daily rainfall; fourier series; gamma distributions; generalized linear model; deviance

Transcription

1 METEOROLOGICAL APPLICATIONS Meteorol. Appl. 16: (29) Published online 17 April 29 in Wiley InterScience ( A comparison of the rainfall patterns between stations on the East and the West coasts of Peninsular Malaysia using the smoothing model of rainfall amounts Jamaludin Suhaila a * and Abdul Aziz Jemain b a Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 8131, Skudai, Johor, Malaysia b School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 436, Bangi, Selangor, Malaysia ABSTRACT: The aim of this study is to model rainfall amounts on a daily basis by fitting a smooth curve to the mean rainfall per rainy day. The rainfall amounts are described as gamma distributionswith smoothingparameters. The smoothing technique used is the approach of the Fourier series. Six rain gauge stations with the daily rainfall series covering the same period ( ) are examined. Different rainfall patterns are observed among the stations, particularly between the east and the west. The stations in the western area have a bimodal pattern of rainfall and are best described with two harmonics, while four harmonics are required for the stations in the eastern area which exhibit a unimodal pattern of rainfall. The resulting curves with fitted smoothing parameters provide a good summary of statistics and useful information for describing the rainfall patterns and climate of the studied stations. Copyright 29 Royal Meteorological Society KEY WORDS daily rainfall; fourier series; gamma distributions; generalized linear model; deviance Received 17 July 28; Revised 15 December 28; Accepted 13 January Introduction Stochastic models of rainfall can be divided into two types. The first type provides a model of rainfall occurrence that simulates a sequence of wet and dry days, while the second type is a model of rainfall amount that simulates the rainfall amount on wet days only. Due to the seasonal variations in rainfall, the parameters of rainfall occurrence and rainfall amounts often change throughout the year. The usual method of handling this variation is to derive separate parameter sets for each month of the year. However, this will cause a large number of parameters to be estimated in the models. A better approach is to smooth the model parameters using the Fourier series as the periodic function, as this can best describe the rainfall pattern and its temporal variation concisely. After smoothing, the parameter sets are ready to be compared between the different gauges and between the periods of record (Jimoh and Webster, 1999). Many studies in the literature have applied the Fourier series as the smoothing function. Jimoh and Webster (1999) applied the Fourier series to smoothen the variations in the stochastic model parameters of daily rainfall in Nigeria, and Woolhiser and Pegram (1979) applied it to US data. The most recent study to compare and investigate the various methods of periodogram analysis was carried out by Kottegoda et al. (24) specifically to * Correspondence to: Jamaludin Suhaila, Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 8131, Skudai, Johor, Malaysia. suhaila@mel.fs.utm.my give some consideration to periodicity and persistence in daily rainfall. The Fourier series was one among several periodic function methods considered in this work. In modelling the rainfall amounts on wet days, gamma distributions have frequently been used in many studies (e.g. Ison et al., 1971; Katz, 1977; Buishand, 1978; Stern and Coe, 1984; Aksoy, 2; May, 24; Unal et al., 24). These parameters of the gamma distributions may change throughout the year. Several examples in the literature have found the Fourier series useful to describe the periodic seasonal fluctuations of parameters in the gamma model. For example, Ison et al. (1971) used the least square estimates of Fourier coefficients to describe the seasonal variability of gamma distribution parameters for rainfall amount for an i-day (i = 1, 2,...,I) wet period. Buishand (1977) fitted a Fourier series to the mean values of gamma distributions. Meanwhile Coe and Stern (1982) and Stern and Coe (1984) applied the Fourier series to the model parameters of gamma distributions for several stations in Africa and Sri Lanka and used this to describe seasonal variations. Important results derived from these models can then be used in agricultural planning (e.g. Stern, 198) and also provide summarizing statistics for the comparison of the rainfall climate between places (e.g. Garbutt et al., 1981). Peninsular Malaysia experiences rainfall that varies seasonally. This seasonal variation is mainly influenced by the southwest (between May and August) and the northeast (between November and February) monsoon winds. During the northeast monsoon, the exposed areas on the east coast of Peninsular Malaysia receive heavy Copyright 29 Royal Meteorological Society

