MODELLING SUMMER EXTREME RAINFALL OVER THE KOREAN PENINSULA USING WAKEBY DISTRIBUTION

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 21: (2001) DOI: /joc.701 MODELLING SUMMER EXTREME RAINFALL OVER THE KOREAN PENINSULA USING WAKEBY DISTRIBUTION JEONG-SOO PARK a, *, HYUN-SOOK JUNG b, RAE-SEON KIM a and JAI-HO OH c a Department of Statistics and Institute of Basic Sciences, Chonnam National Uni ersity, Kwangju, South Korea b Climate Prediction Di ision, Climate Bureau, Korea Meteorological Administration, Seoul, South Korea c Department of En ironmental and Atmospheric Sciences, Pukyong National Uni ersity, Pusan, South Korea Recei ed 21 October 2000 Re ised 6 June 2001 Accepted 7 June 2001 ABSTRACT Attempts to use the Wakeby distribution (WAD) with the method of L-moments estimates (L-ME), on the summer extreme rainfall data (time series of annual maximum of daily and 2-day precipitation) at 61 gauging stations over South Korea have been made to obtain reliable quantile estimates for several return periods. The 90% confidence intervals for the quantiles determined by WAD have been obtained by the bootstrap resampling technique. The isopluvial maps of estimated design values corresponding to selected return periods have been presented. Copyright 2001 Royal Meteorological Society. KEY WORDS: annual maximum precipitation; bootstrap resampling; design value; isopluvial map; method of L-moments estimation; return period 1. INTRODUCTION The modelling of extreme rainfall is essential in the design of water-related structures, in agriculture, in weather modification and for monitoring climate changes (Huff and Angel, 1992). Korean summer rainfall experienced between the months of June and September has been a topic of importance in this sense. Traditionally, hydro-meteorologists have fit various statistical frequency distributions to historical rainfall data to estimate the magnitude of maximum rainfall at the various recurrence intervals. The Wakeby distribution (WAD), defined by Thomas and introduced by Houghton (1978), is defined by the quantile function x(f)= + [1 (1 F) ]/ [1 (1 F) ]/, (1) where F F(x)=P[X x]. This parametrization of Hosking (1986) is different from the one used previously in the literature by Houghton (1978), Landwehr et al. (1978, 1980) and others. The parametrization explicitly exhibits WAD as a generalization of the generalized Pareto distribution (when =0 or =0), and gives estimates of the and parameters that are more stable under small perturbations of the data. It is assumed that + 0. The range of x are x if 0 and 0, (2) x + / / if 0 or =0. (3) * Correspondence to: Department of Statistics and Institute of Basic Sciences, Chonnam National University, 300 Yongbong-dong, Kwangju, South Korea; jspark@chonnam.ac.kr Copyright 2001 Royal Meteorological Society

