Probabilistic interpretationof «Bath's Law»

Abstract

Assuming that, in a catalog, all the earthquakes with magnitude larger than or equal to a cutoff magnitude M c follow the Gutenberg-Richter Law, the compatibility of this hypothesis with «Bath’s Law» is examined. Consi-dering 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*•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.

Key words Assuming that, in a catalog, all the earthquakes with magnitude larger than or equal to a cutoff magnitude M c follow the Gutenberg-Richter Law, the compatibility of this hypothesis with «Bath’s Law» is examined. Consi-dering 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*•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 propose.