Risk Assessment of Droughts in Gujarat Using Bivariate Copulas ...

Water Resour Manage (2012) 26:3301–3327 DOI 10.1007/s11269-012-0073-6

Risk Assessment of Droughts in Gujarat Using Bivariate Copulas Poulomi Ganguli & M. Janga Reddy

Received: 27 May 2011 / Accepted: 17 May 2012 / Published online: 8 June 2012 # Springer Science+Business Media B.V. 2012

Abstract This study presents risk assessment of hydrologic extreme events droughts in Saurashtra and Kutch region of Gujarat state, India. Drought is a recurrent phenomenon and risk assessment of droughts can play an important role in proper planning and management of water resources in the study region. In the study, drought events are characterized by severity and duration, and drought occurrences are modeled by Standardized Precipitation Index (SPI) computed on mean areal precipitation, aggregated at a time scale of 6 months for the period 1900–2008. After evaluating several distribution functions, drought variable— severity is best described by non-parametric kernel density, whereas duration is best fitted by exponential distribution. Considering the extreme nature of drought variables, the upper tail dependence copula families including two Archimedean—Gumbel-Hougaard, BB1 and one elliptical—Student’s t copulas are evaluated for modeling joint distribution of drought variables. On evaluating their performance using various goodness-of-fit measures, Gumbel-Hougaard copula is found to be the best performing copula in modeling the joint dependence structure of drought variables. Also, while comparing with traditional bivariate distributions, the copula based distributions are resulted in better performance as compared to bivariate log-normal and the logistic model for bivariate extreme value distributions. Then joint and conditional return periods of drought characteristics are derived, which can be helpful for risk based planning and management of water resources systems in the study region. Keywords Drought . Standardized precipitation index . Copulas . Hydrological extremes . Tail dependence

1 Introduction Drought refers to a random condition of severe reduction of water supply availability as compared to the normal values, extending along a significant period of time and over a larger P. Ganguli : M. J. Reddy (*) Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India e-mail: [email protected]

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P. Ganguli, M.J. Reddy

region (Rossi 2000). In a large country like India where precipitation varies both in space and time, drought is one of the most frequently occurring natural calamities in different regions of the country. About 33 % of the arable land in India is considered to be droughtprone (i.e., about 14 % of the total area of the country) and a further 35 % can also be affected if rainfall is exceptionally low for an extended period (UNEP 2001). In general, probabilities of droughts are higher in the arid regions (Western India) as compared to other parts of the country. Gujarat, located in the Northwestern part of the country is chronically dry and prone to drought. The state constitute 6.0 % of the total land area in the country and about 50.67 million population (Census 2001) spread over 1,96, 077 km2. The Saurashtra and Kutch region in Gujarat State has arid climate condition with mean precipitation of 300– 600 mm. Rainfall is not only low but also uncertain in the region. The region faces frequent droughts mainly due to scarcity of rainfall. Droughts are complex phenomenon both in terms of definition and causes. It is regional in nature and can vary considerably in space and time. Drought is a multivariate phenomena characterized by severity, duration, intensity and its areal coverage, which are again mutually correlated to each other. Univariate frequency analysis of drought event therefore may lead to over and under estimation of associated risk for water resources management (González and Valdés 2003). Kim et al. (2006) showed that discrepancies in estimated return period can result in high or low quantiles from univariate frequency analysis of drought. Considering stream flow based drought index Shiau and Shen (2001) derived joint probability density function (PDF) of drought using product of conditional distribution of drought severity for a given drought duration and the marginal distribution of drought duration. Yang and Nadarajah (2006) modeled drought magnitude as the product of two independent random variables—one of them assumed to be exponential and the other one comes from either exponential, gamma, Weibull, Pareto or log-normal families. Kim et al. (2006) applied multivariate kernel density estimator for bivariate drought characterization using Palmer Drought Severity Index (PDSI) on droughts in Conchos River Basin, Mexico. Nadarajah (2007) used bivariate gamma distribution for investigating drought and non-drought periods at Nebraska by deriving exact distributions of the sum, their ratio and corresponding moment properties. Nadarajah (2009) applied bivariate Pareto distribution for studying joint distribution of drought variables at Nebraska and derived return periods for pair-wise combination of drought characteristics. Vangelis et al. (2011) analyzed severity of drought conditions using Reconnaissance Drought Index (RDI), and modelled using bivariate normal distribution. The study was based on the assumption that precipitation and potential evapotranspiration are normally distributed and are negatively correlated and tested for several meteorological stations in Greece at multiple time scales. Santos et al. (2011) investigated regional frequency analysis of droughts in Portugal using monthly precipitation data from 144 rain gauge stations. In their study, drought was modeled using Standardized Precipitation Index (SPI) at multiple time scales (1, 3, 6 and 12 consecutive months) and three spatially defined regions were identified using L-moments analysis. Then drought magnitude maps of the region were developed using kriging technique for various return periods. In hydrology and water resources management- modeling dependence of hydrological variables is having greater importance, where uncertainty and risk analysis plays a significant role in decision making. An inappropriate model can lead to inaccurate assessment of risks. Traditionally, the multivariate distributions are developed as an extension to their univariate distributions. It requires the multivariate distribution to follow same kind of marginal distributions, in which complexity of the distribution increases with increase in number of variables. Recently, copulas have been used in hydrology as they offer greater flexibility in modeling dependence. Copulas are parametrically specified joint distributions

