Probabilistic Solution to Stochastic Overland Flow Equation

Abstract

In this paper, the second in a series of two, the theory developed in the companion paper is applied to the case of the stochastic overland flow equation, and a numerical solution method is presented for the resulting Fokker-Planck equation (FPE), which describes the evolution of the probability-density function (PDF) of overland flow depth at the downstream section of a hillslope. The derived FPE is evaluated for two different approximations to the diffusion coefficient of the FPE. The Monte Carlo analysis of stochastic overland flow equation is then performed using the random rainfall sequences, generated by a compound filtered Poisson process model for the stochastic rainfall, in order to provide a benchmark for the results obtained from the FPEs. When compared to the Monte Carlo simulation based PDFs and their ensemble average, the second approximation to the diffusion coefficient gives a good fit in terms of the shape of the PDF and the ensemble average of the overland flow depth. Therefore, the theory proposed here is quite promising for obtaining the ensemble averages of nonlinear hydrological processes.