Abstract

This work aims  to study the uncertainties in the rainfall-runoff process using a stochastic approach derived from the deterministic hydrological model based on the least action principle (ModHyPMA). The stochastic formulation of ModHyPMA allows for consideration of both the dynamics and stochastic nature of the hydrological phenomenon. The main assumption is that  uncertainties in the hydrological process are modelled as Gaussian white noise. It is assumed that hydrological systems are nonlinear dynamical systems that can be described by stochastic differential equations (SDE). From this SDE, we deduce the associated Fokker-Planck equation (FPE). The FPE is a partial differential equation that cannot be solved analytically due to its complexity. We therefore investigated a numerical solution to this equation by using the finite differences and finite volumes methods. The results show that the stochastic model improves the simulations of discharges in  Ouémé at Savè Basin (NSE = 0.89, R2 = 0.90, RMSE = 113 and MAE = 76) compared to the deterministic model (NSE = 0.78, R2 = 0.78, RMSE = 123 and MAE = 51). The plots of the solutions (the density probability of discharges) always coincide when the investigated numerical solutions are compared, except when the number of meshes is very small (100 meshes). The two solutions are convergent. This numerical solution provides information about the distribution of discharges in the Ouémé at Savè Basin.

Key words: Uncertainty, stochastic approach, Fokker-Planck equation (FPE), ModHyPMA, numerical solutions, density probability.