Carnot群上次调和算子的三球面定理及频域波形反演的优化
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摘要
本文主要结构分为两部分．首先研究了Carnot群上次调和函数△Gu≥0的下解的Hadamard三球面定理，并基于基本解的表示和和最大值原理给出了证明．其次本文研究了一类偏微分方程的应用问题，基于声学逼近方法对弹性波动方程的研究，探讨了频域反演的目标函数中某些项的计算技巧．
第一部分主要研究Carnot群上次调和算子的三球面定理，共分两章．
第一章介绍了微分方程极大值原理和Hadamard三球面定理的发展背景和选题的实际意义，以及有关Carnot群的一些基本概念及性质．
第二章主要利用Carnot群上调和函数基本解的表示和次平均值性质，证明了Carnot群上次调和函数的最大值原理，给出了Hadamard三球面定理，得到了函数M(r)=max_(|x~(-1)og|)=r~(u(ξ))是关于|x~(-1)og|~(2-Q)的凸函数的结论．
第二部分主要研究频域波形反演的优化，也分两章．
第三章介绍微分方程反问题的研究背景、实际意义．
第四章基于弹性理论导出了非均匀介质中压力场的波动方程，建立了目标函数，给出了目标函数中数据空间协方差矩阵C_D与模型协方差距阵C_M的关系，给出了用正演模拟相应Green函数的方法来计算弗雷歇矩阵F从而计算出数据误差梯度方向(?)的数学推导过程，最后给出了迭代算法．
The paper is made up of two parts. Firstly, Hadamard's three spheres theorem of Sub-Solution for Sub-Laplacian operation△_Gu≥0is obtained,and is proved on the basis of fundamental solution and Maximum principle of sub-Laplacian operator. Secondly,application of a class of Partial Differential Equation is studied. Based on the acoustic approximation of the elastic wave equation, the calculating skills of some terms of the objective function in frequencydomaininversion are discussed.
The first part consists of two chapters,in which we discuss the Hadamard's three-spheres-theorem of Sub-Laplacian operators on Carnot group.
In the first chapter, we briefly introduce the development of the Maximum principle of Differential Equation and Hadamard's three spheres theorem, the meaning of selecting this question, and some concepts about Carnot group.
In the second chapter, based on the fundamental solution of Sub-Laplacian operation and the Sub-mean property, Maximum principle and Hadamard's three spheres theorem on Carnot group are proved, and we show that maximal value function M(r)=max_(|x~(-1)og|)=r~(u(ξ)) is a convex function with respect to |x~(-1)og|~(2-Q).
The second part consists of two chapters also, in which we discuss an optimalizationfor waveform inversion of the frequency-domain.
In the third chapter, we briefly introduce the development of and background of the inverse problem of Partial Differential Equation.
In the fourth chapter, based on the elasticity theory , we establish the wave equation for the pressure wavefield in the nonhomogeneous medium. Then the relation of covariance matrix C_D in the data space with the model covariance matrixC_M of the objetive function is found. Finally we show the deduction that the gradient direction of the data misfit (?) can be computed by forward modeling Green function to avoid the formidable calculation of the frechet F, and we show the iteratively algorithm.
第一部分主要研究Carnot群上次调和算子的三球面定理，共分两章．
第一章介绍了微分方程极大值原理和Hadamard三球面定理的发展背景和选题的实际意义，以及有关Carnot群的一些基本概念及性质．
第二章主要利用Carnot群上调和函数基本解的表示和次平均值性质，证明了Carnot群上次调和函数的最大值原理，给出了Hadamard三球面定理，得到了函数M(r)=max_(|x~(-1)og|)=r~(u(ξ))是关于|x~(-1)og|~(2-Q)的凸函数的结论．
第二部分主要研究频域波形反演的优化，也分两章．
第三章介绍微分方程反问题的研究背景、实际意义．
第四章基于弹性理论导出了非均匀介质中压力场的波动方程，建立了目标函数，给出了目标函数中数据空间协方差矩阵C_D与模型协方差距阵C_M的关系，给出了用正演模拟相应Green函数的方法来计算弗雷歇矩阵F从而计算出数据误差梯度方向(?)的数学推导过程，最后给出了迭代算法．
The paper is made up of two parts. Firstly, Hadamard's three spheres theorem of Sub-Solution for Sub-Laplacian operation△_Gu≥0is obtained,and is proved on the basis of fundamental solution and Maximum principle of sub-Laplacian operator. Secondly,application of a class of Partial Differential Equation is studied. Based on the acoustic approximation of the elastic wave equation, the calculating skills of some terms of the objective function in frequencydomaininversion are discussed.
