Journal of Hydrology 470–471 (2012) 158–171

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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 s u m m a r y

Article history:Received 2 April 2012Received in revised form 6 July 2012Accepted 22 August 2012Available online 31 August 2012This manuscript was handled by AndrasBardossy, Editor-in-Chief, with theassistance of Sheng Yue, Associate Editor

Keywords:Intensity–duration–frequency (IDF)CopulaStorm eventBivariate frequency analysis

0022-1694/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.jhydrol.2012.08.045

⇑ Corresponding author. Tel: +60 603 89215784; faE-mail addresses: tqah@ukm.my (N.M. Ariff), a

kamarulz@ukm.my (K. Ibrahim), w_zawiah@ukm.my1 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.

Intensity–duration–frequency (IDF) curves are used in many hydrologic designs for the purpose of watermanagements and flood preventions. The IDF curves available in Malaysia are those obtained from uni-variate analysis approach which only considers the intensity of rainfalls at fixed time intervals. As severalrainfall variables are correlated with each other such as intensity and duration, this paper aims to deriveIDF points for storm events in Peninsular Malaysia by means of bivariate frequency analysis. This isachieved through utilizing the relationship between storm intensities and durations using the copulamethod. Four types of copulas; namely the Ali–Mikhail–Haq (AMH), Frank, Gaussian and Farlie–Gum-bel–Morgenstern (FGM) copulas are considered because the correlation between storm intensity, I, andduration, D, are negative and these copulas are appropriate when the relationship between the variablesare negative. The correlations are attained by means of Kendall’s s estimation. The analysis was per-formed on twenty rainfall stations with hourly data across Peninsular Malaysia. Using Akaike’s Informa-tion Criteria (AIC) for testing goodness-of-fit, both Frank and Gaussian copulas are found to be suitable torepresent the relationship between I and D. The IDF points found by the copula method are compared tothe IDF curves yielded based on the typical IDF empirical formula of the univariate approach. This studyindicates that storm intensities obtained from both methods are in agreement with each other for anygiven storm duration and for various return periods.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Statistical analysis of extreme data is important in various dis-ciplines including hydrology, engineering and environmental sci-ence (Reiss and Thomas, 2007). By performing extreme analysison rainfall data, damages to human lives and properties as the con-sequences from extreme rainfall events such as flood or landslidesmay be reduced or prevented. In Malaysia, extreme analysis onrainfall data has been explored for all sorts of purposes such astracing patterns and trends of daily rainfall during monsoon sea-sons (Suhaila et al., 2010a,b), detecting recent changes in extremerainfall events (Wan Zin et al., 2010) and fitting probability distri-butions to annual maximum rainfalls by implementing variousmethods (Shabri et al., 2011; Wan Zin et al., 2009a,b).

The rainfall intensity–duration–frequency (IDF) curves areessential tools in designing hydraulic structures such as dams,spillways and drainage systems. These hydraulic structures helpto lessen the loss caused by extreme rainfall events. Hence, for

ll rights reserved.

x: +60 603 89254519.zizj@ukm.my (A.A. Jemain),(W.Z. Wan Zin).

countries such as Malaysia where flood is considered as the mostsignificant natural hazard (Sulaiman, 2007), correct rainfall estima-tion as points on the IDF curves is important. According to Kout-soyiannis et al. (1998), IDF curves are able to show themathematical 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 univariaterainfall frequency analysis approach because of its mathematicalsimplicity (Singh and Zhang, 2007). The univariate approach ofrainfall frequency analysis is explained in detail by Chow et al.(1988). In addition, most of the IDF curves are constructed usingthe window-based analysis approach where the durations are pre-determined time intervals. Thus, the ‘durations’ do not representthe actual durations of rainfall events. In other words, a smallerwindow may be a subset of a longer extreme rainfall event whilea bigger sized window could contain several short duration rain-falls and dry periods.

Several attempts have been made to derive joint distributions ofrainfalls characteristics, namely; intensity, depth and duration, astheir random variables. This is to accommodate the complexityof hydrological events. It has come to light that events such asstorms and flood always appear to be multivariate and thus, sin-gle-variable frequency analysis can only provide limited assess-ments of these events (Yue et al., 2001). In earlier studies, severalassumptions have been made due to the mathematical difficulties

Table 1The locations and completeness of rainfall data for stations in Peninsular Malaysia.

Region Code Station Latitude Longitude Completeness ofdata (%)

Northwest N01 Alor Setar 6.116 100.356 97.7N02 Bukit

Bendera5.425 100.269 99.2

N03 Jeniang 5.817 100.633 96.7N04 Sungai

Pinang5.404 100.217 99.0

West W01 Bertam 5.144 102.048 90.9W02 Genting

Klang3.204 101.722 99.0

W03 GuaMusang

4.860 101.963 95.5

W04 KalongTengah

3.438 101.658 98.4

W05 Kampar 4.346 101.157 97.2W06 Teluk

Intan4.026 101.021 95.6

East E01 Dungun 4.756 103.400 97.5E02 Endau 2.641 103.661 97.7E03 Kampung

Dura5.060 102.934 93.0

E04 Kemaman 4.233 103.420 96.2E05 Kepasing 2.967 102.867 93.0E06 Paya

Kangsar3.900 102.430 96.8

Southwest S01 Chinchin 2.290 102.474 99.7S02 Johor

Bahru1.463 103.755 96.9

S03 KotaTinggi

1.733 103.720 95.1

S04 Labis 2.395 103.017 94.4

N.M. Ariff et al. / Journal of Hydrology 470–471 (2012) 158–171 159

in obtaining these joint distributions using standard statisticalmethods. One of the assumptions is the independence assumptionbetween the random variables. However, Cordova and Rodriguez-Iturbe (1985) have proven that this assumption is inappropriateand unrealistic. Their study showed that the correlation betweenrainfall duration and its average intensity gives non-negligible ef-fect on storm surface runoff. Hence, following this discovery, mostrainfall models incorporated the dependence between variableswith some limitations imposed on the marginal distributions ofthe random variables. For instance, the marginals are assumed tobe either normal (Yue, 2000) or possess the same type of probabil-ity distribution such as bivariate exponential (Favre et al., 2002;Goel et al., 2000), bivariate Gamma (Yue et al., 2001), bivariate log-normal (Yue, 2002) and bivariate extreme value distribution (Shi-au, 2003). In reality, extreme rainfall is a complicatedphenomenon and its marginal distributions are not necessarilysimilar or distributed as normal. Other distributions should be con-sidered which may produce better rainfall estimates.

The copula approach is a flexible method that allow morechoices of marginal distributions and dependence structures tobe used in multivariate problems (Kao and Govindaraju, 2008).Copulas are based on Sklar’s theorem (1959) which states thatfor joint distribution, the analysis of the marginals and the depen-dence structure can be done separately. A thorough explanation onthe theory and description of copula is given by Nelsen (2006). Var-ious types of copulas have been used in hydrology over the lastdecade. Among them are the Farlie–Gumbel–Morgenstern (FGM)copula (Favre et al., 2004), the elliptic copulas such as the Gaussiancopula (Renard and Lang, 2007) and the Archimedean copulas (DeMichele and Salvadori, 2003; De Michele et al., 2005; Salvadori andDe Michele, 2004; Zhang and Singh, 2006).

In Malaysia, although the IDF curves play important roles, theonly ones utilized are those obtained through the univariate fre-quency analysis. This paper aims to derive the bivariate or the jointdistribution between intensity and duration of extreme stormevents in Peninsular Malaysia using copulas. Then, the points forIDF curves are derived from the copula obtained and comparedwith the IDF points found based on the empirical formula usedin univariate analysis.

The first section of this paper contains the introduction of thisresearch, followed by a section describing the data used for analy-sis and the definition of storm events. The third section comprisesof the explanation on copula method and the construction of IDFpoints using the copula method. Section four provides the variouscopulas under consideration. The different probability distributionfunctions used in this study are shown in section five. In sectionsix, the construction of IDF curves using the typical empiricalmethod is discussed. The seventh section contains the computationof IDF points based on the copula and empirical methods on thehourly rainfalls data in Peninsular Malaysia and section eight pro-vides the conclusion of the research.

