频率域波动方程多参数全波形反演方法研究
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摘要
随着地震勘探技术研究的不断深入，对于复杂介质中地震波正、反演问题的研究也日益增多。描述这类复杂介质中地震波传播规律的波动方程中通常包含多个反映介质不同性质的物性参数，这些物性参数的同时获取对于较为精确地岩性描述和储层预测有着重要的理论和实际意义。当前，全波形反演方法是一种有效的从波动方程中反演物性参数的方法，但是利用全波形反演方法从波动方程中反演得到的物性参数以速度参数为主，反演过程中其他未知的参数被假设为已知，而多参数同时反演可避免这一问题。考虑到高维复杂方程的多参数同时反演难度较大，本论文选择考虑了地震波在介质中传播时的粘滞吸收作用的波动方程（Stokes方程和粘滞性声波方程）来做多参数同时反演，所选择的方程既可较为准确的描述波在介质中的传播，又不太复杂，同时反演得到的物性参数对了解地下的构造展布和油气预测有重要意义。
在论文介绍多参数反演之前，文中首先对基于Stokes方程和2D声波方程的波场正演模拟进行了分析研究。考虑到本论文的多参数反演应用的是频率域全波形反演方法，而实际观测数据是在时间域记录得到的，在应用频率域全波形反演方法进行反演时，需要将时间域的地震记录通过傅氏变换变换到频率域，为此，文中同时给出了时间域和频率域Stokes方程和2D声波方程的有限差分格式，对得到的时间域和频率域波场进行了转换对比。随后，文中将算法在模型上进行了时间域和频率域的正演模拟，并对时间域波场和频率域波场之间的相互转化做了对比分析，其中基于2D声波方程的波场模拟结果表明：低频情况下，直接在频率域计算的波场与通过傅里叶变换从时间域变换到频率域的波场几乎一致；高频情况下，两者在数值上略有差异。
对基于Stokes方程的波场正演模拟进行研究后，文中在频率域用全波形反演方法对Stokes方程多参数反演问题做了研究。论文首先对各参数对目标函数的敏感性做了分析，结果表明密度和速度参数对目标函数的敏感性大于粘滞系数对目标函数的敏感性。论文随后给出了单参数（速度、密度、粘滞系数）反演结果及相关影响因素分析。针对敏感性分析的结果，文中在速度和粘滞系数双参数同时反演时比较了不同的反演策略和不同的步长选择方法，结果表明双参数同时反演策略和抛物线拟合方式计算步长的方法比较好；在随后的密度和速度、密度和粘滞系数同时反演中采用的是同时反演策略和抛物线拟合方式计算步长的方法；在三参数（密度、速度和粘滞系数）同时反演中，文中首先对不同的反演策略进行了比较分析，其中加入Gardner公式约束的三参数同时反演可得到比较好的结果，同时在该约束下，文中对三参数同时反演中的相关影响因素进行了比较分析。
在频率域Stokes方程多参数全波形反演研究中，模型扰动量通过用对角海森矩阵对梯度做尺度化的方式得到，将该算法应用到2D声波方程速度参数的全波形反演时，由于反演问题维度从一维升到二维，相应计算点数增多、计算量增大，计算时间会变的很长，为找到一个适合的优化方法，论文将常用的优化方法（最速下降法、共轭梯度方法、高斯-牛顿方法、拟牛顿方法等）应用于同一模型的测试，并对反演模型从与理论模型的相对误差和计算时间的角度进行了对比分析，结果表明针对所研究的问题，可根据需要选择合适的方法，该研究为后续研究提供优化方法选择上的参考借鉴。另外，针对已有算法中存在的一些问题，文中将一种新的拟牛顿方法-无记忆拟牛顿方法应用于频率域全波形反演中，论文将该方法用于修改后的Marmousi模型和Overthrust模型测试，分别测试了无噪声数据和有噪声数据的情况，并将得到的反演模型与共轭梯度方法得到的反演模型从每次迭代所需的存储量、每次迭代的计算时间和与理论模型的相对误差的角度进行了比较，结果表明：与共轭梯度方法相比，在计算时间和计算量相同的情况下，无记忆拟牛顿方法计算得到的反演模型与理论模型的相对误差更小；尤其在含噪声数据的反演中，无记忆拟牛顿方法所得反演模型更稳定。
在频率域做波动方程正演的优势之一是可以方便的引入描述波吸收衰减的品质因子Q，从而可以对地震波在地下介质中的传播有较为准确的描述；另外，从这类方程中可以反演得到多个物性参数，如利用频率域全波形反演方法从粘滞性声波方程中同时反演密度、速度和Q参数，这些参数可以为准确的了解地下地质构造分布提供依据。
论文将无记忆拟牛顿方法应用于频率域2D粘滞性声波方程多参数全波形反演中。文中首先对各参数对目标函数的敏感性从偏导数波场和目标函数随参数的变化而变化的角度进行了分析，结果表明密度和速度对目标函数的敏感性优于粘滞系数对目标函数的敏感性。在单参数（速度、密度、Q）反演中，文中对不同地震波信息对反演模型的影响做了对比分析；在双参数（密度和速度、速度和Q、密度和Q）同时反演中，文中结果表明，对于与Q组合的双参数反演，当Q-1参数选择了合适的归一化参数时，待反演的两个参数都可以得到与理论模型接近的反演模型；在三参数（速度、密度和Q）同时反演中，文中对比了两种反演策略，结果表明先反演密度和速度参数、再同时反演密度、速度和Q参数的反演策略得到的反演Q模型与理论模型在结构和数值上更为接近。
Stokes方程和粘滞性声波方程的多参数同时反演研究不仅可以得到多个物性参数，为岩性描述和储层预测提供比较准确的信息，同时反演算法的研究还可以为后续更复杂方程的多参数反演奠定基础，也可为其他领域的相关研究提供借鉴。
With the development of technologies for seismic exploration, researches onseismic forward and inverse problems for complex mediun have increased. Waveequations which can precisely model the propagation properties of seismic wave incomplex medium usually contain more than one physical parameters describing theproperties of the complicated medium, and the accurate inversion of these parametersis theoretically and practically significant to the description of rock features andreservoir prediction. Nowadays, full waveform inversion (FWI) is an efficient methodto obtain parameters from wave equations, and the most commonly invertedparameter is velocity where in the inversion process the other parameters are set to beknown but they are unknown actually. This problem can be avoided if theseparameters are inverted simultaneously. However, the intrinsic difficulties ofmultiparameter inversion make it not easy to implement the simultaneous inversion ofmore than one parameter. Considering the importance and difficulties ofmultiparameter inversion, wave equations, such as Stokes equation and visco-acousticequation, which have taken the viscous property of the medium into considerationwhen the wave propagation is modeled, are used to test the multiparameter inversion.The two wave equations are chosen because they can not only describe the wavepropagation accurately but also not too complex. Besides, the parameters obtained areimportant and helpful for the description of tectonic distribution and the prediction ofoil and gas.
Before the multiparameter inversion is started, the forward modeling of Stokesequation and2D acoustic equation is analyzed first. Since frequency domain FWI isused in the multiparameter inversion, and the observed seismic data are recorded inthe time domain, time domain seismic records should be transformed into the frequency domain by fast Fourier Transformation (FFT) in frequency domain FWI.Based on that, formulas of finite-difference time domain (FDTD) and finite-differencefrequency domain (FDFD) about Stokes equation and2D acoustic equation are given.Wavefields directly computed in frequency domain are compared with that obtainedby transforming the time domain wavefields to frequency domain by FFT. Then theformulas are tested on synthetic models to obtain the time domain and frequencydomain wavefields, and comparisons about the transformations of the wavefieldsbetween the domains are given. The comparison results of2D acoustic wavefieldsshow that, for low frequency, wavefields directly computed in frequency domain arenearly the same as that transformed from the time domain by FFT while for highfrequency there are some differences in the values.
After the forward modeling of Stokes equation is analyzed, multiparameterinversion from Stokes equation by frequency domain FWI is researched. Thesensitivity analysis of the parameters to the misfit function are first given before themultiparameter inversion is implemented, and the results show that the sensitivities ofdensity and velocity to misfit function are higher to that of viscosity coefficient tomisfit function. Then mono-parameter inversion (i.e. velocity, density, viscositycoefficient) is implemented and the influences of some relative factors are analyzed.According to the sensitivity analysis, different inversion strategies and step lengthselection methods are compared in the simultaneous inversion of velocity andviscosity coefficient, and the results show that simultaneous inversion strategy andparabolic step length computing method are the best, which are used in thesimultaneous inversion of density and velocity as well as density and viscositycoefficient. In the three parameter inversion (i.e. density, velocity and viscositycoefficient), different inversion strategies are compared, and the results show that thestrategy constrained by the Gardner formula can obtain acceptable inversion results.With the same Gardner constraint, influences of different factors are compared andanalyzed.
