某些数学方法在地震波反演问题求解中的应用
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摘要
本文以地震波广义散射(包括透射、折射、反射、绕射等)为背景,对其反演问题的发展状况做了一个回顾.文中着重介绍了在地震波逆散射问题研究过程中,各种数学、物理的基本理论和基本假设是如何被应用于非线性偏微分方程这一复杂问题的求解过程的.同时,结合对地震波逆散射问题的认识,简单介绍了与之相关的数学方法及其在应用中存在的问题.
In this paper, the inverse question for seismic waves of generalized scattering has been reviewed. It is emphasized how some basically mathematical and physical theories and basic assumptions are applied to solving nonlinearly partial differential equations, such a complicated problem, in research into inverse scattering of seismic waves. Also, a brief introduction on problems in application of the above-mentioned mathematical methods on inversion of inverse scattering of seismic waves are discussed.
In this paper, the inverse question for seismic waves of generalized scattering has been reviewed. It is emphasized how some basically mathematical and physical theories and basic assumptions are applied to solving nonlinearly partial differential equations, such a complicated problem, in research into inverse scattering of seismic waves. Also, a brief introduction on problems in application of the above-mentioned mathematical methods on inversion of inverse scattering of seismic waves are discussed.
引文
[1] Backus G E, Gilbert J F. Numerical application of a formulism for geophysical inversion [J]. Geophys. J. R. astr. Soc., 1967, 13: 247-276.
[2] Backus G E, Gilbert J F. The resolving power of gross earth data [J]. Geophys. J. R. astr. Soc., 1968, 16: 169-205.
[3] Backus G E, Gilbert J F. Uniqueness in the inversion of inaccurate gross earth data [J]. Phil. Trans Roy. Soc. London, Ser A 1970, 266: 123-192.
[4] Aki K, Richards P G. Quantative Seismology Theory and Methods, Vol Ⅱ[M]. Chapter 12, Freeman and Company.
[5] Cooke D A, Schneider W A. Generalized inversion of reflection seismic data [J]. Geophys., 1983, 48: 665-676.
[6] Hutson V, Pym J S. Application of Functional Analysis and Operator Theory [M]. Academic Press, 1980.
[7] Tarentola A, Valatte B. Generalized nonlinear inverse problems solved using the least square criterion [J]. Review of Geophysics Space Physics, 1982, 20: 219-232.
[8] Cohen J K. et al. Three-dimensional Born inversion with an arbitrary reference [J]. Geophys, 1986, 51: 1552-1558.
[9]杨文采．地球物理反演的理论与方法[M]．北京:地质出版社,1997．
[10] Chernov Lev A. Wave Propagation in a Random Medium [M]. McGRAW-Hill Book Company, INC: New York: 1960.
[11] Winggins R A. The generalized linear inverse problem: Implication of surface waves and free oscillations for earth structure [J]. Rev. Geophys. Space Phys., 1972, 10: 251-285.
[12] Jackson D D. Interpretation of inaccurate, insufficient, and inconsistent data [J]. Geophys. J. R. asre. Soc., 1972, 28: 97-109.
[13] Lsor P. An algorithm for sparse linear equations and sparse least quarts [J]. ACM Tmns. Math. Softw., 1982, 8.
[14]杨文采．非线性波动方程地震反演的方法及问题[J]．地球物理学进展,1992,7(1):9-19．
[15] Schneider W S. Integral formulation for migration in two and three dimensions [J]. Geophysics, 1991, 56(10): 1624-1638.
[16] Waterman P C. Matrix theory of elastic wave scattering [J]. JASA, 1976, 60: 567-580.
[17] Jain D L, Kanwal R P. The Born approximation for the scattering theory of elastic waves by two-dimensional flaws [J]. J. Appl. Phys., 1982, 53(6): 4208-4215.
[18] Devaney A J. Inverse-scattering theory within the Rytov approximation [J]. Optics Letters, 1981, 6(8): 374-376.
[19] Moser T J. Shortest path calculation of seismic rays [J]. Geophysics, 1991, 56(1): 59-67.
