Probability distribution of annual, seasonal and monthly precipitation in Japan

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1 Hydrological Sciences Journal ISSN: (Print) (Online) Journal homepage: Probability distribution of annual, seasonal and monthly precipitation in Japan SHENG YUE & MICHIO HASHINO To cite this article: SHENG YUE & MICHIO HASHINO (2007) Probability distribution of annual, seasonal and monthly precipitation in Japan, Hydrological Sciences Journal, 52:5, , DOI: /hysj To link to this article: Published online: 18 Jan Submit your article to this journal Article views: 2840 View related articles Citing articles: 11 View citing articles Full Terms & Conditions of access and use can be found at

2 Hydrological Sciences Journal des Sciences Hydrologiques, 52(5) October Probability distribution of annual, seasonal and monthly precipitation in Japan SHENG YUE 1 & MICHIO HASHINO 2 1 Hazen and Sawyer, PC, 498 Seventh Ave, New York, NY 10018, USA 0syue@hazenandsawyer.com, or 1H1Hshengyue @hotmail.com 2 Chair of Hydrology and River Hydraulics, Dept. of Civil Engineering, The University of Tokushima, 2-1, Minami-josanjima, Tokushima , Japan michio@ce.tokushima-u.ac.jp Abstract The method of L-moment ratio diagrams and the average weighted distance (AWD) are used to determine the probability distribution type of annual, seasonal and monthly precipitation in Japan. For annual precipitation, the log-pearson type III (LP3) distribution provides the best fit to the observations with the generalized-extreme value (GEV), three-parameter lognormal (LN3) and Pearson type III (P3) distributions as potential alternatives. For seasonal precipitation, the P3 distribution shows the best fit to the observations of spring precipitation; the LP3 the best fit for summer and winter precipitation; and the LN3 the best fit for autumn precipitation with the LP3 as a potential alternative. For monthly precipitation, the P3 distribution fits the precipitation best for January, February, March, May, July, October and December; the LP3 for June; and the LN3 for April, August, September and November. The identified probability distribution types of annual, seasonal and monthly precipitation are basically consistent. Overall, the P3 and LP3 distributions are acceptable distribution types for representing statistics of precipitation in Japan with the LN3 distribution as a potential alternative. Key words Japan; precipitation; L-moment; probability distribution Distribution de probabilité des pluviométries annuelle, saisonnières et mensuelles au Japon Résumé La méthode des diagrammes de rapport des L-moments et la distance moyenne pondérée sont utilisées pour déterminer le type de distribution de probabilité des pluviométries annuelle, saisonnières et mensuelles au Japon. Pour la pluviométrie annuelle, la distribution log-pearson type III (LP3) fournit le meilleur ajustement aux observations, les distributions valeur extrême généralisée (GEV), lognormale à trois paramètres (LN3) et Pearson type III (P3) étant des alternatives possibles. Concernant les pluviométries saisonnières, le meilleur ajustement aux observations est obtenu avec la distribution P3 pour les précipitations de printemps; la distribution LP3 pour les précipitations d été et d hiver; et la distribution LN3 pour les précipitations d automne, avec la distribution LP3 comme alternative possible. Pour les pluviométries mensuelles, le meilleur ajustement est obtenu avec la distribution P3 pour Janvier, Février, Mars, Mai, Juillet, Octobre et Décembre; LP3 pour Juin; et LN3 pour Avril, Août, Septembre et Novembre. Les types de distribution de probabilité identifiés pour les pluviométries annuelle, saisonnières et mensuelles sont cohérents. Dans l ensemble, les distributions P3 et LP3 sont acceptables pour représenter les statistiques des précipitations au Japon, avec la distribution LN3 comme alternative potentielle. Mots clefs Japon; pluviométrie; L-moment; distribution de probabilité 1 INTRODUCTION In order to effectively plan, design and manage water resources engineering, such as urban water supplies, hydropower, irrigation systems, etc., the statistics of precipitation over longer durations, such as annual, seasonal and monthly data, are necessary. They are especially useful in ungauged river basins where streamflow data are unavailable. Regional monthly and seasonal precipitation forecasts are crucial for effective planning and planting of crop types in the forthcoming year, as different crops require different amounts of water. These forecasts are generally based on the statistics of these precipitation series (Brown et al., 1986). However, in comparison to extreme rainfall or storm frequency analyses (Stedinger et al., 1993), the literature on the probability distributions of annual, seasonal and monthly precipitation is relatively sparse. Markovic (1965) investigated the probability distribution of annual precipita- Open for discussion until 1 April 2008