2 1372 J.-S. PARK ET AL. For x(f) to be a valid quantile function we must also impose the conditions 0 and + 0. See Hosking (1986) for more details on WAD. This distribution is defined by five parameters, more than most of the common systems of distributions. This allows for a wider variety of shapes, and so a reasonably good fit to a sample might be expected. Actually, by suitable choice of parameter values of WAD, it is possible to mimic the extreme value, log-normal, generalized Pareto and log-gamma distributions. (See Figure 2 for an example of Wakeby probability function with =34.35, =132.61, =3.00, =18.26, =0.24.) Empirical evidence, in relation to the condition of separation (Matalas et al., 1975), suggests that the distributions of floods are more nearly Wakeby-like with 1 and 0 (i.e., long stretched upper tails) than like any of the other more commonly suggested flood distributions (Houghton, 1978; Landwehr et al., 1978). In addition, WAD provides a plausible description of flood sequences, and it also provides a means for representing the seemingly long, stretched upper tail structures of flood distributions, as well as the tail structures of distributions of other hydrologic phenomena (Landwehr et al., 1980). Thus WAD can credibly be considered a parent flood distribution. Because of the above reasons, WAD is widely and successfully used in hydrology, especially for the modelling of extreme events. Recently, Wilks and McKay (1996) concluded that WAD provided the best representations of extreme snowpack water equivalent values, based on the performance evaluation of a suite of theoretical probability distributions. For estimation of the five parameters of WAD, the method of L-moments estimation (Hosking, 1990) has been used. In this study, attempts to use WAD with the method of L-moments estimates (L-ME), on the extreme rainfall data (time series of annual maximum of daily and 2-day precipitation) at 61 gauging stations over South Korea have been made to obtain reliable quantile estimates for several return periods. The 90% confidence intervals for the quantiles determined by WAD have been obtained by the bootstrap resampling technique. The isopluvial maps of quantiles at selected return periods have been presented for use by planners and designers. 2. DATA Two time series of annual maxima of daily precipitation (AMP1) and annual maxima of 2-day precipitation (AMP2) were constructed using the daily rainfall data of each station. The daily precipitation data were obtained from the Korea Meteorological Administration. In total, 61 stations throughout South Korea were considered (the locations are shown in Figure 4). The lengths of time series for stations varied because the opening years of stations are different. The sample sizes for each of the stations are given in Table I, in which only the stations having more than 30 annual maximum observations are listed. The sample sizes of the stations that are not listed in Table I are between 27 and 29. Korean summer extreme rainfall events frequently extend over two days due to the influence of typhoons. However, since the measurement of daily precipitation at each station is based on the calendar day, the summer rainfall characteristics may not be well captured by using (annual maxima of) daily precipitation only. Thus, in addition to the time series of AMP1, annual maxima were extracted from the 2-day moving sums of precipitation at each of the stations to construct the time series of AMP2. See Matsumoto (1989) and Svensson and Berndtsson (1996) for the characteristics of heavy rainfall over East Asia including the Korean peninsula. Figure 1 shows the typical time series of Korean AMP1 and AMP2 which are drawn based on observations at station 156 (Pusan). Each station s code and name, sample size (N), sample statistics including minimum of AMP, sample median, IQR (interquartile range: sample upper quartile minus sample lower quartile), sample L-skewness and maximum of AMP computed from the time series of AMP1 and AMP2 are presented in Tables I and II, respectively, for the stations having more than 30 observations.

3 KOREAN SUMMER EXTREME RAINFALL 1373 Table I. Station s code and name, sample size (N), minimum of AMP, sample median, interquartile range (IQR), sample L-skewness and maximum of AMP computed from the time series of AMP1 for the stations having more than 30 observations (unit, mm) Site Name N Min Median IQR L-skew Max 090 Sokcho Chuncheon Kangnung Seoul Incheon Ulrungdo Suwon Seosan Cheongju Daejeon Chup ryong Pohang Gunsan Daegu Jeonju Ulsan Kwangju Pusan Tongyong Mokpo Yosu Cheju Seoguipo Jinju Jeong-up L-MOMENTS ESTIMATION OF WAKEBY DISTRIBUTION For estimation of the five parameters of WAD, the method of L-moments estimation (Hosking, 1990) has been used. Since the distribution function F(x) of WAD is not explicitly defined, the maximum likelihood estimates (MLE) of parameters are not easily obtained (see Park and Jeon, 2000, for computing MLE). Thus the method of probability weighted moments (PWM) estimation was introduced by Greenwood et al. (1979) for estimation of parameters of the distributions (like WAD) whose inverse form x=x(f) is explicitly defined. Since the L-moments are simple linear combinations of special cases of PWMs, the method of L-moments estimation (L-ME) can be viewed as equivalent to the method of PWM estimation. L-Moments are more convenient, however, because they are more directly interpretable as measures of the scale and shape of the probability distribution. The main advantages of using L-ME are that the parameter estimates are more reliable than the method of moments estimates, particularly from small samples, and are usually computationally more tractable than MLE. Furthermore, due to the use of linear moments instead of the conventional product moments and being resistant to the presence of outliers (which may be present in the sample due to the occurrence of heavy rainfall and typhoon events), the method is quite robust. The (population) L-moments of WAD are, following Hosking (1986): 1 = + (1+ ) + (1 ) 2 = (1+ )(2+ ) + (1 )(2 ) (4) (5)

