Drought Analysis Using CopulasLu Chen1; Vijay P. Singh, F.ASCE2; Shenglian Guo3; Ashok K. Mishra4; and Jing Guo5

Abstract: Droughts produce a complex set of negative economic, environmental, and social impacts from regional to national scales.The drought impact can be quantified using drought index time series. This study uses the monthly standardized precipitation index(SPI) time series as a drought index. Drought characteristics, namely, drought duration, severity, interval time and minimum SPI values,were determined. Five hundred years of daily rainfall data were simulated for evaluating drought characteristics. Appropriate distributionswere selected for modeling drought duration, interval time, drought severity, and minimum SPI value in different drought states. The droughtepisodes were quantified using multivariate copula methods. Several copulas from the Archimedean and metaelliptical families were appliedto construct four-dimensional joint distributions. The dependence structure in each drought state was investigated and drought probabilitiesand return period were calculated and analyzed based on a four-dimensional copula using the upper Han River basin, China, as a study area.DOI: 10.1061/(ASCE)HE.1943-5584.0000697. © 2013 American Society of Civil Engineers.

CE Database subject headings: Droughts; China; River basins.

Author keywords: Drought characterization; Multivariate distribution; Standardized precipitation index (SPI); Copula.

Introduction

A drought is a natural hazard that results from a deficiency of pre-cipitation as compared with the expected or normal amount, whichcan translate into an insufficient amount of water to meet the de-mands of human activities and the environment (Estrela and Vargas2012). It occurs in virtually all climatic zones, including both highand low rainfall areas. Each year, some part of the world experi-ences a drought and suffers from huge economic losses, and thishas been the case ever since the birth of human civilization. Theimpacts produced by droughts are numerous. Historical droughtshave affected large populations (and represent up to 35% of thoseaffected by natural disasters), often resulting in significant fatalities(50% of the mortality due to natural disasters), whereas 7% ofworld economic losses have been attributed to their occurrence(Below et al. 2007; Núñez et al. 2011). Thus, droughts are of great

importance in the planning and management of water resources(Mishra and Singh 2010).

The Han River, which is divided into three regions: theDanjiangkou Reservoir sub-basin (upper sub-basin), the middlesub-basin, and the lower sub-basin, is a tributary of the YangtzeRiver. This river is the source of water for the middle route of thewell-known South-to-North Water Diversion Project (SNWDP) inChina. The middle route, located between the Danjiangkou Reser-voir in the Han River and Beijing, will transfer 14 billionm3 ofwater annually from the Han River to Beijing by 2030. As someof its water is transferred via the SNWDP, the Han River has animpact on socioeconomic development both in the middle andlower sub-basins and northern China. For this reason, the HanRiver was selected as a case study.

Various indices have been developed to detect and monitordroughts, and commonly, Palmer Drought Severity Index (PDSI)and the standardized precipitation index (SPI) are more frequentlyused indices (Mishra and Singh 2010) for drought characterization.Palmer (1965) proposed a moisture index (Palmer Drought SeverityIndex, PDSI) based on water budget accounting using precipitationand temperature data. McKee et al. (1993) proposed the concept ofstandardized precipitation index (SPI) based on the long-term pre-cipitation record for a desired period. PDSI has several limitations(see Alley 1984; Guttman 1991, 1998). For instance, the soundnessof proposed water balance model is questionable, the temporalscale of PDSI is not clear, and the values of PDSI possess neithera physical (such as required rainfall depth) nor statistical meaning(such as recurrence probability) (Kao and Govindaraju 2010).Due to the limitations of PDSI, Guttman (1998) recommended theuse of SPI as a primary drought index because it is simple, spatiallyinvariant in its interpretation, and probabilistic so that it can be usedin risk and decision analysis. Therefore, the SPI series was used forthis study.

Drought properties are usually investigated separately by uni-variate frequency analysis (e.g., Tallaksen et al. 1997; Fernándezand Salas 1999; Cancelliere and Salas 2004; Serinaldi et al. 2009).Since droughts are complex phenomena, one variable cannot pro-vide a comprehensive evaluation of droughts (Shiau et al. 2007).Separate analysis of drought duration distribution and drought

1State Key Laboratory of Water Resources and Hydropower Engineer-ing Science, Wuhan Univ., Wuhan 430072, China; Dept. of Biologyand Agricultural Engineering, Texas A&M Univ., College Station, TX77843-2117; and College of Hydropower and Information Engineering,Huazhong Univ. of Science and Technology, Wuhan 430074, China(corresponding author). E-mail: chl8505@126.com

2Dept. of Biological and Agricultural Engineering and Dept. of Civiland Environmental Engineering, Texas A&M Univ., College Station,TX 77843-2117. E-mail: vsingh@tamu.edu

3State Key Laboratory of Water Resources and Hydropower Engineer-ing Science, Wuhan Univ., Wuhan 430072, China. E-mail: slguo@whu.edu.cn

