Born序列频散方程和Born-Kirchhoff传播算子
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摘要
传统的Kirchhoff传播算子结构简洁,适用于描述横向均匀介质中波的传播.Ray-Kirchhoff传播算子较为精确地描述了波在非均匀介质中传播的运动学特征,其理论上的先天不足依赖于介质的复杂性.本文通过Born序列逼近波在非均匀介质中传播的大角度波分量,提出一种Born-Kirchhoff传播算子,将传统Kirchhoff传播算子的适用范围扩展至非均匀介质,同时描述波的运动学和动力学特征,其精度取决于Born序列逼近的阶数.利用Born序列频散方程,可以精确分析各阶Born-Kirchhoff传播算子对波长、传播角和非均质性的尺度依赖特征,其中,一阶Born-Kirchhoff传播算子的精度高于传统的相屏传播算子.波数域的Born-Kirchhoff传播算子对于高波数波是奇异的,导致波数域数值计算发散,但其空间域版本是非奇异的,无条件数值稳定,可通过Kirchhoff求和数值实施.本文给出各阶Born-Kirchhoff传播算子及其频散方程,可用于不同程度非均匀介质中的波传播模拟,复杂构造地震成像和速度估计.本文利用零阶和一阶Born-Kirchhoff传播算子计算简单二维模型的合成地震图,并与边界元法进行了比较.
Conventional Kirchhoff propagators are conceptually simple and applicable for wave propagation in laterally homogeneous media.Ray-Kirchhoff propagators are kinematically acceptable in the range of seismic frequencies for heterogeneous media, but theoretically suffer congenital deficiencies.In this paper we present a natural way to extend the conventional Kirchhoff propagators to heterogeneous media.The so-called Born-Kirchhoff propagators are designed in the wavenumber domain under Born-series approximation to account for large-angle waves in strong-contrast media.These wavenumber-domain propagators that usually become singular at high wavenumbers can be transformed into the space domain, which are unconditionally stable with the Kirchhoff-summation implementation.Various orders of the Born-Kirchhoff propagators are formulated with a target-oriented flexibility to handle local complex zones for wave propagation, seismic imaging, and velocity estimation.A complete accuracy analysis is conducted by Born-series dispersion equations to characterize the Born-Kirchhoff propagators'scale-dependence on wavelengths, propagation angles, and heterogeneities.Synthetic seismograms for a simple 2D model are calculated with the zeroth-order and first-order Born-Kirchhoff propagators in comparison with those generated by the boundary-element method.
Conventional Kirchhoff propagators are conceptually simple and applicable for wave propagation in laterally homogeneous media.Ray-Kirchhoff propagators are kinematically acceptable in the range of seismic frequencies for heterogeneous media, but theoretically suffer congenital deficiencies.In this paper we present a natural way to extend the conventional Kirchhoff propagators to heterogeneous media.The so-called Born-Kirchhoff propagators are designed in the wavenumber domain under Born-series approximation to account for large-angle waves in strong-contrast media.These wavenumber-domain propagators that usually become singular at high wavenumbers can be transformed into the space domain, which are unconditionally stable with the Kirchhoff-summation implementation.Various orders of the Born-Kirchhoff propagators are formulated with a target-oriented flexibility to handle local complex zones for wave propagation, seismic imaging, and velocity estimation.A complete accuracy analysis is conducted by Born-series dispersion equations to characterize the Born-Kirchhoff propagators'scale-dependence on wavelengths, propagation angles, and heterogeneities.Synthetic seismograms for a simple 2D model are calculated with the zeroth-order and first-order Born-Kirchhoff propagators in comparison with those generated by the boundary-element method.