Risk Assessment of Droughts in Gujarat Using Bivariate Copulas

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obtained from marginal distribution of any form while maintaining dependence structure between the random variables. Data transformation (such as taking natural logarithm or BoxCox transformation) do not influence copula, as they are invariant to monotonic transformation of random variables. Shiau (2006) and Shiau et al. (2007) investigated bivariate joint distribution of drought properties severity and duration using Archimedean class of copulas and then estimated associated return periods. Song and Singh (2010) modeled joint probability distribution of drought duration, severity and inter-arrival time of drought using trivariate Plackett copula for a case study in China. The parameters of bivariate and trivariate Plackett copulas are estimated using pseudo log-likelihood and genetic algorithm methods. Kao and Govindaraju (2010) proposed copula based joint deficit index (JDI) using precipitation and stream flow marginals with window sizes varying from 1 to 12 months in Indiana watershed. Wong et al. (2010) investigated effect of ENSO phenomenon on nature of multivariate drought frequency using precipitation data from two districts in New South Wales, Australia. Although there are numerous studies carried out for frequency analysis of droughts in different parts of the world, there is a lack of systematic examination of meteorological drought frequencies in specific regions in India. Hence, this study focuses on assessing meteorological drought risk in Saurashtra and Kutch region in Gujarat state. A bivariate copula based approach is presented for analyzing historical drought events.

2 Drought Modeling Using Standardized Precipitation Index (SPI) A wide range of drought indices are available in literature for monitoring and quantification of drought events (Keyantash and Dracup 2002). The Standardized Precipitation Index (SPI) is a common indicator of drought that does not require information about land surface conditions and needs only precipitation data to compute drought properties. It is a normalized score and represents an event departure from the mean, expressed in units of standard deviation. SPI is simple, spatially invariant, and probabilistic in nature and is used to estimate the effect of droughts on various water supplies at different time scales such as 3, 6, 9, or 12 months. Hence facilitates temporal analysis of drought events. The 6-month time scale SPI may be useful for seasonal drought identification, 12-month SPI for medium term droughts and the 24-month SPI for long term drought analysis (Labedzki 2007). The computations of SPI index in a given year i and month j, for a time scale k requires (McKee et al. 1993) the following steps: Computation of cumulative precipitation series Xijk ði ¼ 1; . . . ; nÞ for a particular time period of interest j in a year i, where each term is the sum of precipitation of k-1 past consecutive months. (ii) Fitting of a cumulative probability distribution (usually gamma distribution function) on aggregated monthly precipitation series (e.g., k06 months is adopted in this study). The gamma distribution function is defined as, (i)

gðxÞ ¼

1 xa1 ex=b b a Γ ðaÞ

ð1Þ

where, β is a scale parameter, α is a shape parameter and Г(α) is the ordinary gamma function at α.

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(iii)

As two-parameter gamma function is not defined for zero values, and precipitation distribution may contain zeros, a mixed distribution function (zeros and continuous precipitation amount) is employed and the cumulative distribution function (CDF) is given as, FðxÞ ¼ q þ ð1 qÞGðxÞ

ð2Þ

where, G(x) is the cumulative distribution function estimated for nonzero precipitation; and q is the zero precipitation probability from historical time series. (iv) As precipitation is not normally distributed, an equiprobability transformation (Panofsky and Brier 1958) is carried out from the CDF of mixed distribution to the CDF of the standard normal distribution with zero mean and unit variance, which is given as, SPI ¼ y 1 ðFðxÞÞ

ð3Þ

The transformed probability is the SPI. A positive value of SPI indicates that precipitation is above average and a negative value denotes below average precipitation. Usually, drought accompanies with high temperature, which leads to higher evapotranspiration rates. Another recent development in the field of meteorological drought indices is Reconnaissance Drought Index (RDI) proposed by Tsakiris et al. (2007) is based on the ratio between two aggregated quantiles of precipitation and potential evapotranspiration at a location and can be considered as an extension of SPI. By standardization and normalization it receives the same drought classes as the SPI. However, Tsakiris et al. (2007) showed that drought severity characterization in Mediterranean region could be improved by using RDI as compared to SPI. Pashiardis and Michaelides (2008) analyzed regional droughts by comparing SPI and RDI using historical monthly precipitation and temperature data in Cyprus. They found that both indices respond in a similar fashion and can be used for monitoring droughts. Zarch et al. (2011) investigated impact of droughts in Iran using SPI and RDI for various time scales (such as 3, 6, …, 24 months). Based on the analysis, drought severity map was developed for 1999–2000 and it was found that central, eastern and southeastern parts of Iran faced extremely dry conditions while other parts of the country suffered from severe droughts. In their analysis, it was noted that the correlation between SPI and RDI was significant in 3, 6 and 9 months than longer time scales and both methods showed approximately similar results for the effect of droughts on different regions of Iran. The estimation of PET is however difficult because of involvement of numerous parameters such as surface temperature, air humidity, soil incoming radiation, water vapor pressure and ground-atmosphere latent and sensible heat fluxes (Allen et al. 1998). In many regions of the world, this meteorological data is not available for sufficiently longer time periods. Though drought perception varies significantly among regions with different climatic conditions, precipitation is the primary causal factor for drought occurrences in the present study region. Hence, in this study drought is modelled using SPI-6 time series, which is defined as a 6-month moving average value. 2.1 Drought Definition Using SPI A drought period is assumed as a consecutive number of months where SPI values remains below a threshold of −0.8. Based on SPI range drought period can be classified as moderate drought (−0.8 to −1.2), severe drought (−1.3 to −1.5), extreme drought (−1.6 to −1.9), and exceptional drought (−2 or less) condition (Svoboda et al. 2002).


