Evaluation of trends and multivariate frequency analysis of droughts in three meteorological subdivisions of western India

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1 INTERNATIONAL JORNAL OF CLIMATOLOGY Int. J. Climatol. 34: Published online 8 June 3 in Wiley Online Library wileyonlinelibrary.com DOI:./joc.374 Evaluation of trends and multivariate frequency analysis of droughts in three meteorological subdivisions of western India Poulomi Ganguli and M. Janga Reddy* Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai, India ABSTRACT: This article presents evaluation of trends and multivariate frequency analysis of droughts in three meteorological subdivisions of western India, namely, western Rajasthan, Saurashtra and Kutch and Marathwada regions. These regions are frequently affected by droughts and there is an urgent need for effective planning and management of droughts. Meteorological drought is modelled using Standardized Precipitation Index SPI at a time scale of 6 months over years during Trends in SPI time series are investigated by using nonparametric Mann Kendall trend test for different time windows: entire study period of and then splitting into three time windows of , and For total study period, the long-term trend in SPI time series is found to be in an upward direction for the three regions. However, statistically significant downward trend is observed during for western Rajasthan region, and Saurashtra and Kutch region in the month of June, indicating increase in number of drought occurrences during this period. Further, drought is a multivariate natural calamity characterizing severity, duration and peak; hence, probabilistic assessment of drought characteristics is investigated using copula method. The joint distribution of drought properties is modelled using three fully nested forms of Archimedean copulas: Clayton, Gumbel Hougaard and Frank and one elliptical class of Student s t copula. On performing various statistical tests as well as upper tail dependence test it is found that Student s t copula better represents trivariate drought properties when compared with other copula families. The joint distribution obtained from the copula is utilized for computation of conditional probabilities and joint return periods. The importance of trivariate frequency analysis of drought is elucidated over univariate and bivariate frequency analysis. Overall, the results of the study could provide valuable insight towards regional drought risk management under changing climate. KEY WORDS meteorological drought; SPI; trend detection; copula; frequency analysis Received November ; Revised 4 April 3; Accepted April 3. Introduction Droughts are regional in nature and often characterized by temporary departures from normal precipitation resulting in severe water shortage. Drought is a natural and recurring feature of climate, which occurs in virtually all climate regimes. Impacts of droughts are multifaceted, affecting water resources Shiau, 6, agriculture Wang, 5, ecosystems Reichstein et al., as well as socioeconomic aspects Bardsley et al., 984. Therefore, drought management and monitoring is an important issue Wilhite, 996. According to recent studies Pai et al.,, the majority of droughtprone regions in India are concentrated on the states of Rajasthan, Gujarat, Jammu and Kashmir, Punjab and Haryana, where drought probabilities are more than %. Frequent drought is also observed in parts of Maharashtra except Konkan, Rayalaseema in Andhra Pradesh, * Correspondence to: M. J. Reddy, Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 476, India. mjreddy@civil.iitb.ac.in West Bengal and Orissa states. In the year, drought affected around 56% of total geographical area in the country and agricultural gross domestic product GDP was reduced about 3.% Samra, 4. Drought is multiattribute in nature characterizing several mutually correlated random variables, such as severity, duration, peak and spatial extent. The frequency of droughts at various levels of severity, duration and peak provides the exposure risk of drought in a region. It is critical to understand the nature of drought risk in order to establish comprehensive and integrated drought management strategies. Proper management of droughts requires knowledge of the expected frequency of drought magnitude, which can be achieved by employing probabilistic approaches. However, univariate frequency analysis cannot provide accurate assessment of the probability of occurrence of extremes if underlying hydrological event is described by a set of correlated random variables Chebana and Ouarada, and may lead to over/under estimation of associated drought risk González and Valdés, 3. Hence, multivariate 3 Royal Meteorological Society

2 9 P. GANGLI AND M. J. REDDY statistical approach is having high importance in drought studies. India mainly depends on two monsoon systems: southwest or summer monsoon June to September, which accounts for around 7 8% of annual precipitation in many parts of the country, and northeast or winter monsoon October to December, which accounts for rest of the precipitation. Changes in climate during southwest monsoon period have significant impact on agricultural production and water management in the country. Summer monsoon precipitation exhibits both spatial and temporal fluctuations; consequently drought is one of the most frequently occurring natural calamities in arid and semiarid parts of the country. In this aspect, several studies in India have attempted to study trend in meteorological variables such as precipitation and temperature pattern at various spatial and temporal scales Jain and Kumar, ; Kumar et al.,, but not many studies attempted the analysis of temporal trends and multivariate drought risk at frequently drought-affected regions in India. Recently, copulas have gained wide popularity in modelling multivariate drought properties because of its flexibility in allowing marginal distribution of any form and ability to preserve nonlinear dependence pattern of the correlated random variables when compared with traditional multivariate distributions Salvadori and De Michele, 4; Genest and Favre, 7. Copula-based