4Pacific Northwest National Laboratory, PO Box 999, Richland, WA99352; formerly, Dept. of Biological and Agricultural Engineering andDept. of Civil and Environmental Engineering, Texas A&M Univ.,College Station, TX 77843-2117. E-mail: akm.pce@gmail.com

5Hrdro China Huadong Engineering Corporation, Hangzhou 310014,China. E-mail: guo_j2@ecidi.com

Note. This manuscript was submitted on September 28, 2011; approvedon August 7, 2012; published online on August 18, 2012. Discussion per-iod open until December 1, 2013; separate discussions must be submittedfor individual papers. This paper is part of the Journal of Hydrologic En-gineering, Vol. 18, No. 7, July 1, 2013. © ASCE, ISSN 1084-0699/2013/7-797-808/$25.00.

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severity distributions cannot reveal the significant correlationbetween them. Instead of using traditional univariate analysis fordrought assessment, a better approach for describing drought char-acteristics is to derive the joint distribution of drought variables(Mishra and Singh 2010). For example, Shiau and Shen (2001),Bonaccorso et al. (2003), Kim et al. (2003), González and Valdés(2003), Salas et al. (2005), and Cancelliere and Salas (2010) pro-posed different methods to investigate the joint distribution ofdrought duration and drought severity or intensity. These bivariatedistributions have either complex mathematical derivations or theirparameters are obtained by fitting the observed or generated data(Shiau 2006).

Multivariate distributions using copulas, however, can over-come such difficulties. In recent years, copulas have been used formultivariate hydrological analysis. For example, they have beenused for rainfall frequency analysis (De Michele and Salvadori2003; Grimaldi and Serinaldi 2006; Kao and Govindaraju 2007;Zhang and Singh 2007; Kuhn et al. 2007), flood frequency analysis(Favre et al. 2004; Shiau et al. 2006; Zhang and Singh 2006;Renard and Lang 2007; Wang et al. 2009), drought frequencyanalysis (Shiau 2006; Kao and Govindaraju 2010; Song and Singh2010b, a). Details of the theoretical background and the use of cop-ulas can be found in Nelson (2006) and Salvadori et al. (2007).

For drought frequency analysis, Shiau (2006; Shiau et al. 2007;Shiau and Modarres 2009) modeled the joint distribution ofdrought duration and severity using two-dimensional copulas.Mirakbari et al. (2010) used bivariate copula functions for regionaldrought analysis. The use of multivariate copulas (greater than twovariables) has also emerged recently. Song and Singh (2010b) mod-eled the joint probability distribution of drought duration, severityand inter-arrival time using a trivariate Plackett copula. Song andSingh (2010a) applied several metaelliptical copulas, Gumbel-Hougaard, Ali-Mikhail-Haq, Frank, and Clayton copulas to derivetrivariate joint distributions for drought duration, severity, and in-terval time, and the best-fit copula for trivariate drought analysiswas selected. Serinaldi et al. (2009) investigated four drought char-acteristics, including drought length, mean and minimum SPI val-ues, and drought mean areal extent, and built the correspondingfour-dimensional joint distribution of them. Kao and Govindaraju(2010) proposed a new drought indicator using multivariate empir-ical copulas to compute a probability-based overall water deficitindex from multiple drought-related quantities (or indices).

Until now, most of the work has focused on bivariate cases.Investigators have used many different ways to build bivariatedistributions of drought duration and severity. Actually, droughtevents have some other characteristics, such the minimum SPIvalues in one drought event, and drought interval time, which aremutually correlated. The studies mentioned above have only in-cluded some of the drought characteristics. However, it is importantfor design engineers and water resources planners to know not onlythe frequency of droughts, but also the risk of having droughtsof differing duration, severity, and interval times as well as theminimum SPI value within a drought period. For this purpose,a multivariate distribution needs to be determined. In order to sim-plify inference procedures and to derive flexible multivariate dis-tributions, copulas can be efficiently employed.

The objective of this paper is therefore to employ the Archime-dean and metaelliptical copulas to construct four-dimensional jointdistributions for droughts. The dry events were divided into fourstates, and copula functions were built in each state in order toinvestigate their applicability in drought analysis. The drought riskwas defined and analyzed based on the return period (recurrenceinterval) of drought events, which has become standard practicefor the risk-based design of hydraulic structures.

Definition of Drought and Univariate Variable

Definition of Drought Events

Drought identification based on an SPI series can be carried outby assuming a drought period as a consecutive number of timeintervals where SPI values are less than 0 (Shiau 2006). Fig. 1illustrates the time series of SPI and drought events. Each droughtevent is characterized by four main properties: drought dura-tion Dd, drought severity Sd, the minimum SPI (MSPI) Id, anddrought interval time Ld. The definitions of these variables arediscussed below:

Drought duration Dd is defined as the number of consecutiveintervals (months) where SPI remains below the threshold value 0(Shiau 2006).

