A stochastic differential equation approach to soil moisture

Abstract

The dynamics of water within the unsaturated root zone of the soil are represented by a pair of stochastic differential equations (SDE's), one representing the so-called “surplus” state of the moisture and the other the “deficit” condition. The inputs to the model are the climatically controlled random infiltration events and evapotranspiration which are modeled as a compound Poisson process and a Wiener (Brownian motion) process, respectively. The solutions to these SDE's are not in “close-form” but sample functions are obtained by numerical integration. The moment properties of the soil moisture evolution process have also been derived analytically including the mean, variance, covariance and autocorrelation functions. To illustrate the model, climatic parameters representing the “surplus” and “deficit” cases and properties of clay loam soil have been used to numerically derived the corresponding sample functions. With proper selection of all the parameters, physically realistic sample trajectories can be obtained for the model.