SUMMARY
The complete part of the earthquake frequency–magnitude distribution, above the completeness magnitude mc, is well described by the Gutenberg–Richter law. On the other hand, incomplete data does not follow any specific law, since the shape of the frequency–magnitude distribution below max(mc) is function of mc heterogeneities that depend on the seismic network spatiotemporal configuration. This paper attempts to solve this problem by presenting an asymmetric Laplace mixture model, defined as the weighted sum of Laplace (or double exponential) distribution components of constant mc, where the inverse scale parameter of the exponential function is the detection parameter κ below mc, and the Gutenberg–Richter β-value above mc. Using a variant of the Expectation-Maximization algorithm, the mixture model confirms the ontology proposed by Mignan [2012, https://doi.org/10.1029/2012JB009347], which states that the shape of the earthquake frequency–magnitude distribution shifts from angular (in log-linear space) in a homogeneous space–time volume of constant mc to rounded in a heterogeneous volume corresponding to the union of smaller homogeneous volumes. The performance of the proposed mixture model is analysed, with encouraging results obtained in simulations and in eight real earthquake catalogues that represent different seismic network spatial configurations. We find that k = κ/ln(10) ≈ 3 in most earthquake catalogues (compared to b = β/ln(10) ≈ 1), suggesting a common detection capability of different seismic networks. Although simpler algorithms may be preferred on pragmatic grounds to estimate mc and the b-value, other methods so far fail to model the angular distributions observed in homogeneous space-time volumes. Mixture modelling is a promising strategy to model the full earthquake magnitude range, hence potentially increasing seismicity data availability tenfold, since ca. 90 per cent of earthquake catalogue events are below max(mc).
Non-homogeneous space-time fractional Poisson processes
