Bath's law and the self-similarity of earthquakes

[1] We revisit the issue of the so-called Bath's law concerning the difference D-1 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 D-1 on the magnitude thresholds chosen for the main shocks and the aftershocks and in this way partly explains the large D-1 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 D-1 values predicted by the theoretical model and those observed. Limiting our attention to the average D-1 values, Bath's law does not seem to strongly contradict the Gutenberg-Richter law. Nevertheless, a detailed analysis of the D-1 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 D-1 values and a smaller proportion of high D-1 values than the experimental observations. Thus, Bath'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.