Convergence of Rain Process Models to Point Processes

TL;DR

This paper demonstrates that a threshold-based moisture process with switching dynamics converges to a spike train point process, simplifying the analysis of rainfall patterns through mathematical derivations and convergence proofs.

Contribution

It introduces a threshold model with teleporting boundary conditions that approximates rainfall processes as point processes, providing exact formulas for statistical analysis.

Findings

Rainfall process converges to a spike train point process.

Convergence shown via Fokker-Planck derivation and mean-square analysis.

Model simplifies statistical computations for rainfall data.

Abstract

A moisture process with dynamics that switch after hitting a threshold gives rise to a rainfall process. This rainfall process is characterized by its random holding times for dry and wet periods. On average, the holding times for the wet periods are much shorter than the dry. Here convergence is shown for the rain fall process to a point process that is a spike train. The underlying moisture process for the point process is a threshold model with a teleporting boundary condition. This approximation allows simplification of the model with many exact formulas for statistics. The convergence is shown by a Fokker-Planck derivation, convergence in mean-square with respect to continuous functions, of the moisture process, and convergence in mean-square with respect to generalized functions, of the rain process.