A Copula-based Method for River Ecological Flow Quantification and Algal Bloom Risk Assessment in the Hanjiang River

[Objective] Riverine ecological flow quantification is essential for maintaining riverine ecological health. Traditional methods often overlook the joint risks of hydrological and ecological factors and tend to rely on single-scenario calculations. This study aims to establish a joint risk model based on Copula functions and Bayesian theory to quantify the interdependence between hydrological (flow) and ecological (algal density) factors in the middle and lower reaches of Hanjiang River, thereby achieving scientific quantification of ecological flow under multiple scenarios and providing a basis for early warning and regulation of algal bloom risks in rivers. [Methods] Using flow data from Xiantao station and algal density data from Yuekou cross-section during the 2018 algal bloom outbreak in Hanjiang River,this study first determined the marginal distributions of flow and algal density using the Kolmogorov-Smirnov (K-S) test and Akaike Information Criterion (AIC). The optimal Copula functions were then selected using maximum likelihood estimation and goodness-of-fit test. Finally, a joint distribution model of flow and algal density was established, and the Bayesian conditional probability formula was applied to analyze the probability of algal density exceeding the threshold of algal bloom occurrence under different flow scenarios. The results from Copula functions were compared with those from the Tennant method and Hydrological Inflection-Point (HIP) analysis for validation. [Results] (1) The joint distribution model based on Gaussian Copula function passed the K-S test and effectively captured the significant negative dependence structure between flow and algal density.(2) During the 2018 algal bloom outbreak in Hanjiang River, when the flow exceeded 728.00 m3/s and 1 096.30 m3/s, there was over an 80% probability that algal density would exceed 2 561.10×104 cells/L and 902.93×104 cells/L, respectively. (3) The hydrological inflection point method calculated the initial flow of algal bloom with an error within 100 m3/s, demonstrating higher accuracy than the Tennant method. However, both methods underestimated the overall risk due to their neglect of variable dependency. Through bivariate analysis, the Copula model revealed risk details that traditional methods failed to capture. [Conclusions] (1) The joint risk model based on Copula functions can quantitatively capture the complex dependency between hydrological and ecological factors, overcoming limitations of traditional methods that depend on long-term data and neglect multi-factor interactions. It provides an efficient tool for analyzing ecological flow during a single algal bloom outbreak.(2) The multi-scenario conditional probability analysis demonstrates that risk probabilities of algal density differed significantly across different flow ranges, offering refined flow thresholds for reservoir operations and bloom prevention.(3) In complex hydro-ecological systems, it is necessary to integrate multi-factor models like Copula to avoid risk underestimation. This study provides a new method for multi-factor joint risk evaluation and ecological flow quantification in rivers. Future research can further incorporate multiple variables such as water quality and temperature to establish more complex Copula models, improving prediction accuracy and scenario simulation capabilities.