Drought severity Sd is defined as a cumulative SPI value duringa drought period, Sd ¼

PDdi¼1 SPIi where SPIi means the SPI value

in the ith month (Mishra and Singh 2010).The minimum SPI (MSPI) value, defined as the minimum SPI

value within a drought period (Serinaldi et al. 2009), was used inthis study to describe the peak of drought events.

The drought interval time Ld is defined as the period elapsingfrom the initiation of a drought to the beginning of the next drought(Song and Singh 2010a).

Based on the SPI values, the dry states can be divided into fourstates shown in Table 1 (Mishra et al. 2009).

Distributions of Univariate Drought Variables

Generally, drought duration is fitted as a geometric distribution(Kendall and Dracup 1992; Mathier et al. 1992) by treating itas a discrete random variable. Similarly, Shiau and Shen (2001)computed drought interval time as equal to the sum of droughtduration and nondrought duration, on the assumption that droughtand nondrought durations follow a geometric distribution. AsSklar’s theorem requires the continuity of marginal distributions,the continuous distribution is needed in this study. Two continuous

Fig. 1. Definition sketch of drought events

Table 1. Drought Classification Based on SPI

SPI Class

2.00 and above Extremely wet1.50 to 1.99 Very wet1.00 to 1.49 Moderately wet−0.99 to 0.99 Near normal−1 to −1.49 Moderately dry−1.5 to −1.99 Severely dry−2.00 and less Extremely dry

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distributions used mostly in drought analysis are exponential andgamma distributions. For example, the exponential distributionwas selected for fitting drought duration (Zelenhastic and Salvai1987). The gamma distribution has generally been used to describedrought severity (Shiau 2006). According to five drought statesconsidered, twenty marginal distributions will be determined.Exponential or gamma distributions cannot fit every case well.Thus, the Pearson three type distribution, the generalized Paretodistribution and the generalized extreme distribution, which havealso been used to describe hydrologic variables, were considered.Among these distributions, the one with the smallest root meansquare error (RMSE) value between observed and theoretical prob-abilities will be selected.

Copulas for Multivariate Distributions

The problem of specifying a probability model for dependentmultivariate observations can be simplified by expressing the cor-responding n dimensional joint cumulative distribution (Salvadoriand Michele 2010). Following Sklar (1959) and Nelson (2006),if F1;2; : : : ;nðx1; x2; : : : ; xnÞ is a multivariate distribution functionof n correlated random variables of X1;X2; : : : ;Xn with respec-tive marginal distributions F1ðx1Þ;F2ðx2Þ; : : : ;FnðxnÞ, then it ispossible to write an n-dimensional cumulative distribution func-tion (CDF) with univariate margins, F1ðx1Þ;F2ðx2Þ; : : : ;FnðxnÞ,as follows:

Hðx1; x2; : : : ; xnÞ ¼ C½F1ðx1Þ;F2ðx2Þ; : : : ;FnðxnÞ�¼ Cðu1; : : : ; unÞ ð1Þ

where FkðxkÞ ¼ uk for k ¼ 1; : : : ; n, with Uk ∼ Uð0; 1Þ.Previous studies have indicated that copulas perform well for

bivariate problems, and in particular, several families of Archime-dean copulas, including Gumbel-Hougaard, Frank, and Clayton,have been popular choices for dependence models because oftheir simplicity and generation properties (Nelson 2006). Formultivariables (greater than two), the symmetric Archimedeancopula has only one parameter, which forces that all pairs of var-iables share the same dependence structure. In order to model thedifferent dependence structures, Grimaldi and Serinaldi (2006)applied nested classes of the Archimedean copulas. Hpwever,these copulas can only model n − 1 dependence. Therefore,the Archimedean copulas are not adequate for modeling thedependence of three or more variables, given that different pairsexhibit widely varying degrees of dependence (Genest et al.2007). To the contrary, metaelliptical copulas, which are exten-sions of the multivariate Gaussian distribution, can model arbi-trary pairwise dependencies between variables though acorrelation matrix (Kao and Govindaraju 2008). All these copulasmentioned above were used and compared in this study. Thefour-dimensional symmetric and asymmetric Archimedean andmetaelliptical copulas, used in this study, are given in theappendix.

Fig. 2. Location of meteorological stations in the Hanjiang basin and the middle route of the SNWDP in China (map by Jing Guo)

Table 2. Comparison of Statistical Variables between Historical and Synthetic Data

Stations Historical SyntheticAbsoluteerror

Relativeerror (%) Historical Synthetic

Absoluteerror

Relativeerror (%)

Ankang 2.20 2.20 0.00 0.00 5.42 5.13 0.29 5.35Foping 1.88 1.89 0.01 0.53 4.63 4.53 0.10 2.16Hanzhong 2.35 2.35 0.00 0.00 5.63 5.38 0.25 4.44Lueyang 2.17 2.16 0.01 0.46 5.14 4.76 0.38 7.39Shangzhou 1.88 1.80 0.08 4.26 4.63 4.32 0.31 6.70Shiquan 2.42 2.42 0.00 0.00 5.88 5.51 0.37 6.29Wanyuan 3.37 3.31 0.06 1.78 8.09 7.36 0.73 9.02Xixia 2.36 2.35 0.01 0.42 6.02 5.65 0.37 6.15Zhenan 2.15 2.05 0.10 4.65 5.05 4.69 0.36 7.13

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Return Period for Drought Events

A common approach to hydraulic and hydrologic design isfrequency analysis or the recurrence interval or return period ofhydrologic events (Shiau and shen 2001). In particular, drought re-turn periods provide useful information for proper water use underdrought conditions (Serinaldi et al. 2009). The return period of adrought can be defined as the average elapsed time or mean intervaltime between occurrences of two droughts (Shiau and Shen2001; Serinaldi et al. 2009). In this study, return periods of the

univariate and multivariate drought events were calculated andanalyzed.

Univariate Return Period

Shiau and Shen (2001) calculated the return period of a droughtevent with severity equal to or greater than a certain value ds. Shiau(2006) calculated the return period of a drought event with durationequal to or greater than a certain value dl. Similarly, the returnperiod of drought intensity can be obtained using the same formulaexpressed as

Td ¼EðLdÞ

1 − FDdðxÞ ; Td ¼

EðLdÞ1 − FSdðxÞ

; Td ¼EðLdÞ

1 − FIdðxÞð2Þ

where EðLdÞ expected drought interval time.

Multivariate Return Period

Salas et al. (2005) extend Eq. (2) to a more general case of droughtevents defined in terms of either severity or MSPI and duration. Theinterval time between two drought events E is TE ¼ PNd

j¼1 Ldj ,where Ldj is the interval time between any two droughts in general(i.e., droughts not necessarily characterized by E); and Nd is thenumber of droughts until the next drought event E occurs. Then,the return period T is the expected value of TE, and can beexpressed as

T ¼ EðTEÞ ¼ EðNdÞEðLdÞ ð3Þwhere EðNdÞ ¼ 1=PðEÞ. The multivariate return period can becalculated based on Eq. (3).

Shiau (2006) defined the bivariate joint return period Tand andTor as

Tand ¼EðLdÞ

1 − PðXi ≥ xi;Xj ≥ xjÞ

¼ EðLdÞ1 − FðxiÞ − FðxjÞ þ C½FðxiÞ;FðxjÞ�

ð4Þ

Fig. 3. Observed and simulated cumulative monthly rainfall:(a) observed data; (b) simulated data

Fig. 4. SPI values with different time scales: (a) SPI < 0; (b) normal dry event

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Tor ¼EðLdÞ

1 − PðXi ≥ xi or Xj ≥ xjÞ¼ EðLdÞ

1 − C½FðxiÞ;FðxjÞ�ð5Þ

According to Eq. (3), the trivariate return period can bedefined as

Tand ¼EðLdÞ

1 − PðXi ≥ xi;Xj ≥ xj;Xk ≥ xkÞ¼ EðLdÞ=ð1 − FðxiÞ − FðxjÞ − FðxkÞ þ C½FðxiÞ;FðxjÞ�þ C½FðxiÞ;FðxkÞ� þ C½FðxjÞ;FðxkÞ�− C½FðxiÞ;FðxjÞ;FðxkÞ� ð6Þ

Tor ¼EðLdÞ

1 − PðXi ≥ xi or Xj ≥ xj or Xk ≥ xkÞ

¼ EðLdÞ1 − C½FðxiÞ;FðxjÞ;FðxkÞ�

ð7Þ

Conditional Return Period

Shiau (2006) defined the bivariate conditional return period as

Txijxj ¼EðLdÞ

½1 − FxjðxjÞ�½1 − FxiðxiÞ − FxjðxjÞ þ Cðxi; xjÞ�ð8Þ

where Txijxj = conditional return period for Xi given Xj ≥ xj.The bivariate conditional return period of drought duration,

severity, and MSPI were calculated in this study.

Data

Historical Data

The data used in this study to evaluate drought characteristics werethe daily rainfall data from 1961 to 2007 in the upper Han River,China. The Han River is a left tributary of the Yangtze River with alength of 1,532 km. Daily rainfall data from nine gauge stations,including Ankang, Foping, Hanzhong, Lueyang, Shangzhou,Shiquan, Wanyuan, Xixia, and Zhenan, were used in this study.The locations of these stations were shown in Fig. 2. The averageareal rainfall of this basin was calculated based on these ninestations.

Rainfall Data Generation

Since a drought may last for several months or even years, the his-torical record usually falls short for fully evaluating drought char-acteristics (Mishra et al. 2009). In order to avoid this disadvantage,500 year daily rainfall data were generated based on historical datacharacteristics. The Markov model was applied to produce precipi-tation occurrence, and the two-parameter gamma distribution wasused to generate the precipitation quantity (Chen et al. 2010).

Application

Comparisons between Historical and SyntheticPrecipitation Series

The three-order Markov model was applied to produce precipita-tion occurrence, and the two-parameter gamma distribution wasused to generate the precipitation amount. In order to test the ration-ality of the rainfall simulation model, synthetic data with the same

length of the historical data was generated. Their mean and stan-dard variation values were calculated, and given in Table 2. Theabsolute and relative errors between historical and synthetic dataare also shown in the table. Results demonstrate that the maximumabsolute error of mean values is 0.1 mm, corresponding to themaximum relative error of 4.65%, and the relative error for thestandard deviation is less than 10%. Therefore, the synthetic datacan be applied to the calculation hereafter. This model was used togenerate daily rainfall data from the nine stations with a length of

Table 3. Values of Pearson and Kendall Correlation Coefficients for AllDrought Variables in Different Drought States

Drought statesCorrelationcoefficient Duration Severity MSPI

Intervaltime

Dry event Duration 1.00 0.59 0.34 0.60Severity 0.82 1.00 0.77 0.37MSPI 0.42 0.78 1.00 0.21

Interval time 0.71 0.59 0.27 1.00Near normaldry

Duration 1.00 0.49 0.26 0.35Severity 0.77 1.00 0.82 0.18MSPI 0.33 0.78 1.00 0.10

Interval time 0.37 0.29 0.13 1.00Moderatelydry

Duration 1.00 0.41 0.13 0.02Severity 0.93 1.00 0.88 0.02MSPI 0.15 0.49 1.00 0.01

Interval time 0.01 0.00 −0.04 1.00Severe dry Duration 1.00 0.23 0.09 −0.05

Severity 0.90 1.00 0.97 −0.03MSPI 0.11 0.53 1.00 −0.02

Interval time −0.06 −0.06 −0.01 1.00Extreme dry Duration 1.00 0.24 0.12 −0.03

Severity 0.78 1.00 0.97 −0.03MSPI 0.17 0.74 1.00 −0.03

Interval time −0.02 −0.04 −0.04 1.00

Note: The superdiagonal elements are the Kendall correlation; and thesubdiagonal elements are the Pearson correlation.

Table 4. Selected Marginal Distributions and Their Estimated Parameters

Events Severity MSPI Duration Interval time

Dry event P-III GP Gamma Gamma1.63 0.08 2.73 2.171.47 1.50 0.75 1.431.83

Near normal dry Gamma P-III Gamma Gamma1.22 1.05 4.33 2.311.34 0.53 0.35 1.47

1.52

Moderately dry GP GP Gamma GP1.01 0.99 18.66 −0.031.25 0.44 0.06 11.37

Severe dry GEV GP Gamma Gamma1.65 1.51 61.79 1.010.13 0.40 0.02 22.35

−0.26 0.82

Extreme dry GP GP Gamma Gamma2.02 2.01 60.33 1.180.37 0.46 0.02 34.94

−0.16 0.16

Note: The characters above the numbers in each cell mean the selecteddistributions.

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Fig. 5. Frequency curves of marginal distribution

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500 years. The observed and theoretical cumulative monthly rain-fall is shown in Fig. 3, in which the shape and the cumulativemonthly values of these two series are nearly the same. The SPIseries for different time scales were obtained, as shown in Fig. 4.The monthly SPI values were used for analysis hereafter.

Correlation Analysis

The Pearson and Kendall correlation coefficients for all droughtvariables are given in Table 3. Results confirmed that for droughtevents, normal dry state, all variables showed positive associationand a highly correlated relationship was observed. For the left dry

states, some drought variables showed a small negative correlation,but the negative correlations were small, close to zero. This meansthat the negative correlation between these two variables can benegligible and the fully nested copulas can be used in these cases.

Estimation of Marginal Distributions

The exponential distribution, gamma distribution, Pearson typethree distribution, generalized Pareto distribution, and generalizedextreme distribution were applied to fit every drought variable.Parameters of marginal distributions were estimated by L-moments(Hosking 1990). The distribution with the smallest RMSE valuesbetween observed and theoretical probabilities was selected. Thechosen distributions and their estimated parameters are listed inTable 4. Fig. 5 compares computed and empirical marginal distri-butions of observed drought duration, interval, severity and MSPIin the case of SPI < 0 and normal dry events. The figure demon-strates that the theoretical and empirical values fit well for allmarginal distributions.