[20] Kundu T. Inversion of acoustic material signature of multilayered solids [J]. JASA, 1992, 91(2): 591-600.
[21] Mora P. Nonlinear two-dimensional elastic inversion of multi-offset seismic data [J]. Geophysics, 1987, 52: 1211-1228.
[22]黄联捷,杨文采．声波方程逆散射反演的近似方法[J].地球物理学报,1991,34(5):626-634．
[23]黄联捷,杨文采．参考波速线性变化时声波方程逆散射反演[J]．地球物理学报,1991,34(1):80-98．
[24]杨文采．地震波场反演的BG-逆散射方法[J].地球物理学报,1995,38(5):358-361．
[25] Cohen J K, Bleistein N. An inverse method for determining small variations in propagation speed [J]. J. Appl. Math., 1977, 32: 784-799.
[26] Mora P. Inversion=migration + tomography [J]. Geophysics, 1989, 54(12): 1575-1586.
[27]刘光鼎,陈运泰,陈顒,李幼铭．80年代地球物理进展论文集[M]．科学文献出版社,1989,
[28]栾文贵,李幼铭．我国地球物理反演问题研究的某些进展[J]．地球物理学报,1990,33:501-508．
[29] Hudson J A, Heritage J R. The use of the Born approximation in seismic scattering problems [J]. Geophys. J. R. astr. Soc., 1981, 66: 221-240.
[30] Rubramaniam D R, George V F. A comparison between the Born and Rytov approximations for the inverse backscattering problem [J]. Geophysics, 1989, 54: 864-871.
[31] Beylkin G. The inversion problem and applications of the generalized Radon transform [J]. Comm. Pure & Applied Math., 1984, 37:
[32] Shen Y, Yang W C. Application of the generalized Radon transform to prestack seismic data protections [J]. Chinese J. Geophys, 1992, 35: 97-110.
[33]马兴瑞,陶良,黄文虎．弹性波反演方法及其应用[M]．北京:科学出版社,1999．
[34] Langenberg K J. Introduction to the special issue on inverse problems [J]. Wave Motion, 1989, 11: 99-112.
[35] Clayton R W, Stolt R H. A Born-WKBJ inversion method for acoustic reflection data [J]. Geophysics, 1981, 46(11): 1559-1567.
[36] Feng K. Differential schemes for Hamiltonian formalism and symplectic geometry [J]. J. Compu. Math., 1986, 4(3): 279-289.
[37]高亮,李幼铭,陈旭容,杨孔庆．地震射线辛几何算法初探[J]．地球物理学报,2000,43(3):402-409．
[38] Ivansson S Remark on an earlier proposed iterative tomographic algorithm [J]. Geophys. J. Hay. Astr. Soc., 1983, 75: 855-860.
[39]袁亚湘,孙文瑜．最优化理论与方法[M]．北京:科学出版社,1997．
[40] Cryer C W. Numerical Functional Analysis [M]. Clarendon Press Oxford, 1982.
[41] Powell M J D. On the global convergence of trust region algorithms for unconstrained optimization [J]. Math. Prog., 1984, 29: 297-303.
[42] Sheng Y, Benjamin W W. Global optimization for neural network training [J]. Computer, 1996, (3): 45-54
[43] Coleman T F, Conn A R. On the local convergence of a quasi-Newton method for nonlinear programming problem [J]. SIAN J. Numer. Anal, 1984, 21: 755-769.
[44] Wang S L et al. An efficient numerical method for exterior and interior inverse problem of Helmholtz equation [J] Wave Motion, 19911, 13: 387-399.
[45] Pratt R G et al. Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion [J]. Geophys. J. Int., 133: 341-362.
[46] Paige C C, Saunders M A. ASQR: Sparse linear equations and least square problems, AMC. Transactions Math. 1982, 8: 43-71 and 195-209.
[47] Fletcher R. An exact penalty function for nonlinear for constrained optimization [J].J. Inst. Math. Prog., 1973, (5): 129-150.
[48] Powell M J D. Restart procedure for the conjugate gradient method [J]. Math. Prog., 1977, 12: 241-254.
