New Approach to the Characterization of M max and of the Tail of the Distribution of Earthquake Magnitudes

Abstract

We develop a new method for the statistical estimation of the tail of the distribution of earthquake sizes recorded in the Harvard catalog of seismic moments converted to m W -magnitudes (1977–2004 and 1977–2006). For this, we suggest a new parametric model for the distribution of main-shock magnitudes, which is composed of two branches, the pure Gutenberg-Richter distribution up to an upper magnitude threshold m 1, followed by another branch with a maximum upper magnitude bound M max, which we refer to as the two-branch model. We find that the number of main events in the catalog (N = 3975 for 1977–2004 and N = 4193 for 1977–2006) is insufficient for a direct estimation of the parameters of this model, due to the inherent instability of the estimation problem. This problem is likely to be the same for any other two-branch model. This inherent limitation can be explained by the fact that only a small fraction of the empirical data populates the second branch. We then show that using the set of maximum magnitudes (the set of T-maxima) in windows of duration T days provides a significant improvement, in particular (i) by minimizing the negative impact of time-clustering of foreshock/main shock/aftershock sequences in the estimation of the tail of magnitude distribution, and (ii) by providing via a simulation method reliable estimates of the biases in the Moment estimation procedure (which turns out to be more efficient than the Maximum Likelihood estimation). We propose a method for the determination of the optimal choice of the T value minimizing the mean-squares-error of the estimation of the form parameter of the GEV distribution approximating the sample distribution of T-maxima, which yields T optimal = 500 days. We have estimated the following quantiles of the distribution of T-maxima for the whole period 1977–2006: Q 16%(M max) = 9.3, Q 50%(M max) = 9.7 and Q 84%(M max) = 10.3. Finally, we suggest two more stable statistical characteristics of the tail of the distribution of earthquake magnitudes: The quantile Q T (q) of a high probability level q for the T-maxima, and the probability of exceedance of a high threshold magnitude ρ T (m*) = P{m k ≥ m*}. We obtained the following sample estimates for the global Harvard catalog $$ \hat{Q}_T (q=0.98)=8.6 {\pm}0.2 $$ and $$ \hat{\rho}_T (8)=0.13-0.20. $$ The comparison between our estimates for the two periods 1977–2004 and 1977–2006, where the latter period included the great Sumatra earthquake 24.12.2004, m W = 9.0 confirms the instability of the estimation of the parameter M max and the stability of Q T (q) and ρ T (m*) = P{m k ≥ m*}.