KH积分方法合成地震图
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摘要
本论文系统简洁地介绍Kirchhoff-Helmholtz积分方法(简称KH积分方法)的基本原理、推导了基本计算公式并将其应用到计算反射波、转换波和回折波的地震学问题中。KH积分方法是一种边界积分方法，它从严格的波动方程出发，将体积分转化为边界积分，当用于计算反射或透射波时，KH积分方法把界面上的每个点都看作一个点源，认为每个点源对反射或透射振幅都有一定的贡献，把界面上每个点的贡献相加就得到了反射或透射响应。
本文首先用KH积分方法计算单层水平界面的反射波响应，并与反射率法和有限差分方法的计算结果进行了对比，验证了方法及程序的正确性。在此基础上，进一步模拟了弯曲界面的反射波响应，与有限差分模拟的结果对比表明：KH积分方法在计算不规则界面时有很好的精确度，而且其计算速度明显优于有限差分方法。考虑自由界面的反射，我们应用KH积分方法计算了包含深度震相在内的所有反射波：PP、PS、pPP、pPS、pSP、pSS，通过与反射率法的结果对比发现，六种震相都吻合的很好。
进一步地把KH积分方法推广到计算近震／远震转换波响应，推导出转换波的KH积分公式，分别计算了水平界面与弯曲界面上的近震／远震转换波响应，通过与反射率法、动力学射线追踪及有限差分方法的对比，表明KH方法能够很好地模拟不规则界面上的近震／远震转换波。
最后我们用KH方法分别计算了含有多层界面介质中的反射波响应和速度梯度变化介质中的回折波响应，与反射率方法结果非常相近。
我们的研究表明，KH积分方法是一种较好地模拟横向非均匀介质的合成地震图方法，其计算精度不亚于其他方法，且计算效率高。
In this paper, author systematically and succinctly introduced the basic theory and the basisequations of Kirchhoff-Helmholtz (KH) integral, and applied the method to the seismologicalproblems of reflected wave, converted wave and refracted wave. The KH integral is one of theboundary integral methods, it is derived in terms of wave equation and transforms body integral tothe boundary integral. When computing the reflected wave and transmitted wave, every point overthe surface of the boundary can be regarded as a point source which contributes to the reflectedwave or transmitted wave and the wave responses are the summation of the contribution ofevery point over the boundary.
First, we computed the response of reflected wave on a single horizontal interface using themethod of KH integral. The numberical results are good agreement with those of reflectivirymethod and finite-difference method. Furthermore, the response of reflected wave on a curvedinterface are calculated, comparison with the results from the finite-difference method shows thatKH method can work quickly on curved or irregular interface, its computing speeds is superior tofinite-difference method. In addition, the reflected waves from the free surface are obtained by themethod of KH integral. From computing the six kinds of phases: PP, PS, pPP, pPS, pSP, pSS, myresult indicates that both of the amplitude and the wave form for the model of layered medium arebasically similar to the reflectivity method.
From above basis, the KH integral theory is also applied to compute the conversion waveresponses. With the derivation of the conversion wave of KH integral equation, we simulatted thenear seism conversion wave response and the teleseismic conversion wave response respectively.The results compared with that of the reflectivity method and the dynamic ray tracing givessatisfactory result.
Finally, we try to calculate the theoretical seismogram in the multi-layered medium and thesituation that the material parameters are linear below the surface. The result is consistent with thatfrom the reflectivity method.
Our analysis indicates that KH integral method is a preferable synthetic seismogram method in a laterally inhomogeneous elastic medium. The accuracy of this approximation is excelled othermethods and it has the high computing efficiency also.
