Copula-Based Regional Drought Analysis and Derivation of Severity-Area-Frequency Curve in Lake Urmia Basin

Drought is a natural phenomenon and was described when precipitation is less than expected. Since the precipitation amounts in terms of spatial and temporal characteristics are different from one region to another, so this phenomenon is known as a multivariate phenomenon. This phenomenon often characterized by different variables such as drought duration, severity, intensity and spatial extent. Although site specific analysis can provide useful information on drought occurrences in a limited area, but these results have a fundamental uncertainty to drought risk assessment in a large region. Therefore regional drought analysis, provides a more comprehensive assessment in each region, and is essential for short and long term management of water resources. Meanwhile, the copula functions has been developed as a new advanced technique for modeling the two or multivariate joint probability distribution in different fields such as financial, hydrology, water resources and risk management. So, in this research, regional analysis of drought severity and percent of drought area were performed using the copula functions in Lake Urmia basin, as one of the Iran's drought-prone basin. Such study with emphasis on bivariate analysis of drought severity and drought areal extend were conducted for the first time in the study area. The main objectives of this study are: 1) Modeling drought characteristics in Lake Urmia basin, 2) Evaluation of copula functions in modeling the structure of the region's drought characteristics, and 3) Develop the Severity-Area-Frequency curve using the appropriate copula.
Copula is the stochastic model and based on probability. In other words, copulas are function for modeling the two or multivariate random variables. Copulas can be easily coupled the marginal distributions to multiple distributions. There are many parametric copula families available, that seven copula functions such as archimedean (Clayton, Frank, Gumbel and Joe), extreme value (Galambos), elliptical (Normal) and others (Plackett) were used. The SPI-1 was determined at each station and then, the whole area was divided into small grids with cell size of 2000×2000. Distances between the grid centers with all the selected stations were calculated with a programming code. Finally, the SPI values in each grid were calculated using IDW method. The severity and percentage of drought area variables were determined and used for regional drought modeling in the study area based on drought threshold equal to zero. After determining the best statistical distribution of two variables, the appropriate copula function was conducted based on different goodness of fit tests. Finally, the Severity-Area-Frequency curve for the study area was developed based on the appropriate copula function and conditional return periods.
The correlation between the two variables of percentage of drought area and severity was assessed using different graphical (Kendall plot and Chi plot) and statistical tests (Spearman rand order correlation and Kendal tau). The results showed a positive correlation between the drought severity and percentage of drought area variables. Based on Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) and graphical test, the Lognormal and Beta probability distributions were select as a best fit distribution of severity and percentage of area under drought, respectively. Finally, the Frank copula among other type of copulas was selected as an appropriate copula for modeling joint drought severity and percentage of area under drought for the study area based on Maximum log likelihood, AIC, BIC and RMSE criteria. The S-A-F curve was developed using conditional return periods based on Frank copula. According to S-A-F curve, it can be seen that increase in the percentage of area under drought in the study area led to increase in drought severity and vice versa. For example, drought severity with return period of 20 years and drought with 20 percent areal extend is obtained equal to 0.37.
Copula functions are of great importance in the analysis of drought, due to preserve correlation between variables and not have any limitation to have a same marginal distribution in long-term prediction of drought events. In this study, using best fit copula (Frank copula) and conditional return periods, the relationships between drought severity and percent of area under drought for the study area named S-A-F curve were developed. These curves can be useful for planning and management of drought in the region. Drought risk assessment based on the results of this study can be high priorities for drought monitoring in large areas.