Modeling Subgrid variability of Snow Depth Using the Fokker-Planck Equation Approach Water Resources Research

A physically based subgrid variability model for snow process using the Fokker‐Planck equation (FPE) approach was proposed. This FPE can express the evolution of the probability density function (PDF) of snow depth within a finite area, possibly a grid cell of distributed models or a small basin, whose shape can be irregular. The main advantage of this approach is that it does not rely on a given PDF but dynamically computes the PDF through an advection‐diffusion‐type equation, the FPE, which was derived from point‐scale process‐based governing equations. Snow depth was treated as a random variable, while the snow redistribution and snowmelt rate were treated as the sources of stochasticity. The main challenge in solving this FPE is evaluating the time‐space covariances appearing in the diffusion coefficient. In this study, approximations to evaluate the covariance terms, accounting for snowmelt and snow redistribution, were proposed. The simulated results of the FPE model were validated by the measured time series of snow depth at one site and the spatial distributions of snow depth measured by ground penetrating radar and airborne light detection and ranging (Lidar). It was shown that the point‐observed snow depth fell within the simulated range during most of the 2‐year study period. The simulated PDFs of snow depth within the study area were similar to the observed PDFs of snow depth by ground penetrating radar and Lidar. In summary, these results demonstrate the efficacy of the proposed FPE model representing the subgrid variability of snow depth.