3 864 Sheng Yue & Michio Hashino tion in the western USA and southwestern Canada using the chi-squared statistic to measure the goodness of fit of sample data to selected probability distribution. He concluded that annual precipitation can be best approximated by the 2-parameter lognormal (LN2) and gamma (GAM) distributions. The GAM distribution was frequently used to represent monthly and seasonal precipitation (Ropelewski et al., 1985; Wilks & Eggleston, 1992). Guttman et al. (1993) investigated probability distributions of precipitation for the duration of 1, 2, 3, 6, 12, 24, 36 and 60 months in the continental USA by the method of L-moments, and found that the P3 distribution is acceptable for most of these precipitation series. In Japan, to the authors knowledge, there has not been a detailed study yet regarding the statistical properties of annual, seasonal and monthly precipitation. This study makes an attempt to determine the probability distribution types of annual, seasonal and monthly precipitation across Japan by the method of L-moments. L-moments are rank-based estimates of mean, variance, skewness and kurtosis and are almost unbiased and less sensitive to the outliers in comparison to conventional product moments. The method of L-moments is superior to other approaches in the selection of a regional probability distribution of a variable of interest (Hosking, 1990; Vogel & Fennessey, 1993; Stedinger et al., 1993; Hosking & Wallis, 1997; Yue & Wang, 2004a,b; Yue & Pilon, 2005). The study of the probability distribution type of Japan s precipitation will provide useful information for Japanese hydrological engineers to select a suitable probability distribution of precipitation in their planning, design and management of water resources infrastructures. The remainder of the paper is structured as follows: Section 2 outlines the method of L-moment ratio diagrams; Section 3 applies the approach to identify the probability distribution types of annual, seasonal and monthly precipitation; and the final section summarizes the application results. For the sake of conciseness, the following abbreviations are used: normal NOR; gamma with two parameters GAM; Pearson type III P3; log-pearson type III LP3; 2- and 3-parameter lognormal LN2 and LN3, respectively; generalized extreme value GEV; Gumbel GUM; 2- and 3-parameter Weibull W2 and W3, respectively; generalized Pareto GPA; and generalized logistic GLO. 2 METHOD 2.1 L-moment ratio diagrams L-moments are linear combinations of order statistics that are less sensitive to outliers and virtually unbiased for small samples (Hosking, 1990; Vogel & Fennessey, 1993; Stedinger et al., 1993; Hosking & Wallis, 1993, 1997). The L-moments (λ i, i = 1, 2, 3, 4) of any probability distribution can be defined as: λ 1 = β 0 (1a) λ = (1b) 2 2β1 β0 λ = 6 β + β (1c) 3 β2 6 4 β3 30β λ = 20 β β (1d) 1 0

