各向异性介质Low-rank有限差分法纯qP波叠前平面波最小二乘逆时偏移
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摘要
拟声波最小二乘逆时偏移是一种极具潜力的地震波成像工具,但该方法遭受各向异性拟声波近似的限制,TTI介质正演模拟不稳定、反偏移记录中遭受伪横波二次扰动及数值频散假象,另外拟声波最小二乘逆时偏移还面临计算效率低、收敛速度慢、对速度等模型参数依赖性高等问题.为了克服各向异性拟声波最小二乘逆时偏移的缺陷,在反演框架下,本文借助Low-rank有限差分算法首次提出并实现了TTI介质纯qP波线性正演模拟及纯qP波最小二乘逆时偏移;为了进一步提升反演成像效率,同时改善反演成像方法对模型参数误差的依赖性及对地震数据噪声的适应性,通过引入叠前平面波优化策略,发展了TTI介质纯qP波叠前平面波最小二乘逆时偏移成像方法.在编程实现方法的基础上,通过开展模型成像测试,展示了本方法的优势和潜力:一方面加快了反演成像效率,另一方面也提升了方法的抗噪性,同时还降低了方法对模型参数的依赖性.
Least-squares reverse time migration of pseudo acoustic waves is a potential tool for seismic imaging.However,this method is limited by anisotropic pseudo acoustic wave approximation.Thus forward simulation using this tool for TTI medium is unstable and demigration records contain quadratic perturbation of pseudo SV-waves and numerically dispersion.In addition,the least-squares reverse time migration of pseudo acoustic waves also faces some problems such as low computational efficiency,slow convergence and high dependency on velocity and other model parameters.In order to overcome the drawback of this method,under the framework of inversion,this work realizes the linearized forward modeling of pure qP waves and the pure qP-wave leastsquares reverse time migration for the first time by means of a Low-rank finite difference algorithm.Further,in order to speeding up the inversion efficiency,while improving the dependence of the inversion method on model parameter error and enhancing the adaptability to the seismic data noise,we develop a least-squares reverse time migration imaging method for pure-qP-wave prestack plane-waves in TTI media by introducing the prestack plane-wave optimization strategy.After implementing the algorithm by programming,the advantages and potential of this new approach are demonstrated by model imaging tests.On the one hand,the efficiency of inversion imaging is enhanced;on the other hand,the capability of anti-noise is also improved.Meanwhile,dependence of the inversion method on model parameters is lowered.
Least-squares reverse time migration of pseudo acoustic waves is a potential tool for seismic imaging.However,this method is limited by anisotropic pseudo acoustic wave approximation.Thus forward simulation using this tool for TTI medium is unstable and demigration records contain quadratic perturbation of pseudo SV-waves and numerically dispersion.In addition,the least-squares reverse time migration of pseudo acoustic waves also faces some problems such as low computational efficiency,slow convergence and high dependency on velocity and other model parameters.In order to overcome the drawback of this method,under the framework of inversion,this work realizes the linearized forward modeling of pure qP waves and the pure qP-wave leastsquares reverse time migration for the first time by means of a Low-rank finite difference algorithm.Further,in order to speeding up the inversion efficiency,while improving the dependence of the inversion method on model parameter error and enhancing the adaptability to the seismic data noise,we develop a least-squares reverse time migration imaging method for pure-qP-wave prestack plane-waves in TTI media by introducing the prestack plane-wave optimization strategy.After implementing the algorithm by programming,the advantages and potential of this new approach are demonstrated by model imaging tests.On the one hand,the efficiency of inversion imaging is enhanced;on the other hand,the capability of anti-noise is also improved.Meanwhile,dependence of the inversion method on model parameters is lowered.
引文
Ao R D,Dong L G,Chi B X.2015.Source-independent envelopebased FWI to build an initial model.Chinese J.Geophys.(in Chinese),58(6):1998-2010,doi:10.6038/cjg20150615.
Chen K,Sacchi M D.2017.Elastic least-squares reverse time migration via linearized elastic full-waveform inversion with pseudo-Hessian preconditioning.Geophysics,82(5):S341-S358.
Cheng J B,Kang W,Wang T F.2013.Description of qP-wave propagation in anisotropic media,Part I:Pseudo-pure-mode wave equations.Chinese J.Geophys.(in Chinese),56(10):3474-3486,doi:10.6038/cjg20131022.
