Bath's Law as a Consequence of Magnitude Distribution
Abstract
We revisit the issue of the so-called Bath's law concerning the difference D1 between the magnitude of the mainshock, M0, and the second largest shock, M1, in the same sequence. Various authors, in the past, observed that this difference is approximately equal to 1.2. Feller demonstrated in 1966 that the D1 expected value was about 0.5 given that the difference between the two largest random variables of a sample, N, exponentially distributed is also a random variable with the same distribution. Feller's proof leads to the consequence that the mainshock comes from a sample, which is different from the one of its aftershocks. A mathematical formulation of the problem is developed here, the only assumption being that all the events belong to the same self-similar set of earthquakes following the Gutenberg-Richter magnitude distribution with a constant b-value. Assuming that the number of aftershocks in each aftershock series is known, and not extremely large, this model shows a substantial dependence of D1 on the magnitude thresholds chosen for the mainshock and its largest aftershock. In this way it explains the large D1 values reported in the past. Analysis of the PDE catalog of shallow earthquakes demonstrates a good agreement between the average D1 values predicted by the theoretical model and those observed. Limiting our attention to the average D1 values, Bath's law doesn't seem to strongly contradict the Gutenberg-Richter law. Nevertheless, a detailed analysis of the observed D1 distribution shows that the Gutenberg-Richter hypothesis doesn't 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. A reasonable explanation for this mismatch, which appears a minor effect with respect to what was supposed in the past, seems to consist in the byes (not assumed in the model) that the selection of clustered events produces on the average b-value.
- Publication:
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AGU Fall Meeting Abstracts
- Pub Date:
- December 2002
- Bibcode:
- 2002AGUFMNG62B0950C
- Keywords:
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- 3220 Nonlinear dynamics;
- 3250 Fractals and multifractals;
- 7209 Earthquake dynamics and mechanics;
- 7223 Seismic hazard assessment and prediction;
- 7260 Theory and modeling