[49] Philips D L. A technique for the numerical solution of certain integral equation of the first kind [J]. J. ACM., 1962, 9: 84-97
[50] Scherzer O. Convergence rates of iterated Tikhonov regularized solution of nonlinear Ill-posed problems [J]. Numer Math., 1993, 66: 259-279.
[51] Han B, You J H, Liu J Q. A discrete-type continuation regularization method and its application [J]. J of Harbin Inst. Tech., 1995, E-2(l): 5-9 & 10-14.
[52]陈开舟,党创寅,杨再福．不动点理论和算法[M]．西安电子科技大学出版社,西安:1990.
[53] Wang C S, Tang J, Zhao B. An improvement on D dorm deconvolution: a fast algorithm and the related procedure [J] Geophys, 1991, 56(5): 635-644.
[54] Schechter M. Principles of Functional Analysis [Ml. Academic Press, 1971.
[55]李世雄,刘家琦．小波变换和反演数学基础[M]．北京:地质出版社,1994．
[56] Cowie P A, Vanneste C, Sornette D. Statistical physics model for the spatiotemporal evolution of faults [J]. J. Geophys. Res., 1993, 98: 21809-21821.
[57] Tarantola A, Valette B. Inverse problems=quest for information [J]. J. Geophys. 50: 159-170.
[58] Tarantola A. Linearized inversion of seismic reflection data [J]. Geophys. Prosp., 1884, 32: 998-1015.
[59] Tarantola A. Inversion of seismic reflection data in the acoustic approximation [J]. Geophysics, 1984, 49: 1259-1266.
[60] Tarantola A. A strategy for nonlinear elastic inversion of seismic reflection data [J]. Geophysics, 1986, 51: 1893-1903.
[61] Kirkpatrick S, Gelatt Jr C D, Vecchi M P. Optimization by simulated annealing [J]. Science, 1983, 220(4598): 671-680
[62] Ingber L. Genetic algorithms and very fast simulated reannealing: a comparison [J]. Math. Comput. Modeling, 1992, 16: 87-100.
[63] Dean V B, Michael M M. Temporal information transformed into a spatial code by a neural network with realistic properties [J]. Science, 1995, 267(17): 1028-1031.
[64] Djurdje C, Jacek K. Taboo search: an approach to the multiple minima problem [J]. Science, 1995, 267(3): 664-671.
[65] Sambridge M S, Braun, McQueen H. Geophysical parameterization and interpolation of irregular data using natural neighbors [J]. Geophys. J. Int., 1995, 122: 837-857.
[66] Kennett B L N, Sambridge M S, Williamson P R. Subspace methods for inverse problems with multiple parameter classes [J]. Geophys. J. Int., 1995, 94: 237-247.
[67] Sambridge M S. Geophysical inversion with a neighborhood algorithm-(Ⅰ). Searching a parameter space [J]. Geophys. J. Int., 1999, 138: 479-494.
[68] Sambridge M S. Geophysical inversion with a neighborhood algorithm-(Ⅰ). Appraising the ensemble [J]. Geophys J. Int., 1999, 138: 727-746.
[69] Kennett B L N. Sambridge M. S. Inversion for multiple parameter classes [J]. Geophys. J. Int., 1998, 135: 304-306.
[70] Guide G, Francesco M, Warner M. Practical application of fractal analysis: problems and solutions [J]. Geophys. J. Int., 1998, 132: 275-282.
[71] Sugihara g, May R M. Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series [J]. Nature, Vol. 334, 1990.
[72] Guide G, Francesco M, Matteo C. Measuring the fractal dimensions of ideal and actual objects: implications for application in geology and geophysics [J]. Geophys. J. I., 2000, 142: 108-116.
