A Prior Distribution of the Parameters in the Renewal Model with Lognormal Distribution Used for Estimating the Probability of Recurrent Earthquakes

Abstract

We determined the preliminary prior distribution of parameters in the renewal model with lognormal distribution for time intervals between large/moderate recurrent earthquakes in the world and between small repeating events with identical waveform occurring in the east off northeastern Japan. Using the prior distribution for parameters has great advantage to estimate the probability of the oncoming earthquake in a given period by a Bayesian approach from the dates of a few recurrent events in a sequence. It was supposed that n+1 recurrent events had occurred on a fault or in a certain source area, separated by n time intervals, T_i, of which logarithm, x_i=ln (T_i), followed a normal distribution, N (μ, σ²). We supposed that the prior, π(μ, σ²)was uniform for the mean, μ,and inverse gamma with the shape- and scale- parameters, φ and ζ,for the variance, σ²,respectively. Then it was shown that the random variable φV/ζ obeyed F-distribution with degree of freedom (n-1, 2 φ), where V was the unbiased variance of x_i. We got preliminary results as φ=2.5 and ζ=0.23 from 29 sequences with five or more large/moderate events in the world and 24 sequences with 4 events, respectively, using the maximum likelihood method for the variances, V_k, of samples, and by fitting the theoretical distribution of unbiased standard deviation to the observed one. On the other hand, φ=2.5 and ζ=0.44 were obtained from 70 sequences of small repeating events with identical waveform. The means, ζ/(φ-1), of the prior distribution for σ² were 0.15 for large/moderate events in the world and 0.29 for small repeating events. Large variance for small repeating events suggested that they were more influenced by stress changes in near field relating to larger earthquake occurrences than large/ moderate events. The expectation of standard deviation, σ, calculated from the prior distribution, was 0.36 for large/moderate events, which was fairly close to the mean of two values, 0.24 and 0.5, given by the Earthquake Research Committee of Japan in 2001 and by the Working Group on California Earthquake Probabilities in 1995, respectively.