地形起伏条件下可控源音频大地电磁法2.5维正演研究
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摘要
可控源音频大地电磁法(CSAMT)是一种高效的地球物理勘探方法，在当今地球物理勘探的许多领域都有广泛的应用。
电磁场的2.5维正演就是求解三维偶极子源或者有限长导线在二维地质构造上产生的电磁响应。由于许多地质体的电导率在空间坐标系下是二维或者近似二维变化的，求解2.5维问题就显得十分重要。因为对于这种地质体，没有必要采用三维正演算法进行模拟计算，同时2.5维求解也能节省大量的计算时间。
在本文中，采用视狄拉克函数来等效电磁场中的三维源。通常情况下求解电磁响应，为了避免源点周围的奇异性，总是将总场人为分解为一次场和二次场分别求解，而这种分解对于复杂的地质构造是不能精确实现的。视狄拉克函数的引用使我们可以直接求解总场，消除了场分解产生的误差。为了能够更好的模拟地形起伏变化，在对目标区域剖分的过程中，可以根据实际情况，采用不同形状的四边形单元，并用等参有限元法对剖分形成的离散化目标区域进行数值模拟计算。
在2.5维正演过程中，首先对场方程进行降维处理，即对方程中电磁场的各个分量沿走向方向做傅立叶变换，将求解域从空间域变换到波数域；然后对目标区域进行剖分，针对每一个波数，采用等参有限元法得到离散化电磁场方程，求解方程获得走向方向上的电磁场响应；对走向方向上的电磁场响应进行空间求导，从而获得电磁场其他方向上的分量；最后通过傅立叶反变换，将波数域内的电磁场响应转换到空间域。
在本文中，通过对一些简单模型的正演计算来检验算法的正确性和稳定性。
CSAMT (Controlled Source Audio-frequency Magnetotelluric) method is an efficient electromagnetic method; it is widely used in many domain of today's geophysical exploration.
2.5-D forward problem is to compute the response of a two-dimensional conductivity structure excited by a three dimensional current source or a finite source. It is important because many geological targets are approximately 2-D. In those cases a fully 3-D medium is not needed and the computation cost can be greatly reduced by solving just the 2.5-D problem.
Here I attempt to use a pseudo-delta function to distribute the response of the dipole source current. Using this function we do not need to separate the total response field into primary field and secondary field, which is very difficult for complex structure with topography response. Using the pseudo-delta function, we can get the total field directly, and do not needs to consider the error come from field separation .To deal with the topography response conveniently, I employ an isoparametric finite-element technique, which can partition the target area into elements with various shapes according to the topography.
In the process of 2.5-D forward modeling, I first employ Fourier transform to the field in the invariant conductivity direction, transforming the filed form frequency domain to wave number domain. Then at each of a set of discrete spatial wave numbers, an isoparametric finite-element method is used to obtain the total filed directly. Through space derivation The EM field components of the other two directions can be obtained. At last, the field is transformed back to space domain by inverse Fourier transform.
Also in this paper some simple examples are presented to validate the code.
电磁场的2.5维正演就是求解三维偶极子源或者有限长导线在二维地质构造上产生的电磁响应。由于许多地质体的电导率在空间坐标系下是二维或者近似二维变化的，求解2.5维问题就显得十分重要。因为对于这种地质体，没有必要采用三维正演算法进行模拟计算，同时2.5维求解也能节省大量的计算时间。
在本文中，采用视狄拉克函数来等效电磁场中的三维源。通常情况下求解电磁响应，为了避免源点周围的奇异性，总是将总场人为分解为一次场和二次场分别求解，而这种分解对于复杂的地质构造是不能精确实现的。视狄拉克函数的引用使我们可以直接求解总场，消除了场分解产生的误差。为了能够更好的模拟地形起伏变化，在对目标区域剖分的过程中，可以根据实际情况，采用不同形状的四边形单元，并用等参有限元法对剖分形成的离散化目标区域进行数值模拟计算。
在2.5维正演过程中，首先对场方程进行降维处理，即对方程中电磁场的各个分量沿走向方向做傅立叶变换，将求解域从空间域变换到波数域；然后对目标区域进行剖分，针对每一个波数，采用等参有限元法得到离散化电磁场方程，求解方程获得走向方向上的电磁场响应；对走向方向上的电磁场响应进行空间求导，从而获得电磁场其他方向上的分量；最后通过傅立叶反变换，将波数域内的电磁场响应转换到空间域。
在本文中，通过对一些简单模型的正演计算来检验算法的正确性和稳定性。
CSAMT (Controlled Source Audio-frequency Magnetotelluric) method is an efficient electromagnetic method; it is widely used in many domain of today's geophysical exploration.
