Båth's law and the self-similarity of earthquakes

Abstract

We revisit the issue of the so-called Bath’s law concerning the difference D1 between the magnitude of the main shock and the second largest shock in the same sequence. A mathematical formulation of the problem is developed with the only assumption being that all the events belong to the same self-similar set of earthquakes following the Gutenberg– Richter magnitude distribution. This model shows a substantial dependence of D1 on the magnitude thresholds chosen for the main shocks and the aftershocks and in this way partly explains the large D1 values reported in the past. Analysis of the New Zealand and Preliminary Determination of Epicenters (PDE) catalogs of shallow earthquakes demonstrates a rough agreement between the average D1 values predicted by the theoretical model and those observed. Limiting our attention to the average D1 values, Ba ̊th’s law does not seem to strongly contradict the Gutenberg–Richter law. Nevertheless, a detailed analysis of the D1 distribution shows that the Gutenberg–Richter hypothesis with a constant b-value does not fully explain the experimental observations. The theoretical distribution has a larger proportion of low D1 values and a smaller proportion of high D1 values than the experimental observations. Thus, Ba ̊th’s law and the Gutenberg–Richter law cannot be completely reconciled, although based on this analysis the mismatch is not as great as has sometimes been supposed.