引文
[1] Born M. Optic. Springer Publ. Co. Inc, 1933
[2] Kennett B L N. Seismic waves in laterally varying media. Geophys. J. Roy. Astr. Soc, 1972, 27: 310～325
[3] Hudson J A, Humphryes R F, Mason I M, Kembhavi V K. The scattering of longitudinal elastic waves at a rough free surface. J. Phys. D: Appl. Phys, 1973, 6: 2174～2186
[4] Snieder R, Nolet G. Linearised scattering of surface waves on a spherical earth. J. Geophys, 1987, 61: 55～63
[5] Cohen J K, Bleistein N. Velocity inversion procedure for acoustic waves. Geophysics, 1979, 44 : 1077～ 1087
[6] Clayton R W, Stolt R H. A Born-WKBJ inversion method for acoustic reflection data. Geophysics, 1981, 46:1559 ～1567
[7] Wu R S, Toks(o|¨)z M N. Diffration tomography and multisource holography applied to seismic imaging. Geophysics, 1987, 52: 11～25
[8] Snieder R. Large-scale waveform inversions of surface waves for lateral heterogeneity-I. Theory and numerical examples. J. Geophys. Res, 1988, 93:12055～12066
[9] Coates R T, Chapman C H. Generalized Born scattering of elastic waves in 3-D media. Geophys. J. Int, 1991, 107: 231～263
[10] Knopoff L, Hudson J A. Scattering of elastic waves by small inhomogeneities. J. Acoust. Soc. Am, 1964, 36:338～343
[11] Sato H. Attenuation of body waves and envelope formation of three-component seismograms of small local earthquakes in randomly inhomogeneous lithosphere. J. Geophys. Res, 1984, 89: 1221～1241
[12] Wu R S, Aki K. Elastic wave scattering by a random medium and the small-scale inhomogeneities in the Lithosphere. J. Geophys. Res, 1985, 90: 10261～10273
[13] Flatté S M, Wu R S. Small-scale structure in the Lithosphere and asthenosphere deduced from arrival time and amplitude fluctuations. J. Geophys. Res, 1988, 93:6601～6614
[14] Hudson J A, Heritage J R. The use of the Born approximation in seismic scattering problems. Geophys. J. Roy. Astr. Soc, 1982, 66:221～240
[15] Bleistein N, Cohen J K, Stockwell Jr J W. Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion. New York: Springer-Verlag Inc, 2000
[16] Sato H, Fehler M C. Seismic wave propagation and scattering in the heterogeneous earth. Springer, 1998
[17] Fu L Y, Wu R S, Campillo M. Energy partition and attenuation of regional phases by random free surface. Bulletin of the Seismological Society of America, 2002, 92: 1992～2007
[18] Carter J A, Frazar L N. Accommodating lateral velocity changes in Kirchhoff migration by means of Fermat' s principle. Geophysics, 1984, 49:46～53
[19] Vidale J. Finite-difference calculation of traveltimes. Bull. Seis. Soc. Am, 1988, 78:2062～2076
[20] Keho T H, Beydoun W B. Paraxial ray Kirchhoff migration. Geophysics, 1988, 53 : 1540 ～ 1546
[21] Hill N R. Gaussian beam migration. Geophysics, 1990, 55: 1416～1428
[22] Moser T. Shortest path calculation of seismic rays. Geophysics, 1991, 56:59～67
[23] 徐界,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 C, Liu H, et al. A grid-changeable method for ray tracing. Chinese J. Geophys. (in Chinese), 1996,39(1) :97～102
[24] 张建中,陈世军,徐初伟.动态网格最短路径射线追踪.地球物理学报,2004,47(5) :899～904Zhang J Z, Chen C J, Xu C W. A method of shortest path raytracing with dynamic networks. Chinese J. Geophys. (in Chinese), 2004,47(5) :899～904
[25] Zhang M, Chen B, Li X, et al. A fast algorithm of shortest path ray tracing. Chinese J. Geophys. , 2006, 49: 1467～ 1474
