Journal of Hydrology 470–471 (2012) 158–171Contents lists available at SciVerse ScienceDirect Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol IDF relationships using bivariate copula for storm events in Peninsular Malaysia N.M. Ariff ⇑, A.A. Jemain 1, K. Ibrahim 2, W.Z. Wan Zin 3 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia a r t i c l e i n f o Article history: Received 2 April 2012 Received in revised form 6 July 2012 Accepted 22 August 2012 Available online 31 August 2012 This manuscript was handled by Andras Bardossy, Editor-in-Chief, with the assistance of Sheng Yue, Associate Editor Keywords: Intensity–duration–frequency (IDF) Copula Storm event Bivariate frequency analysis s u m m a r y Intensity–duration–frequency (IDF) curves are used in many hydrologic designs for the purpose of water managements and flood preventions. The IDF curves available in Malaysia are those obtained from univariate analysis approach which only considers the intensity of rainfalls at fixed time intervals. As several rainfall variables are correlated with each other such as intensity and duration, this paper aims to derive IDF points for storm events in Peninsular Malaysia by means of bivariate frequency analysis. This is achieved through utilizing the relationship between storm intensities and durations using the copula method. Four types of copulas; namely the Ali–Mikhail–Haq (AMH), Frank, Gaussian and Farlie–Gumbel–Morgenstern (FGM) copulas are considered because the correlation between storm intensity, I, and duration, D, are negative and these copulas are appropriate when the relationship between the variables are negative. The correlations are attained by means of Kendall’s s estimation. The analysis was performed on twenty rainfall stations with hourly data across Peninsular Malaysia. Using Akaike’s Information Criteria (AIC) for testing goodness-of-fit, both Frank and Gaussian copulas are found to be suitable to represent the relationship between I and D. The IDF points found by the copula method are compared to the IDF curves yielded based on the typical IDF empirical formula of the univariate approach. This study indicates that storm intensities obtained from both methods are in agreement with each other for any given storm duration and for various return periods. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction Statistical analysis of extreme data is important in various disciplines including hydrology, engineering and environmental science (Reiss and Thomas, 2007). By performing extreme analysis on rainfall data, damages to human lives and properties as the consequences from extreme rainfall events such as flood or landslides may be reduced or prevented. In Malaysia, extreme analysis on rainfall data has been explored for all sorts of purposes such as tracing patterns and trends of daily rainfall during monsoon seasons (Suhaila et al., 2010a,b), detecting recent changes in extreme rainfall events (Wan Zin et al., 2010) and fitting probability distributions to annual maximum rainfalls by implementing various methods (Shabri et al., 2011; Wan Zin et al., 2009a,b). The rainfall intensity–duration–frequency (IDF) curves are essential tools in designing hydraulic structures such as dams, spillways and drainage systems. These hydraulic structures help to lessen the loss caused by extreme rainfall events. Hence, for ⇑ Corresponding author. Tel: +60 603 89215784; fax: +60 603 89254519. E-mail addresses:
[email protected] (N.M. Ariff),
[email protected] (A.A. Jemain),
[email protected] (K. Ibrahim),
[email protected] (W.Z. Wan Zin). 1 Tel.: +60 603 89215724; fax: +60 603 89254519. 2 Tel.: +60 603 89213702; fax: +60 603 89254519. 3 Tel.: +60 603 89215790; fax: +60 603 89254519. 0022-1694/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jhydrol.2012.08.045 countries such as Malaysia where flood is considered as the most significant natural hazard (Sulaiman, 2007), correct rainfall estimation as points on the IDF curves is important. According to Koutsoyiannis et al. (1998), IDF curves are able to show the mathematical relationship between rainfall intensity, i, duration, d, and return period, T (the annual frequency of exceedance). The construction of IDF curves is mostly done using univariate rainfall frequency analysis approach because of its mathematical simplicity (Singh and Zhang, 2007). The univariate approach of rainfall frequency analysis is explained in detail by Chow et al. (1988). In addition, most of the IDF curves are constructed using the window-based analysis approach where the durations are predetermined time intervals. Thus, the ‘durations’ do not represent the actual durations of rainfall events. In other words, a smaller window may be a subset of a longer extreme rainfall event while a bigger sized window could contain several short duration rainfalls and dry periods. Several attempts have been made to derive joint distributions of rainfalls characteristics, namely; intensity, depth and duration, as their random variables. This is to accommodate the complexity of hydrological events. It has come to light that events such as storms and flood always appear to be multivariate and thus, single-variable frequency analysis can only provide limited assessments of these events (Yue et al., 2001). In earlier studies, several assumptions have been made due to the mathematical difficulties 048 101. 2010.. The information extracted from the rainfall data is the annual maximum storm intensity for several storm durations. the construction of IDF curves using the typical empirical method is discussed.346 4. Various types of copulas have been used in hydrology over the last decade. 2. Storm duration is defined as the time interval for a storm event and storm intensity is the ratio of storm depth to storm duration. These regions are based on their geographical locations as suggested by many previous researches (Deni et al.144 3. extreme rainfall is a complicated phenomenon and its marginal distributions are not necessarily similar or distributed as normal. De Michele et al. In reality.900 103.2 95.157 101. Goel et al.5 97.9 99. following this discovery.0 4.4 4.233 2. East and Southwest region of Peninsular Malaysia. the only ones utilized are those obtained through the univariate frequency analysis. 3. 2008).438 101. These stations are located both at the edge and the middle of the Peninsular.060 103. most rainfall models incorporated the dependence between variables with some limitations imposed on the marginal distributions of the random variables. Ariff et al. although the IDF curves play important roles. The regions are known as the Northwest.017 94. The different probability distribution functions used in this study are shown in section five.720 95. 1.967 3.658 98. Other distributions should be considered which may produce better rainfall estimates.4 N03 N04 West W01 W02 W03 W04 W05 W06 East E01 E02 E03 E04 E05 E06 Southwest S01 S02 S03 S04 pleteness. West. we will introduce the notations used in this paper for simplicity purposes. the marginals are assumed to be either normal (Yue.5 3. the analysis of the marginals and the dependence structure can be done separately. A thorough explanation on the theory and description of copula is given by Nelsen (2006).722 90.. 1982). However.6 Dungun Endau Kampung Dura Kemaman Kepasing Paya Kangsar 4. The first section of this paper contains the introduction of this research. A brief introduction to copula Before we proceed further. Region Code Station Latitude Longitude Completeness of data (%) Northwest N01 N02 Alor Setar Bukit Bendera Jeniang Sungai Pinang 6. The third section comprises of the explanation on copula method and the construction of IDF points using the copula method. Salvadori and De Michele. The seventh section contains the computation of IDF points based on the copula and empirical methods on the hourly rainfalls data in Peninsular Malaysia and section eight provides the conclusion of the research.395 103. The stations are located in four different regions scattered across Peninsular Malaysia. In Malaysia. Data and definition of storm Hourly rainfall data obtained from stations in Peninsular Malaysia are acquired from the Department of Irrigation and Drainage Malaysia.0 Bertam Genting Klang Gua Musang Kalong Tengah Kampar Teluk Intan 5. Uppercase letters (example: X and Y) are defined as random variables and lowercase letters . bivariate Gamma (Yue et al.2 93. Data from these stations are more than 90% complete for the year 1975–2008 as presented in Table 1. bivariate lognormal (Yue.2 5.217 96.7 93.860 101. Suhaila et al.963 95.474 103.290 1.269 97.661 102.817 5.463 102. This paper focuses on the convective storms for the purpose of IDF curves construction. followed by a section describing the data used for analysis and the definition of storm events. Their study showed that the correlation between rainfall duration and its average intensity gives non-negligible effect on storm surface runoff. Copulas are based on Sklar’s theorem (1959) which states that for joint distribution..7 96.1 2.0 96. The IETD value is chosen such that the serial correlation between the two different storms is minimized (Restrepo-Posada and Eagleson. 