本文首先用KH积分方法计算单层水平界面的反射波响应，并与反射率法和有限差分方法的计算结果进行了对比，验证了方法及程序的正确性。在此基础上，进一步模拟了弯曲界面的反射波响应，与有限差分模拟的结果对比表明：KH积分方法在计算不规则界面时有很好的精确度，而且其计算速度明显优于有限差分方法。考虑自由界面的反射，我们应用KH积分方法计算了包含深度震相在内的所有反射波：PP、PS、pPP、pPS、pSP、pSS，通过与反射率法的结果对比发现，六种震相都吻合的很好。
进一步地把KH积分方法推广到计算近震／远震转换波响应，推导出转换波的KH积分公式，分别计算了水平界面与弯曲界面上的近震／远震转换波响应，通过与反射率法、动力学射线追踪及有限差分方法的对比，表明KH方法能够很好地模拟不规则界面上的近震／远震转换波。
最后我们用KH方法分别计算了含有多层界面介质中的反射波响应和速度梯度变化介质中的回折波响应，与反射率方法结果非常相近。
我们的研究表明，KH积分方法是一种较好地模拟横向非均匀介质的合成地震图方法，其计算精度不亚于其他方法，且计算效率高。
In this paper, author systematically and succinctly introduced the basic theory and the basisequations of Kirchhoff-Helmholtz (KH) integral, and applied the method to the seismologicalproblems of reflected wave, converted wave and refracted wave. The KH integral is one of theboundary integral methods, it is derived in terms of wave equation and transforms body integral tothe boundary integral. When computing the reflected wave and transmitted wave, every point overthe surface of the boundary can be regarded as a point source which contributes to the reflectedwave or transmitted wave and the wave responses are the summation of the contribution ofevery point over the boundary.
First, we computed the response of reflected wave on a single horizontal interface using themethod of KH integral. The numberical results are good agreement with those of reflectivirymethod and finite-difference method. Furthermore, the response of reflected wave on a curvedinterface are calculated, comparison with the results from the finite-difference method shows thatKH method can work quickly on curved or irregular interface, its computing speeds is superior tofinite-difference method. In addition, the reflected waves from the free surface are obtained by themethod of KH integral. From computing the six kinds of phases: PP, PS, pPP, pPS, pSP, pSS, myresult indicates that both of the amplitude and the wave form for the model of layered medium arebasically similar to the reflectivity method.
From above basis, the KH integral theory is also applied to compute the conversion waveresponses. With the derivation of the conversion wave of KH integral equation, we simulatted thenear seism conversion wave response and the teleseismic conversion wave response respectively.The results compared with that of the reflectivity method and the dynamic ray tracing givessatisfactory result.
Finally, we try to calculate the theoretical seismogram in the multi-layered medium and thesituation that the material parameters are linear below the surface. The result is consistent with thatfrom the reflectivity method.
Our analysis indicates that KH integral method is a preferable synthetic seismogram method in a laterally inhomogeneous elastic medium. The accuracy of this approximation is excelled othermethods and it has the high computing efficiency also.
引文
Cerveny V，Molotkov I et al．地震学中的射线方法．(刘福田等译)[M]．1986．北京：地质出版社，1～208
王椿镛．1985．层状不均匀介质中合成地震图的计算及其在地震测深资料解释中的应用[D]．北京：中国地震局地球物理研究所．
温联星，姚振兴．1994．用广义射线和有限差分计算近场理论地震图的混合方法[J]．地球物理学报，37(2)：211～219
谢小碧，姚振兴．1988．二维不均匀介质中点源P-SV波响应的有限差分近似算法[J]．地球物理学报，31(5)：540～555
谢小碧，郑天愉，姚振兴．1992．理论地震图计算方法[J]．地球物理学报，35：790～801
朱良保．1993．Maslov渐近理论地震图[D]．北京：中国地震局地球物理研究所．
Aki K, Richards P G.. Quantitative Seismology Theory and Methods[M]. 1980. San Francisco: W H Freeman, 1～932.
