振幅保真的单程波方程偏移理论
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摘要
本报告综述了近年来发展起来的真振幅单程波方程偏移理论,对各种基于单程波方程的偏移振幅作出了系统的分析介绍.为了得到正确的偏移振幅,叠后偏移之前必须进行球面扩散校正.但是,作为叠后相位移偏移算法的推广,Dubrulle氏的共炮检距偏移除了零炮检距和平层两种特例外,它不能给出正确的偏移振幅.通过分析,我们发现传统的单程波单炮偏移不是保振幅算法.利用真振幅单程波方程分解和校正地表的初值条件,我们可以将常规偏移方法改造为真振幅算法并且证明在高频渐近的意义下,新方法的偏移振幅等价于Kirchhoff反演的结果.进一步研究发现可以利用单平方根算子和双平方根算子输出真振幅共反射角剖面.我们的分析指出,这时正确的成像条件为乘积pUpD*.与真振幅共炮偏移所需的商型成像条件pU/pD相比,共反射角偏移的计算稳定性大为改善并且在实际地震资料处理中有重要的应用.
In this technical report,we review the recent development of true amplitude one-way wave equation migration and summarize the migrated amplitude performance from different one-way wave equations.To produce correct amplitude from postack phase-shift migration,we have to apply accurate geometrical spreading compensation in preprocessing.Although Dubrulle's common-offset phase-shift migration gives correct imaging position,it is not a true amplitude migration except for zero offset or flat reflectors.We have shown that the conventional common-shot one-way wave equation migration does not produce correct migration amplitude.Base on the true amplitude one-way wave equations and the surface boundary condition corrections,we developed true amplitude common-shot migration and have proved that it produces an inversion output that agrees asymptotically to Kirchhoff inversion.In the next step,we proposed a way to obtain true amplitude common-angle image gathers from both single-square-root and double-square-root wave equation migrations.Our analysis indicates that the p-(U)p~*-(D) imaging condition is a good candidate to produce true amplitude angle domain common imaging gathers.Compared to the p-(U)/p-(D) imaging condition which is required for true amplitude common-shot migration,the new approach suggests a more stable way of doing inversion and is more attractive to the real seismic data processing.
In this technical report,we review the recent development of true amplitude one-way wave equation migration and summarize the migrated amplitude performance from different one-way wave equations.To produce correct amplitude from postack phase-shift migration,we have to apply accurate geometrical spreading compensation in preprocessing.Although Dubrulle's common-offset phase-shift migration gives correct imaging position,it is not a true amplitude migration except for zero offset or flat reflectors.We have shown that the conventional common-shot one-way wave equation migration does not produce correct migration amplitude.Base on the true amplitude one-way wave equations and the surface boundary condition corrections,we developed true amplitude common-shot migration and have proved that it produces an inversion output that agrees asymptotically to Kirchhoff inversion.In the next step,we proposed a way to obtain true amplitude common-angle image gathers from both single-square-root and double-square-root wave equation migrations.Our analysis indicates that the p-(U)p~*-(D) imaging condition is a good candidate to produce true amplitude angle domain common imaging gathers.Compared to the p-(U)/p-(D) imaging condition which is required for true amplitude common-shot migration,the new approach suggests a more stable way of doing inversion and is more attractive to the real seismic data processing.
引文
[1]Beylkin G.Imaging of discontinuities in the inverse scatteringproblem by inversion of a generalized radon transform.J.Math.Phys.,1985,26:99~108
[2]Bleistein N.On the imaging of reflectors in the earth.Geophysics,1987,52:931~942
[3]Miller D,Oristaglio M,Beylkin G.A new slant on seismic imaging:migration and integral geometry.Geophysics,1987,52:943~964
[4]Claerbout J F.Toward a unified theory of reflector mapping.Geophysics,1971,36(3):467~481
[5]Gazdag J.Wave equation migration with the phase-shift method.Geophysics,1978,43:1342~1351
[6]Bleistein N,Cohen J K,Stockwell J W.Mathematics ofMultidimensional Seismic Inversion.Springer,2001
[7]Schneider W A.Integral formulation for migration in two and threedimensions.Geophysics,1978,43:49~76
[8]Hanitzsch C.Comparison of weights in prestack amplitude-preservingKirchhoff depth migration.Geophysics,1997,62(6):1812~1816
[9]Zhang Y,Gray S,Young J.Exact and approximate weights forKirchhoff migration.70th Ann.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,2000,Ⅱ:1036~1039
[10]Zhang Y,Karazincir M,Notfors C,Sun J,Hung B.Amplitudepreservingv(z)pre-stack Kirchhoff migration,demigration andmodeling.64th EAGE Annual Conference,2002
[11]Loewenthal D,Lu L,Roberson R,Sherwood J.The wave equationapplied to migration.Geophysics,1976,24:380~399
[12]Zhang Y,Gray S,Young J.True-amplitude common-offset,common-azimuthv(z)migration.Journal of Seismic Exploration,2002,11
[13]Ursin B.Offset-dependent geometrical spreading in a layeredmedium.Geophysics,1990,55(4):492~496
[14]Dubrulle A A.Numerical methods for the migration of constant-offsetsections in homogeneous and horizontally layered media.Geophysics,1983,48:1195~1203
[15]Ekren B O,Ursin B.True-amplitude frequency-wavenumber constant-offset migration.Geophysics,1999,64:915~924
[16]Winbow G,Schneider W.Weights for 3-D controlled amplitudeprestack time migration.69th Annual Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1999.Ⅱ:1110~1113
[17]Zhang Y,Sun J,Gray S,Notfors C,Bleistein N.Towards accurateamplitudes for one-way wavefield extrapolation of 3-D common shotrecords.71st Ann.Mtg.,Soc.Expl.Geophys.(workshop),2001
[18]Zhang Y,Sun J,Gray S,Notfors C,Bleistein N,Zhang G.Trueamplitude migration using common-shot one-way wavefieldextrapolation.64th EAGE Annual Conference,2002
[19]Zhang Y,Zhang G,Bleistein N.Theory of true amplitude common-shot migration.72nd Ann.Mtg.,Soc.Expl.Geophys.,2002.2471~2473
[20]Zhang Y,Zhang G,Bleistein N.Theory of true amplitude one-waywave equations and true amplitude common-shot migration.73rd Ann.Mtg.,Soc.Expl.Geophys.,2003,925~928
[21]Zhang Y,Zhang G,Bleistein N.True amplitude wave equationmigration arising from true amplitude one-way wave equations.InverseProblems,2003,19:1113~1138
[22]Zhang Y,Zhang G,Bleistein N.Theory of true amplitude one-waywave equations and true amplitude common-shot migration.Geophysics,2005,70(3):E1~10
[23]Zhang G.System of coupled equations for up-going and down-goingwaves.Acta Math.Appl.Sinica,1993,16(2):251~263
[24]Wapenaar C P A.Representation of the seismic sources in one-waywave equations.Geophysics,1990,55(6):786~790
[25]Keho T H,Beydoun W B.Paraxial ray Kirchhoff migration.Geophysics,1988,53(12):1540~1546
[26]Zhang G,Zhang S,Wang Y,Liu C.A new algorithm for finite-difference migration of steep dips.Geophysics,1988,53(2):167~175
[27]Zhang G,Zhang Y.An iteratice method for the inversion of the two-dimensional wave equation with a potential.Journal of ComputationalPhysics,1998,147:485~506
[28]Claerbout J F,Doherty S M.Downward continuation of moveoutcorrected seismograms.Geophysics,1972,37(5):741~768
[29]Berkhout A J.Steep dip finite-difference migration.GeophysicsProspecting,1979,27:196~213
[30]Paffenholz J.Sigsbee2 synthetic subsalt dataset:image quality asfunction of migration algorithm and velocity model error.71st Ann.Mtg.,Soc.Expl.Geophys.(Workshop W-5),2001
[31]Xu X,Zhang Y,Pham D,Lambar?G.Anti-leakage Fourier trans-form for seismic data regularization.Geophysics,2004,70(4):V87~95
[32]Xu S,Chauris H,Lambar?G,Noble M.Common-angle migration:astrategy for imaging complex media.Geophysics,2001,66:1877~1894
[33]Sava P C,Fomel S.Angle-domain common-image gathers by wave_field continuation methods.Geophysics,2003,68:1065~1074
[34]Xie X,Wu R.Extracting angle related image from migratedwavefield.72nd Ann.Mtg.,Soc.Expl.Geophys.,2002.1360~1363
[35]Zhang Y,Xu S,Bleistein N,Zhang G.True amplitude angledomain common image gathers from one-way wave equationmigrations.Geophysics,2006.submitted.
[36]Zhang Y,Xu S,Zhang G,Bleistein N.How to obtain trueamplitude common-angle gathers from one-way wave equationmigration?74th Ann.Mtg.,Soc.Expl.Geophys.,2004
[37]Operto M S,Xu S,Lambare G.Can we quantitatively imagecomplex structures with rays?Geophysics,2000,65(4):1223~1238
[38]Thierry P.Migration/Inversion 3D en profondeuràamplitudepr?serv?e:application aux donn?es de sismique r?flexion avantsommation(Ph.D.thesis).l’Universit?Paris 7-Denis Diderot,1997
[39]Thierry P,Operto S,Lambar?.Fast 2-D ray+Born migration/inversion in complex media.Geophysics,1999,64(1):162~181
[40]Xu X.Migration/Inversion en milieu complexe par des m?thodesasymptotiques haute fr?quence(Ph.D.thesis).l’cole des Minesde Paris,2001
[41]Bleistein N,Zhang Y,Xu S,Zhang G,Gray S H.Migration/inversion:think image point coordinates,process in acquisitionsurface coordinates.Inverse Problems,2005,21:1715~1744
[42]Rickett J E,Sava P C.Offset and angle-domain common image-pointgathers for shot-profile migration.Geophysics,2002,67(3):883~889
[43]Zhang Y,Xu S,Zhang G,Bleistein N.Angle domain true amplitudewave equation migration.67th EAGE Annual Conference,2005
[44]Claerbout J F.Imaging the Earth’s Interior.Blackwell ScientificPublications,Inc.,1985
[45]Biondi B.Equivalence of source-receiver migration and shot-profilemigration.Geophysics,2003,68:1340~1347
[46]Wapenaar C P A.Berkhout A J.Fullprestack versus shot recordmigration.57th Annual Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1987.761~764
[47]Bleistein N,Zhang Y,Zhang G.Analysis of the neighborhood of asmooth caustic for true_amplitude one_way wave equations.WaveMotion,2006,43/4:323~338
[48]Zhang Y,Xu S,Zhang G,Bleistein N.Imaging complex salt bodieswith turning-wave one-way wave equation.68th EAGE AnnualConference,2006
[2]Bleistein N.On the imaging of reflectors in the earth.Geophysics,1987,52:931~942
[3]Miller D,Oristaglio M,Beylkin G.A new slant on seismic imaging:migration and integral geometry.Geophysics,1987,52:943~964
[4]Claerbout J F.Toward a unified theory of reflector mapping.Geophysics,1971,36(3):467~481
[5]Gazdag J.Wave equation migration with the phase-shift method.Geophysics,1978,43:1342~1351
[6]Bleistein N,Cohen J K,Stockwell J W.Mathematics ofMultidimensional Seismic Inversion.Springer,2001
[7]Schneider W A.Integral formulation for migration in two and threedimensions.Geophysics,1978,43:49~76
[8]Hanitzsch C.Comparison of weights in prestack amplitude-preservingKirchhoff depth migration.Geophysics,1997,62(6):1812~1816
[9]Zhang Y,Gray S,Young J.Exact and approximate weights forKirchhoff migration.70th Ann.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,2000,Ⅱ:1036~1039
[10]Zhang Y,Karazincir M,Notfors C,Sun J,Hung B.Amplitudepreservingv(z)pre-stack Kirchhoff migration,demigration andmodeling.64th EAGE Annual Conference,2002
[11]Loewenthal D,Lu L,Roberson R,Sherwood J.The wave equationapplied to migration.Geophysics,1976,24:380~399
[12]Zhang Y,Gray S,Young J.True-amplitude common-offset,common-azimuthv(z)migration.Journal of Seismic Exploration,2002,11
[13]Ursin B.Offset-dependent geometrical spreading in a layeredmedium.Geophysics,1990,55(4):492~496
[14]Dubrulle A A.Numerical methods for the migration of constant-offsetsections in homogeneous and horizontally layered media.Geophysics,1983,48:1195~1203
[15]Ekren B O,Ursin B.True-amplitude frequency-wavenumber constant-offset migration.Geophysics,1999,64:915~924
[16]Winbow G,Schneider W.Weights for 3-D controlled amplitudeprestack time migration.69th Annual Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1999.Ⅱ:1110~1113
[17]Zhang Y,Sun J,Gray S,Notfors C,Bleistein N.Towards accurateamplitudes for one-way wavefield extrapolation of 3-D common shotrecords.71st Ann.Mtg.,Soc.Expl.Geophys.(workshop),2001
[18]Zhang Y,Sun J,Gray S,Notfors C,Bleistein N,Zhang G.Trueamplitude migration using common-shot one-way wavefieldextrapolation.64th EAGE Annual Conference,2002
[19]Zhang Y,Zhang G,Bleistein N.Theory of true amplitude common-shot migration.72nd Ann.Mtg.,Soc.Expl.Geophys.,2002.2471~2473
[20]Zhang Y,Zhang G,Bleistein N.Theory of true amplitude one-waywave equations and true amplitude common-shot migration.73rd Ann.Mtg.,Soc.Expl.Geophys.,2003,925~928
[21]Zhang Y,Zhang G,Bleistein N.True amplitude wave equationmigration arising from true amplitude one-way wave equations.InverseProblems,2003,19:1113~1138
[22]Zhang Y,Zhang G,Bleistein N.Theory of true amplitude one-waywave equations and true amplitude common-shot migration.Geophysics,2005,70(3):E1~10
[23]Zhang G.System of coupled equations for up-going and down-goingwaves.Acta Math.Appl.Sinica,1993,16(2):251~263
[24]Wapenaar C P A.Representation of the seismic sources in one-waywave equations.Geophysics,1990,55(6):786~790
[25]Keho T H,Beydoun W B.Paraxial ray Kirchhoff migration.Geophysics,1988,53(12):1540~1546
[26]Zhang G,Zhang S,Wang Y,Liu C.A new algorithm for finite-difference migration of steep dips.Geophysics,1988,53(2):167~175
[27]Zhang G,Zhang Y.An iteratice method for the inversion of the two-dimensional wave equation with a potential.Journal of ComputationalPhysics,1998,147:485~506
[28]Claerbout J F,Doherty S M.Downward continuation of moveoutcorrected seismograms.Geophysics,1972,37(5):741~768
[29]Berkhout A J.Steep dip finite-difference migration.GeophysicsProspecting,1979,27:196~213
[30]Paffenholz J.Sigsbee2 synthetic subsalt dataset:image quality asfunction of migration algorithm and velocity model error.71st Ann.Mtg.,Soc.Expl.Geophys.(Workshop W-5),2001
[31]Xu X,Zhang Y,Pham D,Lambar?G.Anti-leakage Fourier trans-form for seismic data regularization.Geophysics,2004,70(4):V87~95
[32]Xu S,Chauris H,Lambar?G,Noble M.Common-angle migration:astrategy for imaging complex media.Geophysics,2001,66:1877~1894
[33]Sava P C,Fomel S.Angle-domain common-image gathers by wave_field continuation methods.Geophysics,2003,68:1065~1074
[34]Xie X,Wu R.Extracting angle related image from migratedwavefield.72nd Ann.Mtg.,Soc.Expl.Geophys.,2002.1360~1363
[35]Zhang Y,Xu S,Bleistein N,Zhang G.True amplitude angledomain common image gathers from one-way wave equationmigrations.Geophysics,2006.submitted.
[36]Zhang Y,Xu S,Zhang G,Bleistein N.How to obtain trueamplitude common-angle gathers from one-way wave equationmigration?74th Ann.Mtg.,Soc.Expl.Geophys.,2004
[37]Operto M S,Xu S,Lambare G.Can we quantitatively imagecomplex structures with rays?Geophysics,2000,65(4):1223~1238
[38]Thierry P.Migration/Inversion 3D en profondeuràamplitudepr?serv?e:application aux donn?es de sismique r?flexion avantsommation(Ph.D.thesis).l’Universit?Paris 7-Denis Diderot,1997
[39]Thierry P,Operto S,Lambar?.Fast 2-D ray+Born migration/inversion in complex media.Geophysics,1999,64(1):162~181
[40]Xu X.Migration/Inversion en milieu complexe par des m?thodesasymptotiques haute fr?quence(Ph.D.thesis).l’cole des Minesde Paris,2001
[41]Bleistein N,Zhang Y,Xu S,Zhang G,Gray S H.Migration/inversion:think image point coordinates,process in acquisitionsurface coordinates.Inverse Problems,2005,21:1715~1744
[42]Rickett J E,Sava P C.Offset and angle-domain common image-pointgathers for shot-profile migration.Geophysics,2002,67(3):883~889
[43]Zhang Y,Xu S,Zhang G,Bleistein N.Angle domain true amplitudewave equation migration.67th EAGE Annual Conference,2005
[44]Claerbout J F.Imaging the Earth’s Interior.Blackwell ScientificPublications,Inc.,1985
[45]Biondi B.Equivalence of source-receiver migration and shot-profilemigration.Geophysics,2003,68:1340~1347
[46]Wapenaar C P A.Berkhout A J.Fullprestack versus shot recordmigration.57th Annual Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1987.761~764
[47]Bleistein N,Zhang Y,Zhang G.Analysis of the neighborhood of asmooth caustic for true_amplitude one_way wave equations.WaveMotion,2006,43/4:323~338
[48]Zhang Y,Xu S,Zhang G,Bleistein N.Imaging complex salt bodieswith turning-wave one-way wave equation.68th EAGE AnnualConference,2006
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