Application of Copula Functions for Bivariate Analysis of Rainfall and River Flow Deficiencies in the Siminehrood River Basin, Iran

In this study, the frequency analysis of river flow deficiency (FD) of the Siminehrood River in the south of Lake Urmia located in northwest Iran was investigated with regard to rainfall deficiency (RD) during the period of 1992–2013 using copula functions. The main purpose of this study is to provide a comprehensive method for bivariate simulation and forecasting based on marginal distribution and joint behavior of the studied series. For this purpose, the FD and RD values were extracted using the deficiency value method. By preparing deficiency values, 57 different distribution functions were fitted to the studied values, and the generalized extreme value (GEV) distribution was selected as the best marginal distribution function based on the evaluation criteria. Before fitting the copula function, the correlation between the RD and FD values was examined using Kendall’s tau, and a correlation of 70% was obtained. After selecting the marginal distribution function and examining the correlation, the goodness of fit of seven different copula functions was examined for frequency analysis of RD and FD values in the Siminehrood River at the Dashband station. The results indicated that the Clayton copula had the best performance for creating a joint distribution of RD and FD values. It was also determined from the joint analysis of deficiency values that the FD values can be estimated with high accuracy for RD values of more than 0.68 mm. Also, the results indicated that if rainfall in the study area were less than long-term mean for 10-day and 60-day durations, with different return periods and probabilities, different conditions will occur for FD values, which can be used as typical curves for water resources management and allocation in the basin. Finally, the accuracy of the copula-based model and its conditional density in the two phases of simulation and forecasting were investigated. The accuracy of the copula-based model and its conditional density in the simulation phase was confirmed [$R^2=0.87$, root mean square error $({\mathrm{RMSE})=0.1\mathrm{m}}^3/\mathrm{s}$, and nash-sutcliffe efficiency $(\mathrm{NSE})=0.86$]. In the forecasting phase, the forecasting equation based on the proposed method had a RMSE of ${0.14\mathrm{m}}^3/\mathrm{s}$ and NSE of 0.89. By using the violin plot, the model certainty was also confirmed. According to the proposed equation, FD values can be forecasted affected by RD values for 10-day duration with high certainty and accuracy.

In this paper, a method for forecasting and simulation of meteorological and hydrological parameters is presented that considers two parameters simultaneously. This study discusses meteorology and hydrology from a different perspective. Given the current climate change, this study uses deficiency values. The proposed forecasting method provides regional results that can be used in the water resources management in each basin specifically. By implementing this method, it is easy to forecast the desired values in the basin with different probabilities and return periods. In this study, river flow deficiency (FD) affected by rainfall deficiency (RD) can be forecasted. The difference between the proposed method and other methods and models of simulation and forecasting is in the connection of two variables with each other, which makes the results more certain and reliable. This method can be used in the field of basin management and water resources allocation and also water resources monitoring.