On the potential of Bartlett Lewis rectangular pulse models for simulating rainfall in Peninsular

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1 Hydrological Sciences Journal ISSN: (Print) (Online) Journal homepage: On the potential of Bartlett Lewis rectangular pulse models for simulating rainfall in Peninsular Malaysia Ibrahim Suliman Hanaish, Kamarulzaman Ibrahim & Abdul Aziz Jemain To cite this article: Ibrahim Suliman Hanaish, Kamarulzaman Ibrahim & Abdul Aziz Jemain (2013) On the potential of Bartlett Lewis rectangular pulse models for simulating rainfall in Peninsular Malaysia, Hydrological Sciences Journal, 58:8, , DOI: / To link to this article: Published online: 21 Oct Submit your article to this journal Article views: 253 View related articles Full Terms & Conditions of access and use can be found at Download by: [ ] Date: 21 November 2017, At: 21:22

2 1690 Hydrological Sciences Journal Journal des Sciences Hydrologiques, 58 (8) On the potential of Bartlett Lewis rectangular pulse models for simulating rainfall in Peninsular Malaysia Ibrahim Suliman Hanaish 1,2, Kamarulzaman Ibrahim 2,3 and Abdul Aziz Jemain 2 1 Department of Statistics, Faculty of Science, Misurata University, Misurata, Libya henaish@yahoo.com 2 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,43600 UKM Bangi, Selangor, Malaysia 3 Solar Energy Research Institute, Universiti Kebangsaan Malaysia,43600 UKM Bangi, Selangor, Malaysia Received 4 August 2011; accepted 18 February 2013; open for discussion until 1 May 2014 Editor D. Koutsoyiannis; Associate editor C. Onof Citation Hanaish, I.S., Ibrahim, K., and Jemain, A.A., On the potential of Bartlett Lewis rectangular pulse models for simulating rainfall in Peninsular Malaysia. Hydrological Sciences Journal, 58 (8), Abstract The applicability of two versions of the Bartlett Lewis rectangular pulse model, the original and the modified model, is discussed for describing the temporal and spatial variation of rainfall patterns observed at 15 raingauge stations in Peninsular Malaysia over the period ; 17 different sets of moment combinations are fitted to these models based on the generalized method of moments approach. The common statistics included in all sets are the mean, variance, lag-1 autocorrelation and the probability of dry based on the hourly rainfall data. The analysis was carried out on hourly rainfall data from all 15 stations for all months of the year. Two stations, Petaling Jaya and Kemaman, located on the west and east coasts of the Peninsula, respectively, are considered for illustration of the results, taking the months of July and November, which correspond to the driest and wettest months, corresponding to the southwest monsoon (May August) and northeast monsoon (November February), respectively. The best moment combination found for the illustrative results is based on the common statistics, as well as the mean and variance based on 24-h aggregated rainfall data, the inclusion of which successfully improved the model performance; the errors were significantly reduced. It was also found that the performance of the fitted models based on the mean absolute deviate error varies according to the type of Bartlett Lewis model applied: errors are much smaller for the fitted model based on the modified model as compared to the original model. In addition, the fitted statistics: mean, lag-1 autocorrelation and probability of dry are quite well fitted for several aggregated time scales; however, the variances are underestimated in both models for all aggregated time scales, particularly in the case of the original model. The results of extreme value analysis indicate that the modified model failed to reproduce the annual hourly and daily rainfall extremes satisfactorily. Key words Bartlett Lewis rectangular pulse rainfall models; spatio-temporal rainfall pattern Sur le potentiel des modèles à modèles impulsions rectangulaires de Bartlett Lewis à simuler les précipitations de Malaisie péninsulaire Résumé On discute de l applicabilité de deux versions du modèle à impulsions rectangulaires de Bartlett Lewis, l original et le modèle modifié, pour décrire la variabilité temporelle et spatiale de la pluviométrie observée en 15 postes pluviométriques de la Malaisie péninsulaire au cours de la période ; 17 ensembles différents de combinaisons de moments ont été calés sur ces modèles en s appuyant sur la méthode des moments généralisée. Les statistiques communes à tous les jeux sont la moyenne, la variance, l autocorrélation d ordre 1 et la probabilité de temps sec sur la base des données pluviométriques horaires. L analyse a été réalisée sur les données pluviométriques horaires des 15 stations pour tous les mois de l année. Deux stations, Petaling Jaya et Kemaman, respectivement situées sur les côtes Est et Ouest de la péninsule, illustrent les résultats, en prenant les mois de juillet et novembre, qui correspondent respectivement aux mois les plus secs (Mousson du Sud-Ouest de mai à août) et les plus humides (Mousson du Nord-Est de novembre à février. La meilleure combinaison de moments est basée sur les statistiques communes, ainsi que sur les moyennes et les variances calculées sur les précipitations cumulées sur 24 heures, dont l inclusion améliore les performances du modèle, puisque les erreurs sont 2013 IAHS Press

