The limits of predictability of volcanic eruptions from accelerating rates of earthquakes

Volcanic eruptions are commonly preceded by increased rates of earthquakes. Previous studies argue that in some instances these sequences follow the inverse Omori law (IOL) and that this model could be the basis for forecasting the timing of eruption onset. However, the catalogue of pre-eruptive sequences is small, and the performance of the IOL as a forecasting tool remains largely untested. Here, we use simulations to quantify upper limits to the accuracy and bias of forecast eruption times based on the IOL in the 'best-case' scenario that uncertainty only arises from model parameter estimation from single realizations of a stochastic point process. We compare different methods for forecasting based on the IOL, and demonstrate that a maximum-likelihood method yields more accurate and less-biased forecasts than methods currently employed. Even in these idealized conditions, we find that large forecast uncertainty and false alarms are inherent features of the mathematics of the IOL. For example model parameter values and 500-d pre-eruptive sequence durations, at 25 d before the eruption, 10 per cent of the forecasts are more than 8 d early or late if the power-law exponent is known a priori, and more than 18 d early or late if the power-law exponent is unknown. We also evaluate methods for model comparison and estimation of the power-law exponent. These techniques are applied to examples of real pre-eruptive earthquake data sets. We find evidence for systematic deviations from the idealized model, indicating the action of multiple processes and resulting in greater forecast error than in the synthetic examples, especially close to the eruption time.