[73]杨文采．地震道的非线性混沌反演-Ⅰ理论和数值试验[J]．地球物理学报,1993,36(2):222-232．
[74]杨文采．非线性地震道的混沌反演-Ⅱ关于L指数和吸引子[J].地球物理学报,1993,36(3):376-386．
[75]杨磊,张向君,李幼铭．地震道非线性反演的参数反馈控制及效果[J]．地球物理学报,1999,42(5):677- 684．
[76]艾印双,刘鹏程,郑天愉．自适应全局混合反演[J]．中国科学(D),28(2):105-110．
[77]王仁,方开泰．关于均匀分布与试验设计(数论方法)[J]．科学通报,1981,26(2):65-70．
[78]杨文采．波方程的逆散射与迭代线性化反演[J].石油物探,1995,34(3):114-122．
[79] Tikhonove A N. Continuation of potential fields in a layered medium by the regularization method [J]. Earth Physics, 1968, 1968.
[80] Tikhonove A N, Arsennin V Y. Solution of Ill-posed Problems [M]. Jonh Wiley, New York, 1977..
[2] Backus G E, Gilbert J F. The resolving power of gross earth data [J]. Geophys. J. R. astr. Soc., 1968, 16: 169-205.
[3] Backus G E, Gilbert J F. Uniqueness in the inversion of inaccurate gross earth data [J]. Phil. Trans Roy. Soc. London, Ser A 1970, 266: 123-192.
[4] Aki K, Richards P G. Quantative Seismology Theory and Methods, Vol Ⅱ[M]. Chapter 12, Freeman and Company.
[5] Cooke D A, Schneider W A. Generalized inversion of reflection seismic data [J]. Geophys., 1983, 48: 665-676.
[6] Hutson V, Pym J S. Application of Functional Analysis and Operator Theory [M]. Academic Press, 1980.
[7] Tarentola A, Valatte B. Generalized nonlinear inverse problems solved using the least square criterion [J]. Review of Geophysics Space Physics, 1982, 20: 219-232.
[8] Cohen J K. et al. Three-dimensional Born inversion with an arbitrary reference [J]. Geophys, 1986, 51: 1552-1558.
[9]杨文采．地球物理反演的理论与方法[M]．北京:地质出版社,1997．
[10] Chernov Lev A. Wave Propagation in a Random Medium [M]. McGRAW-Hill Book Company, INC: New York: 1960.
[11] Winggins R A. The generalized linear inverse problem: Implication of surface waves and free oscillations for earth structure [J]. Rev. Geophys. Space Phys., 1972, 10: 251-285.
[12] Jackson D D. Interpretation of inaccurate, insufficient, and inconsistent data [J]. Geophys. J. R. asre. Soc., 1972, 28: 97-109.
[13] Lsor P. An algorithm for sparse linear equations and sparse least quarts [J]. ACM Tmns. Math. Softw., 1982, 8.
[14]杨文采．非线性波动方程地震反演的方法及问题[J]．地球物理学进展,1992,7(1):9-19．
[15] Schneider W S. Integral formulation for migration in two and three dimensions [J]. Geophysics, 1991, 56(10): 1624-1638.
[16] Waterman P C. Matrix theory of elastic wave scattering [J]. JASA, 1976, 60: 567-580.
[17] Jain D L, Kanwal R P. The Born approximation for the scattering theory of elastic waves by two-dimensional flaws [J]. J. Appl. Phys., 1982, 53(6): 4208-4215.
[18] Devaney A J. Inverse-scattering theory within the Rytov approximation [J]. Optics Letters, 1981, 6(8): 374-376.
[19] Moser T J. Shortest path calculation of seismic rays [J]. Geophysics, 1991, 56(1): 59-67.
[20] Kundu T. Inversion of acoustic material signature of multilayered solids [J]. JASA, 1992, 91(2): 591-600.
[21] Mora P. Nonlinear two-dimensional elastic inversion of multi-offset seismic data [J]. Geophysics, 1987, 52: 1211-1228.
[22]黄联捷,杨文采．声波方程逆散射反演的近似方法[J].地球物理学报,1991,34(5):626-634．
[23]黄联捷,杨文采．参考波速线性变化时声波方程逆散射反演[J]．地球物理学报,1991,34(1):80-98．
[24]杨文采．地震波场反演的BG-逆散射方法[J].地球物理学报,1995,38(5):358-361．
[25] Cohen J K, Bleistein N. An inverse method for determining small variations in propagation speed [J]. J. Appl. Math., 1977, 32: 784-799.
