TTI介质qP波伪谱法正演模拟
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摘要
基于Tsvankin的VTI介质qP波精确相速度公式,利用坐标变换的方法,导出了三维空间TTI介质qP波的精确相速度公式;结合Thomsen的弱各向异性假设以及泰勒展开式,简化了三维TTI介质qP波的精确相速度公式,建立了近似相速度公式; TTI模型的计算结果表明,近似相速度公式能较好地逼近精确相速度公式。从近似相速度公式出发,推导了三维qP波的频散关系以及时间—波数域波动方程,构建了纯qP波时间—波数域波场递推格式,分析了递推公式的稳定性条件。最后,基于伪谱法对三维均匀(VTI、TTI)模型、Hess VTI模型以及BP 2007 TTI模型进行qP波正演模拟。结果表明,利用所提方法得到的波场快照没有伪横波干扰,波场计算结果稳定。
Based on Tsvankins exact phase velocity for VTI media,we obtain the exact phase velocity of qP-wave for 3D TTI media with the coordinate transformation.Then following Thomsen anisotropic parameters,we gain the approximate formula of phase velocity for 3D TTI media with Taylor series.The phase velocity analysis shows that the approximate formula could well fit the exact formula.From the approximate formula of phase velocity,we derive the qP-wave dispersion relation and wave equation in the time-wavenumber domain for 3D TTI media.The stability of the wave equation is analyzed.Numerical examples demonstrate that the solution of the wave equation proposed in this paper is stable and free of S-wave artifacts.
Based on Tsvankins exact phase velocity for VTI media,we obtain the exact phase velocity of qP-wave for 3D TTI media with the coordinate transformation.Then following Thomsen anisotropic parameters,we gain the approximate formula of phase velocity for 3D TTI media with Taylor series.The phase velocity analysis shows that the approximate formula could well fit the exact formula.From the approximate formula of phase velocity,we derive the qP-wave dispersion relation and wave equation in the time-wavenumber domain for 3D TTI media.The stability of the wave equation is analyzed.Numerical examples demonstrate that the solution of the wave equation proposed in this paper is stable and free of S-wave artifacts.
引文
[1] 黄金强,李振春.基于Low-rank分解的复杂TI介质纯qP波正演模拟与逆时偏移[J].地球物理学报,2017,60(2):704-721.HUANG Jinqiang,LI Zhenchun.Modeling and reverse time migration of pure quasi-P-waves in complex TI media with a low-rank decomposition[J].Chinese Journal of Geophysics,2017,60(2):704-721.
[2] 王娟,李振春,孙小东,等.TTI介质逆时偏移成像[J].石油地球物理勘探,2012,47(4):573-577.WANG Juan,LI Zhenchun,SUN Xiaodong,et al.Reverse time migration in tilt transversely isotropic(TTI) media[J].Oil Geophysical Prospecting,2012,47(4):573-577.
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[4] Alkhalifah T.An acoustic wave equation for aniso-tropic media[J].Geophysics,2000,65(4):1239-1250.
[5] Zhou H,Zhang G,Bloor R.An anisotropic acoustic wave equation for VTI media[C].Extended Abstracts of 68th EAGE Conference & Exhibition,2006,H033.
[6] Zhou H,Zhang G,Bloor R.An anisotropic acoustic wave equation for modeling and migration in 2D TTI media[C].SEG Technical Program Expanded Abstracts,2006,25:194-198.
[7] Zhang L,Rector J W,Hoversten G M.Finite-differ-ence modelling of wave propagation in acoustic tilted TI media[J].Geophysical Prospecting,2005,53(6):843-852.
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[12] Yoon K,Suh S,Ji J,et al.Stability and speedup issues in TTI RTM implementation[C].SEG Technical Program Expanded Abstracts,2010,29:3221-3225.
[13] 张岩,吴国忱.TTI介质qP波逆时偏移中伪横波噪声压制方法[J].地球物理学报,2013,56(6):2065-2076.ZHANG Yan,WU Guochen.Methods of removing pseudo SV-wave artifacts in TTI media qP-wave reverse-time migration[J].Chinese Journal of Geophysics,2013,56(6):2065-2076.
[14] Fowler P J,Du X,Fletcher R P.Coupled equations for reverse time migration in transversely isotropic media[J].Geophysics,2010,75(1):S11-S22.
[15] Fletcher R P,Du X,and Fowler P J.Reverse time migration in tilted transversely isotropic (TTI) media[J].Geophysics,2009,74(6):WCA179-WCA187.
[16] Du X,Bancroft J C,Lines L R.Reverse-time migration for tilted TI media[C].SEG Technical Program Expanded Abstracts,2005,24:1930-1934.
[17] Du X,Bancroft J C,Lines L R.Anisotropic reverse-time migration for tilted TI media[J].Geophysical Prospecting,2007,55(6):853-869.