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Drought length or duration (D) is taken as the number of consecutive intervals (months) where SPI remains below the specified threshold value (SPI0, for 0≤u1 ≤u2 ≤1 and 0≤v1 ≤v2 ≤1. As C(•) is a continuous function, the bivariate copula density c(•) is the double derivative of C with respect to its marginal distributions and is defined as, cðu; vÞ ¼

@ 2 C ðu; vÞ @u@v

ð6Þ

The details of the copula families applied in the present study are described below. 3.1 Archimedean Copulas The Archimedean families of copulas are widely applied in hydrology, because they can be easily generated with several desirable properties, such as, symmetry and associativity and a wide range of copula families belong to this class. This class of copulas are related to Laplace transforms of bivariate distribution functions (Joe 1997). If L denote the class of Laplace transforms, which consists of strictly decreasing differentiable functions: n o L ¼ f : ½0; 1Þ ! ½0; 1jf1 ð0Þ ¼ 1; f1 ð1Þ ¼ 0; ð1Þj fðjÞ 0; j ¼ 1; . . . ; 1

ð7Þ

then bivariate Archimedean copula function C: [0, 1]2 → [0, 1] is defined as (Nelsen 1999), C ðu; vÞ ¼ f½1 ðfðuÞ þ fðvÞÞ u; v 2 ½0; 1

ð8Þ

where ϕ(•) is known as generator of the copula and ϕ[−1](•) is the pseudo inverse of ϕ(•). If ϕ (0) 0 ∞ then the pseudo-inverse describes an ordinary inverse function (i.e., ϕ[−1] 0 ϕ−1) and in this case ϕ is known as strict generator. 3.1.1 Gumbel-Hougaard Copula The expression for Gumbel-Hougaard copula is (Nelsen 1999) n 1=θ o ; 1θ 0; θ2 1 C ðu; v; θ1 ; θ2 Þ ¼ 1 þ uθ1 1 2 þ vθ1 1 2 ð11Þ θ with generator function fðtÞ ¼ tθ1 1 2 . The advantage of two-parameter Archimedean copula is that it can simultaneously capture upper and lower tail dependence. 3.1.3 Elliptical Class of Copula: Student’s t Copula The student’s t copula is an elliptical copula, the CDF expression for which is given as, (Cherubini et al. 2004) t Cr;ϑ ðu; vÞ ¼ tr;ϑ tϑ1 ðuÞ; tϑ1 ðvÞ n oðϑþ2Þ=2 R t1 ðuÞ R t1 ðvÞ ð12Þ x2 2rxyþy2 1 ϑ ϑ p ffiffiffiffiffiffiffiffiffiffi 1 þ ¼ 1 dxdy ϑ > 2; 0 r < 1 2 1 ϑð1r Þ 2 2p

ð1r Þ

where, tϑ(x) is the univariate student’s t distribution with ϑ degrees of freedom, Rx ðϑþ1Þ=2 ðϑþ1Þ=2Þ ð1 þ y2 =ϑÞ dy ϑ 6¼ 0 tϑ ðxÞ ¼ 1 Γpðffiffiffiffi pϑΓ ðϑ=2Þ

Z with Γ ðaÞ ¼

1

ta1 expðtÞdt

ð13Þ ð14Þ

0

where y ¼ tϑ1 ðuÞ; tϑ1 ðvÞ ; and tϑ1 ðÞ denotes the quantile function of a standard univariate tϑ distribution; and r denotes Pearson’s correlation coefficient and appears as a off diagonal element of a correlation matrix P. ϑ and r are two dependence parameters of bivariate student’s t copula. Elliptical class can model both positive and negative dependence. They do not have closed form expressions and are restricted to have radial symmetry. This family of copulas offers symmetric tail dependence. The student’s t copula has both upper and lower tail dependence of same magnitude due to its radial symmetry. 3.2 Estimation of Copula Parameter(s) The commonly used methods for estimation of copula parameter are—method of moments (Genest and Rivest 1993) based on non-parametric dependence measures such as Spearman’s ρ and Kendall’s τ; and maximum pseudo-likelihood method (Genest et al. 1995). If FX,Y(x, y) be a continuous bivariate distribution with copula C with univariate margins FX(x) and FY(y) respectively, then assuming joint CDF of copula C is FX,Y(x, y), the relationship between Kendall’s τ and the underlying copula function is expressed as (Schweizer and Wolff 1981), Z

Z

t ij ¼ 4

FX ðxÞFY ðyÞdFX ;Y ðx; yÞ 1 ¼ 4 2

½0;1

C ðu; vÞdC ðu; vÞ 1 2

½0;1

ð15Þ

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Hence, if there is one-to-one correspondence exists between parameter θ of the copula and one of the rank correlations then substituting the empirical values of the rank correlation into this relationship (such as, b θ¼f b t ij ) will yield the estimate of copula parameter (such as estimation of copula parameter in Gumbel-Hougaard copula in Eq. 10). Maximum pseudo-likelihood method does not require any prior assumptions regarding marginal distribution of the dependent variable. The procedure consists of transforming marginal variables into uniformly distributed vectors using its empirical distribution function. The copula parameters are estimated using maximization of the pseudo log-likelihood function. The empirical distribution function (CDF) from a d-dimensional random vector X ∊ {Xi,1, Xi,2, …, Xi,d} is computed using ranks of the data using following expression, U ¼ Ui;d ¼

Ranked data of Xi;d nþ1

; 8i ¼ 1; 2; :::; n

ð16Þ

where for bivariate case, d ∊ {1, 2}, Ui,d denotes the vector of pseudo-sample or empirical distribution function values and n is the number of observations. The rescaling (n+1) at the denominator instead of n avoids numerical problems at the boundaries of [0,1]2. This empirical distribution function is used as a surrogate for the unknown marginals. Substituting the empirical CDF values into bivariate copula density and applying logarithm in both sides of the expression yields the log-likelihood function of the form (Genest and Favre 2007), n o n n P P Ri Si ln cθ Ui;1 ; Ui;2 ¼ ln cθ nþ1 ; nþ1 ln LU ðθÞ ¼ 8i 2 f1; :::; ng ð17Þ i¼1

i¼1

where, Ri and Si denote ranks of the observed data. The parameter θ can be obtained by maximizing this rank-based pseudo log-likelihood function numerically, b θ ¼ arg maxfln LU ðθÞg