4 1374 J.-S. PARK ET AL. Figure 1. Time series plots of typical AMP1 and AMP2 (unit, mm) of Korea at station 156 (Pusan) (1 ) 3 = (1+ )(2+ )(3+ ) + (1+ ) (1 )(2 )(3 ) (1 )(2 ) 4 = (1+ )(2+ )(3+ )(4+ ) + (1+ )(2+ ) (1 )(2 )(3 )(4 ) r = (1+ ) (r 1 ) (1 ) (r+1+ ) + (1 ) (r 1+ ), r 5. (8) (1+ ) (r+1 ) The sample L-moments are obtained from given observations: see Hosking (1990) for the formulas. Now, analogously to the usual method of moments estimations, the method of L-moments estimation (L-ME) obtains parameter estimates by equating the first p (number of parameters) sample L-moments to the corresponding population L-moments. Since no explicit solution of simultaneous equations is possible in WAD, the equations can be solved by Newton Raphson iteration. Landwehr et al. (1979) derived an algorithm to get the estimates in each of the cases: known and unknown. The Fortran program (PELWAK) provided by Hosking (1997) basically uses the method of Landwehr et al. (1979). First a solution is sought in which all five parameters are estimated, as functions of the first five L-moments. If no solution is found due to convergence failure, is set to zero and a solution is sought in which the other (6) (7)

5 KOREAN SUMMER EXTREME RAINFALL 1375 Table II. Station s code and name, sample size (N), minimum of AMP, sample median, IQR, sample L-skewness and maximum of AMP computed from the time series of AMP2 for the stations having more than 30 observations (unit, mm) Site Name N Min Median IQR L-skew Max 090 Sokcho Chuncheon Kangnung Seoul Incheon Ulrungdo Suwon Seosan Cheongju Daejeon Chup ryong Pohang Gunsan Daegu Jeonju Ulsan Kwangju Pusan Tongyong Mokpo Yosu Cheju Seoguipo Jinju Jeong-up four parameters are estimated as functions of the first four L-moments. If this too is unsuccessful, then a generalized Pareto distribution is fitted instead, using the first three L-moments. Note that, when =0 or =0 (but not simultaneously) in WAD, Equation (1) is reduced to the following quantile function of the generalized Pareto distribution: x(f)= + [1 (1 F) k ]/k. (9) We have used the Fortran programs provided by Hosking (1997) for the L-moments related computations in this study. The parameter estimates (L-ME) of WAD, and Kolmogorov Smirnov s (K S) goodness-of-fit statistic D at each station are given in Tables III and IV, respectively, for AMP1 and AMP2. The p-values of K S Ds are at least 0.64, which shows that WAD is acceptable for each of the stations. (The p-values here are computed, using the formula in Press et al., 1996, p. 618, as if the parameters of WAD are specified. So this computation actually overestimates the p-values. If this estimated p-value is as small as 0.1, then a more accurate way (of using simulation) to compute the true p-value is necessary. See Ross (1990, section 9.2), for this direction. However, we did not take such a more accurate way in this study, because the estimated p-values were at least 0.64.) Figure 2 shows the relative frequency histogram and the probability functions (of several distributions) corresponding to the interval of each of the vertical bars computed from a time series of AMP1 at site 143 (Daegu). Here the generalized extreme value (GEV), two-parameter gamma and WAD are compared. The parameters of these distributions are also estimated by L-ME using the Fortran routines of Hosking (1997). The K S statistic D for GEV, two-parameter gamma and WAD are 0.089, and 0.059, respectively.

6 1376 J.-S. PARK ET AL. Table III. Station s code, parameter estimates (L-ME) of WAD, and K S s statistic D computed from the time series of AMP1 for the stations having more than 30 observations Site K S D The probability function value corresponding to the interval (say, (x 1, x 2 )) of a vertical bar in Figure 2 is computed by x 2 f(x)dx=f(x 1 ) F(x 2 ), (10) x 1 where f(x) is the probability density function. To compute F(x 0 ) of WAD, we used the routine CDFWAK of Hosking (1997). The routine solves the equation x(f)=x 0 for F using Halley s method which is the second-order analogue of Newton Raphson iteration, because F(x) is not explicitly expressed. 4. QUANTILE ESTIMATION The quantile (or design value) corresponding to a return period of T years (abbreviated, T years return value) is defined by a magnitude x(f), with F=1 1/T. Here, Equation (1) is used into which the parameter estimates are inserted. The design values and its 90% confidence intervals corresponding to 10, 20, 50, 100, 200 and 500 years computed from the time series of AMP1 and AMP2 at each station are presented in Tables V and VI, respectively. The confidence interval of return value has been obtained by the bootstrap resampling technique. Denote q(t) asthet years return value. Now take B bootstrap samples of size n (same as the given dataset) with replacement from the given data and obtain estimates of parameters of WAD (by L-ME) on each of samples. The B estimates of q(t) thus generated are ordered by increasing magnitude. Then 100 (1 )% confidence interval of q(t) is the interval between [B /2]-th and [B (1 )/2]-th largest values. We set B=2000 in this study. See Dunn (2001) for more discussion on bootstrap