The gradient is scaled by the diagonal approximate Hessian matrix in themultiparameter inversion from Stokes equation by frequency domain FWI. When this method is extended to the inversion of2D acoustic equation, it seems to be tootime-consuming because the computation grids increases and the computationstorages are large with the dimension extended from one to two. To find a properoptimization method, commonly used optimization methods, such as gradientmethods and Newtonian methods are tested on the same synthetic model. Thereconstructed models by these methods are compared and analyzed from the aspectsof relative error and computation time, and the results show that proper optimizationmethod should be chosen according to the problem to be solved. Then a newquasi-Newton method named memoryless quasi-Newton (MLQN) method is appliedin frequency domain FWI to invert velocity from surface seismic data for the firsttime. This method can attain acceptable results with low computational cost and smallmemory storage requirements. To test the efficiency of the MLQN method in FWI,two synthetic models, a modified Marmousi model and a modified overthrust model,are examined from the surface seismic data with and without white Gaussian noise.For comparison, the conjugate gradient (CG) method is carried out for the samevelocity models with the same parameters. The inverted velocities by the two methodsare compared based on the aspects of memory storage requirements, computation timefor each iteration, and error. By keeping the memory storage requirements andcomputation time in each iteration similar, the reconstructed velocity models obtainedusing the MLQN method are closer to the true velocity models than those obtainedusing the CG method, especially for the noise-added data. The numerical tests showthat the MLQN method is feasible and reliable in FWI.
In this thesis, MLQN method is applied in the multiparameter inversion from2Dvisco-acoustic wave equation by frequency domain FWI. Sensitivity analysis of theparameters to the misfit function is given from the aspects of partial derivationwavefields and the variation of the misfit function with the variation of the parameters,and the results show that density and velocity are more sensitive to the misfit functionthan Q. In the mono-parameter inversion (i.e. velocity, density, Q), different seismicinformation is used in the inversion and their influences to the rebuilt models aregiven. As for the two-parameter inversion (i.e. density and velocity, velocity and Q, density and Q), the results show that for parameter couples with Q, acceptablereconstructed models can obtained once appropriate normed Q-1is selected. In thethree parameter inversion (i.e. density, velocity and Q), two inversion strategies arecompared: the first inversion strategy is that the three parameters are inverted at thesame time; the second inversion strategy is that in the first stage, density and velocityare inverted with Q being the initial model and in the second stage, with the initialdensity and velocity model being the models obtained in the first stage, the threeparameters are inverted simultaneously. The results show that the inverted Q model isin better accordance with the true Q models when the second inversion strategy isused.
The simultaneous multiparameter inversion from Stokes equation andvisco-acoustic equation by frequency domain FWI can not only obtain more than oneparameters, supplying more information for the description of rock property andprediction of reservoirs, but also lay foundations for the research on themultiparameter inversion from more complicated equations. Besides, the research ofmultiparameter inversion from seismic data can also supply reference to the samequestion in other fields.
在论文介绍多参数反演之前，文中首先对基于Stokes方程和2D声波方程的波场正演模拟进行了分析研究。考虑到本论文的多参数反演应用的是频率域全波形反演方法，而实际观测数据是在时间域记录得到的，在应用频率域全波形反演方法进行反演时，需要将时间域的地震记录通过傅氏变换变换到频率域，为此，文中同时给出了时间域和频率域Stokes方程和2D声波方程的有限差分格式，对得到的时间域和频率域波场进行了转换对比。随后，文中将算法在模型上进行了时间域和频率域的正演模拟，并对时间域波场和频率域波场之间的相互转化做了对比分析，其中基于2D声波方程的波场模拟结果表明：低频情况下，直接在频率域计算的波场与通过傅里叶变换从时间域变换到频率域的波场几乎一致；高频情况下，两者在数值上略有差异。
对基于Stokes方程的波场正演模拟进行研究后，文中在频率域用全波形反演方法对Stokes方程多参数反演问题做了研究。论文首先对各参数对目标函数的敏感性做了分析，结果表明密度和速度参数对目标函数的敏感性大于粘滞系数对目标函数的敏感性。论文随后给出了单参数（速度、密度、粘滞系数）反演结果及相关影响因素分析。针对敏感性分析的结果，文中在速度和粘滞系数双参数同时反演时比较了不同的反演策略和不同的步长选择方法，结果表明双参数同时反演策略和抛物线拟合方式计算步长的方法比较好；在随后的密度和速度、密度和粘滞系数同时反演中采用的是同时反演策略和抛物线拟合方式计算步长的方法；在三参数（密度、速度和粘滞系数）同时反演中，文中首先对不同的反演策略进行了比较分析，其中加入Gardner公式约束的三参数同时反演可得到比较好的结果，同时在该约束下，文中对三参数同时反演中的相关影响因素进行了比较分析。
在频率域Stokes方程多参数全波形反演研究中，模型扰动量通过用对角海森矩阵对梯度做尺度化的方式得到，将该算法应用到2D声波方程速度参数的全波形反演时，由于反演问题维度从一维升到二维，相应计算点数增多、计算量增大，计算时间会变的很长，为找到一个适合的优化方法，论文将常用的优化方法（最速下降法、共轭梯度方法、高斯-牛顿方法、拟牛顿方法等）应用于同一模型的测试，并对反演模型从与理论模型的相对误差和计算时间的角度进行了对比分析，结果表明针对所研究的问题，可根据需要选择合适的方法，该研究为后续研究提供优化方法选择上的参考借鉴。另外，针对已有算法中存在的一些问题，文中将一种新的拟牛顿方法-无记忆拟牛顿方法应用于频率域全波形反演中，论文将该方法用于修改后的Marmousi模型和Overthrust模型测试，分别测试了无噪声数据和有噪声数据的情况，并将得到的反演模型与共轭梯度方法得到的反演模型从每次迭代所需的存储量、每次迭代的计算时间和与理论模型的相对误差的角度进行了比较，结果表明：与共轭梯度方法相比，在计算时间和计算量相同的情况下，无记忆拟牛顿方法计算得到的反演模型与理论模型的相对误差更小；尤其在含噪声数据的反演中，无记忆拟牛顿方法所得反演模型更稳定。
在频率域做波动方程正演的优势之一是可以方便的引入描述波吸收衰减的品质因子Q，从而可以对地震波在地下介质中的传播有较为准确的描述；另外，从这类方程中可以反演得到多个物性参数，如利用频率域全波形反演方法从粘滞性声波方程中同时反演密度、速度和Q参数，这些参数可以为准确的了解地下地质构造分布提供依据。
论文将无记忆拟牛顿方法应用于频率域2D粘滞性声波方程多参数全波形反演中。文中首先对各参数对目标函数的敏感性从偏导数波场和目标函数随参数的变化而变化的角度进行了分析，结果表明密度和速度对目标函数的敏感性优于粘滞系数对目标函数的敏感性。在单参数（速度、密度、Q）反演中，文中对不同地震波信息对反演模型的影响做了对比分析；在双参数（密度和速度、速度和Q、密度和Q）同时反演中，文中结果表明，对于与Q组合的双参数反演，当Q-1参数选择了合适的归一化参数时，待反演的两个参数都可以得到与理论模型接近的反演模型；在三参数（速度、密度和Q）同时反演中，文中对比了两种反演策略，结果表明先反演密度和速度参数、再同时反演密度、速度和Q参数的反演策略得到的反演Q模型与理论模型在结构和数值上更为接近。
Stokes方程和粘滞性声波方程的多参数同时反演研究不仅可以得到多个物性参数，为岩性描述和储层预测提供比较准确的信息，同时反演算法的研究还可以为后续更复杂方程的多参数反演奠定基础，也可为其他领域的相关研究提供借鉴。
With the development of technologies for seismic exploration, researches onseismic forward and inverse problems for complex mediun have increased. Waveequations which can precisely model the propagation properties of seismic wave incomplex medium usually contain more than one physical parameters describing theproperties of the complicated medium, and the accurate inversion of these parametersis theoretically and practically significant to the description of rock features andreservoir prediction. Nowadays, full waveform inversion (FWI) is an efficient methodto obtain parameters from wave equations, and the most commonly invertedparameter is velocity where in the inversion process the other parameters are set to beknown but they are unknown actually. This problem can be avoided if theseparameters are inverted simultaneously. However, the intrinsic difficulties ofmultiparameter inversion make it not easy to implement the simultaneous inversion ofmore than one parameter. Considering the importance and difficulties ofmultiparameter inversion, wave equations, such as Stokes equation and visco-acousticequation, which have taken the viscous property of the medium into considerationwhen the wave propagation is modeled, are used to test the multiparameter inversion.The two wave equations are chosen because they can not only describe the wavepropagation accurately but also not too complex. Besides, the parameters obtained areimportant and helpful for the description of tectonic distribution and the prediction ofoil and gas.