2.5-D forward problem is to compute the response of a two-dimensional conductivity structure excited by a three dimensional current source or a finite source. It is important because many geological targets are approximately 2-D. In those cases a fully 3-D medium is not needed and the computation cost can be greatly reduced by solving just the 2.5-D problem.
Here I attempt to use a pseudo-delta function to distribute the response of the dipole source current. Using this function we do not need to separate the total response field into primary field and secondary field, which is very difficult for complex structure with topography response. Using the pseudo-delta function, we can get the total field directly, and do not needs to consider the error come from field separation .To deal with the topography response conveniently, I employ an isoparametric finite-element technique, which can partition the target area into elements with various shapes according to the topography.
In the process of 2.5-D forward modeling, I first employ Fourier transform to the field in the invariant conductivity direction, transforming the filed form frequency domain to wave number domain. Then at each of a set of discrete spatial wave numbers, an isoparametric finite-element method is used to obtain the total filed directly. Through space derivation The EM field components of the other two directions can be obtained. At last, the field is transformed back to space domain by inverse Fourier transform.
Also in this paper some simple examples are presented to validate the code.
引文
[1] 付良魁．电法勘探教程．地质出版社，1983
[2] 付良魁．应用地球物理教程——电法、放射性、地热．地质出版社，1991
[3] 施国良，张国雄．宏观场论．中国地质大学出版社，1987
[4] 刘尔烈 等．有限单元法及程序设计．天津大学出版社，1999
[5] 徐世浙．地球物理中的有限单元法．科学出版社，1994
[6] 王勖成．有限单元法．清华大学出版社，2003
[7] 倪光正，钱秀英．电磁场数值计算．高等教育出版社，1996
[8] Hohmann G W. Electromagnetic scattering by conductors in the earth near a line source of current. Geophysics, 1971,36 (1): 101～131
[9] Martyn J, Unsworth, Bryan J. Electromagnetic induction by a finite eletric diple source over a 2-D earth. Geophysics, 1993, Vol. 58, 198—214
[10] 孟永良罗延钟．电偶源CSAMT法二维正演的有限单元算法．勘查地球物理勘查地球化学文集．地质出版社．1996，第20集，103-114
[11] Yuji Mitsuhata. 2-D electromagnetic modeling by finite-element method with a dipole source and topography. Geophysics, 2000, Vol. 65, 465—475
[12] 阎述，陈明生．电偶源频率电磁测深三维地电模型有限元正演．煤田地质与勘探，2000，第28卷，50—56
[13] 底青云，Martyn Unsworth，王妙月．复杂介质有限元法2.5维可控源音频大地电磁法数值模拟．地球物理学报．2004，第47卷，723—730
[14] 底青云，Martyn Unsworth，王妙月．有限元法2.5维CSAMT数值模拟．地球物理学报，2004，第19卷，317—324
[15] 刘树才，何昭友．刘志新适合地形起伏的二维有限单元数值模拟技术．物探化探计算技术，2005．第27卷第2期
[16] 杨进，傅良魁．迭代有限元法数值模拟若干问题的讨论．地质与勘探，2005，第30卷第2期
[17] Stoyer C.H, Greenfield R.J. Numerical solutions of the response of a two—dimensional earth to an oscillating magnetic dipole source. Geophysics, 1976,41:519—530
[18] Everett M.E, Edwords R.H. Transient marine electro-magnetics: The 2.5-D forward problem. Geophys. J. Internat, 1992, 113: 820—823
[19] Johansen H.K, Sorensen K. Fast Hankel transforms. Geophys. Prosp., 1979,27:876—901
[20] Wannamaker P. E, Stodt B.J, Rijo L. Two-dimensional topographic response in magnetotellurics modeled using finite element. Geophysics, 1986,51:2131—2144