3 Bartlett Lewis rectangular pulse models 1691 considérablement réduites. Il a également été constaté que les performances des modèles ajustés sur la base de la moyenne de la valeur absolue des erreurs varient selon le type de modèle de Bartlett Lewis appliqué. Les erreurs sont beaucoup plus faibles pour l ajustement au modèle modifié que pour l ajustement au modèle d origine. En outre, les statistiques ajustées telles que la moyenne, l autocorrélation d ordre 1 et la probabilité de temps sec rendent assez bien compte de plusieurs échelles de temps agrégées, mais les écarts sont sous-estimés dans les deux modèles pour toutes les échelles temporelles agrégées, ceci particulièrement dans le cas du modèle original. Les résultats de l analyse des valeurs extrêmes indiquent que le modèle modifié ne réussit pas à reproduire de manière satisfaisante les précipitations extrêmes horaires, journalières ou annuelles. Mots clefs modèles de précipitations à impulsions rectangulaires de Bartlett Lewis; profil spatio-temporel de précipitation 1 INTRODUCTION Stochastic models have been used to generate rainfall across a range of different time scales for hydrological applications such as reservoir design, flood studies and design of sewerage systems. The aim of stochastic models is to provide a simplified conceptual representation of the main observable features of rainfall processes in time and rainfall fields in space. The advantage of these models is the ability to describe the rainfall process well and based on a small number of parameters from which other properties of the natural process can be deduced. The limitation of a Poisson process model for representing a storm as a single rainfall cell has led to the use of a cluster-based model. This cluster-based model involves a more complex representation of point precipitation in which each storm is assumed to be an independent cluster of rectangular pulses of rain cells. Many authors, such as Cowpertwait (1994, 2006), Velghe et al. (1994), Khaliq and Cunnane (1996), Gyasi-Agyei (1999), Smithers et al. (2002), Onof et al. (2000, 2005), Marani and Zanetti (2007) and Engida and Esteves (2011) have found that the Bartlett Lewis rectangular pulse models, which are the most recognized cluster-based models introduced by Rodriguez-Iturbe et al. (1987b), can successfully describe rainfall processes for a wide range of temporal scales from one hour upwards for countries such as the UK, New Zealand, Ireland, South Africa, Italy and the USA. Owing to the complexity of the precipitation process, a stochastic approach is likely to be preferable to a purely physical model for the generation of synthetic rainfall series (Smith 1981). These studies indicate that work still needs to be done to understand the performance of rectangular Poisson pulse models, especially for areas that have different types of climate. In particular, not much work has been done on the application of the Bartlett Lewis model for rainfall occurrence in Peninsular Malaysia. Although the Bartlett Lewis model is suitable for use to describe rainfall amounts in short time scales, several studies have pointed out limitations of the model and suggested some improvements. Onof and Wheater (1994), for example, introduced a two-parameter gamma distribution in place of the original Bartlett Lewis model, which considers a single parameter exponential distribution to describe the depth of a cell, in order to better capture extreme events. However, the problem of underestimation of extreme values still persists, particularly for lower aggregation levels, as described by Verhoest et al. (1997). Furthermore, Vandenberghe et al. (2010) found that the models demonstrated a too severe clustering of rain events. In this study, 17 different sets of moment combinations are considered to check the effectiveness for parameter estimation of both Bartlett Lewis (BL) rectangular pulse model versions, original (OBL) and modified (MBL), as well as to study the performance of the models in estimating the observed hourly statistics. Both models are fitted to hourly rainfall data for each month of the year from 15 raingauge stations across Peninsular Malaysia for the period An objective function value is computed for each moment combination in order to decide the best set of statistics. The best set of statistics selected for each type of Bartlett Lewis model is decided based on the minimum value of the objective function. In addition, the goodness of fit of the fitted models for the two stations is compared based on the mean absolute deviate error found for the best set which is selected. For discussion and illustration of the results found, analysis based on two raingauge stations, i.e. Petaling Jaya and Kemaman stations located on the east and west coasts of the Peninsula, respectively, for the months of July and November, are considered. The climatic conditions of the east and west coasts are influenced by the southwest and northeast monsoons, respectively. The choice of these two months is made because July and November represent the driest and wettest months, respectively, in the Peninsula.

