各向异性介质的MT正反演研究
摘要
本文主要研究了各向异性介质的大地电磁正反演问题。本文的目的在于解决二维各向异性介质的大地电磁近似解析解、开发一套可用于模拟各向异性地电结构的大地电磁场及其响应函数的有限元数值模拟程序、以及实现大地电磁一维各向异性反演问题。
本文从最简单的各向同性均匀半空间大地电磁理论开始,依次介绍了一维层状各向同性、二维各向同性及三维各向同性介质情形下的大地电磁理论。在此基础上阐述了一维层状各向异性介质二维各向异性介质情形下的大地电磁理论。
对于一般二维介质的大地电磁场,通常需要利用数值方法求解。数值模拟方法也的确可以计算各种复杂二维地电模型的大地电磁场,但是,现实研究中经常会发现对同一个模型用不同的数值模拟方法得到的结果彼此不一致,这时无法判断哪个程序得到的结果是正确的,即使两种数值方法获得了同样的结果也不能证明该结果就是精确的。这就需要一个可以用解析方法求解的简单模型,可以借助该模型的解析解去判断数值模拟方法的正确性及其精度。因此,本文在二维各向同性无限深断裂模型解析解的基础上,研究了几种具有特殊各向异性电导率结构(对角各向异性、水平各向异性及方位各向异性)的无限深断裂的大地电磁场的近似解析解。
本文仔细分析了二维各向同性与各向异性情形下Maxwell方程的异同,在此基础上利用伽辽金加权余量法推导出了大地电磁二维对称各向异性介质的有限元数值模拟方程。在推导有限元数值模拟方程的过程中,利用矩阵方程的变换性质对单元刚度矩阵做了优化处理,使得在求解总体刚度矩阵方程时需要存储的元素数据量减少到最低限度,有效节省了计算机的存储空间。
在推导完二维对称各向异性介质有限元总体刚度矩阵方程后,借助Matlab平台编制了相应的有限元数值模拟程序,直接利用Matlab语言中的提供的稀疏矩阵存储技术存储总体系数矩阵,最大限度地减少了存储空间和内存空间,提高运算效率。采用Matlab平台中自带的解方程方法求解有限元刚度矩阵方程,程序编制简便且计算结果稳定可靠。
为了检验所设计的有限元程序的正确性,本文利用各向异性无限深断裂的大地电磁场近似解析解检验了本文所编写的有限元程序。在验证完有限元数值模拟程序的正确性之后,本文模拟了几种著名的2D各向异性模型(Reddy&Rankin模型,Pek&Li模型)的大地电磁场及其MT响应,并讨论了电性各向异性对大地电磁场的影响。
在完成二维各向异性正演后,本文进行MT全张量一维各向异性的反演研究,并将所研究的反演方法用于实测大地电磁资料。
最后,对本文的工作进行了总结并展望了未来的研究方向。
This thesis deals with the forward and inversion problem of the magnetotelluric (MT) in a two-dimensional (2-D) medium with anisotropic conductivity structure. The goal of this thesis is to obtain quasi-analytic solution of2-D MT fields on an an axially anisotropic infinite fault, and to develop a code by finite element method which can be used to model the magnetotelluric fields and calculate the MT responses in anisotropic conductivity structures, and to carry out the inversion of MT field data in1-D medium.
From the magnetotelluric theory in the simplest case, i.e. isotropic half space model, the MT theory of the one-dimensional (1-D) isotropic layered media, the2-D isotropic midia and the three-dimensional (3-D) case are introduced in sequence. And then the MT theory of the1-D anisotropic layered media and the2-D model with anisotropic conductivity structure are considered.
For the general2-D earth, the magnetotelluric fields must be calculated by numerical modeling methods. And the numerical modeling method can indeed be used to various calculation of2D model, however, sometimes the numerical results given by different methods differ from each other, then it is difficult to tell which of the methods is giving the correct solution and even if two methods do yield the same numerical solution, this does not prove conclusively that it is, in fact, an accurate one. In this case, it is desirable to assess the correctness and precision of the methods by a simple2D model which can also be solved analytically. Hence, the quasi-analytical solutions of the2-D magnetotelluric fields on the finite fault with special anisotropic condutivity structure (i.e. diagonal anisotropy, horizontal anisotropy and azimuthal anisotropy) are given based on that of the isotropic case
The difference between the Maxwell equations corresponding to the2-D isotropic media and anisotropic ones, respectively, is analysed carefully. Base on this, the numerical modeling equations are derived using the galerkin weighted residual finite element method (FEM). The element stiffness matrix is optimized based on the transformation properties of the matrix equation when the FEM numerical modeling equations are derived, which makes the data storage required in the calculation of the overall stiffness matrix equations reduce to the least level so as to reduce the cost of the memory of the computer effectively.