2003. Storm depth is defined as the accumulated rainfall which begins and ends with at least one wet hour and either contains dry periods with less than 6 h or none at all.. This paper aims to derive the bivariate or the joint distribution between intensity and duration of extreme storm events in Peninsular Malaysia using copulas. 2001). / Journal of Hydrology 470–471 (2012) 158–171 in obtaining these joint distributions using standard statistical methods. 2000). Section four provides the various copulas under consideration. The copula approach is a flexible method that allow more choices of marginal distributions and dependence structures to be used in multivariate problems (Kao and Govindaraju.0 4. 2011). Cordova and RodriguezIturbe (1985) have proven that this assumption is inappropriate and unrealistic.204 102. the elliptic copulas such as the Gaussian copula (Renard and Lang. The definition of storm-event depends greatly on the interevent time definition. Hence.420 102. the points for IDF curves are derived from the copula obtained and compared with the IDF points found based on the empirical formula used in univariate analysis. they would not be considered as two different events but parts of the same storm.404 100.400 103.7 99.641 5. 2004). Among them are the Farlie–Gumbel–Morgenstern (FGM) copula (Favre et al.021 97.026 101. In section six.755 99. 2007) and the Archimedean copulas (De Michele and Salvadori. twenty stations are selected based on their locations and com- Table 1 The locations and completeness of rainfall data for stations in Peninsular Malaysia. Convective storm is defined as short duration storm which has great impacts on small or urban catchments (Palynchuk and Guo. In this study.. 2003). 2008). the dry duration between two individual storm events must at least be equal to the IETD value.8 Chinchin Johor Bahru Kota Tinggi Labis 2. 2004. Hence.867 102. One of the assumptions is the independence assumption between the random variables. 2002. Then. The inter-event time definition (IETD) is defined as the minimum duration of dry period between two consecutive storm events. If not.733 103. The locations of these 20 stations are shown in Fig.. 2006). Suhaila and Jemain.M. 2008).7 99. 2005. Zhang and Singh.356 100.430 96.425 100.756 2. For small urban catchments.. For instance.9 1.116 5. 2000) or possess the same type of probability distribution such as bivariate exponential (Favre et al. 2002) and bivariate extreme value distribution (Shiau. 2009.934 97. Most rainfall stations are located at the edge since at the centre of Peninsular Malaysia lies the Titiwangsa Mountain range which is mostly unoccupied. the IETD is usually taken as 6 h because the time concentration of rainfall which is less than 6 h would make the runoff response of successive storms to appear independent (Palynchuk and Guo.159 N.633 100. / Journal of Hydrology 470–471 (2012) 158–171 NORTHWEST EAST WEST SOUTHWEST Fig. The construction of one-parameter copula can be summarized into four simple steps. F Y ðmÞÞ ¼ H X. In this paper. the correlation between random variables x and y is yielded using Kendall’s s. Map of Peninsular Malaysia and the locations of the twenty hourly rainfall stations under consideration. y in R. This is done by fitting probability distribution functions to random variables X and Y. then U and V are uniformly distributed. the copula parameter is denoted as h. For this study. the copula can be obtained by first finding the marginal distribution functions U and V. there exists a unique copula. U = FX(X) U[0. otherwise: ð2Þ . N 2 1 X sign ðxj xk Þðyj yk Þ j<k with sign ¼ 8 1. Thus. Copula reduces the complexity of deriving the joint distribution of random variables X and Y. CU.1]. the dependence between the random variables is calculated. such that for all x. ðxj xk Þðyj yk Þ ¼ 0 > > : 1. y) observations. yÞ: ð1Þ In a way copula helps to map random observations (x. Kendall’s s is estimated as (Kao and Govindaraju. Once the copula family and function to be used is identified. y) from R2 to a bounded domain [0. u = FX(x) and v = FY(y) are realizations of U and V respectively.Y with marginals FX and FY respectively. (example: x and y) are realizations or specific values of random variables. > > > < xj < xk and yj < yk xj > xk and yj > yk > 0. can be obtained.160 N. Y 6 F Y ðmÞÞ 1 ¼ HX. the marginals of the random variables and the dependence between them can be determined separately one at a time. Ariff et al. For N paired (x. all the parameters for the two probability distributions are approximated and their respective distribution functions. 1] and V = FY(Y) U[0. 1. Thus. Y) is denoted as HX.V. 1] (Nelsen. V 6 mÞ ¼ PðX 6 F 1 X ðuÞ. Next.Y ðx. v Þ ¼ PðU 6 u. If we let U and V be the random variables of these marginals. Unlike the traditional bivariate joint distribution. Copula is defined according to Sklar’s theorem (1959) which states that for continuous random variables X and Y with joint distribution HX. 2007) TN ¼ 1 C U.M. the joint distribution function of a pair of random variables (X.Y ðF 1 X ðuÞ. For a bivariate distribution.Y and marginals FX and FY. 2006). U and V.V ðu. A copula is usually described as either a function which links a multivariate distribution function to its marginals (cumulative distribution functions) or a function composed of uniformly distributed marginals in [0. 1]2. . D. Archimedean copula According to Nelson (2006). Gaussian copula is good for practical applications since it possess several properties of the multivariate normal distribution (Favre et al. 2007. has at least 22 copula functions as its member. 2007).3. 4. The choice of copula family and its function relies on the correlation between the random variables under consideration. From the value of u.V ðu.V ðu.. 2007). Ali–Mikhail– Haq (AMH) and Frank copula. For simplicity purposes. The copula function for the Gaussian copula is (Schmidt. He stated that u(1) = 0 and u1(x) = 0 for x P u(0). 2006) C U.. 2007. v Þ ¼ uv .3. s 2 ½1. Farlie–Gumbel–Morgenstern (FGM) copula FGM copula belongs to the family of copulas with quadratic section. For example. The relationship between h and s for FGM copula is (Huard et al. Kendall’s s is defined in the closed interval [1. This i value is then taken as one of the point on the IDF curves indicating the intensity of storm for return period T and storm duration d. I. CU|V=v. Gaussian copula Gaussian copula is from the elliptic copula family. i. other points of the IDF curves can be determined and the IDF curves can be constructed. The conditional distribution of I given D = d is represented in conditional copula form as the realizations of marginal U for known value of V = v. as their realizations. 1: ð8Þ where U is the cumulative distribution function of a standard normal distribution and UR is the bivariate normal distribution with mean 0 and covariance matrix R. 2006). the conditional Gaussian copula is C UjV¼u ðujV ¼ v Þ ¼ @ UR ðU1 ðuÞ. the relationship between the conditional copula and the selected return period is 1 C UjV¼m ðujV ¼ mÞ ¼ 1 : T ð4Þ Hence. the respective i can be yielded since i ¼ F 1 I (u). 2004).e. (8) is actually the Pearson q. s¼ 2h . A one-to-one relationship between the Pearson q and Kendall’s s under normality is provided by Kruskal (1958) as s¼ 2 p arcsin h.. The value of v is easily obtained since v = FD(d). the Archimedean copula can be generalized as C U.The third step is calculating the copula parameter h. the approximated h and the marginals of the random variables can be inserted into the candidate copula function to obtain the respective copula. v ÞjV¼v : @m ð3Þ Similar to the return period of any conditional bivariate distributions. Finally. Below are two copulas from the Archimedean copula family which are considered in this study. by solving Eqs. is known to be negative (Kao and Govindaraju. v Þ ¼ u1 ðuðuÞÞ þ uðv ÞÞ: 4. Thus. 1: ð10Þ 4. 