7 KOREAN SUMMER EXTREME RAINFALL 1377 Table IV. Station s code, parameter estimates (L-ME) of WAD, and K S s statistic D computed from the time series of AMP2 for the stations having more than 30 observations Site K S D Figure 2. Relative frequency histogram and the probability functions fitted for AMP1 data on site 143 (Daegu) using L-moments estimation (see Table I for the parameter estimates of Wakeby distribution). K S s statistic Ds for gamma, GEV and Wakeby distributions are 0.076, and 0.059, respectively

8 1378 J.-S. PARK ET AL. Table V. Design values (unit, mm) corresponding to various return periods (T=10, 20, 50, 100, 200, 500 years) and its 90% confidence intervals computed from the time series of AMP1 for the stations having more than 30 observations Site T=10 T=20 T=50 T=100 T=200 T= (185, 252) (203, 291) (216, 344) (222, 387) (225, 437) (228, 537) (174, 249) (197, 291) (227, 348) (247, 397) (263, 456) (279, 564) (202, 251) (236, 289) (274, 335) (300, 369) (319, 406) (337, 453) (184, 244) (210, 290) (257, 353) (303, 408) (348, 477) (400, 614) (180, 235) (218, 284) (272, 350) (311, 406) (345, 469) (384, 565) (121, 160) (135, 196) (148, 253) (156, 302) (161, 363) (166, 468) (189, 257) (205, 295) (220, 343) (225, 380) (229, 430) (231, 539) (156, 222) (169, 261) (179, 308) (186, 347) (189, 402) (191, 533) (125, 211) (143, 262) (178, 337) (213, 414) (254, 514) (306, 722) (164, 229) (180, 276) (191, 340) (197, 395) (201, 465) (203, 588) (142, 187) (160, 211) (187, 239) (205, 262) (220, 291) (234, 344) (129, 205) (142, 283) (162, 437) (176, 605) (189, 829) (200, 1321) (130, 203) (140, 245) (156, 301) (168, 351) (185, 421) (201, 580) (119, 154) (133, 182) (154, 222) (174, 257) (198, 300) (234, 389) (155, 201) (180, 240) (210, 300) (229, 352) (241, 414) (252, 518) (185, 246) (209, 302) (230, 390) (241, 468) (248, 562) (254, 727) (148, 198) (169, 241) (198, 313) (218, 380) (235, 465) (254, 627) (194, 234) (216, 274) (236, 339) (245, 399) (252, 473) (256, 612) (170, 257) (194, 309) (221, 382) (241, 445) (255, 529) (268, 689) (148, 183) (169, 223) (189, 295) (200, 365) (208, 455) (214, 611) (162, 202) (175, 233) (187, 275) (193, 309) (198, 350) (203, 430) (202, 237) (217, 259) (232, 284) (238, 309) (241, 340) (243, 403) (192, 282) (233, 336) (277, 410) (299, 483) (317, 569) (334, 711) (176, 238) (201, 268) (221, 310) (231, 348) (237, 391) (243, 457) (121, 186) (133, 224) (149, 279) (161, 328) (175, 396) (190, 534)

