Prior Distribution of b-value in Gutenberg-Richter Law for Aftershock Sequences

Abstract

We determine the prior distributions of b-value in Gutenberg-Richter law from 81 aftershock sequences accompanying with shallow earthquakes of M7.5 or larger in the world and 67 sequences related to large events in and around Japan. We suppose that b-value in the prior population follows gamma distribution, Γ (φ, ζ), where φ and ζ are shape and scale parameters, respectively. The observed b-value for each sequence corresponds to a random sample from the population but has statistical estimation error. In this paper, we estimate φ and the mean of population distribution, bavg=φζ, using the maximum likelihood method. Estimated φ and bavg are 55 and 1.13 for sequences of aftershocks of M5.0 or larger in the world, and 28 and 1.22 for ones of aftershocks of D<2.55, respectively, where D is the magnitude difference between the main shock and aftershock. For sequences in Japan, φ=28 and bavg=0.97 are obtained. Numerical tests of Monte Carlo method show that estimated bavg is fairly close to the input value. However φ is apt to become larger than the input. Then smaller values, such as φ=26 to 31 for sequences in the world and φ=20 to 25 in Japan, might be better than the estimates by the maximum likelihood method. On the assumption that the prior distribution of b-value is Γ (φ, bavg/φ), the maximum likelihood estimate, b, for a set of N earthquakes is giveR with b= (N+φ-1) / (N/bU+φ/bavg), where bU is a estimate of the maximum likelihood method without prior distribution proposed by Utsu. The estimate b is more stable and reliable than bU. In particular, the method proposed here is quite valuable when few event data are available.