2 392 J. SUHAILA AND A. A. JEMAIN rainfall. On the other hand, the areas that are sheltered by the mountain ranges, as shown in Figure 1, are more or less free from its influence. In addition, during the transition period between the monsoons, the inter-monsoon period (March April and September October), maximum rainfall is recorded, particularly for stations in the western areas of Peninsular Malaysia. Since the rainfall in Malaysia is seasonal, the probabilities of occurrence of rain and of rainfall amount do change throughout the year. The parameters both for the Markov Chain probability model of occurrence process and gamma distribution for the amounts of rain may change throughout the year. Rather than making separate analyses using polynomial fitting for each different period of the months, using the Fourier series provides a better application to smooth the model parameters as well as for taking into account the time variations (Coe and Stern, 1982). The intention of this study is to model time of year dependence throughout the year using the Fourier series. The approach taken by Coe and Stern (1982) and Stern and Coe (1984) will be adopted in this study. The second type of stochastic model, which focuses on modelling rainfall amount on wet days, will be considered. A generalized linear model (GLM) will be used to model those rainfall distributions that vary as a function of time of year. The present study will analyze the amounts of rainfall on a daily basis and fit a smooth curve to the mean rainfall per rainy day. In addition, the results from the model will also be used to compare the rainfall patterns among the stations in Peninsular Malaysia, particularly between the east and the west coasts. A comparison will be made based on the number of different harmonics required to describe the rainfall patterns of the rain gauge stations, the differences in the seasonal rainfall peaks between stations and the probability of rainfall amounts exceeding specific threshold values for each station on a particular day or in a particular month. 2. Data Six rain gauge stations in Peninsular Malaysia were selected for this study. The stations are classified into two main areas, the east and the west. In this study, the daily rainfall series from the period 1975 to 24 are studied. A wet day is defined as a day with a rainfall amount of at least 1 mm. A summary of the descriptive statistics for each rain gauge station, along with its latitude and longitude, is given in Table I. The locations of the stations together with their monthly rainfall patterns are shown in Figure 2. Referring to Table I the mean, intensity and coefficient of variation of annual rainfall are all found to be greater in the eastern stations than in the western ones. Each of these stations has different characteristics, particularly in their climatic patterns, which are mainly influenced by their geographical and topographical areas as well as the variations in the monsoon. As shown in Figure 2, the rainfall patterns vary, particularly between the stations in the east and the west coasts of Peninsular Malaysia. The figure depicts a unimodal pattern of rainfall for the stations on the east coast, and a bimodal pattern for the stations on the west coast. The wettest months for Bayan Lepas (west coast) are September, October and April whilst the driest months there are January, February and December. This pattern is similar in Gombak and Padang Katong whereas for the other three stations in the eastern area, the wettest month is December while May, June and July are the driest months. The likely cause of the form the rainfall patterns take is the monsoonal flow that contributes to the heavy rainfall on the east and west coasts at different times of the year. 3. Methodology This section is divided into three main sub-sections. The choice of gamma distributions used in the analysis will be discussed in the first sub-section followed by a discussion of the Fourier fitting in the second sub-section. The last sub-section will focus on the methods used for evaluating the deviance Gamma distribution rainfall on specified days of the year Figure 1. Physical map and selected rainfall stations in Peninsular Malaysia. The model for rainfall amounts used here only describes the distribution of rainfall on days when rain occurs.

3 A COMPARISON OF THE RAINFALL PATTERNS BETWEEN STATIONS OF PENINSULAR MALAYSIA COASTS 393 Figure 2. Map of Peninsular Malaysia with the monthly mean rainfall for each selected station. Table I. Geographical coordinates and a summary of statistics for annual rainfall. Stations Latitude Longitude Mean (mm) Annual rainfall coefficient of variation (%) Intensity (mm day 1 ) Rainy days mean West Gombak 3 16 N E Bayan Lepas 5 17 N 1 16 E Padang Katong 6 27 N 1 11 E East Endau 2 35 N 13 4 E Pekan 3 33 N E Kemaman 4 13 N E This rainfall distribution depends on the time of year and on what has occurred during the previous days. Various distributions have been considered in fitting the rainfall amounts. Other than the gamma distribution, the exponential distribution (Todorovic and Woolhiser, 1975), kappa distribution (Mielke, 1973), Weibull distribution (Sharda and Das, 25), log normal distribution (Kedem et al., 199) and mixed exponential distribution (Woolhiser and Roldan, 1982; Chapman, 1997, 1998; Wilks, 1999) are also used to model the rainfall amounts.

