copula function

The drought characteristics are often highly correlated. But, the univariate drought analysis is not a proper approach since it doesn’t involve the dependence structure of drought characteristics. Therefore, the multivariate drought analysis is used since it considers the dependence structure of drought characteristics in the model. The aim of this study is to analysis of multivariate hydrological drought in the Mehran and Sedij basins. For this reason, the empirical copula was used to compute the cumulative distribution function of the runoff and the joint deficit index. Then, the drought severity, duration and magnitude were extracted and several theoretical copulas belongs to the Archimedean and Elipticalfamilies were fitted to obtain the trivariate distribution of drought variables. Results of the JDI and SPI-12 evaluation in the study area represented that the JDI is a proper index of monitoring hydrological drought and provides a more precise estimation than the SPI-12. Furthere, results of joint return period indicated that the joint trivariate return period is larger than the conditional trivariate return period. So that, the joint or conditional probability with high or low return periods is important in predicting drought events. Because the under-estimation or over-estimation of drought risk have serious impact on environmental resources, soil moisture and water quality. Generally, the multi-dimentional copulas are useful approach in evaluating the complicated and non-linear relationship of variables and constructing a comprehensive index for evaluating drought condition.

Recently, copula functions have attracted great attention of hydrologists as a practical tool for multivariate frequency analysis of climatological phenomena. In this study, we focus on the joint frequency analysis of two dependent characteristics of rainfall, including depth (mm) and duration (hr) using copulas for 522 events recorded in Sangdeh rain gauge station located in Kasiliyan watershed. To join the marginal distributions and constructing the joint distribution, seven copulas including Clyton, Ali-Mikhail-Haq, Farlie-Gumbel-Morgenstern, Frank, Galambos, Gumbel-Hougaard and Placket were used and evaluated. By comparing the mentioned parametric copulas with an empirical copula, we found that the Placket is the the best fitted copula on the considered variables. Finally, the joint probabilities, joint return periods and conditional joint return periods were calculated and plotted. For example, joint probability values for two events with duration of 12 and 24 (hr) given rainfall depth that exceeds 15 (mm) were calculated as 0.2663 and 0.7693, respectively. Also, conditional return period was calculated equal to 9.19 year for an event with depth of 30 (mm), given rainfall duration that exceeds 24 (hr) and equal to 14.94 year for an event with duration of 24 (hr), given rainfall depth that exceeds 30 (mm).