Quantitative evaluation of modeling error in Lg-wave attenuation model
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摘要: 对Lg波衰减模型中建模误差的统计特征进行了系统研究，并建立了地壳二维Lg波衰减模型。由于Lg波振幅可能受到几何扩散函数的强烈影响，合理评估反演过程的误差对于能否使用最小二乘意义下的反演非常重要。通过在川滇及其邻近地区收集的建模误差样本，使用K-S数值检验方法、Q-Q图和正态分布图形检验方法对Lg波衰减层析成像反演的输入数据中建模误差的分布特征进行了统计分析。采用奇异值分解（SVD）和反投影方法，分别获得了川滇地区的QLg模型，定量计算模型的协方差矩阵和分辨率矩阵，定量评估了QLg模型中每个格点的分辨率和误差。结果表明：在一阶近似条件下建模误差服从正态分布；通过开发的数据筛选程序，可以产生一个接近完美正态分布的数据集；与反投影方法相比，利用SVD方法获得的地壳Q值的分辨率更高；在射线覆盖较好的区域，QLg模型的分辨率达到100 km，相对误差小于3%。Abstract: In this study, we performed comprehensive statistical studies for the characteristics of the modeling error in Lg-wave attenuation model and obtained a two-dimensional crustal Lg-wave attenuation model. The amplitudes of Lg-wave can be strongly influenced by the geometrical spreading function and it is essential to reasonably evaluate the error of the inversion process if we use the inverted method under the meaning of least squares. Based on the modeling error samples collected in the Sichuan-Yunnan region and its adjacent areas, we use three statistical test methods, namely the K-S test, the Q-Q plot, and the normal distribution plot, to analyze the distribution characteristics of modeling errors in the input data of Lg- wave attenuation tomography inversion. We used both the SVD-based tomographic method and the back-projection method to invert for the Lg Q model of the Sichuan-Yunnan region and its adjacent area respectively. Then, we calculated the covariance matrix and the resolution matrix of the model and evaluated the resolution and error of each grid point in the Lg Q model quantitatively. Our results indicate that the modeling errors in the input data obeyed a normal distribution under first-order approximation. By using the data screening technique, we generated a new dataset with nearly perfect normally distributed for Q tomography. Compared with the result of back-projection, the SVD-based method could allow for obtaining a finer resolution of crustal Q value in the areas with dense ray path coverages, where the model resolution can be resolved within a range of 100 km with a relative error of less than 3%.
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图 1 研究所用地震事件震中（a）和研究区台站（b）分布图
图中红色矩形框代表研究区域和台站所在区域；黑线为主要地块分界线，灰线为主要断裂带，下同
Figure 1. Distribution of epicenters of events （a） and seismic stations （b） in this study
Red rectangular represents the extent of the study area；the black solid lines indicate the major tectonic block boundaries and the grey dashed lines indicate faults，the same below
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图 2 基于双台法的各个路径上Q值重复测量次数分布
（a） TS方法；（b） RTS方法；（C） RTE方法
Figure 2. Distribution of the number of repeated two-stations based method over individual paths
（a） TS method；（b） RTS method；（c） RTE method
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图 3 建模误差
$\varepsilon$ 样本分布（左）和K-S检测图（右）（a） TS方法；（b） RTS方法；（c）RTE方法。左图中两条黑色直线标记了数据筛选准则|ε-median（ε）|＞2std（ε）的区间，红色线表示最佳的正态概率密度函数值；右图中灰色线代表着95%置信区间的CDF的上下界限，绿色小线段代表着K-S检验的经验CDF和理论CDF之间的最大垂向差值
Figure 3. Distribution of samples of modeling error
$\varepsilon$ （left）and results of K-S test （right）（a） TS method；（b） RTS method ；（c） RTS method. The black lines in the left figure mark the interval of |ε-median（ε）|＞2std（ε），which we use as criteria for data screening and the best-fit theoretical normal PDF of the screened samples is plotted over the distribution map with the red line. The grey lines in the right figure indicate lower and upper 95 percent confidence bounds for the CDF and the green segment indicates the maximum distance from the empirical CDF to the theoretical normal CDF which is used in the K-S test
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图 4 样本ε的Q-Q图
（a） TS方法；（b） RTS方法；（c） RTE方法
Figure 4. Q-Q plots of samples of
$\varepsilon$ （a） TS method；（b） RTS method；（c） RTE method
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图 5 （a） 1°×1°棋盘格结果；（b）研究区棋盘格恢复
Figure 5. （a） The checkboard with 1°×1° grid；（b） The recovery of the original checkboard
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图 6
$ {Q}_{{\rm{Lg}}} $ 模型的Backus-Gilbert扩散函数（a） 基于双台方法的射线路径覆盖图；（b） 用Backus-Gilbert扩散函数来表示研究区域的模型分辨率
Figure 6. Backus-Gilbert spread functions of the tomographic Lg Q model
（a） The path coverage of two-station based method over the studied region；（b） Spread function of the tomographic Q model
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图 7 川滇地区及其邻近地区的Q值模型
（a） 新的层析成像方法结果；（b） 反投影方法结果
Figure 7. The Lg Q model of Sichuan-Yunnan area and its neighborhood
（a） Result from the new tomographic method；（b） Result from the back projection method
表 1 使用TS，RTS和RTE方法估计ε的详细信息
Table 1 Estimated ε information using TS， RTS and RTE methods
方法 $ {\varepsilon } $样本的
数量误差在两个标准差
内的$ \varepsilon $占比ECDF和NCDF的
相关系数K-S测试结果 标准差
std$ （ \varepsilon ） $方差Var$ （ \varepsilon ） $ P值 显著性水平$ \alpha $ TS 8 220 87.09% 99.99% 在置信度为95%下接受零假设 0.158 5 0.025 1 0.693 1 0.05 RTS 3 652 88.12% 99.98% 在置信度为95%下接受零假设 0.081 6 0.006 7 0.324 0 0.05 RTE 2 839 93.59% 99.64% 在置信度为95%下接受零假设 0.104 9 0.011 0 0.280 1 0.05 
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