When the overall stiffness matrix equations for the2-D symmetric anisotropic conductivity media are obtained, the corresponding finite element numerical simulation code is developed in Matlab language. The code applys the sparse matrix storage technology provided by the Matlab platform to store the coefficient matrix, which reduces the strorage and memory at the maximum level and improves the computational efficiency significantly. The programming of the code is simple when the method for solving finite element stiffness matrix equation within the Matlab platform is applied, and the results of the calculation are stable and reliable.
In order to examine the correctness of the FEM code, the quasi-analytical solution are used to check the FEM code in this thesis. After the correctness of the FEM code is confirmed, the magnetotelluric fields and the MT responses over several well-known2-D anisotropic media (Reddy&Rankin model, Pek&Li model) are calcalated, and the effects of the anisotropy on the MT fields are discussed.
When the numerical modeling of MT fields in2-D anisotropic conductivity structures is finished, a inversion method of the whole tensor impedance response to one-dimensional anisotropic structure is considered. And then the inversion method is applied to MT field data.
Finally, the work of this thesis is summarized and some directions of the future research are given.
本文从最简单的各向同性均匀半空间大地电磁理论开始,依次介绍了一维层状各向同性、二维各向同性及三维各向同性介质情形下的大地电磁理论。在此基础上阐述了一维层状各向异性介质二维各向异性介质情形下的大地电磁理论。
对于一般二维介质的大地电磁场,通常需要利用数值方法求解。数值模拟方法也的确可以计算各种复杂二维地电模型的大地电磁场,但是,现实研究中经常会发现对同一个模型用不同的数值模拟方法得到的结果彼此不一致,这时无法判断哪个程序得到的结果是正确的,即使两种数值方法获得了同样的结果也不能证明该结果就是精确的。这就需要一个可以用解析方法求解的简单模型,可以借助该模型的解析解去判断数值模拟方法的正确性及其精度。因此,本文在二维各向同性无限深断裂模型解析解的基础上,研究了几种具有特殊各向异性电导率结构(对角各向异性、水平各向异性及方位各向异性)的无限深断裂的大地电磁场的近似解析解。
本文仔细分析了二维各向同性与各向异性情形下Maxwell方程的异同,在此基础上利用伽辽金加权余量法推导出了大地电磁二维对称各向异性介质的有限元数值模拟方程。在推导有限元数值模拟方程的过程中,利用矩阵方程的变换性质对单元刚度矩阵做了优化处理,使得在求解总体刚度矩阵方程时需要存储的元素数据量减少到最低限度,有效节省了计算机的存储空间。
在推导完二维对称各向异性介质有限元总体刚度矩阵方程后,借助Matlab平台编制了相应的有限元数值模拟程序,直接利用Matlab语言中的提供的稀疏矩阵存储技术存储总体系数矩阵,最大限度地减少了存储空间和内存空间,提高运算效率。采用Matlab平台中自带的解方程方法求解有限元刚度矩阵方程,程序编制简便且计算结果稳定可靠。
为了检验所设计的有限元程序的正确性,本文利用各向异性无限深断裂的大地电磁场近似解析解检验了本文所编写的有限元程序。在验证完有限元数值模拟程序的正确性之后,本文模拟了几种著名的2D各向异性模型(Reddy&Rankin模型,Pek&Li模型)的大地电磁场及其MT响应,并讨论了电性各向异性对大地电磁场的影响。
在完成二维各向异性正演后,本文进行MT全张量一维各向异性的反演研究,并将所研究的反演方法用于实测大地电磁资料。
最后,对本文的工作进行了总结并展望了未来的研究方向。
This thesis deals with the forward and inversion problem of the magnetotelluric (MT) in a two-dimensional (2-D) medium with anisotropic conductivity structure. The goal of this thesis is to obtain quasi-analytic solution of2-D MT fields on an an axially anisotropic infinite fault, and to develop a code by finite element method which can be used to model the magnetotelluric fields and calculate the MT responses in anisotropic conductivity structures, and to carry out the inversion of MT field data in1-D medium.
From the magnetotelluric theory in the simplest case, i.e. isotropic half space model, the MT theory of the one-dimensional (1-D) isotropic layered media, the2-D isotropic midia and the three-dimensional (3-D) case are introduced in sequence. And then the MT theory of the1-D anisotropic layered media and the2-D model with anisotropic conductivity structure are considered.
