New Approach to the Characterization of Mmax and of the Tail of the Distribution of Earthquake Magnitudes
New Approach to the Characterization of Mmax and of the Tail of the Distribution of Earthquake Magnitudes
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表征Mmax和地震震级分布尾部的新方法
DOI：
10.1007/s00024-008-0341-9
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发表时间：
2007
影响因子：
2
通讯作者：
M. Rodkin
中科院分区：
文献类型：
--
作者：
V. Pisarenko;A. Sornette;Didier Sornette;Didier Sornette;M. Rodkin
我们开发了一种新的方法，用于统计估计哈佛地震矩目录中记录的地震规模分布的尾部(1977-2004和1977-2006)。为此，我们提出了一个新的主震震级分布的参数模型，它由两个分支组成，一个是纯古登堡-里希特分布，直到震级阈值M1，另一个分支是震级上限为Mmax的分支，我们称之为双分支模型。我们发现，由于估计问题的内在不稳定性，星表中主要事件的数量(1977-2004年为N=3975,1977-2006年为N=4193)不足以直接估计该模型的参数。这个问题很可能适用于其他任何两个分支机构的模型。这种固有的局限性可以用以下事实来解释：第二个分支中只有一小部分经验数据。然后我们表明，在持续时间为T天的窗口中使用最大震级集(T-极大值集)提供了显著的改进，特别是(I)通过最小化前震/主震/余震序列的时间聚集在估计震级分布的尾部中的负面影响，以及(Ii)通过模拟方法在矩估计过程中提供对偏差的可靠估计(事实证明，这比最大似然估计更有效)。我们提出了一种确定T值最优选择的方法，该方法使GeV分布形状参数估计的均方误差最小，逼近T-极大值的样本分布，得到T_(Max)=0.500天。我们已经估计了1977-2006年整个期间T-极大值分布的以下分位数：Q16%(MMAX)=99.3，Q50%(MMAX)=99.7和Q84%(MMAX)=110.3。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QT(Q)和高门限震级ρT(m*)的超越概率T(m*)O=QP{MK-≥*m*}。我们得到了以下关于全球哈佛星表$$\HAT{q}_T(q=0.98)=8.6{\pm}0.2$$和$$\HAT{\rho}_T(8)=0.13-0.20的样本估计。$$我们对1977-2004年和1977-2006年两个时期的估计进行了比较，其中后一个时期包括2004年12月24日的苏门答腊岛大地震，Mw=9.0证实了参数Mmax的估计的不稳定性以及Qt(Q)和ρT(m*)=≥的稳定性。
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 mW-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 m1, followed by another branch with a maximum upper magnitude bound Mmax, 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 Toptimal = 500 days. We have estimated the following quantiles of the distribution of T-maxima for the whole period 1977–2006: Q16%(Mmax) = 9.3, Q50%(Mmax) = 9.7 and Q84%(Mmax) = 10.3. Finally, we suggest two more stable statistical characteristics of the tail of the distribution of earthquake magnitudes: The quantile QT(q) of a high probability level q for the T-maxima, and the probability of exceedance of a high threshold magnitude ρT (m*) = P{mk ≥ 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, mW = 9.0 confirms the instability of the estimation of the parameter Mmax and the stability of QT(q) and ρT (m*) = P{mk ≥ m*}.