Probabilistic interpretation of ⟪Bath's law⟫

Assuming that, in a catalog, all the earthquakes with magnitude larger than or equal to a cutoff magnitude M follow the Gutenberg-Richter Law, the compatibility of this hypothesis with Bath's Law is examined. Considering the mainshock M-0 and the largest aftershock M-1 of a sequence respectively as the first and the second largest order statistic of a sample of independent and identically distributed exponential random variables, the distribution of M-0, M-1 and of their difference D-1 is evaluated. In particular, it is analyzed as the distribution of D-1 changes when only the sequences with the magnitude of the mainshock above a second threshold M-c* greater than or equal to M-c are considered. It results that the distributions of M-0, M-1 and D-1 depend on the difference M-c* - M-c and on the number of events in the sequence. Moreover, the expected value of D-1 increases with increasing of M-c* - M-c for every value of N. Then it is shown that Bath's Law could be ascribed to selection of data caused by the two thresholds M-c and M-c* and that it has a qualitative agreement with the model proposed.