Analytical Derivation of Steady-State Soil Water Probability Density Function Coupled with Simple Stochastic Point Rainfall Model

In this study, a new stochastic model for the propagation analysis of fluctuations in rainfall to soil water dynamics is proposed. Based on a lumped conceptualization of soil water dynamics with rainfall forcings, which are incorporated by a simple stochastic point rainfall model, a model is derived by using cumulant expansion theory from a stochastic differential equation. The advantage of the model is to provide the probabilistic solution in the form of a probability density function (PDF), from which one can find the ensemble average behavior of the system. Steady-state PDF of soil water is analytically obtained and analyzed for different climate, soil, and vegetation conditions. The city of Daegue in Korea, which represents the driest parts of the Korean, Peninsula, is applied for a case study and the result shows that the analytically derived steady-state soil water PDF can make a good agreement to the numerically obtained steady-state PDF from a lumped conceptualization model of soil water dynamics. From this agreement, the steady-state analysis is thought to be appropriate for the study of soil water dynamics where the seasonality of rainfall is not very significant. It is also shown that the fluctuations in rainfall tend to increase the variance of soil water dynamics, while the change of rainfall amount can shift the mode of PDF. General features for the PDFs as a function of different loss and soil characteristics are the decrease of soil water with loss rate and soil water storage capacity. The major conclusion, however, is that the proposed simplified stochastic soil water dynamic model for dry years in the Korean Peninsula can provide quite a reasonable explanation in the main soil water probabilistic properties when the rainfall variability is the only consideration.