The first part consists of two chapters,in which we discuss the Hadamard's three-spheres-theorem of Sub-Laplacian operators on Carnot group.
In the first chapter, we briefly introduce the development of the Maximum principle of Differential Equation and Hadamard's three spheres theorem, the meaning of selecting this question, and some concepts about Carnot group.
In the second chapter, based on the fundamental solution of Sub-Laplacian operation and the Sub-mean property, Maximum principle and Hadamard's three spheres theorem on Carnot group are proved, and we show that maximal value function M(r)=max_(|x~(-1)og|)=r~(u(ξ)) is a convex function with respect to |x~(-1)og|~(2-Q).
The second part consists of two chapters also, in which we discuss an optimalizationfor waveform inversion of the frequency-domain.
In the third chapter, we briefly introduce the development of and background of the inverse problem of Partial Differential Equation.
In the fourth chapter, based on the elasticity theory , we establish the wave equation for the pressure wavefield in the nonhomogeneous medium. Then the relation of covariance matrix C_D in the data space with the model covariance matrixC_M of the objetive function is found. Finally we show the deduction that the gradient direction of the data misfit (?) can be computed by forward modeling Green function to avoid the formidable calculation of the frechet F, and we show the iteratively algorithm.
引文
[1] Protter M H,Weinberger H F. Maxinum Principles in Differential Equations[M]. Beijing: Science Press, 1986.
[2] Landis E M. A Three-Spheres Theorem[J]. Soviet Math. Dokl., 1963,4: 76-78.
[3] Gerasimov K. The Three Spheres Theorem for a Certain Class of Elliptic Equations of High Order and a Refinement of this Theorem for a Linear Elliptic Equation of The Second Order[J]. Mat. Sib. (NS), 1966, 71(113): 563-585.
[4] Brummelhuis R. Three-Spheres Theorem for Second Order Elliptic Equations[J]. Anal Math., 1995, 65: 179-206.
[5] Bonfiglioli A, Lanconelli E. Subharmonic Functions on Carnot Groups[J]. Math. Ann., 2003, 325: 97-122.
[6] Bonfiglioli A, Lanconelli E. Maximum Principle on Unbounded Domains for Sub-Laplacians: A Potential Theory Approach [J]. Porc. Amer. Math. Soc. 2002, 130: 2295-2304.
[7] Bonfiglioli A, Lanconelli E. Liouville-type Theorems for Real Sub-Laplacians. Manuscripta Math., 2001, 105:111-124.
[8] C. Wang. Subelliptic Harmonic Maps from Carnot Groups[J]. Calc. Var. 2003, 18:95-115.
[9] Folland G B. Subelliptic Estimates and Function Spaes on Nilpotent Groups [J]. Arkiv for Mat., 1975, 13:161-207.
[10] Folland G B. A Fundamental Solution for a SuBelliptic Operator [J]. Bull Amer Math. Soc., 1973, 39: 373-376.
[11] Liu Haifeng, Niu Pengcheng. Nontrivial Solutions for a Class of Non-Divergence Equations on Polarizable Carnot Groups[J]. Appl. Math. J.Chinese Univ. Ser. B, 2006, 21(2): 157-164.
[12] Hormonder L. Hypoelliptic Second Order Differential Equations[J]. Acta Math. 1967, 119:141-171.
[13]熊金城．点集拓扑讲义(第三版)[M]．北京：高等教育出版社，2003．
[14]田畴．李群及其在微分方程中的应用[M]．北京：科学出版社，2001．
[15] Wang Y H, Rao Y. Crosshole seismic waveform tomography- .Strategy for real data application[J]. Geophys J Int, 2006, 166: 1224-1236.
