A New Misfit Function for Multimode Dispersion Curve Inversion of Rayleigh Waves
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摘要: 反演瑞雷波频散曲线能有效地获取横波速度和地层厚度，传统的多模式瑞雷波频散曲线反演需要正确的模式判别.然而，当地层中含有低速软弱夹层或高速硬夹层等复杂结构时，瑞雷波可能会出现"模式接吻"和"模式跳跃"等现象，这些现象极易造成模式误判，进而导致错误的反演结果；同时，传统的频散曲线反演方法需要进行求根运算，进而导致现有的瑞雷波非线性反演速度慢，运算时间长.鉴于此，对传统的Haskell-Thomson频散曲线正演模拟算法进行了改进，提出了一种新颖有效的目标函数.该目标函数直接利用实测频散曲线与迭代更新模型频散函数表面形状进行最佳拟合，无需将多模式频散数据归于特定的模式，可有效避免多模式瑞雷波频散曲线反演模式误识别；同时，该目标函数不需要求根运算，进而大大加快了非线性反演速度.基于粒子群优化算法，利用实际工作中经常遇到的3种典型理论地质模型和某一高速公路路基实测资料进行了理论模型试算和实例分析，检验了本文提出的瑞雷波多模式频散曲线反演新方法的有效性和实用性.Abstract: Inversion of the Rayleigh wave dispersion curve can effectively obtain the shear wave velocity and stratum thickness. And the classical inversion of multimode Rayleigh wave dispersion curve requires correct mode identification. However, there may be "mode-kissing" and "mode-jumping" phenomena in the Rayleigh waves when the stratum contains complex structures such as low-velocity soft intercalations or high-speed stiff sandwich layers. These phenomena can easily lead to mode misidentification, and lead to wrong inversion results. At the same time, the classical dispersion curve inversion method needs to seek the roots, which leads to that the nonlinear inversion of Rayleigh wave is slow and the computation time is long. In view of these, the classical Haskell-Thomson dispersion curve forward modeling algorithm is improved, and a novel and effective misfit function is proposed. The misfit function is directly used to fit the dispersion function surface shape of the iterative updating model by using the measured dispersion curve. It is not necessary to assign the multimode dispersion data to a specific mode, which can effectively avoid the mode misidentification in the inversion of multimode Rayleigh wave dispersion curve. And the misfit function does not require the seeking root operation, thus greatly accelerating the nonlinear inversion speed. In this paper, based on the particle swarm optimization algorithm, three theoretical geological models and a certain roadbed test data often encountered in practical work are used to calculate the theoretical model and analyze the example. And the validity and practicability of the new method of Rayleigh wave multimode dispersion curve inversion are verified.
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图 1 Haskell-Thomson算法频散函数表面图与模型A理论频散曲线的对比
a.改进前的Haskell-Thomson算法绘制的频散函数表面图；b.改进后的Haskell-Thomson算法绘制的频散函数表面图.图中白色实线为模型A的理论频散曲线
Fig. 1. Comparison of the Haskell-Thomson algorithm dispersion function surface and the synthetic dispersion curves of Model A
图 2 基于模型B模拟的理论瑞雷波地震记录及其高分辨率频散能量谱
a.60道高精度理论瑞雷波地震记录；b.由图 2a提取的高分辨率频散能量谱.图中白色实点为根据频散能量极大值提取的瑞雷波频散曲线，该频散数据将作为实测频散曲线进行反演
Fig. 2. The theoretical Rayleigh wave seismic record and the high-resolution dispersion energy spectra based on model B