4 1692 Ibrahim Suliman Hanaish et al. 2 RAINFALL MODELS 2.1 The original BL (OBL) model The original BL model for point rainfall was developed by Rodriguez-Iturbe et al. (1987a). In this model, storms arrive randomly following a Poisson distribution with parameter λ (h -1 ). Within each storm, cells arrive randomly following a Poisson distribution with rate β (h -1 ). The rectangular cells that are generated within a storm (cluster) stop after a given time. This storm duration is assumed exponentially distributed with parameter γ (h -1 ). Each cell arrival is associated with a rectangular pulse which has an exponentially distributed duration with parameter η (h -1 ) and an exponentially distributed depth of mean 1/μ x. The number of cells per storm follows a geometric distribution with mean μ c = 1 + β/γ. Thus, the OBL is a model which can be characterized by the set of parameters (λ, μ x, η, β, γ ), as given by Rodriguez-Iturbe et al. (1987a). Although this model has proved efficient for explaining the rainfall characteristics at all time intervals considered (1 h to 24 h) as explained by several authors (e.g. Rodriguez-Iturbe et al. 1988, Onof 1992), a major deficiency is its inability to reproduce the proportion of dry periods correctly. To overcome this problem, Rodriguez-Iturbe et al. (1988) proposedanewversion of the model, which they called the Modified Bartlett Lewis (MBL) model. 2.2 The modified BL (MBL) model In this model, storms are assumed to arrive following a Poisson process with rate λ (h -1 ). Within each storm, rectangular cells arrive following another Poisson process with rate β (h -1 ). The duration of the storm is distributed according to an exponential distribution with parameter γ (h -1 ). The depth of a cell follows an exponential distribution with parameter 1/μ x and the cell duration η (h -1 ) is distributed according to the gamma distribution with shape parameter α and scale parameter ν. Bothβ and γ are scaled proportionally with respect to the cell duration through the dimensionless parameters κ = β/η and φ = γ/η. The MBL is thus defined by six parameters (λ, μ x, α, ν, κ, φ) as shown schematically in Fig. 1 (adapted from Onof 2006). The analytical expressions of the Bartlett Lewis model, in its original or modified configuration, may be found in the appropriate references such as Rodriguez-Iturbe et al. (1987a) and Onof (1992). These equations relate the statistical properties of the rainfall process in discrete time in the entire Fig. 1 Schematic diagram of the Bartlett Lewis rectangular pulse model. time domain to the model parameters, which are estimated based on the generalized method of moment approach. 3 STUDY AREA AND INPUT DATA Peninsular Malaysia lies entirely in the equatorial zone, between 1 Nand6 N latitude and 100 E and 103 E longitude. It experiences rainfall that varies seasonally with respect to the occurrence of the monsoon winds. This seasonal variation is mainly influenced by the southwest monsoon which occurs between May and August, and the northeast monsoon which blows from November to February. During the northeast monsoon, many areas on the east coast of the Peninsula are expected to receive heavy rainfall. However, areas on the west coast that are sheltered by the mountain ranges which form the backbone of the Peninsula are more or less free from the influence of the northeast monsoon. Also, the island of Sumatra provides shelter for the Peninsula against the southwest monsoons. The transition period between the monsoons, i.e. the inter-monsoon period, occurs in the months March April and September October. In this study, hourly rainfall data are obtained from 15 raingauge stations which were selected based on their geographical location (Fig. 2). The hourly data for the period 1971 to 2008 from the stations can be considered as having good quality since more than 98% of the data are available. Data used in this study were collected from the database of the Malaysian Meteorological Service (MMS). 4 MODEL FITTING As proposed by many authors (e.g. Rodriguez-Iturbe et al. 1987a, Segond et al. 2006), it is impossible

