大步长波场深度延拓的理论
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摘要
波场延拓是地震偏移成像的基础.快速进行目标区波场延拓对石油勘探中急需发展的深部地震勘探和无组合海量地震数据的成像有重要意义.在目标区成像中,目前已有的波场延拓方法,包括基于走时计算的Dix方法和射线追踪方法,以及基于小步长波场递推的方法,在适应复杂介质、计算精度和计算效率的某一方面还不能完全满足实际需要.本文提出一种基于“算子相位”李代数积分的快速计算延拓算子的方法,称为大步长波场延拓方法.在该方法中,指向目标区的波场延拓算子象征的复相位被表示成波数的线性组合.线性组合的系数是层速度函数及其导数的深度积分,计算和存储较为方便.波场延拓算子通过相移算子加校正的方法,利用快速Fourier变换在空间域和波数域予以实现.利用动力学等价关系导出了便于计算的表达式.本文比较了算子主象征函数用一步法展开和用两步法展开的精度,从而说明大步长方法的精度要高于递推方法.在横向和纵向线性变化介质中,将大步长方法的脉冲响应与递推法做了比较,说明大步长延拓算子的走时精度主要取决于相移因子中的横向变速校正项;且在各种近似下,大步长算子发生的频散都非常小.
Wavefield extrapolation is the foundation of seismic migration imaging.Finding a way of quick and object oriented wavefield extrapolation is significant not only for deep seismic exploration but also for imaging with super-tremendous seismic data acquired from non-combination receivers,both highly needed to develop in petroleum prospecting.To image target zones,as far as all present wavefield extrapolation methods are concerned,including travel-time based Dix-formula methods,ray tracing,small-step depth recursion,etc.,the adaptability to complex media,precision and computing efficiency are not satisfying for practical needs in some respects.In this paper,a new technique(so-called large-step wavefield depth extrapolation method),that is based on the Lie algebraic integral of operator's phase to compute depth extrapolation operator efficiently,is proposed.In the method,the complex phase of object oriented wavefield extrapolation operator's symbol is expressed as a linear combination of wavenumbers,and the coefficients of linear combination are all in the form of the integral of interval velocity functions and their derivatives over depth.Moreover,the computation is convenient and needs less storage space.With phase shift plus correction,the wavefield extrapolation operator is implemented using FFT(Fast Fourier Transform) in spatial and wavenumber domains.Here,the kinetic equivalent relationship is resorted to derive the expression convenient for computation.We compare the precision of one-step scheme with that of two-step scheme for expanding an operator's primary symbol function,which illustrates the large-step scheme is more accurate than its recursive counterpart.Besides,in a linearly variant medium laterally and vertically,the point source pulse response of the largestep method is compared with that of depth recursion methods.The numerical examples indicate that travel-time precision of the large-step extrapolation operator mainly depends on the lateral velocity variation modification items in phase shift operator.In addition,in different approximation cases,the dispersion caused by the large-step operator is rather small.
Wavefield extrapolation is the foundation of seismic migration imaging.Finding a way of quick and object oriented wavefield extrapolation is significant not only for deep seismic exploration but also for imaging with super-tremendous seismic data acquired from non-combination receivers,both highly needed to develop in petroleum prospecting.To image target zones,as far as all present wavefield extrapolation methods are concerned,including travel-time based Dix-formula methods,ray tracing,small-step depth recursion,etc.,the adaptability to complex media,precision and computing efficiency are not satisfying for practical needs in some respects.In this paper,a new technique(so-called large-step wavefield depth extrapolation method),that is based on the Lie algebraic integral of operator's phase to compute depth extrapolation operator efficiently,is proposed.In the method,the complex phase of object oriented wavefield extrapolation operator's symbol is expressed as a linear combination of wavenumbers,and the coefficients of linear combination are all in the form of the integral of interval velocity functions and their derivatives over depth.Moreover,the computation is convenient and needs less storage space.With phase shift plus correction,the wavefield extrapolation operator is implemented using FFT(Fast Fourier Transform) in spatial and wavenumber domains.Here,the kinetic equivalent relationship is resorted to derive the expression convenient for computation.We compare the precision of one-step scheme with that of two-step scheme for expanding an operator's primary symbol function,which illustrates the large-step scheme is more accurate than its recursive counterpart.Besides,in a linearly variant medium laterally and vertically,the point source pulse response of the largestep method is compared with that of depth recursion methods.The numerical examples indicate that travel-time precision of the large-step extrapolation operator mainly depends on the lateral velocity variation modification items in phase shift operator.In addition,in different approximation cases,the dispersion caused by the large-step operator is rather small.