Chu C L,Macy B K,Anno P D.2013.Pure acoustic wave propagation in transversely isotropic media by the pseudospectral method.Geophysical Prospecting,61(3):556-567.
Clapp M L,Clapp R G,Biondi B L.2005.Regularized least-squares inversion for 3-D subsalt imaging.∥75th Annual International Meeting.SEG,Expanded Abstracts,1814-1817.
Dai W,Schuster J.2009.Least-squares migration of simultaneous sources data with a deblurring filter.∥79th Annual International Meeting.Houston,Texas:SEG,Expanded Abstracts,2990-2994.
Dai W,Wang X,Schuster G T.2011.Least-squares migration of multisource data with a deblurring filter.Geophysics,76(5):R135-R146.
Dai W,Fowler P,Schuster G T.2012.Multi-source least-squares reverse time migration.Geophysical Prospecting,60(4):681-695.
Dai W,Schuster G T.2013.Plane-wave least-squares reverse-time migration.Geophysics,78(4):S165-S177.
Dong S,Cai J,Guo M,et al.2012.Least-squares reverse time migration:Towards true amplitude imaging and improving the resolution.∥82nd Annual International Meeting.Las Vegas,Nevada:SEG,Expanded Abstracts,1190-1194.
Duan Y T,Guitton A,Sava P.2017.Elastic least-squares reverse time migration.Geophysics,82(4):S315-S325.
Duquet B,Marfurt K J,Dellinger J A.2000.Kirchhoff modeling,inversion for reflectivity,and subsurface illumination.Geophysics,65(4):1195-1209.
Dutta G,Schuster G T.2014.Attenuation compensation for leastsquares reverse time migration using the viscoacoustic-wave equation.Geophysics,79(6):S251-S262.
Fang G,Fomel S,Du Q Z.2014.Lowrank finite difference on a staggered grid and its application on reverse time migration.Journal of China University of Petroleum(in Chinese),38(2):44-51.
Feng Z C,Schuster G T.2017.Elastic least-squares reverse time migration.Geophysics,82(2):S143-S157.
Fomel S,Ying L X,Song X L.2010.Seismic wave extrapolation using lowrank symbol approximation.∥80th Annual International Meeting.Denver,Colorado:SEG,Expanded Abstracts,3092-3096.
Gu B L,Li Z C,Yang P,et al.2017.Elastic least-squares reverse time migration with hybrid l1/l2misfit function.Geophysics,82(3):S271-S291.
Guitton A,Kaelin B,Biondi B.2007.Least-squares attenuation of reverse-time-migration artifacts.Geophysics,72(1):S19-S23.
Guo P,McMechan G A.2018.Compensating Qeffects in viscoelastic media by adjoint-based least-squares reverse time migration.Geophysics,83(2):S151-S172.
Hou J,Symes W W.2015.An approximate inverse to the extended born modeling operator.Geophysics,80(6):R331-R349.
Hou J,Symes W W.2016.Accelerating extended least-squares migration with weighted conjugate gradient iteration.Geophysics,81(4):S165-S179.
Huang J P,Huang J Q,Li Z C,et al.2016.Staggered grid pseudospectral numerical simulation of first-order quasi P-wave equation for TTI media.Oil Geophysical Prospecting(in Chinese),51(3):487-496.
Huang J Q,Li Z C.2017.Modeling and reverse time migration of pure quasi-P-waves in complex TI media with a low-rank decomposition.Chinese J.Geophys.(in Chinese),60(2):704-721,doi:10.6038/cjg20170223.
Huang J Q,Li Z C,Huang J P,et al.2017.Lebedev grid highorder finite-difference modeling and elastic-wave-mode separation for TTI media.Oil Geophysical Prospecting(in Chinese),52(5):915-927.
Huang J Q,Li Z C,Jiang W.2018.An efficient forward modeling with the Low-rank finite-difference algorithm for complex TTImedia and its application in reverse time migration.Oil Geophysical Prospecting(in Chinese),53(2):1198-1209.
Huang Y S,Schuster G T.2012.Multisource least-squares migration of marine streamer and land data with frequency-division encoding.Geophysical Prospecting,60(4):663-680.
Kaplan S T,Routh P S,Sacchi M D.2010.Derivation of forward and adjoint operators for least-squares shot-profile split-step migration.Geophysics,75(6):S225-S235.
Kuehl H,Sacchi M.2002.Robust AVP estimation using leastsquares wave-equation migration.∥72nd Annual International Meeting.SEG,Expanded Abstracts,281-284.