5 Bartlett Lewis rectangular pulse models 1693 South China Sea Kemaman Strait of Malacca Petaling Jaya Fig. 2 Map showing the location of the 15 rainfall stations considered in this study, including Petaling Jaya and Kemaman stations. to use standard techniques for estimating the model parameters on the Bartlett Lewis model, such as the maximum likelihood method, because the complex dependencies in the model prohibit the formulation of a likelihood function. Therefore, the generalized method of moments (GMM) is used for fitting both models. Assume that y is a random variable which denotes the rainfall amount for a particular hour. Let θ = (θ 1, θ 2...θ p ) be the parameter vector for the model, T(y) = (T 1 (y), (T 2 (y)...(t k (y)) be a vector of summary statistics computed from the data and τ(θ) = (τ 1 (θ), τ 2 (θ)...τ ik (θ)) denote a vector of the estimated value of T under the model, calculated based on the analytical expressions. The idea behind the generalized method of moments (GMM) is to choose θ to minimize an objective function given by: S(θ) = k i=1 [ ( w i 1 T ) 2 ( i(y) + 1 τ ) ] 2 i(θ) τ i (θ) T i (y) (1) where (w i ; i = 1,2,..., k) is a collection of positive weights which allow more important weight to be given to fit some sample moments relative to others. Weights are given by w i = 1/var(T i (y)) where var(t i (y)) represents the i-th diagonal elements of the covariance matrix of the summary statistics. Minimization of the objective function S(θ) is done using the Nelder-Mead optimization algorithm. The parameters are estimated using the software of Chandler and Lourmas (2010). 4.1 Selection of moments The set of moments chosen to determine the model parameters should have relatively small sampling errors and are not highly mutually correlated. Several alternative sets of rainfall statistics are used to assess the model performance at the hourly time scales of 1 h, 6 h, 12 h and 24 h. The combination of statistics to be included in the objective function is an open-ended problem. The sample moments usually

6 1694 Ibrahim Suliman Hanaish et al. include features of the rainfall depth and the proportion of dry periods, as described by Onof and Wheater (1993). These statistics can be calculated based on the observed series and used for estimation of the parameters present in θ. The statistics include the mean, variance and lag-1 autocorrelation of rainfall intensities at the time scale of 1 h and multiples thereof. To reproduce the temporal intermittency of rainfall, the proportions of dry periods at the same scales are also introduced. Many authors have applied the Bartlett Lewis model to different sets of aggregation levels for estimating the model parameters. For example, Smithers and Schultze (2000) have studied 13 different sets of moment combinations for estimating the parameters in the Bartlett Lewis model, while Yusof et al. (2007) used one set of moments as suggested by Rodriguez-Iturbe et al. (1987a) toestimatetheparameters for the Neyman-Scott model. However, in this study, the goodness of fit of both models is assessed based on 17 different sets of moment combinations which are summarized in Table 1. The moment combinations include mean (Mean(h)), variance (Var(h)), lag-1 autocorrelation (AC(h)) and probability of dry (Pdry(h)) for different levels of aggregated time scales h = 1, 6, 12, 24, 48 h. The nature of rainfall in Malaysia leads us to consider in our analysis at both lower time scales, i.e. h = 1, 6, 12, and higher time scales, i.e. h = 24, 48. To some extent, the aggregated lower and higher time scales allow for the short and long durations of rainfall experienced in the region. During the monsoon periods, particularly on the east coast of the Peninsula, rainfall often has long duration, sometimes up to 5 days. The occurrence of heavy rainfall of short duration is often observed in the west during the southwest monsoons. It seems that the stations in the east exhibit different rainfall pattern when compared to the stations in the west due to the influence of the northeast and southwest monsoons. Thus, for illustration, we provide a comparison of two raingauge stations, Petaling Jaya, a station on the west coast, and Kemaman, a station on the east coast. These two stations are very much influenced by the monsoons characterized by frontal and convective rainfall. On average, the amount of mean hourly rainfall is greater at Kemaman than Petaling Jaya (Fig. 3); the difference is clearly seen during the northeast monsoon season (November February). In this study, we also consider July and November as examples of months representing dry and wet months during the monsoon seasons, respectively. 4.2 Parameter estimation The parameters of both models are estimated using the generalized method of moments (GMM) and the Nelder-Mead optimization method. Inconsistencies in Table 1 Definition of sets of moment combinations used for estimating model parameters at different levels of aggregation. Level of temporal aggregation of moment used (h) Set Mean Variance Lag-1 autocorrelation Probability of dry S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S