4 394 J. SUHAILA AND A. A. JEMAIN The earliest studies by Nelder and Wedderburn (1972) proposed a GLM to model the observations of those variables that were not from a normal distribution. This allowed for any distributions from the exponential family such as the gamma, Weibull and log normal distributions to be used for the amounts of daily rainfall. In statistics, the GLM is defined as a generalization of ordinary least square regression. It relates the mean of the observations to the explanatory variables through a function called the link function. In order to model the rainfall amounts in the present study, the chosen density function must be positively skewed and should have a variance that increases with the mean (McCullagh and Nelder, 1989). In other words, this study hopes to obtain a model that is able to fit the data with a constant coefficient of variation. Hence, the gamma and the lognormal models are left as the choices. Firth (1988) compared the efficacy of both models and found that the gamma model performed slightly better than the lognormal model (McCullagh and Nelder, 1989). Therefore, the gamma distributions have been chosen as the best-selected distribution for modelling the rainfall amounts on wet days in this present study. Suppose the model is fitted to the T days of the year with day t = t 1,t 2,...,t T and T = 366. The amount of rain on day t at year i is x i (t), i = 1, 2,...,n(t) where n(t) represents the number of years in which day t had rained. The observations follow a gamma distribution with a density function of: f(x)= (k/µ(t)) k x i (t) k 1 exp( kx i (t)/µ(t))/ Ɣ(k), x i (t) >,µ(t)> (1) where µ(t) is the mean rainfall on day t with a condition that day t is wet and 1/ k is the coefficient of variation of the distribution. The mean rain per rainy day may change throughout the year but the coefficient of variation 1/ k is assumed to remain constant for all values of t. Using the gamma density function given in Equation (1), the probability of exceeding the specified daily rainfall amount can be determined. Suppose one wished to find the probability of the amount on day t = 15 that rainfall exceeds 3 mm. Given that µ(t) = and the shape parameter k =.62, these values are substituted in the gamma density function as in Equation (1). The computed value is given as follows: P(X > 3) = 1 P(X 3) = 1 3 (.62/36.11).62 x.62 1 exp(.62x/36.11)/ Ɣ(.62)dx =.385 (2) Based on the given results, it can be said that nearly 39% of the amount that falls on day 15 over a specified period will contribute to more than 3 mm per rainy day Fourier fitting of seasonal variation A multiplicative model with gamma-distributed observations was used to model the rainfall amounts on wet days. The idea was to express ln(µ(t)) as a linear function involving the harmonic components which can be written as ln(µ(t)) = g(t). Sinceg(t) is linear when the parameters are unknown, then this model is a GLM. The independent variables are the functions of the time of year and the dependent variables are the parameters for the gamma distributions for rainfall amounts. There are many types of function g(t) that may be used for modelling the time dependence throughout the year. The Fourier series is often chosen as the smoothing function since it can fit both the unimodal and bimodal seasonal patterns easily (Garbutt et al., 1981; Jimoh and Webster, 1999). Moreover, the fitted curves are able to connect at the beginning and end of the year. The Fourier series is expressed as follows: g(t) = A o + m (A j sin(jt ) + B j cos(jt )) (3) j=1 where j is the harmonic, m is the maximum harmonic required for the series, A j and B j are the parameter coefficients which are often denoted as β j s parameters of the series and t = π(t 183)/183. The estimates for the Fourier series parameters in a GLM are maximum likelihood estimates, produced by iteratively weighted least square. The derivation of the coefficients of the Fourier series is given in Appendix A of this article. The GLM package in the S-plus computer program (Venables and Ripley, 1999) is used for the model fitting where function g(t) is fitted to µ(t) by treating them as the gamma variates with the weight n(t) representing the number of years where day t had rained Evaluating the deviances The performance of the Fourier series in describing seasonal behaviour is described by its deviance. Deviance is the statistical term used to describe or measure the discrepancy or correspondence of fit in the form of log likelihood ratio (see McCullagh and Nelder, 1989). In the present study, the deviances are classified into two components; namely the between-day deviance and the within-day deviance. The between-day deviance represents the contribution of different harmonics and a residual term. The formula is as follows: E B = 2 t n(t)[ln ˆµ(t) ln µ(t)] (4) where ˆµ(t) is the fitted value of µ(t). When one harmonic is fitted, three parameters are estimated: a constant, a sine coefficient and a cosine coefficient. Meanwhile, if two harmonics are fitted, then five parameters are estimated and so on. Each time a model is fitted, the deviance

