Quantification of the variability of the soil moisture field in response to variations in the rainfall field

Characterizing the variability of soil moisture forced by the natural variability of rainfall is crucial for hydrology, as the vadose zone represents the hydrological link between the surface water component and the groundwater component in the hydrological cycle. A diffusion-injection model proposed by Entekhabi and Rodriguez-Iturbe (1994), a linear differential equation, provides a simple way to analyze the spatiotemporal variation of soil moisture in response to the spatiotemporal variation of rainfall in a stochastic framework. The existing theories developed to quantify the spatiotemporal variability of the soil moisture field using the diffusion-injection model are based on the assumption of second-order stationarity of soil moisture perturbations. In reality, however, the random perturbation processes of soil moisture are often nonstationary. The goal of this study is therefore to generalize the existing stochastic theories to the case where the soil moisture perturbations are nonstationary in space and time. To achieve this goal, the Fourier-Stieltjes representation approach and the representation theorem are applied. The variance and the semivariogram are obtained in the Fourier wavenumber and frequency domain to quantify the variability of soil moisture and rainfall fields. The influence of the parameters in the diffusion-injection model and the rainfall model on the variability of soil moisture and rainfall fields is evaluated.