Estimating the mass eruption rate of volcanic eruptions from the plume height using Bayesian regression with historical data: the MERPH model

The mass eruption rate (MER) of an explosive volcanic eruption is a commonly used quantifier of the magnitude of the eruption, and estimating it is importance in managing volcanic hazards. The physical connection between the MER and the rise height of the eruption column results in a scaling relationship between these quantities, allowing one to be inferred from the other. Eruption source parameter datasets have been used to calibrate the relationship, but the uncertainties in the measurements used in the calibration are typically not accounted for in applications. This can lead to substantial over- or under-estimation. Here we apply a simple Bayesian approach to incorporate uncertainty into the calibration of the scaling relationship using Bayesian linear regression to determine probability density functions for model parameters. This allows probabilistic prediction of mass eruption rate given a plume height observation in a way that is consistent with the data used for calibration. By using non-informative priors, the posterior predictive distribution can be determined analytically. The methods and dataset are collected in a python package, called merph, and we illustrate their use in sampling plausible MER--plume height pairs, and in identifying usual eruptions. We discuss applications to ensemble-based hazard assessments and potential developments of the approach.