2007) C UjV¼m ðujV ¼ mÞ ¼ @ C U. Selection of copula C U. Singh and Zhang.. Ariff et al. Repeating this process for various return periods and selected values of storm durations.V ðu. 1] with 1 implying total concordance. copula consists of families with many different copula functions. U1 ðv ÞÞ Z U1 ðuÞ Z U1 ðv Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 1 2p 1 h2 ! 2hsx s2 x2 exp dsdx. In this study. the Archimedean copula. we can solve for the corresponding u.M. Gaussian. 9 s2 22 : 99 ð6Þ The conditional copula is C UjV¼m ðujV ¼ mÞ ¼ u þ huð1 uÞð1 v Þ huv ð1 uÞ: There are various copula families and functions available for bivariate frequency analysis. Zhang and Singh. 1: ð5Þ ð7Þ 4. This relationship differs for different copula. The correlation between storm intensity. 1 representing total discordance and 0 showing zero or no concordance between the random variables (Kao and Govindaraju. : 3 s¼1 s ð13Þ The conditional copula for AMH is C UjV¼t ðujV ¼ mÞ ¼ u hmð1 uÞ 1 : 1 hð1 uÞð1 mÞ 1 hð1 uÞð1 mÞ ð14Þ .1. The FGM copula is used for modelling purposes due to their simple analytical form with its copula function written as (Favre et al. 2004).V ðu. Copulas from the Archimedean family are also easily constructed (Zhang and Singh. 4. (3) and (4) simultaneously for any given values of T and v. 2ð1 h2 Þ h 2 ½1. U1 ðv ÞÞjV¼v : @t ð9Þ h in Eq. Ali–Mikhail–Haq (AHM) copula AMH’s copula function is (Huard et al. The conditional copula can be derived from the copula function for realizations of U given the value of V = v. The Archimedean copula family is preferable in hydrologic analysis because it possess desirable properties such as symmetric and associative (Favre et al. / Journal of Hydrology 470–471 (2012) 158–171 Similar to the Pearson correlation coefficient q. 1 hð1 uÞð1 v Þ h 2 ½1. 2008). Among the copulas which have been applied in hydrology that are appropriate for negatively correlated random variables are the Farlie–Gumbel–Morgenstern (FGM). v Þ ¼ UR ðU1 ðuÞ. we denote them as I and D respectively with lowercase letters.2. The conditional copula is written as (Zhang and Singh. and storm duration. In fact. 2006) ð11Þ with u is a copula generator and u1 is appropriately defined. 2006) C U. v Þ ¼ uv þ huv ð1 uÞð1 v Þ. which is one of the most commonly used copula family due to its mathematical tractability and simplicity.1. s = g(h).161 N. This is done by using the relationship between h and Kendall’s s. X is regarded as the storm intensity and Y is the storm duration. 2004) h 2 ½1.V ðu. This conditional distribution is needed in order to attain the values for the intensity– duration–frequency (IDF) curves which involve conditional distribution of storm intensity given storm duration (Singh and Zhang. 1Þ ð12Þ and 2 lnð1 hÞ 2 h þ 1 for h 2h þ 2 3 lnð1 hÞ h 1 2 0:181726. i and d. it is a weibull . / Journal of Hydrology 470–471 (2012) 158–171 COPULA MORE THAN ONEPARAMETER COPULA ONE-PARAMETER COPULA Parameters and distributions required to construct a bivariate copula function θ – copula parameter i) ii) U. The probability distribution functions for exponential. The exponential distribution is used to represent the probability distribution for storm duration since storms are usually assumed to follow the Poisson process such as in the Neyman–Scott and Bartlett–Lewis model of storms. Thus. 2006) C U.2. gamma and weibull probability distribution can be generalized as f ðxÞ ¼ c ca1 x x exp b bCðaÞ b c ð18Þ with a and c as the shape parameters and b as the scale parameter. associate. A short summary of the four copulas used in this study. 1f0g.162 N. gamma. h hðhÞ hðhuÞhðhmÞ hðxÞ ¼ 1 expðxÞ 5. When both a and c have the value of one. v Þ ¼ 1 hðhÞ ln .. Frank copula Frank copula is written as (Huard et al. easily constructed. possess multivariate normal properties Range for τ: Range for τ: Ali-MikhailHaq (AMH) Frank Range for τ: Range for τ: Fig. weibull or lognormal distribution. If a = 1. Probability distribution functions for storm intensity and duration ð15Þ for h h 2 R{0} and 4 h s ¼ 1 ðD1 ðhÞ 1Þ ð16Þ with Dk ðhÞ ¼ k h k Z 0 h tk kh : dt þ kþ1 expðtÞ 1 The domain for s is ½1. then it becomes a gamma probability distribution function and if c = 1.M. Ariff et al. the inter arrival time of storm cells is taken as exponentially distributed. 2. preferable in hydrological analysis Gauss Farlie-GumbelMorgenstern (FGM) Advantage: simple analytical form Advantage: good for practical applications. The probability distribution of storm intensity is chosen from either the exponential. the conditional Frank copula is C UjV¼m ðujV ¼ mÞ ¼ hðhuÞð1 hðhmÞÞ . hðhÞ hðhuÞhðhmÞ hðxÞ ¼ 1 expðxÞ: ð17Þ A short summary of the four copulas considered in this study is shown in Fig. From Eq. 2. (18) becomes an exponential probability distribution function.V ðu. 4. (3). Eq.3. V – marginal distributions of random variables iii) τ – correlation of the two random variables (to get θ ) Copulas for only positive correlations Copulas suitable for negative correlations (used in this paper due to common practise in hydrological analysis) Copulas with quadraric section Archimedean Elliptic Advantage: symmetric. h 20 5 10 15 20 storm duration. (20) and storm intensity i derived by the copula method through Eq. 2006) 5 10 15 storm duration. / Journal of Hydrology 470–471 (2012) 158–171 163 ie ¼ gðd. KT is defined as the frequency factor for the return period T which depends on the probability distribution function of ie. 1988).N. Hence. 6. . 2008). mm/h 40 50 N01 20 ie ¼ where id and sd are the mean and standard deviation of the storm intensity for a given d. ie is approximated using the design rainfall intensity formula. (21). the Gumbel distribution will be utilized. mm/h 40 40 5 storm intensity. TÞ ¼ id þ sd K T ð21Þ probability distribution function. For rainfall data in Malaysia. Application on storm events in Peninsular Malaysia Data from twenty stations representing four regions in Peninsular Malaysia are used for application purposes. (18).. 3. b and c are estimated using the leastsquare method on ie for a set of given T and d. h Fig.M. The parameters a. Ariff et al. The formula used is known as the Sherman equation (Nhat et al. mm/h 30 20 10 storm intensity. The design rainfall intensity for the annual maximums of storms with duration d and return period T is a function of d and T which can be represented as (Kottegoda and Rosso. similar to Eq. the IDF points produced by the copula method are compared to the IDF curves found using one of the IDF empirical formula. h storm duration. Eq. (3) is written as D¼ jie ij 100%: ie ð23Þ 7. the Gumbel distribution is commonly used and is deemed to be suitable (Amin et al. (21) is applicable to many probability distributions of storm intensity that are employed in hydrologic frequency analysis (Chow et al. The scatter plots of W01 40 30 10 0 0 10 15 20 5 10 15 storm duration. The probability distribution function of lognormal distribution is 1 ðln x lÞ f ðxÞ ¼ pffiffiffiffiffiffiffiffiffi exp 2b x 2pb 2 ! ð19Þ where l is the location parameter and. between storm intensity ie found using Eq. IDF curves using empirical method In this paper.. For the purpose of performing least-square. h E01 S01 20 30 0 0 10 10 20 30 storm intensity. 2008) KT ¼ pffiffiffi T 6 : 0:5772 þ ln ln T 1 p ð22Þ The IDF curves constructed from values of storm intensities ie for various T and d are compared to the IDF points found based on the copula method. in this paper. The percentage difference. b is the scale parameter.. Scatter plots of storm intensities against storm durations for one station from each region of Peninsular Malaysia. 