Calculating Rotational Components of Ground Motions by Finite Difference Method
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摘要: 旋转地震学是一门研究由地震、爆破以及工程振动等引起的地球介质运动的新兴学科.旋转运动的研究由来已久，但是由于缺少高精度的旋转分量地震仪，所以旋转运动的研究大多仅限于理论方面.差分法作为利用平动分量获取旋转分量的一种计算方法，在理论研究方面较为成熟，但是缺乏实际数据的验证.通过在对旋转运动研究现状充分调研的基础上，利用模拟数据和实测的爆炸源数据，对差分法进行了测试和分析.通过对比差分法计算的旋转分量和实测旋转分量的波形图、振幅谱和相位谱等，得出了在一定的误差允许范围内，差分法可以作为求取水平旋转分量（RX，RY）的替代方法的结论；同时，针对爆炸源的高频特性以及密集台阵观测的特点，对现有的差分法进行了改进，提出了一种精度更高的求取旋转分量的差分法.Abstract: Rotational seismology is a new subject to study the motions of the earth's medium caused by earthquakes, blasting and environmental vibration in an all-round way. The study of rotational components of ground motion has a long history. However, there is no high-precision rotational component seismograph, so the study of rotational motions is usually limited to theoretical aspects. As an alternative method of calculating rotational components by translational components, the finite difference method is relatively mature in theoretical research, but its application in practical data is still less. Based on the full investigation of the research that studying the rotational motions, the finite difference method is tested and analyzed by using the simulated data and the recording explosive source data. By comparing the waveforms, amplitude spectra and phase spectra of the rotational components calculated by the finite difference method with thoserecorded, it is concluded that the finite difference method can be used as an effective alternative to calculate the horizontal rotational components (RX, RY) within an error allowable range. At the same time, in view of the high frequency characteristics of the explosive source and the dense array, we have improved the finite difference method and proposed a higher precision finite difference method for calculating rotational components.
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图 2 差分法换算与模拟旋转分量波形对比图（台站间距4 m）
Fig. 2. Comparisons of waveforms between the difference method and the simulated array data (4 m interval between stations)
图 1 差分法换算与模拟旋转分量波形对比图（台站间距1 m）
Fig. 1. Comparisons of wave forms between the difference method and the simulated array data (1 m interval between stations)
图 3 差分法换算与模拟旋转分量振幅谱和相位谱对比图
Fig. 3. Comparisons of amplitude and phase spectra between the difference method and the simulated array data
图 4 震源和地震台站分布位置示意图
Fig. 4. Sketch map of location of the explosive sources and seismograph stations
图 5 差分法换算与实测旋转分量波形对比图
Fig. 5. Comparisons of wave forms between the difference method and the recording array data
图 6 差分法换算与实测旋转分量振幅谱和相位谱对比图
Fig. 6. Comparisons of amplitude and phase⁃spectra between the difference method and the recording array data
图 7 地震台站分布示意图（改进差分法）
Fig. 7. Sketch map of location of the seismograph stations (the improved difference method)
图 8 改进差分法换算与实测旋转分量波形对比图
Fig. 8. Comparisons of wave forms between the improved difference method and the recording array data
图 9 改进差分法换算与实测旋转分量振幅谱和相位谱对比图
Fig. 9. Comparison of amplitude and phase spectra between the improved difference method and the recording array data
表 1 正演地震六分量数据模型参数
Table 1. Model parameters of forward six⁃component simulated array data
介质物理量 纵波速度Vp（m/s） 横波速度Vs（m/s） 密度ρ（kg/m3） 层厚（m） 界面倾斜角度θ（°） 第一层 2 000 1 400 2 600 50 0 第二层 3 000 2 100 2 700 70 0 
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表 2 正演地震六分量数据地震记录参数
Table 2. Seismic record parameters of forward six⁃component simulated array data
物理参数 地震子波主频（Hz） 采样间隔（ms） 地震记录长度（s） 数值 120 0.1 0.1
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表 3 正演地震六分量数据网格参数
Table 3. Grid parameters of forward six-component simulated array data
物理参数 X方向网格点数 Y方向网格点数 Z方向网格点数 X方向网格步长（m） Y方向网格步长（m） Z方向网格步长（m） 数值 100 100 200 1 1 1
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表 4 差分法换算与模拟旋转分量波形相关系数（台站间距1 m）
Table 4. Correlation coefficient of wave forms between the difference method and the simulated array data (1 m interval between stations)
旋转分量 RX RY RZ 波形相关系数 0.986 8 0.989 1 0.271 3
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表 5 差分法换算与模拟旋转分量波形相关系数（台站间距4 m）
Table 5. Correlation coefficient of wave forms between the difference method and the simulated array data (4 m interval between stations)
旋转分量 RX RY RZ 波形相关系数 0.976 0 0.917 7 -0.607 3
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表 6 差分法换算与模拟旋转分量振幅、相位相关系数
Table 6. Correlation coefficients of amplitude and phasespectra between the difference method and the simulated array data
旋转分量 RX RY RZ 振幅谱相关系数 0.990 9 0.999 7 0.984 5 相位谱相关系数 0.750 3 0.157 3 0.016 9
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表 7 差分法换算与实测旋转分量的波形相关系数
Table 7. Correlation coefficient of wave forms between the difference method and the recording array data
旋转分量 RX RY RZ 波形相关系数 0.576 9 0.017 4 -0.131 4
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表 8 差分法换算与实测旋转分量振幅谱和相位谱相关系数
Table 8. Correlation coefficients of amplitude and phasespectra between the difference method and the recording array data
旋转分量 RX RY RZ 振幅相关系数 0.823 6 0.508 4 0.581 2 相位相关系数 0.228 6 -0.095 9 -0.024 3
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表 9 改进差分法换算与实测旋转分量波形相关系数
Table 9. Correlation coefficient of wave forms between the improved difference method and the recording array data
旋转分量 RX RY RZ 波形相关系数 0.719 6 0.199 5 -0.010 2
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表 10 改进差分法换算与实测旋转分量振幅谱和相位谱相关系数
Table 10. Correlation coefficients of amplitude and phase spectra between the improved difference method and the recording array data
旋转分量 RX RY RZ 振幅相关系数 0.856 6 0.841 4 0.670 2 相位相关系数 0.244 6 0.242 7 0.242 7
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