Abstract
The negative binomial distribution is found to show a good fit to the distribution of total number of aftershocks whose magnitudes differ by no more than d from the magnitude of the main shock M0. On the basis of this observation, the probability is derived that the difference Di in magnitude between the main shock and its i-th largest aftershock becomes larger than d.
Pr{Di≥d}=∑i-1k=0(α+k-1k)(αbln10)α{a(10db-1)}k/{αbln10+a(10bd-1)}α+k
where a and α denote the mean of a and the power parameter of the negative binomial distribution, respectively, and a and b indicate thee parameters of the Gutenberg-Richter's magnitude-frequency relation. It is assumed that the Gutenberrg-Richter's relation n(M)dM=a10-b(M-M0)dM holds for aftershocks and that b is constant for every aftershock sequence. we studied the aftershock sequences in the area of Japan and Greece, which have been compiled by Utsu and Papazachos etc., respectively. The distribution of the difference D1 in magnitude between the main shock and its largest aftershock deduced from “negative binomial model” is agreeable with the observation. The value of b deduced from D1 distribution is consistent with the one of aftershock sequences given by Utsu and Papazachos. The observed distributions of D2 and D3 show slight discrepancy from those expected from the model.
