Bivariate Analysis of Drought Risk in West and Northwest of Iran Using PSO Algorithm and Copula Functions

Drought as a long-term water scarcity situation is a challenging issue in water resources management. This phenomenon is one of the expensive and less well-known natural disasters. Monitoring and forecasting droughts, especially the precise timing of its onset and its duration, and also the risk analysis of drought is of particular importance in water resources management, determining suitable cropping pattern and planning to decrease the adverse effects of droughts. The purpose of this study was bivariate analysis of the severity and duration of meteorological droughts in eight stations located in west and northwest of Iran using copula functions, drought risk index and particle swarm optimization (PSO) algorithm. For this purpose, the fitness of 10 different copula functions was examined to create a joint distribution of drought severity and duration variables. The drought risk was also evaluated based on the indices of resiliency, vulnerability, reliability, and drought risk index. For the first time, applying numerous copula functions, calculating the drought risk based on its indices, applying the PSO algorithm to determine optimal weight coefficients of indices, using the SPImod index for extracting the drought characteristics are the important innovations of this research.
In this study, the copula functions were used to create a bivariate drought distribution (severity and duration) in the western and northwestern regions of the country. After calculating the SPImod index values at each station, drought severity and duration variables were extracted. Also, the best copula function at each station was determined after evaluating the fitness of 10 different copula functions based on the evaluation statistics. Drought risk index (DRI) was calculated based on the indices of resiliency, vulnerability, and reliability. PSO algorithm was used to determine the optimum risk. In other previous researches around the world, the risk index has been calculated based on the DRI method and other risk methods, but the PSO algorithm and SPImod index have not been used, also the fewer type of copula functions have been used.
The results showed that there is a high correlation between severity and duration of meteorological drought in the study area. Then, the fitness of some two-dimensional copula families were examined based on the mean square error and maximum log-likelihood statistics to select the best fitted copula for each station. Then the joint distribution of duration and severity of drought were constructed by the selected copula function for every studied station. After constructing the joint distribution based on the selected copula functions at each station, some probability properties of drought, such as joint probabilities, bivariate return periods, conditional joint probabilities and conditional return periods were calculated. Also, the Kendall return period values were calculated and compared with the standard definition of the joint return period. The results showed that at a certain critical probability level, t, the Kendall return period is greater than the corresponding standard joint return period, and this difference increases with increasing t value. To calculate the drought risk value, at first the drought duration for each station was extracted by SPImod index, then the values of vulnerability, reliability and resiliency indices were calculated and the optimal values of coefficients of w1, w2, w3 were obtained equal to 0.08, 0.7 and 0.22, respectively by the PSO algorithm. Then the value of the optimal risk index, which was minimized for the mentioned coefficients of the risk index were calculated for each station. The results showed that the lowest risk was belonged to the Khoramabad station (0.565) and the highest one was related to the Kermanshah station (0.617).