[16] Rao Y, Wang Y H, Morgan J V. Crosshole seismic waveform tomography-Ⅱ.Resolution analysis[J]. Geophys J Int, 2006, 166: 1237-1248.
[17] Tarantola A. Inversion of seismic reflection data in the acoustic approximation [J]. Geophysics. Vol, 1984, 49: 1259-1266.
[18] Tarantola A. Inversion Problem:Theory and Methods for Nodel Parameter Estima-tion[J]. Soc. for Industrial Applied Math., 2005.
[19] Partt R G, Worthington M H. Inverse theory applied to multi-source crosshole tomography: I. acoustic wave-equation method[J].Geophys. Prospect,1990,38:287-310.
[20] Pratt R G, Shin C, Hicks G L. Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion [J]. Geophys J Int, 1998,133:341-362.
[21]杨文采．地球物理反演与地震层析成像[M].北京：地质出版社，1989．
[22] Bleistein N. Mathematical methods for wave phenomena[M]. Alton: Academic Press Inc,1984.
[23] Bleistein N et al. Two and one-galf dimensional Born inversion with arbitrary ref-erence[J]. Geophysics, 1987, 51: 26-36.
[24] Cohen J K et al. Three-dimensional Born inversion with an arbitrary reference [J]. Geophysics, 1986, 51:1552-1558.
[25] Cooke D A, Schneider W A. Generalized linear inversion of reflection seismic data[J]. Geophysics, 1983,48:665-676.
[26] Lines L R et al. Digital filtering with the second moment norm[J]. geophysics, 1983, 48:505-514.
[27] Newton G R. Inversion of reflection data for layered media: a review of exact methods[M]. 2nd ed. Geoph S J R Astr.1981.
[28] Weglein A B. Near field inverse Scattering formalism for the three-dimensional wave equation[J]. J A Coast Soc, 1982, 71:1179-1182.
[29] Kanwal R P. Generalized functions: Theory and Technique. Academic Press, 1983.
[30]徐芝纶．弹性力学[M]．北京：高等教育出版社，2006．
[31]陆明万，罗学富．弹性理论基础[M]．北京：清华大学出版社，施普林格出版 社，2001．
[32]赵彦．逆问题研究关联科学创新．科学时报(中国科学院主办，科学时报出版社 出版)，1999(1494)．
[33] Cannon J R. The One-dimensional Heat Equations [J]. California: Addison-Wasley Publishing Company, 1984.
[34] Romanov V G. Inverse Problems of Mathematical Physics[M]. Utrecht,The Nether-land: VNU Science Press BV, 1987.
[35]刘家琦，匡正，王德明．微分方程反问题及其数值解法．哈尔滨：哈尔滨工业大 学出版社，1984．
[36]苏超伟．偏微分方程逆问题的数值解法及其应用．西安：西北工业大学出版 社．1986．
[37]A．H．吉洪诺夫，B．R．阿尔先宁．不适定问题的解法(王秉忱译)．北京：地质出 版社，1979．
[38] Ramm A G. Multidimensional Inverse Scattering Problems. England: Longman Group UK Limited, 1992.
[2] Landis E M. A Three-Spheres Theorem[J]. Soviet Math. Dokl., 1963,4: 76-78.
[3] Gerasimov K. The Three Spheres Theorem for a Certain Class of Elliptic Equations of High Order and a Refinement of this Theorem for a Linear Elliptic Equation of The Second Order[J]. Mat. Sib. (NS), 1966, 71(113): 563-585.
[4] Brummelhuis R. Three-Spheres Theorem for Second Order Elliptic Equations[J]. Anal Math., 1995, 65: 179-206.
[5] Bonfiglioli A, Lanconelli E. Subharmonic Functions on Carnot Groups[J]. Math. Ann., 2003, 325: 97-122.
[6] Bonfiglioli A, Lanconelli E. Maximum Principle on Unbounded Domains for Sub-Laplacians: A Potential Theory Approach [J]. Porc. Amer. Math. Soc. 2002, 130: 2295-2304.
[7] Bonfiglioli A, Lanconelli E. Liouville-type Theorems for Real Sub-Laplacians. Manuscripta Math., 2001, 105:111-124.
[8] C. Wang. Subelliptic Harmonic Maps from Carnot Groups[J]. Calc. Var. 2003, 18:95-115.