4 Probability distribution of annual, seasonal and monthly precipitation in Japan 865 where β r (r = 0, 1, 2, 3) is the rth probability-weighted moment (PWM), which is given by: r β = E{ X[ F( X )] } (2) r in which F(X) is the cumulative distribution function of a random variable X. In practice, the PWMs must be estimated from a finite sample data. The unbiased PWM estimators are given by the following formula: 1 n r 1 n 1 n r j = 1 n j ˆβ r = x( j) (3) r where n is the sample size and {x (j) } is the ordered vector of observations in descending order, i.e. x (1) x (2) x (n). In addtion to the above PWM approach for estimating L-moments, they can also be calculated directly from a sample by using the approach proposed by Wang (1996). The two approaches will result in the same L-moment estimation, as described by Wang (1996). L-moment ratios L-coefficient of variation (L-cv, τ 2 ), L-coefficient of skewness (L-skewness, τ 3 ) and L-coefficient of kurtosis (L-kurtosis, τ 4 ) are defined by: τ = (4a) 2 λ2 / λ1 τ = (4b) 3 λ3 / λ2 τ = (4c) 4 λ4 / λ2 Hosking & Wallis (1997) demonstrated that if the mean of a distribution exists, then all of its L-moments exist. The L-moments uniquely define a distribution, i.e. no two distributions have the same L-moments. L-moment ratio diagrams are the plots of L-cv (τ 2 ) vs L-skewness (τ 3 ) and L-kurtosis (τ 4 ) vs L-skewness (τ 3 ), which are useful as a guide in the selection of an appropriate distribution for describing a set of variables, since different distributions have a distinct relationship between L-moment ratios. The former is for 2-parameter distributions and the latter is for 3-parameter distributions. By comparing sample L-moment ratios to population values via the L-moment ratio diagrams, a best-fit probability distribution may be obtained. For the NOR and GUM distributions, the values of their (τ 3, τ 4 ) are (0, ) and (0.1699, ), respectively, while their τ 2 can attain various values. Hence, in a τ 3 τ 2 diagram, their theoretical τ 3 τ 2 curves are vertical lines; in a τ 3 τ 4 plot, they are a point determined by coordinates (τ 3, τ 4 ). For the theoretical τ 3 τ 2 plot of 2-parameter distributions: GAM, LN2 and W2, the polynomial approximations τ 2 = f(τ 3 ) developed by Vogel & Wilson (1996) are used here. For the τ 3 τ 4 plot of 3-parameter distributions: GLO, GEV, LN3, P3, W3, GPA and LP3, the polynomial approximations τ 4 = f(τ 3 ) are given by Hosking (1991) and Hosking & Wallis (1997). The L-skewness and L-kurtosis for the W3 distribution are equal to τ 3 and τ 4 of the GEV, respectively. The LP3 discribes a random variable whose logarithms are P3-distributed. Thus, by comparing the observed τ 3 τ 4 relationships of the logarithms of sample data to the theoretical τ 3 τ 4 plots of the P3 distribution, the suitability of the LP3 distribution to represent the sample data can be examined.

5 866 Sheng Yue & Michio Hashino 2.2 Goodness of fit An approach to assess goodness of fit of a probability distribution to sample data is to investigate the observed and theoretical relationships of L-moment ratios via the L-moment diagrams. When the theoretical curve of the L-moment ratio for a probability distribution (a) crosses the centre of the cloud formed from the sample L-moment ratios, i.e. the number of the sample estimates evenly scattered around the theoretical curve, and (b) goes along with the tendency of the cloud, the probability distribution is considered to be suitable for representing the sample data. However, the visual observation is somewhat subjective and it is unable to distinguish which distribution is appropriate when the sample is drawn from a mixture of parent probability distributions (Peel et al., 2001). In this study, we implement the average weighted distance (AWD) method of Kroll & Vogel (2002) to measure the differences between sample and theoretical L-moment ratios. The AWD is defined by: N ni di i= 1 AWD = (5a) N n i= 1 i o o τ 2[ τ 3 ( i)] τ 2 ( i) for a 2-parameter probability distribution d i = (5b) o o τ 4[ τ 3 ( i)] τ 4 ( i) for a 3-parameter probability distribution where N is the number of sites in analysis; n i is the record length at site i; τ o k (i) (k = 2, 3, 4) are the observed or sample L-cv, L-skewness and L-kurtosis, respectively; τ 2 [ τ o 3 ( i)] and τ 4 [ τ o 3 ( i)] are the theoretical L-cv and L-kurtosis values calculated from a probability distribution corresponding to a given sample L-skewness, respectively. A distribution with the smallest AWD value provides the best fit to sample data. Peel et al. (2001) pointed out that the sample average of the L-moment ratios may provide a good indication of an appropriate regional probability distribution if the sample is homogenous. In this study, the sample average or sample mean (SM) of the L-moment ratios is also plotted in the L-moment ratio diagram as a reference, to see which distribution s theoretical L-moment ratio curve the SM is close to. However, we do not intend to select the distribution type of sample data based on the SM value since the goodness of fit of the SM to theoretical L-moment ratio diagram cannot be determined. 3 IDENTIFICATION OF PROBABILITY DISTRIBUTIONS OF ANNUAL, SEASONAL AND MONTHLY PRECIPITATION For this study, there are 22 available meteorological observation stations with long-term monthly precipitation records (about 110-year) across Japan. The data before 1950 were published by the Japanese Central Meteorological Observatory (JCMO, 1954); the data after 1950 by Japan s Meteorological Agency (JMA, 1969, 1972, 1982, 1992, 1998). The station name, location, record length and mean annual precipitation are presented in Table 1. The spatial distribution of these sites across Japan is shown in Fig. 1. According