2. Data and definition of storm

Hourly rainfall data obtained from stations in Peninsular Malay-sia are acquired from the Department of Irrigation and DrainageMalaysia. The stations are located in four different regions scat-tered across Peninsular Malaysia. These regions are based on theirgeographical locations as suggested by many previous researches(Deni et al., 2010; Suhaila and Jemain, 2009; Suhaila et al., 2011).The regions are known as the Northwest, West, East and Southwestregion of Peninsular Malaysia. Most rainfall stations are located atthe edge since at the centre of Peninsular Malaysia lies the Titiw-angsa Mountain range which is mostly unoccupied. In this study,twenty stations are selected based on their locations and com-

pleteness. These stations are located both at the edge and the mid-dle of the Peninsular. Data from these stations are more than 90%complete for the year 1975–2008 as presented in Table 1. The loca-tions of these 20 stations are shown in Fig. 1.

The definition of storm-event depends greatly on the inter-event time definition. The inter-event time definition (IETD) is de-fined as the minimum duration of dry period between two consec-utive storm events. Hence, the dry duration between twoindividual storm events must at least be equal to the IETD value.If not, they would not be considered as two different events butparts of the same storm. The IETD value is chosen such that the se-rial correlation between the two different storms is minimized(Restrepo-Posada and Eagleson, 1982). For small urban catch-ments, the IETD is usually taken as 6 h because the time concentra-tion of rainfall which is less than 6 h would make the runoffresponse of successive storms to appear independent (Palynchukand Guo, 2008). Storm depth is defined as the accumulated rainfallwhich begins and ends with at least one wet hour and either con-tains dry periods with less than 6 h or none at all. Storm duration isdefined as the time interval for a storm event and storm intensity isthe ratio of storm depth to storm duration.

The information extracted from the rainfall data is the annualmaximum storm intensity for several storm durations. This paperfocuses on the convective storms for the purpose of IDF curves con-struction. Convective storm is defined as short duration stormwhich has great impacts on small or urban catchments (Palynchukand Guo, 2008).

3. A brief introduction to copula

Before we proceed further, we will introduce the notations usedin this paper for simplicity purposes. Uppercase letters (example: Xand Y) are defined as random variables and lowercase letters

NORTHWEST

WEST

SOUTHWEST

EAST

Fig. 1. Map of Peninsular Malaysia and the locations of the twenty hourly rainfall stations under consideration.

160 N.M. Ariff et al. / Journal of Hydrology 470–471 (2012) 158–171

(example: x and y) are realizations or specific values of randomvariables. For a bivariate distribution, the joint distribution func-tion of a pair of random variables (X,Y) is denoted as HX,Y with mar-ginals FX and FY respectively. If we let U and V be the randomvariables of these marginals, then U and V are uniformly distrib-uted; U = FX(X) � U[0,1] and V = FY(Y) � U[0,1]. Thus, u = FX(x) andv = FY(y) are realizations of U and V respectively.

A copula is usually described as either a function which links amultivariate distribution function to its marginals (cumulative dis-tribution functions) or a function composed of uniformly distributedmarginals in [0,1] (Nelsen, 2006). Copula is defined according toSklar’s theorem (1959) which states that for continuous randomvariables X and Y with joint distribution HX,Y and marginals FX andFY, there exists a unique copula, CU,V, such that for all x, y in R,

CU;V ðu; vÞ ¼ PðU 6 u;V 6 mÞ ¼ PðX 6 F�1X ðuÞ;Y 6 F�1

Y ðmÞÞ

¼ HX;Y ðF�1X ðuÞ; F

�1Y ðmÞÞ ¼ HX;Y ðx; yÞ: ð1Þ

In a way copula helps to map random observations (x,y) from R2

to a bounded domain [0,1]2. Copula reduces the complexity ofderiving the joint distribution of random variables X and Y. Unlikethe traditional bivariate joint distribution, the marginals of the ran-dom variables and the dependence between them can be deter-mined separately one at a time.

The construction of one-parameter copula can be summarizedinto four simple steps. For this study, the copula parameter is de-noted as h. Once the copula family and function to be used is iden-tified, the copula can be obtained by first finding the marginaldistribution functions U and V. This is done by fitting probabilitydistribution functions to random variables X and Y. Thus, all theparameters for the two probability distributions are approximatedand their respective distribution functions, U and V, can be ob-tained. Next, the dependence between the random variables is cal-culated. In this paper, the correlation between random variables xand y is yielded using Kendall’s s. For N paired (x,y) observations,Kendall’s s is estimated as (Kao and Govindaraju, 2007)

TN ¼N

2

� ��1Xj<k

sign ðxj � xkÞðyj � ykÞ� �

ð2Þ

with

sign ¼

1; xj < xk and yj < yk

xj > xk and yj > yk

0; ðxj � xkÞðyj � ykÞ ¼ 0�1; otherwise:

8>>><>>>:

N.M. Ariff et al. / Journal of Hydrology 470–471 (2012) 158–171 161

Similar to the Pearson correlation coefficient q, Kendall’s s isdefined in the closed interval [�1,1] with 1 implying total concor-dance, �1 representing total discordance and 0 showing zero or noconcordance between the random variables (Kao and Govindaraju,2008).The third step is calculating the copula parameter h. This isdone by using the relationship between h and Kendall’s s, i.e.s = g(h). This relationship differs for different copula. Finally, theapproximated h and the marginals of the random variables canbe inserted into the candidate copula function to obtain the respec-tive copula.

The conditional copula can be derived from the copula functionfor realizations of U given the value of V = v. This conditional distri-bution is needed in order to attain the values for the intensity–duration–frequency (IDF) curves which involve conditional distri-bution of storm intensity given storm duration (Singh and Zhang,2007). In this study, X is regarded as the storm intensity and Y isthe storm duration. For simplicity purposes, we denote them as Iand D respectively with lowercase letters, i and d, as their realiza-tions. The conditional distribution of I given D = d is represented inconditional copula form as the realizations of marginal U forknown value of V = v, CU|V=v. The value of v is easily obtained sincev = FD(d). The conditional copula is written as (Zhang and Singh,2007)

CUjV¼mðujV ¼ mÞ ¼ @

@mCU;V ðu; vÞjV¼v : ð3Þ

Similar to the return period of any conditional bivariate distribu-tions, the relationship between the conditional copula and the se-lected return period is

CUjV¼mðujV ¼ mÞ ¼ 1� 1T: ð4Þ

Hence, by solving Eqs. (3) and (4) simultaneously for any givenvalues of T and v, we can solve for the corresponding u. From thevalue of u, the respective i can be yielded since i ¼ F�1

I (u). This i va-lue is then taken as one of the point on the IDF curves indicatingthe intensity of storm for return period T and storm duration d.Repeating this process for various return periods and selected val-ues of storm durations, other points of the IDF curves can be deter-mined and the IDF curves can be constructed.

4. Selection of copula

There are various copula families and functions available forbivariate frequency analysis. In fact, copula consists of familieswith many different copula functions. For example, the Archime-dean copula, which is one of the most commonly used copula fam-ily due to its mathematical tractability and simplicity, has at least22 copula functions as its member. The choice of copula family andits function relies on the correlation between the random variablesunder consideration. The correlation between storm intensity, I,and storm duration, D, is known to be negative (Kao and Gov-indaraju, 2007; Singh and Zhang, 2007; Zhang and Singh, 2007).Among the copulas which have been applied in hydrology thatare appropriate for negatively correlated random variables arethe Farlie–Gumbel–Morgenstern (FGM), Gaussian, Ali–Mikhail–Haq (AMH) and Frank copula.

