Accuracy and precision of frequency-size distribution scaling parameters as a function of dynamic range of observations: example of the Gutenberg-Richter law b-value for earthquakes

Many natural hazards exhibit inverse power-law scaling of frequency and event size, or an exponential scaling of event magnitude (m) on a logarithmic scale, for example the Gutenberg-Richter law for earthquakes, with probability density function p(m) similar to 10(-bm). We derive an analytic expression for the bias that arises in the maximum likelihood estimate of b as a function of the dynamic range r. The theory predicts the observed evolution of the modal value of 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. The situation is more complicated for a single sample in real catalogues due to their heterogeneity, magnitude uncertainty and the true b-value being unknown. The results explain why the likelihood of large events and the associated hazard is often underestimated in small catalogues with low dynamic range, for example in some studies of volcanic and induced seismicity.