Are volcanic seismic b-values high, and if so when?

The Gutenberg-Richter exponent b is a measure of the relative proportion of large and small earthquakes. It is commonly used to infer material properties such as heterogeneity, or mechanical properties such as the state of stress from earthquake populations. It is 'well known' that the b-value tends to be high or very high for volcanic earthquake populations relative to b = 1 for those of tectonic earthquakes, and that b varies significantly with time during periods of unrest. We first review the supporting evidence from 34 case studies, and identify weaknesses in this argument due predominantly to small sample size, the narrow bandwidth of magnitude scales available, variability in the methods used to assess the minimum or cutoff magnitude Mc, and to infer b. Informed by this, we use synthetic realisations to quantify the effect of choice of the cutoff magnitude on maximum likelihood estimates of b, and suggest a new work flow for this choice. We present the first quantitative estimate of the error in b introduced by uncertainties in estimating Mc, as a function of the number of events and the b-value itself. This error can significantly exceed the commonly-quoted statistical error in the estimated b-value, especially for the case that the underlying b-value is high. We apply the new methods to data sets from recent periods of unrest in El Hierro and Mount Etna. For El Hierro we confirm significantly high b-values of 1.5-2.5 prior to the 10 October 2011 eruption. For Mount Etna the b-values are indistinguishable from b = 1 within error, except during the flank eruptions at Mount Etna in 2001-2003, when 1.5 < b < 2.0. For the time period analysed, they are rarely lower than b = 1. Our results confirm that these volcano-tectonic earthquake populations can have systematically high b-values, especially when associated with eruptions. At other times they can be indistinguishable from those of tectonic earthquakes within the total error. The results have significant implications for operational forecasting informed by b-value variability, in particular in assessing the significance of b-value variations identified by sample sizes with fewer than 200 events above the completeness threshold.