bivariate flood frequency analysis using the copula function: a case study of the litija station on...

HYDROLOGICAL PROCESSESHydrol. Process. (2014)Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/hyp.10145

Bivariate flood frequency analysis using the copula function: acase study of the Litija station on the Sava River

Mojca Sraj,* Nejc Bezak and Mitja BrillyFaculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, SI-1000, Ljubljana, Slovenia

*CUnE-m

Co

Abstract:

As an alternative to the commonly used univariate flood frequency analysis, copula frequency analysis can be used. In this study, 58flood events at the Litija gauging station on the Sava River in Slovenia were analysed, selected based on annual maximum dischargevalues. Corresponding hydrograph volumes and durations were considered. Different bivariate copulas from three families wereapplied and compared using different statistical, graphical and upper tail dependence tests. The parameters of the copulas wereestimated using the method of moments with the inversion of Kendall’s tau. The Gumbel–Hougaard copula was selected as the mostappropriate for the pair of peak discharge and hydrograph volume (Q-V). The same copula was also selected for the pair hydrographvolume and duration (V-D), and the Student-t copula was selected for the pair of peak discharge and hydrograph duration (Q-D). Thedifferences among most of the applied copulas were not significant. Different primary, secondary and conditional return periods werecalculated and compared, and some relationships among them were obtained. Copyright © 2014 John Wiley & Sons, Ltd.

KEY WORDS copula; flood frequency analysis; distributions; tail dependence; statistical and graphical tests

Received 14 August 2013; Accepted 1 January 2014

INTRODUCTION

Copula functions have, in recent years, become a morefrequently used tool in hydrology. Because hydrologicalprocesses are often multidimensional, the consideration ofmore than one variable in analyses is reasonable. Copulasare often a good alternative to the most commonly usedunivariate frequency analyses. Among other applications,copula functions have been used for modelling droughts(Kao and Govindaraju, 2010; Wong et al., 2010; Liuet al., 2011; Reddy and Ganguli, 2012a; Lee et al., 2013;Ma et al., 2013; Wong et al., 2013), rainfall analysis(Singh and Zhang, 2007; Zhang and Singh, 2007a;Gargouri-Ellouze and Chebchoub, 2008; Ghosh, 2010;Vandenberghe et al., 2010; Balistrocchi and Bacchi,2011; Ariff et al., 2012), assessing the risk of damovertopping (De Michele et al., 2005), hyetographanalysis (Grimaldi and Serinaldi, 2006b), flood coinci-dence risk analysis (Chen et al., 2012), geostatisticalmodels (Bardossy, 2006) and, indeed, flood frequencyanalysis (Grimaldi and Serinaldi, 2006a; Zhang andSingh, 2006; Renard and Lang, 2007; Zhang and Singh,2007b; Karmakar and Simonovic, 2009). The first paperon copulas in hydrology was written by De Michele and

orrespondence to: Mojca Sraj, Faculty of Civil and Geodetic Engineering,iversity of Ljubljana, Jamova 2, SI-1000 Ljubljana, Slovenia.ail: [email protected]

pyright © 2014 John Wiley & Sons, Ltd.

Salvadori (2003), and in the next few years, several otherpapers were also published, e.g. Favre et al. (2004);Salvadori and De Michele (2004) and Dupuis (2007). Themain advantage of copula functions over classicalbivariate frequency analyses is that the selection ofmarginal distributions and multivariate dependencemodelling are two separate processes, giving additionalflexibility to the practitioner (Zhang and Singh, 2006;Genest and Favre, 2007; Karmakar and Simonovic,2009). Thus, in the case of flood frequency analysis,different parametric or non-parametric distributions canbe used for modelling discharge, volume and durationvariables. Different papers have emphasized severalaspects of copula analysis, such as marginal distributionselection (Karmakar and Simonovic, 2009), copulaparameter estimation (Salvadori et al., 2007), goodnessof fit testing for copulas (Genest and Remillard, 2008;Genest et al., 2011) and multivariate return periodcalculation (Salvadori et al., 2011; Vandenberghe et al.,2011; Graler et al., 2013). More papers about copulas canbe found on the website www.stahy.org. Recently,Ashkar and Aucoin (2011) showed that classical bivariatedistributions should be given more attention in hydrolog-ical practice. They also indicated that the advantage ofcopula models usually lies in their simple closed form,which allows easy implementation in hydrologicalpractice. With a few examples, they showed that bivariatedistributions can still produce comparable results.

http://www.stahy.org

M. SRAJ, N. BEZAK AND M. BRILLY

The aims of this paper were as follows: (i) to performcopula analysis with all the required steps for three pairsof variables: peak discharge and hydrograph volume(Q-V), hydrograph volume and duration (V-D) and peakdischarge and hydrograph duration (Q-D); (ii) to comparedifferent copulas from different families of copulas; (iii) toevaluate and compare some primary, secondary andconditional return periods; and (iv) to compare univariateand bivariate (multivariate) return periods.

METHODS

Daily discharge data, including local maximums for the1953–2010 period from the Litija hydrological station on theSava River, which is part of the Danube River basin (DravaRiver basin it is also part of theDanubeRiver basin; Bonacciand Oskorus, 2010), were used for analyses (Figure 1). Thedrainage area of the Litija gauging station is 4281 km2.The station is located approximately 25 km east ofLjubljana, the capital city of Slovenia.Annual series of maximum peak discharges with

corresponding hydrograph volumes and durations wereused for analyses. Hence, although the peak dischargesweredefinitely annual maximums, the hydrograph volumes anddurations were not necessarily also annual maximums.Other potential approaches may also be used to select asuitable sample (Serinaldi and Grimaldi, 2011). Analternative to the annual maximum series method is thepeaks over threshold method (Bacova-Mitkova and

Figure 1. Location of the Litij

Copyright © 2014 John Wiley & Sons, Ltd.