ð18Þ

For Student’s t copula a two-step estimation procedure is employed (Mashal and Zeevi 2002). For elliptical family of copula the relation between correlation matrix Pi,j with Kendall’s τ is given by 2 b i;j ð19Þ t i;j ¼ arcsin P p Hence the correlation matrix of Student’s t copula is estimated by substituting the Kendall’s τ estimate in Eq. 19 i.e., bij ¼ sin p bt ij ð20Þ P 2 b is estimated using a numerical search technique The parameter ϑ " # n n o X b b ð21Þ ϑ ¼ arg max ln cθ Ui;1 ; Ui;2 ; ϑ; P ϑ2ð2;1

i¼1

3.3 Goodness-of-Fit Tests for Copulas The copula based joint distribution is compared with rank-based empirical copula (Genest and Favre 2007) calculated from the observed data and is defined as,


n X Ri Si b n ðu; vÞ ¼ 1 u; v C I n i¼1 nþ1 nþ1

3309

ð22Þ

where, I(A) denotes the indicator function of set A taking the value 0, if A is false; and value of 1, if A is true. The Akaike’s information criteria (AIC) (Akaike 1974; Bozdogan 2000) and the root mean square error (RMSE) are used to compute the goodness-of-fit measures between fitted copula and empirical joint distribution. For both the criteria the model which has minimum value is chosen as the best model. The AIC can be defined as, AICðlÞ ¼ nlogðMSEÞ þ 2l

ð23Þ

where l denotes the number of fitted parameters; MSE is the mean square error of the fitted copula model with respect to empirical copula and is expressed as, MSE ¼

n 1 X fCn ðu; vÞ C ðu; v; θÞg2 n l i¼1

ð24Þ

where C(u, v; θ) represents parametric family of copulas.

4 Case Study Drought prone area of Saurashtra and Kutch region in Gujarat state, India is taken up as case study. The Gujarat state is situated between 20°06′ N–24°42′ N latitudes and 68°10′ E to 74° 28′ E longitude. The location map of the study region is given in Fig. 2. The Saurashtra is a semiarid region whereas Kutch falls under arid region. The monthly area-weighted precipitation data of seven rain gauge stations in Saurashtra and Kutch meteorological subdivision is obtained from Indian Institute of Tropical Meteorology, Pune (http://www.tropmet.res.in) for a period of 109 years from January 1900 to December 2008 and is used in the study. The monthly SPI-6 series is calculated and 105 drought events are identified. The SPI-6 series obtained using Eq. 3 for monthly precipitation data is shown in Fig. 3. The figure shows in the year 1987, the SPI value dropped below −3 indicating occurrence of severe drought in the region. The occurrence of drought events with their properties severity and duration are shown in Fig. 4, which indicates that the region has experienced major droughts in the year 1901, 1911 and 1987. Table 1 presents statistical properties of precipitation and drought variables observed during the study period. The annual mean precipitation during study period is 391.1 mm and the standard deviation is quite high due to high fluctuation of annual precipitation from a minimum of 59.25 mm to a maximum of 1033.75 mm. The statistics of drought variables shows that they are characterized by high values of standard deviation and skewness. 4.1 Bivariate Dependence Structure Analysis of Drought Characteristics Figure 5 presents scatter plot of drought characteristics—severity and duration. The skewed nature of drought variables can be observed from the nature of histograms along the axis. Figure 6 presents scaled ranks of severity and duration pair obtained using empirical distribution function (Eq. 16). The higher density of the ranked observations in upper right corner (upper tail) of the scatter plot suggests presence of upper tail dependence between the drought variables. The sample estimates of Pearson correlation coefficient r, Kendall’s τ and Spearman’s ρ are 0.96, 0.81 and 0.92 and their p-values are less than 0.0001(i.e., p 0; l > 0 b b1 lx Gamma fX ðxÞ ¼ l xΓ ðbeÞ ; x 0; l > 0; b > 0; R 1 b1 t Γ ðb Þ ¼ 0 t e dt positively skewed distribution fX ðxÞ ¼

μy and σy are mean and standard deviations of PY; yi Where Y 0 ln P(X) and 1 < μy < 1; μy ¼ σy > 0; σ2y ¼

x>0

Gumbel

parameters

1 ðxuÞ=b be

exp eðxuÞ=b

−∞ < x < ∞;

y2i ny2 n1

λ λ 0 scale parameter, β 0 shape parameter P lnðxÞ

A ¼ lnðhxÞ qffiffiffiffiffiffiffiffiffiffiffiffi n i

1 b ¼ 4A 1þ

b 1 þ 4A 3 ;l ¼ x

−∞ < u < ∞, b>0 u and b are location and scale parameters

negatively skewed distribution Weibull

fX ðxÞ ¼ axa1 b a exp½ðx=b Þa ; x 0; α 0 shape parameter, α, β>0 β 0 scale parameter

n


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Table 3 Mathematical expressions of kernel functions for non-parametric density estimation Kernel name

Expression

Normal

n 2o 1 < x < 1 exp x2 KðxÞ ¼ p1ffiffiffiffi 2p KðxÞ ¼ 34 ð1 x2 Þ 1 x 1

Quadratic

the three copula families considered, the Gumbel-Hougaard copula performs better than the others by capturing the dependence structure of drought variables. The random pairs generated from Gumbel-Hougaard copula (shown using grey dots) are well overlapped with the observed data (shown using black dots) whereas the simulated data from BB1 and Student’s t copula show scattering at higher values of severity and duration. Moreover the nature of upper tail exhibited by simulated samples in Student’s t copula does not have the similar trend as those of observed data. Therefore, Gumbel-Hougaard copula fits the data well as compared to the other two copulas. From Table 5, it can be observed that after BB1 copula in terms of AIC, Gumbel-Hougaard copula performs better than Student’s t for distance based statistics. To examine dependence of variables at extreme levels, the coefficients of tail dependence are also tested for the three copulas. 4.4 Comparing Upper Tail Dependence of the Copula Models The tail dependence coefficient captures the concordance between extreme values in the lower left quadrant tail and upper right quadrant tails of the variables. As the present study focuses on frequency analysis of extreme drought events, study of upper tail dependence exhibited by different copula models is important. If u is a threshold value then upper tail dependence coefficient (TDC) between two variables X and Y, denoted by λU is given as lU ¼ lim fFX ðxÞ > ujFY ðyÞ > ug u!1