models have been applied in several hydrological problems: for multivariate frequency analysis of floods Genest and Favre, 7; Salvadori et al.,, analysis of extreme rainfall events Ghosh, ; Vandenberghe et al.,, multivariate frequency analysis of droughts Serinaldi et al., 9; Kao and Govindaraju,, analysis of sea storms De Michele et al., 7, prediction of stream flows in ungauged basins Samaniego et al.,, geostatistical interpolation Bárdossy and Li, 8 and uncertainty assessment studies pertaining to soil moisture Gao et al., 7, uncertainty assessment in prediction of Indian summer monsoon rainfall Maity and Kumar, 8 and downscaling regional climate simulations Laux et al.,, etc. Copulas are parametrically specified joint distributions obtained from specified marginals, and the copula functions can be used effectively for modelling nonlinear dependence of various practical problems. Most of the studies pertaining to copulas in hydrology focussed on modelling bivariate dependence. Shiau 6 investigated bivariate joint distribution of drought properties severity and duration in Southern Taiwan using Standardized Precipitation Index SPI and copulas. Shiau and Modarres 9 employed Archimedean class of Clayton copulas for modelling drought - Duration-Frequency S-D-F curves for two different climatic regions Abadan and Anzali in Iran owing to its simplified structure. The results inferred that drought severity in humid region might be more severe if high rainfall fluctuations existed in the region. Song and Singh modelled bivariate joint probability distribution of drought properties in Texas using meta-elliptical class of copulas and found meta-gaussian copula performed satisfactorily in modelling the dependence. Lee et al. studied influence of tail shape of four different copula families Gumbel Hougaard, Frank, Clayton and Gaussian for bivariate drought frequency analysis in Canada and Iran. Their study showed that Clayton copula is not an appropriate choice for modelling droughts as dependence between two variables in the upper tail of Calyton copula was very weak, whereas Frank and Gumbel Hougaard copula showed better performance for modelling bivariate drought properties. In recent past, few studies have also employed multivariate copulas in modelling drought properties. Wong et al. investigated effect of El Niño Southern Oscillation Index ENSO phenomenon on nature of multivariate drought frequency using precipitation data from two districts in New South Wales, Australia. The performance of trivariate Gumbel Hougaard and Student s t copula was tested for each climate phase. A little difference was observed between the performance of Gumbel Hougaard and Student s t copula in distancebased goodness-of-fit measures and the latter emerged as better model. Madadgar and Moradkhani investigated joint behaviour of drought properties under climate change using copulas in Oregon. For analysing streamflow-based drought indices, the performance of trivariate Gumbel Hougaard and Student s t copulas is tested to model joint dependence between drought properties severity, duration and intensity. Results showed that both Gumbel Hougaard and Student s t copula performed similarly for historical 9 9 time period. Serinaldi et al. 9 modelled four-dimensional joint distribution of drought properties mean SPI, duration, minimum SPI, drought areal extent and computed joint return periods using Student s t copula. Kao and Govindaraju proposed copula-based Joint Deficit Index using precipitation and stream flow as marginals with window sizes varying from to months in Indiana watershed. The temporal dependence among hydrologic variable is modelled using Student s t copula. In most of the studies, researchers employed either bivariate or symmetric trivariate Archimedean copulas for analysing multivariate drought properties. Although Archimedean copulas fit bivariate joint distributions satisfactorily Savu and Trede, 4, they are extremely restrictive in higher dimensions because of their exchangeable structure resulting in equicorrelated ranks Savu and Trede,. In few studies, simplifications were assumed for deriving return period of drought variables, such as dependence between drought properties or independence of the third variable. Then bivariate copulas were employed to model pair-wise dependence between the variables. The failure to take into account the mutual dependence between relevant hydrologic variable may lead to incomplete probabilistic assessment of drought occurrence De Michele et al., 5; Grimaldi and Serinaldi, 6. Hence, in this study, trivariate drought properties involving severity, duration and peak 3 Royal Meteorological Society Int. J. Climatol. 34:

3 TRENDS AND MLTIVARIATE FREQENCY ANALYSIS OF DROGHTS IN INDIA 93 are simultaneously modelled instead of considering pair-wise dependence between the variables separately. An attempt has been made to evaluate the applicability of three fully nested Archimedean FNA class of copulas such as Clayton, Gumbel Hougaard and Frank, and one elliptical class of Student s t copulas to model joint dependence between drought variables. Then the best copula will be used for multivariate frequency analysis of regional droughts in terms of primary and secondary return periods. The aim of this study is to detect potential trends in long-term time series of SPI in order to seek climate change impact and to analyse multivariate meteorological drought frequency in three drought-prone regions western Rajasthan in Rajasthan, Saurashtra and Kutch region in Gujarat and Marathwada in Maharashtra over years during Meteorological drought condition is modelled using SPI at time scale of 6 months SPI-6. Temporal changes in SPI time series are examined using trend analysis over the entire study period and three shorter time windows of , and Then trivariate drought properties, severity, duration and peak, are investigated. Thus, the objectives of this article are to perform trend analysis of droughts based on SPI time series using nonparametric trend tests; to perform copula-based multivariate frequency analysis of droughts and 3 to investigate the regional pattern of droughts in the three drought-prone regions