[26] Gray S H, Etgen J, Dellinger J, et al. Seismic migration problems and solutions. Geophysics, 2001, 66 : 1622～ 1640
[27] Woodward M J. Wave-equation tomography. Geophysics, 1992, 57:15～26
[28] Stoffa P L, Fakkema J T, de Luna Freire R M, et al. Splitstep Fourier migration. Geophysics, 1990, 55 : 410～421
[29] Fisk M D, McCartor G D. The phase screen method for vector elastic waves. J. Geophys. Res, 1991, 96: 5985～ 6010
[30] Wu R S. Synthetic seismogram in heterogeneous media by one-return approximation. Pure and Applied Geophys, 1996, 145:155～173
[31] Huang L J, Fehler M C, Wu R S. Extended local Born Fourier migration method. Geophysics, 1999, 64:1524 ～1534
[32] Jin S, Wu R S, Peng C. Prestack depth migration using a hybrid pseudo screen propagator. 68th Ann. Internat. Mtg., Soc. Expl. Geophys, Expanded Abstracts, 1998, 1819～1822
[33] Wu R S, Jin S, Xie X B. Seismic wave propagation and scattering in heterogeneous crustal wave guides using screen propagators I. SH waves. Bull. Seism. Soc. Am, 2004, 90:401～413
[34] de Hoop M V, Le Rousseau J H, Wu R S. Generalization of the phase-screen approximation for the scattering of acoustic waves. Wave Motion, 2000, 31:43～70
[35] Fu L Y, Duan Y. Fourier depth migration methods with application to salt-related complex geological structures. 72th Ann. Internat. Mtg. , Soc. Expl. Geophys, Expanded Abstracts, 2002． 895～898
[36] Fu L Y. Comparison of different one-way propagators for wave forward propagation in heterogeneous crustal wave guides. Bull. Seism. Soc. Am, 2006, 96:1091～1113
[37] Fu L Y, Sun W J, Li D P. Degenerate fourier migrators for imaging complex fault zones. Chinese J. Geophys, 2007, 50 : 483～495
[38] Collins M D. A split-step Padé solution for the parabolic equation method. J. Acoust. Soc. Am, 1993, 93: 1736～ 1742
[39] Ristow D, R(u|¨)hl T. Fourier finite-difference migration. Geophysics, 1994, 59 : 1882～1893
[40] Wu R S. Wide-angle elastic wave one-way propagation in heterogeneous media and an elastic wave complex-screen method. J. Geophys. Res, 1994, 99:751～766
[41] Wild A J, Hudson J A. A geometrical approach to the elastic complex screen. J. Geophys. Res, 1998, 103:707～725
[42] Hobbs R W. 3-D modelling of seismic wave propagation using complex elastic screens with application to mineral exploration. Special Volume on Hardrock Geophysics, Soc. Expl. Geophys. Publisher, 2003
[43] Fu L Y. Born dispersion equation and Kirchhoff migration in laterally heterogeneous media. 72th Ann. Internat. Mtg. , Soc. Expl. Geophys, Expanded Abstracts, 2002． 1097～ 1100
[44] Fu L Y. Wavefield interpolation in the Fourier wavefield extrapolation. Geophysics, 2004, 69:257～264
[45] Schuster G T. A hybrid BIE+Born series modeling scheme: Generalized Bom series. J. Acoust. Soc. Am, 1985, 77: 865～ 879
[46] Schuster G T. Solution of the acoustic transmission problem by a perturbed Born series. J. Acoust. Soc. Am, 1985, 77: 880～886
[47] Fu L Y. Rough surface scattering: Comparison of various approximation theories for 2D SH waves. Bull. Seism. Soc. Am, 2005, 95: 646～663
[48] Fu L Y, Bouchon M. Discrete wavenumber solutions to numerical wave propagation in piecewise heterogeneous media--I. Theory of two-dimensional SH case. Geophys. J. Int, 2004, 157:481～498