图 3 传统方法与新方法反演模型正演模拟的频散曲线与实测频散曲线的对比(a)和传统方法与新方法反演模型剖面与模型B的对比(b)
图a中:黑色实点为从图 2b中提取的瑞雷波频散曲线；蓝色虚线为利用传统方法反演模型正演模拟的基阶波频散曲线；红色虚点线为利用新方法反演模型正演模拟的多模式频散曲线，该多模式频散曲线在13 Hz处相速度几乎一样，此现象称为“模式接吻”现象.图b中:绿色虚线表示粒子群优化算法反演时模型参数搜索范围；黑色实线表示真实模型B；蓝色虚线表示传统方法反演获得的横波速度剖面；红色虚点线表示新方法反演获得的模型剖面
Fig. 3. Comparison of the dispersion curves inverted by classical method and new method and the measured dispersion curve (a), and comparison of the S-wave velocity profiles inverted by classical method and new method and the profile of model B (b)
图 4 新方法反演模型的频散函数表面与实测频散曲线的拟合对比
Fig. 4. Comparison of the dispersion function surface of model inverted by new method and the measured dispersion curve
图 5 两种方法对应的目标函数表面形状
a.传统方法的目标函数表面形状；b.新方法的目标函数表面形状.二者中的红色圆点代表真实模型，绿色圆点代表各自反演得到的模型
Fig. 5. The misfit function surfaces of two methods
图 6 基于模型C模拟的理论瑞雷波地震记录及其高分辨率频散能量谱
a.60道高精度理论瑞雷波地震记录；b.由图 6a提取的高分辨率频散能量谱.图中白色实点为根据频散能量极大值提取的瑞雷波频散曲线, 该频散数据将作为实测频散曲线进行反演
Fig. 6. The theoretical Rayleigh wave seismic record and the high-resolution dispersion energy spectra based on the model C
图 7 传统方法与新方法反演模型正演模拟的频散曲线与实测频散曲线的对比(a)和传统方法与新方法反演模型剖面与模型C的对比(b)
图a中:黑色实点为从图 6b中提取的瑞雷波频散曲线；蓝色虚线为利用传统方法反演模型正演模拟的基阶波频散曲线；红色虚点线为利用新方法反演模型正演模拟的多模式频散曲线.图b中:绿色虚线表示粒子群优化算法反演时模型参数搜索范围；黑色实线表示真实模型C；蓝色虚线表示传统方法反演得到的模型剖面；红色虚点线表示新方法反演获得的模型剖面
Fig. 7. Comparison of the dispersion curves inverted by classical method and new method and the measured dispersion curve (a) and comparison of the S-wave velocity profiles inverted by classical method and new method and the profile of model C (b)
图 8 新方法反演模型的频散函数表面与实测频散曲线的拟合对比
Fig. 8. Comparison of the dispersion function surface of model inverted by new method and the measured dispersion curve
图 9 基于模型D模拟的理论瑞雷波地震记录及其高分辨率频散能量谱
a.60道高精度理论瑞雷波地震记录；b.由图 9a提取的高分辨率频散能量谱.图中白色实点为根据频散能量极大值提取的瑞雷波频散曲线，该频散数据将作为实测频散曲线进行反演
Fig. 9. The theoretical Rayleigh wave seismic record and the high-resolution dispersion energy spectra based on the Model D
图 10 传统方法与新方法反演模型正演模拟的频散曲线与实测频散曲线的对比(a)和传统方法与新方法反演模型剖面与模型D的对比(b)
图a中:黑色实点为从图 9b中提取的瑞雷波频散曲线；蓝色虚线为利用传统方法反演模型正演模拟的多模式瑞雷波频散曲线；红色虚点线为利用新方法反演模型正演模拟的多模式频散曲线.图b中:绿色虚线表示粒子群优化算法反演时模型参数搜索范围；黑色实线表示真实模型D；蓝色虚线表示传统方法反演得到的模型剖面；红色虚点线表示新方法反演获得的模型剖面
Fig. 10. Comparison of the dispersion curves inverted by classical method and new method and the measured dispersion curve (a), and comparison of the S-wave velocity profiles inverted by classical method and new method and the profile of model D (b)
图 11 新方法反演模型的频散函数表面与实测频散曲线的拟合对比
Fig. 11. Comparison of the dispersion function surface of model inverted by new method and the measured dispersion curve
图 12 基于模型B模拟的含随机噪声的瑞雷波地震记录及其高分辨率频散能量谱
a.含20%随机噪声的地震记录；b.由图a提取的高分辨率能量谱；c.含40%随机噪声的地震记录；d.由图c提取的高分辨率能量谱；图b和d中白色实心点为根据频散能量极大值提取的瑞雷波频散曲线，该频散数据将作为实测频散曲线进行反演
