Recent progress on full waveform inversion
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摘要: 全波形反演是一种基于声波/弹性/黏弹性波动方程来反演三维地球模型的高分辨率成像方法. 目前该方法已经被广泛应用于油气勘探、地壳与上地幔结构以及地幔对流的研究当中. 使用该方法，可以建立一个统一的理论和算法框架来反演地球内部的多个地震学参数模型，主要包括P波和S波速度、各向异性、黏滞性衰减、密度以及反射系数等. 通过联合解释这些地震学多参数结果，可以更好地约束地球内部的温度变化、物质构成、地幔对流以及水和挥发成分的分布. 目前，关于全波形反演的研究前沿主要包括目标函数的选取、多参数联合反演、模型正则化约束、分辨率和不确定性分析，以及其在新型地震数据，例如背景噪声和线性密集台阵中的应用. 此外，为了更好地解释反演所得到的地震学多参数模型以及探讨相关的地球科学问题，需要多学科之间的交叉合作，包括结合地震学以及岩石矿物实验和地球动力学模拟等的结果. 相关的成果对更好地认识油气储层构造、盆地结构、断层分布以及地幔对流具有重要的科学意义.Abstract: Full waveform inversion is an acoustic/elastic/anelastic wave equation-based high accuracy seismic imaging method for studying the Earth's interior structure. To date, it has been widely used in exploration seismology, studies on crustal and mantle structures at both regional and global scales. With this approach, we are able to build a unified theory and algorithm platform to constrain multi-parameter seismic models for the Earth's interior, including P and S wave velocities, anisotropy, attenuation, density and reflectivity, etc. By jointly interpreting these seismic parameters, we hope to better constrain variations in temperature and composition, mantle convection and distribution of water and volatiles. Recent developments include selection of optimal misfit functions, multi-parameter inversion, model regularization, resolution and uncertainty quantification, as well as its applications to special types of datasets, such as ambient-noise recordings and teleseismic scattered waves recorded by dense linear arrays. Furthermore, in order to better interpret inverted multi-parameters and investigate related problems in Earth sciences, we need collaboration among different disciplines, such as synthesizing results from seismology, mineral physics and geodynamic modeling. These results enable us to better understand reservoirs, basin structures, fault distribution and mantle convection.
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图 1 二维均匀速度模型下构建SH波的敏感核. 从左到右分别是正传波场、伴随波场、相互作用波场和剪切波速度的敏感核. 五角星和方块分别表示震源和接收台站的位置. 时间从下往上分别是8 s、16 s、24 s、32 s 和44 s. 地表使用的是自由表面边界条件（修改自Tromp et al., 2005中的图3）
Figure 1. Construction of an SH sensitivity kernel in a 2D homogeneous velocity model. From left to right are forward wavefield, adjoint wavefield, interaction wavefield and shear wave velocity sensitivity kernel. The star and rectangular represent source and receiver. The time steps from bottom to top are 8, 16, 24, 32 and 44 seconds. A traction free boundary condition is applied to the Earth's surface (modified from Figure 3 in Tromp et al., 2005)
图 2 对比九个不同理论模型下使用三种迭代方法所得到的收敛性，包括最速下降法（黑线），非线性共轭梯度法（绿线）和L-BFGS法（红线）. 不同的理论模型如每个图的标题所示（修改自 Modrak and Tromp, 2016中的图2）
Figure 2. Convergency comparison of three iterative methods for nine synthetic velocity models, including the steepest descent method (black lines), nonlinear conjugate gradient method (green lines) and L-BFGS method (red lines). The 2D synthetic velocity models are shown in the titles of each panel (modified from Figure 2 in Modrak and Tromp, 2016)
图 3 对比不同目标函数的特征. （a）显示的是子波波形. （b）和（c）分别显示的是基于最小二乘波形残差和最优化路径所得到的目标函数随着不同时间移动（s）的特征（修改自Engquist and Froese, 2014中的图1）
Figure 3. Comparison of misfit functions based on least-square waveform differences and optimal transport distances. Panel (a) shows the input source wavelet. Panels (b) and (c) are the misfits as functions of time shifts for the least-squares waveform differences and optimal distances, respectively (modified from Figure 1 in Engquist and Froese, 2014)
