横向各向同性介质拟声波方程及其在逆时偏移中的应用
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摘要
地下岩石的速度各向异性影响地震波的传播与成像.横向各向同性(TI)介质为最普遍的等效各向异性模型.引入TI介质拟声波方程可以避免复杂的弹性波方程求解以及各向异性介质波场分离,以满足对纵波成像的实际需要.本文从垂直横向各向同性(VTI)介质弹性波方程出发,推导出正应力表达的拟声波方程以及相应的纵波分量的表达式,进而分析从频散关系得到的拟声波方程的物理意义,而后将拟声波方程扩展到更一般的倾斜横向各向同性(TTI)介质中.波前快照与群速度平面的对比验证了拟声波方程可以很好地近似描述qP波的运动学特征.在此基础上,将拟声波方程应用在逆时偏移中并与其特例声波近似方程进行对比,讨论了计算效率、稳定性等实际问题.数值试验表明VTI介质情况下采用声波近似方程可以提高计算效率,而TTI介质qP-qSV波方程则在效率相当的情况下可以保证稳定性.SEG/HESS模型和逆冲模型逆时偏移试验验证了本文TI介质拟声波方程的实用性.
Neglecting anisotropy in seismic imaging may result in remarkable positional errors and focusing problem,especially for long-offset,wide azimuth seismic data.A transversely isotropic(TI) medium related to shale formations and periodic thin layers is one of the simplest and most practical approximations for anisotropic media in seismic imaging.For P-wave imaging,we introduce pseudo-acoustic wave equations to avoid solving elastic equations and wavefiled separation in TI media.In this paper,starting from elastic wave equations in VTI media,we present new pseudo-acoustic wave equations and the corresponding expression for qP wave.The equations expressed by wavefield and expressed by stress indicate the physical meaning of second-order coupled equations derived from the dispersion relationship.Then we extend the pseudo-acoustic wave equations to TTI media by tilting the coordinate system.The consistency between snapshots and group velocity surfaces shows their good kinematic approximation for qP wave.We use the pseudo-acoustic wave equations as the basis for TI reverse-time migration and compare qP-qSV wave equation with acoustic approximation equation.Then we also discuss the computational efficiency and stability issue.We show numerical examples to demonstrate that acoustic approximation equations save much computational cost in VTI media;but qP-qSV wave equations involving no acoustic approximation can guarantee the stability.Testing result of reverse-time migration with the SEG/HESS model and overthrust model validated these equations.
Neglecting anisotropy in seismic imaging may result in remarkable positional errors and focusing problem,especially for long-offset,wide azimuth seismic data.A transversely isotropic(TI) medium related to shale formations and periodic thin layers is one of the simplest and most practical approximations for anisotropic media in seismic imaging.For P-wave imaging,we introduce pseudo-acoustic wave equations to avoid solving elastic equations and wavefiled separation in TI media.In this paper,starting from elastic wave equations in VTI media,we present new pseudo-acoustic wave equations and the corresponding expression for qP wave.The equations expressed by wavefield and expressed by stress indicate the physical meaning of second-order coupled equations derived from the dispersion relationship.Then we extend the pseudo-acoustic wave equations to TTI media by tilting the coordinate system.The consistency between snapshots and group velocity surfaces shows their good kinematic approximation for qP wave.We use the pseudo-acoustic wave equations as the basis for TI reverse-time migration and compare qP-qSV wave equation with acoustic approximation equation.Then we also discuss the computational efficiency and stability issue.We show numerical examples to demonstrate that acoustic approximation equations save much computational cost in VTI media;but qP-qSV wave equations involving no acoustic approximation can guarantee the stability.Testing result of reverse-time migration with the SEG/HESS model and overthrust model validated these equations.
引文
[1]Helbig K.Elliptical anisotropy-Its significance and meaning.Geophysics,1983,48(7):825-832.
[2]Muir F,Dellinger J.A practical anisotropic system.SEP,1985,44:55-58.
[3]Thomsen L.Weak elastic anisotropy.Geophysics,1986,51(10):1954-1966.
[4]Cohen J K.Analytic study of the effective parameters fordetermination of the NMO velocity function in transverselyisotropic media.Colorado School of Mines Center for WavePhenomena,1996:CWP-191.
[5]Alkhalifah T.Acoustic approximations for processing intransversely isotropic media.Geophysics,1998,60(2):1550-1566.
[6]Fletcher R P,Du X,Fowler P J.Reverse time migration intilted transversely isotropic(TTI)media.Geophysics,2009,74(6):WCA179-WCA187.
[7]Fowler P J,Du X,Fletcher R P.Coupled equations forreverse time migration in transversely isotropic media.Geophysics,2010,75(1):S11-S22.