7 Bartlett Lewis rectangular pulse models 1695 Fig. 3 Mean hourly rainfall depth for all months for Petaling Jaya and Kemaman stations. Table 2 Boundary constraints for the parameters of the MBL model. Parameter Lower bound Upper bound λ μ x σ/μ x 1 1 α ν κ φ the data would lead to unstable values of the parameter estimates and unrealistic storms. To obtain realistic storms, the parameter space is reduced by fixing the value of ν in the MBL model, see Smithers et al. (2002) for details. Based on their study, Schnorbus and Alila (2004) recommended that the value of ν be between 0.2 and However, in this study, the boundary constraints identified for the parameters of the MBL model that contribute to the stability in the parameter estimates are as given in Table 2.The boundary constraints for the parameters of the OBL model are the same as those for MBL except for the parameter λ which ranges from 0.01 to Model performance The performance of the models is evaluated by assuming that the historical time series are stationary (Cox and Isham 1980). Therefore, we calibrate the results for each month separately, assuming that monthly rainfall forms a stationary time series (Rodriguez-Iturbe et al. 1987b). To assess the model performance under various moment combinations, the results of goodness-of-fit statistics and graphical display involving the comparison between the observed and fitted statistics are considered. Although the MBL and OBL models are both fitted to the data, to simplify our explanation, we illustrate results that are found based on the MBL model for July and November for Petaling Jaya and Kemaman, respectively. Figures 4 6 provide a comparison between the observed and estimated statistics under all sets of moment combinations. The values of observed and estimated 1-h mean matched excellently, indicating that the MBL model performed equally well under all moment combinations. Figures 5 and 6 indicate that the observed and estimated lag-1 autocorrelation and probability of dry differ slightly for every set of moment combinations considered. However, as is apparent in Fig. 4, itis found that 1-h variance is poorly estimated for many sets of moment combinations. Based on these figures, parameter sets 7, 8 and 9 are the candidates with most potential to be considered for the best selection. When a comparison between the observed and estimated 1-h variance is made, sets 7 and 9 perform equally well, better than set 8. However, when other statistics are compared, in some cases, set 8 performs slightly better than the other two sets. For example, the inclusion of 24-h lag-1 autocorrelation in set 7 as an additional statistic, compared to set 8 improved the estimates of the variances; however, the estimates of 1-h lag- 1 autocorrelation and probability of dry became less precise when compared to those found using set 8. Therefore, it is quite difficult to decide on the best moment combination which contributes to the best model, graphically. Since the decision on the best moment combination based on the graphs is inconclusive, as an alternative, the values of the objective functions found based on all sets of moment combinations are compared. These values are computed for all sets of moment combinations under the OBL and MBL models; however, to illustrate how to decide on the best set, the results for the MBL model are provided. The results of goodness-of-fit statistics for all sets of moment combinations listed in Table 1 are given in Table 3 and the best set is determined based on the lowest value of the objective function S(θ). It is clear that set 8 contributed to the best performance for Petaling Jaya and Kemaman for both