Onderka, 2010; Bezak et al., 2014). The graphical three-point method was used for baseflow separation to obtainhydrograph volumes and durations. Table I shows somedescriptive statistics values of the considered variables (peakdischarge, Q; hydrograph volume, V; and hydrographduration,D). First, we performed univariate flood frequencyanalyses. Different parametric distribution functions (i.e.log-Pearson 3, Pearson 3, GEV, GL, Gumbel, normal andlog-normal) were used, and the parameters were estimatedusing the method of moments, method of L-moments andmaximum likelihood method. Statistical and graphical testswere applied to select the distributions that produced the bestfit to the data (Šraj et al., 2012; Bezak et al., 2014). The Rpackage copula was used for the copula analyses (Hofertet al., 2013).The first step of the copula approach was to assess the

dependence between the pairs of considered variables.Chi-plot (Fisher and Switzer, 1985; Fisher and Switzer,2001) and K-plot (Genest and Boies, 2003) were used forthe graphical presentation of dependence. Three correlationcoefficients (Pearson, Kendall and Spearman) were alsocalculated. Pearson’s correlation coefficient measures onlylinear dependence, whereas Kendall’s and Spearman’scorrelation coefficients are based on ranks. In copulaparameter estimation, Kendall’s correlation coefficient wasselected over Spearman’s rho because Kendall’s tau is moreinsensitive to ties in data (Genest and Favre, 2007).Copula functions connect univariatemarginal distribution

functions with the multivariate probability distribution(Sklar, 1959):

a station on the Sava River

Hydrol. Process. (2014)

Table I. Some descriptive statistics of peak discharges (Q),hydrograph volumes (V) and hydrograph durations (D)

Statistics Q V D

Mean 1234.4m3/s 338 · 106/m3 12.98 daysMaximum 2326m3/s 766 · 106/m3 30 daysMinimum 579m3/s 123 · 106/m3 6 daysStandard deviation 391.62m3/s 152 · 106/m3 4.84 daysMedian 1174m3/s 300 · 106/m3 12 daysKurtosis �0.071 0.333 2.327Skewness 0.569 0.943 1.455

COPULA FLOOD FREQUENCY ANALYSIS FOR THE LITIJA STATION IN SLOVENIA

F x1; x2…; xnð Þ ¼ C FX1 x1ð Þ;FX2 x2ð Þ;…;FXn xnð Þf g (1)

where FX1 ;…;FXn are marginal distributions; if these arecontinuous, then the copula function C can be written asfollows:

C u1; u2…; unð Þ ¼ F F�1X1

u1ð Þ;F�1X2

u2ð Þ;…;F�1Xn

unð Þ� �

(2)

A more detailed description of copula functionproperties can be found in Joe (1997), Nelsen (1999)and Salvadori et al. (2007).Some basic characteristics of the chosen Archimedean

[Gumbel–Hougaard, Clayton, Frank, Joe, Ali–Mikhail–Haq (AMH)], extreme value (Galambos, Hüsler–Reiss,Tawn) and elliptical (Normal or Gaussian, Student-t)copulas are shown in Table II, where Φ represents thecumulative distribution function of the standard normaldistribution and tν is the standard t distribution with νdegrees of freedom.

Table II. Basic properti

Copula

Gumbel–Hougaard exp{�((�ln u)θ + (�ln v)θ)1/θ}

Cook–Johnson (Clayton) {u� θ + v� θ� 1}� 1/θ

Frank � 1θ ln 1þ e�θu�1ð Þ e�θv�1ð Þ

e�θ�1

� Joe 1� {(1� u)θ + (1� v)θ� (1�Ali–Mikhail–Haq uv

1�θ 1�uð Þ 1�vð ÞGalambos uvexp{((�ln u)� θ + (�ln v)� θ

Hüsler–Reiss exp � uΦ 1θ þ θ

2 lnuv

� �n o� v

hTawn uvexp �θ lnu lnv

ln uvð Þh i

Normal ∫Φ�1 uð Þ

�∞∫

Φ�1 vð Þ

�∞1

2πffiffiffiffiffiffiffiffiffiffiffiffi1�θ2ð Þp exp �s

�

Student-t ∫t�1ν uð Þ

�∞∫

t�1ν vð Þ

�∞1

2πffiffiffiffiffiffiffiffiffiffiffiffi1�θ2ð Þp 1þ s2�2

ν 1ð�


The parameters of one-parameter copulas were estimatedusing the method of moments with the use of the Kendallcorrelation coefficient (Nelsen, 1999; Genest and Favre,2007). For two-parameter copulas, the maximum likeli-hood method or maximum pseudo-likelihood method(Dupuis, 2007; Zhang and Singh, 2007b) could beselected. After parameter estimation, graphical andstatistical tests were used to determine which copulasproduced a better fit to the data. Graphical tests, whichcompare generated and measured values (Genest andFavre, 2007), were selected, and some tests comparingtheoretical and empirical copulas were conducted. Theempirical copula can be definedwith the following expression(bivariate case):

Cn u; vð Þ ¼ 1n∑n

i¼11

Ri

nþ 1≤ u;

Sinþ 1

≤v

� �(3)

where 1 is an indicator function (Salvadori et al., 2007) andRi and Si denote ranks of the ordered sample. We used theCramér-von Mises test (Genest and Remillard, 2008)defined by Genest et al. (2009):

SIn ¼ n∫ Cn uð Þ � Cθn uð Þf g2dCn uð Þ (4)

The test based on Kendall’s distribution was used todistinguish among the different copulas. The Kolmogo-rov–Smirnov test can be calculated using the followingexpression (Genest and Favre, 2007):

τn ¼ffiffiffin

pmaxi¼0;1;0≤j≤n�1 Kn

j

n

� �� Kθn

jþ i

n

� ��

� (5)

where the variableWi is defined by the following equation(Genest and Favre, 2007):

es of applied copulas

Cθ(u,v) θ ∈

[1,∞)

[�1,∞)\{0}

(�∞,∞)\{0}

u)θ(1� v)θ}1/θ [1,∞)[�1, 1)

)� 1/θ} [0,∞)

Φ 1θ þ θ

2 lnu v

� �n oiu ¼ � lnu;v ¼ � lnv [0,∞)

[0,1]

2�2θstþt2

2 1�θ2ð Þdsdt [�1, 1]

θstþt2

�θ2Þ�νþ2

2

dsdt [�1, 1]


;


Wi:n ¼ nn� 1

i� 1

!∫1

0wkθ wð Þ Kθ wð Þf gi�1 1� Kθ wð Þf gn�idw

Kθ wð Þ ¼ w� w ln w

(6)

where kθ(w) is the corresponding density (Genest andFavre, 2007). The drawback of this test is that there is nodifference between extreme-value copulas (Genest andFavre, 2007). To distinguish between extreme-valuecopulas, the Cramér-von Mises II statistical test wasapplied, which was defined by Genest et al. (2011)

SIIn ¼ ∫1

0n An wð Þ � Aθn wð Þj j2dw (7)

where An(w) is the non-parametric estimator and Aθn isthe parametric estimator of the Pickands dependencefunction A (Genest et al., 2011).Tail dependence coefficients can provide additional

information about the adequacy of the chosen copulamodels. More information about tail dependence can befound in Nelsen (1999) and Salvadori et al. (2007). Frahmet al. (2005) introduced the upper tail dependence estimator,whichwas also used by Poulin et al. (2007) and Ganguli andReddy (2013).The next step of the copula frequency analysis approach

was to calculate some conditional, primary and secondaryreturn periods for the bivariate case. Yue and Rasmussen(2002) introduced some basic concepts of bivariate returnperiods. Joint (primary) return periods called OR and ANDwere used (Salvadori et al., 2007; Graler et al., 2013):