ð27Þ

Table 4 Performance of different probability distributions in modeling marginal drought variables Drought variables Severity

Duration

Distribution

RMSE

AIC

Normal kernel

0.0153

−877.23

Quadratic kernel

0.0162

−865.83

Gamma

0.0667

−564.54

Log-normal

0.0474

−636.22

Exponential

0.0586

−593.92

Gumbel Weibull

0.1349 0.0599

−416.72 −587.15

Normal kernel

0.1370

−417.43

Quadratic kernel

0.1477

−401.60

Gamma

0.0765

−535.80

Log-normal

0.0655

−568.35

Exponential

0.0505

−624.86

Gumbel

0.1149

−450.37

Weibull

0.0698

−555.00

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(a)

0.4

0.8 Histogram Normal Kernel

FS(s)

fS(s)

0.3

0.2

0.1

0

0

5

10 15 Severity

20

0.4

0

25

(b)

0.3

0

5

10 15 Severity

20

25

1

0.8 Histogram Exponential PDF

0.2

FD(d)

fD(d)

Observed Normal Kernel

0.6

0.2

0.4

0.1

0

1

Observed Exponential CDF

0.6

0.4

0

5 10 Duration (months)

0.2

15

0


15

Fig. 7 Fitting marginal distribution of drought variables: a severity fitted with non-parametric normal kernel, and b duration fitted using exponential distribution

If λU ∊ (0, 1], then FX(x) and FY(y) are said to show upper tail dependence or extremal dependence in upper tail. If λU 00 then the variables are asymptotically independent in the upper tail. In terms of copula Eq. 27 can be expressed as (Nelsen et al. 2008) ðu;uÞ ðu;uÞ ¼ 2 lim 1C ¼ 2 d C ð1 Þ lU ¼ lim 12uþC 1u 1u 0

u!1

u!1

ð28Þ

where the function δC(•) is the diagonal section of copula C and given by δC(u) 0 C(u, u) for every u ∊ [0, 1].

Table 5 Parameters and goodness-of-fit measures of the fitted copula models Copula Families

Copula parameter

RMSE

AIC

Gumbel-Hougaard

θ05.38

0.0981

−485.54

BB1

θ1 01.58, θ2 01.86

0.0912

−498.81

Student’s t

ϑ02.11, r00.958

0.0976

−484.70

*

best estimator is shown as bold, second best estimator is shown as italics


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Fig. 8 Comparison of observed data (black dots) and thousand simulated random data (grey dots) using (a) Gumbel-Hougaard copula, (b) BB1 copula, (c) Student’s t copula

The upper tail dependence coefficient λU measures the concordance between extremely high values of X and Y. The expression for upper tail dependence exhibited by different copula families and their corresponding sample estimates are presented in Table 6. As discussed by Frahm et al. (2005) the model error could be large in a parametric TDC estimate, so, non-parametric TDC estimates are also evaluated. 4.5 Estimation of Non-parametric Tail Dependence Coefficient The non-parametric estimates of tail dependence coefficients are based on empirical copula (Eq. 22). To study non-parametric tail dependence coefficient several methods have been suggested in literature, such as, estimator based on logarithm of the diagonal section of the (Coles et al. 1999), estimator based on the slope of the secant along the copula copula lLOG U diagonal lSEC (Joe et al. 1992), and Capéraá-Fougéres-Genest estimator lCFG (Capéraá et al. U U 1997; Frahm et al. 2005). The lCFG estimator assumes that the underlying copula can be U approximated by an extreme-value copula. Studies have shown that the estimator performs well even if the copula does not belong to extreme value class as discussed by Frahm et al. , for computation of other estimators, a threshold is required. If {(u1, (2005). Except lCFG U v1), …, (un, vn)} are random samples obtained from Copula C(•), the bivariate upper tail is given by, dependence coefficient lCFG U "

b lCFG U

n 1X ¼ 2 2 exp log n i¼1

(sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)#

, 1 1 1 log log log ui vi max ðui ; vi Þ2

ð29Þ

As Gumbel-Hougaard copula is an extreme-value copula, in order to compare the results obtained using lCFG , estimator lLOG is also considered. If the diagonal section of copula C U U (u, u) is differentiable for u ∊ (1−ε, 1) for any ε>0 then lU ¼ 2 lim u!1

1 C ðu; uÞ dC ðu; uÞ log C ðu; uÞ ¼ 2 lim ¼ 2 lim u!1 u!1 1u du logðuÞ

The log-estimator is based on Eq. 30 and can be expressed as bn 1 k ; 1 k log C LOG n n b lU ¼ 2 log 1 kn

ð30Þ

ð31Þ

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Table 6 Coefficient of upper tail dependence for different copula models Copula

λU

b lU

Gumbel-Hougaard

2−21/θ

0.862

BB1

1=θ2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2tϑþ1

Student’s t

ðϑþ1Þð1ρÞ ð1þρÞ

0.550 0.812

where k ∊ {1, …, n−1} represents the threshold to be selected. The selection of threshold is performed through a plateau-finding algorithm as described in Frahm et al. (2005). In first step, the curve of b lk is smoothed by a kernel function. A kernel smoother defines a set of weights {Wi(x), i01, 2, …, n} for each x and can be expressed as, b f ðxÞ ¼

n X

Wi ðxÞyi

ð32Þ

i¼1

where yi are the observations to be smoothed. In Eq. (32) the weight sequence {Wi(x), i01, 2, …, n} describes the shape of the weight function, using a kernel density function of specified bandwidth, which adjusts the size and form of the weights near x. For a given band width, the weight sequence is defined as, Kð

