A modified form of Bath's law

Bath's law states that the differences in magnitudes between mainshocks and their largest aftershocks are approximately constant, independent of the magnitudes of mainshocks. In our modified form of Bath's law we introduce the notion of the inferred "largest" aftershock from an extrapolation of the Gutenberg-Richter frequency-magnitude statistics of the aftershock sequence of a given mainshock. To illustrate the application of this modified law we consider 10 large earthquakes that occurred in California between 1987 and 2003 with magnitudes equal to or greater than m(ms) greater than or equal to 5.5. The mean difference in magnitudes between these mainshocks and their largest detected aftershocks is 1.16 with a standard deviation sigma(Deltam) = 0.46 (Bath's law). Our estimated mean difference in magnitudes between the mainshocks and the inferred "largest" aftershocks is 1.11 with sigma(Deltam*) = 0.29. The scaling associated with the modified Bath's law implies that the stress transfer responsible for the occurrence of aftershocks is a self-similar process. We also estimate the partitioning of energy during a mainshock-aftershock sequence and find that about 96% of the energy dissipated in a sequence is associated with the mainshock and the rest is due to aftershocks. We suggest that the observed partitioning of energy could play a crucial role in explaining the physical origin of Bath's law.