[8]McMechan G A.Migration by extrapolation of time-dependent boundary values.Geophysical Prospecting,1983,31(3):413-420.
[9]Whitmore D N.Iterative Depth imaging by back timepropagation.53rd Annual International Meeting,SEGExpanded Abstracts,1983:827-830.
[10]Baysal E,Kosloff D D,Sherwood J W C.Reverse timemigration.Geophysics,1983,48(11):1514-1524.
[11]董良国,马在田,曹景忠等.一阶弹性波方程交错网格高阶差分解法.地球物理学报,2000,43(3):411-419.Dong L G,Ma Z T,Cao J Z,et al.A staggered-grid high-order difference method of one-order elastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(3):411-419.
[12]张美根,王妙月,李小凡等.各向异性弹性波场的有限元数值模拟.地球物理学进展,2002,17(3):384-389.Zhang M G,Wang M Y,Li X F,et al.Finite elementforward modeling of anisotropic elastic waves.Progress inGeophysics(in Chinese),2002,17(3):384-389.
[13]孙文博,孙赞东.基于伪谱法的VSP逆时偏移及其应用研究.地球物理学报,2010,53(9):2196-2203.Sun W B,Sun Z D.VSP reverse time migration based on thepseudo-spectral method and its applications.Chinese J.Geophys.(in Chinese),2010,53(9):2196-2203.
[14]王童奎,李瑞华,李小凡等.横向各向同性介质中地震波场谱元法数值模拟.地球物理学进展,2007,22(3):778-784.Wang T K,Li R H,Li X F,et al.Numerical spectralelement modeling for seismic wave propagation intransversely isotropic medium.Progress in Geophysics(inChinese),2007,22(3):778-784.
[15]王童奎,付兴深,朱德献等.谱元法叠前逆时偏移研究.地球物理学进展,2008,23(3):681-685.Wang T K,Fu X S,Zhu D X,et al.Spectral-elementmethod for prestack reverse-time migration.Progress inGeophysics(in Chinese),2008,23(3):681-685.
[16]Chang W F,McMechan G A.Reverse-time migration ofoffset vertical seismic profiling data using thr excitation-timeimaging condition.Geophysics,1986,51(1):67-84.
[17]Claerbout J F.Toward a unified theory of reflector mapping.Geophysics,1971,36(3):467-481.
[18]李博,刘红伟,刘国峰等.地震叠前逆时偏移算法的CPU/GPU实施对策.地球物理学报,2010,53(12):2938-2943.Li B,Liu H W,Liu G F,et al.Computational strategy ofseismic pre-stack reverse time migration on CPU/GPU.Chinese J.Geophys.(in Chinese),2010,53(12):2938-2943.
[19]刘红伟,刘洪,邹振等.地震叠前逆时偏移中的去噪与存储.地球物理学报,2010,53(9):2171-2180.Liu H W,Liu H,Zou Z,et al.The problems of denoise andstorage in seismic reverse time migration.Chinese J.Geophys.(in Chinese),53(9):2171-2180.
[20]刘红伟,李博,刘洪等.地震叠前逆时偏移高阶有限差分算法及GPU实现.地球物理学报,2010,53(7):1725-1733.Liu H W,Li B,Liu H,et al.The algorithm of high orderfinite difference pre-stack reverse time migration and GPUimplementation.Chinese J.Geophys.(in Chinese),2010,53(7):1725-1733.
[21]吴国忱.TI介质qP波地震深度偏移成像方法研究[博士论文].上海:同济大学,2006.Wu G C.Seismic depth-migration imaging method of qP wavein TI media[Ph.D.thesis](in Chinese).Shanghai:TongjiUniversity,2006.
[22]朱兆林.倾斜横向各向同性介质中P-S波处理方法研究[博士论文].上海:同济大学,2008.Zhu Z L.The processing method of P-S wave in titledtransversely isotropic media[Ph.D.thesis](in Chinese).Shanghai:Tongji University,2008.
[23]杜启振,秦童.横向各向同性介质弹性波多分量叠前逆时偏移.地球物理学报,2009,52(3):801-807.Du Q Z,Qin T.Multicomponent prestack reverse-timemigration of elastic waves in transverse isotropic medium.Chinese J.Geophys.(in Chinese),2009,52(3):801-807.
[24]陈沫.横向各向同性介质地震波场逆时偏移.岩性油气藏,2009,21(4):78-81.Chen M.The seismic wave field reverse-time migration intransversely isotropic media.Lithologic Reservoirs(inChinese),2009,21(4):78-81.
