Probabilistic enhancement of the Failure Forecast Method using a stochastic differential equation and application to volcanic eruption forecasts

TL;DR

This paper presents a stochastic extension of the Failure Forecast Method for volcanic eruptions, incorporating uncertainty and providing probabilistic forecasts that improve prediction accuracy over traditional methods.

Contribution

The authors develop a doubly stochastic differential equation model that enhances the Failure Forecast Method by including uncertainty and mean reversion, offering probabilistic eruption forecasts.

Findings

The new model improves forecasting skill compared to statistical regression.

It provides a complete posterior probability distribution for eruption timing.

Application to historical data demonstrates increased prediction accuracy.

Abstract

We introduce a doubly stochastic method for performing material failure theory based forecasts of volcanic eruptions. The method enhances the well known Failure Forecast Method equation, introducing a new formulation similar to the Hull-White model in financial mathematics. In particular, we incorporate a stochastic noise term in the original equation, and systematically characterize the uncertainty. The model is a stochastic differential equation with mean reverting paths, where the traditional ordinary differential equation defines the mean solution. Our implementation allows the model to make excursions from the classical solutions, by including uncertainty in the estimation. The doubly stochastic formulation is particularly powerful, in that it provides a complete posterior probability distribution, allowing users to determine a worst case scenario with a specified level of confidence. We apply the new method on historical datasets of precursory signals, across a wide range of possible values of convexity in the solutions and amounts of scattering in the observations. The results show the increased forecasting skill of the doubly stochastic formulation of the equations if compared to statistical regression.