Correlation between Rainfall and Runoff in Fuchun River Basin Based on Copula Functions and Kernel Density Estimation - YANG Sheng-mei, ZHU De-kang, CHENG Xiang, LI Bo, ZHU Yan-ze, MA Wen-sheng (长江科学院院报)
[Objective] Rainfall and runoff are two important hydrological variables in river basins, exhibiting the characteristic of random distribution. In-depth analysis of the relationship between rainfall and runoff holds significant importance for watershed flood risk management, water resource scheduling, and hydraulic engineering planning and design. [Methods] This study utilized the advantages of Copula functions in describing dependence relationships among random variables. First, a non-parametric kernel density estimation method was introduced, and four types of kernel density functions were used to characterize the marginal distributions of rainfall and runoff variables in the Fuchun River Basin. Subsequently, a bivariate Copula function was employed to establish a joint distribution model. Simulation performance for both marginal and joint distributions was validated using root mean square error (RMSE) and Euclidean distance. [Results] (1) By comparing the RMSEs between the estimated results using four kernel density functions (Gaussian, Uniform, Triangle, Epanechnikov) and the empirical frequencies of rainfall and runoff in the river basin, the Gaussian was found to have the smallest errors. The Gaussian was selected to estimate the marginal distributions of hydrological variables in the Fuchun River Basin, demonstrating higher simulation accuracy without relying on any distribution assumption.(2) By estimating the Kendall and Spearman rank correlation coefficients of the bivariate functions of Gaussian-Copula, t-Copula, Clayton-Copula, Frank-Copula, and Gumbel-Copula, and comparing them with the Kendall and Spearman rank correlation coefficients of the original observed data, it was found that Gaussian-Copula and Gumbel-Copula were closer to the observed data.(3) By calculating the Euclidean distance, the fitting performance of the Copula functions was evaluated. The Gumbel-Copula function was further selected as the optimal Copula function to describe the dependence structure between rainfall and runoff in the river basin. It revealed that the rainfall and runoff variables in the upper tail of the joint distribution were highly sensitive to changes, indicating strong correlation between annual rainfall and runoff extreme values in the river basin.(4) Further calculation of the upper tail correlation coefficient yielded a value of 0.758 3, indicating a 75.83% probability of both the annual rainfall and annual runoff reaching extreme values simultaneously. When an extreme value of rainfall occurs in a specific year in the river basin, runoff could be estimated based on the dependence relationship between rainfall and runoff in joint distribution established in this study. This provided a reference for flood risk management. [Conclusion] The Gaussian kernel function demonstrates excellent simulation performance for the marginal distributions of rainfall and runoff variables in the Fuchun River Basin, and the Gumbel-Copula function shows high goodness-of-fit for the joint distribution of rainfall and runoff. The findings of this study offer substantial implications for flood risk management and water resource scheduling in river basins, and provide a theoretical foundation for further research on rainfall-runoff stochastic simulation using Copula functions in the Fuchun River Basin. Additionally, they offer practical value for the calculation and analysis of hydrological variables and for the planning and design of hydraulic engineering in river basins.