Probabilistic solution to stochastic overland flow equation using the cumulant expansion method

In this study, overland flow is considered as a stochastic system which is subject to random forcing in the form of rainfall, and the procedure for the development of probabilistic solutions to nonlinear stochastic differential equations is illustrated. The overland flow equation in this study is an ordinary differential equation (ODE), whose state variable is the flow depth at the lower end of the hillslope; this is the result of areal averaging of the sheet flow equation under the kinematic wave approximation.; Unlike a linear differential equation, a nonlinear differential equation brings in higher moments of the state variable when the ensemble average is sought. Consequently one ends with a closure problem. In order to overcome this difficulty in the original space, the stochastic overland flow equation is transformed into a stochastic Liouville equation, which is basically the continuity equation in the phase space. The resultant equation has the form of a linear stochastic partial differential equation. An ensemble average form of the stochastic Liouville equation is then obtained by using cumulant expansion theory. By Van Kampen's lemma, the ensemble average of the state variable represents the probability density function of the state variable of the original stochastic ODE. The ensemble-averaged equation has the form of a Fokker-Planck equation that describes the evolution of the probability density function of the state variable.; For the validation of the model, a random generation of rainfall sequences was performed by using a Poisson-type stochastic rainfall model to provide inflows for the Monte Carlo simulation of the solution to the stochastic overland flow equation. The ensemble of rainfall realizations was also used in the estimation of the parameters which appeared as the coefficients of the Fokker-Planck equation. These coefficients are basically the mean function and the integrated autocovariance function of the rainfall. Comparisons between the new theory and numerical Monte Carlo simulation results were satisfactory in terms of both the probability density function and the mean. Therefore, the ensemble averaging approach using the cumulant expansion method showed good promise for the stochastic modeling of nonlinear hydrologic processes.