基于MPI的频率域航空电磁法有限元二维正演并行计算研究
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摘要
航空电磁法(Airborne Electromagnetic Method)是航空物探中常用的观测方法之一,广泛的应用于地质普查、资源勘探和环境监测等领域。本文针对吊舱式直升机频率域航空电磁系统IMPULSE的各方面特点进行深入分析,提出了二维正反演研究的重要性。频率域航空电磁法二维正演问题实际上是,求解三维场源在二维地电断面中引起的电磁响应,也被称为2.5维问题。利用电磁场沿走向以场源中心呈对称分布的特点,可通过沿走向的傅立叶变换将空间域2.5维问题简化为波数域二维问题进行求解。
本文首先按照场分离的思路,对一次场和二次场进行分离,通过将背景电导率分布设为全空气介质,使用解析解计算一次场,而二次场则使用有限元法求解。二次场求解过程中,使用伽辽金加权余量法和抛物线等参单元法进行处理。在求解大型线性方程组时,没有对整体刚度矩阵进行集成,而是选择波前法求解,从而避免的对整体刚度矩阵的存储,提高了计算效率。在完成公式推导后,根据得到的算法设计流程图,使用Fortran语言编写了串行代码。精度检验的结果显示,该频率域航空电磁二维正演算法具有相当的可靠性。随后,对8组不同的地下异常体模型和观测装置组合进行正演计算,并对得到的电磁响应结果进行分析。
鉴于频率域航空电磁法多维正反演问题的复杂性,并为将来的反演计算打下基础,本文尝试对提出的正演算法进行并行化。Message Passing Interface(MPI)是用于并行编程的一种主要标准和规范,通过对其深入研究和对提出的频率域航空电磁法有限元正演模拟算法的分析,选择了域分解算法设计和SPMD并行编程思路。通过在小型网络机群的测试,正演计算的效率得到了大幅提升。
AEM(Airborne Electromagnetic Method)is one of the most commonly used branch of airborne geophysical method, which is widely used in geological survey, recourse exploration, environment monitoring and other related fields. Based on a thoroughly investigation in the features of all aspects of IMPULSE -one of the most popular frequency domain airborne electromagnetic systems, the importance of 2D modelling and inversion of AEM data was revealed. The 2D forward problem of AEM is actually to calculate the response of 2D geoelectrical model to a 3D source, which is also named 2.5D problem. During the practical procedure of calculating, the 2.5D Problem could be reduced to 2D problem through taking a Fourier transform along the strike direction because of its symmetry character on the strike.
By using a free space as the distribution of background conductivity, the primary and secondary fields could be calculated separately, one by analytical solution and the other by FEM (Finite Element Method). Galerkin weighted residual method and isoparametric element with quadratic basis were both employed in calculating the secondary field .The system matrix for coupled equations is solved using the frontal solution, which is similar to Gaussian elimination except that the system matrix is never actually assembled and essentially reduce the computation cost. A serial code was developed by Fortran after finishing the formulating. The accuracy and efficiency of the code was tested by an simple layered model by comparing to the analytical solution which turned out to be that the 2D modelling algorithm is considerable reliable. Subsequently, the response of 8 different set of model and configurations were calculated by the code and analyzed for features.
As the complexity of multi-dimensional frequency AEM modelling and inversion, parallelized attempt was made on the forward code for inversion consideration in the future. MPI(Message Passing Interface)is one of the most important standard in parallel programming, which leads to a domain decomposition algorithm and a SPMD(Single Program Multi data) idea. A small Cluster was built on which the parallel code was tested and turn out to increase the modelling process dramatically.