[49] Fu L Y, Mu Y G, Yang H J. Forward problem of nonlinear Fredholm integral equation in reference medium via velocityweighted wavefield function. Geophysics, 1997, 62: 650～656
[50] Fu L Y, Wu R S. Infinite boundary element absorbing boundary for wave propagation simulations. Geophysics, 2000, 65:625～637
[51] Berkhout A J, Wapenaar C P A. One way versions of the Kirchhoff integral. Geophysics, 1989, 54:460～467
[52] Hilterman F J. Three dimensional seismic modeling. Geophysics, 1970, 35:1020～1037
[53] Trorey A W. A simple theory for seismic diffractions. Geophysics, 1970, 35:762～784
[54] French W S. Computer migration of oblique seismic reflection profiles. Geophysics, 1975, 40 : 6 ～ 16
[55] Schneider W A. Integral formulation for migration in two and three dimensions. Geophysics, 1978, 43:49～76
[56] Safar M H. On the lateral resolution achieved by Kirchhoff migration. Geophysics, 1985, 50: 1091～1099
[57] Yilmaz O. Seismic data processing. Soc. Expl. Geophys. Publisher, 1987
[58] Hubral P, Schleicher J, Tygel M, Hanitzsch C. Determination of Fresnel zones from traveltime measurements. Geophysics, 1993, 58: 703～712
[59] Schleicher J, Hubral P, Tygel M, et al. Minimum apertures and Fresnel zones in migration and demigration. Geophysics, 1997, 62:183～194
[60] Bleistein N, Cohen J K,. Stockwell Jr J W. Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion. New York: Springer-Verlag Inc, 2000
[61] Huang L J, Fehler M C. Accuracy analysis of the split-step Fourier propagator: implications for seismic modeling and migration. Bull. Seism. Soc. Am, 1998, 88:18～29
[62] Sánchez-sesma F J, Campillo M. Topographic effects for incident P, SV and Rayleigh waves. Tectonophysics, 1993, 218:113～125
[2] Kennett B L N. Seismic waves in laterally varying media. Geophys. J. Roy. Astr. Soc, 1972, 27: 310～325
[3] Hudson J A, Humphryes R F, Mason I M, Kembhavi V K. The scattering of longitudinal elastic waves at a rough free surface. J. Phys. D: Appl. Phys, 1973, 6: 2174～2186
[4] Snieder R, Nolet G. Linearised scattering of surface waves on a spherical earth. J. Geophys, 1987, 61: 55～63
[5] Cohen J K, Bleistein N. Velocity inversion procedure for acoustic waves. Geophysics, 1979, 44 : 1077～ 1087
[6] Clayton R W, Stolt R H. A Born-WKBJ inversion method for acoustic reflection data. Geophysics, 1981, 46:1559 ～1567
[7] Wu R S, Toks(o|¨)z M N. Diffration tomography and multisource holography applied to seismic imaging. Geophysics, 1987, 52: 11～25
[8] Snieder R. Large-scale waveform inversions of surface waves for lateral heterogeneity-I. Theory and numerical examples. J. Geophys. Res, 1988, 93:12055～12066
[9] Coates R T, Chapman C H. Generalized Born scattering of elastic waves in 3-D media. Geophys. J. Int, 1991, 107: 231～263
[10] Knopoff L, Hudson J A. Scattering of elastic waves by small inhomogeneities. J. Acoust. Soc. Am, 1964, 36:338～343
[11] Sato H. Attenuation of body waves and envelope formation of three-component seismograms of small local earthquakes in randomly inhomogeneous lithosphere. J. Geophys. Res, 1984, 89: 1221～1241
[12] Wu R S, Aki K. Elastic wave scattering by a random medium and the small-scale inhomogeneities in the Lithosphere. J. Geophys. Res, 1985, 90: 10261～10273
[13] Flatté S M, Wu R S. Small-scale structure in the Lithosphere and asthenosphere deduced from arrival time and amplitude fluctuations. J. Geophys. Res, 1988, 93:6601～6614