Before the multiparameter inversion is started, the forward modeling of Stokesequation and2D acoustic equation is analyzed first. Since frequency domain FWI isused in the multiparameter inversion, and the observed seismic data are recorded inthe time domain, time domain seismic records should be transformed into the frequency domain by fast Fourier Transformation (FFT) in frequency domain FWI.Based on that, formulas of finite-difference time domain (FDTD) and finite-differencefrequency domain (FDFD) about Stokes equation and2D acoustic equation are given.Wavefields directly computed in frequency domain are compared with that obtainedby transforming the time domain wavefields to frequency domain by FFT. Then theformulas are tested on synthetic models to obtain the time domain and frequencydomain wavefields, and comparisons about the transformations of the wavefieldsbetween the domains are given. The comparison results of2D acoustic wavefieldsshow that, for low frequency, wavefields directly computed in frequency domain arenearly the same as that transformed from the time domain by FFT while for highfrequency there are some differences in the values.
After the forward modeling of Stokes equation is analyzed, multiparameterinversion from Stokes equation by frequency domain FWI is researched. Thesensitivity analysis of the parameters to the misfit function are first given before themultiparameter inversion is implemented, and the results show that the sensitivities ofdensity and velocity to misfit function are higher to that of viscosity coefficient tomisfit function. Then mono-parameter inversion (i.e. velocity, density, viscositycoefficient) is implemented and the influences of some relative factors are analyzed.According to the sensitivity analysis, different inversion strategies and step lengthselection methods are compared in the simultaneous inversion of velocity andviscosity coefficient, and the results show that simultaneous inversion strategy andparabolic step length computing method are the best, which are used in thesimultaneous inversion of density and velocity as well as density and viscositycoefficient. In the three parameter inversion (i.e. density, velocity and viscositycoefficient), different inversion strategies are compared, and the results show that thestrategy constrained by the Gardner formula can obtain acceptable inversion results.With the same Gardner constraint, influences of different factors are compared andanalyzed.
The gradient is scaled by the diagonal approximate Hessian matrix in themultiparameter inversion from Stokes equation by frequency domain FWI. When this method is extended to the inversion of2D acoustic equation, it seems to be tootime-consuming because the computation grids increases and the computationstorages are large with the dimension extended from one to two. To find a properoptimization method, commonly used optimization methods, such as gradientmethods and Newtonian methods are tested on the same synthetic model. Thereconstructed models by these methods are compared and analyzed from the aspectsof relative error and computation time, and the results show that proper optimizationmethod should be chosen according to the problem to be solved. Then a newquasi-Newton method named memoryless quasi-Newton (MLQN) method is appliedin frequency domain FWI to invert velocity from surface seismic data for the firsttime. This method can attain acceptable results with low computational cost and smallmemory storage requirements. To test the efficiency of the MLQN method in FWI,two synthetic models, a modified Marmousi model and a modified overthrust model,are examined from the surface seismic data with and without white Gaussian noise.For comparison, the conjugate gradient (CG) method is carried out for the samevelocity models with the same parameters. The inverted velocities by the two methodsare compared based on the aspects of memory storage requirements, computation timefor each iteration, and error. By keeping the memory storage requirements andcomputation time in each iteration similar, the reconstructed velocity models obtainedusing the MLQN method are closer to the true velocity models than those obtainedusing the CG method, especially for the noise-added data. The numerical tests showthat the MLQN method is feasible and reliable in FWI.
In this thesis, MLQN method is applied in the multiparameter inversion from2Dvisco-acoustic wave equation by frequency domain FWI. Sensitivity analysis of theparameters to the misfit function is given from the aspects of partial derivationwavefields and the variation of the misfit function with the variation of the parameters,and the results show that density and velocity are more sensitive to the misfit functionthan Q. In the mono-parameter inversion (i.e. velocity, density, Q), different seismicinformation is used in the inversion and their influences to the rebuilt models aregiven. As for the two-parameter inversion (i.e. density and velocity, velocity and Q, density and Q), the results show that for parameter couples with Q, acceptablereconstructed models can obtained once appropriate normed Q-1is selected. In thethree parameter inversion (i.e. density, velocity and Q), two inversion strategies arecompared: the first inversion strategy is that the three parameters are inverted at thesame time; the second inversion strategy is that in the first stage, density and velocityare inverted with Q being the initial model and in the second stage, with the initialdensity and velocity model being the models obtained in the first stage, the threeparameters are inverted simultaneously. The results show that the inverted Q model isin better accordance with the true Q models when the second inversion strategy isused.
The simultaneous multiparameter inversion from Stokes equation andvisco-acoustic equation by frequency domain FWI can not only obtain more than oneparameters, supplying more information for the description of rock property andprediction of reservoirs, but also lay foundations for the research on themultiparameter inversion from more complicated equations. Besides, the research ofmultiparameter inversion from seismic data can also supply reference to the samequestion in other fields.
引文
[1]杨文采.地球物理反演的理论与方法[M].北京:地质出版社,1997.
[2]李世雄,刘家琦.小波变换和反演数学基础[M].北京:地质出版社,1994.
[3]姚姚.地球物理反演基本理论与应用方法[M].武汉:中国地质大学出版社,2002.
[4]宋海斌,张关泉.层状介质弹性参数反演问题研究综述[J].地球物理学进展,1998,13(4):67-78.
[5] LIU C, GAO F, FENG X, et al. Incorporating attenuation effects into frequency-domain fullwaveform inversion from zero-offset VSP data from Stokes equation[J]. Journal ofGeophysics and Engineering,2013,10(3):035004.
[6] VIRIEUX J, OPERTO S. An overview of full-waveform inversion in explorationgeophysics[J]. Geophysics,2009,74(6): WCC1-WCC26.
[7] TARANTOLA A. Inversion of seismic reflection data in the acoustic approximation[J].Geophysics,1984,49(8):1259-1266.
[8] GAUTHIER O, VIRIEUX J, TARANTOLA A. Two-dimensional nonlinear inversion ofseismic waveforms:numerical results[J]. Geophysics,1986,51(7):1387-1403.
[9] PICA A, DIET J P, TARANTOLAR A. Nonlinear inversion of seismic reflection data in alaterally invariant medium[J]. Geophysics,1990,55(3):284-292.
[10] CRASE E, PICAT A, NOBLE M, et al. Robust elastic nonlinear waveforminversion:Application to real data[J]. Geophysics,1990,55(5):527-538.
[11] MORA P. Nonlinear two-dimensional elastic inversion of multioffset seismic data[J].Geophysics,1987,52(9):1211-1228.
[12] MORA P. Elastic wave-field inversion of reflection and transmission data[J]. Geophysics,1988,53(6):750-759.
[13] TARANTOLA A. A strategy for nonlinear elastic inversion of seismic reflection data[J].Geophysics,1986,51(10):1893-1903.
[14] VIGH D, STARR E W. Comparisons for Waveform Inversion, Time Domain or FrequencyDomain?[J]. SEG Las Vegas2008Annual Meeting,2008,1890-1894.
[15] PRATT R G, WORTHINGTON M H. Inverse theory application to multi-source cross-holetomography.Part1: acoustic wave equation method[J]. Geophysical Prospecting,1990,38(3):287-310.
[16] PRATT R G. Inverse theory application to multi-source cross-hole tomography.Part2:elastic wave equation method[J]. Geophysical Prospecting,1990,38(3):311-329.
[17] PRATT R G, C.SHIN, G.J.HICKS. Gauss-Newton and full Newton methods infrequency-space seismic waveform inversion[J]. Geophysical Journal International,1998,133(2):341-362.
[18] PRATT R G. Seismic waveform inversion in the frequency domain，Part1：Theory andverification in a physical scale model[J]. Geophysics,1999,64(3):888-901.
[19] PRATT R G, SHIPP R M. Seismic waveform inversion in the frequency domain，Part2：Fault delineation in sediments using crosshole data[J]. Geophysics,1999,64(3):902-914.
[20] SHIN C, CHA Y H. Waveform inversion in the Laplace domain[J]. Geophysical JournalInternational,2008,173(3):922–931.
[21] PYUN S, SHIN C, LEE H.3D elastic full waveform inversion in the Laplace domain[J].SEG Las Vegas2008Annual Meeting,2008,1976-1980.
[22] CHA Y H, SHIN C.2D Laplace-domain waveform inversion using adaptive finite elementmethod[J]. SEG Houston2009International Exposition and Annual Meeting,2009,2263-2268.
[23] KOO N H, SHIN C, CHA Y H, et al. Sequentially ordered single-frequency2-D acousticwaveform inversion in the Laplace-Fourier domain[J]. SEG Houston2009InternationalExposition and Annual Meeting,2009,2248-2252.