TORu;v ¼

μ1� Cu;v u; vð Þ;

TANDu;v ¼ μ

1� u� vþ Cu;v u; vð Þ(8)

where μ is the mean interarrival time of the two consecutiveevents. Two conditional cases were also considered(Vandenberghe et al., 2011):

TU>ujV≤v ¼ μ

1� Cu;v u; vð Þu

;

TU>ujV>v ¼ μ1� u

11� u� vþ Cu;v u; vð Þ

(9)

The secondary return period, called Kendall’s returnperiod, is defined as follows (Salvadori et al., 2011):

T>x ¼ μ

1� Kc tð Þ (10)

where KC is Kendall’s distribution, associated withtheoretical copula function Cθ (Table II). This notation


of the return period is more similar to univariate returnperiods because the critical layer, the function of thecritical level t that is associated with the Kendall returnperiod, partitions the return period space into threeregions: sub-critical region, super-critical region andcritical layer (Salvadori et al., 2011). On the basis ofsimulations, Salvadori et al. (2011) provided an algorithmto calculate KC. The only condition is that copula functionbe available in the parametric form (Table II). For theArchimedean copulas, KC can be calculated withanalytical expressions (Vandenberghe et al., 2011).

RESULTS AND DISCUSSION

Fifty-eight flood events at the Litija gauge station on theSava River were considered in this study. The events wereselected based on the annual maximum discharge seriesmethod. After baseflow separation, the correspondinghydrograph volume and duration values were estimated.First, univariate flood frequency analyses were performed(Figure 2). The Weibull plotting position formula was usedfor data presentation (Figure 2). In all three cases,distribution parameters were estimated with the method ofL-moments, which produced better test results than themethod of moments or the maximum likelihood method(Šraj et al., 2012; Bezak et al., 2014). Graphical andstatistical test results indicated that the log-Pearson 3distribution was the most appropriate for modellingdischarge peaks and hydrograph durations, whereas thePearson 3 distribution was selected for hydrograph volumes(Šraj et al., 2012). Cumulative distribution functions andparameters of the selected marginal distributions arepresented in Table III. After univariate analyses, copulafunctions from different families of copulas were selected tomodel the pairs of variables.To assess the dependence between the pairs of considered

variables, correlation coefficients were calculated. ThePearson, Kendall and Spearman correlation coefficientvalues were 0.52, 0.39 and 0.54 for Q-V, 0.68, 0.48 and0.63 for V-D and �0.15, �0.08 and �0.14 for Q-D. Thecorrelation between the components of the pair V-D washigher than for the pairQ-V. One of the possible reasons forthis difference may be related to the baseflow separationmethod. Grimaldi and Serinaldi (2006a) investigated theinfluence of the threshold value of baseflow separation onthe correlation coefficient value and found that thecorrelation for V-D was higher than for the pair Q-V atlower threshold values (for some analysed stations) ofbaseflow separation but that the situation was reversed forhigher threshold values. In contrast, Karmakar andSimonovic (2008, 2009) and Reddy and Ganguli (2012b)obtained higher correlation values for the pair Q-V. Theseresults can lead to the conclusion that the dependence


Figure 2. Results of the univariate frequency analyses for the method of L-moments

Table III. Cumulative distribution functions and estimated parameters for the selected marginal distribution functions of individualvariables

Variable Distribution CDF Parameters

Q log-Pearson 3 FY yð Þ ¼ ∫y

0

1Γ αð Þ

y�cβ

� �α�1e� y�cð Þ=βdy; y= log x α= 182.88 β =� 0.01 c= 4.99

V Pearson 3 FX xð Þ ¼ ∫x

c

1βΓ αð Þ

x�cβ

� �α�1e� x�cð Þ=βdx α= 2.53 β = 9.82 * 107 c= 8.87 * 107

D log-Pearson 3 FY yð Þ ¼ ∫y

0

1Γ αð Þ

y�cβ

� �α�1e� y�cð Þ=βdy; y= log x α= 6.90 β = 0.06 c= 0.70


between different pairs of variables depends not only on theselected method of baseflow separation but also on the basincharacteristics (i.e. hydrograph properties). In our case, thelarge hydrograph duration values are mostly the conse-quence of hydrographs that were composited from severalsmaller hydrographs as the consecutive peaks weredependent. The correlation coefficient for Q-D was muchsmaller than for the other two pairs (Q-V and V-D). Thisfinding is consistent with findings from previous studies.Grimaldi and Serinaldi (2006a), Karmakar and Simonovic(2009) and Reddy and Ganguli (2012b) also found that thecorrelation coefficient forQ-Dwas smaller than for the othertwo pairs.Graphical representations of the dependence of differ-

ent variable pairs are shown with the use of Chi-plots andK-plots (Figure 3). As we can see, the variables Q and Vare dependent, but the dependence is not strong. If twoindependent variables were observed, the majority ofevents would be located in the area defined by theconfidence intervals in the Chi-plot (Fisher and Switzer,1985; Fisher and Switzer, 2001). In our case, fewer than20% of the events lie in this area. For the K-plot, we canassume that if we had an independent case, the eventswould lie on the line x = y. If the events are located above


the line, a positive dependence is indicated, and viceversa, such that if the points lie under the line, a negativedependence is indicated. The results shown in Figure 3 yieldconclusions similar to those obtained by the calculatedcorrelation coefficients. Furthermore, theChi-plot (Figure 3)shows that in approximately one-third of events, one value ishigher (Q or V), and one value is lower (Q or V) than themedian value (negative lambda values). Thus, high-peakdischarge does not necessarily mean high hydrographvolume. As mentioned previously, the reason could be thebaseflow separation method. The analysis for the pair V-Dindicates that the dependence between the variables ispositive (Figure 3). Fewer than 15% of the events lie in thearea defined by the confidence intervals in the Chi-plot,which is less than for the pair Q-V. The ties in the sampleV-D are the consequence of the integer values of thehydrograph durations due to the baseflow separationmethodology and the time scale of the data used (daily timeseries). However, a dependence assessment for the pairQ-Dshowed that more than 85% of the events lie in theconfidence area defined in the Chi-plot and that almost alldata points lie near the line x= y in the K-plot (Figure 3). Onthe basis of these findings, we can conclude that thedependence betweenQ andD is very weak or hardly exists.