Berryhill J R. 1979. Wave-equation damming [J]. Geophysics, 44:1329～1344
Bouchon M. 2003. A review of the discrete wavenumber method [J]. Pure. Appl. Geophy, 160:445～465
Bouchon M, Aki K. 1977. Discrete wave-number representation of seismic-source wave fields [J]. Bull. Seismol. Soc. Am, 67:259～277
Carter J A, Frazer L N. 1983. A method for modeling reflection data from media with lateral velocity changes [J]. J. geophys. Res, 88:6469～647
Cerveny V, Molotkov I A. Ray method in seismology[M]. 1977. Karlova University: Praha,79～86
Cerveny V. 1983. Synthetic body wave seismograms for laterally varying layered structures by Gaussian beam method[J]. Geophy. J. R. astr. Soc, 73:389～426
Chapman C H. 1978. A new method for computing synthetic seismograms [J]. Geophy. J. R. astr. Soc, 54:481～518
Chapman C.H. 1985. The computation of body wave synthetic seismograms in laterally homogeneous media [J]. Rev. Geophys, 23(2): 105-163
Chapman C H, Drumnood R. 1982. Body wave seismograms in inhomogeneous media using Maslov asymptotic theory [J]. Bull. Seismol. Soc. Am ,72(6):277-317
Chen X F. 1990. Seismograms synthesis for multi-layered media with irregular interfaces by global generalized reflection/transmission matrices method.Part I.Theory of 2-D SH case[J]. Bull.Seism.Soc.Am,80:1696-1724
Cormier V F, Richards P G.. 1977. Full wave theory applied to a discontinuous velocity increase:the inner core boundary [J]. J. Geophys, 43:3-22
Deregowski S M, Brown S M. 1983. A theory of acoustic diffractors applied to 2D models [J]. Geophys. Prospect, 31:293-333
Dmowska R. Advances in wave propagation in heterogeneous earth [M]. 2007. London: Academic Press, 1 -606
Frazer L N, Sen M K. 1985. Kirchhoff-Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium- I . Theory [J]. Geophys. J. R. astr. Soc, 80:121-147
Fuchs K, Muller G.. 1971. Computation of synthetic seismograms with the reflectivity method and comparison with observation [J]. Geophy.J. R. astr. Soc, 23(2):417-433
Helmberger D V. 1968. Crust-mantle transition in the Bering sea [J]. Bull. Seism. Soc. Amer, 58:179-214
Helmberger D V. 1974. Generalized ray theory for shear dislocations [J]. Bull. Seism. Soc. Amer, 64:45-54
Hilterman F J. 1970. Three dimensional seismic modeling [J]. Geophysics, 35:1020-1037
Igel H, Mora P, Riollet B. 1995. Anisotropic wave propagation through finite-difference grids [J]. Geophysics,60:1203-1216
Kennett B L N. 1983. Seismic waves propagation in a stratified media[M]. Cambridge: Cambridge University, 342
Lamb H. 1904. On the propagation of tremors over the surface of an elastic solid[J]. Philos.Trans.R.Soc.London, A203:1 -42
Levander R. 1988. Fourth-order finite-difference P-SV seismograms [J]. Geophysics,53: 1425-1436
Scott P, Helmberger D. 1983. Applications of the Kirchhoff-Helmholtz integral to problems in seismology[J]. Geophys.J.R.astr.Soc, 72:237-254
Sen M K, Frazer L N. 1987. Synthetic seismograms using multifold path integrals- II .Theory[J]. Geophys. J. R. astr. Soc,88:647-671
Smith W D. 1975. The application of finite element analysis to body-wave propagation problems [J]. Geophys. J. R. Astr. Soc,,42:747-768
Wu R S. 1989. The perturbation method in elastic wave scattering [J]. Pure. Appl. Geophy, 131:605-637
Zhang W, Chen X F. 2006. Traction image method for irregular free surface boundaries in finite difference seismic wave simulation [J]. Geophys. J. Int,167:337-353
Ziolkowski R W , Deschamps G A. 1980. The Maslov method and the asymptotic Fourier transform:caustic analysis. Electro. Lab. Sci. Rep. No.80-9, University of Illinois at Urbana- Champain.
王椿镛．1985．层状不均匀介质中合成地震图的计算及其在地震测深资料解释中的应用[D]．北京：中国地震局地球物理研究所．
温联星，姚振兴．1994．用广义射线和有限差分计算近场理论地震图的混合方法[J]．地球物理学报，37(2)：211～219
谢小碧，姚振兴．1988．二维不均匀介质中点源P-SV波响应的有限差分近似算法[J]．地球物理学报，31(5)：540～555
谢小碧，郑天愉，姚振兴．1992．理论地震图计算方法[J]．地球物理学报，35：790～801
朱良保．1993．Maslov渐近理论地震图[D]．北京：中国地震局地球物理研究所．
Aki K, Richards P G.. Quantitative Seismology Theory and Methods[M]. 1980. San Francisco: W H Freeman, 1～932.