2006) and is written as aT k ðd þ bÞc ð20Þ with ie as the intensity of storm in millimetres per hour (mm/h). j. denoted as D. d as storm duration in hours (h) and T as the return period of storm in years. The frequency factor for Gumbel distribution is (Guo. mm/h 20 storm intensity. In Eq. 20 5949.13 1. using the estimated h and marginal distributions U and V. 3 also shows that for short storms.45 0. The s values are found not to be in the domain of s for the Ali–Mikhail–Haq (AMH) and Farlie– Gumbel–Morgenstern (FGM) copulas. that is the Kendall’s s.00a 5904.45a 5897.75 0.55 0.48 5.19 5931.85 16.34 6. It can also be observed that there is a slight variation on the characteristics of storms when different regions are compared especially for short duration storms. Hence.25 3.84 15.48 0.11 5. The most appropriate probability distributions fitted to the storm intensities and durations of these stations are shown in Table 2.48 4.07 1.07 8. the values for h and AIC of both the Gaussian and Frank copula are presented in Table 3. From the probability distributions selected.49 5. weibull and lognormal distribution as well as pairs of (L-skewness.42.44 4. / Journal of Hydrology 470–471 (2012) 158–171 storm intensity versus duration for a selected station in each region is displayed in Fig.77 0. resent the relationship between I and D is done with the help of Akaike’s Information Criteria (AIC).10a 5709.42a 5563. the dependence level between the storm intensities and durations are moderate. (10) and (16). 3.48 5.30a 5876.76 2.82 Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential Exponential 5.60 0. the smaller will the storm intensity be. In other words.08 5752. This is because storm intensity is the rate of storm depth which is yielded by taking the ratio of storm depth to storm duration.27a Smaller value of AIC.21 6.75 0.82 0.42 0. The AIC is given as (Zhang and Singh. L-kurtosis) of the storm intensities. c or l b Dist b Gamma Gamma Gamma Gamma Weibull Weibull Lognormal Lognormal Gamma Weibull Lognormal Gamma Gamma Lognormal Weibull Gamma Weibull Weibull Lognormal Lognormal 2.37 5.53 5. c = Shape parameter. Thus.52 5506.09 5.34a 5790.31 1.20a 5963. it is clear that the larger the storm duration.50 5793.29 5. gamma. gamma.46 0. 4.82 0.54 5871. storms with not more than 12 h.51 4.55 1.82 5501.98 5895.71 4.77 0. The estimated correlations between I and D.50 5.56 0.47 5. 2007) AIC ¼ 2 logðmaximized likelihood for the copulaÞ þ 2ðno: of fitted parametersÞ ð24Þ or AIC ¼ N logðMSEÞ þ 2ðno: of fitted parametersÞ: Fig.24 5.50 13. are also given in Table 2.49 5929.48 0.69 0.66 2. Since the dependence level is not high.77 5906.83 9. Table 3 Values of h and Akaike’s Information Criteria (AIC) for Gaussian and Frank copula. Thus.08 0.82 0.50 5. The L-moment ratio diagram of Fig.53 0.29 5.56 2. it is also inaccurate to reduce the joint distribution of random variables I and D to the product of the marginal distributions since the dependence level is not low and s is not close to 0.60 0.42 7.52a 5963. It can also be observed that the difference between the AIC values of . For all the 20 stations.37 4.01 6.55 0.55 5922. The comparison of both copulas to determine which type of copula is better to rep- Table 2 Distribution for storm intensities and durations as well as the Kendall’s s for the correlations between the two random variables.56a 5966. Ariff et al.56a 5502. Fig. The values are neither close to 1 nor 0. From this figure. The parameter.32a 5911.86 0.77 5929.11 5.35a 5744. U = FI(I) and V = FD(D).92 8. as displayed in Eqs.70 0.53a 5887.74 0. Station N01 N02 N03 N04 W01 W02 W03 W04 W05 W06 E01 E02 E03 E04 E05 E06 S01 S02 S03 S04 a Gaussian Frank h AIC h AIC 0. It can be seen that the values of Kendall’s s for all twenty stations considered are negative which is in accordance with Fig.37 5.M. both copulas are fitted to the paired I and D by using their copula functions which are Eqs.61 0.31 3.44 2.35 2.04a 5504.17 6. Fig.84 0.77 5908.90 2.61 a.81 8.164 N. The values of s for all the stations lie within the range of 0. for instance.81 0.43 4. weibull and lognormal distributions were fitted to storm intensities of the twenty stations.27 5946.52 5567.58 8.50 0.75 0.15 6.70 5.54 0.82 0.43 4.64 6 s 6 0.07 5974.45 5.58 1.85 2.67a 5923.04 5708. it is unsuitable to use regression method to represent their relationship.48 2.82 4. b = scale parameter and l = location parameter.46 5. ð25Þ The copula which shows a smaller value of AIC is chosen to represent the joint distribution of I and D.75 0.81 5946.37a 5940.46 5.54 0.61 0.54 8.71 8.92 5963. 4 illustrates the L-moments ratio diagram for the four distributions with the sample L-skewness and L-kurtosis of storm intensities for each of the twenty stations.44 4.62 0. L-moment ratio diagram for the exponential.45 0. for both copulas are estimated using their respective relationship to Kendall’s s.46 5926.40 4.61 0.44 1.62 0. only the Frank and Gaussian copulas will be further considered for the construction of bivariate distributions of storm intensities and durations.58a 5929.70 14. (8) and (15) respectively. 4 and Table 2 imply that there may be dissimilarities among the best fitted storms’ distribution according to different regions.05 8.64 0. there is a higher correlation between storm intensities and durations.70 2.89 0. Station N01 N02 N03 N04 W01 W02 W03 W04 W05 W06 E01 E02 E03 E04 E05 E06 S01 S02 S03 S04 Intensity (mm/h) s Duration (h) Dist a. On the other hand.62 0.50 5. Table 3 indicates that the Frank copula provides a comparatively smaller value of AIC for most of the twenty stations.88a 5930.48 16.80 0.55a 5942.53 5.53 4. 3.65 0.42 0.83 15.86 7.45 7. it can be seen that there is a negative correlation between the two variables.16 1.46 5979. we may obtain the marginal distribution functions. h.57 0.60 0. The exponential. 22 11.17 9.49 21.06 39.79 71.41 53.95 11.90 41.42 34.06 7.23 22.23 2.68 13.44 21.62 13.80 19.04 25.87 18.34 11.29 19.44 31.44 18.18 10.92 5.25 17.09 49.41 W02 2 5 10 25 50 100 23.79 43.93 12.10 14.69 5.85 24.45 24.75 17.18 9.61 11.93 6.95 21.06 16.79 22.01 13.52 9.08 14.13 16.28 17.05 19.50 8.64 7.05 21.18 12.69 25.19 6.37 28.08 14.91 22.10 23.62 6.14 22.33 1.25 22.65 0.65 5.30 6.46 19.95 13.69 40.36 14.53 11.06 9.44 9.32 11.55 16.64 21.62 0.83 35.88 52.35 11.95 20.31 10.68 10.94 14.19 9.39 23.77 17.97 11.67 12.16 9.75 29.11 12.74 11.94 17.56 13.72 13.60 16.54 7.31 7.30 6.00 1.60 6.79 26.20 33.36 21.09 15.46 22.88 43.89 17.24 22.71 26.54 15.18 24.38 56.22 11.23 9.57 14.87 13.70 33.85 12.83 7.11 45.59 26.27 39.62 12.86 19.86 5.97 9.46 22.19 7.20 15.09 29.72 6.06 17.18 21.67 21.44 15.29 18.26 2.01 21.58 20.99 18.13 19.87 6.30 14.30 1.99 60.56 7.58 10.69 22.25 13.53 23.62 15.29 10.40 8.34 13.89 (continued on next page) .44 6.38 7.13 5.65 13.17 3.00 11.02 8.93 23.05 22.70 11.71 17.08 16.84 8.76 24.25 13.36 14.52 18.08 28.67 18.22 12.12 23.00 23.00 20.46 12.41 30.07 5.86 18.25 10.62 31.05 31.29 6.20 26.09 36.92 13.55 37.89 13.40 8.84 22.65 17.14 21.93 30.34 8.84 3.41 21.20 10.90 17.58 9.77 41.37 26.14 7.67 18.16 18.75 48.07 29.93 11.31 9.70 19.51 25.88 20.19 38.88 12.62 16.30 4.28 23.80 5.65 14.96 15.89 17.46 0.88 6.05 7.22 14.39 50.51 30.62 19.87 4.32 12.78 W01 2 5 10 25 50 100 22.26 19.77 7.76 9.02 22.62 18.21 9.29 3.98 5.37 6.45 17.92 11.57 23.08 22.68 43.14 20.78 39.71 17.64 30.69 38.65 4.20 7.30 16.43 11.12 18.59 10.53 23.54 19.12 10.46 14.32 13.16 16.29 10.32 8.37 14.98 10.55 9.74 24.95 31.77 20.21 11.10 26.38 31.07 14.57 13.69 18.59 20.27 9.22 52.99 12.24 3.49 15.58 6.76 18.67 23.78 53.69 25.50 21.03 10.63 19.77 6.63 5.02 8.28 44.27 17.47 22.43 13.53 15.53 20.89 9.59 14.20 13.70 5.96 19.03 22.50 8.24 7.58 27.87 28.91 10.31 12.50 31.20 28.48 7.48 15.57 25.90 16.19 14.79 12.13 36.95 11.79 3.97 19.41 10.90 5.62 26.70 11.81 43.02 4.57 39.25 17.94 5.97 4.92 49.47 2.00 26.53 14.43 9.03 32.04 26.22 40.88 35.50 30.00 13.07 29.52 15.38 2.78 8.52 12.26 15.38 24.89 12.52 9.08 25.01 7.12 15.59 21.87 33.90 7.68 19.58 21.97 14.47 15.54 12.62 6.30 8.03 0.42 0.32 29.40 22.70 28.72 14.62 18. Station T (years) Storms intensities (mm/h) 1-h Storms 3-h Storms 6-h Storms 9-h Storms 12-h Storms ie i D ie i D ie i D ie i D ie i D N01 2 5 10 25 50 100 22.39 8.95 32.71 4.79 28.33 14.53 45.16 5.22 2.88 22.72 28.83 0.63 9.11 7.84 9.90 25.55 12.56 15.10 39.27 9.31 26.56 16.04 3.46 26.84 16.86 30.23 21.49 22.27 19.05 11.80 7.22 10.73 32.34 4.12 9.89 15.73 11.03 12.60 27.32 8.57 20.54 30.07 4.81 10.41 20.64 12.82 15.60 19.23 3.91 17.94 37.57 53.84 38.62 18.95 32.92 26.85 13.27 22.31 12.73 22.11 11.48 27.55 11.81 19.62 17.46 36.40 4.31 9.54 11.92 13.31 20.97 22.15 20.68 8.95 20.37 18.55 31.27 28.88 45.10 52.08 18.02 11.76 11.79 14.01 13.05 N02 2 5 10 25 50 100 25.78 8.34 14.77 15.91 8.29 29.57 13.83 16.42 33.61 12.12 15.31 0.52 15.85 62.65 6.78 13.42 5.14 42.23 13.88 8.19 6.83 14.64 9.84 7.86 12.25 27.33 19.97 45.08 9.35 6.83 24.24 10.46 25.84 19.43 50.68 18.34 12.57 9.42 19.35 36.30 39.38 10.82 13.64 5.68 10.89 15.49 20.83 26.67 10.32 13.41 11.61 2.72 32.96 