New approach to the characterization of M max and of the tail of the distribution of earthquake magnitudes

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 Q84%( 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 rho(T) (m*) = P{m(k) C m*}. We obtained the following sample estimates for the global Harvard catalog (Q) over cap (T)(q = 0: 98) = 8.6 +/- 0.2 and (rho) over cap (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 Mmax and the stability of Q(T)(m*) and rho(T) (m*) = P{m(k) >= m*}.