Fig. 12. Rayleigh wave seismic records with random noise and their high-resolution dispersion energy spectra based on model B
图 13 含不同随机噪声数据反演模型剖面与模型B的对比
图中绿色虚线表示粒子群优化算法反演时模型参数搜索范围；黑色实线表示真实模型B；红色实线表示无噪声数据反演得到的模型剖面；洋红色点划线表示含20%随机噪声数据反演获得的模型剖面.蓝色点虚线表示含40%随机噪声数据反演获得的模型剖面
Fig. 13. Comparison of the S-wave velocity profiles inverted by different random noise data and the profile of model B
图 14 不同层数设置时反演模型剖面与模型C的对比
图中绿色虚线表示粒子群优化算法反演时模型参数搜索范围；黑色实线表示真实模型C；红色实线表示设置层数为四层时反演得到的模型剖面；洋红色点划线表示设置层数为五层时反演获得的模型剖面.蓝色点虚线表示设置层数为六层时反演获得的模型剖面
Fig. 14. Comparison of the S-wave velocity profiles with different layer setting and the profile of model C
图 15 某高速公路路基瑞雷波勘探实例
a.野外实测的24道瑞雷波地震记录；b.由图 15a中的多道瑞雷波炮集地震记录提取的f-k域频散能量谱.图中白色实线为提取的f-k域实测瑞雷波频散曲线
Fig. 15. Exploration case of Rayleigh wave of a highway roadbed
图 16 传统方法与新方法反演模型正演模拟的频散曲线与实测频散曲线的对比(a)和传统方法与新方法反演的横波速度剖面与钻孔资料的对比(b)
图a中:黑色实点为从图 15中提取得到的实测频散数据变换到f-v域的频散曲线; 红色虚点线为利用新方法反演模型正演模拟获得的多模式频散曲线；蓝色虚线表示利用传统方法反演模型正演模拟获得的基阶波频散曲线.图b中:绿色虚线为粒子群优化算法反演时模型参数搜索范围；带有圆圈的实线表示钻孔资料剖面；红色虚点线表示新方法反演的横波速度剖面；蓝色虚线表示传统方法反演得到的横波速度剖面
Fig. 16. Comparison of the dispersion curves inverted by classical method and new method and the measured dispersion curve (a), and comparison of the S-wave velocity profiles inverted by classical method and new method and the borehole data (b)
图 17 新方法反演模型频散函数表面与实测频散曲线对比拟合情况
Fig. 17. Comparison of the dispersion function surface of model inverted by new method and the measured dispersion curve
表 1 模型A:三层含低速软弱夹层地质模型参数
Table 1. Model A: a three-layer model with a soft layer trapped between two stiff layers
层序号 VS(m/s) VP(m/s) ρ(g/cm3) h(m) 1 220 437 1.8 6 2 160 285 2.0 3 均匀半空间 400 794 2.1 
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表 2 模型B:两层速度递增型地质模型参数及反演搜索范围
Table 2. Model B: a two-layer model characterized by S-wave velocities increasing with depth and search space in the inversion
层序号 模型参数 搜索范围 VS(m/s) VP(m/s) ρ(g/cm3) h(m) VS(m/s) h(m) 1 150 298 1.8 5 100~300 1~10 均匀半空间 450 802 2.1 ∞ 200~3 000 ∞
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表 3 模型C:四层含低速软夹层地质模型参数及反演搜索范围
Table 3. Model C: a four-layer model with a soft layer trapped between two stiff layers and search space in the inversion
层序号 模型参数 搜索范围 VS(m/s) VP(m/s) ρ(g/cm3) h(m) VS(m/s) h(m) 1 200 490 2.0 5 120~300 2~8 2 160 392 2.0 5 100~300 2~8 3 260 637 2.0 5 100~500 2~8 均匀半空间 380 931 2.0 ∞ 200~800 ∞
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表 4 模型D:六层含高速硬夹层地质模型参数及反演搜索范围
Table 4. Model D: a six-layer model with a stiff layer sandwiched between two soft layers and search space in the inversion
层序号 模型参数 搜索范围 VS(m/s) VP(m/s) ρ(g/cm3) h(m) VS(m/s) h(m) 1 200 490 2.0 2.5 150~400 0.1~5 2 260 637 2.0 2.5 10~400 0.1~5 3 120 294 2.0 2.5 50~600 0.1~5 4 240 588 2.0 3 100~600 0.1~5 5 260 637 2.0 3 150~600 0.1~5 6 370 906 2.0 ∞ 200~600 ∞
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