4.1. Farlie–Gumbel–Morgenstern (FGM) copula

FGM copula belongs to the family of copulas with quadratic sec-tion. The FGM copula is used for modelling purposes due to theirsimple analytical form with its copula function written as (Favreet al., 2004)

CU;V ðu;vÞ ¼ uv þ huvð1� uÞð1� vÞ; h 2 ½�1;1�: ð5Þ

The relationship between h and s for FGM copula is (Huard et al.,2006)

s ¼ 2h9; s 2 �2

929

� �: ð6Þ

The conditional copula is

CUjV¼mðujV ¼ mÞ ¼ uþ huð1� uÞð1� vÞ � huvð1� uÞ: ð7Þ

4.2. Gaussian copula

Gaussian copula is from the elliptic copula family. Gaussian cop-ula is good for practical applications since it possess several proper-ties of the multivariate normal distribution (Favre et al., 2004). Thecopula function for the Gaussian copula is (Schmidt, 2006)

CU;V ðu; vÞ ¼ URðU�1ðuÞ;U�1ðvÞÞ

¼Z U�1ðuÞ

�1

Z U�1ðvÞ

�1

1

2pffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� h2

p� exp

2hsx� s2 �x2

2ð1� h2Þ

!dsdx;

h 2 ½�1;1�: ð8Þ

where U is the cumulative distribution function of a standard nor-mal distribution and UR is the bivariate normal distribution withmean 0 and covariance matrix R. Thus, the conditional Gaussiancopula is

CUjV¼uðujV ¼ vÞ ¼ @

@tURðU�1ðuÞ;U�1ðvÞÞjV¼v : ð9Þ

h in Eq. (8) is actually the Pearson q. A one-to-one relationship be-tween the Pearson q and Kendall’s s under normality is provided byKruskal (1958) as

s ¼ 2p

arcsin h; s 2 ½�1;1�: ð10Þ

4.3. Archimedean copula

According to Nelson (2006), the Archimedean copula can begeneralized as

CU;V ðu; vÞ ¼ u�1ðuðuÞÞ þuðvÞÞ: ð11Þwith u is a copula generator and u�1 is appropriately defined. Hestated that u(1) = 0 and u�1(x) = 0 for x P u(0). The Archimedeancopula family is preferable in hydrologic analysis because it possessdesirable properties such as symmetric and associative (Favre et al.,2004). Copulas from the Archimedean family are also easily con-structed (Zhang and Singh, 2006). Below are two copulas from theArchimedean copula family which are considered in this study.

4.3.1. Ali–Mikhail–Haq (AHM) copulaAMH’s copula function is (Huard et al., 2006)

CU;V ðu; vÞ ¼uv

1� hð1� uÞð1� vÞ ; h 2 ½�1;1Þ ð12Þ

and

s ¼ 1� 23

lnð1� hÞh2 h2 � 2hþ h

lnð1� hÞ þ 1� �

for s

2 �0:181726;13

� �: ð13Þ

The conditional copula for AMH is

CUjV¼tðujV ¼ mÞ

¼ u1� hð1� uÞð1� mÞ 1� hmð1� uÞ

1� hð1� uÞð1� mÞ

� �: ð14Þ

(used in this paper due to common practise in hydrological analysis)

COPULA

ONE-PARAMETER COPULA

MORE THAN ONE-PARAMETER COPULA

Parameters and distributions required to construct a bivariate copula function i) θ – copula parameter ii) U, 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

Copulas with quadraric section

Elliptic Archimedean

Farlie-Gumbel-Morgenstern (FGM)

Gauss

Advantage: simple analytical form

Range for τ:

Advantage: good for practical applications, possess multivariate normal properties

Range for τ:

Advantage: symmetric, associate, easily constructed, preferable in hydrological analysis

Ali-Mikhail-Haq (AMH)

Frank

Range for τ: Range for τ:

Fig. 2. A short summary of the four copulas used in this study.

162 N.M. Ariff et al. / Journal of Hydrology 470–471 (2012) 158–171

4.3.2. Frank copulaFrank copula is written as (Huard et al., 2006)

CU;V ðu; vÞ ¼ �1h

lnhðhÞ

hðhÞ � hðhuÞhðhmÞ ; hðxÞ ¼ 1� expð�xÞ ð15Þ

for h h 2 R{0} and

s ¼ 1� 4hðD1ð�hÞ � 1Þ ð16Þ

with

Dkð�hÞ ¼ k

hk

Z h

0

tk

expðtÞ � 1dt þ kh

kþ 1:

The domain for s is ½�1;1�f0g. From Eq. (3), the conditionalFrank copula is

CUjV¼mðujV ¼ mÞ ¼ hðhuÞð1� hðhmÞÞhðhÞ � hðhuÞhðhmÞ ; hðxÞ ¼ 1� expð�xÞ: ð17Þ

A short summary of the four copulas considered in this study isshown in Fig. 2.

5. Probability distribution functions for storm intensity andduration

The probability distribution of storm intensity is chosen fromeither the exponential, gamma, weibull or lognormal distribution.The exponential distribution is used to represent the probabilitydistribution for storm duration since storms are usually assumedto follow the Poisson process such as in the Neyman–Scott andBartlett–Lewis model of storms. Thus, the inter arrival time ofstorm cells is taken as exponentially distributed. The probabilitydistribution functions for exponential, gamma and weibull proba-bility distribution can be generalized as

f ðxÞ ¼ cbCðaÞ

xb

� �ca�1

exp � xb

� �c� �ð18Þ

with a and c as the shape parameters and b as the scale parameter.When both a and c have the value of one, Eq. (18) becomes an expo-nential probability distribution function. If a = 1, then it becomes agamma probability distribution function and if c = 1, it is a weibull

N.M. Ariff et al. / Journal of Hydrology 470–471 (2012) 158–171 163

probability distribution function. The probability distribution func-tion of lognormal distribution is

f ðxÞ ¼ 1xffiffiffiffiffiffiffiffiffi2pb

p exp �ðln x� lÞ2

2b

!ð19Þ

where l is the location parameter and, similar to Eq. (18), b is thescale parameter.

6. IDF curves using empirical method

In this paper, the IDF points produced by the copula method arecompared to the IDF curves found using one of the IDF empiricalformula. The formula used is known as the Sherman equation(Nhat et al., 2006) and is written as

ie ¼aTk

ðdþ bÞcð20Þ

with ie as the intensity of storm in millimetres per hour (mm/h), das storm duration in hours (h) and T as the return period of storm inyears. The parameters a, j, b and c are estimated using the least-square method on ie for a set of given T and d. For the purpose ofperforming least-square, ie is approximated using the design rainfallintensity formula. The design rainfall intensity for the annual max-imums of storms with duration d and return period T is a function ofd and T which can be represented as (Kottegoda and Rosso, 2008)

010

2030

40

N01

storm duration, h

stor

m in

tens

ity, m

m/h

5 10 15 20

5 10 15 20

010

2030

40

E01

storm duration, h

stor

m in

tens

ity, m

m/h

Fig. 3. Scatter plots of storm intensities against storm duration

ie ¼ gðd; TÞ ¼ �id þ sdKT ð21Þ

where �id and sd are the mean and standard deviation of the stormintensity for a given d. In Eq. (21), KT is defined as the frequency factorfor the return period T which depends on the probability distributionfunction of ie. Eq. (21) is applicable to many probability distributionsof storm intensity that are employed in hydrologic frequency analy-sis (Chow et al., 1988). For rainfall data in Malaysia, the Gumbel dis-tribution is commonly used and is deemed to be suitable (Amin et al.,2008). Hence, in this paper, the Gumbel distribution will be utilized.The frequency factor for Gumbel distribution is (Guo, 2006)

KT ¼ �ffiffiffi6p

p0:5772þ ln ln

TT � 1

� �� � : ð22Þ

The IDF curves constructed from values of storm intensities ie forvarious T and d are compared to the IDF points found based onthe copula method. The percentage difference, denoted as D, be-tween storm intensity ie found using Eq. (20) and storm intensityi derived by the copula method through Eq. (3) is written as

D ¼ jie � ijie

100%: ð23Þ

7. Application on storm events in Peninsular Malaysia

Data from twenty stations representing four regions in Peninsu-lar Malaysia are used for application purposes. The scatter plots of

5 10 15 20

5 10 15 20

010

2030

4050

W01

storm duration, h

stor

m in

tens

ity, m

m/h

010

2030

40

S01

storm duration, h

stor

m in

tens

ity, m

m/h

s for one station from each region of Peninsular Malaysia.

Table 2Distribution for storm intensities and durations as well as the Kendall’s s for thecorrelations between the two random variables.

