Analytical derivation of steady state soil water probability distribution function under rainfall forcing using cumulant expansion theory

A new stochastic model for the propagation analysis of fluctuations in rainfall to soil water dynamics is proposed with major focus on its probabilistic structure. This model is derived by using cumulant expansion theory from a stochastic differential equation with stochastic rainfall forcing, and has the advantage of providing the probabilistic solution in the form of a probability distribution function, from which one can obtain the ensemble average behavior of the system. Steady state probability distribution function for soil water is obtained analytically and analyzed for different climater and soil conditions.