Li Q Y,Huang J P,Li Z C,et al.2016.Mean-residual normalized cross-correlation least-squares reverse time migration and its application.Chinese J.Geophys.(in Chinese),59(8):3006-3015,doi:10.6038/cjg20160823.
Li Q Y,Huang J P,Li Z C.2017.Source-independent least-squares reverse time migration using student′s t distribution.Chinese J.Geophys.(in Chinese),60(12):4790-4800,doi:10.6038/cjg20171220.
Li Z C,Guo Z B,Tian K.2014.Least-squares reverse time migration in visco-acoustic medium.Chinese J.Geophys.(in Chinese),57(1):214-228,doi:10.6038/cjg20140118.
Li Z C,Huang J Q,Huang J P,et al.2017.Fast least-squares reverse time migration based on plane-wave encoding for VTImedia.Chinese J.Geophys.(in Chinese),60(1):240-257,doi:10.6038/cjg20170120.
Li Z N,Li Z C,Wang P,et al.2017.Reverse time migration of multiples based on different-order multiple separation.Geophysics,82(1):S19-S29.
Liu X J,Liu Y K.2016.Least-squares reverse-time migration of surface-related multiples.Chinese J.Geophys.(in Chinese),59(9):3354-3365,doi:10.6038/cjg20160919.
Liu Y J,Symes W W,Li Z C.2013.Multisource least-squares extended reverse-time migration with preconditioning guided gradient method.∥83rd Annual International Meeting.Houston,Texas:SEG,Expanded Abstracts,3709-3715.
Liu Y K,Liu X J,Zhang Y B.2018.Migration of seismic multiple reflections.Chinese J.Geophys.(in Chinese),61(3):1025-1037,doi:10.6038/cjg2018L0368.
Nemeth T,Wu C J,Schuster G T.1999.Least-squares migration of incomplete reflection data.Geophysics,64(1):208-221.
Nocedal J.1980.Updating quasi-Newton matrices with limited storage.Mathematics of Computation,35(151):773-782.
Sheng X,Zhou H B.2014.Accurate simulations of pure quasi-P-waves in complex anisotropic media.Geophysics,79(6):T341-T348.
Song G J,Huang R,Tian J D,et al.2016.A new QP-wave equation for 2DVTI media.∥86th Annual International Meeting.Dallas,Texas:SEG,Expanded Abstracts,3977-3981.
Song X L,Fomel S,Ying L X,et al.2011.Lowrank finite-differences for wave extrapolation.∥81st Annual International Meeting.SEG,Expanded Abstracts,3372-3376.
Song X L,Alkhalifah T.2013.Modeling of pseudoacoustic P-waves in orthorhombic media with a low-rank approximation.Geophysics,78(4):C33-C40.
Song X L,Fomel S,Ying L X.2013.Lowrank finite-differences and lowrank Fourier finite-differences for seismic wave extrapolation in the acoustic approximation.Geophysical Journal International,193(2):960-969.
Stanton A,Sacchi M D.2017.Elastic least-squares one-way waveequation migration.Geophysics,82(4):S293-S305.
Sun J Z,Fomel S,Zhu T Y,et al.2016.Q-compensated leastsquares reverse time migration using low-rank one-step wave extrapolation.Geophysics,81(4):S271-S279.
Tang Y X.2009.Target-oriented wave-equation least-squares migration/inversion with phase-encoded Hessian.Geophysics,74(6):WCA95-WCA107.
Tarantola A.1984.Linearized inversion of seismic reflection data.Geophysical Prospecting,32(6):998-1015.
Vigh D,Starr E W.2008.3Dprestack plane-wave,full-waveform inversion.Geophysics,73(5):VE135-VE144.
Wong M,Biondi B L,Ronen S.2010.Joint least-squares inversion of up-and down-going signal for ocean bottom data sets.∥80th Annual International Meeting.SEG,Expanded Abstracts,2752-2756.
Wong M,Ronen S,Biondi B L.2011.Least-squares reverse time migration/inversion for ocean bottom data:A case study.∥81st Annual International Meeting.San Antonio,Texas:SEG,Expanded Abstracts,2369-2373.
Wong M,Biondi B L,Ronen S.2015.Imaging with primaries and free-surface multiples by joint least-squares reverse time migration.Geophysics,80(6):S223-S235.