[24] MARFURT K J. Accuracy of finite-difference and finite-element modeling of the scalar andelastic wave equations[J]. Geophysics,1984,49(5):533-549.
[25] VIRIEUX J. W-wave propagation in heterogeneous media: Velocity-stress finite-differencemethod[J]. Geophysics,1984,49(11):1933-1957.
[26] VIRIEUX J. P-SV wave propagation in heterogeneous media:Velocity-stressfinite-difference method[J]. Geophysics,1986,51(4):889-901.
[27] LEVANDER A R. Fourth-order finite-difference P-SV seismograms[J]. Geophysics,1988,53(11):1425-1436.
[28] PRATT R G. Frequency-domain elastic wave modeling by finite differences:A tool forcrosshole seismic imaging[J]. Geophysics,1990,55(5):626-632.
[29] JO C H, SHIN C, SUH J H. An optimal9-point, finite-difference, frequency-space,2-Dscalar wave extrapolator[J]. Geophysics,1996,61(2):529-537.
[30] SHIN C, SOHN H. A frequency-space2-D scalar wave extrapolator using extended25-point finite-difference operator[J]. Geophysics,1998,63(1):289-296.
[31] STEKL I, PRATT R G. Accurate viscoelastic modeling by frequency-domain finitedifferences using rotated operators[J]. Geophysics,1998,63(5):1779-1794.
[32] MIN D J, SHINZ C, KWON B-D, et al. Improved frequency-domain elastic wave modelingusing weighted-averaging difference operators[J]. Geophysics,2000,65(3):884-895.
[33] MIN D J, YOO H S, SHIN C, et al. Weighting-averaging finite-element method for scalarwave equation in the frequency domain[J]. Journal of Seismic Exploration,2002,11:197-222.
[34] HUSTEDT B, OPERTO S, VIRIEUX J. Mixed-grid and staggered-grid finite-differencemethods for frequency-domain acoustic wave modeling[J]. Geophysical JournalInternational,2004,157(3):1269-1296.
[35] OPERTO S, VIRIEUX J, AMESTOY P, et al.3D finite-difference frequency-domainmodeling of visco-acoustic wave propagation using a massively parallel direct solver: Afeasibility study[J]. Geophysics,2007,72(5): SM195-SM211.
[36] BROSSIER R, VIRIEUX J, OPERTO S. Parsimonious finite-volume frequency-domainmethod for2D P-SV-wave modelling[J]. Geophysical Journal International,2008,175(2):541-559.
[37] OPERTO S, VIRIEUX J, RIBODETTI A, et al. Finite-difference frequency-domainmodeling of viscoacoustic wave propagation in2D tilted transversely isotropic (TTI)media[J]. Geophysics,2009,74(5): T75-T95.
[38]殷文,印兴耀,吴国忱.高精度频率域弹性波方程有限差分方法及波场模拟[J].地球物理学报,2006,49(2):561-568.
[39]吴国忱,梁楷. VTI介质频率-空间域准P波正演模拟[J].石油地球物理勘探,2005,40(5):535-545.
[40]梁锴,吴国忱,印兴耀. T T I介质qP波方程频率-空间域加权平均有限差分算子[J].石油地球物理勘探,2007,42(5):516-525.
[41]吴国忱,罗彩明,梁楷.TTI介质弹性波频率-空间域有限差分数值模拟[J].吉林大学学报（地球科学版）,2007,37(5):1023-1033.
[42]刘璐,刘洪,刘红伟.优化15点频率-空间与有限差分正演模拟[J].地球物理学报,2013,56(2):644-652.
[43] GU B, LIANG G, LI Z. A21-point finite difference scheme for2D frequency-domainelastic wave modelling[J]. Exploration Geophysics,2013,44(3):156-166.
[44]卞爱飞,於文辉,周华伟.频率域全波形反演方法研究进展[J].地球物理学进展,2010,25(3):982-993.
[45] BROSSIER R, OPERTO S, VIRIEUX J. Which data residual norm for robust elasticfrequency-domain Full Waveform Inversion?[J]. Geophysics,2010,75(3): R37–R46.
[46] SHIN C, MIN D J. Waveform inversion using a logarithmic wavefield[J]. Geophysics,2006,71(3): R31–R42.
[47] SHIN C, PYUN S, BEDNAR J B. Comparison of waveform inversion, part1: conventionalwavefield vs logarithmic wavefield[J]. Geophysical Prospecting,2007,55(4):449-464.
[48] BEDNAR J B, SHIN C, PYUN S. Comparison of waveform inversion, part2: phaseapproach[J]. Geophysical Prospecting,2007,55(4):465-475.
[49] PYUN S, SHIN C, BEDNAR J B. Comparison of waveform inversion, part3: amplitudeapproach[J]. Geophysical Prospecting,2007,55(4):477-485.
[50] SHI Y M, ZHAO W Z, CAO H. Nonlinear process control of wave-equation inversion andits application in the detection of gas[J]. Geophysics,2007,72(1): R9-R18.
[51] HU W, ABUBAKAR A, HABASHY T M. Simultaneous multifrequency inversion offull-waveform seismic data[J]. Geophysics,2009,74(2): R1-R14.
[52] BAE H S, PYUN S, CHUNG W, et al. Frequency-domain acoustic-elastic coupledwaveform inversion using the Gauss-Newton conjugate gradient method[J]. GeophysicalProspecting,2012,60(3):413-432.
[53] RAVAUT C, OPERTO S, IMPROTA L, et al. Multiscale imaging of complex structuresfrom multifold wide-aperture seismic data by frequency-domain full-waveform tomography:application to a thrust belt[J]. Geophysical Journal International,2004,159(3):1032-1056.
[54] SOURBIER F, OPERTO S, VIRIEUX J. FWT2D: a massively parallel program forfrequency-domain Full-Waveform Tomography of wide-aperture seismic data-Part1:algorithm[J]. Computer&Geosciences,2009,35(3):487-495.
[55] SOURBIER F, OPERTO S, VIRIEUX J. FWT2D:Amassively parallel program forfrequency-domain full-waveform tomography of wide-aperture seismic data—Part2Numerical examples and scalability analysis[J]. Computer&Geosciences,2009,35(3):496-514.
[56]刘国峰,刘洪,孟小红,等.频率域波形反演中与频率相关的影响因素分析[J].地球物理学报,2012,55(4):1345-1353.
[57] JEONG W, LEE H Y, MIN D J. Full waveform inversion strategy for density in thefrequency domain[J]. Geophysical Journal International,2012,188(3):1221-1242.
[58]朱童,李小凡,汪文帅,等.粒子群-梯度算法在频率域地震波形反演中的应用[J].地球物理学进展,2013,28(1):180-189.
[59] NOCEDAL J, WRIGHT S J. Numerical Optimization [M]. New York: Springer-Verlag NewYork,Inc,2006.
[60]刘璐,刘洪,张衡.基于修正拟牛顿公式的全波形反演[J].地球物理学报,2013,56(7):2447-2451.
[61] LIN Y, ABUBAKAR A, HABASHY T M. Seismic full-waveform inversion using truncatedwavelet representations[J]. SEG Extended Abstracts,2012,
[62] MA Y, HALE D. Quasi-Newton full-waveform inversion with a projected Hessian matrix[J].Geophysics,2012,77(5): R207-R216.
[63] MA Y, HALE D, GONG B, et al. Image-guided sparse-model full waveform inversion[J].Geophysics,2012,77(4): R189-R198.
[64] BROSSIER R, OPERTO S, VIRIEUX J. Seismic imaging of complex onshore structures by2D elastic frequency-domain full-waveform inversion[J]. Geophysics,2009,74(6):WCC105-WCC118.
[65] MALINOWSKI M, OPERTO S, RIBODETTI A. High-resolution seismic attenuationimaging from wide-aperture onshore data by visco-acoustic frequency-domainfull-waveform inversion[J]. Geophysical Journal International,2011,186(3):1179-1204.
[66] GUITTON A, AYENI G, D AZ E. Constrained full-waveform inversion by modelreparameterization[J]. Geophysics,2012,77(2): R117-R127.
[67] ANAGAW A Y, SACCHI M D. Full waveform inversion with simultaneous sources usingthe full Newton Method[J]. SEG,2012,1-5.
[68] M TIVIER L, BROSSIER R, VIRIEUX J, et al. Toward Gauss-Newton and Exact NewtonOptimization for Full Waveform Inversion[J].74th EAGE Conference&Exhibition,Extended Abstracts,2012, P016.
[69] BROSSIER R. Two-dimensional frequency-domain visco-elastic full waveforminversion:Parallel algorithms,optimization and performance[J]. Computers&Geosciences,2011,37(4):444-455.
[70] K HN D, NIL DE D, KURZMANN A, et al. On the influence of model parametrization inelastic full waveform tomography[J]. Geophysical Journal International,2012,191(1):325-345.