9 KOREAN SUMMER EXTREME RAINFALL 1379 Table VI. Design values (unit, mm) corresponding to various return periods (T=10, 20, 50, 100, 200, 500 years) and its 90% confidence intervals computed from the time series of AMP2 for the stations having more than 30 observations Site T=10 T=20 T=50 T=100 T=200 T= (259, 400) (297, 510) (328, 686) (339, 836) (347, 1031) (352, 1377) (269, 364) (316, 417) (353, 498) (368, 567) (375, 644) (382, 754) (252, 322) (293, 381) (342, 460) (377, 527) (411, 604) (444, 729) (248, 322) (291, 389) (350, 486) (392, 569) (430, 666) (473, 824) (237, 316) (293, 385) (374, 481) (433, 562) (490, 656) (552, 803) (151, 194) (166, 227) (181, 274) (188, 310) (194, 355) (200, 438) (240, 371) (275, 461) (310, 589) (324, 707) (336, 850) (347, 1119) (211, 332) (237, 407) (261, 508) (271, 593) (278, 697) (287, 924) (172, 261) (197, 322) (233, 412) (259, 502) (281, 614) (309, 828) (207, 324) (237, 381) (277, 457) (312, 523) (342, 614) (381, 810) (189, 223) (207, 241) (222, 262) (228, 278) (230, 295) (232, 317) (174, 232) (195, 284) (217, 375) (228, 455) (234, 544) (239, 716) (179, 255) (202, 301) (221, 363) (230, 421) (236, 494) (241, 617) (177, 209) (197, 239) (211, 282) (219, 319) (224, 360) (229, 420) (211, 262) (240, 314) (266, 388) (279, 448) (288, 511) (296, 614) (213, 285) (228, 358) (240, 484) (246, 618) (251, 796) (254, 1136) (196, 261) (221, 312) (259, 392) (286, 475) (306, 568) (324, 758) (254, 309) (291, 363) (326, 434) (343, 488) (355, 543) (364, 621) (226, 373) (276, 449) (345, 558) (395, 662) (431, 793) (471, 1019) (180, 229) (206, 287) (237, 389) (257, 490) (277, 626) (297, 878) (217, 291) (254, 343) (296, 417) (321, 480) (345, 553) (370, 677) (251, 306) (266, 345) (283, 400) (293, 459) (301, 538) (306, 709) (246, 348) (280, 403) (325, 475) (352, 538) (375, 609) (397, 734) (239, 293) (264, 313) (281, 340) (287, 362) (290, 386) (291, 419) (151, 231) (173, 269) (205, 323) (233, 374) (260, 443) (287, 574)

10 1380 J.-S. PARK ET AL. Figure 3. Histogram of B=2000 bootstrap estimates of the design value (unit, mm) corresponding to 10-year return period at site 159 (Pusan) for AMP1. The 90% confidence interval of the design value is (194.4, 234.8). The average and standard deviation of these bootstrap estimates are and 12.28, respectively. The solid line fitted to the histogram is the normal probability function Figure 4. Isopluvial map of the estimated design values (unit, mm) corresponding to 10-year return period for AMP1. Locations of stations are marked by small circles

11 KOREAN SUMMER EXTREME RAINFALL 1381 confidence intervals for predicted rainfall quantiles. Figure 3 is a histogram of B=2000 bootstrap estimates of the 10 years return value at site 159 (Pusan) for AMP1. This figure illustrates that the bootstrap estimates are normally distributed. Isopluvial maps of the estimated design values corresponding to selected return periods (10, 50, 100 years) for AMP1 and AMP2 are presented in Figures 4 6 and Figures 7 9, respectively. These maps are drawn using the values of 61 stations. The highest return values are centred at sites in the southwestern part of the Korean peninsula. This is due to record-breaking rainfall by the influence of Typhoon Agnes between 30 August and 4 September During this period, the stations (Jangheung, Haenam, Goheung) around Mokpo had almost half of the annual total precipitation in a single day; between 394 and 548 mm. The high return values of 50 and 100 years for AMP1 in the mid-western part of the Korean peninsula mainly result from the intensive rainfall from 31 July to 18 August Actually, the convergence zone of the monsoon front stayed over the middle part of China, Japan and the Korean peninsula (Lee et al., 1999), and this condition was maintained for 3 weeks due to a stationary planetary wave over Asia (Yun et al., 1999). Overall, the distribution of return values for AMP2 is more similar to the climatological features of annual total precipitation than that of AMP1. 5. SUMMARY AND DISCUSSION We have modelled Korean summer extreme rainfall using the Wakeby distribution and the method of L-ME. The design values corresponding to various return periods and its confidence intervals have been obtained for two time series of annual maximum of daily and 2-day precipitation, respectively. The isopluvial maps of the design values have been presented. Figure 5. Isopluvial map of the estimated design values (unit, mm) corresponding to 50-year return period for AMP1

12 1382 J.-S. PARK ET AL. Figure 6. Isopluvial map of the estimated design values (unit, mm) corresponding to 100-year return period for AMP1 Figure 7. Isopluvial map of the estimated design values (unit, mm) corresponding to 10-year return period for AMP2

13 KOREAN SUMMER EXTREME RAINFALL 1383 Figure 8. Isopluvial map of the estimated design values (unit, mm) corresponding to 50-year return period for AMP2 Figure 9. Isopluvial map of the estimated design values (unit, mm) corresponding to 100-year return period for AMP2