Zhan G,Pestana R C,Stoffa P L.2012.Decoupled equations for reverse time migration in tilted transversely isotropic media.Geophysics,77(2):T37-T45.
Zhan G,Pestana R C,Stoffa P L.2013.An efficient hybrid pseudospectral/finite-difference scheme for solving the TTIpure P-wave equation.Journal of Geophysics and Engineering,10(2):025004.
敖瑞德,董良国,迟本鑫.2015.不依赖子波、基于包络的FWI初始模型建立方法研究.地球物理学报,58(6):1998-2010,doi:10.6038/cjg20150615.
程玖兵,康玮,王腾飞.2013.各向异性介质qP波传播描述I:伪纯模式波动方程.地球物理学报,56(10):3474-3486,doi:10.6038/cjg20131022.
方刚,Fomel S,杜启振.2014.交错网格Lowrank有限差分及其在逆时偏移中的应用.中国石油大学学报(自然科学版),38(2):44-51.
黄建平,黄金强,李振春等.2016.TTI介质一阶qP波方程交错网格伪谱法正演模拟.石油地球物理勘探,51(3):487-496.
黄金强,李振春.2017.基于Low-rank分解的复杂TI介质纯qP波正演模拟与逆时偏移.地球物理学报,60(2):704-721,doi:10.6038/cjg20170223.
黄金强,李振春,黄建平等.2017.TTI介质Lebedev网格高阶有限差分正演模拟及波型分离.石油地球物理勘探,52(5):915-927.
黄金强,李振春,江文.2018.TTI介质Low-rank有限差分法高效正演模拟及逆时偏移.石油地球物理勘探,53(6):1198-1209.
李庆洋,黄建平,李振春等.2016.去均值归一化互相关最小二乘逆时偏移及其应用.地球物理学报,59(8):3006-3015,doi:10.6038/cjg20160823.
李庆洋,黄建平,李振春.2017.基于Student's t分布的不依赖子波最小二乘逆时偏移.地球物理学报,60(12):4790-4800,doi:10.6038/cjg20171220.
李振春,郭振波,田坤.2014.黏声介质最小平方逆时偏移.地球物理学报,57(1):214-228,doi:10.6038/cjg20140118.
李振春,黄金强,黄建平等.2017.基于平面波加速的VTI介质最小二乘逆时偏移.地球物理学报,60(1):240-257,doi:10.6038/cjg20170120.
刘学建,刘伊克.2016.表面多次波最小二乘逆时偏移成像.地球物理学报,59(9):3354-3365,doi:10.6038/cjg20160919.
刘伊克,刘学建,张延保.2018.地震多次波成像.地球物理学报,61(3):1025-1037,doi:10.6038/cjg2018L0368.
Chen K,Sacchi M D.2017.Elastic least-squares reverse time migration via linearized elastic full-waveform inversion with pseudo-Hessian preconditioning.Geophysics,82(5):S341-S358.
Cheng J B,Kang W,Wang T F.2013.Description of qP-wave propagation in anisotropic media,Part I:Pseudo-pure-mode wave equations.Chinese J.Geophys.(in Chinese),56(10):3474-3486,doi:10.6038/cjg20131022.
Chu C L,Macy B K,Anno P D.2013.Pure acoustic wave propagation in transversely isotropic media by the pseudospectral method.Geophysical Prospecting,61(3):556-567.
Clapp M L,Clapp R G,Biondi B L.2005.Regularized least-squares inversion for 3-D subsalt imaging.∥75th Annual International Meeting.SEG,Expanded Abstracts,1814-1817.
Dai W,Schuster J.2009.Least-squares migration of simultaneous sources data with a deblurring filter.∥79th Annual International Meeting.Houston,Texas:SEG,Expanded Abstracts,2990-2994.
Dai W,Wang X,Schuster G T.2011.Least-squares migration of multisource data with a deblurring filter.Geophysics,76(5):R135-R146.
Dai W,Fowler P,Schuster G T.2012.Multi-source least-squares reverse time migration.Geophysical Prospecting,60(4):681-695.
Dai W,Schuster G T.2013.Plane-wave least-squares reverse-time migration.Geophysics,78(4):S165-S177.
Dong S,Cai J,Guo M,et al.2012.Least-squares reverse time migration:Towards true amplitude imaging and improving the resolution.∥82nd Annual International Meeting.Las Vegas,Nevada:SEG,Expanded Abstracts,1190-1194.