Estimation of Joint Distributions

Four-dimensional Archimedean and metaelliptical copulas weretested to determine the best-fit copula for modeling the dependenceamongst the four drought characteristics. For the Archimedeanfamily, three widely used copulas, including Gumbel-Hougaard,Frank, and Clayton, were used; for the metaelliptical copulas, nor-mal and Student copulas were used. A pseudolikelihood techniqueinvolving the ranks of the data was used for estimating parametersof the four-variate symmetric and asymmetric Archimedean copu-las. The estimated parameters of both symmetric and asymmet-ric Archimedean copulas are given in Table 5. The inversion ofKendall’s tau method (Genest et al. 2007) was used to estimate

Table 5. Estimated Parameters of Symmetric and AsymmetricArchimedean Copulas

Dry events Family

Symmetric Asymmetric

θ θ1 θ2 θ3

Drought (SPI < 0) Gumbel 1.66 1.40 1.40 2.80Frank 4.62 3.38 3.38 12.50Clayton 1.27 0.54 0.55 6.78

Near normal Gumbel 1.32 1.18 1.26 2.53Frank 2.70 1.38 1.39 14.95Clayton 0.79 0.28 0.28 8.51

Moderately dry Gumbel 1.07 1.00 1.02 2.83Frank 2.69 0.17 0.17 21.52Clayton 0.39 0.11 0.11 14.97

Severe dry Gumbel 1.03 1.00 1.00 6.26Frank 0.66 0.001 0.10 35Clayton 0.27 0.04 0.06 59.53

Extreme dry Gumbel 1.03 1.00 1.00 8.81Frank 0.66 0.001 0.26 30.0Clayton 0.28 0.04 0.06 66.26

Table 6. Total Correlation Analysis of Four-Dimensional Joint Distribution

Number Classifications Copulas ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ν

1 Drought event Normal 0.92 0.73 0.50 0.45 0.30 0.70t 0.88 0.59 0.51 0.32 0.26 0.79 18.95

2 Normal dry Normal 0.93 0.62 0.27 0.34 0.15 0.42t 0.90 0.33 0.25 0.16 0.12 0.50 8.67

3 Moderately dry Normal 0.92 0.51 0.03 0.16 0.008 0.03t 0.97 −0.17 0.003 −0.12 0.01 0.05 3.26

4 Severe dry Normal 0.98 0.30 −0.04 0.12 −0.03 −0.06t 0.85 0.60 0.28 0.14 0.06 0.47 8.19

5 Extreme dry Normal 0.98 0.30 −0.05 0.15 −0.05 −0.03t 0.99 −0.09 −0.31 −0.07 −0.30 0.01 10.95

Table 7. RMSE and AIC Values of Different Copulas

Events Family

Archimedean Metaelliptical

Gumbel Frank Clayton

Normal StudentA B A B A B

Drought RMSE 0.048 0.028 0.031 0.022 0.052 0.049 0.013 0.022AIC −8897 −10482 −10179 −11189 −8662 −8841 −12721 −11177

Near RMSE 0.068 0.05 0.055 0.037 0.055 0.045 0.018 0.036Normal AIC −7321 −8154 −7899 −8975 −7899 −8441 −10931 −9043Moderately RMSE 0.053 0.099 0.056 0.034 0.079 0.028 0.032 0.041Dry AIC −2906 −2284 −2852 −3342 −2511 −3534 −3396 −3148Severe RMSE 0.103 0.024 0.095 0.019 0.086 0.013 0.025 0.034Dry AIC −1144 −1874 −1184 −1992 −1235 −2183 −1847 −1690Extreme RMSE 0.102 0.02 0.092 0.021 0.083 0.014 0.022 0.043Dry AIC −637 −1089 −666 −1076 −695 −1189 −1057 −867

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parameters of the normal copula. The estimated parameters of nor-mal copula are listed in Table 6. Parameters of the Student copulawere estimated by the maximum pseudolikelihood method, andtheir values are also given in Table 6.

In order to select an appropriate copula, the root mean squareerror (RMSE) and the Akaike information criterion (AIC) wereused (Zhang and Singh 2006). The RMSE and AIC values ofthe Archimedean and metaelliptical copulas are shown in Table 7which indicates that the asymmetric copulas gave a better fit thanthe symmetric copula did in the Archimedean family. Generallymetaelliptical copulas fitted better than the Archimedean copulasdid, except for the Clayton copula. The RMSE and AIC valuesof the normal copula exhibited a better fit than the Student copuladid. Fig. 6 compares observed and theoretical joint probabilities,which indicates that the observed and theoretical values fitted eachother well.

For drought events with SPI values less than 0 and near nor-mal dry state, the normal copula gave a better fit than others.For the remaining three states, the Clayton copula fitted better.

The theoretical probabilities of moderately, severely and extremelydry states calculated by both the normal and Clayton copulas areshown in Fig. 6. It is indicated from the RMSE and AIC values inTable 7 and the fitting results in Fig. 6 that the difference betweennormal and Clayton copulas is small. Therefore, the normal copulais an appropriate copula for all of the dry events in the upperHan River basin.

