退化的Fourier偏移算子及其在复杂断块成像中的应用
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摘要
波动方程宽角抛物逼近得到的通常是非常系数的单程波传播算子,其系数是速度横向变化的函数,因此需要利用有限差分(FD)进行数值实施.通过对Lippmann-Schwinger单程波动积分方程的退化核逼近,本文研究了一类宽角退化算子的偏移成像.这种退化偏移算子只用快速Fourier变换进行波场延拓,将常规的Fourier分裂步地震偏移方法(SSF)推广适应强速度横向变化介质和大角度传播波场.退化的Fourier偏移算子通过在两个分裂步项之间作波数域线性插值来实现波场延拓,每延拓一层需要比常规的SSF地震偏移方法多一次快速Fourier变换(FFT).通过SEG/EAGE盐丘模型和实际地震资料的应用表明,退化Fourier偏移算子能很好地对盐下的陡倾角断层和实际地震剖面上的复杂小断块和大断裂地质构造成像.
Wide-angle parabolic approximations to wave equation usually lead to non-constant differential migrators with their coefficients varying with lateral velocity variations, and hence require a finite-difference numerical implementation. With degenerate approximations to the one-way Lippmann-Schwinger integral equation, we study a category of wide-angle degenerate migrators that permit the implementation of seismic migration using fast Fourier transforms alone. The computationally efficient degenerate migrators extend the split-step Fourier (SSF) method to steeper propagation angles in large-contrast media at the cost of one more Fourier transform for each extrapolation step. Advancing wavefields by the degenerate migrators is actually a linear interpolation between two split-step terms in the wavenumber domain. When applied to image the subsalt dipping faults in the SEG/EAGE salt model and fault blocks/zones in real datasets, the seismic migration by degenerate migrators gives well positioned images for fault-related complex geological structures.
Wide-angle parabolic approximations to wave equation usually lead to non-constant differential migrators with their coefficients varying with lateral velocity variations, and hence require a finite-difference numerical implementation. With degenerate approximations to the one-way Lippmann-Schwinger integral equation, we study a category of wide-angle degenerate migrators that permit the implementation of seismic migration using fast Fourier transforms alone. The computationally efficient degenerate migrators extend the split-step Fourier (SSF) method to steeper propagation angles in large-contrast media at the cost of one more Fourier transform for each extrapolation step. Advancing wavefields by the degenerate migrators is actually a linear interpolation between two split-step terms in the wavenumber domain. When applied to image the subsalt dipping faults in the SEG/EAGE salt model and fault blocks/zones in real datasets, the seismic migration by degenerate migrators gives well positioned images for fault-related complex geological structures.
引文
[1]Claerbout J F.Toward a unified theory of reflector mapping.Geophysics,1971,36(2):467～481
[2]马在田.高阶有限差分偏移.石油地球物理勘探,1982,17(1):6～15Ma Z T.Thefinite-difference migration of higher-order equation.Oil Geophysical Prospecting(in Chinese),1982,17(1):6～15
[3]张关泉.利用低阶偏微分方程组的大倾角差分偏移.地球物理学报,1986,29(2):273～282Zhang G Q.Steep dipfinite-difference migration usingthe systemof lower-order partial differential equations.Chinese J.Geophys.(in Chinese),1986,29(2):273～282
[4]Hale D.Stable explicit depth extrapolation of seismic wavefields.Geophysics,1991,56:1770～1777
[5]RistowD,Rhl T.Fourier finite-difference migration.Geophysics,1994,59:1882～1893
[6]Stolt R H.Migration by Fourier transform.Geophysics,1978,43(1):23～48
[7]Gazdag J.Wave equation migration with the phase-shift method.Geophysics,1978,43:1342～1351
[8]Stoffa P L,Fakkema J T,de Luna Freire R M.Split-step Fourier migration.Geophysics,1990,55:410～421
[9]Wu RS.Synthetic seismogramin heterogeneous media by one-return approximation.Pure and Applied Geophys,1996,148:155～173