5 A COMPARISON OF THE RAINFALL PATTERNS BETWEEN STATIONS OF PENINSULAR MALAYSIA COASTS 395 is calculated. If a model is correct then the deviance has a distribution that is approximately a multiple of a χ 2 distribution. The models with different numbers of harmonics are then compared by considering reductions in deviance. The difference in deviances for the models can be used as a guideline to determine the number of harmonics as well as the number of parameters required for the model. The values of deviance from the models are then used to construct the analysis of the deviance table, which is quite similar to the analysis of the variance table with normally distributed data. The mean deviances for each harmonic are calculated as deviance/degree of freedom. The ratios of two mean deviances have approximate F -distributions (Hogg and Craig, 25) where the denominator is referred to as the mean residual of the between-day deviance (residual of between-day deviance/degrees of freedom for residual term). The significance of the F -values of each harmonic is then compared to the critical values of F -distribution for the specified level of significance with the specific degrees of freedom. The maximum number of harmonics required for the model can be determined when there are no further harmonics that reduce the deviance significantly. On the other hand, the within-day deviance is given in the following expression: E W = 2 t n(t)[ln µ(t) ln x(t)] (5) where ln x(t) = n(t) / ln x i (t) n(t). The formula of withinday deviance can be used in the estimation of k values i=1 that are required to complete the gamma model. However, it can only be used if the problem of the inaccurate recording of small amounts of rain is ignored and the solution is given as the following: 2n ( ) ln ˆk Ɣ (ˆk) Ɣ(k ) = E W (6) where n = t 2 n(t)., but if there are inaccuracies in t 1 recording small amounts in the data set they will cause an overestimation of k. Therefore, it is suggested that the value of k be estimated based on between-day deviances because it is robust against these inaccuracies (Stern and Coe, 1984). The maximum likelihood estimate for the between-day estimate is as follows: 2n ( ) ln(n(t)ˆk) Ɣ (n(t)ˆk) Ɣ(n(t)k ) = E RB (7) where E RB is the residual between-day deviances. Given that n(t)k is large (>1), then Equation (7) may be approximated as follows: k = T/E RB (8) 4. Results and discussion This section of this paper is divided into four main sub-sections. In the first sub-section, the number of harmonics required for each station will be identified. The second sub-section consists of a discussion of the estimated k values. The third sub-section will identify the seasonal rainfall peaks, the dry as well as the wet periods and the dates of maximum and minimum rainfall. The lower and upper quartiles will be used as the criteria to separate the period of time between the lowest and the highest mean rainfalls. The fourth sub-section will discuss the estimated probability of amounts that exceed a certain threshold value on a particular day. In addition, this study could determine which particular day contributes to the highest percentage and the percentages of that contribution. The present study only considers the models that are fitted to the mean rainfall per rainy day as independent observations on the time of the year, irrespective of the state of the previous day Identifying the number of harmonics Table II shows the analysis of deviance produced by fitting the Fourier series to the data from Gombak station in the western area. There were 4729 rainy days in the record period. The ratios of mean deviance had approximate F -distributions and were based on the probability values (P -value) given in Table II, the results indicate that two harmonics are required to model the mean rainfall per rainy day for this station. No further harmonics are required in the model since they do not reduce the deviance significantly. The results have been plotted in Figure 3(a). Also shown in the figure are the observed values of the mean rainfall per rainy day and it seems that the observed values are fitted well by the two harmonics. In contrast to Table II, the results of the analysis of deviance for Kemaman station in Table III give four harmonics. The results from the table show that the deviances were still much larger for two and three harmonics indicating that the model still did not fit very well. When four harmonics were applied, the deviance Table II. Analysis of deviance for studying the effect of increasing the number of harmonics fitted to the mean rainfall per rainy day at Gombak. Source Degrees of freedom Deviance Mean deviance F P- value Between day One harmonics Two harmonics Three harmonics Four harmonics Five harmonics Residual Within days Total

6 396 J. SUHAILA AND A. A. JEMAIN Table III. Analysis of deviance for studying the effect of increasing the number of harmonics fitted to the mean rainfall per rainy day at Kemaman. Source Degrees of freedom Deviance Mean deviance F P-value Between day One harmonics Two harmonics Three harmonics Four harmonics Five harmonics Residual Within days Total was reduced significantly indicating that this term is still needed but no further harmonics are required. As a result, four harmonics are sufficient to model the mean rainfall per rainy day at this station. No obvious lack of fit was shown in Figure 3(f). The results of all the rain gauge stations with the fitted harmonics are shown in Figure 3. As shown in Figures 3(a) 3(f), two harmonics are sufficient to describe the seasonal variation in the model parameters for stations in the western area, while four harmonics are sufficient for stations in the eastern area. In this study, two harmonics are required for the western stations that display a bimodal rainfall pattern, while four harmonics are usually adequate for the eastern stations with a unimodal rainfall pattern Estimated k values In order to complete the gamma model, the estimated k values are required. Using Equation (8) in Section 3.3 with T = 366, the estimated k values for Gombak station can be computed as k = 366/ =.71. The results for the other stations are indicated in Figure 3. These values usually affect the proportion of extreme observations, as they will increase as k decreases. Similarly, the coefficient of variation will also increase as k decreases. In this study, the shape parameter of the gamma distribution, k, was found to be slightly smaller for stations in the eastern area than in the western area. This indicates that a slightly larger variation was