1000 1500 20001e+

083e

+08

5e+

087e

+08

Scatter plot

Q [m3/s]

V [m

3]

−1.0 −0.5 0.0 0.5 1.0

−1.

0−

0.5

0.0

0.5

1.0

Chi−plot

λ

χ

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

K plot

W1:n

H

xxxxx

xxxx

xxxxxxxxxxxx

xxxxx

x

xxxxx

xxxx

xxx

xxxx x

xx

x xx

xx x x

x xx

x

x

1e+08 3e+08 5e+08 7e+08

1015

2025

30

Scatter plot

V [m3]

D [d

ays]

−1.0 −0.5 0.0 0.5 1.0

−1.

0−

0.5

0.0

0.5

1.0

Chi−plot

λ

χ

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

K plot

W1:n

H

xxxxxxxxxxx

x

xxxxxxx

xxxxxx

xxxxxx

xxxxx

xxx

xxx

xx

xx

xx

xx x

xx

xx x

xx

1000 1500 2000

1015

2025

30

Scatter plot

Q [m3/s]

D [d

ays]

−1.0 −0.5 0.0 0.5 1.0

−1.

0−

0.5

0.0

0.5

1.0

Chi−plot

λ

χ

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

K plot

W1:n

H

xxxxxxxx

xxxxxxxx

xxx

xxxxxxxx

xxxxx

xxxxxx

xxxx

xx

x xx x

xx x

xx

xx

xx

x

Figure 3. Assessment of dependence between peak discharges – hydrograph volumes (Q-V), hydrograph volumes – hydrograph durations (V-D) andpeak discharges – hydrograph durations (Q-D)


This conclusion is consistent with that obtained by thecorrelation coefficient calculation.In the next step, different bivariate copulas from three

families were applied to all three variable pairs (Table II).The parameters of one-parameter copulas were estimatedusing the method of moments with the use of the Kendallcorrelation coefficient. For two-parameter copulas, themaximum likelihood method or maximum pseudo-likelihood method could be used. The considered copulaswere compared with different statistical, graphical andupper tail dependence tests. On the basis of the correlationcoefficient results, the AMH copula could not be appliedfor the pair Q-V because it can only be used for Kendall’scorrelation coefficient values between �0.18 and 0.33


(Nelsen, 1999). All other copulas from Table II wereconsidered; the results of the Cramér-von Mises test SInand Kolmogorov–Smirnov test τn for the pair Q-V arepresented in Table IV. p-values were calculated based onthe parametric bootstrap or multipliers (Kojadinovicet al., 2011) procedure with 10 000 runs. On the basisof a graphical test (Figure 4), ν= 6 was selected for theStudent-t copula. To distinguish between the individualextreme value copula functions, the Cramér von-Mises teststatistic SIIn was also calculated. The results and corres-ponding p-values were 0.032 (0.235), 0.033 (0.220), 0.034(0.189) and 0.032 (0.381) for the Gumbel–Hougaard,Galambos, Hüsler–Reiss and Tawn copulas, respectively.On the basis of the presented statistical and graphical tests,


Table IV. Some statistical test results for the pairs of peak discharge – hydrograph volume, hydrograph volume –hydrograph duration and peak discharge – hydrograph duration

Pair Q-V V-D Q-D

Copula SIn (p-value) τn SIn (p-value) SIn (p-value)

Gumbel–Hougaard 0.021 (0.29) 0.697 0.022 (0.12) /Clayton 0.034 (0.05) 0.772 0.041 (0.01) 0.028 (0.19)Frank 0.019 (0.39) 0.705 0.023 (0.11) 0.029 (0.14)Joe 0.028 (0.26) 0.902 0.033 (0.09) /Ali–Mikhail–Haq / / / 0.033 (0.07)Galambos 0.021 (0.29) 0.697 0.022 (0.11) /Hüsler–Reiss 0.021 (0.27) 0.697 0.022 (0.12) /Tawn 0.021 (0.29) 0.697 / /Normal 0.020 (0.34) 0.704 0.022 (0.13) 0.028 (0.17)Student-t 0.020 (0.33) 0.727 0.022 (0.14) 0.025 (0.25)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Gumbel−Hougaard

u

v

xx

x

xx

x

xx

x

x

x

xx

x

x

x

x

xx

x

x

xx

x

x

x

x

x

xx

x x

xx

x

x

x

x

x

x

x

x

x

x

x

xx

x

xx

x

x

x

x

x

x

x

x

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Frank

u

v

xx

x

xx

x

xx

x

x

x

xx

x

x

x

x

xx

x

x

xx

x

x

x

x

x

xx

x x

xx

x

x

x

x

x

x

x

x

x

x

x

xx

x

xx

x

x

x

x

x

x

x

x

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Tawn

u

v

xx

x

xx

x

xx

x

x

x

xx

x

x

x

x

xx

x

x

xx

x

x

x

x

x

xx

x x

xx

x

x

x

x

x

x

x

x

x

x

x

xx

x

xx

x

x

x

x

x

x

x

x

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0 1000 2000 3000 4000

2.0e

+08

6.0e

+08

1.0e

+09

1.4e

+09

Gumbel−Hougaard

Q [m3/s]

V [m

3]

xx

x

xx

x

xx

x

x

x

x xxx

x

x

x x

x

x

xxxx

xxx

xx

x x

x x

x

x

x

x

x

x

x

x

x

x x

xx

xxxxx

xx

x

xx

x

0 1000 2000 3000 4000

2.0e

+08

6.0e

+08

1.0e

+09

1.4e

+09

0 1000 2000 3000 4000

2.0e

+08

6.0e

+08

1.0e

+09

1.4e

+09

Frank

Q [m3/s]

V [m

3]

xx

x

xx

x

xx

x

x

x

x xxx

x

x

x x

x

x

xxxx

xxx

xx

x x

x x

x

x

x

x

x

x

x

x

x

x x

xx

xxxxx

xx

x

xx

x

0 1000 2000 3000 4000

2.0e

+08

6.0e

+08

1.0e

+09

1.4e

+09

0 1000 2000 3000 4000

2.0e

+08

6.0e

+08

1.0e

+09

1.4e

+09

Tawn

Q [m3/s]