12.36 9.77 10.25 7.47 10.37 39.06 13.33 12.46 10.68 6.91 11.76 63.47 3.22 16.52 26.10 14.12 14.94 29.74 32.02 49.94 W06 2 5 10 25 50 100 24.96 26.69 26.39 11.30 19.01 25.58 19.51 1.67 14.16 22.07 14.14 15.32 32.80 17.80 29.59 21.66 4.60 18.79 17.36 37.58 8.75 16.81 10.04 10.17 13. Ariff et al.64 4.39 18.40 32.96 1.39 16.82 16.48 14.26 30.54 13.74 7.31 2.11 34.75 5.08 8.38 7.53 15.29 14.33 30.88 33.35 16.75 15.37 72.28 2.43 17.56 30.99 8.36 14.72 11.49 52.22 30.54 55.34 6.48 9.15 15.81 0.70 15.19 74.90 6.99 9.56 21.54 28.04 E01 2 5 10 25 50 100 19.07 21.15 7.46 6.02 18.32 8.23 19.95 13.68 12.46 9.58 17.80 12.79 16.20 43.59 11.89 W03 2 5 10 25 50 100 28.30 14.13 17.96 15.89 69.45 23.19 13.11 14.24 10.33 8.64 33.68 11.05 0.51 15.83 0.29 16.34 9.17 5.07 18.39 12.65 47.53 22.71 34.88 11.57 35.39 21.46 23.61 4.08 7.88 8.01 17.64 15.50 31.92 6.21 9.74 45.80 1.02 6.02 26.44 12.21 24.94 9.93 29.71 14.49 34.77 41.11 50.03 9.39 14.89 30.45 17.67 14.30 20.03 18.09 12.69 44.69 4.15 18.52 21.20 29.52 12.89 20.78 16.49 13.24 9.75 15.83 27.64 14.23 38.02 20.20 11.03 11.95 N04 2 5 10 25 50 100 21.86 35.22 30.28 10.86 26.38 20.45 12.85 4.95 3.66 39.95 14.17 43.52 8.47 15.27 8.43 7.65 18.61 39.15 20.18 10.53 6.04 19.26 2.57 9.58 8.77 35.22 24.79 9.64 27.43 46.87 7.38 11.08 27.01 2.96 10.6 16.97 25.16 10.02 4.04 24.42 W04 2 5 10 25 50 100 29.38 4.81 21.94 19.16 10.86 0.24 9.98 18.00 21.62 8.69 18.08 18.39 28.65 20.07 4.93 4.97 18.55 3.50 21.02 10.85 18.77 11.21 9.31 19.33 29.15 24.35 22.59 38.66 19.24 12.49 10.14 16.45 8.38 24.79 18.72 26.70 8.44 14.18 32.08 23.73 65.69 16.62 15.32 10.06 26.36 22.44 50.93 43.36 12.33 18.97 5.35 1.33 13.27 29.18 7.66 11.63 8.29 24.79 12.45 15.83 36.19 20.33 6.48 9.48 20.39 8.42 27.68 30.45 27.07 10.18 10.65 13.89 20.02 23.94 15.23 33.44 10.1 9.26 25.61 19.39 11.59 18.23 8.04 13.52 9.07 45.76 9.45 34.16 3.27 N03 2 5 10 25 50 100 27.66 37.96 34.88 10.52 7.92 39.65 16.54 26.62 42.60 15.48 29.36 9.35 23.94 16.35 12. / Journal of Hydrology 470–471 (2012) 158–171 Table 4 Comparisons between the storm intensities obtained based on the typical IDF empirical formula and the copula method.76 37.30 10.26 33.13 23.17 61.10 15.92 24.55 34.22 14.93 6.88 15.62 19.79 39.26 22.45 9.33 8.32 15.09 15.18 8.69 22.53 15.27 22.43 14.26 14.03 0.58 10.70 3.60 11.37 47.93 18.52 10.09 13.69 22.42 10.02 23.25 58.64 20.01 11.38 28.90 16.32 10.28 9.98 39.01 12.13 8.63 12.45 26.14 9.15 20.83 12.61 18.54 2.77 17.84 24.69 15.88 3.34 6.41 11.65 27.06 26.20 38.82 17.14 8.98 9.55 18.49 11.02 18.50 13.08 24.25 33.99 20.78 3.64 16.32 10.43 14.95 35.68 14.83 13.63 35.90 27.38 13.05 40.11 48.48 43.35 26.96 25.19 10.25 36.05 23.M.68 11.18 58.73 14.63 26.56 9.30 10.75 12.22 13.93 8.01 14.99 6.94 9.13 34.36 19.98 28.165 N.70 16.83 20.46 17.00 32.59 16.60 37.79 42.88 14.36 21.56 18.82 8.31 9.36 5.71 13.08 44.71 11.19 10.55 12.53 18.56 11.67 46.43 16.53 30.44 0.18 22.50 5.67 7.46 23.51 15.61 19.39 18.30 4.77 38.53 9.60 26.57 4.68 25.98 31.31 22.18 36.87 1.38 21.09 36.94 15.64 17.55 10.61 13.11 W05 2 5 10 25 50 100 29.67 37.64 16.02 1.21 3.73 33.69 11.73 34.16 28.74 13.54 18. 30 14.59 22.06 16.21 16.68 41.10 3. ie.25 30.98 17.12 39.53 11.83 22.34 36.99 17.06 13.18 24.36 10.43 1.30 28.68 25.04 20.94 24.26 17.40 30.23 9.25 9.82 32.28 19.40 10.52 5.64 20.14 56.58 7.96 32.50 12. for the storm intensities.50 11.90 8.06 S03 2 5 10 25 50 100 25.32 18.98 2.12 39.27 21.77 5.81 61.59 29.54 20. 10.45 14.08 7.29 36. By solving Eqs.28 15.70 5.58 14. b and c.95 6. 25.72 15.20 18.27 56.03 18.31 12.33 5.41 22.69 12.38 12.81 6.35 22.54 11.67 34.97 15.41 7. a.80 44.18 23.41 5.97 29.30 17.41 45.09 15.77 10.06 25.71 11.62 7.58 20.M. (20).27 20.18 27.28 32.39 11.03 8.81 20.21 34. 3.52 4.51 13.81 6.96 4.32 3.30 0.23 16.65 14.32 39.80 19.82 7.72 18. the storm durations which are taken into account are 1.64 15.02 10. This implies that both copulas are equally suitable in fitting the observed data.00 13.94 9.99 16.17 21.76 30.24 14.54 25.38 7.35 S04 2 5 10 25 50 100 31.45 51.01 9.37 17.55 35.37 21.13 14.95 19.77 10.05 15.09 19.38 11.77 0. j.85 8.47 20.48 4.92 3.71 80.07 16.63 39.96 17.93 20.23 18.37 9.85 11.52 2.05 29.36 21.05 4.42 23.15 42.29 8.07 12.70 21.08 8.62 4.29 49.24 E03 2 5 10 25 50 100 28.47 33.25 9.16 27.81 E06 2 5 10 25 50 100 20.08 24.87 12.06 1.61 34.22 3.76 10.27 62.05 19.54 10.92 18.94 17.62 2.76 25.33 15.24 10.53 9.51 14.41 31.20 6.20 9.43 28.83 27.36 1.42 13.36 6.85 8.78 15.42 23.91 0.85 13.74 12.90 16.25 2.05 6.72 7.07 21.91 5.34 32.60 48.42 13.14 3.14 10.16 7.88 6.72 6.42 10.77 28.02 2.47 17. With this reason and the fact that it is easier to construct an Archimedean copula.95 13.36 15.58 23.37 13.78 76.83 67.79 12.23 15.82 23.94 40. are shown in Table 4 along with the percentage of differences between i and ie.22 21.27 0.83 12.48 19.98 30.21 39.99 12.45 18.03 14. compared to Gaussian copula which falls under the elliptic copula family.37 36.03 15.28 28.83 29.90 14.08 47.84 11.78 13.22 9.26 9.31 15.98 17.84 9.64 18.60 10.01 51.89 5.16 33.12 2.66 9.34 28.59 25.33 15.27 22.65 23.78 5.74 57.20 17.39 5.65 15.04 22.17 32.84 43. / Journal of Hydrology 470–471 (2012) 158–171 Table 4 (continued) Station T (years) Storms intensities (mm/h) 1-h Storms 3-h Storms 6-h Storms 9-h Storms 12-h Storms ie i D ie i D ie i D ie i D ie i D E02 2 5 10 25 50 100 23.68 17.09 29.17 6.22 2. Since.83 12.27 8.84 26. which in this case is the Frank copula.46 3.91 16.166 N.01 0.88 13.47 17.44 12.65 35.70 25.82 41.31 18.60 3.73 4.08 17.00 1. The values for return period.52 16.75 43.49 24.59 15.13 10.52 1.01 18.22 11.56 41.25 53.00 8.81 22.59 24.05 1.24 28.55 29.03 6.91 20.43 13.83 5.90 3.69 19.29 10.98 20.35 10.55 19.91 27.78 41.98 5.38 44.72 26.76 25.29 22.09 11.99 12.40 12.90 16.71 42.83 29.89 8.86 14.39 8.15 32.33 13.45 15.11 22.41 25.93 8.37 50.25 12.73 14.25 15.87 0.01 1.37 24.17 6.74 8.55 69.52 29.30 3.38 7.94 4.55 11.30 25.28 10.15 22.19 48.32 14.68 13.37 21.47 6. this paper focuses on convective storms which is short duration storms.66 32.00 13.06 18.21 16.01 14.25 22.33 31.80 11.80 12.39 16.27 9.92 17.05 20.56 11.66 40.33 42.06 29.97 32.34 6.33 17.63 43.60 35.90 15. 9 and 12 h.19 24.41 31.63 1.55 57.68 37.88 10.08 44.10 0.59 9.26 4.68 31.68 11.28 16.38 15.40 13.88 5.16 9.98 19.19 10.39 49.28 4.55 19.55 6.70 11.21 40.78 17.54 44.59 13.10 13.02 33.58 29.43 33.50 19.23 34.27 29.45 S01 2 5 10 25 50 100 26.71 14. i.96 30.49 8.48 31.19 39.01 19.52 5.66 13.05 13.66 13.66 21.58 10.00 34.69 27.35 6.32 9.51 5.49 65.14 11.54 12. d.02 11.20 12.86 37.95 24.36 8.51 23.87 19.36 28.76 10.10 19.22 21.78 15. The estimated parameters of Eq.61 40.37 12.54 1.57 48.70 5.16 19.82 8.99 37.81 11. The points on the IDF curves are yielded using the conditional Frank copula shown in Eq.78 12.12 4.18 24.94 1.51 9.17 22.98 53.23 6. (4) and (17) using these return periods and storm durations.59 10.96 11.68 21.68 5.13 21.87 35.23 13.25 4.23 4.20 17.25 36.73 33.97 36.08 31. the Frank copula is chosen to represent the twenty rainfall stations in Peninsular Malaysia.09 15. for all twenty stations.69 11.02 18.13 24.63 15.06 29.89 23.71 2.95 10.84 20.64 62.80 4.87 20.34 2.25 24.92 12.18 11.12 16.15 18.43 22.03 43.24 59.06 8.91 7.21 24.31 97.60 19.42 8.01 13.12 23.88 10.49 10.97 21.44 9.19 4.67 20.78 12.15 11.08 30.66 17.17 9.54 11.04 6.51 16.90 6.30 29.00 21.48 41.80 13.62 7.67 53.08 5. 5 provides the visual comparisons between the IDF curves drawn based on Eq.90 0.17 32.07 9.75 28.49 22.35 11.19 14.64 63.46 18.40 31.61 13.78 8.49 8.08 16.03 15.26 6.21 97.45 63.57 7.16 11.67 16.19 13.68 27.49 11.89 20.37 0. and return periods.64 20.87 5. 50 and 100 years.20 4.89 15. .16 21.42 27.22 37.28 10.03 18.77 33.38 2.68 8.66 14.25 10.83 19.82 13.85 8.17 24.46 9.93 7.29 10.73 29.42 10.38 13.04 15. D.43 22.90 20.28 11.76 78.70 28.69 17.29 17.87 17.39 35.69 19.82 E05 2 5 10 25 50 100 20.56 20.72 18.01 33.37 33.29 13.45 33.18 7.19 17.26 31.16 8.66 7.70 12.84 24.67 7.51 12.00 15.83 5.56 9.67 13.44 5.84 9.45 27.00 15.03 6.87 32.96 17. considered in this analysis are 2.16 12.25 4.15 20.17 13.49 7. T.53 16.85 28.99 20.82 5.38 12.33 4.64 20.51 21. T.81 18.54 20.02 18.09 3.50 30.50 18.69 15.89 28.46 27.08 9.01 17.63 49.98 10.91 8.33 19.47 18.20 39.65 11.14 7.63 35.21 15.61 23.34 Gaussian and Frank copula is relatively small compared to the AIC values of individual copula at each station.46 21.00 20.23 24.91 17.52 13.92 35.07 31.81 6.82 20.60 5.61 34.07 15.63 20.06 3.05 52.88 18.61 22.48 22.74 0.92 13.92 29.02 52.52 5. the corresponding intensities.65 22.63 7.93 3.97 92.00 5.28 6.73 13.65 10.43 13.05 8.00 16.23 16.76 25.30 31.56 S02 2 5 10 25 50 100 29.23 17.55 41.09 8.69 36.06 19.20 1.52 2.82 19.75 7.01 20.29 59.07 E04 2 5 10 25 50 100 23.36 10.37 20.83 28.67 6.48 45.71 18.55 27.99 47.67 36.33 7.23 34.83 18.07 6. (17).04 25.89 5.89 7. are obtained and tabulated in Table 4.33 24.40 9.55 24.38 10.36 13.50 7.20 36.28 15.79 8. The values of ie for all storm durations.06 33.87 12.85 31.60 15.92 16.31 28.82 13.51 0.34 27. 