observed in the mean rainfall per rainy day for stations in the eastern area than in the western area. The results of k values produced here are consistent with the summary statistics given in Table I. In Table I, the coefficient of variation for stations in the eastern area is found to be larger than for stations in the western area, which indicates that large variations can be observed in the annual rainfall for the eastern stations. Similar results are produced here using the estimated k values where the variability of the mean rainfall per rainy day for the eastern stations was slightly larger than for the western stations. In this situation, the proportion of extreme observations in the eastern area is expected to increase as k decreases. The differences in terms of the number of harmonics required for each region and their corresponding estimated k values may be due to the effects of monsoon changes and their geographical locations. During the northeast monsoon, the exposed areas in the eastern part of the Peninsula receive heavy rainfall. Meanwhile, the areas that are sheltered by the mountain range (Main Range) are more or less free from its influence. This explains the reason why more extreme amounts are found in the eastern area than in the western area Identifying the seasonal rainfall peaks Figures 3(a), 3(b) and 3(c) describe the fitted curves for the stations in the western area while Figures 3(d), 3(e) and 3(f) describe the stations in the eastern area. The fitted values for stations in the western area range from approximately 1 to 21 mm per day. The two highest peaks are found during the months of April to May and September to October. On the other hand, the stations in the eastern area record the lowest amount of 12 mm and the highest of 37 mm per day. The highest peak is recorded in December, which is during the period of the northeast monsoon. When comparing both regions, the heaviest rainfall is observed at the stations in the eastern area. None of the stations in the western area recorded rainfall of more than 3 mm per day. However, the mean rainfall of 3 mm per day was a common value recorded by stations in the eastern area particularly during the northeast monsoon season. Hence, to decide which threshold values might indicate an environmental disaster such as floods is quite difficult. Due to the different climatic conditions, threshold values that are considered extreme in one location may not be extreme at another location. Therefore, to make the comparison of rainfall patterns between both regions much easier, this study uses the quartile thresholds. The lower (25th percentile) and upper (75th percentile) quartiles of the fitted data for each station were computed and are shown as dash lines in Figure 3. In this study, the values that are smaller than the lower quartile could be considered as the lowest mean, and the period of the lowest mean could possibly be regarded as the dry period. Meanwhile, values, which are greater than the

7 A COMPARISON OF THE RAINFALL PATTERNS BETWEEN STATIONS OF PENINSULAR MALAYSIA COASTS 397 (a) (d) (b) (e) (c) (f) Figure 3. Observed and fitted mean rainfall for rainy days for each station and the coefficient of the Fourier series. (a) Gombak with two harmonics and estimated k =.71, A = 2.76,A 1 =.36,B 1 =.11,A 2 =.24,B 2 =.114; (b) Bayan Lepas with two harmonics and estimated k =.6, A = 2.741,A 1 =.11,B 1 =.197,A 2 =.146,B 2 =.6; (c) Padang Katong with two harmonics and estimated k =.83, A = 2.622,A 1 =.4,B 1 =.122,A 2 =.13,B 2 =.12; (d) Endau with four harmonics and estimated k =.55, A = 2.98,A 1 =.34,B 1 =.386,A 2 =.19,B 2 =.166, A 3 =.3,B 3 =.7,A 4 =.11,B 4 =.22; (e) Pekan with four harmonics and estimated k =.51, A = 2.838,A 1 =.13,B 1 =.32,A 2 =.64,B 2 =.16, A 3 =.91,B 3 =.41, A 4 =.178,B 4 =.16; (f) Kemaman with four harmonics and estimated k =.55, A = 2.797,A 1 =.11,B 1 =.228,A 2 =.18, B 2 =.135, A 3 =.178,B 3 =.22,A 4 =.163,B 4 =.7. upper quartile, are considered as the highest mean rainfall and this period could be noted as the wet period. Table IV gives all the details of the dry and wet periods in terms of the low and high mean rainfall per rainy day at each station, with the dates and values for maximum and minimum rainfall. This study uses these rainfall values in comparing the rainfall patterns between stations Seasonal rainfall peaks for western stations Gombak (Figure 3(a)) which is located in the western area is very well fitted with the two harmonics, which indicates that it has two peaks in the rainfall pattern. The wet periods in terms of the high mean rainfall for this station are from early April to early May and mid August to late October with the highest amount for each period recorded on 2 April and 27 September respectively over the duration of the 3 years. The study notes that these events occurred during the two inter-monsoon seasons (March April and September October), which are known to bring heavy rainfall, particularly to stations in the western area. On the other hand, the lowest mean rainfall was recorded from late November to late February. The study notices that these months are in the period of the northeast monsoon season and that Gombak station is among the stations in the western area, an area that is sheltered by the mountain ranges and thus free from the influence of this monsoon. Hence, this period can be considered as the dry period for the Gombak station. The patterns of rainfall for the other two stations in the western area are also bimodal. As shown in Figures 3(b) and 3(c), the period from late November to late February can also be considered as the driest period for Bayan Lepas and Padang Katong stations. The minimum rainfall at both these stations, which was approximately 11 mm per day, was recorded in early January. The Padang Katong station has the same wet period as the Gombak station. However, for the Bayan Lepas station, it is clearly shown in Figure 3(b) that the mean rainfall per rainy