Station Intensity (mm/h) Duration (h) s

Dist a, c or l b Dist b

N01 Gamma 2.35 5.50 Exponential 5.49 �0.55N02 Gamma 2.76 5.34 Exponential 5.50 �0.55N03 Gamma 2.31 6.44 Exponential 5.53 �0.61N04 Gamma 3.13 4.70 Exponential 5.48 �0.48W01 Weibull 1.58 14.85 Exponential 5.47 �0.54W02 Weibull 1.70 16.08 Exponential 5.50 �0.62W03 Lognormal 2.48 0.46 Exponential 5.48 �0.62W04 Lognormal 2.53 0.44 Exponential 5.51 �0.57W05 Gamma 4.16 4.83 Exponential 4.11 �0.45W06 Weibull 1.85 15.89 Exponential 5.46 �0.64E01 Lognormal 2.25 0.43 Exponential 5.48 �0.42E02 Gamma 3.07 4.86 Exponential 4.09 �0.42E03 Gamma 1.90 7.60 Exponential 5.37 �0.61E04 Lognormal 2.31 0.50 Exponential 5.45 �0.45E05 Weibull 1.56 13.71 Exponential 5.29 �0.53E06 Gamma 2.44 4.84 Exponential 5.46 �0.50S01 Weibull 1.55 15.48 Exponential 5.37 �0.61S02 Weibull 1.44 16.75 Exponential 5.37 �0.60S03 Lognormal 2.66 0.56 Exponential 4.11 �0.54S04 Lognormal 2.24 0.82 Exponential 5.48 �0.61

a, c = Shape parameter, b = scale parameter and l = location parameter.

Table 3Values of h and Akaike’s Information Criteria (AIC) for Gaussian and Frank copula.

Station Gaussian Frank

h AIC h AIC

N01 �0.77 5929.20a �6.45 5929.27N02 �0.75 5963.20 �7.05 5946.37a

N03 �0.82 5949.92 �8.29 5940.52a

N04 �0.70 5963.98 �5.17 5963.53a

W01 �0.75 5895.55 �6.58 5887.88a

W02 �0.84 5922.30a �8.07 5930.54W03 �0.82 5876.50 �8.42 5871.34a

W04 �0.80 5793.82 �7.40 5790.56a

W05 �0.69 5501.10a �4.83 5502.04W06 �0.86 5709.19 �9.43 5708.67a

E01 �0.60 5931.52 �4.82 5923.04a

E02 �0.65 5506.81 �4.54 5504.58a

E03 �0.82 5946.77 �8.70 5929.45a

E04 �0.62 5906.08 �5.15 5897.35a

E05 �0.75 5752.77 �6.53 5744.00a

E06 �0.74 5908.46 �5.92 5904.32a

S01 �0.82 5926.07 �8.81 5911.56a

164 N.M. Ariff et al. / Journal of Hydrology 470–471 (2012) 158–171

storm intensity versus duration for a selected station in each re-gion is displayed in Fig. 3. From this figure, it can be seen that thereis a negative correlation between the two variables. This is becausestorm intensity is the rate of storm depth which is yielded by tak-ing the ratio of storm depth to storm duration. Thus, it is clear thatthe larger the storm duration, the smaller will the storm intensitybe. Fig. 3 also shows that for short storms, for instance, storms withnot more than 12 h, there is a higher correlation between stormintensities and durations. It can also be observed that there is aslight variation on the characteristics of storms when different re-gions are compared especially for short duration storms.

The exponential, gamma, weibull and lognormal distributionswere fitted to storm intensities of the twenty stations. Fig. 4 illus-trates the L-moments ratio diagram for the four distributions withthe sample L-skewness and L-kurtosis of storm intensities for eachof the twenty stations. The most appropriate probability distribu-tions fitted to the storm intensities and durations of these stationsare shown in Table 2. The L-moment ratio diagram of Fig. 4 and Ta-ble 2 imply that there may be dissimilarities among the best fittedstorms’ distribution according to different regions. From the prob-ability distributions selected, we may obtain the marginal distribu-tion functions, U = FI(I) and V = FD(D).

The estimated correlations between I and D, that is the Kendall’ss, are also given in Table 2. It can be seen that the values of Ken-dall’s s for all twenty stations considered are negative which isin accordance with Fig. 3. The values of s for all the stations liewithin the range of �0.64 6 s 6 �0.42. The values are neither closeto �1 nor 0. In other words, the dependence level between thestorm intensities and durations are moderate. Since the depen-dence level is not high, it is unsuitable to use regression methodto represent their relationship. On the other hand, it is also inaccu-rate to reduce the joint distribution of random variables I and D tothe product of the marginal distributions since the dependence le-vel is not low and s is not close to 0. The s values are found not tobe in the domain of s for the Ali–Mikhail–Haq (AMH) and Farlie–Gumbel–Morgenstern (FGM) copulas. Thus, only the Frank andGaussian copulas will be further considered for the constructionof bivariate distributions of storm intensities and durations. Theparameter, h, for both copulas are estimated using their respectiverelationship to Kendall’s s, as displayed in Eqs. (10) and (16).Hence, using the estimated h and marginal distributions U and V,both copulas are fitted to the paired I and D by using their copulafunctions which are Eqs. (8) and (15) respectively. The comparisonof both copulas to determine which type of copula is better to rep-

Fig. 4. L-moment ratio diagram for the exponential, gamma, weibull and lognormaldistribution as well as pairs of (L-skewness, L-kurtosis) of the storm intensities.

S02 �0.81 5974.52 �8.01 5966.42a

S03 �0.77 5567.46 �6.71 5563.55a

S04 �0.77 5979.21 �8.49 5942.27a

a Smaller value of AIC.

resent the relationship between I and D is done with the help ofAkaike’s Information Criteria (AIC). The AIC is given as (Zhangand Singh, 2007)

AIC ¼ �2 logðmaximized likelihood for the copulaÞþ 2ðno: of fitted parametersÞ ð24Þ

or

AIC ¼ N logðMSEÞ þ 2ðno: of fitted parametersÞ: ð25Þ

The copula which shows a smaller value of AIC is chosen to rep-resent the joint distribution of I and D. For all the 20 stations, thevalues for h and AIC of both the Gaussian and Frank copula are pre-sented in Table 3.

Table 3 indicates that the Frank copula provides a compara-tively smaller value of AIC for most of the twenty stations. It canalso be observed that the difference between the AIC values of

Table 4Comparisons between the storm intensities obtained based on the typical IDF empirical formula and the copula method.

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 22.42 18.57 17.19 15.29 12.64 17.34 11.22 8.45 24.74 9.22 6.43 30.22 7.97 5.39 32.435 27.38 25.41 7.17 18.67 17.50 6.26 13.70 12.08 11.84 11.25 9.61 14.63 9.73 8.32 14.50

10 31.84 30.13 5.37 21.71 21.10 2.83 15.94 14.65 8.07 13.09 11.79 9.92 11.32 10.34 8.7125 38.88 36.22 6.84 26.51 26.26 0.96 19.46 18.41 5.40 15.98 14.89 6.80 13.82 13.15 4.8850 45.22 40.74 9.89 30.83 30.45 1.23 22.63 21.69 4.17 18.59 17.59 5.38 16.08 15.56 3.26

100 52.59 45.25 13.96 35.86 34.80 2.95 26.32 25.38 3.58 21.62 20.68 4.31 18.70 18.32 2.05

N02 2 25.53 21.08 17.44 17.19 14.63 14.88 12.48 10.04 19.55 10.18 7.77 23.67 8.77 6.57 25.065 30.95 27.95 9.68 20.83 19.40 6.87 15.13 13.65 9.77 12.34 11.00 10.88 10.63 9.60 9.65

10 35.79 32.69 8.68 24.09 22.90 4.97 17.50 16.13 7.83 14.28 13.13 8.01 12.29 11.59 5.6625 43.39 38.81 10.54 29.20 27.95 4.30 21.22 19.69 7.18 17.31 16.08 7.06 14.90 14.29 4.0750 50.18 43.37 13.58 33.78 32.09 4.99 24.54 22.79 7.14 20.02 18.59 7.11 17.23 16.54 4.02

100 58.05 47.82 17.62 39.07 36.44 6.75 28.38 26.30 7.32 23.15 21.46 7.30 19.93 19.08 4.27

N03 2 27.42 22.20 19.03 17.36 14.62 15.78 12.01 9.52 20.70 9.49 7.02 26.01 7.98 5.70 28.625 33.23 29.55 11.08 21.04 19.23 8.60 14.55 12.88 11.48 11.50 10.01 12.98 9.68 8.49 12.20