[21] 胡建德．电法勘探中的数学模型．数学的实践与认识，2004，第34卷第2期
[22] 胡建德，阎述，陈明生．线电流源声频大地电磁测深的二维正演计算及响应特点．现代地质，1997，第11卷第2期
[23] 史明娟，徐世浙，刘斌．大地电磁二次插值函数的有限元法正演模拟．地球物理学报，1997．第40卷第3期
[24] 安晓卫，邵伟平．二维问题的有限元网格的自动剖分．沈阳工业学院学报，1994．12，第13卷第4期
[25] Wu E R..Techniques to avoid duplicate nodes and relax restrictions on the super element numbering in a mesh generator comp. struct. 1982, 15:419-422
[26] Chave A. D. Numerical integration of related Hankel transform by quadrature and continued fraction expansion. Geophysics, 48: 1671—1686
[27] Fox, R. C, Hohmann, G. W, Killpack, J. J.,topographic effects in resistivity and induced polarization surveys. Geophysics, 45:75-93
[28] 阎述，陈明生．线源频率域电磁测深二维正演(一)．煤田地质与勘探，1999，27(5)：60～62
[29] 阎述，陈明生，线源频率域电磁测深二维正演(二)．煤田地质与勘探，1999．27(6)：56～59
[30] Coggon, J.H. Electromagnetic and electrical modeling by the finite-element method. Geophysics, 36:132-155
[31] Oden J.T, G.F. Carey.An introduction to the finite-element method. Prentice-Hall, Inc.
[32] Partha S. Routh, Douglas W. oldenburg. Inversion of Controlled Source audio—frequency magnetotedurics data for a horizontally layered earth. Geophysics, 1999,64 (6): 1689～1697
[33] Leppin, M., Electromagnetic modeling of 3-D sources over 2-D inhomogeneties in tire time domain. Geophysics, 1992,57:994-1003
[34] Goldman, Y, Hubans, C, Nicoletis, S. A finite element solution for the transient electromagnetic response of an arbitrary two-dimensional resistivity distribution. Geophysics, 1986,51:1450-1461
[35]Basokur A T , Rasmussen T M , Kaya C .Comparison of induced polarization and controlled source audio magnetotellurics methods for massive chalcopyrite exploration in a volcanic area., Geophysics . 1997, 62:1087—1096
[36]Mitsuhata Y, Toshihito U , Hiroshi A . 2. 5 D inversion of frequency domain electromagnetic data generated by a grounded2wire source. Geophysics . 2002, 67(6): 1753—1768
[37]Wannamaker P E . Tensor CSAMT survey over the sulphur springs thermal area , Valles Caldera , NewMexico, U. S. A. , Part I : Implications for structure of the western caldcra. Geophysics , 1997a ,62(4) : 451-465
[38]Sugeng, F, Raiche, A, and Xiong, Z. Modeling topographic effects using an isoparametric finite-element method: 13th Workshop on Electromagnetic Induction in the Earth. 1996, Abstracts, 109-110
[2] 付良魁．应用地球物理教程——电法、放射性、地热．地质出版社，1991
[3] 施国良，张国雄．宏观场论．中国地质大学出版社，1987
[4] 刘尔烈 等．有限单元法及程序设计．天津大学出版社，1999
[5] 徐世浙．地球物理中的有限单元法．科学出版社，1994
[6] 王勖成．有限单元法．清华大学出版社，2003
[7] 倪光正，钱秀英．电磁场数值计算．高等教育出版社，1996
[8] Hohmann G W. Electromagnetic scattering by conductors in the earth near a line source of current. Geophysics, 1971,36 (1): 101～131
[9] Martyn J, Unsworth, Bryan J. Electromagnetic induction by a finite eletric diple source over a 2-D earth. Geophysics, 1993, Vol. 58, 198—214
[10] 孟永良罗延钟．电偶源CSAMT法二维正演的有限单元算法．勘查地球物理勘查地球化学文集．地质出版社．1996，第20集，103-114
[11] Yuji Mitsuhata. 2-D electromagnetic modeling by finite-element method with a dipole source and topography. Geophysics, 2000, Vol. 65, 465—475
[12] 阎述，陈明生．电偶源频率电磁测深三维地电模型有限元正演．煤田地质与勘探，2000，第28卷，50—56