8 398 J. SUHAILA AND A. A. JEMAIN Table IV. Periods of wet and dry with the dates and the maximum and minimum rainfall values. Stations Dry period Date at minimum Value at minimum (mm) Wet period Date at maximum Value at maximum (mm) West Gombak 27 November 25 February 9 January April 5 May 2 April August 26 October 27 September 17.5 Bayan Lepas 26 November 25 February 6 January July 24 October 13 September April 4 May 25 April 15 Padang Katong 25 November 25 February 9 January August 23 October 2 September 16 East 13 April 14 May Endau 26 June 24 August 26 July November 15 February 21 December April 12 May 28 June 12 August 2 July November 18 January 1 December 34 Pekan 2 October 19 October 28 February 17 March 12 April 22 May 15 July 22 July 6 March 21 March Kemaman 25 September 18 October 2 May October 13 January 7 December 35.7 day in the first peak is less than the values of the upper quartile. The amount goes up again in late July until it reaches the highest peak on 13 September with the maximum per day rainfall of approximately 21 mm observed. In terms of the mean rainfall, the wet period for the Bayan Lepas station is in between late July and late October. Thus, it can be said that the rainfall pattern for the Bayan Lepas station is strongly affected by the mixture of the southwest monsoon and the second intermonsoon seasons, while it is less impacted by the first inter-monsoon season Seasonal rainfall peaks for eastern stations A different rainfall pattern can be seen for stations in the eastern area of the Peninsula. Four harmonics are sufficient to describe the rainfall patterns at Endau, Pekan and Kemaman. In the previous section, the study noted that these stations display a unimodal rainfall pattern. As shown in Figures 3(d), 3(e) and 3(f), the mean rainfall per rainy day at these three stations changes rapidly throughout the year. The highest peak for Endau station is on 21 December with approximately 37 mm of rainfall per day. In addition, the wet period for this station is from mid-november to mid-february with an average rainfall of more than 3 mm recorded per day. The amount starts to drop after mid-february, when it is less than 2 mm, until it reaches the lowest amount at nearly 12 mm recorded on 26 July. The dry period for Endau station, as indicated in Table IV, is found to be from mid-april until mid-may and from late June to late August. These months, as noted previously, are during the southwest monsoon season. As shown in Figures 3(e) and 3(f), mean rainfalls of more than 2 mm per day were recorded in early November to mid-january. The maximum rainfall at Pekan station is nearly 34 mm while approximately 36 mm is the maximum rainfall received by Kemaman station across the period of 3 years. Both the amounts were recorded in early December. This amount starts to drop after mid-january, with less than 2 mm per day recorded. However, the amount begins to increase again in late February to mid-march at both of the stations and the cycle is repeated several times until it reaches the highest peak in early December. Referring to Table IV, the dry periods for these stations frequently occurs during the periods of the southwest monsoon and the two intermonsoon seasons, while the wet period occurs during the northeast monsoonal flow Comparing the probability of rainfall amounts that exceed the threshold value As mentioned in the previous sub-section, a threshold of 3 mm is a common value recorded at the stations in the eastern area. Referring to Figure 3(b), a mean rainfall of 3 mm is often observed during the month of December. Hence, by using the gamma density function described in Section 3.1, together with the estimated shape parameter, k, and the fitted mean rainfall values on a particular day, the estimated probability of amounts that exceed a certain threshold value on that day can be determined. Two findings that can be observed are firstly what the percentage is of the amount of rain that falls on a particular day that contributes to the threshold values, and secondly the particular day of the year that has the highest probability of exceeding the threshold values. This approach represents another way of making a comparison between the stations. The probability of the amount exceeding 3 mm was calculated for the three stations in the eastern area and the results are given in Table V. The probability of the amount exceeding 3 mm for the Endau station varies between 34 and 39%. A five-day duration, which is

9 A COMPARISON OF THE RAINFALL PATTERNS BETWEEN STATIONS OF PENINSULAR MALAYSIA COASTS 399 Table V. Probability of amounts that contribute to more than 3 mm in December. Date Endau Stations Pekan Kemaman Bold cells indicate the largest probability at a particular day at each station in the eastern area. from 19 December to 23 December, gives the highest probability with 39% of the amount that falls on these days contributing to more than 3 mm per day. The highest probability for Pekan station is observed during the second week of December. However, the percentage of the amount exceeding 3 mm starts to drop from mid-december to end December where only 3% of the amount on 31 December exceeds 3 mm per day. A similar pattern describes the rainfall data from Kemaman station. The highest probability of the amount exceeding 3 mm was computed in early December, with nearly 39%. The probability then begins to decrease until it reaches the lowest at 28% recorded at the end of December. These results indicate that the rainfall patterns between the three stations in the eastern area are different. The lowest probability is observed in early December at Endau station while the highest probability is found in mid-december. For the other two stations, the