Båth's law and the self‐similarity of earthquakes

説明

<jats:p>We revisit the issue of the so‐called Båth's law concerning the difference <jats:italic>D</jats:italic><jats:sub>1</jats:sub> 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 <jats:italic>D</jats:italic><jats:sub>1</jats:sub> on the magnitude thresholds chosen for the main shocks and the aftershocks and in this way partly explains the large <jats:italic>D</jats:italic><jats:sub>1</jats:sub> 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 <jats:italic>D</jats:italic><jats:sub>1</jats:sub> values predicted by the theoretical model and those observed. Limiting our attention to the average <jats:italic>D</jats:italic><jats:sub>1</jats:sub> values, Båth's law does not seem to strongly contradict the Gutenberg–Richter law. Nevertheless, a detailed analysis of the <jats:italic>D</jats:italic><jats:sub>1</jats:sub> distribution shows that the Gutenberg–Richter hypothesis with a constant <jats:italic>b</jats:italic>‐value does not fully explain the experimental observations. The theoretical distribution has a larger proportion of low <jats:italic>D</jats:italic><jats:sub>1</jats:sub> values and a smaller proportion of high <jats:italic>D</jats:italic><jats:sub>1</jats:sub> values than the experimental observations. Thus, Båth'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.</jats:p>

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