Berryhill J R. 1979. Wave-equation damming [J]. Geophysics, 44:1329～1344
Bouchon M. 2003. A review of the discrete wavenumber method [J]. Pure. Appl. Geophy, 160:445～465
Bouchon M, Aki K. 1977. Discrete wave-number representation of seismic-source wave fields [J]. Bull. Seismol. Soc. Am, 67:259～277
Carter J A, Frazer L N. 1983. A method for modeling reflection data from media with lateral velocity changes [J]. J. geophys. Res, 88:6469～647
Cerveny V, Molotkov I A. Ray method in seismology[M]. 1977. Karlova University: Praha,79～86
Cerveny V. 1983. Synthetic body wave seismograms for laterally varying layered structures by Gaussian beam method[J]. Geophy. J. R. astr. Soc, 73:389～426
Chapman C H. 1978. A new method for computing synthetic seismograms [J]. Geophy. J. R. astr. Soc, 54:481～518
Chapman C.H. 1985. The computation of body wave synthetic seismograms in laterally homogeneous media [J]. Rev. Geophys, 23(2): 105-163
Chapman C H, Drumnood R. 1982. Body wave seismograms in inhomogeneous media using Maslov asymptotic theory [J]. Bull. Seismol. Soc. Am ,72(6):277-317
Chen X F. 1990. Seismograms synthesis for multi-layered media with irregular interfaces by global generalized reflection/transmission matrices method.Part I.Theory of 2-D SH case[J]. Bull.Seism.Soc.Am,80:1696-1724
Cormier V F, Richards P G.. 1977. Full wave theory applied to a discontinuous velocity increase:the inner core boundary [J]. J. Geophys, 43:3-22
Deregowski S M, Brown S M. 1983. A theory of acoustic diffractors applied to 2D models [J]. Geophys. Prospect, 31:293-333
Dmowska R. Advances in wave propagation in heterogeneous earth [M]. 2007. London: Academic Press, 1 -606
Frazer L N, Sen M K. 1985. Kirchhoff-Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium- I . Theory [J]. Geophys. J. R. astr. Soc, 80:121-147
Fuchs K, Muller G.. 1971. Computation of synthetic seismograms with the reflectivity method and comparison with observation [J]. Geophy.J. R. astr. Soc, 23(2):417-433
Helmberger D V. 1968. Crust-mantle transition in the Bering sea [J]. Bull. Seism. Soc. Amer, 58:179-214
Helmberger D V. 1974. Generalized ray theory for shear dislocations [J]. Bull. Seism. Soc. Amer, 64:45-54
Hilterman F J. 1970. Three dimensional seismic modeling [J]. Geophysics, 35:1020-1037
Igel H, Mora P, Riollet B. 1995. Anisotropic wave propagation through finite-difference grids [J]. Geophysics,60:1203-1216
Kennett B L N. 1983. Seismic waves propagation in a stratified media[M]. Cambridge: Cambridge University, 342
Lamb H. 1904. On the propagation of tremors over the surface of an elastic solid[J]. Philos.Trans.R.Soc.London, A203:1 -42
Levander R. 1988. Fourth-order finite-difference P-SV seismograms [J]. Geophysics,53: 1425-1436
Scott P, Helmberger D. 1983. Applications of the Kirchhoff-Helmholtz integral to problems in seismology[J]. Geophys.J.R.astr.Soc, 72:237-254
Sen M K, Frazer L N. 1987. Synthetic seismograms using multifold path integrals- II .Theory[J]. Geophys. J. R. astr. Soc,88:647-671
Smith W D. 1975. The application of finite element analysis to body-wave propagation problems [J]. Geophys. J. R. Astr. Soc,,42:747-768
Wu R S. 1989. The perturbation method in elastic wave scattering [J]. Pure. Appl. Geophy, 131:605-637
Zhang W, Chen X F. 2006. Traction image method for irregular free surface boundaries in finite difference seismic wave simulation [J]. Geophys. J. Int,167:337-353
Ziolkowski R W , Deschamps G A. 1980. The Maslov method and the asymptotic Fourier transform:caustic analysis. Electro. Lab. Sci. Rep. No.80-9, University of Illinois at Urbana- Champain.