14 1384 J.-S. PARK ET AL. Parida (1999) used the four-parameter Kappa distribution (Hosking, 1994) with L-ME in modelling Indian summer monsoon rainfall. Even though the four-parameter Kappa distribution (K4D) is robust and potentially useful for modelling extreme rainfall, the parameter space is restricted to ensure the existence of the L-moments and the uniqueness of parameters. Therefore, the routine provided by Hosking (1997) sometimes fails to get the valid estimates for some datasets, because no unique solution exists inside of the restricted region. The precise reason of this failure is, according to Hosking (1997), that the L-skewness and L-kurtosis pair ( 3, 4 ) lies above the generalized-logistic line, which suggests that L-moments are not consistent with any K4D with h 1. In this case, neither L-ME of the parameters and quantile estimates are obtainable. Actually, in fitting K4D to time series of AMP1 of Korea, we have experienced this problem at 19 out of 61 stations. This is one of the reasons why WAD instead of K4D is used in this study. Park and Jeon (2000) developed a computer program for computing MLE of parameters of WAD. Unlike the L-ME routine, a convergence problem does not hinder these methods. Actually in fitting WAD with L-ME to time series of AMP1, we have experienced this problem at 19 out of 61 stations, even though parameters are still estimated by setting some parameters equal to zero. Thus MLE can sometimes be a better estimate than L-ME, probably for larger samples. A study comparing these estimates of WAD, especially in estimating design values corresponding to high return periods, is ongoing. ACKNOWLEDGEMENTS This research was performed for the Greenhouse Gas Research Center, one of the Critical Technology-21 Programs, funded by the Ministry of Science and Technology of Korea. The authors would like to thank JRM Hosking and IBM TJ Watson Research Center for sending us the research report. The authors would also like to thank the reviewer for providing constructive comments. REFERENCES Dunn PK Bootstrap confidence intervals for predicted rainfall quantiles. International Journal of Climatology 21: Greenwood JA, Landwehr JM, Matalas NC Probability weighted moments: definition and relation to parameters of several distributions expressible in inverse form. Water Resources Research 15: Hosking JRM The theory of probability weighted moments. Research Report RC12210, IBM TJ Watson Research Center, Yorktown Heights, NY. Hosking JRM L-moments: Analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society, Series B 52: Hosking JRM The four-parameter Kappa distribution. IBM Journal of Research and De elopment 38: Hosking JRM LMOMENTS: Fortran routines for L-moment computations and regional frequency analysis, Version 3.02 (March 1997). Available at the L-moments homepage: Houghton JC Birth of a parent: The Wakeby distribution for modeling flood flows. Water Resources Research 14: Huff FA, Angel JR Rainfall Frequency Atlas of the Midwest. Illinois State Water Survey, Bulletin 71: Champaign, IL. Landwehr JM, Matalas NC, Wallis JR Some comparisons of flood statistics in real and log space. Water Resources Research 14: Landwehr JM, Matalas NC, Wallis JR Estimation of parameters and quantiles of Wakeby distributions. Water Resources Research 15: Landwehr JM, Matalas NC, Wallis JR Quantile estimation with more or less floodlike distributions. Water Resources Research 16: Lee DK, Wi TK, Lee SW, Lee CH, Ha JC Large-scale characteristics of heavy rainfall events in Proceedings of Spring Conference of Korea Meteorological Society, April 1999, Seoul: Matalas NC, Slack JR, Wallis JR Regional skew in search of a parent. Water Resources Research 11: Matsumoto J Heavy rainfalls over East Asia. International Journal of Climatology 9: Parida BP Modelling of Indian summer monsoon rainfall using a four-parameter Kappa distribution. International Journal of Climatology 19: Park JS, Jeon JW Maximum likelihood estimation of Wakeby distribution. Technical Report, Dept of Statistics, Chonnam N University, Kwangju, Korea. Press WH, Teukolsky SA, Vetterling WT, Flannery BP Numerical Recipes in Fortran77: The Art of Scientific Computing (2nd edn). Cambridge University Press: New York. Ross SM A Course in Simulation. Macmillan Publishing Co: New York. Svensson C, Berndtsson R Characterization of extreme rainfall in an East Asian monsoon-climate catchment in the upper reaches of the Hui river, China. International Journal of Climatology 16: Wilks DS, McKay M Extreme-value statistics for snowpack water equivalent in the northeastern United States using the cooperative observer network. Journal of Applied Meteorology 35: Yun WT, Byun YH, Park CK Analysis of the characteristics of heavy rainfall in Proceedings of Spring Conference of Korea Meteorological Society, April 1999, Seoul: