Nonhomogeneous Poisson model for volcanic eruptions

Abstract

A simple Poisson process is more specifically known as a homogeneous Poisson process since the rateλ was assumed independent of time t. The homogeneous Poisson model generally gives a good fit to many volcanoes for forecasting volcanic eruptions. If eruptions occur according to a homogeneous Poisson process, the repose times between consecutive eruptions are independent exponential variables with meanθ=1/λ. The exponential distribution is applicable when the eruptions occur “at random” and are not due to aging, etc. It is interesting to note that a general population of volcanoes can be related to a nonhomogeneous Poisson process with intensity factorλ(t). In this paper, specifically, we consider a more general Weibull distribution, WEI (θ, β), for volcanism. A Weibull process is appropriate for three types of volcanoes: increasing-eruption-rate (β>1), decreasing-eruption-rate (β<1), and constant-eruption-rate (β=1). Statistical methods (parameter estimation, hypothesis testing, and prediction intervals) are provided to analyze the following five volcanoes: Also, Etna, Kilauea, St. Helens, and Yake-Dake. We conclude that the generalized model can be considered a goodness-of-fit test for a simple exponential model (a homogeneous Poisson model), and is preferable for practical use for some nonhomogeneous Poisson volcanoes with monotonic eruptive rates.