highest probability is recorded in early December and this probability starts to fall in mid-december until the lowest probability is achieved at the end of December. Even though these three stations are located in the same region and are affected by the same monsoon, other factors such as the topographical and geographical effects also play important roles in influencing the results. Besides comparing the days in the same month, a comparative illustration between the months at each station can also be made. For example, the probability of the amount exceeding 2 mm was computed on day 15 of each month at each station. The results are given in Table V1. The bold cells indicate the highest probability of the amount exceeding 2 mm at each station. The lowest probability of the amount exceeding 2 mm occurs during December, January and February at all stations in the western area. Meanwhile, April, September and October record the highest probability at Gombak and Padang Katong stations. However, for Bayan Lepas station, the greatest probability is recorded during the second inter-monsoon season (September October) with 34% of the amount that falls on the fifteenth day of September exceeding 2 mm per day. In the east region, approximately 5% of the rainfall amount collected on the fifteenth day of December exceeds 2 mm. The lowest percentage is observed in July, which is considered the driest period for stations in the eastern area. These results are consistent with the earlier findings in the previous section. 5. Conclusions The rainfall distribution at six rain gauge stations in Peninsular Malaysia is compared via the smoothing techniques of the Fourier series. The results produced by using the Fourier series can describe the seasonal variation of the model parameters as well as determining the number of harmonics required for each station. In addition, the seasonal rainfall peaks as well as the dates for the maximum and minimum rainfall and the probability of amounts falling that exceed certain threshold values are determined and then compared among the stations. The fitted curves provide a good summary of statistics and give useful information for describing the rainfall patterns of the studied stations. Two harmonics are required for the stations with a bimodal rainfall pattern. On the other hand, a unimodal rainfall pattern is best described with four harmonics. In terms of wet and dry periods, April, May, September and October can be considered the wettest months in the western area, while the driest month was January. In contrast, the maximum rainfall for stations in the eastern area is recorded in mid-december at Endau station and in early December at Kemaman and Pekan stations. In addition, May, June and July are the driest months at these three stations. Unfortunately, there are several limitations to this study. Firstly, the study did not consider the events of the previous days, which could have been dry or wet events, and how these results could be affected by that.

10 4 J. SUHAILA AND A. A. JEMAIN Table VI. Probability of amounts that contribute to more than 2 mm (e.g. on the fifteenth day of each month). Stations January February March April May June July August September October November December Gombak Bayan Lepas Padang Katong Endau Pekan Kemaman Bold cells indicate the largest probability at a particular month at each station. Secondly, the issue of the probability of rainy events has not yet been investigated. Hence, some results such as the pattern of the fitted probability curves, mean length of rainy spell and dry spell, as well as the proportion of rainy days exceeding certain thresholds could not be compared. However, the study is still in its preliminary stage and the findings from this study so far are sufficient to describe the rainfall pattern based on the mean rainfall per rainy day. For upcoming studies, several issues that have been listed above will be incorporated into future analysis. In conclusion, in order to give a good description and useful summary of the rainfall patterns in Peninsular Malaysia, both the amounts of rain and the probability of rain should be considered together. Appendix. The amount of rain on a wet day follows a gamma distribution. The total rainfall, x(t), on n(t) wet days also has a gamma distribution with a density function of: ( ) 1 k n(t)k f n(t) (x) = x(t) n(t)k 1 Ɣ(n(t)k) µ(t) ( exp kx(t) ) (A.1) µ(t) where x(t) = n(t) x i (t). i=1 The log likelihood is given as follows: series is derived as follows: l = β j t l β j = t x(t) µ 2 (t) n(t) µ 2 (t) µ(t) β j n(t) µ(t) µ(t) β j (x(t) µ(t)) µ(t) β j = (A.4) The above equation can be simplified in the matrix form, which is given by: l β = DT W(x(t) µ(t)) (A.5) [ ] µ(t) where D = β and W is a diagonal matrix of j T p weights given by ( ) n(t) W = diag µ 2 with T T entries (A.6) (t) The Fisher information for β is: [ ] 2 l I(β) = E β j β r = t n(t) µ 2 (t) µ(t) µ(t) β j β r = D T WD. (A.7) The maximum likelihood estimate of the coefficients of the Fourier series β j is expressed as follows: l = t {n(t)k ln k ln Ɣ(n(t)k)}+ t (n(t)k 1) ˆβ (l) = ˆβ (l 1) + I(ˆβ (l 1) ) 1 U( ˆβ (l 1) ) (A.8) ln x(t) t n(t)k ln(µ(t)) k t x(t) µ(t) (A.2) The Fourier series for the parameters of a gamma model may be expressed as: µ(t) = exp(g(t)) (A.3) The derivative of the log likelihood equation given in Equation A.2 with respect to parameter β j of the Fourier where l represents the number of iterations and U(β) is the score matrix given in Equation A.5 while I(β) as in Equation A.7. Since Equation A.8 is nonlinear in β, it is solved iteratively through the Newton Raphson procedure. References Aksoy H. 2. Use of gamma distribution in hydrological analysis. Turkish Journal of Engineering Environmental Sciences 24: Buishand TA Stochastic Modeling of Daily Rainfall Sequences. Veenman and Zonen: Wageingen.