[13] 底青云，Martyn Unsworth，王妙月．复杂介质有限元法2.5维可控源音频大地电磁法数值模拟．地球物理学报．2004，第47卷，723—730
[14] 底青云，Martyn Unsworth，王妙月．有限元法2.5维CSAMT数值模拟．地球物理学报，2004，第19卷，317—324
[15] 刘树才，何昭友．刘志新适合地形起伏的二维有限单元数值模拟技术．物探化探计算技术，2005．第27卷第2期
[16] 杨进，傅良魁．迭代有限元法数值模拟若干问题的讨论．地质与勘探，2005，第30卷第2期
[17] Stoyer C.H, Greenfield R.J. Numerical solutions of the response of a two—dimensional earth to an oscillating magnetic dipole source. Geophysics, 1976,41:519—530
[18] Everett M.E, Edwords R.H. Transient marine electro-magnetics: The 2.5-D forward problem. Geophys. J. Internat, 1992, 113: 820—823
[19] Johansen H.K, Sorensen K. Fast Hankel transforms. Geophys. Prosp., 1979,27:876—901
[20] Wannamaker P. E, Stodt B.J, Rijo L. Two-dimensional topographic response in magnetotellurics modeled using finite element. Geophysics, 1986,51:2131—2144
[21] 胡建德．电法勘探中的数学模型．数学的实践与认识，2004，第34卷第2期
[22] 胡建德，阎述，陈明生．线电流源声频大地电磁测深的二维正演计算及响应特点．现代地质，1997，第11卷第2期
[23] 史明娟，徐世浙，刘斌．大地电磁二次插值函数的有限元法正演模拟．地球物理学报，1997．第40卷第3期
[24] 安晓卫，邵伟平．二维问题的有限元网格的自动剖分．沈阳工业学院学报，1994．12，第13卷第4期
[25] Wu E R..Techniques to avoid duplicate nodes and relax restrictions on the super element numbering in a mesh generator comp. struct. 1982, 15:419-422
[26] Chave A. D. Numerical integration of related Hankel transform by quadrature and continued fraction expansion. Geophysics, 48: 1671—1686
[27] Fox, R. C, Hohmann, G. W, Killpack, J. J.,topographic effects in resistivity and induced polarization surveys. Geophysics, 45:75-93
[28] 阎述，陈明生．线源频率域电磁测深二维正演(一)．煤田地质与勘探，1999，27(5)：60～62
[29] 阎述，陈明生，线源频率域电磁测深二维正演(二)．煤田地质与勘探，1999．27(6)：56～59
[30] Coggon, J.H. Electromagnetic and electrical modeling by the finite-element method. Geophysics, 36:132-155
[31] Oden J.T, G.F. Carey.An introduction to the finite-element method. Prentice-Hall, Inc.
[32] Partha S. Routh, Douglas W. oldenburg. Inversion of Controlled Source audio—frequency magnetotedurics data for a horizontally layered earth. Geophysics, 1999,64 (6): 1689～1697
[33] Leppin, M., Electromagnetic modeling of 3-D sources over 2-D inhomogeneties in tire time domain. Geophysics, 1992,57:994-1003
[34] Goldman, Y, Hubans, C, Nicoletis, S. A finite element solution for the transient electromagnetic response of an arbitrary two-dimensional resistivity distribution. Geophysics, 1986,51:1450-1461
[35]Basokur A T , Rasmussen T M , Kaya C .Comparison of induced polarization and controlled source audio magnetotellurics methods for massive chalcopyrite exploration in a volcanic area., Geophysics . 1997, 62:1087—1096
[36]Mitsuhata Y, Toshihito U , Hiroshi A . 2. 5 D inversion of frequency domain electromagnetic data generated by a grounded2wire source. Geophysics . 2002, 67(6): 1753—1768
[37]Wannamaker P E . Tensor CSAMT survey over the sulphur springs thermal area , Valles Caldera , NewMexico, U. S. A. , Part I : Implications for structure of the western caldcra. Geophysics , 1997a ,62(4) : 451-465
[38]Sugeng, F, Raiche, A, and Xiong, Z. Modeling topographic effects using an isoparametric finite-element method: 13th Workshop on Electromagnetic Induction in the Earth. 1996, Abstracts, 109-110