Inversion of fault dip angle based on the random sampling consistency combined with the grid search algorithm
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摘要: 采用小地震的丛集性活动资料定量地研究地下断层的三维几何信息对地震危险性评估具有至关重要的意义。若所使用的资料存在高比率的离群值，将会使断层几何参数的估值产生较大偏差。为了提高断层几何参数估值的稳健性，本文将随机抽样一致性（RANSAC）与网格搜索（GS）相结合，给出了随机抽样一致性-网格搜索（RANSAC-GS）估算方法。在数值模拟试验中，对模拟观测值加入1%，5%，10%和20%的离群值，分别采用GS和RANSAC-GS两种方法估算了断层面倾角，并从反演参数的残差、反演模型计算值与观测值的密合度、目标函数及相关度等方面证明了RANSAC-GS方法即使在高比率离群值的情况下，依然能够给出稳健的参数估值。最后，利用2008年1月至2012年12月鄂尔多斯地区小震重定位结果，以震源点到断层面距离最小为准则，采用RANSAC-GS方法反演获得太谷断裂的断层面倾角为52.5°，与前人的结果有较好的一致性，在此基础上对大地测量形变观测给出合理的阐释，从而证明了本文方法的有效性。
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关键词:
- 随机抽样一致性-网格搜索（RANSAC-GS） /
- 断层面参数 /
- 倾角 /
- 离群值 /
- 太谷断裂
Abstract: It is of great significance to quantitatively study the geometric parameters of deep fault by using the cluster activity data of small earthquakes. If there is a high ratio of outliers, it will have a serious impact on the estimated parameters. Therefore, in order to improve the robustness of parameter estimation, this paper combined the random sampling consistency （RANSAC） algorithm with the grid search （GS） algorithm, and gave the RANSAC-GS algorithm. In the numerical simulation tests, 1%, 5%, 10% and 20% outliers were added to the simulated observations, respectively. At the same time, both GS and RANSAC-GS were used to compute the dip angle of the fault. The robustness of the RANSAC-GS was proved by the residuals of the inversion parameters, the tightness between the calculated values and the observations, the objective function, and the correlation, even if in the cases with a high ratio of outliers. And then we took the Taigu fault as an example to verify the RANSAC-GS algorithm. Basing on the fine locations of small earthquakes in the Ordos area from January of 2008 to December of 2012, we obtained the dip angle of Taigu fault to be 52.5° by RANSAC-GS algorithm with the minimum distance from the source point to the fault surface as the criterion. The dip angle result agrees well with the previous results, suggesting the effectiveness of the RANSAC-GS algorithm.-
Keywords:
- RANSAC-GS /
- fault plane parameters /
- dip angle /
- outliers /
- Taigu fault
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图 1 离群值比率为1% （a），5% （b），10% （c）和20% （d）的情形下模拟数据与断层面的空间关系
横轴 “水平距离” 为震源到断层地表轨迹的水平距离，下同
Figure 1. Spatial relationship between simulated data and fault plane with the outlier ratio of 1% （a），5% （b），10% （c） and 20% （d）
The distance on horizontal axis is the horizontal distance from the epicenter to fault surface traces, the same below
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图 2 GS与RANSAC-GS反演所得倾角残差的对比
Figure 2. Comparison of dip angle residual obtained from inversion by GS with that by RANSAC-GS algorithm
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图 3 离群值比率为 1% （a），5% （b），10% （c） 和20% （d）时观测值残差在各区间所占百分比直方图
横轴为由反演模型计算的地震震源深度和由模拟观测数据计算所得地震震源深度的残差区间，纵轴为各残差区间内震源点个数占总震源数的百分比
Figure 3. Histogram of percentage of error within each residual interval with the outlier ratio of 1% （a），5% （b），10% （c） and 20% （d）
The horizontal axis is the residual interval of the focal depth calculated by inversion model and that of the focal depth calculated by simulated observation data，and the vertical axis is the percentage of the number of focal points in each residual interval to the total number of focal points
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图 4 不同离群值比率下基于RANSAC-GS方法所得断层面倾角的线性拟合结果
Figure 4. The linear fitting result of fault dip angle based on RANSAC-GS with different outlier ratios
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图 5 研究区域构造背景及断裂分布
Figure 5. The tectonic settings and fracture distribution in the studied area
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图 6 利用式（3）所得目标函数计算出的太谷断层倾角估计结果
（a） 余震震源分布，图中小圆圈为地震震源，小圆圈中间带 “＋” 表示被RANSAC选中的内部点，不同颜色为不同迭代次选中的点，直线与横轴的夹角即为该参考点的断层面倾角，下同；（b） 目标函数随倾角的变化；（c） 基于式（4）所得目标函数的残差分析；（d） 拟合相关度分析
Figure 6. The estimation result of Taigu fault dip with objective function calculated by equation （3）
（a） Source distribution of aftershocks，where the small circles are the seismic source，and the circles with cross indicate the internal point selected by RANSAC，and the crosses with the different colors are the points selected by different iterations，and the angle between the straight line and the horizontal axis is the dip of the fault plane for the reference point，which are the same below；（b） Variation of the objective function with the dip；（c） The residual analysis of the objective function obtained from equation （4）；（d） The fitting correlation analysis
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图 7 利用式（2）所得目标函数计算出的太谷断层倾角估计结果
（a） 余震震源分布图；（b） 目标函数随倾角的变化；（c） 目标函数残差分析；（d） 拟合相关度分析
Figure 7. The estimation result of Taigu fault dip with objective function calculated by equation （2）
（a） Source distribution of aftershocks；（b） Variation of the objective function with the dip；（c） The residual analysis of the objective function；（d） The fitting correlation analysis
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图 8 利用式（6）所得目标函数计算出的太谷断层倾角估计结果
（a） 余震震源分布；（b） 目标函数随倾角的变化；（c） 目标函数残差分析；（d） 拟合相关度分析
Figure 8. The estimation result of Taigu fault dip with objective function calculated by equation （6）
（a） Source distribution of aftershocks；（b） Variation of the objective function with the dip；（c） The residual analysis of the objective function；（d） The fitting correlation analysis
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