[18] Liu F Q,Morton S A,Jiang S S,et al.Decoupled wave equations for P and SV waves in an acoustic VTI media[C].SEG Technical Program Expanded Abstracts,2009,28:2844-2848.
[19] Pestana R C,Stoffa P L.Time evolution of the wave equation using rapid expansion method[J].Geophy-sics,2010,75(4):121-131.
[20] Pestana R C,Ursin B,Stoffa P L.Separate P-and SV-wave equations for VTI media[C].SEG Technical Program Expanded Abstracts,2011,30:163-167.
[21] Chu C,Macy B K,Anno P D.Approximation of pure acoustic seismic wave propagation in TTI media[J].Geophysics,2011,76(5):WB97-WB107.
[22] Zhan G,Pestana R C,Stoffa P L.Decoupled equations for reverse time migration in tilted transversely isotropic media[J].Geophysics,2012,77(2):37-45.
[23] 王伟国,熊水金,徐华宁,等.TTI介质各向异性伪谱法逆时偏移[J].石油地球物理勘探,2012,47(4):566-572.WANG Weiguo,XIONG Shuijin,XU Huaning,et al.Reverse-time migration using pseudo-spectral method for tilted TI media[J].Oil Geophysical Prospecting,2012,47(4):566-572.
[24] 黄建平,黄金强,李振春,等.TTI介质一阶qP波方程交错网格伪谱法正演模拟[J].石油地球物理勘探,2016,51(3):487-496.HUANG Jianping,HUANG Jinqiang,LI Zhenchun,et al.Staggered grid pseudo-spectral numerical simulation of first-order quasi P-wave equation for TTI media[J].Oil Geophysical Prospecting,2016,51(3):487-496.
[25] 郭成锋,杜启振,张明强,等.改进的TTI介质纯P波方程正演模拟与逆时偏移[J].地球物理学报,2017,60(1):258-270.GUO Chengfeng,DU Qizhen,ZHANG Mingqiang,et al.Numerical simulation and reverse time migration using an improved pure P-wave equation in tilted transversely isotropic media[J].Chinese Journal of Geophysics,2017,60(1):258-270.
[26] Tsvankin I.P-wave signatures and notation for tranversely isotropic media:An overview[J].Geophysics,1996,61(2):467-483.
[27] Thomsen L.Weak elastic anisotropy[J].Geophysics,1986,51(10):1954-1966.
[28] 郝重涛,姚陈.任意空间取向TI介质中体波速度特征[J].地球物理学报,2007,50(2):546-555.HAO Chongtao,YAO Chen.Analysis of body-wave velocity characteristic for TI medium with arbitrary spatial orientation[J].Chinese Journal of Geophysics,2007,50(2):546-555.
[29] 姚振岸,孙成禹,邓小凡,等.任意空间取向TI介质弹性波速度特征分析[J].石油地球物理勘探,2017,52(4):715-733.YAO Zhen’an,SUN Chengyu,DENG Xiaofan,et al.Velocity characteristic analysis of elastic wave for TI media with arbitrary spatial orientation[J].Oil Geophysical Prospecting,2017,52(4):715-733.
[30] Cerjan C,Kosloff D,Kosloff R,et al.A nonreflecting boundary condition for discrete acoustic and elastic wave equation[J].Geophysics,1985,50(4):705-708.
[2] 王娟,李振春,孙小东,等.TTI介质逆时偏移成像[J].石油地球物理勘探,2012,47(4):573-577.WANG Juan,LI Zhenchun,SUN Xiaodong,et al.Reverse time migration in tilt transversely isotropic(TTI) media[J].Oil Geophysical Prospecting,2012,47(4):573-577.
[3] Alkhalifah T.Acoustic approximations for processing in transversely isotropic media[J].Geophysics,1998,63(2):623-631.
[4] Alkhalifah T.An acoustic wave equation for aniso-tropic media[J].Geophysics,2000,65(4):1239-1250.
[5] Zhou H,Zhang G,Bloor R.An anisotropic acoustic wave equation for VTI media[C].Extended Abstracts of 68th EAGE Conference & Exhibition,2006,H033.
[6] Zhou H,Zhang G,Bloor R.An anisotropic acoustic wave equation for modeling and migration in 2D TTI media[C].SEG Technical Program Expanded Abstracts,2006,25:194-198.
[7] Zhang L,Rector J W,Hoversten G M.Finite-differ-ence modelling of wave propagation in acoustic tilted TI media[J].Geophysical Prospecting,2005,53(6):843-852.
[8] Zhang H,Zhang Y.Reverse time migration in 3D heterogeneous TTI media[C].SEG Technical Program Expanded Abstracts,2008,27:2196-2200.
[9] Zhang Y,Zhang H.A stable TTI reverse time migration and its implementation[C].SEG Technical Program Expanded Abstracts,2009,28:2794-2798.
[10] Fletcher R P,Du X,Fowler P J.A new pseudo-acoustic wave equation for TI media[C].SEG Technical Program Expanded Abstracts,2008,27:2082-2086.
