A simplified stochastic model for infiltration into a heterogeneous soil forced by random precipitation

A simplified stochastic infiltration model is presented, aimed at representing the rainfall-runoff transformation in the presence of heterogeneity in the soil and with precipitation as a random variable with complex temporal evolution. Such a model is based on a simple water mass balance of a surface soil layer, considered as a non-linear reservoir. The explicit inclusion of spatial heterogeneity allows the model to be used in sub-grid parametrizations at a variety of scales, from the distributed modelling of the hydrological response of small watersheds to the representation of surface mass fluxes in General Circulation Models. An approximate solution procedure is developed, which allows the estimation of statistical moments of the soil effective saturation and runoff inside discrete time steps where the hydraulic saturated conductivity and the rainfall intensity are taken as random variables with known probability density functions. As a first test of the proposed model, two different simulations, relative to two soils with different hydraulic conductivity distributions, are presented and discussed. A year long record of hourly averaged rainfall intensities, as measured by a tipping bucket gauge in Central Italy, is taken as the main input. The main finding is that the non-linear nature of the soil filter is such that, for random precipitation intensity, the coefficient of variation of the runoff is always higher than that of precipitation. Such a non-linear variability enhancement, due mainly to the threshold character of the soil mass balance equation, tends to be slightly dampened by the variability of the hydraulic saturated conductivity.