Wi ðxÞ ¼ P n

xxi h

Kð

Þ

xxi h

Þ

where

n P

Wi ðxi Þ ¼ 1

i¼1

ð33Þ

i¼1

The box kernel with bandwidth h ¼ b0:005nc , h ∊ N is chosen as suggested by Frahm et al. (2005). The analytical expression for box-kernel estimator is given by following window function, 1; j xj 1=2; KðxÞ ¼ ð34Þ 0; otherwise where K(x) defines a unit interval centered at the origin. Thus the kernel smoothed map of k7!b lk leads to the means of 2b+1 successive points of b b l1 ; . . . ; ln to a new smoothed map of e l1 ; . . . ; e ln2b : In next step, a vector pk ¼ e lkþm1 ; k ¼ 1; . . . ; n 2b m þ 1 is defined with a plateau of length m ¼ lk ; . . . ; e pffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2b . Then the algorithm stops at the first plateau pk, whose elements fulfill the condition kþm1 X

e e li lk 2σ

ð35Þ

i¼kþ1

ln2b . Then the tail dependence coefficient where σ represents the standard deviation of e l1 ; . . . ; e is estimated as arithmetic mean of the vector corresponding to the plateau, m 1 X e b lkþi1 ðkÞ ¼ lLOG U m i¼1

ð36Þ

If no plateau fulfills the stopping condition, the estimate of tail dependence coefficient is set as zero.


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Bivariate random numbers of sample size 1,000 are generated from each copula family and the corresponding b lCFG and b lLOG values are compared in Table 7. For each copulas, the U U CFG LOG b b computation of lU and lU is repeated for ten different runs, b ln;i ; i ¼ 1; . . . ; 10 and the b b lU and standard deviation σ b b lU of ten runs. From estimates are compared by mean μ Table 7, it can be observed that Gumbel-Hougaard copula provides the best estimate for non-parametric tail dependence as compared to other copula families. Though there is little difference exist between Student’s t and Gumbel-Hougaard copulas in terms of distance based goodness-of-fit measures as well as non-parametric tail dependence coefficients b lCFG , the later one is selected for final model construction, U since Gumbel-Hougaard has less number of parameters than the Student’s t copula model. Figure 9 displays the surface plots of joint probability density function (PDF) (Fig. 9a), joint cumulative distribution function (CDF) (Fig. 9b) and contour plots of joint CDF (Fig. 9c) for Gumbel-Hougaard copula at different probability levels. The peak of the density plot at lower left corner of the unit square indicates presence of strong positive dependence between the drought variables.

5 Comparison of Copula-Based Joint CDF with Traditional Bivariate Distributions The copula based joint CDF is compared with bivariate log-normal and bivariate logistic models. A positive random variable X is said to be log-normally distributed with parameters μX (mean) and σX (standard deviation) if Y 0 ln X is normally distributed with mean μY and standard deviation σY. The PDF of bivariate log-normal distribution is given as (Yue 2000) h 2i 1 pffiffiffiffiffiffiffiffi exp q2 f ðx1 ; x2 Þ ¼ 1 < ρ < 1 ð37Þ 1ρ2 2px x σ σ 1 2 Y1 Y2

where, "

# lnðx1 Þ μY1 2 lnðx1 Þ μY1 lnðx2 Þ μY2 lnðx2 Þ μY2 2 1 q¼ exp 2ρ þ 1 ρ2 σY1 σY1 σY2 σY2

ð38Þ μYi and σYi are mean and standard deviation of Yi{i01, 2} respectively, ρ is the linear correlation coefficient (Pearson’s correlation) between the variables. The CDF of the expression can be computed by numerically integrating its PDF. The general form of logistic model for bivariate extreme value distributions is originally proposed by Gumbel (1961) as follows n o 1=m ð39Þ FX ;Y ðx; yÞ ¼ exp ½ð ln FX ðxÞÞm þ ð ln FY ðyÞÞm where FX(x) and FY(y) are the marginal distributions of random variables X and Y respectively, which must be one of the three distributions: type I or Gumbel, type II or Fréchet, type III or Weibull. The parameter m ∊ [1, ∞] describes the association between two random variables X and Y with m01 representing the stochastic independence case and m → ∞ represents complete dependence case. The association parameter m is computed as (Gumbel and Mustafi 1967) 1 m ¼ pffiffiffiffiffiffiffiffiffiffiffi 1ρ

ð40Þ

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Table 7 Comparison of non-parametric upper tail dependence coefficients estimated using b lCFG and b lLOG for U U various copula models Sample estimate

bb μ lU σ b b lU *

Gumbel - Hougaard b lCFG U

b lLOG U

BB 1 b lCFG U

Student’s t b lLOG U

b lCFG U

b lLOG U

0.8580

0.8543

0.740

0.595

0.8576

0.8165

0.0048

0.0255

0.0086

0.0039

0.0159

0.0303

Best estimates are shown in italics

In the past, the bivariate logistic model has been used for flood frequency analysis considering Gumbel marginal distributions (Yue 2001), Weibull and mixed Weibull distributions (Escalante 2007). In this study, Weibull distributions are adopted for marginal distributions of bivariate logistic model. The performance of the copula based joint distributions viz., Gumbel-Hougaard, BB1 and Student’s t copula and traditional bivariate distributions such as log-normal and logistic models (with Weibull marginal distributions) are compared with the empirical joint