[71] RAO Y, WANG Y. Fracture effects in seismic attenuation images reconstructed bywaveform tomography[J]. Geophysics,2009,74(4): R25-R34.
[72] BARNES C, CHARARA M, TSUCHIYA T. Feasibility study for an anisotropic fullwaveform inversion of cross-well seismic data[J]. Geophysical Prospecting,2008,56(6):897-906.
[73] PLESSIX R E, CAO Q. A parameterization study for surface seismic full waveforminversion in an acoustic vertical transversely isotropic medium[J]. Geophysical JournalInternational,2011,185(1):539–556.
[74] LEE H Y, KOO J M, MIN D J. Frequency-domain elastic full waveform inversion for VTImedia[J]. Geophysical Journal International,2010,183(2):884-904.
[75] GHOLAMI Y, BROSSIER R, OPERTO S, et al. Which parameterization is suitable foracoustic vertical transverse isotropic full waveform inversion? Part1: Sensitivity andtrade-off analysis[J]. Geophysics,2013,78(2): R81–R105.
[76] GHOLAMI Y, BROSSIER R, OPERTO S, et al. Which parameterization is suitable foracoustic vertical transverse isotropic full waveform inversion? Part2: Synthetic and realdata case studies from Valhall[J]. Geophysics,2013,78(2): R107–R124.
[77] PRIEUX V, BROSSIER R, OPERTO S E, et al. Multiparameter full waveform inversion ofmulticomponent ocean-bottom-cable data from the Valhall field. Part1: imagingcompressional wave speed, density and attenuation[J]. Geophysical Journal International,2013,194(3):1640-1664.
[78] PRIEUX V, BROSSIER R, OPERTO S E, et al. Multiparameter full waveform inversion ofmulticomponent ocean-bottom-cable data from the Valhall field. Part2: imagingcompressive-wave and shear-wave velocities[J]. Geophysical Journal International,2013,194(3):1665–1681.
[79] OPERTO S, GHOLAMI Y, BROSSIER R, et al. A guided tour of multi-parameter fullwaveform inversion with multi-component data: from theory to practice[J]. The LeadingEdge,2013,32(9):1040-1054.
[80] BARROS L D, DIETRICH M, VALETTE B. Full waveform inversion of seismic wavesreflected in a stratified porous medium[J]. Geophysical Journal International,2010,182(3):1543-1556.
[81] WARNER M, RATCLIFFE A, NANGOO T, et al. Anisotropic3D full-waveforminversion[J]. Geophysics,2013,78(2): R59–R80.
[82] PRIEUX V, LAMBAR G, OPERTO S, et al. Building starting models for full waveforminversion from wide-aperture data by stereotomography[J]. Geophysical Prospecting,2013,61(s1):109-137.
[83] SHIN C, HA W. A comparison between the behavior of objective functions for waveforminversion in the frequency and Laplace domains[J]. Geophysics,2008,73(5): VE119–VE133.
[84] SHIN C, CHA Y H. Waveform inversion in the Laplace–Fourier domain[J]. GeophysicalJournal International,2009,177(3):1067–1079.
[85] PRATT R G, HOU F, BAUER K, et al. Waveform tomography images of velocity andinelastic attenuation from the Mallik2002crosshole seismic surveys[J]. Bulletin-Geological Survey of Canada,2005,585(1-14.
[86] WANG Y, RAO Y. Crosshole seismic waveform tomography–I. Strategy for real dataapplication[J]. Geophysical Journal International,2006,166(3):1224–1236.
[87] RAO Y, WANG Y, MORGAN J V. Crosshole seismic waveform tomography–II.Resolution analysis[J]. Geophysical Journal International,2006,166(3):1237-1248.
[88] PRATT R G, PLESSIX R E, W.A.MULDER. Seismic waveform tomography the effect ofalyering and anisotrop[J]. EAGE63rd Conference&Technical Exhibition,2001, P092.
[89] FICHTNER A, TRAMPERT J, CUPILLARD P, et al. Multiscale full waveform inversion[J].Geophysical Journal International,2013,194(1):534-556.
[90] BRENDERS A J, PRATT R G. Full waveform tomography for lithospheric imaging: Resultsfrom a blind test in a realistic crustal model[J]. Geophysical Journal International,2007,168(1):133-151.
[91] BRENDERS A J, PRATT R G. Efficient waveform tomography for lithospheric imaging:implications for realistic, two-dimensional acquisition geometries and low-frequency data[J].Geophysical Journal International,2007,168(1):152–170.
[92] JAISWAL P, ZELT C A, DASGUPTA R, et al. Seismic imaging of the Naga thrust usingmultiscale waveform inversion[J]. Geophysics,2009,74(6): WCC129–WCC140.
[93] BLEIBINHAUS F, HILBERG S. Shape and structure of the Salzach Valley, Austria, fromseismic traveltime tomography and full waveform inversion[J]. Geophysical JournalInternational,2012,189(3):1701–1716.
[94] BAETEN G, MAAG J W D, PLESSIX R E-E, et al. The use of low frequencies in afull-waveform inversion and impedance inversion land seismic case study[J]. GeophysicalProspecting,2013,61(4):701-711.
[95] DESSA J X, OPERTO S, KODAIRA S, et al. Multiscale seismic imaging of the easternNankai trough by full waveform inversion[J]. gEOPHYSICAL Research Letters,2004,31(18):1-4.
[96] MORGAN J, WARNER M, BELL R, et al. Next-generation seismic experiments:wide-angle, multi-azimuth, three-dimensional, full-waveform inversion[J]. GeophysicalJournal International,2013,195(3):1657-1678.
[97] OPERTO S, RAVAUT C, IMPROTA L, et al. Quantitative imaging of complex structuresfrom dense wideaperture seismic data by multiscale traveltime and waveform inversions:A case study[J]. Geophysical Prospecting,2005,52(6):625–651.
[98] TAKOUGANG E M T, CALVERT A J. Application of waveform tomography to marineseismic reflection data from the Queen Charlotte Basin of western Canada[J]. Geophysics,2011,76(2): B55–B70.
[99] TAKOUGANG E M T, CALVERT A J. Seismic velocity and attenuation structures of theQueen Charlotte Basin from full-waveform tomography of seismic reflection data[J].Geophysics,2012,77(3): B107–B124.
[100] SMITHYMAN B R, CLOWES R M. Waveform tomography of field vibroseis data using anapproximate2D geometry leads to improved velocity models[J]. Geophysics,2012,77(1):R33–R43.
[101]陈小宏,牟永光.二维地震资料波动方程非线性反演[J].地球物理学报,1996,39(3):401-408.
[102]张霖斌,姚振兴.层状介质的声波波动方程反演[J].地球物理学进展,2000,15(2):22-29.
[103]许琨,王妙月.声波方程频率域有限元参数反演[J].地球物理学报,2001,44(4):852-864.
[104]许琨,王妙月.利用地质规则块体建模方法的频率域有限元弹性波速度反演[J].地球物理学报,2004,47(4):708-717.
[105]丁继才,常旭,刘伊克.反射地震数据的逐层波形反演[J].地球物理学报,2007,50(2):574-580.
[106]胡光辉,贾春梅,夏洪瑞,等.三维声波全波形反演的实现与验证[J].石油物探,2013,52(4):417-425.
[107] N.H.瑞克,许云译,吴律校.粘弹性介质中的地震波[M].北京:地质出版社,1981.
[108]刘财,谢靖,韩道范,等.非均匀粘弹性介质中波传播的几个基本问题[J].长春科技大学学报,2000,30(专辑):93-96.
[109]刘财,谢靖,韩道范,等.用Stokes方程的差分方法制作合成地震记录[J].石油地球物理勘探,2002,37(3):230-236.
[110] CERJAN C, KOSLOF D, KOSLOFF R, et al. A nonreflecting boundary condition fordiscrete acoustic and elastic wave equations[J]. Geophysics,1985,50(4):705-708.
[111] OPERTO S, VIRIEUX J, SOURBIER F. Documentation of FWT2D program (version4.8):Frequency-domain full-waveform modeling/inversion of wide-aperture seismic data forimaging2D acoustic media (Technical report N0007-SEISCOPE project).2007.
[112]刘财,陈业全,刘洋,等.勘探地震资料处理新方法及新技术[M].北京:科学出版社,2006.
[113] VIGH D, STARR E W, KAPOOR J. Developing earth models with full waveforminversion[J]. The Leading Edge,2009,28(4):432-435.
[114] KENNETT B L N, SAMBRIDGE M S, WILLIAMSON P R. Subspace methods for largeinverse problems with multiple parameter classes[J]. Geophysical Journal,1988,94(2):237-247.
[115] SAMBRIDGE M S, TARANTOLA A, KENNETT B L N. An alternative strategy fornon-linear inversion of seismic waveforms[J]. Geophysical Prospecting,1991,39(6):723-736.
