Enhancing the Failure Forecast Method using a noisy mean-reverting process

Abstract

The Failure Forecast Method (FFM) for volcanic eruptions is a classical tool in the interpretation of monitoring data as potential precursors, providing quantitative predictions of the eruption onset. The basis of FFM is a fundamental law for failing materials: dX/dt=AXα, where X is the rate of the precursor signal, and α≥1, A are model parameters. The solution X is a power law of exponent 1/(1-α) diverging at time tf, called failure time. The model represents the potential cascading of precursory signals leading to the final rupture of materials, with tf a good approximation to the eruption onset te. We generalize this approach by incorporating a stochastic noise in the original equation, and extending the uncertainty quantification beyond previous efforts. Embedding noise in the model can enable the FFM equation to have greater forecasting skill by focusing on averages and moments. Sudden changes in the power law properties are indeed possible, and this is particularly critical when the method is applied to calderas like Campi Flegrei (Italy) which are prone to prolonged unrest and ambiguous monitoring signals. In our model, the prediction is thus perturbed inside a range that can be tuned on previously observed variations, producing probabilistic forecasts. In more detail, the change of variables η=X1-α implies dη/dt=(1-α)A, i.e. a straight line which hits zero at tf. The most efficient graphical and computational methods indeed rely on the regression analysis of inverse rate plots. We re-define η with dηt=γ[(1-α)A(t-t0)+ηt0-ηt]dt+σdWt, called Hull-White model in financial mathematics. Parameter σ defines the strength of the noise, and γ the rapidity of the mean-reverting. We test the new method on historical datasets of precursory signals already studied with the classical FFM, including tilt, line-length, and fault movement at Mt. St. Helens 1981-82, seismic signals registered from Bezymyanny 1960, and surface movement of Mt. Toc1960-63.