6 Probability distribution of annual, seasonal and monthly precipitation in Japan 867 Table 1 Information on observation stations. Station name Latitude ( ) Longitude ( ) Observation period Mean (mm) Missing data Sapporo Akita Niigata Kanazawa Sakai Ishinomaki Nagano Utsunomiya Kofu Tokyo Nagoya Hamamatsu May 1945 (Tokyo data used) Osaka Wakayama Hiroshima April 1989 (Osaka data used) Tokushima Kochi Fukuoka Kagoshima Naze Naha-Okinawa Ishigakijima , December 1946 Fig. 1 Spatial distribution of meteorological stations across Japan.

7 868 Sheng Yue & Michio Hashino to Geography and Climate ( a major feature of Japan s climate is the clear-cut temperature changes between the four seasons: spring (March May), summer (June August), autumn (September November) and winter (December February). The distribution types of annual, seasonal and monthly precipitation are investigated using the method of L-moment ratio diagrams. 3.1 Probability distribution of annual precipitation Figure 2 illustrates the observed and theoretical relationships of τ 3 τ 2 for the 2-parameter distributions: NOR, GUM, LN2, GAM and W2; Fig. 3 the τ 3 τ 4 relationships for the 3-parameter distributions: GLO, GEV, LN3, P3, W3 and GPA; Fig. 4 for LP3 distribution. Figure 2 indicates that none of the 2-parameter distributions can represent the observations well. Figures 3 and 4 show that the GEV, LN3, P3 and LP3 distributions all seem to be suitable for describing the observations. It is almost impossible to assess which one shows a best fit to the observations based solely on the L-moment ratio diagrams. In order to find the best-fit distribution to the observations, equation (5a) was used to compute the AWD values for the above mentioned distribution types, except for the NOR and GUM distributions since their sample values are evidently far away from their theoretical values. The calculated AWD values are listed in Table 2. The orders of the probability distributions ranked according to their AWD values are also presented in Table 2. The LP3 provides a best fit to the observations, whose AWD is marked by a shaded bold italic number. The GEV and LN3 distributions might also be suitable for representing annual precipitation since the AWD values of the GEV and LN3 distributions are close to that of the LP3. The AWD values for the GEV and LN3 distributions are highlighted by bold numbers. Fig. 2 L-moment ratio diagrams of L-cv vs L-skewness for annual precipitation.

8 Probability distribution of annual, seasonal and monthly precipitation in Japan 869 Fig. 3 L-moment ratio diagrams of L-kurtosis vs L-skewness for annual precipitation. Fig. 4 L-moment ratio diagrams of L-kurtosis vs L-skewness for the suitability of the LP3 distribution. For reference, sample average or mean (SM) of the L-moments ratios was also plotted in Figs 2 4. However, we may not draw a conclusion based solely on the SM value since it represents only one value and the goodness of fit of sample data against the theoretical L-moment ratio curve cannot be evaluated. For 2-parameter distributions, Fig. 2 indicates that the SM is close to the GAM distribution, which is a special form of the P3 distribution. For 3-parameter distributions, Figs 3 and 4 indicate that the