8 1696 Ibrahim Suliman Hanaish et al. 1-hour variance July Kemaman 1-hour variance November Kemaman 1-hour variance July Petaling Jaya 1-hour variance November Petaling Jaya Fig. 4 Comparison between the observed and estimated 1-h variance based on all sets of moment combinations for Petaling Jaya and Kemaman rainfall stations for July and November. July and November because the value of the objective function is the lowest. This set includes the common statistics which are the mean, variance, lag- 1 autocorrelation, probability of dry based on the hourly rainfall data and the 24-h mean and variance. These findings imply that incorporating daily statistics improves the model performance. The further discussion on the performance of the fitted models and extreme value analysis involves set Comparing the performance of the MBL and OBL models In this section, the OBL and MBL models are compared in terms of the mean absolute deviate error and the reproduction of the fitting statistics based on the best set, set 8. As suggested by Velghe et al. (1994), an efficient way to measure the overall performance of the model in reproducing the historical rainfall statistics is to use the mean absolute deviate error (MAD): m MAD = 1 m 1 X i est X i obs 100 (2) i=1 where X i est stands for the value of the ith estimated statistics, X i obs is the corresponding observed value and m is the number of statistics evaluated. A comparison of mean absolute deviate errors found for July and November for Petaling Jaya and Kemaman, based on both models, is shown in Fig. 7. The mean absolute deviate errors for the MBL model are much smaller than those for OBL model and the difference is due to the estimation of variances as given in Tables 5 and 6.

9 Bartlett Lewis rectangular pulse models 1697 Fig. 5 Comparison between the observed and estimated 1-h lag-1 autocorrelation based on all sets of moment combinations for Petaling Jaya and Kemaman rainfall stations for July and November. Both models reproduce the observed values of 1- h mean, lag-1 autocorrelation and probability of dry very well (Tables 5 and 6). The OBL model reproduces the probability of dry excellently and this is contrary to some studies mentioned in the literature. From Tables 5 and 6 it is clear that both models underestimated the observed 1-h and 24-h variances. The discrepancy between the observed and estimated values is very substantial for the case of 24-h variance, particularly when the OBL model is considered. For further assessment of the goodness of fit, simulated statistics for various aggregated time scales obtained based on a set of 38 years of simulated data of the same length as the observed rainfall data using the MBL model with parameters derived from set 8 are considered and also provided in Tables 5 and 6. It is found that the MBL model generally simulates most of the observed statistics very well, but the variance is underestimated for all aggregated time scales considered. In general, based on the properties examined so far, the MBL model gives a better performance. 5 RESULTS AND DISCUSSION 5.1 Sensitivity of parameters From the results of the estimated parameters of the MBL model given in Table 4, it is found that the estimated values of the storm arrival parameter λ are generally quite stable, except for a relatively much higher value of λ observed at Kemaman in the month

10 1698 Ibrahim Suliman Hanaish et al. 1-hour probability of dry July Kemaman 1-hour probability of dry November Kemaman 1-hour probability of dry July Petaling Jaya 1-hour probability of dry November Petaling Jaya Fig. 6 Comparison between the observed and estimated 1-h probability of dry based on all sets of moment combinations for Petaling Jaya and Kemaman rainfall stations for July and November. of November. Meanwhile, the values of the mean cell intensity μ x and the parameters α and ν of the gamma distribution, which describe the cell duration, varied considerably with respect to the different sets of moment combinations. This result is contrary to the work by Khaliq and Cunnane (1996) which performed a sensitivity analysis of the parameters of MBL model using five different sets of moments for Ireland data. They state that parameters λ and μ x are the most stable, while α and ν are the least stable. Moreover, a few more studies have also obtained results that are different from those reported here. For example, Onof and Wheater (1993) found that the estimated parameters of MBL model particularly α and ν determined based on UK data using two different set of moments, are unstable, with the exception of parameters λ and μ x. In contrast to these results, Velghe et al. (1994) reported that when using five different sets of moments to derive the MBL parameters for USA data, it was found that there is not much difference in the values of the parameters from set to set, but a large difference is noted for the parameter ν only. It is not surprising to observe different findings, as reported in these two studies and our study, due to the different climate conditions in the UK and the USA, as well as the tropical climate experienced in the Peninsula. The effect of the monsoons on the nature of rainfall in the Peninsula brings about many interesting findings. It is found that Kemaman experiences the highest rate of the storm arrival λ during the month of November, while Petaling Jaya experiences the highest rate of the storm arrival in July. Also, for Kemaman and Petaling Jaya, the values of storm duration φ are found to be higher for the months of November and July, respectively. The higher value of