Fig. 9 Surface plots of (a) Joint PDF (fS(s)fD(d)c[FS(s), FD(d)]), (b) Joint CDF (C[FS(s), FD(d)]), where fS(s) and fD(d) denotes PDF, andFS(s) and FD(d) denotes CDF of severity and duration respectively; (c) Contour plot of joint CDF superimposed on historical drought events


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distribution of observed data (using Gringorten’s plotting position formula), which is expressed as (Yue 2001) FX ;Y ðx; yÞ ¼ PðX x; Y yÞ ¼

No: of ðxj xi and yj yi Þ0:44 nþ0:12

8i; j ¼ 1; 2; . . . ; n

ð41Þ

The associated results for each distribution fit are presented in Table 8. The Table shows that copula based distribution provides better fit to the observed data as compared to traditional multivariate distributions. Also, the Kolmogorov-Smirnov (KS) goodness-of-fit test is conducted to assess whether the proposed models can represent the observed data well or not. The critical value of KS test statistic Dmax(α) with sample size n0105 and α05 % significance level is 0.133. The maximum deviation (dmax) observed between empirical joint distribution and joint CDF of Gumbel-Hougaard copula is 0.126. Among the three copulas considered, GumbelHouggard copula offers minimum AIC value. Hence, the results suggest that joint CDF obtained from Gumbel-Hougaard copula yields better estimate as compared to bivariate parametric distributions having identical marginals.

6 Bivariate Frequency Analysis Using Copulas 6.1 Computation of Joint Return Period In hydrology and water resources engineering, the return period is defined as the average elapsed time between occurrences of an event with a certain magnitude or greater. But when the concept of return period is applied to droughts/low flows—it is the average time between events with a certain magnitude or less (Haan 1977). In univariate context, the return period of droughts in terms of drought severity and duration can be expressed as, TD ¼

E ðId Þ 1 FD ðdÞ

ð42Þ

TS ¼

E ðId Þ 1 FS ðsÞ

ð43Þ

where E(Id) is the expected or mean inter-arrival time of drought events which can be estimated from observed drought data, FD(d) and FS(s) denotes CDF of univariate drought duration and severity respectively. But as drought events are characterized by joint behavior of mutually correlated random variables, univariate frequency analysis may lead to over/underestimation of associated risk of the events. Thus risk (exceedance probability) should be defined in terms of joint behavior of the specific drought events (Salvadori 2004). The two cases of bivariate return periods can be computed: either drought duration or severity exceeding a specific value, TDS (i.e., D ≥ d or S ≥ s)or by 0 drought duration and severity exceeding a specific value, TDS ði:e:; D d and S sÞ . These two types of return periods can be computed using copula-based approach (Shiau 2003), which are given by

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Table 8 Comparison of performance of traditional bivariate distribution and copula based distribution models Models

Joint distribution

Parameters

Traditional model

log normal

μ b ¼ ½0:794 0:605

Copula

GumbelHougaard

AIC

0.511

−279.71

bx ¼ 3:531ð2:89; 4:32Þ , a b b x ¼ 1:013ð0:89; 1:16Þ by ¼ 2:733ð2:30; 3:24Þ , a b b y ¼ 1:198ð1:05; 1:37Þ

0.196

−553.67

0.126

−687.35

CovðX ; Y Þ ¼

Logistic model

KSstatistics

0:75 0:62 0:62 0:55

m04.92 b θ ¼ 5:38

BB1

θ1 01.58, θ2 01.86

0.147

−624.28

Student’s t

ϑ02.11, r00.958

0.126

−677.46

by , b bx , b b x and a b y represents shape and scale parameters of the univariate Weibull distribution parameters a for severity and duration respectively. Bracketed values denote 95 % confidence interval of estimated parameters using MLE. Best estimate is shown in bold

*

TDS ¼

E ð Id Þ E ðId Þ E ðId Þ ¼ ¼ PðD d or S sÞ 1 FDS ðd; sÞ 1 C ðFD ðdÞ; FS ðsÞÞ

0

TDS ¼

E ðI d Þ PðDd and SsÞ

¼ ¼

E ðI d Þ 1FD ðdÞFS ðsÞþFDS ðd;sÞ E ðI d Þ 1FD ðdÞFS ðsÞþC ðFD ðdÞ;FS ðsÞÞ

ð44Þ

ð45Þ

where the mean inter-arrival time estimated from observed data is 11.9 months. As various combinations of severity and duration lead to same return period, the iso-lines of joint return periods can be derived. Figure 10 presents iso-lines of joint return period evaluated using Eqs. 44 and 45 respectively. The contour lines for specific joint return periods in which either severity or duration exceeded TDS has no bounds (Fig. 10a); whereas the joint return periods in which both severity and duration 0 exceeded TDS are described by horizontal and vertical axes (Fig. 10b). It is noticed 0 that for TDS case, there are four observed drought events with more than 100 year return periods. The higher density of scatter points in less than one-in-5 year drought events indicates frequent water deficit in the region. For example, drought in 1911 had severity of 20.24 and duration 11 months, the univariate return period of this drought event using severity is 69.5 and duration 75.5 years; whereas the bivariate 0 return periods TDS (Eq. 44) and TDS (Eq. 45) estimated as 63.5 and 84.3 years respectively. While comparing with univariate return periods estimated using drought 0 severity and duration separately, TDS resulted in smaller return period, whereas TDS resulted in higher return period. Also in many cases, it is observed that the values of 0 TDS are smaller than that of TDS for a given severity. For example, the bivariate joint 0 return periods TDS and TDS for 9 months drought event in the year 1987 with severity 22 are 34.4 and 210 years respectively.