9 870 Sheng Yue & Michio Hashino Table 2 The AWD values of probability distributions and their ranks. Period AWD values LN2 GAM W2 GLO GEV W3 LN3 GPA P3 LP3 Year Spring Summer Autumn Winter Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Period Ranks of PDs from best fit to worst fit Year LP3 GEV LN3 P3 W3 GLO LN2 GAM GPA W2 Spring P3 W3 LN3 GEV LP3 GLO LN2 GAM GPA W2 Summer LP3 GEV LN3 P3 W3 GLO GAM LN2 W2 GPA Autumn LN3 LP3 GEV P3 W3 GLO GAM LN2 W2 GPA Winter LP3 P3 W3 GEV LN3 GAM GPA GLO LN2 W2 Jan. P3 LN3 LP3 GEV W3 GPA GLO GAM W2 LN2 Feb. P3 LP3 LN3 GEV W3 GAM GLO GPA LN2 W2 Mar. P3 GEV LN3 W3 LP3 W2 GLO GAM GPA LN2 Apr. LN3 GEV W3 P3 LP3 GLO W2 GAM LN2 GPA May P3 GEV LN3 LP3 W3 GLO GAM LN2 GPA W2 Jun. LP3 P3 LN3 GEV W3 GAM GLO GPA W2 LN2 Jul. P3 LN3 GEV W3 W2 LP3 GAM GLO GPA LN2 Aug. LN3 GEV P3 W2 LP3 W3 GAM GPA GLO LN2 Sep. LN3 GEV P3 LP3 W3 GLO W2 GAM GPA LN2 Oct. P3 LN3 GEV LP3 W3 GAM W2 GLO GPA LN2 Nov. LN3 P3 GEV LP3 W3 W2 GLO GAM GPA LN2 Dec. P3 LN3 GEV LP3 W3 W2 GLO GAM GPA LN2 Note: shaded bold italic: best-fit distribution; bold regular: alternative distribution. SM is close to LN3, P3 and LP3, which is basically consistent with the conclusion drawn based on the AWD. These visual inspection results are also given in Table Probability distributions of seasonal precipitation For seasonal precipitation, Figs 5 7, respectively, compare the theoretical and observed τ 3 τ 2 relationships of the 2-parameter distributions, τ 3 τ 4 relationships of the 3-parameter distributions and the LP3 distribution. Figure 5 illustrates that none of the 2-parameter distributions can approximate the observations well. From Figs 6 and 7, it

10 Probability distribution of annual, seasonal and monthly precipitation in Japan 871 Table 3 Distribution type to which the SM is close. Period Probability distribution 2-parameter 3-parameter Year GAM LN3, P3, LP3 Spring GAM LN3, P3, LP3 Summer GAM P3, LP3, GEV Autumn GAM LN3, LP3, P3 Winter GAM W3, LP3 January GUM, GAM P3 February GAM LN3 March GAM P3, LN3, GEV April GAM LN3, GEV May GAM, GUM LN3, P3, GEV June GUM, GAM P3. LP3 July GUM, GAM P3 August GAM P3, LN3 September GAM, GUM LN3, GEV October GAM, GUM P3 November GUM P3 December GUM LN3, GEV, P3 Fig. 5 L-moment ratio diagrams of L-cv vs L-skewness for seasonal precipitation. seems that the GEV, LN3, P3 and LP3 distributions all are possible candidates for representing spring, summer and autumn precipitation; and W3 and LP3 are possible candidates for winter precipitation. It is impossible to discern which distribution among these distribution types is the best one to represent the observations from the L-moment ratio diagrams alone. The AWD values and the ranks of the distributions to fit the observations according to the AWD values are presented in Table 2. Table 2

11 872 Sheng Yue & Michio Hashino Fig. 6 L-moment ratio diagrams of L-kurtosis vs L-skewness for seasonal precipitation. L-Skewness L-Skewness Fig. 7 L-moment ratio diagrams of L-kurtosis vs L-skewness for the LP3 distribution. indicates that the P3 distribution is the best one for representing spring precipitation. The LP3 distribution is the best one for summer precipitation with the GEV as a possible alternative because there is no great difference in the AWD value between the LP3 and GEV distributions. The LN3 distribution is the best for autumn precipitation