[26] Mora P. Inversion=migration + tomography [J]. Geophysics, 1989, 54(12): 1575-1586.
[27]刘光鼎,陈运泰,陈顒,李幼铭．80年代地球物理进展论文集[M]．科学文献出版社,1989,
[28]栾文贵,李幼铭．我国地球物理反演问题研究的某些进展[J]．地球物理学报,1990,33:501-508．
[29] Hudson J A, Heritage J R. The use of the Born approximation in seismic scattering problems [J]. Geophys. J. R. astr. Soc., 1981, 66: 221-240.
[30] Rubramaniam D R, George V F. A comparison between the Born and Rytov approximations for the inverse backscattering problem [J]. Geophysics, 1989, 54: 864-871.
[31] Beylkin G. The inversion problem and applications of the generalized Radon transform [J]. Comm. Pure & Applied Math., 1984, 37:
[32] Shen Y, Yang W C. Application of the generalized Radon transform to prestack seismic data protections [J]. Chinese J. Geophys, 1992, 35: 97-110.
[33]马兴瑞,陶良,黄文虎．弹性波反演方法及其应用[M]．北京:科学出版社,1999．
[34] Langenberg K J. Introduction to the special issue on inverse problems [J]. Wave Motion, 1989, 11: 99-112.
[35] Clayton R W, Stolt R H. A Born-WKBJ inversion method for acoustic reflection data [J]. Geophysics, 1981, 46(11): 1559-1567.
[36] Feng K. Differential schemes for Hamiltonian formalism and symplectic geometry [J]. J. Compu. Math., 1986, 4(3): 279-289.
[37]高亮,李幼铭,陈旭容,杨孔庆．地震射线辛几何算法初探[J]．地球物理学报,2000,43(3):402-409．
[38] Ivansson S Remark on an earlier proposed iterative tomographic algorithm [J]. Geophys. J. Hay. Astr. Soc., 1983, 75: 855-860.
[39]袁亚湘,孙文瑜．最优化理论与方法[M]．北京:科学出版社,1997．
[40] Cryer C W. Numerical Functional Analysis [M]. Clarendon Press Oxford, 1982.
[41] Powell M J D. On the global convergence of trust region algorithms for unconstrained optimization [J]. Math. Prog., 1984, 29: 297-303.
[42] Sheng Y, Benjamin W W. Global optimization for neural network training [J]. Computer, 1996, (3): 45-54
[43] Coleman T F, Conn A R. On the local convergence of a quasi-Newton method for nonlinear programming problem [J]. SIAN J. Numer. Anal, 1984, 21: 755-769.
[44] Wang S L et al. An efficient numerical method for exterior and interior inverse problem of Helmholtz equation [J] Wave Motion, 19911, 13: 387-399.
[45] Pratt R G et al. Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion [J]. Geophys. J. Int., 133: 341-362.
[46] Paige C C, Saunders M A. ASQR: Sparse linear equations and least square problems, AMC. Transactions Math. 1982, 8: 43-71 and 195-209.
[47] Fletcher R. An exact penalty function for nonlinear for constrained optimization [J].J. Inst. Math. Prog., 1973, (5): 129-150.
[48] Powell M J D. Restart procedure for the conjugate gradient method [J]. Math. Prog., 1977, 12: 241-254.
[49] Philips D L. A technique for the numerical solution of certain integral equation of the first kind [J]. J. ACM., 1962, 9: 84-97
[50] Scherzer O. Convergence rates of iterated Tikhonov regularized solution of nonlinear Ill-posed problems [J]. Numer Math., 1993, 66: 259-279.
[51] Han B, You J H, Liu J Q. A discrete-type continuation regularization method and its application [J]. J of Harbin Inst. Tech., 1995, E-2(l): 5-9 & 10-14.
[52]陈开舟,党创寅,杨再福．不动点理论和算法[M]．西安电子科技大学出版社,西安:1990.