[11] 梁锴,吴国忱,印兴耀,等.三维TTI介质波动方程分解[J].石油地球物理勘探,2009,44(1):19-27.LIANG Kai,WU Guochen,YING Xinyao,et al.Wave equation decomposition in 3-D TTI medium[J].Oil Geophysical Prospecting,2009,44(1):19-27.
[12] Yoon K,Suh S,Ji J,et al.Stability and speedup issues in TTI RTM implementation[C].SEG Technical Program Expanded Abstracts,2010,29:3221-3225.
[13] 张岩,吴国忱.TTI介质qP波逆时偏移中伪横波噪声压制方法[J].地球物理学报,2013,56(6):2065-2076.ZHANG Yan,WU Guochen.Methods of removing pseudo SV-wave artifacts in TTI media qP-wave reverse-time migration[J].Chinese Journal of Geophysics,2013,56(6):2065-2076.
[14] Fowler P J,Du X,Fletcher R P.Coupled equations for reverse time migration in transversely isotropic media[J].Geophysics,2010,75(1):S11-S22.
[15] Fletcher R P,Du X,and Fowler P J.Reverse time migration in tilted transversely isotropic (TTI) media[J].Geophysics,2009,74(6):WCA179-WCA187.
[16] Du X,Bancroft J C,Lines L R.Reverse-time migration for tilted TI media[C].SEG Technical Program Expanded Abstracts,2005,24:1930-1934.
[17] Du X,Bancroft J C,Lines L R.Anisotropic reverse-time migration for tilted TI media[J].Geophysical Prospecting,2007,55(6):853-869.
[18] Liu F Q,Morton S A,Jiang S S,et al.Decoupled wave equations for P and SV waves in an acoustic VTI media[C].SEG Technical Program Expanded Abstracts,2009,28:2844-2848.
[19] Pestana R C,Stoffa P L.Time evolution of the wave equation using rapid expansion method[J].Geophy-sics,2010,75(4):121-131.
[20] Pestana R C,Ursin B,Stoffa P L.Separate P-and SV-wave equations for VTI media[C].SEG Technical Program Expanded Abstracts,2011,30:163-167.
[21] Chu C,Macy B K,Anno P D.Approximation of pure acoustic seismic wave propagation in TTI media[J].Geophysics,2011,76(5):WB97-WB107.
[22] Zhan G,Pestana R C,Stoffa P L.Decoupled equations for reverse time migration in tilted transversely isotropic media[J].Geophysics,2012,77(2):37-45.
[23] 王伟国,熊水金,徐华宁,等.TTI介质各向异性伪谱法逆时偏移[J].石油地球物理勘探,2012,47(4):566-572.WANG Weiguo,XIONG Shuijin,XU Huaning,et al.Reverse-time migration using pseudo-spectral method for tilted TI media[J].Oil Geophysical Prospecting,2012,47(4):566-572.
[24] 黄建平,黄金强,李振春,等.TTI介质一阶qP波方程交错网格伪谱法正演模拟[J].石油地球物理勘探,2016,51(3):487-496.HUANG Jianping,HUANG Jinqiang,LI Zhenchun,et al.Staggered grid pseudo-spectral numerical simulation of first-order quasi P-wave equation for TTI media[J].Oil Geophysical Prospecting,2016,51(3):487-496.
[25] 郭成锋,杜启振,张明强,等.改进的TTI介质纯P波方程正演模拟与逆时偏移[J].地球物理学报,2017,60(1):258-270.GUO Chengfeng,DU Qizhen,ZHANG Mingqiang,et al.Numerical simulation and reverse time migration using an improved pure P-wave equation in tilted transversely isotropic media[J].Chinese Journal of Geophysics,2017,60(1):258-270.
[26] Tsvankin I.P-wave signatures and notation for tranversely isotropic media:An overview[J].Geophysics,1996,61(2):467-483.
[27] Thomsen L.Weak elastic anisotropy[J].Geophysics,1986,51(10):1954-1966.
[28] 郝重涛,姚陈.任意空间取向TI介质中体波速度特征[J].地球物理学报,2007,50(2):546-555.HAO Chongtao,YAO Chen.Analysis of body-wave velocity characteristic for TI medium with arbitrary spatial orientation[J].Chinese Journal of Geophysics,2007,50(2):546-555.
[29] 姚振岸,孙成禹,邓小凡,等.任意空间取向TI介质弹性波速度特征分析[J].石油地球物理勘探,2017,52(4):715-733.YAO Zhen’an,SUN Chengyu,DENG Xiaofan,et al.Velocity characteristic analysis of elastic wave for TI media with arbitrary spatial orientation[J].Oil Geophysical Prospecting,2017,52(4):715-733.
[30] Cerjan C,Kosloff D,Kosloff R,et al.A nonreflecting boundary condition for discrete acoustic and elastic wave equation[J].Geophysics,1985,50(4):705-708.