[25]Tsvankin I.Seismic Signatures and Analysis of ReflectionData in Anisotropic Media.Elsevier Science,2001.
[26]Dellinger J.Anisotropic seismic wave propagation.StanfordUniversity,1991.
[27]Alkhalifah T.An acoustic wave equation for anisotropicmedia.Geophysics,2000,65(4):1239-1250.
[28]Zhou H B,Zhang G Q,Bloor R.An anisotropic acousticwave equation for modeling and migration in 2DTTI media.76th Annual International Meeting,SEG,ExpandedAbstracts,2006:194-198.
[29]Fletcher R,Du X,Fowler P J.A new pseudo-acoustic waveequation for TI media.78th Annual International Meeting,SEG,Expanded Abstracts,2008:2082-2086.
[30]Duveneck E,Milcik P,Bakker P M.Acoustic VTI waveequations and their application for anisotropic reverse-timemigration.78th Annual International Meeting,SEG,Expanded Abstracts,2008:2186-2190.
[31]Cerveny V.Seismic Ray Theory.Cambridge:CambridgeUniversity Press,2001.
[32]Grechka V,Zhang L,Rector J W.Shear waves in acousticanisotropic media.Geophysics,2004,69(2):576-582.
[33]Zhang Y,Zhang H.A stable TTI reverse time migration andits implementation.79th Annual International Meeting,SEGExpanded Abstracts,2009:2794-2798.
[34]Jin S W,Jiang F,Ren Y Q.Comparison of isotropic,VTIand TTI reverse time migration:an experiment on BPanisotropic benchmark dataset.80th Annual InternationalMeeting,SEG Expanded Abstracts,2010:3198-3202.
[35]Youn O K,Zhou H W.Depth imaging with multiples.Geophysics,2001,66(1):246-255.
[36]Fei T,Dellinger J A,Murphy G E,et al.Anisotropic true-amplitude migration.68th Annual International Meeting,SEG Expanded Abstracts,1998:1677-1679.
[2]Muir F,Dellinger J.A practical anisotropic system.SEP,1985,44:55-58.
[3]Thomsen L.Weak elastic anisotropy.Geophysics,1986,51(10):1954-1966.
[4]Cohen J K.Analytic study of the effective parameters fordetermination of the NMO velocity function in transverselyisotropic media.Colorado School of Mines Center for WavePhenomena,1996:CWP-191.
[5]Alkhalifah T.Acoustic approximations for processing intransversely isotropic media.Geophysics,1998,60(2):1550-1566.
[6]Fletcher R P,Du X,Fowler P J.Reverse time migration intilted transversely isotropic(TTI)media.Geophysics,2009,74(6):WCA179-WCA187.
[7]Fowler P J,Du X,Fletcher R P.Coupled equations forreverse time migration in transversely isotropic media.Geophysics,2010,75(1):S11-S22.
[8]McMechan G A.Migration by extrapolation of time-dependent boundary values.Geophysical Prospecting,1983,31(3):413-420.
[9]Whitmore D N.Iterative Depth imaging by back timepropagation.53rd Annual International Meeting,SEGExpanded Abstracts,1983:827-830.
[10]Baysal E,Kosloff D D,Sherwood J W C.Reverse timemigration.Geophysics,1983,48(11):1514-1524.
[11]董良国,马在田,曹景忠等.一阶弹性波方程交错网格高阶差分解法.地球物理学报,2000,43(3):411-419.Dong L G,Ma Z T,Cao J Z,et al.A staggered-grid high-order difference method of one-order elastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(3):411-419.
[12]张美根,王妙月,李小凡等.各向异性弹性波场的有限元数值模拟.地球物理学进展,2002,17(3):384-389.Zhang M G,Wang M Y,Li X F,et al.Finite elementforward modeling of anisotropic elastic waves.Progress inGeophysics(in Chinese),2002,17(3):384-389.
[13]孙文博,孙赞东.基于伪谱法的VSP逆时偏移及其应用研究.地球物理学报,2010,53(9):2196-2203.Sun W B,Sun Z D.VSP reverse time migration based on thepseudo-spectral method and its applications.Chinese J.Geophys.(in Chinese),2010,53(9):2196-2203.
[14]王童奎,李瑞华,李小凡等.横向各向同性介质中地震波场谱元法数值模拟.地球物理学进展,2007,22(3):778-784.Wang T K,Li R H,Li X F,et al.Numerical spectralelement modeling for seismic wave propagation intransversely isotropic medium.Progress in Geophysics(inChinese),2007,22(3):778-784.