12 Probability distribution of annual, seasonal and monthly precipitation in Japan 873 with the LP3 as a potential alternative. The LP3 distribution is the best for winter precipitation. These results are basically consistent with the identified distribution for representing annual precipitation, which is expected since annual precipitation is the summation of precipitation over four seasons, and the distribution type of a summation of several variables that follow a same probability distribution would not change. The SM value for different seasons is plotted in Figs 5 7. Visual inspection results for the distributions to which the SM is close are summarized in Table 3. For 2-parameter distributions, the SM is close to the GAM distribution for all the four seasons. For 3-parameter distributions, the SM is close to LN3, P3 and LP3 for spring; P3, LP3 and GEV for summer; LN3, LP3 and P3 for autumn; and W3 and LP3 for winter. The LP3 is the only distribution among the best ones for all seasons, and the P3 is among the best for three out of four seasons. These results are basically consistent with the distribution type selected based on the AWD. 3.3 Probability distributions of monthly precipitation Figures 8 10, respectively, compare the theoretical and observed τ 3 τ 2 relationships of the 2-parameter distributions, τ 3 τ 4 relationships of the 3-parameter distributions and the LP3 distribution of monthly precipitation. Figure 8 indicates that none of the 2-parameter distributions may be suitable for modelling monthly precipitation series. From Figs 9 and 10, it is hard to judge which distribution is the best for representing monthly precipitation series. Table 2 presents the AWD values and the ranks of the distributions to fit the observations according to the AWD values for the 12 months. It indicates that the P3 distribution is the best one for describing precipitation statistics of January, February, March, May, July, October and December. The LP3 distribution is best for June. The LN3 distribution is best for April, August, September and November. The GEV distribution is a potential alternative for April, May, August and September precipitation. The LN3 can be a potential alternative for July and the P3 can be a potential alternative for November. These alternatives are marked in bold format. The AWD value provides an objective methodology for selection of a distribution type over a region. The probability distribution types of monthly precipitation are basically consistent with the identified distribution types for annual and seasonal precipitation. The SM value is also plotted in Figs Visual inspection results for the distributions to which the SM is close are presented in Table 3. For 2-parameter distributions, the SM is close to the GAM distribution for February, March, April, May, August, September and October; and to the GUM distribution for January, June, July, November and December. For 3-parameter distributions, the SM is close to P3 for January, March, June, July, August, October and November; and to LN3 for February, April, May, September and December. These results are also consistent with the distributions selected based on the AWD. 4 CONCLUSIONS This study employed the method of L-moments and the average weighted distance (AWD) criterion to identify the regional probability distributions of annual, seasonal and monthly precipitation across Japan. The LP3 distribution provides the best fit to

13 874 Sheng Yue & Michio Hashino Fig. 8 L-moment ratio diagrams of L-cv vs L-skewness for monthly precipitation. the observations of annual precipitation with the GEV and LN3 distributions as alternative candidates. For seasonal precipitation, the P3 distribution is the best one for spring precipitation; the LP3 for summer with the GEV distribution as a potential alternative; the LN3 for autumn with the LP3 as a potential alternative; and the LP3 for winter. For monthly precipitation, the P3 distribution provides the best fit to the observations for January, February, March, May, July, October and December; the LP3 for June; and the LN3 for April, August, September and November. The identified distribution types of annual, seasonal and monthly precipitation are basically consistent, which is to be expected as seasonal and annual precipitations are the

14 Probability distribution of annual, seasonal and monthly precipitation in Japan 875 Fig. 9 L-moment ratio diagrams of L-kurtosis vs L-skewness for monthly precipitation. summations of monthly precipitation. The results based on the sample mean (SM) of the L-moment ratios are consistent with those by the AWD values. Overall, the P3 and LP3 distributions are acceptable distribution types to represent the statistics of precipitation for monthly to annual durations, with the LN3 and GEV distributions as potential alternatives. The distribution types of Japan s annual, seasonal and monthly precipitation identified in the present study will be useful for Japanese hydrological engineers to choose the distribution type of precipitation in their future water resources engineering practices.

15 876 Sheng Yue & Michio Hashino Fig. 10 L-moment ratio diagrams of L-kurtosis vs L-skewness for LP3. Acknowledgement The authors would like to express their thanks to Dr Efi Foufoula- Georgiou and the anonymous reviewer for their constructive comments that helped improve the quality of the paper. REFERENCES Brown, B. G., Katz, R. W. & Murphy, A. H. (1986) On the economic value of seasonal-precipitation forecasts: the fallowing/planting problems. Bull. Am. Met. Soc. 67, Guttman, N. B., Hosking, J. R. M. & Wallis, J. R. (1993) Regional precipitation quantile values for the continental United States computed from L-moments. J. Climate 6,

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