Fig. 6. Observed and theoretical joint probabilities for the four drought characteristics: (a) TId jSd ; (b) TSdjId ; (c) TSdjDd; (d) TDd jSd ; (e) TId jDd

; (f) TDd jId

Table 8. Results of Joint and Conditional Distributions

FðxÞ Dd Sd Id Ld

Jointprobabilities

Conditionalprobabilities

0.1 0.25 0.24 0.65 0.95 0.01 0.140.3 0.68 0.57 1.22 1.82 0.10 0.330.5 1.21 0.95 1.78 2.67 0.25 0.490.7 1.98 1.40 2.49 3.76 0.46 0.660.9 3.58 2.03 3.81 5.78 0.78 0.860.99 6.84 2.57 6.24 9.55 0.97 0.98

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Table 9. Joint Return Periods (Years) of Drought Events E

T FðxÞ Dd Sd Id

Sd > sd, Id > id Dd > dd, Sd > sd Dd > dd, Id > id Dd > dd, Sd > sd, Id > id

Fðx; yÞ Tand Tor Fðx; yÞ Tand Tor Fðx; yÞ Tand Tor Fðx; y; zÞ Tand Tor

5 0.38 1.44 0.88 0.72 0.32 5.55 4.55 0.27 6.13 4.22 0.21 6.85 3.94 0.21 6.95 3.9210 0.69 2.45 1.94 1.38 0.63 12.24 8.45 0.59 15.15 7.46 0.54 19.80 6.69 0.53 19.99 6.5920 0.85 3.31 2.95 1.83 0.81 26.51 16.06 0.78 36.39 13.79 0.75 55.47 12.20 0.74 55.84 11.8150 0.94 4.34 4.27 2.21 0.92 72.66 38.11 0.90 113.35 32.07 0.89 211.55 28.35 0.88 216.57 26.81100 0.97 5.08 5.25 2.39 0.96 154.81 73.85 0.95 265.09 61.62 0.94 576.19 54.75 0.94 594.73 50.73

Fig. 7. Conditional return periods of drought events: (a) TIdjSd ; (b) TSdjId ; (c) TSd jDd; (d) TDd jSd ; (e) TId jDd

; (f) TDd jId

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Drought Probability Analysis

In this study, drought events were defined by drought duration, se-verity, interval time and MSPI. It is necessary to know the occur-rence probabilities of arbitrary drought events. Table 8 gives Dd,Sd, Id, and Ld values corresponding to different values of FðxÞ.The joint probabilities of some drought events E ¼ fDd ≤ dd;Sd ≤ sd; Id ≤ id;Ld ≤ ldg were also given in Table 8. Taking FðxÞequal to 0.1, for example, the corresponding magnitudes in Table 8for drought duration, severity, MSPI, and time interval are 0.25,0.24, 0.65, and 0.95, respectively. The joint probability of thisdrought event is 0.01. Based on this model, the joint probability ofany drought event can be obtained. The joint probability increaseswhen marginal probabilities increase. From this point of view, thecalculated results seem justified.

The conditional probabilities defined in Eq. (9) were calculated,as given in Table 8. The probability of events E ¼ fDd ≤ dd;Sd ≤ sd; Id ≤ idg under the condition Ld ≤ ld can be defined as

PðDd ≤ dd; Sd ≤ sd; Id ≤ idjLd ≤ ldÞ

¼ PðDd ≤ dd; Sd ≤ sd; Id ≤ id;Ld ≤ ldÞPðLd ≤ ldÞ

ð9Þ

The conditional probability defined in Eq. (9) was calculated,as given in Table 8. When FðxÞ ¼ 0.99, the corresponding magni-tudes for drought duration, severity, MSPI, and interval time inTable 8 are 6.84, 2.57, 6.24, and 9.55, respectively. The conditionalprobability of event E ¼ fDd ≤ dd; Sd ≤ sd; Id ≤ idg under thecondition Ld ≤ ld is 0.98. Thus, the conditional probability of anydrought event can be obtained from Eq. (9).

Return Period Analysis

The average drought interval time estimated from both observeddata and theoretical distributions was 3.09 months. Therefore,the calculated value of 3.09 months was used hereafter. First, theunivariate return period was analyzed based on Eq. (2). Given thereturn values T of 5, 10, 20, 50, and 100 years, the marginal prob-abilities FðxÞ were derived. The magnitudes corresponding to thereturn periods defined by separate values of drought duration, se-verity and MSPI were calculated by solving the inverse function F,as given in Table 9.

The joint return period is related to marginal probabilities andjoint probability. The marginal probabilities have been calculatedbefore for analyzing the univariate return period. The joint proba-bilities of drought variables are also given in Table 9. The bivariateand trivariate return periods Tand and Tor were then calculated, asshown in Table 9, which shows that the univariate return period islarger than the joint return period Tor and less than the joint returnperiod Tand. The trivariate joint return period Tand is larger thanbivariate one in the same row in Table 9, but the trivariate jointreturn period Tor is less than the bivariate one. This is because thatadding one variable in the model makes the exceedance probabil-ities PðX1 > x1;X2 > x2;X3 > x3Þ smaller than the two bivariateone PðX1 > x1;X2 > x2Þ.