10 38.43 34.77 9.52 24.33 22.65 6.92 16.83 15.15 9.94 13.30 11.95 10.14 11.19 10.31 7.8725 46.57 41.67 10.52 29.49 27.73 5.97 20.39 18.39 9.81 16.12 14.59 9.48 13.56 12.72 6.1950 53.85 46.88 12.94 34.10 32.09 5.90 23.58 21.21 10.04 18.64 16.80 9.88 15.68 14.69 6.34

100 62.27 52.04 16.43 39.43 36.82 6.64 27.27 24.49 10.19 21.56 19.31 10.43 18.13 16.87 6.95

N04 2 21.92 19.72 10.03 16.95 14.53 14.31 13.78 10.57 23.28 12.07 8.61 28.65 10.95 7.59 30.755 26.54 26.62 0.30 20.53 20.02 2.46 16.68 14.97 10.29 14.62 12.53 14.28 13.26 11.25 15.19

10 30.67 31.18 1.65 23.72 23.93 0.87 19.28 18.08 6.24 16.89 15.26 9.68 15.33 13.79 10.0225 37.14 36.90 0.64 28.73 29.25 1.81 23.35 22.52 3.54 20.46 19.14 6.42 18.56 17.38 6.3450 42.92 41.07 4.32 33.20 33.36 0.47 26.98 26.22 2.84 23.64 22.45 5.07 21.45 20.45 4.70

100 49.61 45.20 8.90 38.37 37.48 2.32 31.19 30.13 3.39 27.33 26.08 4.58 24.80 23.86 3.78

W01 2 22.98 19.69 14.30 15.84 13.39 15.50 11.73 8.67 26.11 9.69 6.34 34.56 8.42 5.13 39.025 28.26 26.33 6.84 19.49 18.47 5.23 14.43 12.69 12.08 11.92 9.93 16.71 10.35 8.47 18.21

10 33.05 30.63 7.32 22.79 22.04 3.29 16.88 15.42 8.64 13.94 12.32 11.58 12.11 10.71 11.5225 40.65 35.92 11.64 28.03 26.93 3.93 20.76 19.25 7.25 17.14 15.61 8.91 14.89 13.75 7.6250 47.54 39.68 16.53 32.77 30.73 6.24 24.27 22.46 7.45 20.05 18.36 8.40 17.41 16.27 6.58

100 55.60 43.30 22.12 38.33 34.52 9.92 28.39 25.96 8.56 23.44 21.42 8.63 20.36 19.06 6.41

W02 2 23.60 21.48 9.00 16.62 14.70 11.55 12.53 9.65 22.97 10.46 7.03 32.77 9.16 5.62 38.655 27.95 27.50 1.62 19.69 19.03 3.33 14.84 13.12 11.58 12.39 10.24 17.32 10.85 8.68 20.01

10 31.76 31.46 0.95 22.37 22.06 1.42 16.86 15.37 8.87 14.08 12.26 12.91 12.33 10.61 13.9625 37.62 36.37 3.30 26.50 26.29 0.80 19.97 18.44 7.67 16.67 14.91 10.58 14.60 13.10 10.3150 42.75 39.88 6.70 30.11 29.70 1.38 22.69 21.00 7.48 18.95 17.05 10.03 16.60 15.06 9.23

100 48.58 43.21 11.06 34.22 33.20 2.99 25.79 23.84 7.55 21.54 19.40 9.94 18.86 17.19 8.89

W03 2 28.88 21.56 25.36 17.49 13.62 22.16 11.67 9.05 22.45 9.02 6.99 22.51 7.46 5.93 20.545 35.79 30.59 14.52 21.68 18.14 16.30 14.46 11.94 17.45 11.17 9.43 15.64 9.24 8.16 11.68

10 42.10 38.02 9.69 25.50 21.83 14.39 17.01 14.01 17.62 13.14 11.09 15.65 10.87 9.66 11.0925 52.17 49.25 5.61 31.60 27.88 11.76 21.08 17.15 18.66 16.29 13.45 17.44 13.47 11.74 12.8350 61.37 58.89 4.04 37.17 33.69 9.35 24.80 20.08 19.03 19.16 15.53 18.95 15.85 13.52 14.70

100 72.18 69.51 3.71 43.72 40.69 6.93 29.17 23.73 18.64 22.53 18.02 20.05 18.64 15.58 16.42

W04 2 29.83 22.05 26.08 18.46 14.31 22.46 12.53 9.68 22.77 9.79 7.59 22.49 8.16 6.52 20.125 36.93 31.66 14.27 22.86 19.57 14.38 15.52 13.09 15.67 12.12 10.44 13.89 10.11 9.12 9.82

10 43.41 39.44 9.16 26.87 23.89 11.08 18.24 15.61 14.43 14.25 12.44 12.68 11.88 10.92 8.1425 53.76 50.99 5.16 33.27 30.94 7.02 22.59 19.53 13.55 17.64 15.39 12.79 14.71 13.49 8.3450 63.19 60.79 3.79 39.11 37.53 4.02 26.55 23.25 12.41 20.74 18.07 12.88 17.30 15.76 8.89

100 74.27 71.46 3.79 45.97 45.26 1.54 31.21 27.91 10.57 24.38 21.36 12.40 20.33 18.48 9.11

W05 2 29.77 24.74 16.88 20.97 18.15 13.47 15.81 13.69 13.44 13.20 11.79 10.69 11.56 10.94 5.395 35.77 32.55 8.99 25.20 24.37 3.31 19.00 18.88 0.61 15.86 16.58 4.52 13.89 15.54 11.83

10 41.09 37.69 8.30 28.96 28.71 0.86 21.83 22.45 2.85 18.23 19.84 8.88 15.96 18.67 16.9725 49.38 44.11 10.68 34.79 34.61 0.51 26.22 27.44 4.65 21.90 24.39 11.36 19.18 23.02 19.9950 56.73 48.78 14.02 39.97 39.20 1.93 30.13 31.57 4.78 25.16 28.18 12.00 22.04 26.64 20.89

100 65.18 53.35 18.15 45.93 43.82 4.60 34.62 35.95 3.83 28.91 32.31 11.76 25.32 30.63 20.94

W06 2 24.48 21.01 14.18 16.33 14.62 10.46 11.78 9.83 16.52 9.56 7.24 24.34 8.21 5.78 29.575 29.00 25.94 10.55 19.35 18.02 6.90 13.96 12.64 9.45 11.33 9.95 12.25 9.72 8.44 13.23

10 32.98 29.23 11.36 22.00 20.35 7.53 15.87 14.39 9.35 12.89 11.57 10.22 11.06 10.03 9.2725 39.08 33.35 14.67 26.07 23.62 9.41 18.81 16.69 11.23 15.27 13.62 10.80 13.10 12.01 8.3650 44.43 36.30 18.31 29.64 26.32 11.20 21.38 18.57 13.14 17.36 15.22 12.36 14.90 13.51 9.34

100 50.52 39.10 22.6 33.71 29.18 13.41 24.31 20.64 15.1 19.74 16.92 14.32 16.94 15.07 11.04

E01 2 19.85 15.05 24.18 13.75 10.54 23.33 10.22 7.53 26.31 8.46 6.18 26.96 7.36 5.50 25.295 24.75 22.70 8.28 17.15 15.77 8.05 12.75 11.19 12.19 10.55 9.22 12.65 9.18 8.24 10.30

10 29.25 28.66 2.01 20.26 20.06 0.98 15.07 14.16 6.03 12.47 11.62 6.83 10.85 10.38 4.3525 36.48 37.28 2.21 25.27 26.72 5.76 18.79 18.93 0.75 15.55 15.46 0.57 13.53 13.77 1.7750 43.11 44.49 3.22 29.86 32.57 9.07 22.20 23.38 5.31 18.38 19.12 4.02 15.99 17.00 6.30

100 50.94 52.29 2.65 35.29 39.10 10.81 26.24 28.60 9.02 21.72 23.54 8.38 18.90 20.96 10.89

(continued on next page)

