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

机译：表征M（max）和地震震级分布尾部的新方法

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摘要

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*}.
机译：我们开发了一种新的统计方法，该方法可以对哈佛地震目录中记录的转换为m（W）量级（1977-2004和1977-2006）的地震规模分布的尾部进行统计估计。为此，我们建议一种用于主震幅度分布的新参数模型，该模型由两个分支组成，即纯Gutenberg-Richter分布，直至最大幅度阈值m（1），然后是另一个具有最大上限的分支幅度边界M-max，我们称为两分支模型。我们发现目录中的主要事件数量（1977-2004年为N = 3975，1977-2006年为N = 4193）不足以直接估计此模型的参数，这是因为估计问题固有的不稳定性。对于任何其他两分支模型，此问题可能都是相同的。这种固有的局限性可以通过以下事实来解释：只有一小部分经验数据填充第二分支。然后，我们表明，在持续时间T天的窗口中使用最大震级集（T最大值集）会带来显着改善，特别是（i）通过最大程度地减少前震/主震/余震时间簇的负面影响（ii）通过模拟方法提供矩量估计程序中偏差的可靠估计（事实证明比最大似然估计更有效）。我们提出了一种确定T值的最佳选择的方法，该方法可最小化GEV分布的形状参数估算值的均方误差，该误差近似于T-最大值的样本分布，从而产生T-最优= 500天。我们估计了1977-2006年整个期间T-最大值的分布的以下分位数：Q（16％）（M-max）= 9.3，Q（50％）（M-max）= 9.7和Q84％（ M-max）= 10.3。最后，我们提出了地震震级分布尾部的两个更稳定的统计特征：T极大值的高概率水平q的分位数Q（T）（q）和超高阈值震级的概率rho（T）（m *）= P {m（k）C m *}。对于上限（T）（q = 0：98）= 8.6 +/- 0.2和上限（T）（8）= 0.13-0.20的（rho），我们获得了以下全球哈佛目录（Q）的样本估计值。我们对1977-2004年和1977-2006年这两个时期的估计值之间的比较，其中后一个时期包括苏门答腊岛大地震，2004年12月24日，m（W）= 9.0，证实了参数Mmax估计的不稳定性和Mmax的稳定性。 Q（T）（m *）和rho（T）（m *）= P {m（k）> = m *}。

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