5.33 25. Ariff et al. acquired by the typical IDF empirical formula are given in Table 5.62 36.79 17.64 16.18 13.82 21.56 31.68 33.93 17.39 17.81 13.89 7.64 36.24 7.20 23.99 8. Fig.14 55.16 8.34 41.49 18.98 7.55 16.01 10.12 19.36 10.02 31.98 2.39 4.25 46.64 48.15 10.59 11.52 8.93 12.64 26.55 35.93 21.75 6.98 8.67 14.40 2.47 14.28 2.17 32.64 11.09 41.98 10.41 0.06 35.89 10.07 9.65 13.56 0.89 2.25 2.60 16. 6.60 30.61 6.95 12.00 3.12 18.05 15.08 33.65 7.14 25.32 5.25 16.75 10.90 36.27 40.64 8.72 12.75 11.59 24.30 17.23 18.19 15.95 13.97 12.99 10.53 15.70 24.18 32.18 15.87 24. Table 6 also indicates that the size of the range for IDF points of storm intensities obtained from the copula method. 8. T.55 0. 5.69 0. the storm events which are used in both methods are the same since we consider similar definition of storms in both analyses.54 0.00 36. between storm intensity and duration as well as estimating the copula parameter. 50 and 100 are 11.51 0. The storm intensities of each category are assumed to have storm duration equal to the value which represents them. Eq. values of ie are mostly higher than i for all values of T and d.00 1. however. i and D for each d and T are given in Table 6. the range’s size of i decreases as the storm duration increases.97 31.49.15 mm and 12.67 47. 71%. 50 and 100 years.00 1.87 (20) and the points of IDF curves obtained using the copula method. 3. The IDF points found from the latter method considers properties of actual storm characteristics such as storm intensities. we can regard that the IDF points found by the copula method is somewhat in agreement with the IDF points obtained based on the typical IDF empirical formula. Meanwhile.44 mm and 7.00 0. the average range sizes for i are 28.05 and 37.51 0.00 1.19 0.53 mm. i. Briefly. 0.63%. 13. Furthermore.66 0.16 mm.21 0. Nevertheless.00 1. The only exception is the station Kota Tinggi (S03).00 1. 1.22 0.26 24.00 1.90 mm.25 0. 34. 9. 11.00 1.51.24.12. The differences in percentage are largest for the 2-year return period of most storm durations for all stations with the mean of D is 23. 66% and 59% of D respectively.85%. 50 and 100 years are 7.73. half of the values for D are less than 10 with most of them are from storm intensities for the return periods 10. less than 50 for most storm durations and return periods considered in this study. The IDF curves are then build by approximating the parameters a.71 0. 27.22 0.24 mm. more than 50% of D for these four return periods are less than 10 with each recording 66%. 7.20 0. 1.36 mm. a small error in the intensity will translate into significant error in the differences (Singh and Zhang.00 1.62%. 5 and 10) and higher (T = 25. 8.57 0. while the average of the minimum D for storms with 2-year return period is 9.23 0. depths and durations.44 24. The average values of the range size for i with respect to return periods 2.49 0.23 0.21 0. 25.58 0. where the percentage difference of the return period of 50 and 100 years for 1-h storms are 56.05 mm and 21.69. / Journal of Hydrology 470–471 (2012) 158–171 Table 5 Parameters a.23 0. the average of the maximum D is about the same for T = 2 and T = 100 years which are 38. by calculating the correlation.96.29 1.23. 54% of D for T = 5 years is in the range of 10–20. and return period.00 1.03.52 0. Table 6 shows that although the percentage of difference between i and ie are smaller for higher return periods.96.12 25.79 37. The typical IDF curves using empirical derivations have not much theoretical background.22 0. This downward trend is not so prominent for ie with the average values for all stations and return periods are 28.27 32. D.50 29. IDF points obtained based on the copula method are supported by a sound copula theory. 50 and 100) return periods.53 0. 8.00 1.22 0. the copula method fit probability distributions to represent storm intensity and duration for each station.e.28 respectively. The difference usually increases as the storm duration increases from 3-h to 9-h and drops at 12-h storms.26 mm and 26. 50 and 100 years. Ariff et al. 12.76 0.21 0. T = 25.57 42. 6. D. j.N. 9 and 12 h respectively.26 and 34.56. 2007). are slightly larger compared to those from the 167 typical IDF empirical formula.10 mm.18 0.99 mm.23 for higher return periods.12 37.00 1.68 mm.81.77 32. This is most likely due to the fact that although the two methods handle the data differently.96.00 1.64% and 63. the copula theory can be applied to obtain the joint distribution for the two random variables in copula form. On the other hand. 20. 0.M.72 0.26% and 10. It can be observed from Table 4 that stations in Peninsular Malaysia have values of percentage difference between i and ie. 3. since most of the differences are small. 10. the storms which are categorized with respect to the storm duration for the typical IDF empirical formula are analyzed separately to get the design rainfall intensity for various return periods.05. the results of both methods which are the storm intensities i and ie are consistent because both methods are different storm models which represent the same storm events. 9 and 12 h.65 0. the storm duration is considered as a continuous random variable in the copula method while it is used as a discrete and fixed value in the IDF empirical formula with representative values are used to define groups or categories of storm intensities. However.00 1. is also not apparent. 10.54 41. 12.21 0. s.78 mm.00 1.24 0. 32. all the storms are combined irrespective of the storm duration to obtain the joint distribution for storm intensity and duration.51 mm.91 and 1. 25.48 40.71 31. The range for both i and ie increase as the return period increases for all d hour storm durations. i.24 mm.00 1.52 mm. 6. In short. The IDF points are then yielded based on the conditional distribution of storm intensity given storm duration d. In a way.52 0. 10.49 mm.72 for smaller return periods and 31.21 mm. It can be seen that the average of the minimum values of D are very small for T = 5. Eighty three percentage of D are less than 20 which implies that the percentage of difference between i and ie are usually small. the mean and standard deviation of each category of storm intensities with fix duration d are computed separately in order to obtain ie for each storm duration d and various return periods T. these results in the difference between the size range of D for smaller (T = 2. d. j.33 and 28. The averages for the rest of the maximum values of D are 29. Furthermore.43%. the reason behind the pattern for the percentage of differences. the sizes of their range are in average larger than those for T = 2. the typical IDF empirical formula regards the storm intensity.00 1. 5 and 10 years. The average sizes for the range of D are 29.23 0.00 1.61 and 38. ie.63 25. 25.60 mm for d = 1. The mean of the percentage difference between i and ie for return period 5%. As for the typical IDF empirical formula.53 0. (20).21 mm while the corresponding values for ie are 7. 29. 13. In fact. This can be seen from the mean values of D according to storm duration 1. The range of ie. 16.21 0.22 0. Station a j b c N01 N02 N03 N04 W01 W02 W03 W04 W05 W06 E01 E02 E03 E04 E05 E06 S01 S02 S03 S04 28. Thus.77 40. b and c of the typical IDF empirical formula. 10. 3.81 and 13.78 mm. as a function of fix storm duration. 11.74 0. Hence. 8. 6. 10. It can be seen from the results in Tables 4 and 6 that the differences between the storm intensities obtained from either the copula method or the typical IDF empirical formula does not follow any particular pattern. 50 and 100 years. 25. 17.25 28. As a consequence to that.51 28. Generally.70. 12. 9 and 12 h which are 12. ie. The difference is that the storm intensity and duration in the copula method are both regarded as random variables while only the storm intensity is taken as a random variable in the IDF empirical formula.37 0. b and c through the nonlinear least squares method performed on the Sherman equation. Then. the copula method is able to provide the . This is believed to be due to the small value of storm intensity.00 1.54 0. h. For the copula approach. 