11 A COMPARISON OF THE RAINFALL PATTERNS BETWEEN STATIONS OF PENINSULAR MALAYSIA COASTS 41 Buishand TA Some remarks on the use of daily rainfall models. Journal of Hydrology 36: Chapman TG Stochastic models for daily rainfall in the Western Pacific. Mathematics and Computers in Simulation 43: Chapman TG Stochastic modelling of daily rainfall: the impact of adjoining wet days on the distribution of rainfall amounts. Environmental Modelling & Software 13: Coe R, Stern RD Fitting models to daily rainfall data. Journal of Applied Meteorology 21: Firth D Multiplicative errors: lognormal or gamma? Journal of the Royal Statistical Society Series B 5: Garbutt DJ, Stern RD, Dennett MD, Elston J A comparison of the rainfall climate of eleven places in West Africa using a twopart model of daily rainfall. Archive for Meteorology Geophysic Bioclimatology Series B 29: Hogg RV, McKean JW, Craig AT. 25. Introduction to Mathematical Statistics, 6th edn. Pearson Prentice Hall: Upper Saddle River, NJ. Ison NT, Feyerherm AM, Dean BL Wet period precipitation and the gamma distribution. Journal of Applied Meteorology 1: Jimoh OD, Webster P Stochastic modeling of daily rainfall in Nigeria: intra-annual variation of model parameters. Journal of Hydrology 222: Katz RW Precipitation as chain-dependent process. Journal of Applied Meteorology 16: Kedem B, Chiu LS, Karni Z An analysis of the threshold method for measuring area-average rainfall. Journal of Applied Meteorology 29: 3 2. Kottegoda NT, Natale L, Raiteri E. 24. Some considerations of periodicity and persistence in daily rainfalls. Journal of Hydrology 296: May W. 24. Variability and extremes of daily rainfall during the Indian summer monsoon in the period Global and Planetary Change 44: McCullagh P, Nelder JA Generalized Linear Models. Chapman and Hall: London. Mielke PW Another family of distributions for describing and analyzing precipitation data. Journal of Applied Meteorology 12: Nelder JA, Wedderburn RWM Generalized linear models. Journal of the Royal Statistical Society Series A 135: Sharda VN, Das PK. 25. Modelling weekly rainfall data for crop planning in a sub-humid climate of India. Agricultural Water Management 76: Stern RD Analysis of daily rainfall at Samaru Nigeria using a simple two-part model. Archive for Meteorology Geophysic Bioclimatology Series B 28: Stern RD, Coe R A model fitting analysis of daily rainfall data. Journal of Royal Statistical Society Series A 147: Todorovic P, Woolhiser DA A Stochastic model of n-day precipitation. Journal of Applied Meteorology 14(1): Unal NE, Aksoy H, Akar T. 24. Annual and monthly rainfall data generation schemes. Stochastic Environmental Research and Risk Assessment 18(4): Venables WN, Ripley BD Modern Applied Statistics with S- PLUS, 3rd edn. Springer-Verlag, Inc: New York. Wilks DS Interannual variability and extreme-value characteristics of several stochastic daily precipitation models. Agricultural and Forest Meteorology 93: Woolhiser DA, Pegram GGS Maximum likelihood estimation of Fourier coefficients to describe seasonal variations of parameters in stochastic daily precipitation models. Journal of Applied Meteorology 18(1): Woolhiser DA, Roldan J Stochastic daily precipitation models 2. A comparison of distribution of amounts. Water Resources Research 18(5):