loading page

Bath's law derived from magnitude distribution
  • Bogdan Felix Apostol
Bogdan Felix Apostol
Institute for Earth'sc Physics, Magurele-Bucharest

Corresponding Author:[email protected]

Author Profile


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).