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6.2 Conditional Return Period of Drought Events The conditional drought return periods can be derived using bivariate copulas. The conditional return periods of drought duration given severity exceeding certain threshold s′can be expressed as (Shiau 2003) TDjSs ¼

TS E ðId Þ 1 ¼ PðD d and S sÞ 1 FS ðsÞ 1 FS ðsÞ FD ðdÞ þ FD;S ðd; sÞ

ð46Þ

Similarly, conditional return period of drought severity given duration exceeding a certain threshold d′ can be expressed as TSjDd ¼

Td E ðId Þ 1 ¼ PðD d and S sÞ 1 Fd ðdÞ 1 FS ðsÞ FD ðdÞ þ FD;S ðd; sÞ

ð47Þ

where TD|S≥s denotes the conditional return period for drought duration D given S ≥ s; TS|D≥d represents conditional return period for drought severity S given D ≥ d. Figure 11 presents conditional return periods of drought events TD|S≥s (Fig. 11a) and TS|D≥d (Fig. 11b) defined by Eq. 46 and Eq. 47, respectively for various percentile levels (75th, 80th, 90th and 95th) of severity and duration. From Fig. 11, it can be observed that conditional return period of duration given severity (TD|S≥s), and severity given duration (TS|D≥d) increases with increase in severity and duration respectively. With the increase in severity and duration at each percentile level, the trends of skewness in return period are also increasing. For example, conditional return period of drought severity exceeding 4.56 given duration less than 4 months is 28 years. Thus, these derived joint and conditional return periods of drought severity and duration can be useful in risk based planning and management of water resources systems in drought affected areas.

TSD 14

(b)

10

100 50

Duration (months)

12 10 8

25

5 15

2

6 4

0 0

10 15 Severity

10 25 8

15 10

6

100 50

5

2 5

100 50

4

5 2

T'SD 14 12

Duration (months)

(a)

20

2 2 1 1

25 5

10 15 Severity

20

Fig. 10 Bivariate return period of drought severity and duration (a) drought duration or severity exceeding a 0 specific value TDS (b) drought duration and severity exceeding a specific value TDS

3324


(a)

4

10

s' > 3.4 (75th percentile) s' > 4.3 (80th percentile) s' > 9.8 (90th percentile)

3

10 TD|S s ≥

s' > 13.0 (95th percentile)

2

10

1

10

(b)

0

2

4


10

12

14

4

10

d'> 3 (75th percentile) d'> 3.5 (80th percentile) d'> 7 (90th percentile) 3

d'> 8 (95th percentile)

TS|D d ≥

10

2

10

1

10

0

5

10

15

20

25

Severity

Fig. 11 Conditional return period (in years) (a) duration given drought severity TD|S≥s, (b) severity given drought duration (in months) TS|D≥d

7 Summary and Conclusions The study presents a semi-parametric copula based approach for frequency analysis of extreme drought events in Saurashtra and Kutch region in Gujarat state, India. The Standardized Precipitation Index (SPI 6) is used for characterizing drought events for the time period 1900–2008. Considering the extreme behavior of drought variables (severity and duration) and their dependence, this study applied two Archimedean copulas – Gumbel-Hougaard and BB1, and one elliptical – Student’s t copula for modeling the joint dependence of drought characteristics. The specific conclusions drawn from the study are given below:

&

Estimation of marginal distribution of drought variables has a great impact on joint exceedance probabilities of two variables. For modeling marginal behavior of drought variables several distributions from the family of parametric and nonparametric distributions are investigated and found that the drought severity is best described by non-


&

&

&

&

3325

parametric kernel density estimator, while the duration is best represented by exponential distribution. For modeling joint behavior of drought variables, three copula families are applied and their performance is assessed by statistical and graphical goodness-of-fit tests. The goodness-of-fit tests as well as tail dependence test reveals that Gumbel-Hougaard copula is the best representing model for describing joint dependence structure of drought variables. Based on comparison of the copula based models with bivariate log-normal and the logistic model for bivariate extreme value distribution (with Weibull marginals), it is found that the copula-based joint distribution (in general), Gumbel-Hougaard copula (in particular) best representing the drought characteristics of study region. The bivariate joint return periods of drought events are investigated considering two cases: either drought duration or severity exceeding a specific value, TDS (i.e., D ≥ d or S ≥ s) 0 or by drought duration and severity exceeding a specific value, TDS ði:e:; D d and S sÞ: It is noted that the contour lines for specific joint return periods in which either severity or duration exceeded (TDS) have no bounds; whereas the joint return 0 period in which both severity and duration exceeded TDS are described by horizontal and vertical axes. The conditional return period curves of different conditional values of severity and duration show that there is increase in trend for return periods at higher percentile values of severity and durations.

Appendix A1 Simulation from Bivariate Archimedean Copulas Simulation of random pair (u, v) from distribution function Copula C(•) is performed using following steps (Joe 1997): 1. Simulate sequence of two independent uniform random variates u and q from U (0, 1). 2. Evaluate the inverse of conditional distribution function at v, i.e., u;v;θÞ is the first derivative of the bivariate v 0 C−1(q|u), where C ðvju; θÞ ¼ @C ð@u marginal of copula function, C(u, v; θ). 3. Then the pair (u, v) are uniformly distributed random variables drawn from the respective copula families C(u, observations from copulas can be obtained v; θ). The simulated

by computing ðx; yÞ ¼ FX1 ðuÞ; FY1 ðvÞ , where FX1 ðÞ and FY1 ðÞ are the inverse of cumulative distribution functions of X and Y respectively. A2 Simulation from Bivariate Student’s t Copula 1. Generate random vector z1, z2 with ϑ degrees of freedom and Pi,j as correlation matrix after calibration of Student’s t copula, i.e., (z1, z2) ≈ td(ϑ, Pi,j); where td(•) is multivariate t distribution. 2. Set (u, v) 0 (tϑ(z1), tϑ(z2))′, where tϑ(•) is the univariate Student’s t distribution with ϑ degrees of freedom. 3. Simulated observations from copula based joint distribution can be obtained by com

puting ðx; yÞ ¼ FX1 ðuÞ; FY1 ðvÞ .

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