图 4 三维俯冲板块导致的地幔对流. （a）和（b）分别显示中美洲和卡斯凯迪亚俯冲带的反演结果. 绿色块体显示的是剪切波速度扰动大于1.5%的结果. 黄色箭头表示通过伴随方位各向异性成像所得到的地幔对流情况[（a）和（b）分别修改自 Zhu et al., 2020a中的图7 和 Zhu et al., 2020c中的图5 ]
Figure 4. Three dimensional subducting slabs and induced mantle flows. Panels (a) and (b) are results for the Middle American and Cascadian subduction zones. Green bodies represent regions with shear wave velocity perturbations greater than 1.5%. Yellow arrows represent horizontal mantle flows constrained by azimuthal anisotropy adjoint tomography [Panels (a) and (b) are modified from Figure 7 in Zhu et al., 2020a and Figure 5 in Zhu et al., 2020c, respectively]
图 5 对比三个全球尺度的上地幔地震波衰减成像结果. （a-c）分别来自于QRLW8（Gung and Romanowicz, 2004）、SEMUCB-UMQ（Karaoglu and Romanowicz, 2018）和QRFSI12（Dalton et al., 2008）. 每一个图的下方第一行和第二行分别显示的是各个深度的剪切模量衰减值及其扰动量，其中SEMUCB-UMQ是全波形衰减反演的结果（修改自Karaoglu and Romanowicz, 2018中的图9 ）
Figure 5. Comparison of three global scale upper mantle seismic attenuation tomography models. Panels (a-c) are results from QRLW8 (Gung and Romanowicz, 2004)，SEMUCB-UMQ (Karaoglu and Romanowicz, 2018) and QRFSI12 (Dalton et al., 2008). The first and second lines in each panel represent absolute and relative perturbations of shear modulus attenuation (modified from Figure 9 in Karaoglu and Romanowicz, 2018 )
图 6 不同正则化约束下二维弹性波全波形反演所得到的P波速度结果. （a-c）分别表示基于Tikhonov、全变差和改进的全变差正则化的结果（修改自 Lin and Huang, 2014中的图10）
Figure 6. P wave velocity models from an elastic full waveform inversion with different regularization schemes. Panels (a-c) are results constrained with Tikhonov, Total variation and modified Total variation regularization, respectively (modified from Figure 10 in Lin and Huang, 2014)
图 7 使用点扩散函数分析二维全波形反演所得到的不同位置上的分辨率. （a-i）分别显示点扩散函数在右下角图中不同区域的结果（修改自 Fichtner and van Leeuwen, 2015中的图5）
Figure 7. Using point-spread functions to analyze resolution at different locations in an 2D full waveform inversion. Panels (a-i) illustrate results at different locations as shown in the bottom right panel (modified from Figure 5 in Fichtner and van Leeuwen, 2015)
图 8 散射波对于不同模型参数组合的辐射样式. （a, b）分别表示散射波对于声波速度和密度组合所得到的辐射样式. （c, d）分别表示声波速度和波阻抗组合的结果. 蓝色和白色五角星分别表示激发震源和散射点. 绿色曲线表示由射线加上波恩近似所得到的理论波前（修改自 Operto et al., 2013中的图2）
Figure 8. Radiation patterns of scattered waves for different combinations of model parameters. Panels (a, b) show the radiation patterns for P wave velocity and density, respectively. Panels (c, d) are results for the combination of P wave velocity and impedance. Blue and white stars denote exciting source and scattering point. Green curves represent the amplitudes of scattered waves from Ray+Born approximation (modified from Figure 2 in Operto et al., 2013)
图 9 通过使用背景噪声信号改进南加州地壳剪切波速度模型. 从左到右分别是M16（只使用天然地震记录）、M21（结合天然地震记录和背景噪声互相关函数），以及两者之间的差（Diff）（修改自 Wang et al., 2019中的图9）
Figure 9. Comparison of shear wave velocity models by incorporating ambient noise cross correlation functions. From left to right are model M16 (only constrained by earthquake records), M21(further constrained by ambient noise cross correlation functions) and differences between M16 and M21 (modified from Figure 9 in Wang et al., 2019)
图 10 对比不同南加州地壳模型在二维LARSE-I（a）和LARSE-II（b）剖面上的P波速度结果. CVM-S4.26、CVM-S4和CVM-H11.9分别是三个南加州地震研究中心的标准公共地壳模型. Lutter等（1999, 2004）以及Fuis等（2003）分别是两个二维主动源地壳速度反演结果（修改自 Lee et al., 2014中的图8）
Figure 10. Comparisons of P wave velocities in different Southern Californian velocity models for 2D LARSE-I and LARSE-II cross sections. Models CVM-S4.26, CVM-S4 and CVM-H11.9 are three 3D reference crustal velocity models from the Southern California Earthquake Center (SCEC). Lutter et al. (1999, 2004) and Fuis et al. (2003) are 2D crustal velocity models constrained by active source experiments (modified from Figure 8 in Lee et al., 2014)
图 11 使用通过地幔转换带的SH波改进俯冲板块伴随层析成像结果. （a）显示的是二维剖面和所使用的地震台站的位置. （b）对比初始模型（左）和迭代改进之后的模型（右）对SH地幔转换波的影响. 黑色和红色地震图分别是实际观测和模拟计算所得到的结果. （c）和（d）分别是初始模型和迭代改进之后的模型在（a）中所示的二维剖面上的剪切波速度扰动（修改自 Tao et al., 2018中的图6）
Figure 11. Imaging subducting slabs by using SH triplication waveforms. Panel (a) shows the locations of 2D vertical cross sections and stations. Panel (b) compares observed (black) and synthetic (red) SH waveforms from the starting (left) and updated models (right). Panels (c) and (d) show shear wave velocity perturbations for the starting and updated models on the 2D vertical cross section shown in Panel (a) (modified from Figure 6 Tao et al., 2018)
图 12 对比三个全球尺度地幔剪切波成像模型在6个二维剖面上的结果. 从左到右分别是来自于GLAD-M25（Lei et al., 2020）、TX2015（Lu and Grand, 2016）和SEMUCB-WM1（French and Romanowicz, 2015）. （a-f）分别对应于以下的热点:（a）阿法尔州;（b）百慕大群岛和加那利群岛;（c）佛得角和哈加尔高原;（d）冰岛和艾费尔高原;（e）复活节岛 和加拉帕戈斯群岛；（f） 马里昂县和凯尔盖朗群岛（修改自 Lei et al., 2020中的图15）
Figure 12. Comparisons of shear wave velocity perturbations in three global-scale mantle tomography models. From left to right are results from GLAD-M25 (Lei et al., 2020)，TX2015 (Lu and Grand, 2016) and SEMUCB-WM1 (French and Romanowicz, 2015). Panels (a) to (f) show results for the following hotspots: (a) Afar; (b) Bermuda and Canary; (c) Cape Verde and Hoggar; (d) Iceland and Eifel; (e) Easter and Galapagos and (f) Marion and Kergulen (modified from Figure 15 in Lei et al., 2020)
图 13 对比全波形反演在北海Valhall油田勘探当中的应用. 左图和右图分别是使用反射波到时层析成像和全波形反演所得到的结果. （a-c）以及（h-j）分别是在175 m、500 m和1000 m所得到的P波速度结果. （d-g）以及（k-n）分别是相应虚线位置的二维剖面结果（修改自 Operto et al., 2015中的图4和图11）
Figure 13. Comparison of velocity models for the ocean bottom node data from the Valhall oilfield. The left panel and right panel are models from reflection travel time tomography and full waveform inversion, respectively. (a-c) and (h-j) in each panel are P wave velocities at 175 m, 500 m and 1000 m. (d-g) and (k-n) demonstrate results in corresponding vertical profiles as shown by dashed lines (modified from Figures 4 and 11 in Operto et al., 2015)
图 14 在二维均匀速度模型下对比直达波和反射波的梯度. （a）和（b）分别是直达波和反射波的梯度，其中反射界面如图（b）中的紫色直线所示. （c）直达波+反射波的梯度. （d）改进的联合全波形梯度结果（修改自 Zhou et al., 2015中的图3）
Figure 14. Gradients for direct and reflected waves in a 2D homogeneous velocity model. Panels (a) and (b) are gradients for direct and reflected P waves. The reflector is shown as the purple line in Panel (b). Panels (c) and (d) are gradients for direct+reflected waves, and joint full waveform inversion gradient (modified from Figure 3 in Zhou et al., 2015)
图 15 使用基于目标函数梯度的马尔科夫链蒙特卡罗方法反演二维Marmousi模型. （a-c）分别表示平均模型、标准差模型以及平均与真实模型之间的差. （d）显示6个采样点的一维边界后验概率分布情况（修改自 Zhao and Sen, 2021中的图9和10）
Figure 15. Results for the 2D Marmousi model from a gradient based Markov chain Monte Carlo sampling. Panels (a) to (c) are results for the mean velocity model, standard deviation and differences between the mean and true velocity models. Panel (d) shows the 1D marginal posteriori probability density functions at six different locations (shown in Panel a) (modified from Figures 9 and 10 in Zhao and Sen, 2021)
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