Bath's law derived from magnitude distribution

The empirical Bath’s law is derived from the statistical
Gutenberg-Richter distribution in magnitude difference of pairs of
earthquakes. The derivation of the statistical Gutenberg-Richter
distributions in energy and magnitude is presented, as resulting from a
geometric-growth model of energy accumulation in the focal region. It is
shown that the most suitable framework of understanding the origin of
the Bath’s law is the extension of the statistical distributions to
pairs of earthquakes, where the difference in magnitude is allowed to
take negative values. If the seismic activity which accompanies a main
shock is viewed as a relaxation process, then we need to include both
the aftershocks and the foreshocks in this accompanying seismic
activity, and to view it as fluctuations in magnitude. The extension of
the magnitude difference to negative values leads to a vanishing mean
value of the fluctuations and to accepting the standard deviation as a
measure of these fluctuations. It is suggested that the standard
deviation of the magnitude difference is the average difference in
magnitude between the main shock and its largest aftershock (foreshock),
thus providing an insight into the nature and the origin of the Bath’s
law. The geometric-growth model of energy accumulation in the focal
region induces a lower bound to the magnitudes of the largest
aftershocks (foreshocks), such that the (average) reference value
$\Delta M=1.2$ between the magnitudes of the main shock
and the largest accompanying seismic event corresponds to the smallest
aftershock (foreshock) in the whole set of the largest aftershocks
(foreshocks).