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We develop a new statistical method to obtain the probability distributions of magnitude of the largest aftershock and the number of major events, using the negative binomial model by Okada and the modified Omori formula. In Gutenberg-Richter's law n(M)dM=a10-bM dM for M<Mm, we suppose that the a-value follows a gamma distribution, F(a, Qo), which is justified by the fact that a negative binomial law holds for the number of aftershocks. Here, Mm is the magnitude of main shock. The b-value is considered to be the same for all aftershock sequences. If we observe k aftershocks larger than a threshold, Mm-d, after the main shock, the a-value for the entire aftershock period distributes as ƒ¡{ƒ¿+ƒÈ,ƒÐ0/(1 +qƒÐ0ƒÅ(d))}. Here, q is portion of aftershocks occurring in the observation period, which is calculated with the modified Omori formula, and ƒÅ(d)=(10bd -1)/(b•~ln 10). On the basis of this formula, it is easy to derive analytically the distributions of the magnitude difference between main shock and the largest aftershock and the number of major aftershocks. The parameters have been determined from aftershock data of shallow earthquakes with 6.0 in magnitude or larger in inland and coastal area of Japan from 1926 through 1995. Applying the method at the hypothetical times of 0, 24 and 72 hours after the main shock, we see that the present method is more useful just after main shock occurrence and in the early stage of aftershock activity than the conventional probabilistic method.

Probabilistic Prediction of Major Aftershocks by Negative Binomial Model (1)