V [m

3]

xx

x

xx

x

xx

x

x

x

x xxx

x

x

x x

x

x

xxxx

xxx

xx

x x

x x

x

x

x

x

x

x

x

x

x

x x

xx

xxxxx

xx

x

xx

x

0 1000 2000 3000 4000

2.0e

+08

6.0e

+08

1.0e

+09

1.4e

+09

Figure 4. Comparison between the observed and simulated values (Q-V) for three chosen copula functions (Gumbel–Hougaard, Frank and Tawn)


the Gumbel–Hougaard, Frank and Tawn copulas wereselected for further analyses. Figure 4 shows the comparisonbetween simulated (sample size 10000) and measured


values (Q-V). The upper part of Figure 4 shows acomparison between the simulated values and the pairs ofranks, whereas the lower part presents the graphical


Figure 5. Comparison between level curves for empirical and fitted theoretical Gumbel–Hougaard, Frank and Tawn copulas (Q-V)

Figure 6. Comparison between the observed and simulated values (V-D) for three chosen copula functions (Gumbel–Hougaard, Frank and Normal)


Copyright © 2014 John Wiley & Sons, Ltd. Hydrol. Process. (2014)


goodness-of-fit test for the entire copula model (includingmarginal distributions). The comparison between empiricaland theoretical copula functions for the pairQ-V is presentedin Figure 5. The solid thick line represents the empiricalcopula, and the dashed thin line shows the chosen theoreticalcopula function. Identifying the differences among the threechosen copula functions is difficult; however, all threecopulas produced a good graphical fit to the empiricalcopula (sample). Additional information about the adequacyof the chosen copula models was obtained with the taildependence test. The characteristics of the Frank copula arethe lower (λL) and upper (λU) tail dependence parameterswith value 0 (Nelsen, 1999), whereas for the Gumbel–Hougaard and Tawn copulas, values of λL = 0 andλU=0.468, respectively, were calculated. The empirical taildependence coefficient λCFGU =0.439was also estimated. Theresults of the first statistical tests (Table IV) could also beconfirmed with the upper tail coefficient values (λU), whichwere 0 and 0.618 for the Clayton and Joe copulas,

Figure 7. Comparison between observed and simulated values (Q-D) for t


respectively. Given that the Frank copula could underesti-mate the risk of an event because it cannot model the taildependence efficiently (Poulin et al., 2007), we selected theGumbel–Hougaard copula as the most appropriate formodelling the pair Q-V. This finding is in accordance withZhang and Singh (2006), Poulin et al. (2007) and Karmakarand Simonovic (2009).Because the AMH copula can only be used for

Kendall’s correlation coefficient values between �0.18and 0.33 and the Tawn copula for values in the range of[0; 0.418], both copulas were excluded from furtheranalysis for the pair V-D. The Cramér-von Mises test SInresults for the other copulas applied to the pair V-D arepresented in Table IV. On the basis of the graphical test,ν= 8 was selected for the Student-t copula. We can seethat Clayton copula was rejected with the chosensignificance level of 0.05. The test results and corre-sponding p-values for the Cramér von-Mises test SIIn forthe Gumbel–Hougaard, Galambos and Hüsler–Reiss

hree chosen copula functions (Clayton, Ali–Mikhail–Haq and Student-t)


Table V. Comparison of different conditional, primary and secondary return periods (years) for selected copulas for the median (firstvalue) and maximum (second value) values of peak discharge, hydrograph volume and hydrograph duration and for two t values

(Kendall’s return period)

Pair CopulaTORu;v TAND

u;v TU>u|V≤ v TU>u|V>v Kendall’s RP

Med Max Med Max Med Max Med Max t= 0.9 t = 0.99

Q-V Gumbel 1.2 40.7 2.8 132.8 3.1 96.7 2.8 2764.8 24.0 258.3Frank 1.5 32.1 2.7 1035.5 3.4 58.8 5.3 71 938.8 56.2 5027.7Tawn 1.5 40.7 2.8 133.0 3.1 96.6 5.5 9236.8 24.1 258.5

V-D Gumbel 1.6 43.6 2.8 111.7 4.4 187.8 5.4 6300.1 19.8 207.8Frank 1.6 32.6 2.7 801.4 5.0 75.9 5.2 45218.2 44.2 3681.9Normal 1.6 36.4 2.8 224.6 4.4 100.9 5.4 12 674.7 28.5 543.8

Q-D Clayton 1.3 35.2 4.5 5700.3 2.0 70.3 8.8 396 020 224.6 24154.6AMH 1.3 35.1 4.6 8125.7 1.9 70.1 9.0 564 522 318.3 37174.7Student-t 1.3 36.8 4.5 708.6 2.0 77.1 8.8 49 227 105.6 1431.4

Figure 8. Joint return period values for OR and AND cases for the analysed pairs


Copyright © 2014 John Wiley & Sons, Ltd. Hydrol. Process. (2014)


copulas were 0.014 (0.412), 0.015 (0.358) and 0.017(0.282), respectively. On the basis of these test results, theGumbel–Hougaard, Frank and Normal (Gauss) copulaswere selected for further analyses for the pair V-D(Figure 6). Tail dependence coefficients provided addi-tional information about the adequacy of the chosencopula models. The Normal and Frank copulas had lowerand upper tail coefficients of 0. In addition, λL= 0 andλU= 0.565 were calculated for the Gumbel–Hougaardcopula from the Archimedean family. Additionally, theempirical tail dependence coefficient λCFGU was calculated,which was 0.570. The Clayton copula was rejected by theCramér-von Mises test, and the upper tail dependencecoefficient (λU) was 0 (significance level 0.05). Incontrast, the upper tail dependence coefficient for theJoe copula was 0.707. On the basis of the same reason asthat for the pair Q-V (underestimation of the risk by the

500 1000 1500 2500

010

020

030

040

050

0

Q>q | V<=v (Q−V; Gumbel)

Q [m3/s]

T [y

ears

]

V=2*10^8 m3V=4*10^8 m3V=6*10^8 m3V=7*10^8 m3

1e+08 2e+0

010

020

030

040

050

0

V>V | D<=d

V

T [y

ears

]

1200 1600 2000 2400

010

020

030

040

050

0

Q>q | V>v (Q−V; Gumbel)

Q [m3/s]

T [y

ears

]

V = 2*10^8 m3V = 4*10^8 m3V = 6*10^8 m3V = 7*10^8 m3

1e+08 2e+0

010

020

030

040

050

0

V>v | D>d

V

T [y

ears

]

