Estimating the likelihood of an eruption from a volcano with missing onsets in its record

Historical eruption records are often incomplete, a problem which is exacerbated in dealing with catalogs derived from geologic records. We examine the problem of estimating the true (adjusted for missing observations) parameters and hence the hazard in a Weibull (or gamma) renewal model, which is commonly used to model time series of volcanic onsets. Robust regression, robust estimation using repeated medians, and results from the theory of inverses of thinned renewal processes failed to provide consistent estimates from simulated data. Hence we adopted a hidden Markov model framework, where the hidden state is a reflection of the number of missing onsets. This also allows for the completeness level of the record to be estimated, and offers a means of determining where in the record the missing observations are likely to be found. Tested on data from the Holocene record of Mt Taranaki, the preliminary estimates of completeness are 86–87% complete (record 7 ka — present) and 78–80% complete beyond that. These figures were independently verified using a model of tephra dispersal. The estimated present hazard is approximately 20% higher than estimated without allowing for missing data.