本文首先按照场分离的思路,对一次场和二次场进行分离,通过将背景电导率分布设为全空气介质,使用解析解计算一次场,而二次场则使用有限元法求解。二次场求解过程中,使用伽辽金加权余量法和抛物线等参单元法进行处理。在求解大型线性方程组时,没有对整体刚度矩阵进行集成,而是选择波前法求解,从而避免的对整体刚度矩阵的存储,提高了计算效率。在完成公式推导后,根据得到的算法设计流程图,使用Fortran语言编写了串行代码。精度检验的结果显示,该频率域航空电磁二维正演算法具有相当的可靠性。随后,对8组不同的地下异常体模型和观测装置组合进行正演计算,并对得到的电磁响应结果进行分析。
鉴于频率域航空电磁法多维正反演问题的复杂性,并为将来的反演计算打下基础,本文尝试对提出的正演算法进行并行化。Message Passing Interface(MPI)是用于并行编程的一种主要标准和规范,通过对其深入研究和对提出的频率域航空电磁法有限元正演模拟算法的分析,选择了域分解算法设计和SPMD并行编程思路。通过在小型网络机群的测试,正演计算的效率得到了大幅提升。
AEM(Airborne Electromagnetic Method)is one of the most commonly used branch of airborne geophysical method, which is widely used in geological survey, recourse exploration, environment monitoring and other related fields. Based on a thoroughly investigation in the features of all aspects of IMPULSE -one of the most popular frequency domain airborne electromagnetic systems, the importance of 2D modelling and inversion of AEM data was revealed. The 2D forward problem of AEM is actually to calculate the response of 2D geoelectrical model to a 3D source, which is also named 2.5D problem. During the practical procedure of calculating, the 2.5D Problem could be reduced to 2D problem through taking a Fourier transform along the strike direction because of its symmetry character on the strike.
By using a free space as the distribution of background conductivity, the primary and secondary fields could be calculated separately, one by analytical solution and the other by FEM (Finite Element Method). Galerkin weighted residual method and isoparametric element with quadratic basis were both employed in calculating the secondary field .The system matrix for coupled equations is solved using the frontal solution, which is similar to Gaussian elimination except that the system matrix is never actually assembled and essentially reduce the computation cost. A serial code was developed by Fortran after finishing the formulating. The accuracy and efficiency of the code was tested by an simple layered model by comparing to the analytical solution which turned out to be that the 2D modelling algorithm is considerable reliable. Subsequently, the response of 8 different set of model and configurations were calculated by the code and analyzed for features.
As the complexity of multi-dimensional frequency AEM modelling and inversion, parallelized attempt was made on the forward code for inversion consideration in the future. MPI(Message Passing Interface)is one of the most important standard in parallel programming, which leads to a domain decomposition algorithm and a SPMD(Single Program Multi data) idea. A small Cluster was built on which the parallel code was tested and turn out to increase the modelling process dramatically.
引文
安晓卫,邵伟平.二维问题的有限元网格的自动剖分[J].沈阳工业学院学报,1994,13(4)
程信尧.HDY202型补偿式航电仪试制总结报告[R].物化探研究报导,1982.
傅良魁.电法勘探教程[M].北京:地质出版社,1983.
傅良魁.应用地球物理教程-电法、放射性、地热.北京[M]:地质出版社, 1991.
黄浩平,王维中.时间域航空电磁法的反演[J].地球物理学报,1990,33:87~97.
江乐玉,雷宛.地球物理数据处理教程[M].北京:地质出版社, 2006.
韩悦慧.基于神经网络的时间域航空电磁数据CDI成像与反演初探[D].长春:吉林大学,2009.
雷栋,胡祥云等.航空电磁法的发展现状,地质找矿论丛[J],2006 ,21(1)
李金铭,罗延钟.电法勘探新进展[M],北京:地质出版社,1996.
李金铭.地电场与电法勘探[M].北京:地质出版社,2005.
李文杰.频率域航空电磁数据处理技术研究[D].北京:中国地质大学,2008.
刘尔烈,等.有限单元法及程序设计[M].天津:天津大学出版社, 1999
刘桂芬.回线源层状大地航空瞬变电磁场的理论计算[D].长春:吉林大学,2008.
刘国兴.电法勘探原理与方法[M].北京:地质出版社, 2005.
刘崧.谱激电法[M],武汉:中国地质大学出版社, 1998.
罗延钟,张胜业,王卫平.时间域航空电磁法一维正演研究[J].地球物理学报,2003,46(5):719~724.
孟庆敏.频率域航空电磁法层状反演及应用研究[D].北京:中国地质大学,2005.
孟庆敏.江西省上高-高安地区航空物探(电/磁)综合测量(试生产)成果报告[R].廊
坊:物化探研究所研究报告,1995.
满延龙.国外航空电磁法现状和我国现阶段航电的发展[J].物探与化探,1994,18(3):174~178.
牛之琏.时间域电磁法原理[M].长沙:中南工业大学出版社,1992.
欧介甫.胶东地区航空物探(电/磁)在金矿预测中的应用研究[R].廊坊:物化探研究所研究报告,1993.
彭国伦. Fortran 95程序设计[M].北京:中国电力出版社, 2002
朴化荣.电磁测深法原理[M].北京:地质出版社,1990.