D = 5D = 1D = 2D = 2

Figure 9. Conditional return period values for the


Frank copula), we decided to select the Gumbel–Hougaard copula as the most appropriate for modellingthe pair V-D.For the pairQ-D, only the Clayton, Frank, AMH, Normal

and Student-t copulas from the Archimedean and ellipticalfamilies could be applied. All other copulas (Table II) couldnot be used because of the negatively dependent data. ν=3was selected for the Student-t copula. The Cramér-vonMises testSIn results are presented in Table IV. Given that theClayton, AMH and Student-t copulas have different taildependence parameters, they were chosen for furtheranalyses for the pair Q-D. Figure 7 shows the comparisonamong three selected copulas (10 000 simulated values andmeasured values). The empirical tail dependence coefficientλCFGU for the pairQ-Dwas 0.068. The upper tail dependencecoefficients (λU) for the Clayton, Frank, AMH and Normalcopulas were 0, and for the Student-t copula, λU=0.087,

8 5e+08

(V−D; Gumbel)

[m3]

D = 5 daysD = 10 daysD = 15 daysD = 20 days

5 10 15 20 30

020

4060

8010

0

Q>q | D<=d (Q−D; Student−t)

D [days]

T [y

ears

]

Q = 500 m3/sQ = 1000 m3/sQ = 1250 m3/sQ = 1500 m3/s

8 5e+08

(V−D; Gumbel)

[m3]

days5 days0 days5 days

500 1000 1500 2500

010

020

030

040

050

0

Q>q | D>d (Q−D; Student−t)

Q [m3/s]

T [y

ears

]

D = 10 daysD = 15 daysD = 20 daysD = 25 days

TU>u|V>v and TU>u|V≤ v cases for analysed pairs



meaning that the Student-t copula produced a better fit to ourdata sample. Therefore, the Student-t copula was selected asthe most appropriate for frequency analysis via copulas forthe pair Q-D.For each pair of variables (Q-V, V-D and Q-D), three

copula functions were selected for further analyses based onthe statistical and graphical goodness-of-fit test results. Forthese selected copulas, different primary, secondary andconditional return periodswere calculated (Table V). For thecalculation of the secondary Kendall’s return period, weused 107 simulations. Given the tail dependence indices andthe comparison of the return period values (Table V), themost appropriate copulas were selected for each pair ofvariables. Figure 8 shows the joint return (primary) periodsfor the OR and AND cases. Two conditional return periodsdefinedwith Equation (9) are presented in Figure 9. Figure 9shows the results for the copulas that were selected as themost appropriate for frequency analysis, whereas Figure 8shows the results for some copula functions that could alsobe applied to data from the Litija station on the Sava River.Table V shows that different relationships between thereturn periods could be obtained. The return period TU>u|

V>v is always higher than TANDu;v , and the primary return

periodTORu;v is always lower than the conditional return period

TU>u|V≤ v. Furthermore, the differences between the returnperiods are much more expressive for maximum values ofvariables than for median values. Salvadori et al. (2007)showed that the relationship between univariate and primary(bivariate) return periods can be writ ten asTORu;v < TUNI < TAND

u;v . In our case, univariate return periodsfor all three marginal distributions were calculated (log-Pearson 3 forQ andD and Pearson 3 for V), and for medianvalues ofQ,V andD, return periods of 2.0, 1.9 and 2.1 years,respectively, were obtained. For the maximum values of Q,V and D, the calculated values of the return periods were69.5, 56.4 and 70.5 years, respectively (Šraj et al., 2012). Aswe can see, these results are in accordance with the findingsof Salvadori et al. (2007). Copulas that could underestimatethe risk (Clayton, Frank; Poulin et al., 2007) also producedhigher secondary Kendall’s return period values for all threeconsidered variable pairs (Table V). By comparing theresults for the return periods of 10 and 100 years for the pairsQ-V, V-D andQ-D, we can also conclude that the theoreticalrelationship between the return periods holds. The corre-sponding discharge, hydrograph volume and hydrographduration values for a return period of 10years (TUNI) were1768m3/s, 546*106m3 and 19.2 days, respectively. If afixed value of the return period is selected for the joint returnperiods ORorAND (TOR

u;v ; TANDu;v ), different combinations of

variables are suitable for individual conditions (Figure 8).However, we should bear inmind that the direct comparisonof different conditional and joint return period values isdifficult due to the different physical meanings of individualreturn periods.


CONCLUSIONS

In the present work, a detailed case study of copula approachflood frequency analysis was performed using an extensiveselection of bivariate copulas as well as different statistical,graphical and upper tail dependence tests for comparisonamong them. The parameters of one-parameter copulaswereestimated using the method of moments with the use of theKendall correlation coefficient. Different primary, second-ary and conditional return periods were evaluated andcompared, and the relationship between univariate andbivariate return periods was presented.The following conclusions may be drawn from this study.

(i) The correlation coefficient for Q-D is much smaller thanfor the other two pairs of variables (Q-V and V-D).Consistent with these results, the correlation for thepairs of peak discharge – hydrograph volume (Q-V) andhydrograph volume – hydrograph duration (V-D) wasstatistically significant, whereas the correlation for the pairpeak discharge – hydrograph duration (Q-D) was statisti-cally non-significant. (ii) On the basis of graphical andstatistical test results and upon the observation of the uppertail dependence coefficient, the Gumbel–Hougaard copulafrom the Archimedean family of copulas was selected asthe most appropriate copula for modelling peak dischargesand hydrograph volumes (Q-V) as well as hydrographvolumes and hydrograph durations (V-D). Given thatKendall’s correlation coefficient is negative for the pairpeak discharge – hydrograph duration (Q-D), only fivepresented copulas remain for modelling this pair ofvariables. The Student-t copula (ν=3) was selected asthe most appropriate for the pair peak discharge –hydrograph duration (Q-D). (iii) The presented statistical,graphical and upper tail dependence tests showed that thedifferences between some copulas are small and notsignificant, and thus some additional copulas could beselected as the most appropriate for individual pairs ofvariables. With the exception of the Clayton copula in thecase of hydrograph volume and hydrograph duration (V-D)modelling, none of the selected copulas was rejected by theCramér-von Mises test with the chosen significance levelof 0.05. (iv) On the basis of our results, differentrelationships between the return periods were obtained.The conditional return period TU>u|V>v was always higherthan primary return period TAND

u;v , and the primary returnperiod TOR

u;v was always lower than the conditional returnperiod TU>u|V≤ v. The univariate return period was higherthan primary return period TOR

u;v and lower than the returnperiod TAND

u;v . These relationships are in accordance with thefindings of Salvadori et al. (2007). The differencesbetween the return periods are much more expressive formaximum values of variables than for median values. (v)Copulas with λU = 0 (e.g. Frank, Clayton) generally gavehigher estimated return period values (TAND

u;v , TU>u|V>v,



and Kendall’s return period) than some other copulafunctions. This finding is consistent with the results ofPoulin et al. (2007).The results of the study indicate that the copula flood

frequency approach can be useful to determine occurrenceprobabilities of different pairs of variables of flood events.These various occurrence combinations can be useful formore reliable risk assessments associated with floodcontrol and hydrologic design, given that extreme eventssuch as floods are multivariate, and therefore, the risk ismore likely to be a function of several variables. Thepresented methodology can also be followed as anexample for other similar case studies.

