Exact solution of a stochastic model of rain-induced shallow landslides in hollows

D’Odorico & Fagherazzi (2003) have proposed a probabilistic model of rainfall-triggered shallow landslides in hollows. Their model describes the long-term evolution of colluvial deposits through a probabilistic soil mass balance at a point. The model accounts for hollow infilling, expressed as a deterministic function of the deposit thickness, and soil erosion by shallow landslides, modeled as a time-dependent stochastic point process related to the occurrence of triggering precipitation events. Further building blocks of the model are: an infinite-slope stability analysis; a steady-state kinematic wave model of hollow groundwater hydrology; and a statistical model relating intensity, duration, and frequency of extreme precipitation. The authors provide an analytical solution to their model under the simplifying assumption that the occurrence rate of landslide-triggering rain events is independent of the colluvial deposit thickness. We present an exact solution to the stochastic landslide model for the general case where the triggering rain event occurrence rate depends on the soil thickness and hence on time.