[116]马中高,解吉高.岩石的纵、横波速度与密度的规律研究[J].地球物理学进展,2005,20(4):905-910.
[117] SIRGUE L, PRATT R G. Efficient waveform inversion and imaging: A strategy forselecting temporal frequencies[J]. Geophysics,2004,69(1):231-248.
[118] KIM Y, CHO H, MIN D, et al. Comparison of frequency-selection strategies for2Dfrequency-domain acoustic waveform inversion[J]. Pure and Applied Geophysics,2011,168(10):1715-1727.
[119]何玉梅,郑天愉.利用地震波形反演研究震源破裂时空过程[J].地球物理学报,1998,41(2):281-289.
[120]吴建平,明跃红,王椿镛.川滇地区速度结构的区域地震波形反演研究[J].地球物理学报,2006,49(5):1369-1376.
[121]李少华,王彦宾,梁子斌.甘肃东南部地壳速度结构的区域地震波形反演[J].地球物理学报,2012,55(4):1186-1197.
[122] DENNIS J E, SCHNABEL R B. Numerical Methods for Unconstrained Optimization andNonlinear Equations [M]. Philadelphia: Society for Industrial and Applied Mathematics,1996.
[123]高凤霞,刘财,冯晅,等.几种优化方法在频率域全波形反演中的应用效果及对比分析研究[J].地球物理学进展,2013,28(4):2060-2068.
[124] SHANNO D F. Conjugate gradient methods with inexact searches[J]. Mathmatics ofOperations Research,1978,3(3):244-256.
[125]尉继英.一种求解无约束极值问题的无记忆拟牛顿算法[J].计算数学,1990,(3):259-269.
[126]徐成贤,陈志平,李乃成.近代优化方法[M].北京:科学出版社,2002.
[127] HABASHY T M, ABUBAKAR A. A general framework for constraint minimization for theinversion of electromagenic measurements[J]. Progress In Electromagnetics Research,2004,46(265-312.
[128] RAO Y, WANG Y H. The Strategies for Attenuation Inversion with WaveformTomography[J].70th EAGE Conference&Exhibition,2008, H006.
[129] KAMEI R, PRATT R G. Inversion strategies for visco-acoustic waveform inversion[J].Geophysical Journal International,2013,194(2):859-884.
[130] SONG Z-M, WILLIAMSON P R, PRATT R G. Frequency-domain acoustic-wave modelingand inversion of crosshole data: Part II-Inversion method, synthetic experiments andreal-data results[J]. Geophysics,1995,60(3):796-809.
[2]李世雄,刘家琦.小波变换和反演数学基础[M].北京:地质出版社,1994.
[3]姚姚.地球物理反演基本理论与应用方法[M].武汉:中国地质大学出版社,2002.
[4]宋海斌,张关泉.层状介质弹性参数反演问题研究综述[J].地球物理学进展,1998,13(4):67-78.
[5] LIU C, GAO F, FENG X, et al. Incorporating attenuation effects into frequency-domain fullwaveform inversion from zero-offset VSP data from Stokes equation[J]. Journal ofGeophysics and Engineering,2013,10(3):035004.
[6] VIRIEUX J, OPERTO S. An overview of full-waveform inversion in explorationgeophysics[J]. Geophysics,2009,74(6): WCC1-WCC26.
[7] TARANTOLA A. Inversion of seismic reflection data in the acoustic approximation[J].Geophysics,1984,49(8):1259-1266.
[8] GAUTHIER O, VIRIEUX J, TARANTOLA A. Two-dimensional nonlinear inversion ofseismic waveforms:numerical results[J]. Geophysics,1986,51(7):1387-1403.
[9] PICA A, DIET J P, TARANTOLAR A. Nonlinear inversion of seismic reflection data in alaterally invariant medium[J]. Geophysics,1990,55(3):284-292.
[10] CRASE E, PICAT A, NOBLE M, et al. Robust elastic nonlinear waveforminversion:Application to real data[J]. Geophysics,1990,55(5):527-538.
[11] MORA P. Nonlinear two-dimensional elastic inversion of multioffset seismic data[J].Geophysics,1987,52(9):1211-1228.
[12] MORA P. Elastic wave-field inversion of reflection and transmission data[J]. Geophysics,1988,53(6):750-759.
[13] TARANTOLA A. A strategy for nonlinear elastic inversion of seismic reflection data[J].Geophysics,1986,51(10):1893-1903.
[14] VIGH D, STARR E W. Comparisons for Waveform Inversion, Time Domain or FrequencyDomain?[J]. SEG Las Vegas2008Annual Meeting,2008,1890-1894.
[15] PRATT R G, WORTHINGTON M H. Inverse theory application to multi-source cross-holetomography.Part1: acoustic wave equation method[J]. Geophysical Prospecting,1990,38(3):287-310.
[16] PRATT R G. Inverse theory application to multi-source cross-hole tomography.Part2:elastic wave equation method[J]. Geophysical Prospecting,1990,38(3):311-329.
[17] PRATT R G, C.SHIN, G.J.HICKS. Gauss-Newton and full Newton methods infrequency-space seismic waveform inversion[J]. Geophysical Journal International,1998,133(2):341-362.
[18] PRATT R G. Seismic waveform inversion in the frequency domain，Part1：Theory andverification in a physical scale model[J]. Geophysics,1999,64(3):888-901.
[19] PRATT R G, SHIPP R M. Seismic waveform inversion in the frequency domain，Part2：Fault delineation in sediments using crosshole data[J]. Geophysics,1999,64(3):902-914.
[20] SHIN C, CHA Y H. Waveform inversion in the Laplace domain[J]. Geophysical JournalInternational,2008,173(3):922–931.
[21] PYUN S, SHIN C, LEE H.3D elastic full waveform inversion in the Laplace domain[J].SEG Las Vegas2008Annual Meeting,2008,1976-1980.
[22] CHA Y H, SHIN C.2D Laplace-domain waveform inversion using adaptive finite elementmethod[J]. SEG Houston2009International Exposition and Annual Meeting,2009,2263-2268.
[23] KOO N H, SHIN C, CHA Y H, et al. Sequentially ordered single-frequency2-D acousticwaveform inversion in the Laplace-Fourier domain[J]. SEG Houston2009InternationalExposition and Annual Meeting,2009,2248-2252.
[24] MARFURT K J. Accuracy of finite-difference and finite-element modeling of the scalar andelastic wave equations[J]. Geophysics,1984,49(5):533-549.
[25] VIRIEUX J. W-wave propagation in heterogeneous media: Velocity-stress finite-differencemethod[J]. Geophysics,1984,49(11):1933-1957.
[26] VIRIEUX J. P-SV wave propagation in heterogeneous media:Velocity-stressfinite-difference method[J]. Geophysics,1986,51(4):889-901.
[27] LEVANDER A R. Fourth-order finite-difference P-SV seismograms[J]. Geophysics,1988,53(11):1425-1436.
[28] PRATT R G. Frequency-domain elastic wave modeling by finite differences:A tool forcrosshole seismic imaging[J]. Geophysics,1990,55(5):626-632.
[29] JO C H, SHIN C, SUH J H. An optimal9-point, finite-difference, frequency-space,2-Dscalar wave extrapolator[J]. Geophysics,1996,61(2):529-537.
[30] SHIN C, SOHN H. A frequency-space2-D scalar wave extrapolator using extended25-point finite-difference operator[J]. Geophysics,1998,63(1):289-296.
[31] STEKL I, PRATT R G. Accurate viscoelastic modeling by frequency-domain finitedifferences using rotated operators[J]. Geophysics,1998,63(5):1779-1794.
[32] MIN D J, SHINZ C, KWON B-D, et al. Improved frequency-domain elastic wave modelingusing weighted-averaging difference operators[J]. Geophysics,2000,65(3):884-895.
[33] MIN D J, YOO H S, SHIN C, et al. Weighting-averaging finite-element method for scalarwave equation in the frequency domain[J]. Journal of Seismic Exploration,2002,11:197-222.
[34] HUSTEDT B, OPERTO S, VIRIEUX J. Mixed-grid and staggered-grid finite-differencemethods for frequency-domain acoustic wave modeling[J]. Geophysical JournalInternational,2004,157(3):1269-1296.
[35] OPERTO S, VIRIEUX J, AMESTOY P, et al.3D finite-difference frequency-domainmodeling of visco-acoustic wave propagation using a massively parallel direct solver: Afeasibility study[J]. Geophysics,2007,72(5): SM195-SM211.
[36] BROSSIER R, VIRIEUX J, OPERTO S. Parsimonious finite-volume frequency-domainmethod for2D P-SV-wave modelling[J]. Geophysical Journal International,2008,175(2):541-559.