11 Bartlett Lewis rectangular pulse models 1699 Table 3 Minimum value of the objective function S(θ) for the MBL model for each moment combination considered. The best fit values are highlighted in bold. Set Minimum of S(θ) for July and November Petaling Jaya Kemaman July November July November S1 1.07E E E E-06 S2 5.22E E E E-06 S3 7.55E E E E-05 S4 2.69E E E E-06 S5 4.24E E E E-09 S6 4.59E E E E-07 S7 2.52E E E E-06 S8 1.20E E E E-09 S9 2.69E E E E-04 S E E E E+03 S E E E E-05 S E E E E-01 S E E E E-06 S E E E E+01 S E E E E-05 S E E E E+01 S E E E E-01 φ for the month of November contributes many heavy rainfall episodes that often occur in the east coast areas such as Kemaman. As a result, the northeast monsoon, which occurs from November to February, causes floods in various areas on the east coast. It is also found that all values of the mean cell intensity μ x are high, particularly for Petaling Jaya, in the month of July. It is clear from the results that this high value of μ x is associated with a small value of φ, indicating that the occurrence of convective rainfall is quite common in July. This situation is expected since July is one of the months when the southwest monsoon occurs. The occurrence of heavy rain of short duration, which is a characteristic of convective rainfall, often induces flash floods in the city area such as Petaling Jaya. In general, the estimated parameter values are found to be larger for Kemaman than Petaling Jaya, particularly for November. This implies that the effect of the northwest monsoon is more severe than the southwest monsoon, due to a longer flood season in the east coast than the west coast. 5.2 Extreme value analysis To evaluate the model performance, the effect on the reproduction of the distribution of daily or sub daily extremes are also assessed based on comparisons between the observed and simulated daily and hourly annual maxima for Petaling Jaya and Kemaman. One hundred sets of simulated rainfall series were generated, in which each set has a record length equal to that of the observed data. To study how well the Gumbel distribution fits the extreme data, a visual representation of the annual maximum rainfall is displayed on the Gumbel plots. The Gumbel reduced variate plot provides a test of model ability to produce simulated annual extremes within the observed record. In these plots, the vertical axis represents annual maximum rainfall and the horizontal axis represents the reduced variate of the Gumbel distribution which is given by: Y(T) = ln( ln(1 1/T)) (3) where T is the return period, in years, assigned according to the Gringorten plotting position formula (Gringorten 1963), as given by: 20 MBL OBL Mean Absolute Deviate Error % July Petaling Jaya November Petaling Jaya July Kemaman November Kemaman Months Fig. 7 Mean absolute deviate error for July and November for Petaling Jaya and Kemaman rainfall stations.

12 1700 Ibrahim Suliman Hanaish et al. Fig. 8 Comparison of the observed and simulated annual hourly rainfall maxima (a) and observed and simulated annual daily rainfall maxima (b) based on the MBL model when cell intensities are assumed to follow gamma, exponential or pareto distributions for Petaling Jaya. The corresponding plots for annual hourly rainfall maxima (c) and annual daily rainfall maxima (d) for Kemaman. T = N K (4) where N is the number of years of data available and K 1 is the rank of the observation in the ordered set of maxima (K 1 = 1 for the largest). In this study, 95%

13 Bartlett Lewis rectangular pulse models 1701 Table 4 Estimated parameters of the MBL model found using different moment combinations for July and November for Petaling Jaya and Kemaman. The best fit values are highlighted in bold. Petaling Jaya Kemaman λ μ x α ν κ φ λ μ x α ν κ φ July July S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S November November S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S Table 5 Summary statistics for the observed data, the fitted OBL and MBL models and simulated MBL model based on set 8 for July and November for Petaling Jaya. Statistics Mean1 Mean24 Var1 Var24 ACF1(1) Pdry1 July Observed MBL estimated OBL estimated MBL simulated November Observed MBL estimated OBL estimated MBL simulated confidence intervals around the observed maxima are calculated by (X ± 2σ ) (David and Nagaraja 2003), where X is the observed rainfall maxima and σ 2 is the variance of each observed maxima, given by: σ 2 = n + 1 k 2 [ ] 2 (5) α 2 k 2 (n + 2) ln( k 2 n+2 )