11.00 1. 10. For storm duration 1.98% respectively.25 mm. it is found easier to perform the copula method since it involves the combination of storm intensities irrespective of storm duration as opposed to the IDF empirical formula which requires storm intensities to be categorized with respect to storm duration.mm/h 10 50 25 100 50 10 5 2 10 100 10 60 60 50 30 40 20 50 0 Intensities. have to be analyzed individually to obtain the mean.h 2 0 15 5 0 60 5 2 5 10 50 15 0 5 60 5 2 10 Durations. Thus.h 100 30 40 Intensities.mm/h 2510 100 10 20 50 15 E02 E01 100 10 Durations. 5. 25. T = 2.h 0 50 25 30 40 10 20 Intensities. Comparisons between the IDF curves obtained from the typical IDF empirical formula and IDF points from the copula method. . No extra computation is needed for the copula method but more categories of storm intensities.h Fig.mm/h 10 100 20 50 30 40 50 20 Intensities. 5.mm/h 10 Durations. standard deviation and design rainfall intensity of each category.mm/h 50 30 40 20 Intensities.h 60 10 20 30 40 50 60 2 Durations.168 N. Note: The numbers at the beginning of each curve indicates the respective values of return period. the typical IDF empirical formula will take considerably more time and is less efficient compared to general form for the joint distribution of storm intensity and duration which is more informative and pertinent for further analysis.mm/h 100 25 N03 N02 N01 0 5 10 Durations.h N04 0 2 10 20 5 10 5 0 0 0 25 30 40 10 20 2 Intensities.M. corresponding to the fix storm durations considered. inadvertently the copula method lessen the computational effort for future analysis compared to the IDF empirical method which does not provide additional information on storm events and any further research 60 5 10 15 0 5 Durations.mm/h 10 5 0 50 15 W04 50 30 40 10 20 50 25 5 2 Durations. Ariff et al.h 15 0 5 10 15 Durations.h Intensities.h 10 60 100 25 10 0 0 10 50 60 5 50 30 40 Intensities. for a large set or range of storm duration.h 10 25 10 5 5 10 60 60 50 30 40 50 0 Intensities.mm/h 0 15 W01 Durations.mm/h 50 15 15 10 60 0 10 100 50 25 10 W05 0 10 20 2 5 5 Durations. Hence. computing the moments and producing the quantile functions of storm characteristics could be performed based on the probability statement given by the general form of the joint distribution between storm intensity and storm duration.mm/h 100 W03 Intensities.mm/h 10 20 5 2 15 W02 100 25 10 Durations. 10. 50 and 100 years. Moreover.h 15 50 100 50 30 40 Intensities.mm/h 50 25 10 25 2 10 5 0 0 0 5 2 50 30 40 Intensities.h W06 5 10 5 0 0 Durations. Hence. other analysis on storm events such as obtaining the conditional distribution of storm duration given storm intensity. / Journal of Hydrology 470–471 (2012) 158–171 would have to be done separately. (17) and (9). Eqs.169 N. the Frank copula has an additive advantage since it is a member of Archimedean copula family which can be easier constructed.mm/h 10 20 30 40 50 60 50 25 10 0 Intensities. the Frank copula is preferable in hydrological analysis compared to the Gaussian copula which falls under the elliptic copula family.h 10 15 Durations. The values of Kendall’s s obtained imply that the dependence level between I and D are neither very high nor very low.h Durations. The storm intensity. However. The conditional distribution of Frank copula is also less complicated compared to the conditional Gaussian copula. There is no exact pattern for the differences . the nonlinear least squares method applied under the IDF empirical formula could sometime fail to approximate the parameters of the Sherman equation used in the analysis if the initial values provided are unsuitable.mm/h 100 0 2 10 20 30 40 50 60 5 0 Intensities. finding the IDF points with the copula method prove to be more advantageous for researchers and hydrologists to extend their understandings of storm events. Singh and Zhang. Thus.mm/h 25 10 50 0 15 10 60 50 30 40 50 0 Intensities. are negatively correlated which implies that shorter storms have higher intensities. The Frank copula is deemed appropriate to represent the relationship between storm intensity and duration in Peninsular Malaysia based on the AIC values which are mostly smaller than the AIC values of Gaussian copula. It is found that both copulas results in relatively similar AIC values when used to fit the relationship between storm intensities and durations. 8. and duration. D. several conclusions are drawn.h Intensities. Based on this relationship.mm/h E05 E04 E03 100 0 5 10 15 0 5 Durations. 2007).h Fig. The difference between the IDF curves obtained based on the typical IDF empirical formula and the points on IDF points acquired using the copula method depends on the given storm duration d and return period T. I. This finding agrees with previous literatures done in other countries (Kao and Govindaraju. Hence.h 25 10 5 0 10 20 5 2 100 20 Intensities.mm/h 50 40 25 10 100 10 20 30 40 60 S03 30 10 Durations.h 0 25 10 10 5 50 30 40 5 100 20 25 10 10 Intensities. rainfall variables such as intensity and duration are related to each other. Gaussian. However.mm/h 50 0 Intensities.M. (continued) the copula method. Ali–Mikhail– Haq (AMH) and Frank copula. The four copulas are the Farlie–Gumbel–Morgenstern (FGM). 5.h 100 20 100 50 30 40 30 5 15 20 40 25 10 10 Durations. four copulas which are commonly applied in hydro- logical analysis are considered in this study. only the Frank and Gaussian copula have domains of s which are suitable for analyzing the storm events in Peninsular Malaysia and these two copulas are further considered in this analysis. In addition to that.mm/h 60 50 50 Durations. The IDF curves currently available in Malaysia are done using the univariate analysis and empirical formula. Conclusions Intensity–duration–frequency (IDF) curves are important especially for a country like Malaysia where flood is regarded by the Ministry of Natural Resources and Environment as the most significant natural hazard. Ariff et al. Based on this study.h 5 2 10 50 25 5 2 0 0 10 50 60 S04 Intensities. 2007.mm/h 10 S01 100 5 5 2 Durations. Hence. / Journal of Hydrology 470–471 (2012) 158–171 60 50 2 10 0 15 5 E06 60 S02 2 0 10 15 0 5 50 5 2 15 0 5 10 15 Durations. This paper derives IDF points using bivariate frequency analysis by utilizing the copula method. 64 32.05–24.. M. A Generalized Pareto intensity–duration model of storm rainfall exploiting 2-Copulas.15 15.30–20..C.. Palynchuk. A. In conclusion.62–62. R. 44 (2).. Hydrol. 2006. Suhaila. Res.27 2. Evin.94 5. A.94–34.70–36. A.96 11.18–53.59 13. UKM-ST-06-FRGS0181-2010 and UKM-GGPM-PI-028-2011. Kao.C. S. the percentage difference.52–47.R.C.77 0. Furthermore.95–31. M.73 2..M.90 0. M. Asian Pacific FRIEND.. 40 (1). 93–102. These differences are very much influenced by two monsoon seasons. there are great benefits in using the bivariate copula method as opposed to the univariate IDF empirical formula in practice.45 15. Rosso.61 26. Thomas. bivariate frequency analysis of storm events is imperative for hydraulic applications in order to represent the complexity of actual hydrologic events.31 3. Stat.16 18. This is due to the fact that all storms are combined irrespective of the storm duration for the copula approach and thus no extra computation is needed for analysis while the storms are categorized with respect to storm duration for the typical IDF empirical method and each storm category has to be analyzed individually. Multivariate hydrological frequency analysis using copulas.81–23.18 16.05 29. W. 2003. Geophys. W06410.66–15. Musy. B. W01101.05 1.07–18.40–29. 1307..30–63.57–31. 118–135. 2006. Bobée. Manetas. S.44 18.17–46. A. De Michele.69 2. Thus. 228 (1–2).S.45–22. 10.A. Kozonis.61–11.95 4.64 12. Applied Hydrology.61 0. Perreault..87–50.36–19.27–11.. 897–912.Z. 49.W.09–15.43–36.54–32. Cordova. Mathur. Guo.42 14..03 28. 2007. 4067.18–52. S. J.69 9.12–25. Maidment.. 56–67. Bayesian copula selection. Y. which occurred in Peninsular Malaysia from November to March and from May to September each year.. A derived flood frequency distribution for correlated rainfall intensity and duration.61–32. Goel.. Finance.85–16. J. Y.13–29.65–18.34–65. R. N.27 1.49–19.90–33. W. Salvadori. J.T.N.64 0.71 23.. d (hour)/T (years) 1 3 6 9 12 2 5 10 25 50 100 19. Probabilistic structure of storm surface runoff considering the dependence between average intensity and storm duration of rainfall events. C. 814–861. Applied statistics for civil and environmental engineers. A mathematical framework for studying rainfall intensity–duration–frequency relationships.68 10. B.90 13.20 10. R.95–24.55 0. 