According to Eq. (8), the values of bivariate conditional returnperiod of drought duration, severity and MSPI were calculated asplotted in Fig. 7, which shows that the conditional return periodincreases when the values of drought variables increase. The de-rived conditional return periods of drought duration, severity andMSPI can be used to evaluate the risk of a specific water resourcessystem. For instance, based on the derived conditional distribution,water resources managers are informed that the probabilities for a

drought severity greater than 1.0 and 2.0 given drought durationexceeding 1 months are equal to 6.65 and 12.85 years, respectively.

Conclusions

In this study, a drought is defined by drought duration, severity,MSPI, and interval time. The upstream Han River is selected asa case study. The exponential distribution, gamma distribution,generalized Pareto distribution, Pearson three type distribution,generalized Pareto distribution, and generalized extreme dis-tribution are applied to fit univariate data. The Archimedean andmetaelliptical copulas are used to establish the joint multivariatedistributions. The joint probabilities and return period are thenestimated. The main conclusions of this study are1. The established marginal distributions of the four drought

variables fit the empirical data well, and can be used fordrought analysis;

2. Five copula functions are constructed for different droughtstates. The RMSE and AIC values are used to select the ap-propriate copula. Results of fitting indicate that it is applicableto use copulas for multivariate drought analysis. The normalcopula basically fits every state well, therefore this one isselected for computing probabilities and return period analy-sis; and

3. The drought risk is estimated based on joint probabilities andreturn periods, which give important information for watermanagement and planning.

Appendix. Copula Functions

The equations of several copulas used in this study were given inthe following.1. Gumbel copula

a. Asymmetric Gumbel copula

Cðu1; u2; u3; u4Þ¼ exp½−ðð− ln u4Þθ1 þ f½ð− ln u1Þθ3 þ ð− ln u2Þθ3 �θ2=θ3

þ ð− ln u3Þθ2gθ1=θ2Þ1=θ1 �

where θ1, θ2, and θ2 are parameters of copulas,1 < θ1 ≤ θ2 ≤ θ3

b. Symmetric Gumbel copula

Cðu1; u2; u3; u4Þ ¼ expf−½ð− ln u1Þθ þ ð− ln u2Þθþ ð− ln u3Þθ þ ð− ln u4Þθ�1=θg

where θ is parameter of copulas, θ > 1.2. Frank copula

a. Asymmetric Frank copula

Cðu1;u2;u3;u4Þ

¼−1θ1

ln

�1þ 1

e−θ1 − 1ðe−θ1u4 − 1Þ

��1− 1

ð1− e−θ2Þ

·

�1−

�1− 1

ð1− e−θ3Þ ð1− e−θ3u1Þð1− e−θ3u2Þ�θ2=θ3

�

× ð1− e−θ2u3Þ�

where θ1, θ2, and θ2 are parameters of copulas,0 < θ1 ≤ θ2 ≤ θ3.

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b. Symmetric Frank copula

Cðu1; u2; u3; u4Þ ¼−1θ1

ln

�1þ

Q4i¼1ðe−θ1ui − 1Þðe−θ1 − 1Þ3

�

where θ is parameter of copulas, θ ∈ R.3. Clayton copula

a. Asymmetric Clayton copula

Cðu1; u2; u3; u4Þ ¼ ðu−θ14 þ ððu−θ31 þ u−θ32 − 1Þθ2=θ3

þ u−θ23 − 1Þθ1=θ2 − 1Þ−1=θ1

where θ1, θ2, and θ2 are parameters of copulas,0 < θ1 ≤ θ2 ≤ θ3.

b. Symmetric Clayton copula where

Cðu1; u2; u3; u4Þ ¼ ðu−θ1 þ u−θ2 þ u−θ3 þ u−θ4 − 3Þ−1=θ

where θ = parameter of copulas, θ > 0.4. Normal copula

CN

�u;

X�¼

ZΦ−1ðu1Þ

−∞· · ·

ZΦ−1ðunÞ

−∞1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2πÞnjP jp

× exp

�− 1

2x 0 X−1 x

�dx

whereP

= correlation matrix.5. Student t copula

Ct

�u;

X; v�¼

Zt−1ðu1Þ

−∞· · ·

Zt−1ðunÞ

−∞Γðvþn

v ÞΓðv

2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðπvÞnjP jp

×

�1þ 1

vx 0 X−1 x

�ðvþnÞ=2dx

whereP

= correlation matrix; and v = degree of freedom.

Acknowledgments

The study is financially supported byMinistry of Science and Tech-nology of China (2009BAC56B02) and Chinese Natural ScienceFoundation (51079100 and 51190094).

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