[53] Wang C S, Tang J, Zhao B. An improvement on D dorm deconvolution: a fast algorithm and the related procedure [J] Geophys, 1991, 56(5): 635-644.
[54] Schechter M. Principles of Functional Analysis [Ml. Academic Press, 1971.
[55]李世雄,刘家琦．小波变换和反演数学基础[M]．北京:地质出版社,1994．
[56] Cowie P A, Vanneste C, Sornette D. Statistical physics model for the spatiotemporal evolution of faults [J]. J. Geophys. Res., 1993, 98: 21809-21821.
[57] Tarantola A, Valette B. Inverse problems=quest for information [J]. J. Geophys. 50: 159-170.
[58] Tarantola A. Linearized inversion of seismic reflection data [J]. Geophys. Prosp., 1884, 32: 998-1015.
[59] Tarantola A. Inversion of seismic reflection data in the acoustic approximation [J]. Geophysics, 1984, 49: 1259-1266.
[60] Tarantola A. A strategy for nonlinear elastic inversion of seismic reflection data [J]. Geophysics, 1986, 51: 1893-1903.
[61] Kirkpatrick S, Gelatt Jr C D, Vecchi M P. Optimization by simulated annealing [J]. Science, 1983, 220(4598): 671-680
[62] Ingber L. Genetic algorithms and very fast simulated reannealing: a comparison [J]. Math. Comput. Modeling, 1992, 16: 87-100.
[63] Dean V B, Michael M M. Temporal information transformed into a spatial code by a neural network with realistic properties [J]. Science, 1995, 267(17): 1028-1031.
[64] Djurdje C, Jacek K. Taboo search: an approach to the multiple minima problem [J]. Science, 1995, 267(3): 664-671.
[65] Sambridge M S, Braun, McQueen H. Geophysical parameterization and interpolation of irregular data using natural neighbors [J]. Geophys. J. Int., 1995, 122: 837-857.
[66] Kennett B L N, Sambridge M S, Williamson P R. Subspace methods for inverse problems with multiple parameter classes [J]. Geophys. J. Int., 1995, 94: 237-247.
[67] Sambridge M S. Geophysical inversion with a neighborhood algorithm-(Ⅰ). Searching a parameter space [J]. Geophys. J. Int., 1999, 138: 479-494.
[68] Sambridge M S. Geophysical inversion with a neighborhood algorithm-(Ⅰ). Appraising the ensemble [J]. Geophys J. Int., 1999, 138: 727-746.
[69] Kennett B L N. Sambridge M. S. Inversion for multiple parameter classes [J]. Geophys. J. Int., 1998, 135: 304-306.
[70] Guide G, Francesco M, Warner M. Practical application of fractal analysis: problems and solutions [J]. Geophys. J. Int., 1998, 132: 275-282.
[71] Sugihara g, May R M. Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series [J]. Nature, Vol. 334, 1990.
[72] Guide G, Francesco M, Matteo C. Measuring the fractal dimensions of ideal and actual objects: implications for application in geology and geophysics [J]. Geophys. J. I., 2000, 142: 108-116.
[73]杨文采．地震道的非线性混沌反演-Ⅰ理论和数值试验[J]．地球物理学报,1993,36(2):222-232．
[74]杨文采．非线性地震道的混沌反演-Ⅱ关于L指数和吸引子[J].地球物理学报,1993,36(3):376-386．
[75]杨磊,张向君,李幼铭．地震道非线性反演的参数反馈控制及效果[J]．地球物理学报,1999,42(5):677- 684．
[76]艾印双,刘鹏程,郑天愉．自适应全局混合反演[J]．中国科学(D),28(2):105-110．
[77]王仁,方开泰．关于均匀分布与试验设计(数论方法)[J]．科学通报,1981,26(2):65-70．
[78]杨文采．波方程的逆散射与迭代线性化反演[J].石油物探,1995,34(3):114-122．
[79] Tikhonove A N. Continuation of potential fields in a layered medium by the regularization method [J]. Earth Physics, 1968, 1968.
[80] Tikhonove A N, Arsennin V Y. Solution of Ill-posed Problems [M]. Jonh Wiley, New York, 1977..
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