[14] Hudson J A, Heritage J R. The use of the Born approximation in seismic scattering problems. Geophys. J. Roy. Astr. Soc, 1982, 66:221～240
[15] Bleistein N, Cohen J K, Stockwell Jr J W. Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion. New York: Springer-Verlag Inc, 2000
[16] Sato H, Fehler M C. Seismic wave propagation and scattering in the heterogeneous earth. Springer, 1998
[17] Fu L Y, Wu R S, Campillo M. Energy partition and attenuation of regional phases by random free surface. Bulletin of the Seismological Society of America, 2002, 92: 1992～2007
[18] Carter J A, Frazar L N. Accommodating lateral velocity changes in Kirchhoff migration by means of Fermat' s principle. Geophysics, 1984, 49:46～53
[19] Vidale J. Finite-difference calculation of traveltimes. Bull. Seis. Soc. Am, 1988, 78:2062～2076
[20] Keho T H, Beydoun W B. Paraxial ray Kirchhoff migration. Geophysics, 1988, 53 : 1540 ～ 1546
[21] Hill N R. Gaussian beam migration. Geophysics, 1990, 55: 1416～1428
[22] Moser T. Shortest path calculation of seismic rays. Geophysics, 1991, 56:59～67
[23] 徐界,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 C, Liu H, et al. A grid-changeable method for ray tracing. Chinese J. Geophys. (in Chinese), 1996,39(1) :97～102
[24] 张建中,陈世军,徐初伟.动态网格最短路径射线追踪.地球物理学报,2004,47(5) :899～904Zhang J Z, Chen C J, Xu C W. A method of shortest path raytracing with dynamic networks. Chinese J. Geophys. (in Chinese), 2004,47(5) :899～904
[25] Zhang M, Chen B, Li X, et al. A fast algorithm of shortest path ray tracing. Chinese J. Geophys. , 2006, 49: 1467～ 1474
[26] Gray S H, Etgen J, Dellinger J, et al. Seismic migration problems and solutions. Geophysics, 2001, 66 : 1622～ 1640
[27] Woodward M J. Wave-equation tomography. Geophysics, 1992, 57:15～26
[28] Stoffa P L, Fakkema J T, de Luna Freire R M, et al. Splitstep Fourier migration. Geophysics, 1990, 55 : 410～421
[29] Fisk M D, McCartor G D. The phase screen method for vector elastic waves. J. Geophys. Res, 1991, 96: 5985～ 6010
[30] Wu R S. Synthetic seismogram in heterogeneous media by one-return approximation. Pure and Applied Geophys, 1996, 145:155～173
[31] Huang L J, Fehler M C, Wu R S. Extended local Born Fourier migration method. Geophysics, 1999, 64:1524 ～1534
[32] Jin S, Wu R S, Peng C. Prestack depth migration using a hybrid pseudo screen propagator. 68th Ann. Internat. Mtg., Soc. Expl. Geophys, Expanded Abstracts, 1998, 1819～1822
[33] Wu R S, Jin S, Xie X B. Seismic wave propagation and scattering in heterogeneous crustal wave guides using screen propagators I. SH waves. Bull. Seism. Soc. Am, 2004, 90:401～413
[34] de Hoop M V, Le Rousseau J H, Wu R S. Generalization of the phase-screen approximation for the scattering of acoustic waves. Wave Motion, 2000, 31:43～70
[35] Fu L Y, Duan Y. Fourier depth migration methods with application to salt-related complex geological structures. 72th Ann. Internat. Mtg. , Soc. Expl. Geophys, Expanded Abstracts, 2002． 895～898
[36] Fu L Y. Comparison of different one-way propagators for wave forward propagation in heterogeneous crustal wave guides. Bull. Seism. Soc. Am, 2006, 96:1091～1113
[37] Fu L Y, Sun W J, Li D P. Degenerate fourier migrators for imaging complex fault zones. Chinese J. Geophys, 2007, 50 : 483～495