of India.. Methodology.. Drought modelling using SPI In this study, drought is modelled using SPI, which is normal quantile transformation applied to a fitted parametric distribution of precipitation time series for a particular time scale e.g. 6 months. This study adopted 6-month SPI SPI-6; it compares the precipitation for that period with the same 6-month period over the historical record. For example, a 6-month SPI at the end of October compares the precipitation total for the May to October period with all the past totals for that same period. The aggregated precipitation data are fitted to Gamma distribution function. As two-parameter gamma function is undefined for zero values, but precipitation distribution may contain zeros, a mixed distribution function which can account zeros and real values of precipitation is employed, and the corresponding cumulative distribution function CDF is given by McKee et al., 993 F X x = q + q G X x where q is the zero precipitation probability obtained from historical time series; G X x is the CDF of Gamma distribution estimated for nonzero precipitation. The CDF of Gamma distribution is given as G X x = x g X x dx = x β ξ Ɣ ξ x ξ e x/ β dx, where g X x is Gamma probability density function PDF, ξ is a shape parameter and β is a scale parameter of Gamma distribution; Ɣξ is the Gamma function at ξ, Ɣ ξ = t ξ e t dt for ξ>. As precipitation is not normally distributed, an equiprobability transformation is carried out from the CDF of mixed distribution to the CDF of the standard normal distribution with zero mean and unit variance. This transformed probability gives the SPI for given accumulation time scale. i.e. Z = ψ F X x, where ψ is the CDF of standard normal distribution and ψ is the inverse of standard normal CDF. The drought condition is identified when the SPI value falls below a threshold limit, which is taken as percentile value of SPI Svoboda et al.,. In specific, the following drought properties can be determined for each identified drought period: Drought length or duration D is taken as the number of consecutive intervals months where SPI remains below the specified threshold value. As the drought event is defined at monthly time scale, the minimum duration of drought is month. Drought severity S is the cumulative values of SPI within the drought duration. For convenience, severity of drought event i is taken to be positive, which is given by McKee et al., 993 D S i = SPI i,t i =,..., n t= where n is the number of observed data points. Drought peak P is the absolute value of the minimum value taken by the SPI over the duration of the drought. Figure presents identification of drought properties using SPI for two consecutive drought events over a particular time period... Detecting drought trends using nonparametric trend analysis... Mann Kendall trend test For detection of statistically significant trend in SPI time series, Mann Kendall MK trend test Mann, 945; Kendall, 975 is employed with correction for autocorrelation Hamed and Rao, 998 and ties Hirsch et al., 98, as SPI time series are generally autocorrelated. For this study, the lagged autocorrelation is reported for i =,,.. up to n/4 lag of ranks of the observations, where n is the length of the time series. For seasonal MK trend test, an overall test statistic ZMK Seasonal is computed across m seasons by summing individual season s test statistics Z MK Hirschet al., 98. The statistical significance of trends is examined at significance levels of α =.5 and.. The magnitude of temporal trends 3 Royal Meteorological Society Int. J. Climatol. 34:

4 94 P. GANGLI AND M. J. REDDY 3.. SPI. Drought Events Threshold level t i t e t P Non drought duration D n -3. Drought severity D Si= SPI t= it, Drought duration D Time Interval months Figure. Illustration of drought properties severity S, duration D and peak P. Drought events and are shown by shaded region in the plot. t i and t e show initiation and termination of the drought events, respectively. in SPI time series is estimated by Sen s slope estimator Sen, Dependence measures The qualitative dependence among drought variables is analysed using graphical diagnostic tools such as Chi plots and Kendall plots, whereas quantitative assessment is performed using Pearson s linear correlation r and two nonparametric dependence measures Spearman s rank correlation ρ and Kendall s τ. Details about Chi and Kendall plots for assessing dependence are available in Genest and Favre Multivariate dependence modelling of droughts using copula.4.. Definition and properties Copulas are joint distribution functions of standard uniform random variates, which are capable of capturing dependence between two or more random variables. If X ={X,...,X d } be a random vector with continuous marginal CDFs F,..., F d. By following Sklar theorem Sklar, 959, the relation between the joint CDF H X and copula C can be written as Nelsen, 6 H X = C {F x,..., F d x d ; θ} X R d 3 where the function C : [,] d [,] is called a d-dimensional copula, with association parameter set θ. More details on theoretical background and properties of various copula families can be found in Nelsen 6 and Joe 997. In the following, brief details of copulas used in this study are presented..4.. Fully nested Archimedean copulas The FNA copula joins two or more ordinary bivariate or higher dimensional Archimedean copulas by another Archimedean copula Savu and Trede,. Table presents expressions for the CDF and associated parameter ranges of trivariate FNA copula families used in this study. More details about FNA class of copulas can be found in Grimaldi and Serinaldi 6 and Ganguli and Reddy Elliptical class of Student s t copula This family of copula belongs to a class of elliptical copulas. The Student s t copula is specified by multivariate Student s t-distribution. If R d for x R d denotes a symmetric shape parameter matrix i.e. is correlation matrix of multiple variables in d dimension, then trivariate Student s t copula for u ={u,u,u 3 } [,] 3 with ϑ degrees of freedom is defined as Mashal and Zeevi, C u; ϑ, = t d ϑ, tϑ u, tϑ u, tϑ u 3 = tϑ u tϑ u t ϑ u 3 Ɣ ϑ + d / Ɣ ϑ/ϑπ d/ / + y T y/ϑ ϑ+d/ dy dy dy 3, ϑ, σ ij < 4 where d = 3; y ={y,y,y 3 }; ϑ and σ i,j, i, j ={,,3} with the elements σ i,j are the parameters of trivariate Student s t copula. The shape parameter matrix is 3 Royal Meteorological Society Int. J. Climatol. 34:

5 TRENDS AND MLTIVARIATE FREQENCY ANALYSIS OF DROGHTS IN INDIA 95 Table. Trivariate Archimedean copulas and associated properties. Copulas Expression Conditions Clayton Gumbel Hougaard Frank { exp θ log [ ] u θ + u θ θ /θ θ θ + u 3 [ log u θ + log u θ ] } θ /θ + log u 3 θ θ [ ] θ /θ e θ u. e θ u { c c. e θ u 3 }, where c = e θ and c = e θ θ <θ, θ, θ <θ, θ [, θ <θ, θ, θ and θ represent dependence parameters; u, u and u 3 denote marginal CDFs. interpreted as positive-definite correlation matrix. The density of trivariate t copula explicitly is given by, c u; ϑ, = Ɣ ϑ + 3 / [ Ɣ ϑ/ ] [ ] 3 Ɣ ϑ + / / [ 3 i= + y i /ϑ ] ϑ+3/ ϑ+/ + y T y/ϑ In Equation 4 y i = tϑ u i denotes the quantile function of a standard univariate t ϑ distribution with ϑ degrees of freedom Estimation of copula parameters The parameters of copula functions are estimated using maximum pseudo-likelihood MPL method. The MPL method does not require any prior assumptions regarding marginal distribution of the dependent variables. Also, it provides computational convenience and flexibility over other methods. sing MPL method the marginal variables are transformed into uniformly distributed vectors using its empirical distribution and then copula parameters are estimated using maximization of pseudo log-likelihood function. For random vector X {X i,, X i,, X i,3 } the empirical distribution ECDF is estimated using expression, i,d = n { } X j,d X i,d n + j = 5 i =,,..., n, j i, d = 3 6 where { } being a logical indicator function of set and taking the value of either if is false or if is true. On substituting the empirical CDFs into copula density yields log-likelihood function of the form Genest and Favre, 7; Aussenegg and Cech, n L θ = ln [ { }] c θ i,, i,, i,3 = i= n i= { { }} Ri, ln c θ n +, R i, n +, R i,3 n + i {,..., n} 7 where c θ is the copula density, R i,d denotes ranks of the observed data, with average rank used for tied observation. The parameter θ can be obtained by maximizing this rank-based pseudo log-likelihood function numerically, θ = arg max {L θ} 8 For estimating parameters of Student s t copula, a twostep transformation procedure is employed. The elements σ i,j of shape parameter matrix are estimated using the relationship σ i,j = sin π τ i,j, i, j {,,3}, where τi,j is the pair-wise Kendall s dependence measure. However, sometimes it may be possible that the matrix is not positive definite using above transformation. In that case, eigenvalue decomposition Rousseeuw and Molenberghs, 993 is employed to transform correlation matrix into positive definite McNeil et al., 5. Then for estimating ϑ, numerical search technique is employed, which can be expressed as: [ n ϑ = argmax ln { } ] c θ i,, i,, i,3 ϑ, σi,j 9 ϑ, ] i= To avoid trapping at local optimal solution while using a gradient-based search technique, a real-coded genetic algorithm R-GA is employed to obtain optimal parameters of the copula functions Selection of appropriate copula family The appropriate copula model in trivariate case is selected by minimizing distance between parametric copulas and the empirical copula. The empirical copula C n is obtained from pseudo-observations { i,, i,, i,3 } using expression Genest and Favre, 7 C n u = n = n n i= n { } Û i, u, Û i, u, Û i,3 u 3 i= { Ri, n + u, R i, n + u, } R i,3 n + u 3 u = {u, u, u 3 } [, ] 3 As empirical copula is defined on a lattice l, the distance statistics are defined with discrete norms. 3 Royal Meteorological Society Int. J. Climatol. 34:

6 96 P. GANGLI AND M. J. REDDY Anderson Darling AD and integrated Anderson Darling IAD statistics are also used as a distance measures between empirical copula and fitted copulas. The copula family that results in minimum values of AD and IAD statistics will be the best fitted copula. The expressions for these distance measures are given below Ané and Kharoubi, 3: AD = max i n, j n, k n Ĉn i n, j n, k n C i θ n, j n, k n [ C θ i n, j n, k n C i θ n, j n, k n ] i n n n [Ĉn n, j n, k n C i θ n, j n, k n IAD = [ ] i= j = k= C i θ n, j n, k n C i θ n, j n, k n where i, j and k represent order statistics of the random variable u, u and u 3. The AD and IAD statistics emphasize deviations at the tails the corners of the unit square of the distribution by applying C θ in, j n, k n ] [ C in θ, j n, n k as the weight function. Apart from these distance-based statistics for selection of appropriate copula families the entropy-based statistic is also employed, which measures uncertainty of the distribution. The entropy offers a distance measure based on copula density; whereas the previous statistics are computed using CDFs. For trivariate distribution, the entropy of f c X, X {x,x,x 3 } is equal to the sum of the entropies of individual marginal distribution plus the entropy of the copula distribution function. Therefore, the discrete entropy of the copula model can be expressed as Ané and Kharoubi, 3 E [ f c X ] = E [ f X x ] + E [ f X x ] + E [ f X3 x 3 ] + E { c θ [ FX x, F X x, F X3 x 3 ]} 3 where E[f X x] = f X xln[f X x]dx and E { c θ [ FX x, F X x, F X3 x 3 ]} = n n i= j = k= n i c θ n, j n, k [ i ln c θ n n, j n, k ] n For visual inspection, a graphical comparison between observed data and simulated samples from selected copula model is also performed. This also offers qualitative assessment in finding a suitable copula model Tail dependence test of fitted copula model The upper tail dependence coefficient TDC captures the concordance between extreme values in the upper right quadrant tails of the distribution. If u be a threshold value then upper tail dependence between two random variables X and Y, denoted as λ is given by Nelsen, 6 λ = lim u {F X x > u F Y y > u} 4 ] sing copula the above equations can also be expressed as Nelsen et al., 8 u + C u, u λ = lim u u C u, u = lim = δ u C 5 u where the function δ C is the diagonal section of copula C and given by δ C u = C u,u for every u [,]. The estimate λ measures the concordance between extremely low values and extremely high