__M. Murru__^{*}, R. Console^{*}, A.M. Lombardi^{*}, D. Rhoades^{·}

** ^{*}** Istituto Nazionale di Geofisica e Vulcanologia,
Via di Vigna Murata, 605, I-00143 Rome, Italy. (Console@ingv.it;
Lombardi@ingv.it; murru@ingv.it)

^{·}^{ }Institute of Geological and Nuclear Sciences,
P.O. Box 30-368, Lower Hutt, New Zealand (D.Rhoades@gns.cri.nz)

** **

**Abstract.** We revisit the issue of the so called Bath’s
law concerning the difference *D*_{1 }between the magnitude
of the mainshock, *M*_{0}, and the second largest shock, *M*_{1},
in the same sequence, considered by various authors, in the past, approximately
equal to 1.2. *Feller** *demonstrated
in 1966 that the *D*_{1} 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 assumption that the mainshock comes from a
sample, which is different from the one of its aftershocks.

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 mainshocks
and the aftershocks, and in this way partly explains the large* D*_{1 }values reported in the past. Analysis of the New
Zealand and 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 doesn’t seem to strongly contradict
the Gutenberg-Richter law. Nevertheless, a detailed
analysis of the observed* D*_{1 }distribution shows that the Gutenberg-Richter
hypothesis with a constant *b*-value doesn’t 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.