[10]Huang L J,Fehler M C,Wu R S.Extended local Born Fouriermigration method.Geophysics,1999,64:1524～1534
[11]Le Rousseau J H,de Hoop M V.Modeling and imaging with the scalar generalized-screen algorithms in isotropic media.Geophysics,2001,66:1551～1568
[12]Fu L Y.Broadband constant-coefficient propagators.Geophysical Prospecting,2005,53:299～310
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[15]Kessinger W.Extended split-step Fourier migration.62th Ann.Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1992.917～920
[16]Wu R S,Jin S.Windowed GSP(generalized screen propagators)migration applied to SEG-EAEG salt model data.67th Ann.Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1997.1746～1749
[17]Fu L Y,Duan Y.Fourier depth migration methods with applicationto salt-related complex geological structures.72nd Annual International Meeting,Society of Exploration Geophysicists,Expanded Abstracts,2002.895～898
[18]Fu L Y.Wavefield interpolation in the Fourier wavefield extrapolation.Geophysics,2004,69:257～264
[19]J B Chen,H Liu.Optimization approximation with separable variables forthe one-way wave operator.Geophys.Res.Lett.2004,31,L06613
[20]Liu L N,Zhang J F.3D wavefield extrapolation with optimumsplit-step Fourier method.Geophysics,2006,71:95～108
[21]Fu L Y.Comparison of different one-way propagatorsfor waveforward propagation in heterogeneous crustal wave guides.Bulletin ofthe Seismological SocietyofAmerica,2006,96:1091～1113
[22]Fu L Y,Mu YG,Yang HJ.Forward problemof nonlinear Fredholm integral equationin reference mediumvia velocity-weighted wavefield function.Geophysics,1997,62:650～656
[23]Fu L Y.Numerical study of generalized Lippmann-Schwinger integral equationincludingsurfacetopography.Geophysics,2003,68:665～671
[24]Fu L Y.Seismogramsynthesis for piecewise heterogeneous media.Geophys.J.Int.,2002,150:800～808
[25]Fu L Y,Bouchon M.Discrete wavenumber solutions to numerical wave propagation in piecewise heterogeneous media—I.Theory of two-dimensional SHcase.Geophys.J.Int.,2004,157:481～498
[26]Tappert F D.The parabolic approximation method.In:Keller J B,Papadakis J Seds.Wave Propagation and Underwater Acoustics.New York:Springer-Verlag,1977.224～287
[27]Thomson DJ,Chapman N R.A wide-angle split-step algorithmfor the parabolic equation.J.Acoust.Soc.Am,1983,74:1848～1854
[28]Thomson DJ.Wide-angle parabolic equation solutions to two range-dependent benchmark problems.J.Acoust.Soc.Am,1990,87:1514～1520
[29]Collins M D.Asplit-step Pad?solution for the parabolic equation method.J.Acoust.Soc.Am,1993,93:1736～1742
[30]Trefethen L N,Halpern L.Well-posedness of one-way equations and absorbing boundary conditions.Math.Comput,1986,147:421～435
[31]Bamberger A,Engquist B,Halpern L.Higher order paraxial wave equation approximations in heterogeneous media,SIAMJ.Appl.Math,1988,48:129～154
[32]Halpern L,Trefethen L N.Wide-angle one-way wave equations.J.Acoust.Soc.Am,1988,84:1397～1404
[33]Aminzadeh F,Burkhard N,Kunz T,Nicoletis L.3-D modeling project:3rdreport.The Leading Edge,1995,14:125～128
[2]马在田.高阶有限差分偏移.石油地球物理勘探,1982,17(1):6～15Ma Z T.Thefinite-difference migration of higher-order equation.Oil Geophysical Prospecting(in Chinese),1982,17(1):6～15
[3]张关泉.利用低阶偏微分方程组的大倾角差分偏移.地球物理学报,1986,29(2):273～282Zhang G Q.Steep dipfinite-difference migration usingthe systemof lower-order partial differential equations.Chinese J.Geophys.(in Chinese),1986,29(2):273～282
[4]Hale D.Stable explicit depth extrapolation of seismic wavefields.Geophysics,1991,56:1770～1777
[5]RistowD,Rhl T.Fourier finite-difference migration.Geophysics,1994,59:1882～1893