引文
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[2]Claerbout J F,Doherty S M.Downward continuation of moveout-corrected seismograms.Geophysics,1972,37(5):741~768
[3]齐民友,徐超江,王维克.现代偏微分方程引论.武汉:武汉大学出版社,2005Qi M Y,Xu C J,Wang W K.Introduction of Modern PartialDifferential Equation(in Chinese).Wuhan:Wuhan UniversityPress,2005
[4]Dix C H.Seismic velocities from surface measurements.Geophysics,1955,20(1):68~86
[5]Hubral P.Time migration-some ray theoretical aspects.Geophys.Prosp.,1975,25:738~745
[6]Larner K L,Hatton L,Gibson B S,et al.Depth migration of imagedtime section.Geophysics,1981,46(5):734~750
[7]Sun C,Martinez R.3D Kirchhoff PS-wave prestack time migrationforV(z)and VTI media.73rd Ann.Internat.Mtg.,Soc.Expl.Geophys..Expanded Abstracts,2003.957~960
[8]Jiao J,Martinez R.Horizon-based residual depth and time migrationvelocity analysis.73rd Ann.Internat.Mtg.,Soc.Expl.Geophys..Expanded Abstracts,2003.2108~2111
[9]Castle R J.Wave-equation migration in the presence of lateralvelocity variations.Geophysics,1982.47(7):1001~1011
[10]王华忠,徐蔚亚,徐兆涛等.DMO和叠前时间偏移的共同起点.石油地球物理勘探,2002,37(3):224~229Wang H Z,Xu W Y,Xu Z T,et al.The same start of DMO andprestack time migration.Oil Geophysical Prospecting(in Chinese),2002,37(3):224~229
[11]Bleistein N,Cohen J K,Stockwell J W.Mathematics ofMultidimensional Seismic Imaging,Migration and Inversion.NewYork:Spinger-Verlag,2001
[12]Cerveny V,Molotkov I A,Psencik I.Ray Methods in Seismology.Prague:Unverzita Karlova,1977
[13]杨长春,刘兴材,李幼铭等.地震叠前深度偏移方法流程及应用.地球物理学报,1996,39(3):409~415Yang C C,Liu X C,Li Y M,et al.A methodology for seismicprestack depth migration.Chinese J.Geophys.(in Chinese),1996,39(3):409~415
[14]徐升,杨长春,刘洪等.射线追踪的微变网格方法.地球物理学报,1996,39(1):97~102Xu S,Yang C H,Liu H,et al.A grid-changeable method for raytracing.Chinese J.Geophys.(in Chinese),1996,39(1):97~102
[15]高亮,李幼铭,陈旭荣等.地震射线辛几何算法初探.地球物理学报,2000,43(3):402~410Gao L,Li Y M,Chen X R,et al.An attempt to seismic ray tracingwith Symplectic algorithm.Chinese J.Geophys.(in Chinese),2000,43(3):402~410
[16]陈景波,秦孟兆.射线追踪、辛几何算法与波场的数值模拟.计算物理,2001,18(6):481~486Chen J B,Qin M Z.Ray tracing,Symplectic algorithms and wavefield simulation.Chinese J.Computational Physics(in Chinese),2001,18(6):481~486
[17]秦孟兆,陈景波.Maslov渐近理论与辛几何算法.地球物理学报,2000,43(4):522~533Qin M Z,Chen J B.Maslov asymptotic theory and Symplecticalgorithm.Chinese J.Geophys.(in Chinese),2000,43(4):522~533
[18]张仁和.水下声道中的反转点会聚区(II)广义射线理论.声学学报,1982,7(2):75~87Zhang R H.Turning-point convergence-zones in underwater soundchannel(Ⅱ),a generalized ray theory.Acta Acoustic(in Chinese),1982,7(2):75~87
[19]Zhang R H,Liu H,He Y.The WKBZ adiabatic mode approach tosound propagation gradually range-dependent channels.AnnualReport of State Key Lab ofAcoustics,1992,2:9~12
[20]Zhang R H,He Y.Long range pulse propagation in ocean channels.Journal of Sound and Vibration,1995,179(2):313~325
[21]刘洪,罗明秋,李幼铭.共反射点轨迹的Hamilton方法.地球物理学报,1999,42(5):685~694Liu H,Luo M Q,Li Y M.Hamilton method in describing common-reflection-point trajectory.Chinese J.Geophys.(in Chinese),1999,42(5):685~694
[22]刘洪,罗明秋,李幼铭.横向线性变速介质中共炮检距延拓.地球物理学报,1999,42(6):809~817Liu H,Luo M Q,Li Y M.Common offset continuation in laterallinear varied velocity media.Chinese J.Geophys.(in Chinese),1999,42(6):809~817
[23]马在田.高阶方程偏移的分裂算法.地球物理学报,1983,26(4):377~388Ma Z T.A splitting-up method for solution of higher-order migrationequation by finite-difference scheme.Chinese J.Geophys.(inChinese),1983,26(4):377~388
[24]Zhang G Q.A new algorithm for finite-difference migration of steepdips.Geophysics,1988,53(2):167~175
[25]Ristow D,Ruhl T.Fourier finite-difference migration.Geophysics,1994,59(12):1882~1893
[26]Ristow D,Ruhl T.3-D implicit finite-difference migration by multi-way splitting.Geophysics,1997,62(2):554~567
[27]李幼铭.知识创新工程重大项目在大庆油田的技术成果要览.地球物理学进展,2003,18(1):5~18Li Y M.Summary of key project technique achievements inknowledge innovation engineering for Daqing Oilfield.Progress inGeophysics(in Chinese),2003,18(1):5~18
[28]杨辉,刘洪,李幼铭.单程波方程偏移算子辛格式.地球物理学报,2003,46(4):533~538Yang H,Liu H,Li Y M.High order Symplectic scheme for one-waywave equation operator.Chinese J.Geophys.(in Chinese),2003,46(4):533~538
[29]刘礼农,刘洪,李幼铭.SEG/EAGE盐丘和推覆体模型的波动方程三维叠前深度偏移成像.地球物理学报,2004,47(2):312~320Liu L N,Liu H,Li Y M.The wave-equation 3-D prestack depthmigration of SEG/EAGE salt and overthrust model.Chinese J.Geophys.(in Chinese),2004,47(2):312~320
[30]Wu R S.Wide-angle elastic wave one-way propagation inheterogeneous media and an elastic wave complex-screen method.Journal of Geophysical Research,1994,99(B1):751~766
[31]Wu R S.Wave propagation,scattering and imaging using dual-domain one-way and one-return propagators.Pure and AppliedGeophysics,2003,160(3):509~539
[32]Zhang J F,Li Y M.Numerical simulation of elastic wave propagationin inhomogeneous media.Wave Motion,1997,25(1):109~125
[33]Clearbout J F.Multidimensional recursive filters via a helix.Geophysics,1998,63(5):1532~1541
[34]Zhang G Q,Zhang Y,Zhou H B.Helical finite-difference schemesfor 3-D depth migration.70th Ann.Internat.Mtg.,Soc.Expl.Geophys..Expanded Abstracts,2000.862~865
[35]罗明秋,朱国同,刘洪等.适用于三维隐式叠前深度偏移中矩阵求逆的混合算法.地球物理学报,2003,46(5):684~689Luo M Q,Zhu G T,Liu H,et al.A hybrid method of matrixinversion suited for 3D implicit prestack depth migration.Chinese J.Geophys.(in Chinese),2003,46(5):684~689
[36]Wen Y W,Ng M,Ching W,et al.A note on the stability of Toeplitzmatrix inversion formulas.Applied Mathematics Letter,2004,17(8):903~907
[37]王华忠,马在田,曹景忠.优化系数傍轴近似方程三维一步法偏移.石油地球物理勘探,1998,33(2):170~184Wang H Z,Ma Z T,Cao J Z.Three dimensional one-pass migrationusing paraxial approximate equation with optimized coefficients.OilGeophysical Prospecting(in Chinese),1998,33(2):170~184
[38]宋健.高维函数和流形在低维可视空间中的最优表达.科学通报,2001,46(12):977~984Song J.The optimization expression of functions and manifolds in highdimensions by ones in low dimensions.Chin.Sci.Bull.(inChinese),2001,46(12):977~984
[39]刘洪,刘礼农,杨辉.波场延拓算子的最佳乘积基函数展开方法研究.见:张忠杰等主编.中国大陆地球深部结构与动力学研究.北京:科学出版社,2004.483~490Liu H,Liu L N,Yang H.The optimized product base function ofwavefield extrapolation operator.In:Zhang Z J eds.Research onDeep Structure and Geodynamics of China Continental(in Chinese).Beijing:Science Press,2004.483~490
[40]Chen J B,Liu H.Optimization approximation with separablevariables for the one-way wave operator.Geophysical Research Letter,2004,31:L06613
[41]Feng K.On difference schemes and Symplectic geometry.In:Feng Ked.Sympletic Differential Geometry and Differential Equations.Beijing:Science Press,1985
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