Duan Y T,Guitton A,Sava P.2017.Elastic least-squares reverse time migration.Geophysics,82(4):S315-S325.
Duquet B,Marfurt K J,Dellinger J A.2000.Kirchhoff modeling,inversion for reflectivity,and subsurface illumination.Geophysics,65(4):1195-1209.
Dutta G,Schuster G T.2014.Attenuation compensation for leastsquares reverse time migration using the viscoacoustic-wave equation.Geophysics,79(6):S251-S262.
Fang G,Fomel S,Du Q Z.2014.Lowrank finite difference on a staggered grid and its application on reverse time migration.Journal of China University of Petroleum(in Chinese),38(2):44-51.
Feng Z C,Schuster G T.2017.Elastic least-squares reverse time migration.Geophysics,82(2):S143-S157.
Fomel S,Ying L X,Song X L.2010.Seismic wave extrapolation using lowrank symbol approximation.∥80th Annual International Meeting.Denver,Colorado:SEG,Expanded Abstracts,3092-3096.
Gu B L,Li Z C,Yang P,et al.2017.Elastic least-squares reverse time migration with hybrid l1/l2misfit function.Geophysics,82(3):S271-S291.
Guitton A,Kaelin B,Biondi B.2007.Least-squares attenuation of reverse-time-migration artifacts.Geophysics,72(1):S19-S23.
Guo P,McMechan G A.2018.Compensating Qeffects in viscoelastic media by adjoint-based least-squares reverse time migration.Geophysics,83(2):S151-S172.
Hou J,Symes W W.2015.An approximate inverse to the extended born modeling operator.Geophysics,80(6):R331-R349.
Hou J,Symes W W.2016.Accelerating extended least-squares migration with weighted conjugate gradient iteration.Geophysics,81(4):S165-S179.
Huang J P,Huang J Q,Li Z C,et al.2016.Staggered grid pseudospectral numerical simulation of first-order quasi P-wave equation for TTI media.Oil Geophysical Prospecting(in Chinese),51(3):487-496.
Huang J Q,Li Z C.2017.Modeling and reverse time migration of pure quasi-P-waves in complex TI media with a low-rank decomposition.Chinese J.Geophys.(in Chinese),60(2):704-721,doi:10.6038/cjg20170223.
Huang J Q,Li Z C,Huang J P,et al.2017.Lebedev grid highorder finite-difference modeling and elastic-wave-mode separation for TTI media.Oil Geophysical Prospecting(in Chinese),52(5):915-927.
Huang J Q,Li Z C,Jiang W.2018.An efficient forward modeling with the Low-rank finite-difference algorithm for complex TTImedia and its application in reverse time migration.Oil Geophysical Prospecting(in Chinese),53(2):1198-1209.
Huang Y S,Schuster G T.2012.Multisource least-squares migration of marine streamer and land data with frequency-division encoding.Geophysical Prospecting,60(4):663-680.
Kaplan S T,Routh P S,Sacchi M D.2010.Derivation of forward and adjoint operators for least-squares shot-profile split-step migration.Geophysics,75(6):S225-S235.
Kuehl H,Sacchi M.2002.Robust AVP estimation using leastsquares wave-equation migration.∥72nd Annual International Meeting.SEG,Expanded Abstracts,281-284.
Li Q Y,Huang J P,Li Z C,et al.2016.Mean-residual normalized cross-correlation least-squares reverse time migration and its application.Chinese J.Geophys.(in Chinese),59(8):3006-3015,doi:10.6038/cjg20160823.
Li Q Y,Huang J P,Li Z C.2017.Source-independent least-squares reverse time migration using student′s t distribution.Chinese J.Geophys.(in Chinese),60(12):4790-4800,doi:10.6038/cjg20171220.
Li Z C,Guo Z B,Tian K.2014.Least-squares reverse time migration in visco-acoustic medium.Chinese J.Geophys.(in Chinese),57(1):214-228,doi:10.6038/cjg20140118.
Li Z C,Huang J Q,Huang J P,et al.2017.Fast least-squares reverse time migration based on plane-wave encoding for VTImedia.Chinese J.Geophys.(in Chinese),60(1):240-257,doi:10.6038/cjg20170120.
Li Z N,Li Z C,Wang P,et al.2017.Reverse time migration of multiples based on different-order multiple separation.Geophysics,82(1):S19-S29.
Liu X J,Liu Y K.2016.Least-squares reverse-time migration of surface-related multiples.Chinese J.Geophys.(in Chinese),59(9):3354-3365,doi:10.6038/cjg20160919.