[6]Stolt R H.Migration by Fourier transform.Geophysics,1978,43(1):23～48
[7]Gazdag J.Wave equation migration with the phase-shift method.Geophysics,1978,43:1342～1351
[8]Stoffa P L,Fakkema J T,de Luna Freire R M.Split-step Fourier migration.Geophysics,1990,55:410～421
[9]Wu RS.Synthetic seismogramin heterogeneous media by one-return approximation.Pure and Applied Geophys,1996,148:155～173
[10]Huang L J,Fehler M C,Wu R S.Extended local Born Fouriermigration method.Geophysics,1999,64:1524～1534
[11]Le Rousseau J H,de Hoop M V.Modeling and imaging with the scalar generalized-screen algorithms in isotropic media.Geophysics,2001,66:1551～1568
[12]Fu L Y.Broadband constant-coefficient propagators.Geophysical Prospecting,2005,53:299～310
[13]Claerbout J F.Imaging the Earth s Interior.Blackwell Scientific Publication,1985
[14]Gazdag J,Sguazzero P.Migration of seismic data by phase shift plus interpolation.Geophysics,1984,49:124～131
[15]Kessinger W.Extended split-step Fourier migration.62th Ann.Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1992.917～920
[16]Wu R S,Jin S.Windowed GSP(generalized screen propagators)migration applied to SEG-EAEG salt model data.67th Ann.Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1997.1746～1749
[17]Fu L Y,Duan Y.Fourier depth migration methods with applicationto salt-related complex geological structures.72nd Annual International Meeting,Society of Exploration Geophysicists,Expanded Abstracts,2002.895～898
[18]Fu L Y.Wavefield interpolation in the Fourier wavefield extrapolation.Geophysics,2004,69:257～264
[19]J B Chen,H Liu.Optimization approximation with separable variables forthe one-way wave operator.Geophys.Res.Lett.2004,31,L06613
[20]Liu L N,Zhang J F.3D wavefield extrapolation with optimumsplit-step Fourier method.Geophysics,2006,71:95～108
[21]Fu L Y.Comparison of different one-way propagatorsfor waveforward propagation in heterogeneous crustal wave guides.Bulletin ofthe Seismological SocietyofAmerica,2006,96:1091～1113
[22]Fu L Y,Mu YG,Yang HJ.Forward problemof nonlinear Fredholm integral equationin reference mediumvia velocity-weighted wavefield function.Geophysics,1997,62:650～656
[23]Fu L Y.Numerical study of generalized Lippmann-Schwinger integral equationincludingsurfacetopography.Geophysics,2003,68:665～671
[24]Fu L Y.Seismogramsynthesis for piecewise heterogeneous media.Geophys.J.Int.,2002,150:800～808
[25]Fu L Y,Bouchon M.Discrete wavenumber solutions to numerical wave propagation in piecewise heterogeneous media—I.Theory of two-dimensional SHcase.Geophys.J.Int.,2004,157:481～498
[26]Tappert F D.The parabolic approximation method.In:Keller J B,Papadakis J Seds.Wave Propagation and Underwater Acoustics.New York:Springer-Verlag,1977.224～287
[27]Thomson DJ,Chapman N R.A wide-angle split-step algorithmfor the parabolic equation.J.Acoust.Soc.Am,1983,74:1848～1854
[28]Thomson DJ.Wide-angle parabolic equation solutions to two range-dependent benchmark problems.J.Acoust.Soc.Am,1990,87:1514～1520
[29]Collins M D.Asplit-step Pad?solution for the parabolic equation method.J.Acoust.Soc.Am,1993,93:1736～1742
[30]Trefethen L N,Halpern L.Well-posedness of one-way equations and absorbing boundary conditions.Math.Comput,1986,147:421～435
[31]Bamberger A,Engquist B,Halpern L.Higher order paraxial wave equation approximations in heterogeneous media,SIAMJ.Appl.Math,1988,48:129～154
[32]Halpern L,Trefethen L N.Wide-angle one-way wave equations.J.Acoust.Soc.Am,1988,84:1397～1404
[33]Aminzadeh F,Burkhard N,Kunz T,Nicoletis L.3-D modeling project:3rdreport.The Leading Edge,1995,14:125～128
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