N.M. Ariff et al. / Journal of Hydrology 470–471 (2012) 158–171 165

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 23.28 18.54 20.36 16.06 13.03 18.88 11.90 9.42 20.89 9.83 7.93 19.30 8.54 7.28 14.775 28.73 25.60 10.89 19.82 18.59 6.22 14.69 13.94 5.10 12.13 12.03 0.87 10.54 11.18 5.99

10 33.68 30.29 10.07 23.24 22.55 2.98 17.22 17.16 0.34 14.23 14.95 5.05 12.36 13.96 12.9125 41.57 36.21 12.90 28.68 27.97 2.48 21.25 21.76 2.38 17.56 19.11 8.84 15.26 17.93 17.5050 48.74 40.54 16.82 33.63 32.20 4.24 24.92 25.58 2.63 20.59 22.64 9.94 17.89 21.30 19.05

100 57.15 44.81 21.60 39.43 36.46 7.53 29.22 29.63 1.40 24.14 26.48 9.68 20.98 25.01 19.24

E03 2 28.21 22.06 21.80 16.64 13.72 17.60 16.64 8.39 22.78 8.29 5.90 28.81 6.79 4.63 31.805 34.61 29.92 13.54 20.42 18.36 10.08 20.42 11.68 12.42 10.16 8.77 13.76 8.32 7.27 12.62

10 40.39 35.63 11.79 23.83 21.83 8.38 23.83 13.92 10.59 11.86 10.66 10.17 9.71 9.02 7.1225 49.55 43.29 12.63 29.23 27.07 7.38 29.23 17.10 10.42 14.55 13.23 9.07 11.92 11.37 4.6250 57.83 49.14 15.02 34.12 31.67 7.16 34.12 19.89 10.75 16.98 15.39 9.40 13.91 13.28 4.51

100 67.49 55.00 18.52 39.82 36.78 7.64 39.82 23.15 10.99 19.82 17.83 10.03 16.23 15.41 5.07

E04 2 23.55 16.84 28.50 15.01 11.33 24.52 10.43 7.81 25.19 8.28 6.26 24.39 6.98 5.49 21.335 29.55 26.02 11.94 18.83 17.25 8.41 13.09 11.78 10.00 10.38 9.50 8.52 8.76 8.38 4.28

10 35.08 33.38 4.87 22.36 22.25 0.49 15.55 15.05 3.20 12.33 12.08 2.00 10.40 10.66 2.5325 44.02 44.25 0.52 28.06 30.22 7.70 19.51 20.41 4.65 15.47 16.27 5.15 13.05 14.30 9.6050 52.27 53.45 2.25 33.32 37.41 12.28 23.16 25.56 10.38 18.37 20.33 10.67 15.49 17.81 15.00

100 62.06 63.70 2.64 39.56 45.59 15.25 27.50 31.76 15.49 21.81 25.37 16.34 18.39 22.22 20.82

E05 2 20.76 18.08 12.90 14.47 12.12 16.28 10.82 7.75 28.38 8.99 5.64 37.20 7.84 4.58 41.605 25.30 24.31 3.92 17.64 16.84 4.54 13.18 11.47 13.00 10.95 8.96 18.20 9.55 7.66 19.87

10 29.39 28.37 3.47 20.49 20.17 1.56 15.31 14.01 8.52 12.72 11.19 12.06 11.10 9.74 12.2025 35.82 33.37 6.84 24.97 24.75 0.89 18.66 17.58 5.83 15.50 14.25 8.07 13.53 12.58 6.9650 41.60 36.94 11.21 29.01 28.33 2.32 21.68 20.58 5.08 18.01 16.82 6.60 15.71 14.93 4.93

100 48.32 40.37 16.45 33.69 31.92 5.24 25.18 23.85 5.29 20.91 19.68 5.91 18.25 17.55 3.81

E06 2 20.23 16.65 17.72 13.98 11.53 17.52 10.38 7.85 24.36 8.58 6.08 29.16 7.46 5.16 30.785 24.77 22.91 7.52 17.12 16.16 5.62 12.71 11.36 10.55 10.51 9.15 12.89 9.13 8.01 12.35

10 28.87 27.17 5.89 19.95 19.55 2.01 14.81 13.87 6.30 12.24 11.30 7.73 10.65 9.99 6.2025 35.34 32.62 7.70 24.43 24.34 0.36 18.13 17.54 3.25 14.99 14.37 4.12 13.03 12.78 1.9150 41.19 36.66 11.00 28.47 28.17 1.06 21.13 20.70 2.01 17.47 17.04 2.43 15.19 15.20 0.05

100 48.00 40.66 15.29 33.18 32.08 3.29 24.62 24.19 1.74 20.36 20.09 1.33 17.70 17.96 1.45

S01 2 26.41 21.34 19.17 16.09 13.88 13.75 10.78 8.64 19.85 8.36 6.03 27.80 6.93 4.67 32.595 31.90 27.68 13.23 19.43 18.07 7.04 13.03 11.88 8.81 10.09 8.98 11.03 8.37 7.44 11.05

10 36.80 31.99 13.07 22.42 21.04 6.17 15.03 13.99 6.93 11.65 10.85 6.83 9.65 9.22 4.5225 44.45 37.48 15.69 27.08 25.30 6.57 18.15 16.87 7.08 14.07 13.31 5.41 11.66 11.51 1.2750 51.29 41.48 19.12 31.25 28.87 7.59 20.94 19.29 7.91 16.23 15.29 5.81 13.45 13.33 0.94

100 59.17 45.33 23.39 36.05 32.67 9.35 24.16 22.01 8.89 18.72 17.46 6.75 15.52 15.28 1.56

S02 2 29.64 23.40 21.06 17.70 14.87 15.95 11.67 9.00 22.85 8.95 6.20 30.77 7.36 4.78 35.145 36.15 31.34 13.31 21.59 20.21 6.39 14.23 13.00 8.68 10.92 9.72 10.98 8.98 8.02 10.71

10 42.01 36.75 12.51 25.08 24.06 4.09 16.54 15.69 5.14 12.69 12.04 5.06 10.44 10.19 2.4025 51.24 43.64 14.83 30.60 29.61 3.22 20.18 19.48 3.46 15.47 15.20 1.77 12.73 13.09 2.8050 59.55 48.67 18.27 35.56 34.21 3.78 23.45 22.72 3.10 17.98 17.82 0.90 14.79 15.45 4.44

100 69.20 53.51 22.68 41.32 39.05 5.49 27.25 26.41 3.09 20.90 20.76 0.65 17.19 18.06 5.06

S03 2 25.40 23.97 5.64 17.68 13.91 21.33 13.19 8.87 32.80 10.95 7.01 35.97 9.54 6.23 34.725 30.65 36.25 18.26 21.33 20.06 5.98 15.92 12.89 19.00 13.21 10.40 21.25 11.52 9.35 18.83

10 35.33 46.64 31.99 24.59 25.16 2.33 18.35 15.93 13.17 15.23 12.90 15.25 13.28 11.65 12.2625 42.63 62.78 47.27 29.67 33.71 13.61 22.14 20.70 6.51 18.37 16.67 9.25 16.02 15.06 6.0050 49.14 76.97 56.64 34.20 42.05 22.96 25.52 25.30 0.88 21.18 20.18 4.74 18.46 18.17 1.60

100 56.65 92.89 63.98 39.42 52.27 32.59 29.42 31.15 5.88 24.41 24.55 0.56 21.28 22.00 3.35

S04 2 31.78 20.66 34.99 17.43 11.21 35.69 10.73 6.51 39.34 7.88 4.61 41.45 6.27 3.70 40.985 41.37 32.84 20.61 22.68 16.37 27.84 13.96 9.38 32.82 10.25 6.85 33.17 8.16 5.66 30.73

10 50.49 43.81 13.23 27.69 20.89 24.55 17.04 11.59 32.02 12.51 8.49 32.18 9.97 7.07 29.0525 65.71 61.76 6.02 36.03 28.86 19.90 22.18 15.12 31.85 16.28 10.95 32.77 12.97 9.14 29.5050 80.21 78.31 2.37 43.98 37.08 15.69 27.08 18.61 31.27 19.87 13.23 33.43 15.83 11.01 30.48

100 97.90 97.61 0.30 53.68 47.63 11.28 33.05 23.19 29.82 24.26 16.09 33.67 19.32 13.27 31.34

166 N.M. Ariff et al. / Journal of Hydrology 470–471 (2012) 158–171

Gaussian and Frank copula is relatively small compared to the AICvalues of individual copula at each station. This implies that bothcopulas are equally suitable in fitting the observed data. With thisreason and the fact that it is easier to construct an Archimedeancopula, which in this case is the Frank copula, compared to Gauss-ian copula which falls under the elliptic copula family, the Frankcopula is chosen to represent the twenty rainfall stations in Penin-sular Malaysia.