1958.. Threshold analysis of rainstorm depth and duration statistics at Toronto.S. Hence. S.44–39. It is believed that modelling the real events of storms will have a more significant contribution on current hydrologic designs which are mostly based on the univariate analysis approach.T.85 0. K. A.75–31.46–27. This is probably because both storm intensities represent the same storm events since similar definition of storms are considered in both analyses.. D.27 6. Spatial trends of dry spells over Peninsular Malaysia during monsoon seasons. N. Climatol. Thiemonge. Res.. R.91 6. D. W02415.02 11.64–35. J.12 24.96 24.. Data Anal.57 20...70–10.67 5. Takara. The typical IDF empirical formula does not have much theoretical background while the IDF points based on the copula method are supported by a sound copula theory. Ordinal measures of association. 2005. Utmost appreciations to the Ministry of Higher Education (MOHE) and Universiti Kebangsaan Malaysia (UKM) for the allocation of research Grants. 30 (4). Petaccia.K.89 9.. 2008.56–33.R.D. Kottegoda. Malaysia.170 N. Trivariate statistical analysis of extreme rainfall events via the Plackett family of copulas. I.H.71 41.39–41...79 6. 43 (6). Theor. J. An Introduction to Copulas.. Deni. 809–822. Bivariate statistical approach to check adequacy of dam spillway.M. Urban Hydrology and Hydraulic Design... Two-site modeling of rainfall based on the Neyman–Scott process.95–28. 2008. Water Resources Publications. 2006. the Northeast and Southwest Monsoon.23–28.14–23.40–29..48 1.B. 755–763. Morgenthaler. Water Resour... Wiley-Blackwell.27–30. Nelsen. 38 (12).M. Rodriguez-Iturbe.20 19... Hydrol. D.S.37 19.01–26. J. Appl. Water Resour. Govindaraju. Renard..59–22..84 10.55–28. 2002. Dis.71–20.23 20.46–35.27–29. R. J.12–27.82 7. between them also varies from one region to another. G.38–18. L. Springer Verlag.64 13. Acknowledgements The authors would like to thank the Department of Irrigation and Drainage Malaysia for providing hourly rainfalls data for the use of the study.72–28.25–56.19–80. Eng.58 8. 2010.88 11.87–33. Res.C... Desa. A.23–25. 53– 57. Hydrol. The storm intensities yielded based on both methods indicate that they depend on the geographical locations of the rainfall stations since the distribution of storm intensity and its pattern vary for each region.75–41. Prev. Birkhauser. Res. 357–371.34 Range of storm intensities (mm) ie but they are generally small.83 18.18 0.78 24. storm intensities found based on both methods are believed to be in agreement with each other.50 0.99 0. 2007.16–13. B..97 17. Use of a Gaussian copula for multivariate extreme value analysis: some case studies in hydrology. Hence.02 0. McGraw-Hill. 1985. Ann. A. / Journal of Hydrology 470–471 (2012) 158–171 Table 6 Range of the storm intensities based on the typical IDF empirical formula and the copula method as well as their percentage differences. D. R. The regionalization of storms in Peninsular Malaysia as well as their characteristics will be included in future papers. 21 (5).45–31. Canada. Kao.04 15... R.22–16.49 35..39 13. Statist. Hydrology and Other Fields..54 7. New York. Ariff et al. Am.00–97. J.47–43. Water Resour. Water Resour.88–13. 348 (3–4).96 1.09–32.98 33.75–20.. .86 11. R. Daud.52–25. Res. 206 (1– 2). Salvadori. S..10–97. L.S. 108 (D2).84 0. 2007.M.54–18. G. Wan Zin. Res. 50. Reiss.47–22. Adv.77–24. Guo. M.. 2000. Tachikawa.88–31.65–15.32 i 2 5 10 25 50 100 15.99 0.Z..37 28..M. Hence. Assoc.90 16.16–39. Water Resour.Y.61 10.82 0. 99 (3). Chow..34–32. Favre.51–13. N. Comput.07–21. El Adlouni. 2008.32–42. the copula method is more meaningful for researchers and hydrologists to extend their study on storm events. Favre. Kurothe. Inst. Hydrol.56 8.43 0.78 36.21 48. J. M.67 9. Kruskal. Favre. L.74 22. Establishment of intensity–duration– frequency curves for precipitation in the monsoon area of Vietnam. 535–545. References Amin. B. 572p.M.28–30.98 10. Mays.34 0.30–78.69–41.85–31. D. Nhat.60 4.. 2008. C.. Huard.C..M. Canossi. V.. This may be caused by the different characteristics of storms in each region.. Z.66–19.. 1988. De Michele. 51 (2).57–32.63 D 2 5 10 25 50 100 5. Jemain..18 13. copula method is found to be easier to perform for a large set or range of storm duration compared to the IDF empirical formula.82 10. Koutsoyiannis. Govindaraju. 1998. 2006.17 2.05–30. Res. LLC.64–34. Lang. 2004. Although the two methods provide consistent results. G.62–34..C.. The copula method is also worthwhile to be used in storm analysis since it provides the general form for the joint distribution of storm intensity and duration which is pertinent for further analysis. Vogel. the typical IDF empirical approach is less efficient and more time consuming.31 39.23 14. On the probabilistic structure of storm surface runoff. Rosso.25 27.45 0. Statistical Analysis of Extreme Values: With Applications to Insurance.49–24.. Water Resour.00–32. 2007. Ministry of Natural Resources and Environment. Appl. G.. 368 (1– 4).. Deni.A. Wan Zin.B. Bobee. Hydrol. J.M. . T. Jamaludin.D. pp. Wan Zin. 303–314. 96 (3). 2011.. J. IDF curves using the Frank Archimedean copula.. 11. J.. 2007. 332 (1–2). 2009. The best fitting distribution of annual maximum rainfall in Peninsular Malaysia based on methods of Lmoment and LQ-moment. A.W.. Res. Sains Malaysiana 38 (5). A. Recent changes in extreme rainfall events in Peninsular Malaysia: 1971–2005. W. 99 (3). Zhang. 2001. Jemain.. A... W12511.. S. M. Publ. Investigating the impacts of adjoining wet days on the distribution of daily rainfall amounts in Peninsular Malaysia. W. 171 Suhaila. C. Flood and Drought Management in Malaysia.M. 12.. Climatol. Jemain. 1–18.M. 197–206.. Comparing rainfall patterns between regions in Peninsular Malaysia via a functional data analysis technique. 2003.. Spatial patterns and trends of daily rainfall regime in Peninsular Malaysia during the southwest and northeast monsoons: 1975–2004. 2000. Joint probability distribution of annual maximum storm peaks and amounts as represented by daily rainfalls. Regional analysis of annual maximum rainfall using TL-moments method.. Frequency analysis via copulas: theoretical aspects and applications to hydrological events. A. W. Environmetrics 13 (8).. 2010.. A review of bivariate gamma distributions for hydrological application. S. Phys. Theor. Return period of bivariate distributed extreme hydrological events. S. Ouarda. 2009b.Z. L.. L. Hydrol. 411. T. 811–819. Environ. Deni. P. A.. Bivariate rainfall frequency distributions using Archimedean copulas.A. Suhaila. J. 315–326. N. W... Eng.Z. 1–10.. J. J. 751–760. J. 337–344. 2006. Jemain. Res. Trends in Peninsular Malaysia rainfall data during the southwest monsoon and northeast monsoon seasons: 1975–2004. Risk A 17 (1).. Daud. Zin.Z. B. V. 2010b.A. Wan Zin. 150. J.Z.M. 2009a. Ibrahim.T. 651–662. Zhang. Jemain. Sulaiman.Z. Bivariate flood frequency analysis using the copula method. K.. Singh. 55 (1–4).. 40 (12)..H. A comparative study of extreme rainfall in Peninsular Malaysia: With reference to partial duration and annual extreme series.. Meteorol.. 42–57.. Shiau. J. Climatol. A.F.. 2007. A. Yue. W. 93–109. Theor. Eagleson.. Atmos. A. Theor. 45 (2). Hydrol. Identification of independent rainstorms. 2004. Suhaila.. / Journal of Hydrology 470–471 (2012) 158–171 Restrepo-Posada. A. S. M. 17–25. Sains Malaysiana 39 (4). 11. Schmidt. Hydrol. Fonctions de repartition a n dimensions et leurs marges. Ariff et al.N.P. Jemain. S.A.. De Michele. Water Resour. S.M. Ariff. P. Singh. Eng. Yue. Appl... Stat.A. The bivariate lognormal distribution for describing joint statistical properties of a multivariate storm event. Ibrahim. Wan Zin. Hydrol. K. 1–18.A.. Suhaila.A. J. Inst. Appl..S... A.J... Hamdan. Sklar. Deni. 4–5. Singh. 303–319. Chapter Forthcoming in Risk Books: Copulas from Theory to Applications in Finance. Paris 8 (1). 533–542. 2011.. 1982. Hydrol. 246 (1–4). W.M.P.. Coping with Copulas. Salvadori.. Zhang. Jemain. Sayang. L. Sci.Z. J. Wan Zin. V. J. Hydrol. Stoch.. V. Suhaila. Univ. S. 2006. J.. Z. Yue.. Speech Text.P. J. 2010a.. Hydrol... Climatol.. 2002. Shabri. 1959. Jemain.