Liu Y J,Symes W W,Li Z C.2013.Multisource least-squares extended reverse-time migration with preconditioning guided gradient method.∥83rd Annual International Meeting.Houston,Texas:SEG,Expanded Abstracts,3709-3715.
Liu Y K,Liu X J,Zhang Y B.2018.Migration of seismic multiple reflections.Chinese J.Geophys.(in Chinese),61(3):1025-1037,doi:10.6038/cjg2018L0368.
Nemeth T,Wu C J,Schuster G T.1999.Least-squares migration of incomplete reflection data.Geophysics,64(1):208-221.
Nocedal J.1980.Updating quasi-Newton matrices with limited storage.Mathematics of Computation,35(151):773-782.
Sheng X,Zhou H B.2014.Accurate simulations of pure quasi-P-waves in complex anisotropic media.Geophysics,79(6):T341-T348.
Song G J,Huang R,Tian J D,et al.2016.A new QP-wave equation for 2DVTI media.∥86th Annual International Meeting.Dallas,Texas:SEG,Expanded Abstracts,3977-3981.
Song X L,Fomel S,Ying L X,et al.2011.Lowrank finite-differences for wave extrapolation.∥81st Annual International Meeting.SEG,Expanded Abstracts,3372-3376.
Song X L,Alkhalifah T.2013.Modeling of pseudoacoustic P-waves in orthorhombic media with a low-rank approximation.Geophysics,78(4):C33-C40.
Song X L,Fomel S,Ying L X.2013.Lowrank finite-differences and lowrank Fourier finite-differences for seismic wave extrapolation in the acoustic approximation.Geophysical Journal International,193(2):960-969.
Stanton A,Sacchi M D.2017.Elastic least-squares one-way waveequation migration.Geophysics,82(4):S293-S305.
Sun J Z,Fomel S,Zhu T Y,et al.2016.Q-compensated leastsquares reverse time migration using low-rank one-step wave extrapolation.Geophysics,81(4):S271-S279.
Tang Y X.2009.Target-oriented wave-equation least-squares migration/inversion with phase-encoded Hessian.Geophysics,74(6):WCA95-WCA107.
Tarantola A.1984.Linearized inversion of seismic reflection data.Geophysical Prospecting,32(6):998-1015.
Vigh D,Starr E W.2008.3Dprestack plane-wave,full-waveform inversion.Geophysics,73(5):VE135-VE144.
Wong M,Biondi B L,Ronen S.2010.Joint least-squares inversion of up-and down-going signal for ocean bottom data sets.∥80th Annual International Meeting.SEG,Expanded Abstracts,2752-2756.
Wong M,Ronen S,Biondi B L.2011.Least-squares reverse time migration/inversion for ocean bottom data:A case study.∥81st Annual International Meeting.San Antonio,Texas:SEG,Expanded Abstracts,2369-2373.
Wong M,Biondi B L,Ronen S.2015.Imaging with primaries and free-surface multiples by joint least-squares reverse time migration.Geophysics,80(6):S223-S235.
Zhan G,Pestana R C,Stoffa P L.2012.Decoupled equations for reverse time migration in tilted transversely isotropic media.Geophysics,77(2):T37-T45.
Zhan G,Pestana R C,Stoffa P L.2013.An efficient hybrid pseudospectral/finite-difference scheme for solving the TTIpure P-wave equation.Journal of Geophysics and Engineering,10(2):025004.
敖瑞德,董良国,迟本鑫.2015.不依赖子波、基于包络的FWI初始模型建立方法研究.地球物理学报,58(6):1998-2010,doi:10.6038/cjg20150615.
程玖兵,康玮,王腾飞.2013.各向异性介质qP波传播描述I:伪纯模式波动方程.地球物理学报,56(10):3474-3486,doi:10.6038/cjg20131022.
方刚,Fomel S,杜启振.2014.交错网格Lowrank有限差分及其在逆时偏移中的应用.中国石油大学学报(自然科学版),38(2):44-51.
黄建平,黄金强,李振春等.2016.TTI介质一阶qP波方程交错网格伪谱法正演模拟.石油地球物理勘探,51(3):487-496.
黄金强,李振春.2017.基于Low-rank分解的复杂TI介质纯qP波正演模拟与逆时偏移.地球物理学报,60(2):704-721,doi:10.6038/cjg20170223.
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