14 1702 Ibrahim Suliman Hanaish et al. Table 6 Summary statistics for the observed data, the fitted models and simulates MBL model based on set 8 for July and November for Kemaman. Statistics Mean1 Mean24 Var1 Var24 ACF1(1) Pdry1 July Observed MBL estimated OBL estimated MBL simulated November Observed MBL estimated OBL estimated MBL simulated where n is the sample size, k 2 is the rank of the observed maxima in the ascending order, α = 1.282/sd, where sd is the standard deviation of the rainfall maxima. The model performance is acceptable if the simulated data are within 95% confidence intervals around the observed maxima. 5.3 Annual performance The Gumbel reduced variate plots that are provided in Fig. 8(a) (d) show that the fitted MBL model based on the best moment combination fails to reproduce the annual hourly and daily maxima when the rain cell intensities are assumed to follow either exponential, gamma or pareto distributions. It is found that the simulated maxima based on the assumed distributions underestimate the observed maxima for the two stations because the simulated values fail to fall within 95% prediction intervals around the observed maxima. 6 CONCLUSION The main objective of the present study is to select a stochastic Bartlett Lewis rectangular pulse model that can accurately describe rainfall processes in Peninsular Malaysia. Two versions of Bartlett Lewis rectangular pulse rainfall models are investigated, the original Bartlett Lewis (OBL) and the modified Bartlett Lewis (MBL). In addition, the purpose of this study is to identify the best moment combination which can be used for data generation at different time scales. Hourly rainfall data available at 15 raingauge stations were fitted to OBL and MBL models using the method of generalized moments. Since the results found could be attributed to two different locations in the Peninsula, i.e. the east coast and west coast, which are influenced by the monsoons, rainfall data for the months of July and November observed for Petaling Jaya and Kemaman stations, located on the west and east coasts, respectively, are considered for illustration of the findings. July and November are considered as examples of the dry and wet periods that are affected by the southwest and northeast monsoons, which blow from May to August and November to February, respectively. Seventeen combinations of moments are selected to check the effectiveness on the model parameters, and to study the performance of the models in estimating the observed hourly statistics. The contribution of each set of moment combinations on the model fits is decided based on graphical display comparing the observed and estimated statistics. The best moment combination for Petaling and Kemaman is set 8 which includes the hourly rainfall statistics and an additional statistics of mean and variance based on 24-h aggregated rainfall data. The use of mean and variance based on rainfall of duration of 24-h estimated from the daily values successfully improved the model performance for July and November. The inclusion of 24-h lag-1 autocorrelation in set 7 as an additional statistic compared to the statistics in set 8 improved the estimates of the variances but not estimates of the 1-h lag-1 autocorrelation and probability of dry. The performance of both models is studied in terms of mean absolute deviate error for set 8 in reproducing the estimated statistics. It is found that the mean absolute deviate errors are much smaller for the fitted MBL model than the fitted OBL model. It is also found that the MBL model generally simulates most of the observed statistics very well, but the variance is underestimated for all aggregated time scales considered. Therefore, in general, based on the properties examined so far, the MBL model gives the best performance.

15 Bartlett Lewis rectangular pulse models 1703 The results of extreme value analysis indicate that the MBL model failed to reproduce annual hourly and daily rainfall maxima successfully when rainfall cell intensities are assumed to follow either exponential, gamma or pareto distributions. The characteristics of rainfall in this country contribute to the presence of many extremely large values of rainfall amount in the data set. Verhoest et al. (2010), although not using real data, found that the MBL model fails to capture extremes. In our study, data which involve many extreme values are used and the findings support the ideas suggested by Verhoest et al. (2010). Further work is required to successfully model rainfall events which involve extremes. Acknowledgements The authors wish to thank Prof. Demetris Koutsoyiannis and Prof. Christian Onof for their positive and constructive comments on this research. 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