values of random variables, respectively. If λ,], then F X x and F Y y are said to show upper tail dependence or extremal dependence. In this study, λ CFG estimator Capéraá et al., 997; Frahm et al., 5 and λ LOG estimator Frahm et al., 5 are employed to study nonparametric upper TDC. If {u,v,...,u n,v n } be random sample obtained from Copula C, then bivariate upper TDC using λ CFG is given by Frahm et al., 5 λ CFG = exp [ n { n log log i= u i log }] log max u i, v i v i / 6 The λ LOG estimator is expressed as Frahm et al., 5 λ LOG = log Ĉ n k n, k n log k 7 n where k {,..., n } represents the threshold to be selected. The selection of threshold is performed through a plateau-finding algorithm as described in Frahm et al. 5. In first step, the curve of λ k is smoothened by nonparametric box kernel with bandwidth b =.5n, b N. Thus, the kernel smoothened map of k λ k leads to the means of b + successive points of λ,..., λ n to a new smoothed map of λ,..., λ n b. In second step, a vector p k = λ k,..., λ k+m, k =,..., n b m + is defined with a plateau of length m = n b. Then the algorithm stops at the first plateau p k, whose elements fulfil the condition Frahm et al., 5 k+m i=k+ λ i λ k ε 8 where ε represents the standard deviation of λ,..., λ n b. Then the upper TDC is estimated as arithmetic mean of the vector corresponding to the plateau, λ LOG k = m λ k+i 9 m i= If no plateau fulfils the stopping condition, the estimate of TDC is set as zero and the procedure is repeated with a different set of parameters. 3 Royal Meteorological Society Int. J. Climatol. 34:

7 TRENDS AND MLTIVARIATE FREQENCY ANALYSIS OF DROGHTS IN INDIA Copula-based conditional probability of droughts The conditional probability of drought severity given drought duration and peaks exceeding certain thresholds d and p, respectively, is expressed as P S s D d, P p F S s F SD s, d F SP s, p +F SDP s, d, p = F D d F P p +F DP d, p F S s C SD s, d C SP s, p +C SDP s, d, p = F D d F P p +C DP d, p.6. Multivariate frequency analysis of droughts.6.. Copula-based joint primary return period As drought events are characterized by joint behaviour of mutually correlated random variables, univariate frequency analysis may lead to over/underestimation of associated risk of the event. For multivariate return period, different combinations of probabilities may produce the same return period. Hence, exceedance probability of drought should be defined in terms of joint behaviour of the specific events Salvadori and De Michele, 4. For multivariate case, in which X, X..., X d exceeds their respective thresholds {X > x,..., X d > x d } the joint return period is computed using inclusive probability OR and AND case of all the events Salvadori and De Michele, 4, which are known as primary return periods. The two cases of trivariate return periods can be computed: drought duration or severity or peak exceeding a specific value, TDSP i.e. D d or S s or P p, and drought duration and severity and peak exceeding a specific value, TDSP i.e. D d and S s and P p. The return period can be computed using copula-based approach and is given by Shiau, 3: T DSP = T DSP = = = = ζ F DSP d, s, p = ζ C DSP d, s, p ζ PDSP ζ F D d F S s F P p + F DS d, s + F SP s, p + F DP d, p F DSI d, s, p ζ F D d F S s F P p + C SD s, d + C SP s, p + C DP d, p C DSP d, s, p ζ P DSP where ζ = N n, N = total length of SPI time series years and n = total number of observed extremes here, drought events out of N years. PDSP = P D d S s P p denotes joint probability of occurrence of any one of the drought variables, i.e. either severity or duration or peak and PDSP = P D d S s P p denotes joint probability of occurrence of all the three drought variables simultaneously. C SD s,d, C DP d,p andc SP s,p are the joint distributions obtained from bivariate copula for severity duration, duration peak and severity peak combinations, respectively..6.. Copula-based joint secondary return period Salvadori and De Michele 4 presented the secondary return period for analysing frequency of supercritical drought events. Events with equal probability of exceedances form an iso-surface or critical layer at probability level p,]. For a d-dimensional distribution F = C θ the critical layer is defined using expression Salvadori et al., : = { F X = p, X R d} 3 The iso-surface partitions R d into three nonoverlapping regions: the subcritical region, which includes events with where FX < p; critical region, where FX = p and the supercritical region, which includes events with FX > p. The secondary return period is defined as the average time between occurrences of two supercritical events. The return period of supercritical event for multivariate random variables, including severity, duration and peak, associated with critical probability level can be computed as Salvadori et al., : T DSP = ζ K CDSP 4 where K CDSP is the Kendall s distribution of the supercritical events associated with copula C θ at critical probability level. sing function K CDSP multivariate information can be projected into a single dimension. As explicit expression of K CDSP is not available for all copula families, K CDSP values are estimated through Monte Carlo simulation using following procedure at probability level p,]: Generate a random sample X,..., X m, where m n and n = sample length, from copula C θ and compute their associated rank vectors R,..., R m Compute the transformed data V m m m j = i= R j R i m d = m j = l= R jl R il, i = j {,...,m} for every R ={R,...,R d } Approximate K CDSP. by K CDSP p = m Vi p. 3 Royal Meteorological Society Int. J. Climatol. 34:

8 98 P. GANGLI AND M. J. REDDY state ranges between and 4 m below ground level bgl, even more than 4 m bgl has also been recorded CGWB,. ARABIAN SEA INDIAN INDIA Saurashtra & Kutch 3 Marathwada SRI LANKA BAY OF BENGAL OCEAN CHINA BANGLA DESH Andaman & Nicobar Islands Figure. Meteorological subdivisions in India chosen for the study: western Rajasthan region, Saurashtra and Kutch region and 3 Marathwada region. 3. Application 3.. Study area and data In this study, detection of trends and multivariate frequency analysis of droughts is carried out with application to three drought-prone regions in India, namely, western Rajasthan region in Rajasthan, Saurashtra and Kutch region in Gujarat and Marathwada region in Maharashtra. Figure presents location of three study regions western Rajasthan 4.5 N 3 N, 69.5 E 76 E, Saurashtra and Kutch N 4 N, 69 E 7 E and Marathwada 8 N 9.5 N, 75 E 77.5 E Western Rajasthan The state of Rajasthan in India has variable topographic features, which is located between latitudes 3 3 Nand 3 N and longitudes 69 3 E and 78 7 E. The Thar Desert lies in northwestern part of the state. The climate is characterized by low, highly variable and ill-distributed rainfall, high wind speed, high evaporation losses and extremes of seasonal temperatures. Rainfall is not only low but also uncertain in the state. Rainfall distribution is highly variable both in space and time. Annual rainfall ranges between and 5 mm in western part of the state RACP,. The weather is classified into four seasons such as pre-monsoon, monsoon, post-monsoon and the winter. The average temperature during summer varies between 5 and 46 C and during winter between 8 and 8 C. Almost 5% of the area in the state comprises western arid region comprising Barmer and Jodhpur districts, which contributes only % of the total water resources RACP,. The groundwater level in the 3... Saurashtra and Kutch Gujarat Saurashtra and Kutch region is situated in Gujarat. The state of Gujarat is situated between 6 Nand 4 4 N latitude and 68 E and 74 8 E longitude, which comprises three distinct geographical regions: the peninsula, the region is known as Saurashtra, comprising plateau with low hills. Saurashtra comprises an area of about 6 km and its topography resembles an inverted saucer; Kutch on the northeast is dry and rocky and contains Rann desert of Kutch, the greater Rann in the north and little Rann in the east and 3 the mainland. The Saurashtra and Kutch are two water scarce regions in Gujarat. The Saurashtra is a semiarid region, whereas Kutch falls under arid region. The precipitation in Saurashtra varies from 5 to 7 mm, the highest being in the central portion, whereas the rainfall is very erratic in Kutch region with an average annual rainfall of about 9 mm Marathwada Maharashtra Marathwada region is situated in Maharashtra 5 4 N N latitudes and 7 3 E 8 3 E longitudes and comprises about km geographical area ENVIS,. Most part of the state lies in rain shadow belt of Western Ghat where annual average precipitation ranges between 6 and 7 mm Dept. of Agriculture Maharashtra,. Deficient rainfall leads to drought in places such as western Maharashtra and Marathwada regions in Maharashtra. Marathwada region comprises eight districts. The region in the part of upper Godavari basin with shape is roughly triangular with east west maximum extent about 394 km and north south extent about 33 km. Climate in this region is dry and moderately extreme in nature with annual precipitation about 7 mm source: Specific/ Data used The monthly area-weighted precipitation data from January 896 to December 5 are used for analysis of droughts in three regions western Rajasthan, Saurashtra and Kutch and Marathwada regions data source: IITM Pune, Analysis of drought climatology For modelling droughts in the three regions, 6-month SPI SPI-6 time series are computed and used for drought analysis. The SPI-6 measures the precipitation deficit or wetness for the past 6 months and identifies seasonal drought conditions. Soil moisture conditions respond to precipitation anomalies on relatively shorter time scale, and short-term droughts 3 6 months can affect a growing season of crops. The SPI-6 can monitor these drought 3 Royal Meteorological Society Int. J. Climatol. 34:

9 TRENDS AND MLTIVARIATE FREQENCY ANALYSIS OF DROGHTS IN INDIA 99 a 5 SPI Year b 4 SPI-6 c SPI Year Year Figure 3. Historical time series of SPI-6 for a Western Rajasthan, b Saurashtra and Kutch and c Marathwada regions during conditions quite well and it can help in proper planning of agriculture and water management in the drought-prone regions. Figure 3 shows monthly SPI-6 time series for the three regions during The drought conditions are identified based on criteria that when SPI-6 value for a particular time period falls less than or equal to percentile value. The SPI threshold value at percentile level comes out to be.74,.8 and.8 for western Rajasthan, Saurashtra and Kutch and Marathwada regions, respectively. The western Rajasthan region experienced 77 years under drought out of years of study period, and major droughts occurred during the years of 899 9, 9 9, 95 96, 9 9, 95 96, 98 99, 947, , 3 and 4 Figure 3a. The Saurashtra and Kutch region experienced 8 years under drought, and major droughts during the years of 899 9, 9 9, 94 95, 9 9, 98 99, 9, 93 94, 939, 948, , 974, 986 and 987 Figure 3b. Whereas the Marathwada region experienced 76 years under drought, and major droughts occurred during the years of 897, 899 9, 9, 95, 97 98, 9 93, 93 94, 98 99, 9 9, 94, 99 93, , 94, 97, , 984, , and 3 Figure 3c. Table presents comparative summary statistics of precipitation and drought properties observed during the study period The annual mean precipitations during study period are estimated as 43.6 mm for western Rajasthan region, 383. mm for Saurashtra and Kutch region and mm for Marathwada region Table. It is noted that the coefficient of variation of precipitation for Saurashtra and Kutch region is maximum. The drought events are characterized by trivariate properties, namely, severity, duration and peak. During the study period, the total number of historical drought events identified for the three regions as: 93 events at western Rajasthan region, 6 events at Saurashtra and Kutch region and 9 events at Marathwada region Table. It can be noted that the Saurashtra and Kutch region has experienced maximum number of years under drought. Number of droughts during study period is further subdivided into summer and winter droughts, and analysed separately. A summer drought is defined as an event starting between April and September, whereas the drought events starting between October and March are classified as winter droughts. The majority of drought events are summer droughts Table. Maximum drought duration is observed for western Rajasthan region as 8 months during August 968 to January 97, 4 months for Saurashtra and Kutch region during June 9 to July 9 and months for Marathwada region during June 9 to May 9, and April 97 to March 3 Royal Meteorological Society Int. J. Climatol. 34:

10 9 P. GANGLI AND M. J. REDDY Table. Summary statistics of regional precipitation and drought properties for the three regions during Climatic variables Statistics Western Rajasthan Saurashtra and Kutch Marathwada Precipitation Mean annual precipitation mm Standard deviation mm Coefficient of variation % Drought Number of droughts Number of years under drought Number of summer droughts Number of winter droughts Drought severity Mean Maximum Standard deviation Skewness Kurtosis Drought duration months Mean Maximum 8 4 Standard deviation Skewness Kurtosis Drought peak Mean Maximum Standard deviation Skewness Kurtosis In all the regions, drought is characterized by high values of skewness and kurtosis. In particular, western Rajasthan region has the highest value of severity, duration and peak during study period Table Analysis of drought trends in three regions Precipitation during monsoon season is important for agricultural production in the three regions. To assess any significant changes in frequency of drought occurrences, trend analysis is performed on SPI-6 time series using a nonparametric test modified MK trend test. Statistical significance of the upward or downward trend in SPI time series is examined for individual monsoon months as well as for the total monsoon period June to September and the magnitude of temporal trend is estimated by Sen s slope approach. Trends are analysed over four different time periods: long-term trend is computed for the complete - year time series 896 5; to quantify whether trends appear particularly severe during a particular time interval, short-term trends are analysed for three time windows: 36 years , 35 years and 39 years The statistical significance of upward/downward trends is evaluated using MK test statistics at significance levels of α =. and.5. Table 3 lists corresponding test statistics and magnitude of trend obtained using Sen s slope method. For entire time period 896 5, significant upward trend is observed in SPI time series i.e. long-term change towards wetter condition in the months of June and July also for total monsoon period for western Rajasthan region; in the month of June for Saurashtra and Kutch region and in the month of August also for total monsoon period for Marathwada region. Downward trend in SPI-6 time series is observed during second time window of for the month of June Table 3 at western Rajasthan statistically significant at α =. as well as Saurashtra and Kutch region statistically significant at α =.5. The decrease in trend of SPI time series indicates increase in number of drought occurrences. No significant upward or downward trend is observed for other time periods Analysing trivariate dependence between drought variables Initially, qualitative dependence between drought variables is analysed using graphical diagnostic tools Chi and Kendall plots for pair-wise dependent drought variables for the three regions. Figure 4 presents Chi and Kendall plots of drought variables for western Rajasthan, Saurashtra and Kutch and Marathwada regions. A strong deviation from the control limit is observed for all the three cases. Most of the points fall outside the confidence band in the Chi plots. In case of Kendall plots, a strong deviation is observed from the centre of the main diagonal, indicating strong positive association between drought variables. The observation pairs in the upper portion of the Kendall plot diverge from the diagonal line, which indicates presence of upper tail dependence between drought variables. By comparing the Kendallplot behaviour of all three regions, the maximum deviations from the diagonal line are observed for western Rajasthan region for severity peak and duration peak pairs when compared with the other two regions. The quantitative dependence among drought variables is analysed using Pearson s correlation coefficient r and two nonparametric dependence measures such as Kendall s τ and Spearman s ρ along with their p-value. 3 Royal Meteorological Society Int. J. Climatol. 34:

11 TRENDS AND MLTIVARIATE FREQENCY ANALYSIS OF DROGHTS IN INDIA 9 Table 3. Mann Kendall test statistics and magnitude of trend indicator Sen s slope parameter during each month of monsoon season and for total monsoon season for the three regions. Region Time period June July August September June to September Z MK b Z MK b Z MK b Z MK b ZMK Seasonal b Western Rajasthan Saurashtra and Kutch e e e e e Marathwada e e Z MK and b indicate Mann Kendall test statistics and slope estimated using Sen s slope estimator, respectively. Bold and bold italic numbers indicate that trend is statistically significant at and 5% significance level, respectively, as tested by modified Mann Kendall trend test. Figure 4. Depiction of pair-wise dependence pattern of drought variables using Chi plots and Kendall plots for a Western Rajasthan, b Saurashtra and Kutch region and c Marathwada region. The first, second and third rows depict the dependence patterns for severity duration, severity peak and duration peak pairs, respectively. Table 4 presents corresponding dependence measures for all three regions. The p-value estimates of the drought variable pairs are found to be less than. i. e. p <.. The dependencies between drought variables are found to be statistically significant at % significance level as checked by a standard two-tailed t-test Marginal distribution fit of drought variables Drought events may be characterized by a few large and many small events Hisdal et al.,. Past literature suggests that drought severity is usually modelled by Gamma Shiau et al., 7; Shiau and Modarres, 9; Song and Singh, ; Lee et al.,, extreme value Dalezios et al., ; Mishra and Desai, 5 and log-normal Clausen and Pearson, 995; Burn et al., 4; Fleig et al., 6 distributions, whereas duration can be modelled by exponential Shiau et al., 7; Shiau and Modarres, 9; Lee et al.,, Weibull Song and Singh, ; Wong et al., and log-normal Wong et al., distributions. In this study, for fitting drought variables several parametric families of distributions, such as Gamma, log-normal, exponential and Weibull distributions, are evaluated. The parameters of the distributions are estimated using maximum likelihood method. Table 5 presents the distributions, associated parameters, AIC criteria and Kolmogorov Smirnov K-S distance statistics for fitting marginal distributions of drought variables. The best fitted distribution for each variate is selected by employing minimum AIC statistics and K-S distance statistics. Distributions with minimum AIC and K-S distance statistics are preferred to model drought variables. As observed from 3 Royal Meteorological Society Int. J. Climatol. 34:

12 9 P. GANGLI AND M. J. REDDY Table 4. Association among drought variables for the three regions during Region Dependence measure duration Duration peak peak Western Rajasthan Pearson s r e e e 7 Kendall s τ.8.4e e e 9 Spearman s ρ.9 3.e e e 35 Gujarat Pearson s r e e e 5 Kendall s τ.8.63e e e 9 Spearman s ρ.9.3e e.85.e 3 Marathwada Pearson s r.96.5e e e Kendall s τ e e e Spearman s ρ.94.8e e.8. Table 5. Estimated parameters and comparison of performance of marginal distribution fit of drought variables for the three regions AIC = Akaike information criteria, d * = K-S statistic. Region Drought variable Distribution Shape Scale AIC d * Western Rajasthan Gamma Log-normal Weibull Duration Gamma Log-normal Weibull Exponential Gamma Log-normal Weibull Saurashtra and Kutch Gamma Log-normal Weibull Duration Gamma Log-normal Weibull Exponential Gamma Log-normal Weibull Marathwada Gamma Log-normal Weibull Duration Gamma Log-normal Weibull Exponential Gamma Log-normal Weibull AIC = n*log MSE + *m, wheren = length of observation; m = number of parameters of the distribution; MSE = mean square error; best fitted distribution is marked in bold letters. Table 5, drought characteristics severity, duration and peak are best modelled by log-normal, exponential and log-normal distributions, respectively. Figure 5 shows marginal distribution fit of drought variables for the three regions, which indicates good correspondence between theoretical distributions and empirical distributions, where the empirical distributions are estimated by using Gringorten s plotting position formula Copula-based joint dependence modelling of drought variables The fully nested form of Clayton, Gumbel Hougaard and Frank copulas and one elliptical class of Student s t copula copula families are chosen to model drought properties. The copula functions are fitted with MPL method, where parameters are estimated using R-GA. The GA parameters used involve, for nested Archimedean class of copula: population size of 5, generations of 5, single point crossover with crossover rate of.8 and Gaussian mutation function with mutation rate of. and selection strategy as stochastic uniform, whereas for Student s t copula: generations of and remaining parameters are same as above. The performance of the copula families is compared using distancebased statistics AD, IAD and entropy measures. Table 6 presents estimated copula parameters and performance 3 Royal Meteorological Society Int. J. Climatol. 34:

13 TRENDS AND MLTIVARIATE FREQENCY ANALYSIS OF DROGHTS IN INDIA 93 a b c.4 Marginal PDF Marginal CDF.4 Marginal PDF Marginal CDF.5 Marginal PDF Marginal CDF f S s. F S s.5 f d d. F d d.5 f i i.5 F I i f S s. F S s.5 f d d.5 F d d.5 f i i.5 F I i f S s. F S s.5 f d d.5 F d d.5 f i i.5 F I i Figure 5. Marginal distribution fit for drought variables: severity, duration and peak at a western Rajasthan, b Saurashtra and Kutch and c Marathwada regions. Histograms in the plot represent density histograms. Table 6. Estimated copula parameters and comparison of performance of different copula families fitted for the three regions. Regions Copula family Parameters AD IAD Entropy Western Rajasthan Clayton θ =.53, θ = Gumbel Hougaard θ =.54, θ = Frank θ = 8.49, θ = Student s t ϑ =., σ, =.95, σ,3 =.79, σ,3 = Saurashtra and Kutch Clayton θ =.33, θ = Gumbel Hougaard θ =.98, θ = Frank θ = 6.67, θ = Student s t ϑ =., σ, =.95, σ,3 =.7, σ,3 = Marathwada Clayton θ =.9, θ = Gumbel Hougaard θ =.96, θ = Frank θ = 5.83, θ = Student s t ϑ =., σ, =.95, σ,3 =.66, σ,3 = Best estimate is shown in bold letters. measures of copulas. It is noted that for all the cases, trivariate Student s t copula with ϑ =. degrees of freedom performed the best for characterizing drought properties when compared with other copula families Table 6. The poor performance of FNA copula can be attributed to the fact that there is restriction in the application of fully nested copula families as the correlation between two pairs should be identical and lower than the third pair such as rank-based correlation of τ S,D τ P,D = τ S,P. For example, for western Rajasthan the Kendall s τ dependence values from 5 simulated random samples from FNA copula class of Gumbel Hougaard family are as follows: τ S,D =.78, τ D,P =.37 and τ S,P =.38, and for Clayton copula family: τ S,D =.56, τ D,P =.55 and τ S,P =.58. Whereas the Kendall s τ dependence values for Student s t copula are τ S,D =.8, τ D,P =.56 and τ S,P =.76. Figure 6 presents 5 simulated random data from Student s t copula for western Rajasthan, Saurashtra and Kutch and Marathwada regions. It can be observed that the generated data from Student s t copula overlapped well with the observed drought variables, indicating Student s t copula satisfactorily fitted the observed drought properties. At the same time, Kendall s τ value from simulated drought data closely resembles to that of observed sample. To test the performance of the Student s t copula in modelling upper tail dependence, pair-wise TDCs for both parametric and nonparametric estimates are computed and used for evaluation of copula model param efficacy. The parametric upper TDC λ of Student s t copula is computed using relationship λ param ϑ+ σ i,j = t ϑ+ i,j, i, j {,, 3} and +σ i,j corresponding nonparametric upper TDC is evaluated using nonparametric estimators CFG estimator λ CFG and 3 Royal Meteorological Society Int. J. Climatol. 34: