地震逆散射波场和算子的谱分解
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摘要
本文对地震逆散射的研究,旨在于为抑制层间多次波和地震波场多重散射对一次反射干扰效应提供理论依据.这对薄互层地层滤波的高频恢复、保幅弹性反演、衍射地震勘探及海洋地震勘探中的干扰消除皆具重要意义.本文基于上下行波分解及弹性波互易定理,导出横向变速介质条件下线性预测算子的表达式和反射数据的广义谱分解方程.文中先由上覆地层广义反射透射矩阵的元素定义线性预测算子,并将其表示成一系列单程波算子的线性组合,之后将横向变速介质条件下线性预测方程表达为反射数据与线性预测算子及其逆的乘积.对该方程的求解可获得上覆地层的线性预测算子,从而可借以求出相应的反射透射算子.本文先将水平层状介质条件下垂直入射的一维线性预测方程推广到斜入射的情况,以此为参照,导出横向非均匀介质条件下反射数据的地震逆散射广义谱分解方程.文中也揭示了单程波地震逆散射算子、反射透射算子的性态.本文还针对水平层状介质条件,给出斜入射的数值结果.
The research goal of seismic inverse scattering in this paper is to provide theoretical base for attenuating interior multiple and for suppressing the perturbation effect of seismic wavefield multiple scattering on the primary.This is of great significance for high-frequency recovery of finely layer's stratigraphic filtering,amplitude preserving elastic wave inversion,diffraction seismic exploration,and interference subtraction in oceanic seismic exploration.Based on up-and down-going wave decomposition and elastic wave power reciprocity theorem,this paper derives,in the lateral velocity varying case,linear prediction operator's expression and the generalized spectral factorization equation for reflection dataset.We express the linear prediction operator,which is defined by the elements of overlaid layer's generalized reflection/transmission matrices,as a linear combination of a series of one-way wave operators.And the linear prediction equation,under the condition of lateral velocity varying,is expressed as a connecting relation between reflection dataset,the product of linear prediction operator and its inversion.Solving the equation,the linear prediction operator of overlaid layers,then corresponding reflection/transmission operators,can be obtained.Then the one-dimensional linear prediction equation of vertical incident wave,under horizontally layered condition,is generalized to oblique incident case.Using the equation as a reference,the generalized spectral factorization equation of seismic inverse scattering for reflection dataset in a lateral inhomogeneous medium is deduced.The properties of the one-way wave operator and reflection/transmission operators of seismic inverse scattering are revealed.Numerical results for horizontally layered media with inclined incidence are given.
The research goal of seismic inverse scattering in this paper is to provide theoretical base for attenuating interior multiple and for suppressing the perturbation effect of seismic wavefield multiple scattering on the primary.This is of great significance for high-frequency recovery of finely layer's stratigraphic filtering,amplitude preserving elastic wave inversion,diffraction seismic exploration,and interference subtraction in oceanic seismic exploration.Based on up-and down-going wave decomposition and elastic wave power reciprocity theorem,this paper derives,in the lateral velocity varying case,linear prediction operator's expression and the generalized spectral factorization equation for reflection dataset.We express the linear prediction operator,which is defined by the elements of overlaid layer's generalized reflection/transmission matrices,as a linear combination of a series of one-way wave operators.And the linear prediction equation,under the condition of lateral velocity varying,is expressed as a connecting relation between reflection dataset,the product of linear prediction operator and its inversion.Solving the equation,the linear prediction operator of overlaid layers,then corresponding reflection/transmission operators,can be obtained.Then the one-dimensional linear prediction equation of vertical incident wave,under horizontally layered condition,is generalized to oblique incident case.Using the equation as a reference,the generalized spectral factorization equation of seismic inverse scattering for reflection dataset in a lateral inhomogeneous medium is deduced.The properties of the one-way wave operator and reflection/transmission operators of seismic inverse scattering are revealed.Numerical results for horizontally layered media with inclined incidence are given.
引文
[1]Prosser R T.Formal solutions of inverse scattering problems:Ⅰ.J.Math.Phys.,1969,10(8):1819-1822
[2]Prosser R T.Formal solutions of inverse scattering problems:Ⅱ.J.Math.Phys.,1976,17(11):1775-1779
[3]Prosser R T.Formal solutions of inverse scattering problems:Ⅲ.J.Math.Phys.,1980,21(11):2648-2653
[4]Prosser R T.On the solutions of the Gel’fand-Levitan equation.J.Math.Phys.,1984,25:1924-1929
[5]Claerbout J F.Fundamentals of Geophysical Data Processing.England:Blackwell Scientific Publications,1985
[6]Claerbout J F,Fomel S.Image Estimation by Example:Geophysicalsoundings image construction-multidimensional autoregression.PaloAlto:Stanford University,2002
[7]罗明秋,刘洪,李幼铭.基于螺旋坐标的地震波场隐式辛算法.地球物理学报,2001,44(3):379-388Luo M Q,Liu H,Li Y M.Seismic wave modeling with implicitsymplectic method based on spectral factorization on helix.ChineseJ.Geophys.(in Chinese),2001,44(3):379-388
[8]罗明秋,刘洪,李幼铭.适用于波场成像的谱法LU分解.地球物理学报,2003,46(3):421-427Luo M Q,Liu H,Li Y M.LU decomposition with spectralfactorization in seismic imaging.Chinese J.Geophys.(in Chinese),2003,46(3):421-427
[9]Newton R G.Scattering Theory of Wave and Particles.New York:Springer-Verlag,1982
[10]Tarantola A.Inversion Problem Theory and Methods for ModelParameter Estimation.Philadelphia:Society for Industrial andApplied Mathematics,2005
[11]Miller D,Oristaglio M L,Beylkin G.A new slant on seismicimaging-migration and integral geometry.Geophysics,1987,52(7):943-964
[12]Hormander L.The analysis of linear partial differential operators,I,II.Springer-Verlag,1983
[13]Hormander L.The analysis of linear partial differential operators,III,IV.Springer-Verlag,1985
[14]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 University Press,2005
[15]De Hoop M V.Generalization of the Bremmer coupling series.J.Math.Phys.,1996,37:3246-3282
[16]Malcolm A E,De Hoop M V.A method for inverse scattering basedon the generalized Bremmer coupling series.Inverse Problems,2005,21:1137-1167
[17]Zhang Y,Zhang G,Bleistein N.Theory of true amplitude one-waywave equations and true-amplitude common-shot migration.Geophysics,2005,70(3):E1-E10
[18]Chen X F.Seismogram synthesis for multi-layered media withirregular interfaces by global generalized reflection/transmissionmatrices method:Ⅰ.Theory of two-dimensional SH case.Bull.Seism.Soc.Am.,1990,80(6):1696-1724
[19]Chen X F.Seismogram synthesis for multi-layered media withirregular interfaces by global generalized reflection/transmissionmatrices method:Ⅱ.Application for 2D SH case.Bull.Seism.Soc.Am.,1995,85(4):1094-1106
[20]Chen X F.Seismogram synthesis for multi-layered media withirregular interfaces by global generalized reflection/transmissionmatrices method:Ⅲ.Theory of 2D P-SV case.Bull.Seism.Soc.Am.,1996,86(2):389-405
[21]Fu L Y,Bouchon M.Discrete wavenumber solutions to numericalwave propagation in piecewise heterogeneous media-Ⅰ:Theory oftwo-dimensional SH case.Geophys.J.Int.,2004,157:481-498
[22]Yokoi Toshiaki.Reflection and transmission operator for irregularinterfaces derived from the indirect boundary element method.Geophys.J.Int.,2002,148:88-102
[23]Berkhout A J.Seismic Migration,Imaging of Acoustic Energy byWave Field Extrapolation:Theoretical Aspects.Amsterdam:ElsevierScience Publishing Co.,Inc.,1982
[24]Berkhout A J,Verschuur D J.Removal of internal multiples with thecommon-focus-point(CFP)approach:Part 1-explanation of thetheory.Geophysics,2005,70(3):V45-V60
[25]Verschuur D J,Berkhout A J.Removal of internal multiples with thecommon-focus-point(CFP)approach:Part 2-application strategiesand data examples.Geophysics,2005,70(3):V61-V72
[26]Matson K H,Weglein A B,Young C Y,et al.Comparing theinterface and point scatter methods for attenuating internal multiples:a study with synthetic data-PartⅡ.68th Ann.Int.SEG Mtg.,Expanded Abstr..1998.1523-1526
[27]Matson K.An overview of wavelet estimation using free-surfacemultiple remove.The Leading edge,2000,19(1):50-55
[28]Weglein A,Foster D,Matson K,et al.An inverse-scattering sub-series for predicting the spatial location of reflectors without theprecise reference medium and wave velocity.71th Ann.Int.SEGMtg.,Expanded Abstr..2001.2108-2111
[29]丁科,宋守根,谢忠球.逆散射理论的发展及应用前景.地球物理学进展,2005,20(3):661-666Ding K,Song S G,Xie Z Q.Development and future application ofinverse scattering theory.Progress in Geophysics(in Chinese),2005,20(3):661-666
[30]De Hoop M V,Le Rousseau J H,Wu R S.Generalization of thephase-screen approximation for the scattering of acoustic waves.WaveMotion,2000,31:43-70
[31]Wang Y H.Approximations to the Zoeppritz equations and their usein AVO analysis.Geophysics,1999,64(6):1920-1927
[32]谢桂生,刘洪,李幼铭等.界面起伏条件下反射/透射算子+单程波方程的地震波模拟.地球物理学报,2005,48(5):1172-1178Xie G S,Liu H,Li Y M,et al.Seismic modeling by reflectiontransmission operator+one-way wave equation.Chinese J.Geophys.(in Chinese),2005,48(5):1172-1178
[33]Wapenaar K.Short note:Synthesis of an inhomogeneous medium fromits acoustic transmission response.Geophysics,2003,68(5):1756-1759
[2]Prosser R T.Formal solutions of inverse scattering problems:Ⅱ.J.Math.Phys.,1976,17(11):1775-1779
[3]Prosser R T.Formal solutions of inverse scattering problems:Ⅲ.J.Math.Phys.,1980,21(11):2648-2653
[4]Prosser R T.On the solutions of the Gel’fand-Levitan equation.J.Math.Phys.,1984,25:1924-1929
[5]Claerbout J F.Fundamentals of Geophysical Data Processing.England:Blackwell Scientific Publications,1985
[6]Claerbout J F,Fomel S.Image Estimation by Example:Geophysicalsoundings image construction-multidimensional autoregression.PaloAlto:Stanford University,2002
[7]罗明秋,刘洪,李幼铭.基于螺旋坐标的地震波场隐式辛算法.地球物理学报,2001,44(3):379-388Luo M Q,Liu H,Li Y M.Seismic wave modeling with implicitsymplectic method based on spectral factorization on helix.ChineseJ.Geophys.(in Chinese),2001,44(3):379-388
[8]罗明秋,刘洪,李幼铭.适用于波场成像的谱法LU分解.地球物理学报,2003,46(3):421-427Luo M Q,Liu H,Li Y M.LU decomposition with spectralfactorization in seismic imaging.Chinese J.Geophys.(in Chinese),2003,46(3):421-427
[9]Newton R G.Scattering Theory of Wave and Particles.New York:Springer-Verlag,1982
[10]Tarantola A.Inversion Problem Theory and Methods for ModelParameter Estimation.Philadelphia:Society for Industrial andApplied Mathematics,2005
[11]Miller D,Oristaglio M L,Beylkin G.A new slant on seismicimaging-migration and integral geometry.Geophysics,1987,52(7):943-964
[12]Hormander L.The analysis of linear partial differential operators,I,II.Springer-Verlag,1983
[13]Hormander L.The analysis of linear partial differential operators,III,IV.Springer-Verlag,1985
[14]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 University Press,2005
