Probabilistic Enhancement of the Failure Forecast Method Using a Stochastic Differential Equation and Application to Volcanic Eruption Forecasts

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.