Bivariate Copula Modeling of Senegal River Floods: Mathematical Foundations and a Simulation-Based Empirical Study

Abstract This article develops a mathematical framework for the bivariate analysis of flood frequency in the Senegal River recorded at the Bakel station, using peak flow and flood volume as event characteristics. It clarifies the distinction between annual maxima and peaks over threshold (POT), states the assumptions required for marginal standardization and copula identification, derives the Gumbel--Hougaard conditional laws and tail-dependence coefficient, and proves the formulas and ordering properties of AND and OR return periods. A marked-Poisson formulation converts event probabilities into return periods in years. The empirical workflow is illustrated with a reproducible synthetic sample of 100 hydrologically plausible flood events distributed over 1990--2025. Peak excesses are fitted by a Generalized Pareto Distribution (GPD) and volumes by a shifted Gamma (Pearson III) distribution. The simulated sample yields Kendall's $\widehat\tau=0.604$, a Gumbel--Hougaard parameter $\widehat\theta=2.523$, and upper-tail dependence $\widehat\lambda_U=0.684$. Among four one-parameter copulas, Gumbel--Hougaard has the lowest AIC and then consequently outperforms.