Crossing properties for geophysical systems forced by Poisson noise

Threshold crossings are crucial in many geophysical processes, including soil water balance, vegetation stress, landslides, particle erosion and transport. We focus here on the crossing properties of systems with additive or multiplicative stochastic impulses having the form of a marked Poisson process. We derive exact expressions for the probability of excursions of very short duration above/below a generic threshold. These expressions, together with mean first passage times, allow for approximating the full probability density function of excursion times with two‐parameter distributions. Such approximations are effective for a wide range of stochastic processes covering relevant geophysical applications, as confirmed by the two examples provided regarding the dynamics of soil moisture content and water table depth.