[15]De Hoop M V.Generalization of the Bremmer coupling series.J.Math.Phys.,1996,37:3246-3282
[16]Malcolm A E,De Hoop M V.A method for inverse scattering basedon the generalized Bremmer coupling series.Inverse Problems,2005,21:1137-1167
[17]Zhang Y,Zhang G,Bleistein N.Theory of true amplitude one-waywave equations and true-amplitude common-shot migration.Geophysics,2005,70(3):E1-E10
[18]Chen X F.Seismogram synthesis for multi-layered media withirregular interfaces by global generalized reflection/transmissionmatrices method:Ⅰ.Theory of two-dimensional SH case.Bull.Seism.Soc.Am.,1990,80(6):1696-1724
[19]Chen X F.Seismogram synthesis for multi-layered media withirregular interfaces by global generalized reflection/transmissionmatrices method:Ⅱ.Application for 2D SH case.Bull.Seism.Soc.Am.,1995,85(4):1094-1106
[20]Chen X F.Seismogram synthesis for multi-layered media withirregular interfaces by global generalized reflection/transmissionmatrices method:Ⅲ.Theory of 2D P-SV case.Bull.Seism.Soc.Am.,1996,86(2):389-405
[21]Fu L Y,Bouchon M.Discrete wavenumber solutions to numericalwave propagation in piecewise heterogeneous media-Ⅰ:Theory oftwo-dimensional SH case.Geophys.J.Int.,2004,157:481-498
[22]Yokoi Toshiaki.Reflection and transmission operator for irregularinterfaces derived from the indirect boundary element method.Geophys.J.Int.,2002,148:88-102
[23]Berkhout A J.Seismic Migration,Imaging of Acoustic Energy byWave Field Extrapolation:Theoretical Aspects.Amsterdam:ElsevierScience Publishing Co.,Inc.,1982
[24]Berkhout A J,Verschuur D J.Removal of internal multiples with thecommon-focus-point(CFP)approach:Part 1-explanation of thetheory.Geophysics,2005,70(3):V45-V60
[25]Verschuur D J,Berkhout A J.Removal of internal multiples with thecommon-focus-point(CFP)approach:Part 2-application strategiesand data examples.Geophysics,2005,70(3):V61-V72
[26]Matson K H,Weglein A B,Young C Y,et al.Comparing theinterface and point scatter methods for attenuating internal multiples:a study with synthetic data-PartⅡ.68th Ann.Int.SEG Mtg.,Expanded Abstr..1998.1523-1526
[27]Matson K.An overview of wavelet estimation using free-surfacemultiple remove.The Leading edge,2000,19(1):50-55
[28]Weglein A,Foster D,Matson K,et al.An inverse-scattering sub-series for predicting the spatial location of reflectors without theprecise reference medium and wave velocity.71th Ann.Int.SEGMtg.,Expanded Abstr..2001.2108-2111
[29]丁科,宋守根,谢忠球.逆散射理论的发展及应用前景.地球物理学进展,2005,20(3):661-666Ding K,Song S G,Xie Z Q.Development and future application ofinverse scattering theory.Progress in Geophysics(in Chinese),2005,20(3):661-666
[30]De Hoop M V,Le Rousseau J H,Wu R S.Generalization of thephase-screen approximation for the scattering of acoustic waves.WaveMotion,2000,31:43-70
[31]Wang Y H.Approximations to the Zoeppritz equations and their usein AVO analysis.Geophysics,1999,64(6):1920-1927
[32]谢桂生,刘洪,李幼铭等.界面起伏条件下反射/透射算子+单程波方程的地震波模拟.地球物理学报,2005,48(5):1172-1178Xie G S,Liu H,Li Y M,et al.Seismic modeling by reflectiontransmission operator+one-way wave equation.Chinese J.Geophys.(in Chinese),2005,48(5):1172-1178
[33]Wapenaar K.Short note:Synthesis of an inhomogeneous medium fromits acoustic transmission response.Geophysics,2003,68(5):1756-1759
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