[37] OPERTO S, VIRIEUX J, RIBODETTI A, et al. Finite-difference frequency-domainmodeling of viscoacoustic wave propagation in2D tilted transversely isotropic (TTI)media[J]. Geophysics,2009,74(5): T75-T95.
[38]殷文,印兴耀,吴国忱.高精度频率域弹性波方程有限差分方法及波场模拟[J].地球物理学报,2006,49(2):561-568.
[39]吴国忱,梁楷. VTI介质频率-空间域准P波正演模拟[J].石油地球物理勘探,2005,40(5):535-545.
[40]梁锴,吴国忱,印兴耀. T T I介质qP波方程频率-空间域加权平均有限差分算子[J].石油地球物理勘探,2007,42(5):516-525.
[41]吴国忱,罗彩明,梁楷.TTI介质弹性波频率-空间域有限差分数值模拟[J].吉林大学学报（地球科学版）,2007,37(5):1023-1033.
[42]刘璐,刘洪,刘红伟.优化15点频率-空间与有限差分正演模拟[J].地球物理学报,2013,56(2):644-652.
[43] GU B, LIANG G, LI Z. A21-point finite difference scheme for2D frequency-domainelastic wave modelling[J]. Exploration Geophysics,2013,44(3):156-166.
[44]卞爱飞,於文辉,周华伟.频率域全波形反演方法研究进展[J].地球物理学进展,2010,25(3):982-993.
[45] BROSSIER R, OPERTO S, VIRIEUX J. Which data residual norm for robust elasticfrequency-domain Full Waveform Inversion?[J]. Geophysics,2010,75(3): R37–R46.
[46] SHIN C, MIN D J. Waveform inversion using a logarithmic wavefield[J]. Geophysics,2006,71(3): R31–R42.
[47] SHIN C, PYUN S, BEDNAR J B. Comparison of waveform inversion, part1: conventionalwavefield vs logarithmic wavefield[J]. Geophysical Prospecting,2007,55(4):449-464.
[48] BEDNAR J B, SHIN C, PYUN S. Comparison of waveform inversion, part2: phaseapproach[J]. Geophysical Prospecting,2007,55(4):465-475.
[49] PYUN S, SHIN C, BEDNAR J B. Comparison of waveform inversion, part3: amplitudeapproach[J]. Geophysical Prospecting,2007,55(4):477-485.
[50] SHI Y M, ZHAO W Z, CAO H. Nonlinear process control of wave-equation inversion andits application in the detection of gas[J]. Geophysics,2007,72(1): R9-R18.
[51] HU W, ABUBAKAR A, HABASHY T M. Simultaneous multifrequency inversion offull-waveform seismic data[J]. Geophysics,2009,74(2): R1-R14.
[52] BAE H S, PYUN S, CHUNG W, et al. Frequency-domain acoustic-elastic coupledwaveform inversion using the Gauss-Newton conjugate gradient method[J]. GeophysicalProspecting,2012,60(3):413-432.
[53] RAVAUT C, OPERTO S, IMPROTA L, et al. Multiscale imaging of complex structuresfrom multifold wide-aperture seismic data by frequency-domain full-waveform tomography:application to a thrust belt[J]. Geophysical Journal International,2004,159(3):1032-1056.
[54] SOURBIER F, OPERTO S, VIRIEUX J. FWT2D: a massively parallel program forfrequency-domain Full-Waveform Tomography of wide-aperture seismic data-Part1:algorithm[J]. Computer&Geosciences,2009,35(3):487-495.
[55] SOURBIER F, OPERTO S, VIRIEUX J. FWT2D:Amassively parallel program forfrequency-domain full-waveform tomography of wide-aperture seismic data—Part2Numerical examples and scalability analysis[J]. Computer&Geosciences,2009,35(3):496-514.
[56]刘国峰,刘洪,孟小红,等.频率域波形反演中与频率相关的影响因素分析[J].地球物理学报,2012,55(4):1345-1353.
[57] JEONG W, LEE H Y, MIN D J. Full waveform inversion strategy for density in thefrequency domain[J]. Geophysical Journal International,2012,188(3):1221-1242.
[58]朱童,李小凡,汪文帅,等.粒子群-梯度算法在频率域地震波形反演中的应用[J].地球物理学进展,2013,28(1):180-189.
[59] NOCEDAL J, WRIGHT S J. Numerical Optimization [M]. New York: Springer-Verlag NewYork,Inc,2006.
[60]刘璐,刘洪,张衡.基于修正拟牛顿公式的全波形反演[J].地球物理学报,2013,56(7):2447-2451.
[61] LIN Y, ABUBAKAR A, HABASHY T M. Seismic full-waveform inversion using truncatedwavelet representations[J]. SEG Extended Abstracts,2012,
[62] MA Y, HALE D. Quasi-Newton full-waveform inversion with a projected Hessian matrix[J].Geophysics,2012,77(5): R207-R216.
[63] MA Y, HALE D, GONG B, et al. Image-guided sparse-model full waveform inversion[J].Geophysics,2012,77(4): R189-R198.
[64] BROSSIER R, OPERTO S, VIRIEUX J. Seismic imaging of complex onshore structures by2D elastic frequency-domain full-waveform inversion[J]. Geophysics,2009,74(6):WCC105-WCC118.
[65] MALINOWSKI M, OPERTO S, RIBODETTI A. High-resolution seismic attenuationimaging from wide-aperture onshore data by visco-acoustic frequency-domainfull-waveform inversion[J]. Geophysical Journal International,2011,186(3):1179-1204.
[66] GUITTON A, AYENI G, D AZ E. Constrained full-waveform inversion by modelreparameterization[J]. Geophysics,2012,77(2): R117-R127.
[67] ANAGAW A Y, SACCHI M D. Full waveform inversion with simultaneous sources usingthe full Newton Method[J]. SEG,2012,1-5.
[68] M TIVIER L, BROSSIER R, VIRIEUX J, et al. Toward Gauss-Newton and Exact NewtonOptimization for Full Waveform Inversion[J].74th EAGE Conference&Exhibition,Extended Abstracts,2012, P016.
[69] BROSSIER R. Two-dimensional frequency-domain visco-elastic full waveforminversion:Parallel algorithms,optimization and performance[J]. Computers&Geosciences,2011,37(4):444-455.
[70] K HN D, NIL DE D, KURZMANN A, et al. On the influence of model parametrization inelastic full waveform tomography[J]. Geophysical Journal International,2012,191(1):325-345.
[71] RAO Y, WANG Y. Fracture effects in seismic attenuation images reconstructed bywaveform tomography[J]. Geophysics,2009,74(4): R25-R34.
[72] BARNES C, CHARARA M, TSUCHIYA T. Feasibility study for an anisotropic fullwaveform inversion of cross-well seismic data[J]. Geophysical Prospecting,2008,56(6):897-906.
[73] PLESSIX R E, CAO Q. A parameterization study for surface seismic full waveforminversion in an acoustic vertical transversely isotropic medium[J]. Geophysical JournalInternational,2011,185(1):539–556.
[74] LEE H Y, KOO J M, MIN D J. Frequency-domain elastic full waveform inversion for VTImedia[J]. Geophysical Journal International,2010,183(2):884-904.
[75] GHOLAMI Y, BROSSIER R, OPERTO S, et al. Which parameterization is suitable foracoustic vertical transverse isotropic full waveform inversion? Part1: Sensitivity andtrade-off analysis[J]. Geophysics,2013,78(2): R81–R105.
[76] GHOLAMI Y, BROSSIER R, OPERTO S, et al. Which parameterization is suitable foracoustic vertical transverse isotropic full waveform inversion? Part2: Synthetic and realdata case studies from Valhall[J]. Geophysics,2013,78(2): R107–R124.
[77] PRIEUX V, BROSSIER R, OPERTO S E, et al. Multiparameter full waveform inversion ofmulticomponent ocean-bottom-cable data from the Valhall field. Part1: imagingcompressional wave speed, density and attenuation[J]. Geophysical Journal International,2013,194(3):1640-1664.
[78] PRIEUX V, BROSSIER R, OPERTO S E, et al. Multiparameter full waveform inversion ofmulticomponent ocean-bottom-cable data from the Valhall field. Part2: imagingcompressive-wave and shear-wave velocities[J]. Geophysical Journal International,2013,194(3):1665–1681.
[79] OPERTO S, GHOLAMI Y, BROSSIER R, et al. A guided tour of multi-parameter fullwaveform inversion with multi-component data: from theory to practice[J]. The LeadingEdge,2013,32(9):1040-1054.
[80] BARROS L D, DIETRICH M, VALETTE B. Full waveform inversion of seismic wavesreflected in a stratified porous medium[J]. Geophysical Journal International,2010,182(3):1543-1556.
[81] WARNER M, RATCLIFFE A, NANGOO T, et al. Anisotropic3D full-waveforminversion[J]. Geophysics,2013,78(2): R59–R80.
