Application of copula functions and intelligent algorithms for analysis of meteorological drought of Shahrood

Drought, on the contrary to other natural events like floods, earthquake and storms, occurs as a creeping and hidden phenomenon. In other words, it takes weeks or months to detect drought. Planning and development of water resources systems under drought conditions requires estimation of joint and conditional probability of duration and intensity of drought. Copula functions, which can be used for joint analysis of two or more variables, calculate the correlation between these variables, too. It should be noted that for construction of joint distributions, there is no limitation in selection of marginal distributions. In this research, to monitor meteorological drought in Shahrood synoptic station, Iran, two drought characteristics (i. e. intensity and duration) were analyzed jointly by using historical precipitation records and also copula functions. The Shahrood synoptic station, located in Semnan province, Iran, has longitude of 54◦58' E, latitude of 36◦25' N and height of 1325 m above mean sea level. The Semnan province has a variety of climates (from hot and dry to Caspian type). Since the standardized precipitation index (SPI) is known as famous index in studying the droughts, therefore, monthly rainfall data were obtained for the period of 1951-2010 and monthly SPI values were used to characterize drought intensity and duration. Some mono-variate distribution functions were separately fitted on drought intensity and duration. As a result, marginal distributions of Gamma and exponential were used for statistical analysis of duration and intensity of droughts. Then, to do the joint analysis, five copula functions (Clayton, Plackett, Galambos, Gumbel-Huggard, and Frank), which are usually considered in hydrological studies, were fitted on the data and their performance was evaluated by such criteria as root mean square error (RMSE), Akaike information criterion (AIC) and Nash-Sutcliffe efficiency (NSE). Copulas are the functions which connect multivariate distribution functions to their one-dimensional marginal distribution functions. In this research, the best copula function was selected and its parameter was estimated by three methods of maximum log-likelihood (MLE), firefly algorithm (FF) and big bang-big crunch (BB-BC) algorithm. By using the selected copula function, the joint and conditional probability and return period of intensity and duration of drought were calculated. In this paper, bivariate analysis of intensity and duration of drought in Shahrood synoptic station, for statistical period of 1951-2010, was performed, using copula functions. The parameter of selected objective function was compared by three methods. The results showed that joint and conditional probabilities of drought occurrence for duration of 8 months and intensity of 6.92 are 0.0038 and 0.073, respectively. The return periods for these conditions are 1106.47 and 236.21 years. The obtained results revealed that among the studied copula functions, Galambos was the most appropriate for bivariate analysis of drought intensity and duration in Shahrood synoptic station. This function was selected because it had the highest maximum log-likelihood (-443.8199), the least root mean square error (0.0683), the least value of Akaike information criterion (889.6399) and the highest Nash-Sutcliffe efficiency (0.9347). To show the good fit of Galambos copula function on duration and intensity variables of drought, the graph of empirical copula function was drawn with respect to theoretical copula function (Galambos), based on three methods of MLE, FF and BB-BC. The results showed that the points on these graphs could be fitted by the 45-degree line. Among the three criteria that were used to evaluate the copula functions, the maximum log-likelihood criterion estimated the objective function (RMSE) equal to 0.0683. While, the parameters which were optimized by firefly algorithm and big bang-big crunch algorithm, estimated the objective function equal to 0.0409. Therefore, the intelligent algorithms (i.e. firefly and big bang-big crunch algorithms) gave better results and thus are recommended to minimize the objective function and evaluate the copula functions. One of the reasons for using two different intelligent algorithms for optimizing the objective function is to get reliable results in the optimization process. In general, it could be concluded that the gained information from application of copula functions and intelligent algorithms in this research could provide accurate description of drought in the studied region, before its happening. This kind of information is usable in water resources management. When a drought is happening, this analyzed information can reduce the cost of damages in the region.