Frequency-size parameters as a function of dynamic range – the Gutenberg-Richter b-value for earthquakes

The Gutenberg-Richter b-value represents the relative proportion of small to large earthquakes in a scale-free population and is an important parameter used in earthquake hazard assessment. Discussion of the amount of data required to obtain a robust b-value has been extensive and is ongoing. To complement these analyses, we show the effect of the b-value with changes to the dynamic range – the difference between minimum magnitude (or magnitude of completeness) and maximum magnitude, which is inherently linked to the sample size, but not proportionately correlated. Additionally, we show that biases in high b-values are due to the bias in the mean magnitude of a catalogue, which asymptotically converges from below.

We derive and analytic expression for the bias that arises in the maximum likelihood estimate of b as a function of dynamic range r. Our theory predicts the observed evolution of the modal value of the mean magnitude in multiple random samples of synthetic catalogues at different r, including the bias to high b at low r and the observed trend to an asymptotic limit with no bias. In the case of a single sample in real catalogues, the situation is substantially more complicated due to the heterogeneity, magnitude uncertainty and lack of knowledge of the true b-value. We summarise that these results explain why the likelihood of large events and the associated hazard is often underestimated in small catalogues with low r, for example in some studies of volcanic and induced seismicity.