[82] PRIEUX V, LAMBAR G, OPERTO S, et al. Building starting models for full waveforminversion from wide-aperture data by stereotomography[J]. Geophysical Prospecting,2013,61(s1):109-137.
[83] SHIN C, HA W. A comparison between the behavior of objective functions for waveforminversion in the frequency and Laplace domains[J]. Geophysics,2008,73(5): VE119–VE133.
[84] SHIN C, CHA Y H. Waveform inversion in the Laplace–Fourier domain[J]. GeophysicalJournal International,2009,177(3):1067–1079.
[85] PRATT R G, HOU F, BAUER K, et al. Waveform tomography images of velocity andinelastic attenuation from the Mallik2002crosshole seismic surveys[J]. Bulletin-Geological Survey of Canada,2005,585(1-14.
[86] WANG Y, RAO Y. Crosshole seismic waveform tomography–I. Strategy for real dataapplication[J]. Geophysical Journal International,2006,166(3):1224–1236.
[87] RAO Y, WANG Y, MORGAN J V. Crosshole seismic waveform tomography–II.Resolution analysis[J]. Geophysical Journal International,2006,166(3):1237-1248.
[88] PRATT R G, PLESSIX R E, W.A.MULDER. Seismic waveform tomography the effect ofalyering and anisotrop[J]. EAGE63rd Conference&Technical Exhibition,2001, P092.
[89] FICHTNER A, TRAMPERT J, CUPILLARD P, et al. Multiscale full waveform inversion[J].Geophysical Journal International,2013,194(1):534-556.
[90] BRENDERS A J, PRATT R G. Full waveform tomography for lithospheric imaging: Resultsfrom a blind test in a realistic crustal model[J]. Geophysical Journal International,2007,168(1):133-151.
[91] BRENDERS A J, PRATT R G. Efficient waveform tomography for lithospheric imaging:implications for realistic, two-dimensional acquisition geometries and low-frequency data[J].Geophysical Journal International,2007,168(1):152–170.
[92] JAISWAL P, ZELT C A, DASGUPTA R, et al. Seismic imaging of the Naga thrust usingmultiscale waveform inversion[J]. Geophysics,2009,74(6): WCC129–WCC140.
[93] BLEIBINHAUS F, HILBERG S. Shape and structure of the Salzach Valley, Austria, fromseismic traveltime tomography and full waveform inversion[J]. Geophysical JournalInternational,2012,189(3):1701–1716.
[94] BAETEN G, MAAG J W D, PLESSIX R E-E, et al. The use of low frequencies in afull-waveform inversion and impedance inversion land seismic case study[J]. GeophysicalProspecting,2013,61(4):701-711.
[95] DESSA J X, OPERTO S, KODAIRA S, et al. Multiscale seismic imaging of the easternNankai trough by full waveform inversion[J]. gEOPHYSICAL Research Letters,2004,31(18):1-4.
[96] MORGAN J, WARNER M, BELL R, et al. Next-generation seismic experiments:wide-angle, multi-azimuth, three-dimensional, full-waveform inversion[J]. GeophysicalJournal International,2013,195(3):1657-1678.
[97] OPERTO S, RAVAUT C, IMPROTA L, et al. Quantitative imaging of complex structuresfrom dense wideaperture seismic data by multiscale traveltime and waveform inversions:A case study[J]. Geophysical Prospecting,2005,52(6):625–651.
[98] TAKOUGANG E M T, CALVERT A J. Application of waveform tomography to marineseismic reflection data from the Queen Charlotte Basin of western Canada[J]. Geophysics,2011,76(2): B55–B70.
[99] TAKOUGANG E M T, CALVERT A J. Seismic velocity and attenuation structures of theQueen Charlotte Basin from full-waveform tomography of seismic reflection data[J].Geophysics,2012,77(3): B107–B124.
[100] SMITHYMAN B R, CLOWES R M. Waveform tomography of field vibroseis data using anapproximate2D geometry leads to improved velocity models[J]. Geophysics,2012,77(1):R33–R43.
[101]陈小宏,牟永光.二维地震资料波动方程非线性反演[J].地球物理学报,1996,39(3):401-408.
[102]张霖斌,姚振兴.层状介质的声波波动方程反演[J].地球物理学进展,2000,15(2):22-29.
[103]许琨,王妙月.声波方程频率域有限元参数反演[J].地球物理学报,2001,44(4):852-864.
[104]许琨,王妙月.利用地质规则块体建模方法的频率域有限元弹性波速度反演[J].地球物理学报,2004,47(4):708-717.
[105]丁继才,常旭,刘伊克.反射地震数据的逐层波形反演[J].地球物理学报,2007,50(2):574-580.
[106]胡光辉,贾春梅,夏洪瑞,等.三维声波全波形反演的实现与验证[J].石油物探,2013,52(4):417-425.
[107] N.H.瑞克,许云译,吴律校.粘弹性介质中的地震波[M].北京:地质出版社,1981.
[108]刘财,谢靖,韩道范,等.非均匀粘弹性介质中波传播的几个基本问题[J].长春科技大学学报,2000,30(专辑):93-96.
[109]刘财,谢靖,韩道范,等.用Stokes方程的差分方法制作合成地震记录[J].石油地球物理勘探,2002,37(3):230-236.
[110] CERJAN C, KOSLOF D, KOSLOFF R, et al. A nonreflecting boundary condition fordiscrete acoustic and elastic wave equations[J]. Geophysics,1985,50(4):705-708.
[111] OPERTO S, VIRIEUX J, SOURBIER F. Documentation of FWT2D program (version4.8):Frequency-domain full-waveform modeling/inversion of wide-aperture seismic data forimaging2D acoustic media (Technical report N0007-SEISCOPE project).2007.
[112]刘财,陈业全,刘洋,等.勘探地震资料处理新方法及新技术[M].北京:科学出版社,2006.
[113] VIGH D, STARR E W, KAPOOR J. Developing earth models with full waveforminversion[J]. The Leading Edge,2009,28(4):432-435.
[114] KENNETT B L N, SAMBRIDGE M S, WILLIAMSON P R. Subspace methods for largeinverse problems with multiple parameter classes[J]. Geophysical Journal,1988,94(2):237-247.
[115] SAMBRIDGE M S, TARANTOLA A, KENNETT B L N. An alternative strategy fornon-linear inversion of seismic waveforms[J]. Geophysical Prospecting,1991,39(6):723-736.
[116]马中高,解吉高.岩石的纵、横波速度与密度的规律研究[J].地球物理学进展,2005,20(4):905-910.
[117] SIRGUE L, PRATT R G. Efficient waveform inversion and imaging: A strategy forselecting temporal frequencies[J]. Geophysics,2004,69(1):231-248.
[118] KIM Y, CHO H, MIN D, et al. Comparison of frequency-selection strategies for2Dfrequency-domain acoustic waveform inversion[J]. Pure and Applied Geophysics,2011,168(10):1715-1727.
[119]何玉梅,郑天愉.利用地震波形反演研究震源破裂时空过程[J].地球物理学报,1998,41(2):281-289.
[120]吴建平,明跃红,王椿镛.川滇地区速度结构的区域地震波形反演研究[J].地球物理学报,2006,49(5):1369-1376.
[121]李少华,王彦宾,梁子斌.甘肃东南部地壳速度结构的区域地震波形反演[J].地球物理学报,2012,55(4):1186-1197.
[122] DENNIS J E, SCHNABEL R B. Numerical Methods for Unconstrained Optimization andNonlinear Equations [M]. Philadelphia: Society for Industrial and Applied Mathematics,1996.
[123]高凤霞,刘财,冯晅,等.几种优化方法在频率域全波形反演中的应用效果及对比分析研究[J].地球物理学进展,2013,28(4):2060-2068.
[124] SHANNO D F. Conjugate gradient methods with inexact searches[J]. Mathmatics ofOperations Research,1978,3(3):244-256.
[125]尉继英.一种求解无约束极值问题的无记忆拟牛顿算法[J].计算数学,1990,(3):259-269.
[126]徐成贤,陈志平,李乃成.近代优化方法[M].北京:科学出版社,2002.
[127] HABASHY T M, ABUBAKAR A. A general framework for constraint minimization for theinversion of electromagenic measurements[J]. Progress In Electromagnetics Research,2004,46(265-312.
[128] RAO Y, WANG Y H. The Strategies for Attenuation Inversion with WaveformTomography[J].70th EAGE Conference&Exhibition,2008, H006.
[129] KAMEI R, PRATT R G. Inversion strategies for visco-acoustic waveform inversion[J].Geophysical Journal International,2013,194(2):859-884.
[130] SONG Z-M, WILLIAMSON P R, PRATT R G. Frequency-domain acoustic-wave modelingand inversion of crosshole data: Part II-Inversion method, synthetic experiments andreal-data results[J]. Geophysics,1995,60(3):796-809.