For the general2-D earth, the magnetotelluric fields must be calculated by numerical modeling methods. And the numerical modeling method can indeed be used to various calculation of2D model, however, sometimes the numerical results given by different methods differ from each other, then it is difficult to tell which of the methods is giving the correct solution and even if two methods do yield the same numerical solution, this does not prove conclusively that it is, in fact, an accurate one. In this case, it is desirable to assess the correctness and precision of the methods by a simple2D model which can also be solved analytically. Hence, the quasi-analytical solutions of the2-D magnetotelluric fields on the finite fault with special anisotropic condutivity structure (i.e. diagonal anisotropy, horizontal anisotropy and azimuthal anisotropy) are given based on that of the isotropic case
The difference between the Maxwell equations corresponding to the2-D isotropic media and anisotropic ones, respectively, is analysed carefully. Base on this, the numerical modeling equations are derived using the galerkin weighted residual finite element method (FEM). The element stiffness matrix is optimized based on the transformation properties of the matrix equation when the FEM numerical modeling equations are derived, which makes the data storage required in the calculation of the overall stiffness matrix equations reduce to the least level so as to reduce the cost of the memory of the computer effectively.
When the overall stiffness matrix equations for the2-D symmetric anisotropic conductivity media are obtained, the corresponding finite element numerical simulation code is developed in Matlab language. The code applys the sparse matrix storage technology provided by the Matlab platform to store the coefficient matrix, which reduces the strorage and memory at the maximum level and improves the computational efficiency significantly. The programming of the code is simple when the method for solving finite element stiffness matrix equation within the Matlab platform is applied, and the results of the calculation are stable and reliable.
In order to examine the correctness of the FEM code, the quasi-analytical solution are used to check the FEM code in this thesis. After the correctness of the FEM code is confirmed, the magnetotelluric fields and the MT responses over several well-known2-D anisotropic media (Reddy&Rankin model, Pek&Li model) are calcalated, and the effects of the anisotropy on the MT fields are discussed.
When the numerical modeling of MT fields in2-D anisotropic conductivity structures is finished, a inversion method of the whole tensor impedance response to one-dimensional anisotropic structure is considered. And then the inversion method is applied to MT field data.
Finally, the work of this thesis is summarized and some directions of the future research are given.
引文
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Xiong, Z.,1989. Electromagnetic fields of electric dipoles embedded in a stratified anisotropic earth. Geophysics 54,1643-1646.
Yin, C.,2000. Geoelectrical inversion for 1D anisotropic models and inherent non-uniqueness. Geophys. J. Int.140,11-23.
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Yoshino, T., Matsuzaki, T., Yamashita, S., Katsura, T.,2006. Hydrous olivine unable to account for conductivity anomaly at the top of the asthenosphere. Nature 443,973-976.
Zyserman, F. I., Guarracino, L., Santos, J. E.,1999. A hybridized mixed finite element domain decomposed method for two dimensional magnetotelluric modelling. Earth Planets Space 51,297-306.
陈乐寿,刘任,王天生,1989.大地电磁测深资料处理与解释.石油工业出版社,北京.
陈乐寿,王光锷,1990.大地电磁测深法.地质出版社,北京.
邓建中,葛仁杰,程正兴,1985.计算方法.西安交通大学出版社,西安.
晋光文,1987.大地电磁视电阻率和阻抗相位资料的联合反演.地震地质9,81-86.
林长佑,武玉霞,杨长福等,1996.水平层状对称各向异性介质的大地电磁资料反演.地球物理学报39(增刊),342-348.
秦林江,杨长福,2012.大地电磁全张量响应的一维各向异性反演.地球物理学报55,693-701.
徐翠薇,孙绳武,2002.计算方法引论.高等教育出版社,北京.
徐世浙,1994.地球物理中的有限单元法.科学出版社,北京.
徐世浙,赵生凯,1985.二维各向异性地电断面大地电磁场的有限元法解法.地震学报,80-90.
杨长福,1997.大地电磁二维对称各向异性介质的有限元数值模拟.西北地震学报19,28-34.
杨长福,林长佑,孙崇赤等,2005.二维对称各向异性介质大地电磁反演.地震学报27,339-345.
杨长福和林长佑,1997.甘肃及邻近地区的各向异性介质MT反演及该区地壳深部应力场和形变带.内陆地震11,143-147.
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