The points on the IDF curves are yielded using the conditionalFrank copula shown in Eq. (17). The values for return period, T,considered in this analysis are 2, 5, 10, 25, 50 and 100 years. Since,

this paper focuses on convective storms which is short durationstorms, the storm durations which are taken into account are 1,3, 6, 9 and 12 h. By solving Eqs. (4) and (17) using these return peri-ods and storm durations, the corresponding intensities, i, are ob-tained and tabulated in Table 4.

The estimated parameters of Eq. (20), a, j, b and c, for the stormintensities, ie, acquired by the typical IDF empirical formula are gi-ven in Table 5. The values of ie for all storm durations, d, and returnperiods, T, are shown in Table 4 along with the percentage of differ-ences between i and ie, D, for all twenty stations. Fig. 5 provides thevisual comparisons between the IDF curves drawn based on Eq.

Table 5Parameters a, j, b and c of the typical IDF empirical formula.

Station a j b c

N01 28.27 0.22 1.00 0.55N02 32.79 0.21 1.00 0.57N03 37.44 0.21 1.00 0.66N04 24.51 0.21 1.00 0.37W01 28.50 0.23 1.00 0.54W02 29.48 0.18 1.00 0.51W03 40.54 0.23 1.00 0.72W04 41.00 0.23 1.00 0.69W05 36.77 0.20 1.00 0.51W06 32.26 0.19 1.00 0.58E01 24.25 0.24 1.00 0.53E02 28.77 0.23 1.00 0.54E03 40.97 0.22 1.00 0.76E04 31.12 0.25 1.00 0.65E05 25.63 0.22 1.00 0.52E06 25.12 0.22 1.00 0.53S01 37.57 0.21 1.00 0.71S02 42.71 0.22 1.00 0.74S03 31.67 0.21 1.00 0.52S04 47.49 0.29 1.00 0.87

N.M. Ariff et al. / Journal of Hydrology 470–471 (2012) 158–171 167

(20) and the points of IDF curves obtained using the copula meth-od. The range of ie, i and D for each d and T are given in Table 6.

It can be observed from Table 4 that stations in PeninsularMalaysia have values of percentage difference between i and ie,D, less than 50 for most storm durations and return periods consid-ered in this study. The only exception is the station Kota Tinggi(S03), where the percentage difference of the return period of 50and 100 years for 1-h storms are 56.64% and 63.98% respectively.Eighty three percentage of D are less than 20 which implies thatthe percentage of difference between i and ie are usually small. Infact, half of the values for D are less than 10 with most of themare from storm intensities for the return periods 10, 25, 50 and100 years; more than 50% of D for these four return periods are lessthan 10 with each recording 66%, 71%, 66% and 59% of D respec-tively. 54% of D for T = 5 years is in the range of 10–20. The meanof the percentage difference between i and ie for return period5%, 10, 25, 50 and 100 are 11.63%, 8.85%, 8.43%, 9.26% and10.62%. The differences in percentage are largest for the 2-year re-turn period of most storm durations for all stations with the meanof D is 23.51. This is believed to be due to the small value of stormintensity. Hence, a small error in the intensity will translate intosignificant error in the differences (Singh and Zhang, 2007). Thedifference usually increases as the storm duration increases from3-h to 9-h and drops at 12-h storms. This can be seen from themean values of D according to storm duration 1, 3, 6, 9 and 12 hwhich are 12.96, 8.73, 11.81, 13.81 and 13.23.

Table 6 shows that although the percentage of difference be-tween i and ie are smaller for higher return periods; T = 25, 50and 100 years; the sizes of their range are in average larger thanthose for T = 2, 5 and 10 years. It can be seen that the average ofthe minimum values of D are very small for T = 5, 10, 25, 50 and100 years; 1.70, 1.24, 0.69, 0.91 and 1.05; while the average ofthe minimum D for storms with 2-year return period is 9.12. How-ever, the average of the maximum D is about the same for T = 2 andT = 100 years which are 38.61 and 38.28 respectively. The averagesfor the rest of the maximum values of D are 29.03, 29.96, 32.26 and34.96. Thus, these results in the difference between the size rangeof D for smaller (T = 2, 5 and 10) and higher (T = 25, 50 and 100)return periods. The average sizes for the range of D are 29.49,27.33 and 28.72 for smaller return periods and 31.56, 34.05 and37.23 for higher return periods. Table 6 also indicates that the sizeof the range for IDF points of storm intensities obtained from thecopula method, i, are slightly larger compared to those from the

typical IDF empirical formula, ie. The range for both i and ie increaseas the return period increases for all d hour storm durations. Theaverage values of the range size for i with respect to return periods2, 5, 10, 25, 50 and 100 years are 7.24 mm, 8.78 mm, 10.51 mm,13.68 mm, 17.05 mm and 21.21 mm while the corresponding val-ues for ie are 7.78 mm, 10.25 mm, 12.49 mm, 16.16 mm,20.26 mm and 26.10 mm. On the other hand, the range’s size of idecreases as the storm duration increases. For storm duration 1,3, 6, 9 and 12 h, the average range sizes for i are 28.24 mm,11.99 mm, 10.52 mm, 7.44 mm and 7.21 mm. This downwardtrend is not so prominent for ie with the average values for all sta-tions and return periods are 28.90 mm, 12.53 mm, 11.36 mm,12.15 mm and 12.60 mm for d = 1, 3, 6, 9 and 12 h respectively.Generally, values of ie are mostly higher than i for all values of Tand d.

It can be seen from the results in Tables 4 and 6 that the differ-ences between the storm intensities obtained from either the cop-ula method or the typical IDF empirical formula does not followany particular pattern. As a consequence to that, the reason behindthe pattern for the percentage of differences, D, is also not appar-ent. Nevertheless, since most of the differences are small, we canregard that the IDF points found by the copula method is some-what in agreement with the IDF points obtained based on the typ-ical IDF empirical formula. This is most likely due to the fact thatalthough the two methods handle the data differently, the stormevents which are used in both methods are the same since we con-sider similar definition of storms in both analyses. The difference isthat the storm intensity and duration in the copula method areboth regarded as random variables while only the storm intensityis taken as a random variable in the IDF empirical formula. Further-more, the storm duration is considered as a continuous randomvariable in the copula method while it is used as a discrete andfixed value in the IDF empirical formula with representative valuesare used to define groups or categories of storm intensities. Thestorm intensities of each category are assumed to have storm dura-tion equal to the value which represents them. In a way, the typicalIDF empirical formula regards the storm intensity, ie, as a functionof fix storm duration, d, and return period, T. For the copula ap-proach, all the storms are combined irrespective of the storm dura-tion to obtain the joint distribution for storm intensity andduration. Meanwhile, the storms which are categorized with re-spect to the storm duration for the typical IDF empirical formulaare analyzed separately to get the design rainfall intensity for var-ious return periods. Briefly, the copula method fit probability dis-tributions to represent storm intensity and duration for eachstation. Then, by calculating the correlation, s, between stormintensity and duration as well as estimating the copula parameter,h, the copula theory can be applied to obtain the joint distributionfor the two random variables in copula form. The IDF points arethen yielded based on the conditional distribution of storm inten-sity given storm duration d. As for the typical IDF empirical for-mula, the mean and standard deviation of each category of stormintensities with fix duration d are computed separately in orderto obtain ie for each storm duration d and various return periodsT. The IDF curves are then build by approximating the parametersa, j, b and c through the nonlinear least squares method performedon the Sherman equation, i.e. Eq. (20). In short, the results of bothmethods which are the storm intensities i and ie are consistent be-cause both methods are different storm models which representthe same storm events.