[9] Folland G B. Subelliptic Estimates and Function Spaes on Nilpotent Groups [J]. Arkiv for Mat., 1975, 13:161-207.
[10] Folland G B. A Fundamental Solution for a SuBelliptic Operator [J]. Bull Amer Math. Soc., 1973, 39: 373-376.
[11] Liu Haifeng, Niu Pengcheng. Nontrivial Solutions for a Class of Non-Divergence Equations on Polarizable Carnot Groups[J]. Appl. Math. J.Chinese Univ. Ser. B, 2006, 21(2): 157-164.
[12] Hormonder L. Hypoelliptic Second Order Differential Equations[J]. Acta Math. 1967, 119:141-171.
[13]熊金城．点集拓扑讲义(第三版)[M]．北京：高等教育出版社，2003．
[14]田畴．李群及其在微分方程中的应用[M]．北京：科学出版社，2001．
[15] Wang Y H, Rao Y. Crosshole seismic waveform tomography- .Strategy for real data application[J]. Geophys J Int, 2006, 166: 1224-1236.
[16] Rao Y, Wang Y H, Morgan J V. Crosshole seismic waveform tomography-Ⅱ.Resolution analysis[J]. Geophys J Int, 2006, 166: 1237-1248.
[17] Tarantola A. Inversion of seismic reflection data in the acoustic approximation [J]. Geophysics. Vol, 1984, 49: 1259-1266.
[18] Tarantola A. Inversion Problem:Theory and Methods for Nodel Parameter Estima-tion[J]. Soc. for Industrial Applied Math., 2005.
[19] Partt R G, Worthington M H. Inverse theory applied to multi-source crosshole tomography: I. acoustic wave-equation method[J].Geophys. Prospect,1990,38:287-310.
[20] Pratt R G, Shin C, Hicks G L. Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion [J]. Geophys J Int, 1998,133:341-362.
[21]杨文采．地球物理反演与地震层析成像[M].北京：地质出版社，1989．
[22] Bleistein N. Mathematical methods for wave phenomena[M]. Alton: Academic Press Inc,1984.
[23] Bleistein N et al. Two and one-galf dimensional Born inversion with arbitrary ref-erence[J]. Geophysics, 1987, 51: 26-36.
[24] Cohen J K et al. Three-dimensional Born inversion with an arbitrary reference [J]. Geophysics, 1986, 51:1552-1558.
[25] Cooke D A, Schneider W A. Generalized linear inversion of reflection seismic data[J]. Geophysics, 1983,48:665-676.
[26] Lines L R et al. Digital filtering with the second moment norm[J]. geophysics, 1983, 48:505-514.
[27] Newton G R. Inversion of reflection data for layered media: a review of exact methods[M]. 2nd ed. Geoph S J R Astr.1981.
[28] Weglein A B. Near field inverse Scattering formalism for the three-dimensional wave equation[J]. J A Coast Soc, 1982, 71:1179-1182.
[29] Kanwal R P. Generalized functions: Theory and Technique. Academic Press, 1983.
[30]徐芝纶．弹性力学[M]．北京：高等教育出版社，2006．
[31]陆明万，罗学富．弹性理论基础[M]．北京：清华大学出版社，施普林格出版 社，2001．
[32]赵彦．逆问题研究关联科学创新．科学时报(中国科学院主办，科学时报出版社 出版)，1999(1494)．
[33] Cannon J R. The One-dimensional Heat Equations [J]. California: Addison-Wasley Publishing Company, 1984.
[34] Romanov V G. Inverse Problems of Mathematical Physics[M]. Utrecht,The Nether-land: VNU Science Press BV, 1987.
[35]刘家琦，匡正，王德明．微分方程反问题及其数值解法．哈尔滨：哈尔滨工业大 学出版社，1984．
[36]苏超伟．偏微分方程逆问题的数值解法及其应用．西安：西北工业大学出版 社．1986．
[37]A．H．吉洪诺夫，B．R．阿尔先宁．不适定问题的解法(王秉忱译)．北京：地质出 版社，1979．
[38] Ramm A G. Multidimensional Inverse Scattering Problems. England: Longman Group UK Limited, 1992.