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We present two models using monitoring data in the production of volcanic eruption forecasts. The first model enhances the well-established failure forecast method introducing an SDE in its formulation. In particular, we developed new method for performing short-term eruption timing probability forecasts, when the eruption onset is well represented by a model of a significant rupture of materials. The method enhances the well-known failure forecast method equation. We allow random excursions from the classical solutions. This provides probabilistic forecasts instead of deterministic predictions, giving the user critical insight into a range of failure or eruption dates. Using the new method, we describe an assessment of failure time on present-day unrest signals at Campi Flegrei caldera (Italy) using either seismic count and ground deformation data. The new formulation enables the estimation on decade-long time windows of data, locally including the effects of variable dynamics. The second model establishes a simple method to update prior vent opening spatial maps. The prior reproduces the two-dimensional distribution of past vent distribution with a Gaussian Field. The likelihood relies on a one-dimensional variable characterizing the chance of material failure locally, based, for instance, on the horizontal ground deformation. In other terms, we introduce a new framework for performing short-term eruption spatial forecasts by assimilating monitoring signals into a prior (“background”) vent opening map. To describe the new approach, first we summarize the uncertainty affecting a vent opening map pdf of Campi Flegrei by defining an appropriate Gaussian random field that replicates it. Then we define a new interpolation method based on multiple points of central symmetry, and we apply it on discrete GPS data. Finally, we describe an application of the Bayes’ theorem that combines the prior vent opening map and the data-based likelihood product-wise. We provide examples based on either seismic count and interpolated ground deformation data collected in the Campi Flegrei volcanic area.

In this study, we use a doubly stochastic model to develop a short-term eruption forecasting method based on precursory signals. The method enhances the Failure Forecast Method (FFM) equation, which represents the potential cascading of signals leading to failure. The reliability of such forecasts is affected by uncertainty in data and volcanic system behavior and, sometimes, a classical approach poorly predicts the time of failure. To address this, we introduce stochastic noise into the original ordinary differential equation, converting it into a stochastic differential equation, and systematically characterize the uncertainty. Embedding noise in the model can enable us to have greater forecasting skill by focusing on averages and moments. In our model, the prediction is thus perturbed inside a range that can be tuned, producing probabilistic forecasts. Furthermore, our 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 verify the new method on simple historical datasets of precursory signals already studied with the classical FFM. The results show the increased forecasting skill of our doubly stochastic formulation. We then present a preliminary application of the method to more recent and complex monitoring signals.

Episodes of slow uplift and subsidence of the ground, called bradyseism, characterize the recent dynamics of the Campi Flegrei caldera (Italy). In the last decades two major bradyseismic crises occurred, in 1969/1972 and in 1982/1984, with a ground uplift of 1.70 m and 1.85 m, respectively. Thousands of earthquakes, with a maximum magnitude of 4.2, caused the partial evacuation of the town of Pozzuoli in October 1983. This was followed by about 20 years of overall subsidence, about 1 m in total, until 2005. After 2005 the Campi Flegrei caldera has been rising again, with a slower rate, and a total maximum vertical displacement in the central area of ca. 70 cm. The two signals of ground deformation and background seismicity have been found to share similar accelerating trends. The failure forecast method can provide a first assessment of failure time on present‐day unrest signals at Campi Flegrei caldera based on the monitoring data collected in [2011, 2020] and under the assumption to extrapolate such a trend into the future. In this study, we apply a probabilistic approach that enhances the well‐established method by incorporating stochastic perturbations in the linearized equations. The stochastic formulation enables the processing of decade‐long time windows of data, including the effects of variable dynamics that characterize the unrest. We provide temporal forecasts with uncertainty quantification, potentially indicative of eruption dates. The basis of the failure forecast method is a fundamental law for failing materials: ẇ-α ẅ = A, where ẇ is the rate of the precursor signal, and α, A are model parameters that we fit on the data. The solution when α >1 is a power law of exponent 1/(1 − α) diverging at time Tf , called failure time. In our case study, Tf is the time when the accelerating signals collected at Campi Flegrei would diverge if we extrapolate their trend. The interpretation of Tf as the onset of a volcanic eruption is speculative. It is important to note that future variations of monitoring data could either slow down the increase so far observed, or suddenly further increase it leading to shorter failure times than those here reported. Data from observations at all locations in the region were also aggregated to reinforce the computations of Tf reducing the impact of observation errors.