[38] Collins M D. A split-step Padé solution for the parabolic equation method. J. Acoust. Soc. Am, 1993, 93: 1736～ 1742
[39] Ristow D, R(u|¨)hl T. Fourier finite-difference migration. Geophysics, 1994, 59 : 1882～1893
[40] Wu R S. Wide-angle elastic wave one-way propagation in heterogeneous media and an elastic wave complex-screen method. J. Geophys. Res, 1994, 99:751～766
[41] Wild A J, Hudson J A. A geometrical approach to the elastic complex screen. J. Geophys. Res, 1998, 103:707～725
[42] Hobbs R W. 3-D modelling of seismic wave propagation using complex elastic screens with application to mineral exploration. Special Volume on Hardrock Geophysics, Soc. Expl. Geophys. Publisher, 2003
[43] Fu L Y. Born dispersion equation and Kirchhoff migration in laterally heterogeneous media. 72th Ann. Internat. Mtg. , Soc. Expl. Geophys, Expanded Abstracts, 2002． 1097～ 1100
[44] Fu L Y. Wavefield interpolation in the Fourier wavefield extrapolation. Geophysics, 2004, 69:257～264
[45] Schuster G T. A hybrid BIE+Born series modeling scheme: Generalized Bom series. J. Acoust. Soc. Am, 1985, 77: 865～ 879
[46] Schuster G T. Solution of the acoustic transmission problem by a perturbed Born series. J. Acoust. Soc. Am, 1985, 77: 880～886
[47] Fu L Y. Rough surface scattering: Comparison of various approximation theories for 2D SH waves. Bull. Seism. Soc. Am, 2005, 95: 646～663
[48] Fu L Y, Bouchon M. Discrete wavenumber solutions to numerical wave propagation in piecewise heterogeneous media--I. Theory of two-dimensional SH case. Geophys. J. Int, 2004, 157:481～498
[49] Fu L Y, Mu Y G, Yang H J. Forward problem of nonlinear Fredholm integral equation in reference medium via velocityweighted wavefield function. Geophysics, 1997, 62: 650～656
[50] Fu L Y, Wu R S. Infinite boundary element absorbing boundary for wave propagation simulations. Geophysics, 2000, 65:625～637
[51] Berkhout A J, Wapenaar C P A. One way versions of the Kirchhoff integral. Geophysics, 1989, 54:460～467
[52] Hilterman F J. Three dimensional seismic modeling. Geophysics, 1970, 35:1020～1037
[53] Trorey A W. A simple theory for seismic diffractions. Geophysics, 1970, 35:762～784
[54] French W S. Computer migration of oblique seismic reflection profiles. Geophysics, 1975, 40 : 6 ～ 16
[55] Schneider W A. Integral formulation for migration in two and three dimensions. Geophysics, 1978, 43:49～76
[56] Safar M H. On the lateral resolution achieved by Kirchhoff migration. Geophysics, 1985, 50: 1091～1099
[57] Yilmaz O. Seismic data processing. Soc. Expl. Geophys. Publisher, 1987
[58] Hubral P, Schleicher J, Tygel M, Hanitzsch C. Determination of Fresnel zones from traveltime measurements. Geophysics, 1993, 58: 703～712
[59] Schleicher J, Hubral P, Tygel M, et al. Minimum apertures and Fresnel zones in migration and demigration. Geophysics, 1997, 62:183～194
[60] Bleistein N, Cohen J K,. Stockwell Jr J W. Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion. New York: Springer-Verlag Inc, 2000
[61] Huang L J, Fehler M C. Accuracy analysis of the split-step Fourier propagator: implications for seismic modeling and migration. Bull. Seism. Soc. Am, 1998, 88:18～29
[62] Sánchez-sesma F J, Campillo M. Topographic effects for incident P, SV and Rayleigh waves. Tectonophysics, 1993, 218:113～125
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