ACKNOWLEDGEMENTS

We wish to thank the Environmental Agency of theRepublic of Slovenia for data provision. The results of thestudy are part of the Slovenian national research projectJ2-4096. The study was partially supported by theSlovenian Research Agency (ARRS). The critical anduseful comments of two anonymous reviewers improvedthis work, for which the authors are very grateful.

REFERENCES

Ariff NM, Jemain AA, Ibrahim K, Zin WZW. 2012. IDF relationshipsusing bivariate copula for storm events in Peninsular Malaysia. Journalof Hydrology 470: 158–171. DOI: 10.1016/j.jhydrol.2012.08.045.

Ashkar F, Aucoin F. 2011. A broader look at bivariate distributionsapplicable in hydrology. Journal of Hydrology 405: 451–461. DOI:10.1016/j.jhydrol.2011.05.043.

Bacova-Mitkova V, Onderka M. 2010. Analysis of extreme hydrologicalevents on the danube using the peak over threshold method. Journal ofHydrology and Hydromechanics 58: 88–101. DOI: 10.2478/v10098-010-0009-x.

Balistrocchi M, Bacchi B. 2011. Modelling the statistical dependence ofrainfall event variables through copula functions. Hydrology and EarthSystem Sciences 15: 1959–1977. DOI: 10.5194/hess-15-1959-2011.

Bardossy A. 2006. Copula-based geostatistical models for groundwaterquality parameters. Water Resources Research 42. DOI: 10.1029/2005wr004754.

Bezak N, Brilly M, Šraj M. 2014. Comparison between the peaks overthreshold method and the annual maximum method for flood frequencyanalyses. Hydrological Sciences Journal-Journal Des SciencesHydrologiques. DOI: 10.1080/02626667.2013.831174.

Bonacci O, Oskorus D. 2010. The changes in the lower Drava River waterlevel, discharge and suspended sediment regime. Environmental EarthSciences 59: 1661–1670. DOI: 10.1007/s12665-009-0148-8.

Chen L, Singh VP, Guo SL, Hao ZC, Li TY. 2012. Flood coincidence riskanalysis using multivariate copula functions. Journal of HydrologicEngineering 17: 742–755. DOI: 10.1061/(asce)he.1943-5584.0000504.

De Michele C, Salvadori G. 2003. A Generalized Pareto intensity-durationmodel of storm rainfall exploiting 2-Copulas. Journal of GeophysicalResearch-Atmospheres 108. DOI: 10.1029/2002jd002534.

De Michele C, Salvadori G, Canossi M, Petaccia A, Rosso R. 2005.Bivariate statistical approach to check adequacy of dam spillway.Journal of Hydrologic Engineering 10: 50–57. DOI: 10.1061/(asce)1084-0699(2005)10:1(50).

Dupuis DJ. 2007. Using copulas in hydrology: benefits, cautions, andissues. Journal of Hydrologic Engineering 12: 381–393. DOI: 10.1061/(asce)1084-0699(2007)12:4(381).


Favre AC, El Adlouni S, Perreault L, Thiemonge N, Bobee B. 2004.Multivariate hydrological frequency analysis using copulas. WaterResources Research 40. DOI: 10.1029/2003wr002456.

Fisher NI, Switzer P. 1985. Chi-plots for assessing dependence.Biometrika 72: 253–265. DOI: 10.1093/biomet/72.2.253.

Fisher NI, Switzer P. 2001. Graphical assessment of dependence: is apicture worth 100 tests? American Statistician 55: 233–239. DOI:10.1198/000313001317098248.

Frahm G, Junker M, Schmidt R. 2005. Estimating the tail-dependencecoefficient: properties and pitfalls. Insurance Mathematics & Econom-ics 37: 80–100. DOI: 10.1016/j.matheco.2005.05.008.

Ganguli P, Reddy MJ. 2013. Probabilistic assessment of flood risks usingtrivariate copulas. Theoretical and Applied Climatology 111: 341–360.DOI: 10.1007/s00704-012-0664-4.

Gargouri-Ellouze E, Chebchoub A. 2008. Modelling the dependencestructure of rainfall depth and duration by Gumbel’s copula.Hydrological Sciences Journal-Journal Des Sciences Hydrologiques53: 802–817. DOI: 10.1623/hysj.53.4.802.

Genest C, Boies JC. 2003. Detecting dependence with Kendall plots.American Statistician 57: 275–284. DOI: 10.1198/0003130032431.

Genest C, Favre AC. 2007. Everything you always wanted to know aboutcopulamodeling but were afraid to ask. Journal of Hydrologic Engineering12: 347–368. DOI: 10.1061/(asce)1084-0699(2007)12:4(347).

Genest C, Remillard B. 2008. Validity of the parametric bootstrap forgoodness-of-fit testing in semiparametric models. Annales De L InstitutHenri Poincare-Probabilites Et Statistiques 44: 1096–1127. DOI:10.1214/07-aihp148.

Genest C, Remillard B, Beaudoin D. 2009. Goodness-of-fit tests forcopulas: a review and a power study. Insurance Mathematics &Economics 44: 199–213. DOI: 10.1016/j.insmatheco.2007.10.005.

Genest C, Kojadinovic I, Neslehova J, Yan J. 2011. A goodness-of-fit testfor bivariate extreme-value copulas. Bernoulli 17: 253–275. DOI:10.3150/10-bej279.

Ghosh S. 2010. Modelling bivariate rainfall distribution and generatingbivariate correlated rainfall data in neighbouring meteorologicalsubdivisions using copula. Hydrological Processes 24: 3558–3567.DOI: 10.1002/hyp.7785.

Graler B, van den Berg MJ, Vandenberghe S, Petroselli A, Grimaldi S, DeBaets B, Verhoest NEC. 2013. Multivariate return periods in hydrology:a critical and practical review focusing on synthetic design hydrographestimation. Hydrology and Earth System Sciences 17: 1281–1296. DOI:10.5194/hess-17-1281-2013.

Grimaldi S, Serinaldi F. 2006a. Asymmetric copula in multivariate floodfrequency analysis. Advances in Water Resources 29: 1155–1167. DOI:10.1016/j.advwatres.2005.09.005.