[15]王童奎,付兴深,朱德献等.谱元法叠前逆时偏移研究.地球物理学进展,2008,23(3):681-685.Wang T K,Fu X S,Zhu D X,et al.Spectral-elementmethod for prestack reverse-time migration.Progress inGeophysics(in Chinese),2008,23(3):681-685.
[16]Chang W F,McMechan G A.Reverse-time migration ofoffset vertical seismic profiling data using thr excitation-timeimaging condition.Geophysics,1986,51(1):67-84.
[17]Claerbout J F.Toward a unified theory of reflector mapping.Geophysics,1971,36(3):467-481.
[18]李博,刘红伟,刘国峰等.地震叠前逆时偏移算法的CPU/GPU实施对策.地球物理学报,2010,53(12):2938-2943.Li B,Liu H W,Liu G F,et al.Computational strategy ofseismic pre-stack reverse time migration on CPU/GPU.Chinese J.Geophys.(in Chinese),2010,53(12):2938-2943.
[19]刘红伟,刘洪,邹振等.地震叠前逆时偏移中的去噪与存储.地球物理学报,2010,53(9):2171-2180.Liu H W,Liu H,Zou Z,et al.The problems of denoise andstorage in seismic reverse time migration.Chinese J.Geophys.(in Chinese),53(9):2171-2180.
[20]刘红伟,李博,刘洪等.地震叠前逆时偏移高阶有限差分算法及GPU实现.地球物理学报,2010,53(7):1725-1733.Liu H W,Li B,Liu H,et al.The algorithm of high orderfinite difference pre-stack reverse time migration and GPUimplementation.Chinese J.Geophys.(in Chinese),2010,53(7):1725-1733.
[21]吴国忱.TI介质qP波地震深度偏移成像方法研究[博士论文].上海:同济大学,2006.Wu G C.Seismic depth-migration imaging method of qP wavein TI media[Ph.D.thesis](in Chinese).Shanghai:TongjiUniversity,2006.
[22]朱兆林.倾斜横向各向同性介质中P-S波处理方法研究[博士论文].上海:同济大学,2008.Zhu Z L.The processing method of P-S wave in titledtransversely isotropic media[Ph.D.thesis](in Chinese).Shanghai:Tongji University,2008.
[23]杜启振,秦童.横向各向同性介质弹性波多分量叠前逆时偏移.地球物理学报,2009,52(3):801-807.Du Q Z,Qin T.Multicomponent prestack reverse-timemigration of elastic waves in transverse isotropic medium.Chinese J.Geophys.(in Chinese),2009,52(3):801-807.
[24]陈沫.横向各向同性介质地震波场逆时偏移.岩性油气藏,2009,21(4):78-81.Chen M.The seismic wave field reverse-time migration intransversely isotropic media.Lithologic Reservoirs(inChinese),2009,21(4):78-81.
[25]Tsvankin I.Seismic Signatures and Analysis of ReflectionData in Anisotropic Media.Elsevier Science,2001.
[26]Dellinger J.Anisotropic seismic wave propagation.StanfordUniversity,1991.
[27]Alkhalifah T.An acoustic wave equation for anisotropicmedia.Geophysics,2000,65(4):1239-1250.
[28]Zhou H B,Zhang G Q,Bloor R.An anisotropic acousticwave equation for modeling and migration in 2DTTI media.76th Annual International Meeting,SEG,ExpandedAbstracts,2006:194-198.
[29]Fletcher R,Du X,Fowler P J.A new pseudo-acoustic waveequation for TI media.78th Annual International Meeting,SEG,Expanded Abstracts,2008:2082-2086.
[30]Duveneck E,Milcik P,Bakker P M.Acoustic VTI waveequations and their application for anisotropic reverse-timemigration.78th Annual International Meeting,SEG,Expanded Abstracts,2008:2186-2190.
[31]Cerveny V.Seismic Ray Theory.Cambridge:CambridgeUniversity Press,2001.
[32]Grechka V,Zhang L,Rector J W.Shear waves in acousticanisotropic media.Geophysics,2004,69(2):576-582.
[33]Zhang Y,Zhang H.A stable TTI reverse time migration andits implementation.79th Annual International Meeting,SEGExpanded Abstracts,2009:2794-2798.
[34]Jin S W,Jiang F,Ren Y Q.Comparison of isotropic,VTIand TTI reverse time migration:an experiment on BPanisotropic benchmark dataset.80th Annual InternationalMeeting,SEG Expanded Abstracts,2010:3198-3202.
[35]Youn O K,Zhou H W.Depth imaging with multiples.Geophysics,2001,66(1):246-255.
[36]Fei T,Dellinger J A,Murphy G E,et al.Anisotropic true-amplitude migration.68th Annual International Meeting,SEG Expanded Abstracts,1998:1677-1679.
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