values from two methods increase, particularly for the high return periods (Tables 10 and 11). In the sites of Isfahan (Fig. 3), Eskandari (Fig. 4), Pole-Zamankahn (Fig. 5), Tiran (Fig. 6), Meimeh (Fig. 7), Koohpaieh (Fig. 8), Maqsoodbeig (Fig. 9), and Mahiar (Fig. 10), fitted distributions from both at-site and regional methods have good harmony with the observed data and their variations, but it seems that the at-site estimates could predict quantiles for the high return periods better than the regional estimates. Ac- FIG. 5. Quantile plot for site Pole-Zamankhan. FIG. 6. Quantile plot for site Tiran. FIG. 3. Quantile plot for site Isfahan. FIG. 7. Quantile plot for site Meimeh.

8 APRIL 2007 E S L A M I A N A N D F E I Z I 501 FIG. 8. Quantile plot for site Koohpaieh. FIG. 12. Quantile plot for site Pole-Mazraeh. FIG. 9. Quantile plot for site Maqsoodbeig. FIG. 13. Quantile plot for site Garmaseh. FIG. 10. Quantile plot for site Mahiar. FIG. 14. Quantile plot for site Khajoo. FIG. 11. Quantile plot for site Pol-Kalleh. FIG. 15. Quantile plot for site Ziar.

9 502 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 46 FIG. 16. Quantile plot for site Khomeini-Shahr. FIG. 18. Quantile plot for site Paiabe-Sad. cording to the above-mentioned sections, the distance of the regional estimation line from the observation one has been enlarged by increasing the return periods, especially for the greater-than-50-yr return period. Contrary to these sites, for some of the other sites in this region, such as Pol-Kalleh (Fig. 11), Pole-Mazrae (Fig. 12), Garmaseh (Fig. 13), Khajoo (Fig. 14), Ziar (Fig. 15), and Khomeini-Shahr (Fig. 16), regional estimates have better predictions of quantiles for the high return periods. For example, for the Khomeini-Shahr site, which has the shortest record length in this region, the differences of regional and at-site estimates are large. Furthermore, the regional estimates appear to be more adequate than the at-site predictions. An important point that has been shown in Fig. 10 is the high differences between at-site and regional estimates of quantiles for the return periods of more than 20 yr. Therefore, the regional rainfall predictions of Mahiar are beyond the confidence intervals. According to this, in addition to the outlier manner of this site shown in the L-C L-C s diagram, we can conclusively say this site is heterogeneous from the other sites of this region. For the remaining sites of this region, namely, Damaneh (Fig. 17), Paiabe-Sad (Fig. 18), Firoozabad (Fig. 19), and Shahrokh (Fig. 20), the estimates from both regional and at-site methods are almost similar, and both methods have good harmony with the observed value for the low and high return periods. 4. Conclusions According to the results from applying L-moments to Iranian rainfall data and associated ratio diagrams, it is found that the method presented is a useful and robust tool for confirming either similarities or differences in the regional frequency analysis of the rainfall events. Furthermore, the robustness of regional frequency analysis based on L-moments for parameter estimation allows one to obtain relatively accurate quantiles from these distributions that will provide reliable estimates of rainfall for most cases. It is found for stations having a record length of less than 30 yr (8 of 18 sites), based on at-site maximum likelihood estimates, we obtain about 15% 20% underestimation for the greater-than-200-yr return period (Feizi 2003). It also seems that, for some cases, quantiles estimated from the regional method could not give the reliable prediction of such events, especially in samples having the L-moments ratio with a relatively large distance from the regional weighted mean of L-moments ratio. FIG. 17. Quantile plot for site Damaneh. FIG. 19. Quantile plot for site Firoozabad.

10 APRIL 2007 E S L A M I A N A N D F E I Z I 503 REFERENCES FIG. 20. Quantile plot for site Shahrokh. To overcome this problem, it might be necessary to accomplish some changes in values of homogeneity and dissimilarity s. Another important outcome of this study is that the homogeneity test and s are necessary for testing homogeneity and similarity of the sites in one region but it may not be enough to test fully for homogeneity. On the other hand, application of these s without using the L-moments ratio diagrams, especially the L-C L-C s diagram, may result in incorrect homogeneity status. The final point is that for regions with a large number of sites, we usually have one distribution that passes the goodness-of-fit test; however, for the small region, there might be more than one distribution that could pass the goodness-of-fit test. Acknowledgments. We thank Ms. Tara Troy from Princeton University for her valuable editorial help. Dalen, J., 1987: Algebraic bounds on standardized sample moments. Stat. Probab. Lett., 5, Feizi, H., 2003: Analytical comparing of parameter estimation using L-moments and maximum likelihood estimates for some station of Isfahan region. M. S. dissertation, Isfahan University of Technology, 115 pp. Greenwood, J. A., J. Landwehr, N. C. Matalas, and J. R. Wallis, 1979: Probability weighted moments: Definition and relation to parameters of several distribution expressible in inverse form. Water Resour. Res., 15, Hosking, J. R. M., 1990: L-moments: Analysis and estimation of distributions using linear combinations of order s. Roy. Stat. Soc. London, 52, , and J. R. Wallis, 1987: Parameter and quantile estimations for the generalized Pareto distribution. Technometrics, 29, , and, 1988: The 4-parameter kappa distribution. Research Rep. RC 13412, IBM Research Division, Yorktown Heights, NY., and, 1991: Regional frequency analysis using L- moments. Research Rep., Watson Research Center, IBM Research Division, Yorktown Heights, NY., and, 1993: Some al useful in regional frequency analysis. Water Resour. Res., 29, ,, and E. F. Wood, 1985: Estimation of generalized extreme value distribution by the method of probability weighted moments. Technometrics, 27, Rao, C. R., and K. H. Hamed, 1999: Flood Frequency Analysis. CRC Press, 330 pp. Wald, A., and J. Wolfowitz, 1943: An exact test for randomness in the nonparametric case based on serial correlation. Ann. Math. Stat., 14, Wallis, J. R., 1988: Catastrophes, computing and containment: Living with our restless habitat. Speculations Sci. Technol., 11,