Grimaldi S, Serinaldi F. 2006b. Design hyetograph analysis with 3-copulafunction. Hydrological Sciences Journal-Journal Des SciencesHydrologiques 51: 223–238. DOI: 10.1623/hysj.51.2.223.

Hofert M, Kojadinovic I, Maechler M, Yan J. 2013. Copula: multivariatedependence with copulas. R package version 0.999-7, URL: http://CRAN.R-project.org/package=copula (13.8.2013).

Joe H. 1997. Multivariate models and dependence concepts. Chapman &Hall: London, New York; 399.

Kao SC, Govindaraju RS. 2010. A copula-based joint deficit index fordroughts. Journal of Hydrology 380: 121–134. DOI: 10.1016/j.jhydrol.2009.10.029.

Karmakar S, Simonovic SP. 2008. Bivariate flood frequency analysis:Part1. Determination of marginals by parametric and nonparametrictechniques. Journal of Flood Risk Management 1: 190–200. DOI:10.1111/j.1753-318X.2008.00022.x.

Karmakar S, Simonovic SP. 2009. Bivariate flood frequency analysis.Part 2: a copula-based approach with mixed marginal distributions.Journal of Flood Risk Management 2: 32–44. DOI: 10.1111/j.1753-318X.2009.01020.x.

Kojadinovic I, Yan J, Holmes M. 2011. Fast large-sample goodness-of-fittests for copulas. Statistica Sinica 21: 841–871.

Lee T, Modarres R, Ouarda T. 2013. Data-based analysis of bivariatecopula tail dependence for drought duration and severity. HydrologicalProcesses 27: 1454–1463. DOI: 10.1002/hyp.9233.

Liu CL, Zhang Q, Singh VP, Cui Y. 2011. Copula-based evaluations ofdrought variations in Guangdong, South China. Natural Hazards 59:1533–1546. DOI: 10.1007/s11069-011-9850-4.


http://CRAN.R-project.org/package=copula

http://CRAN.R-project.org/package=copula


Ma MW, Song SB, Ren LL, Jiang SH, Song JL. 2013. Multivariatedrought characteristics using trivariate Gaussian and Student tcopulas. Hydrological Processes 27: 1175–1190. DOI: 10.1002/hyp.8432.

Nelsen RB. 1999. An Introduction to Copulas. Springer: New York; 272.Poulin A, Huard D, Favre AC, Pugin S. 2007. Importance of taildependence in bivariate frequency analysis. Journal of HydrologicEngineering 12: 394–403. DOI: 10.1061/(asce)1084-0699(2007)12:4(394).

Reddy MJ, Ganguli P. 2012a. Application of copulas for derivation ofdrought severity-duration-frequency curves. Hydrological Processes26: 1672–1685. DOI: 10.1002/hyp.8287.

Reddy MJ, Ganguli P. 2012b. Bivariate flood frequency analysis of upperGodavari River flows using archimedean copulas. Water ResourcesManagement 26: 3995–4018. DOI: 10.1007/s11269-012-0124-z.

Renard B, Lang M. 2007. Use of a Gaussian copula for multivariateextreme value analysis: some case studies in hydrology. Advances inWater Resources 30: 897–912. DOI: 10.1016/j.advwatres.2006.08.001.

Salvadori G, De Michele C. 2004. Frequency analysis via copulas:theoretical aspects and applications to hydrological events. WaterResources Research, 40. DOI: 10.1029/2004wr003133.

Salvadori G, De Michele C, Kottegoda NT, Rosso R. 2007. Extremes inNature an Approach using Copulas. Springer: Dordrecht; 292.

Salvadori G, De Michele C, Durante F. 2011. On the return period anddesign in a multivariate framework. Hydrology and Earth SystemSciences 15: 3293–3305. DOI: 10.5194/hess-15-3293-2011.

Serinaldi F, Grimaldi S. 2011. Synthetic design hydrographs based ondistribution functions with finite support. Journal of HydrologicEngineering 16 : 434–446. DOI: 10.1061/(asce)he.1943-5584.0000339.

Singh VP, Zhang L. 2007. IDF curves using the Frank Archimedeancopula. Journal of Hydrologic Engineering 12: 651–662. DOI:10.1061/(asce)1084-0699(2007)12:6(651).


Sklar A. 1959. Fonction de re’partition a’ n dimensions et leurs marges, vol.8. Publications de L’Institute de Statistique, Universite’ de Paris: Paris;229–231.

Šraj M, BezakN, BrillyM. 2012. The influence of the choice of method on theresults of frequency analysis of peaks, volumes and durations of floodwavesof the Sava River in Litija. Acta Hydrotehnica 25: 41–58. (In Slovenian).

Vandenberghe S, Verhoest NEC, De Baets B. 2010. Fitting bivariatecopulas to the dependence structure between storm characteristics: adetailed analysis based on 105 year 10 min rainfall. Water ResourcesResearch, 46. DOI: 10.1029/2009wr007857.

Vandenberghe S, Verhoest NEC, Onof C, De Baets B. 2011. Acomparative copula-based bivariate frequency analysis of observedand simulated storm events: a case study on Bartlett-Lewis modeledrainfall. Water Resources Research 47. DOI: 10.1029/2009wr008388.

Wong G, Lambert MF, Leonard M, Metcalfe AV. 2010. Drought analysisusing trivariate copulas conditional on climatic states. Journal of HydrologicEngineering 15: 129–141. DOI: 10.1061/(asce)he.1943-5584.0000169.

Wong G, van Lanen HAJ, Torfs P. 2013. Probabilistic analysis ofhydrological drought characteristics using meteorological drought.Hydrological Sciences Journal-Journal Des Sciences Hydrologiques58: 253–270. DOI: 10.1080/02626667.2012.753147.

Yue S, Rasmussen P. 2002. Bivariate frequency analysis: discussion ofsome useful concepts in hydrological application. HydrologicalProcesses 16: 2881–2898. DOI: 10.1002/hyp.1185.

Zhang L, Singh VP. 2006. Bivariate flood frequency analysis using thecopula method. Journal of Hydrologic Engineering 11: 150–164. DOI:10.1061/(asce)1084-0699(2006)11:2(150).

Zhang L, Singh VP. 2007a. Bivariate rainfall frequency distributions usingArchimedean copulas. Journal of Hydrology 332: 93–109. DOI:10.1016/j.jhydrol.2006.06.033.

Zhang L, Singh VP. 2007b. Trivariate flood frequency analysis using theGumbel–Hougaard copula. Journal of Hydrologic Engineering 12:431–439. DOI: 10.1061/(asce)1084-0699(2007)12:4(431).


bivariate flood frequency analysis using the copula function: a case study of the litija station on...

Documents