The typical IDF curves using empirical derivations have notmuch theoretical background; however, IDF points obtained basedon the copula method are supported by a sound copula theory. TheIDF points found from the latter method considers properties of ac-tual storm characteristics such as storm intensities, depths anddurations. Furthermore, the copula method is able to provide the

168 N.M. Ariff et al. / Journal of Hydrology 470–471 (2012) 158–171

general form for the joint distribution of storm intensity and dura-tion which is more informative and pertinent for further analysis.Hence, other analysis on storm events such as obtaining the condi-tional distribution of storm duration given storm intensity, com-puting the moments and producing the quantile functions ofstorm characteristics could be performed based on the probabilitystatement given by the general form of the joint distribution be-tween storm intensity and storm duration. Thus, inadvertentlythe copula method lessen the computational effort for future anal-ysis compared to the IDF empirical method which does not provideadditional information on storm events and any further research

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would have to be done separately. Moreover, for a large set orrange of storm duration, it is found easier to perform the copulamethod since it involves the combination of storm intensities irre-spective of storm duration as opposed to the IDF empirical formulawhich requires storm intensities to be categorized with respect tostorm duration. No extra computation is needed for the copulamethod but more categories of storm intensities, correspondingto the fix storm durations considered, have to be analyzed individ-ually to obtain the mean, standard deviation and design rainfallintensity of each category. Hence, the typical IDF empirical formulawill take considerably more time and is less efficient compared to

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N.M. Ariff et al. / Journal of Hydrology 470–471 (2012) 158–171 169

the copula method. In addition to that, the nonlinear least squaresmethod applied under the IDF empirical formula could sometimefail to approximate the parameters of the Sherman equation usedin the analysis if the initial values provided are unsuitable. Hence,finding the IDF points with the copula method prove to be moreadvantageous for researchers and hydrologists to extend theirunderstandings of storm events.

8. Conclusions

Intensity–duration–frequency (IDF) curves are important espe-cially for a country like Malaysia where flood is regarded by theMinistry of Natural Resources and Environment as the most signif-icant natural hazard. The IDF curves currently available in Malaysiaare done using the univariate analysis and empirical formula. How-ever, rainfall variables such as intensity and duration are related toeach other. This paper derives IDF points using bivariate frequencyanalysis by utilizing the copula method. Based on this study, sev-eral conclusions are drawn.

The storm intensity, I, and duration, D, are negatively correlatedwhich implies that shorter storms have higher intensities. Thisfinding agrees with previous literatures done in other countries(Kao and Govindaraju, 2007; Singh and Zhang, 2007). Based on thisrelationship, four copulas which are commonly applied in hydro-

logical analysis are considered in this study. The four copulas arethe Farlie–Gumbel–Morgenstern (FGM), Gaussian, Ali–Mikhail–Haq (AMH) and Frank copula. The values of Kendall’s s obtainedimply that the dependence level between I and D are neither veryhigh nor very low. Thus, only the Frank and Gaussian copula havedomains of s which are suitable for analyzing the storm events inPeninsular Malaysia and these two copulas are further consideredin this analysis.

The Frank copula is deemed appropriate to represent the rela-tionship between storm intensity and duration in PeninsularMalaysia based on the AIC values which are mostly smaller thanthe AIC values of Gaussian copula. It is found that both copulas re-sults in relatively similar AIC values when used to fit the relation-ship between storm intensities and durations. However, the Frankcopula has an additive advantage since it is a member of Archime-dean copula family which can be easier constructed. Hence, theFrank copula is preferable in hydrological analysis compared tothe Gaussian copula which falls under the elliptic copula family.The conditional distribution of Frank copula is also less compli-cated compared to the conditional Gaussian copula, Eqs. (17) and(9).

The difference between the IDF curves obtained based on thetypical IDF empirical formula and the points on IDF points acquiredusing the copula method depends on the given storm duration dand return period T. There is no exact pattern for the differences

Table 6Range of the storm intensities based on the typical IDF empirical formula and the copula method as well as their percentage differences.

d (hour)/T (years) 1 3 6 9 12

Range of storm intensities (mm)ie 2 19.85–31.78 13.75–20.97 10.22–16.64 7.88–13.20 6.27–11.56

5 24.75–41.37 17.12–25.20 12.71–20.42 10.09–15.86 8.16–13.8910 28.87–50.49 19.95–28.96 14.81–23.83 11.65–18.23 9.65–15.9625 35.34–65.71 24.43–36.03 18.13–29.23 14.07–21.90 11.66–19.1850 41.19–80.21 28.47–43.98 20.94–34.12 16.23–25.16 13.45–22.04

100 48.00–97.90 33.18–53.68 24.16–39.82 18.72–28.91 15.52–25.32

i 2 15.05–24.74 10.54–18.15 6.51–13.69 4.61–11.79 3.70–10.945 22.70–36.25 15.77–24.37 9.38–18.88 6.85–16.58 5.66–15.54

10 27.17–46.64 19.55–28.71 11.59–22.45 8.49–19.84 7.07–18.6725 32.62–62.78 23.62–34.61 15.12–27.44 10.95–24.39 9.14–23.0250 36.30–78.31 26.32–42.05 18.57–31.57 13.23–28.18 11.01–26.64

100 39.10–97.61 29.18–52.27 20.64–35.95 16.09–32.31 13.27–30.63

D 2 5.64–34.99 10.46–35.69 13.44–39.34 10.69–41.45 5.39–41.605 0.30–20.61 2.46–27.84 0.61–32.82 0.87–33.17 4.28–30.73

10 0.95–31.99 0.49–24.55 0.34–32.02 2.00–32.18 2.40–29.0525 0.52–47.27 0.36–19.90 0.75–31.85 0.57–32.77 1.27–29.5050 2.25–56.64 0.47–22.96 0.88–31.27 0.90–33.43 0.05–30.48

100 0.30–63.98 1.54–32.59 1.40–29.82 0.56–33.67 1.45–31.34

170 N.M. Ariff et al. / Journal of Hydrology 470–471 (2012) 158–171

but they are generally small. Hence, storm intensities found basedon both methods are believed to be in agreement with each other.This is probably because both storm intensities represent the samestorm events since similar definition of storms are considered inboth analyses. Although the two methods provide consistent re-sults, there are great benefits in using the bivariate copula methodas opposed to the univariate IDF empirical formula in practice. Thetypical IDF empirical formula does not have much theoretical back-ground while the IDF points based on the copula method are sup-ported by a sound copula theory. The copula method is alsoworthwhile to be used in storm analysis since it provides the gen-eral form for the joint distribution of storm intensity and durationwhich is pertinent for further analysis. Furthermore, copula meth-od is found to be easier to perform for a large set or range of stormduration compared to the IDF empirical formula. This is due to thefact that all storms are combined irrespective of the storm durationfor the copula approach and thus no extra computation is neededfor analysis while the storms are categorized with respect to stormduration for the typical IDF empirical method and each storm cat-egory has to be analyzed individually. Thus, the typical IDF empir-ical approach is less efficient and more time consuming. Hence, thecopula method is more meaningful for researchers and hydrolo-gists to extend their study on storm events.

The storm intensities yielded based on both methods indicatethat they depend on the geographical locations of the rainfall sta-tions since the distribution of storm intensity and its pattern varyfor each region. Hence, the percentage difference, D, between themalso varies from one region to another. This may be caused by thedifferent characteristics of storms in each region. These differencesare very much influenced by two monsoon seasons, the Northeastand Southwest Monsoon, which occurred in Peninsular Malaysiafrom November to March and from May to September each year.The regionalization of storms in Peninsular Malaysia as well astheir characteristics will be included in future papers.

In conclusion, bivariate frequency analysis of storm events isimperative for hydraulic applications in order to represent thecomplexity of actual hydrologic events. It is believed that model-ling the real events of storms will have a more significant contribu-tion on current hydrologic designs which are mostly based on theunivariate analysis approach.

Acknowledgements

The authors would like to thank the Department of Irrigationand Drainage Malaysia for providing hourly rainfalls data for the

use of the study. Utmost appreciations to the Ministry of HigherEducation (MOHE) and Universiti Kebangsaan Malaysia (UKM)for the allocation of research Grants, UKM-ST-06-FRGS0181-2010and UKM-GGPM-PI-028-2011.

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