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We introduce a doubly stochastic method for performing material failuretheory based forecasts of volcanic eruptions. The method enhances the wellknown Failure Forecast Method equation, introducing a new formulation similarto the Hull-White model in financial mathematics. In particular, we incorporatea stochastic noise term in the original equation, and systematicallycharacterize the uncertainty. The model is a stochastic differential equationwith mean reverting paths, where the traditional ordinary differential equationdefines the mean solution. Our implementation allows the model to makeexcursions from the classical solutions, by including uncertainty in theestimation. The doubly stochastic formulation is particularly powerful, in thatit provides a complete posterior probability distribution, allowing users todetermine a worst case scenario with a specified level of confidence. We applythe new method on historical datasets of precursory signals, across a widerange of possible values of convexity in the solutions and amounts ofscattering in the observations. The results show the increased forecastingskill of the doubly stochastic formulation of the equations if compared tostatistical regression.

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