钱立新.频率域航空电磁法的一维反演及在航空电阻率填图中的应用[D].廊坊:物化探研究所,1990.
汤井田,何继善.可控源音频大地电磁法及其应用[M].长沙:中南大学出版社,2005.
沈金松,孙文博.2.5维电磁响应的有限元模拟与波数取值研究[J].物探化探计算技术, 2008 ,30(2).
王卫平,王守坦.直升机频率域航空电磁系统在均匀半空间上方的电磁响应特征与
探测深度[J].地球物理学报,2003,24(3):285~288.
王卫平,周锡华,王守坦,等.吊舱式直升机频率域电磁系统性能及应用效果[J].地球物理学进展,2008,23(3).
王卫平,吴成平.频率域航空电磁法软件研究现状与发展趋势[J].工程地球物理学报, 2010,7(3).
王勖成.有限单元法[M].北京:清华大学出版社, 2003
徐世浙.地球物理中的有限单元法[M],北京:科学出版社, 1994 .
薛琴舫.场论[M].北京:地质出版社, 1978.
杨进.应用地球物理数值模拟的迭代有限元法[M],北京:地质出版社,1996.
张敬华.吉林省四平-长春地区航空物探(电/磁)综合测量成果报告[R].廊坊:物化探研究所研究报告,1992.
雍蔚华.频率域航空电磁法资料的反演解释方法及其应用研究[D].北京:中国地质大学,1993.
周道卿.频率域航空电磁资料实用化处理解释系统研究[D].北京:中国地质大学,2006.
张舒,褚艳利.2009,GPU高性能运算之CUDA[M].北京:水利水电出版社
郑圣谈.高频电磁场模拟及其反演[D].长春:吉林大学,2007
Anderson, W. L. A hybrid fast Hankel transform algorithm for electromagnetic modeling, Geophysics , 1989, 54(2): 263~266.
Ahl, A. Automatic 1D inversion of multifrequency airborne electromagnetic datawith arti- -ficial neural networks: discussion and a case study[J]. GeophysicalProspecting, 2003, v51: 89~97.
Alan D. Chave. Numerical integration of related Hankel transforms by quadrature and continued fraction expansion[J].Geophysics, 1983, 48(12):1671-1686.
Anderson W.L. Numerical integration of related Hankel transforms of order 0 and 1 by adaptive digital filtering[J]. Geophysics, 1979, 44(7):1287~1305.
Anderson W.L. A hybrid fast Hankel transform algorithm for electromagnetic modeling [J] .Geophysics, 1989, 54(2):263~266.
Avdeev, D, B.et al. Three-dimensional frequency-domain modeling of airborne electromagnetic responses [J]. Exploration Geophysics, 1998, v29:111-119.
Commer ,M. and Newman, G.A. New advances in three-dimensional controlled-source electromagnetic inversion[J]. Geophysics ,2008,172(2):513~535.
D.Guptasarma, B.Singh. New digital linear filters for Hankel J0 and J1 transforms[J]. Geophysical prospecting, 1997, 45:745~762.
Dmitriev, V.I. Electromagnetic fields in inhomogeneous media[R]. Computer Center, Moscow University, 1969: 132.
David Fountain, Richard Smith.55 Years of AEM[C].50 Years of Fugro. Fountain, D. Airborne electromagnetic system–50 years of development [J] .Exploration Geophysics, 1998, 29:1-11.
Fraser,D.C. Resistivity mapping with an airborne multi-coil electromagnetic system[J]. Geophysics, 1978, 43:144~172.
Fraser, D.C. A new multi-coil aerial electromagnetic prospecting system [J].Geophysics, 1972, 27:518~537.
Ganquan Xie, Jianhua Li.et al. Oristaglio. 3-D electromagnetic modeling and nonlinear inversion [J]. Geophysics, 2000, 65:804-822.
Gianzero, S., Chemali, R. and Su, S.-M. Induction, resistivity and MWD tools in horizontal wells, Log Analyst, 1990, 3l: 158~171.
Goldman, M.M. and Stayer, C.H. Finite-difference calculations of the transient field of an axially symmetric earth for vertical magnetic dipole excitation[J].
Geophysics, , 1983 48: 953~963.
Goldman, M.M., Tabarovsky, L. and Rabinovich, M. On the influence of three-dimensional structures in the interpretation of transient electromagnetic sounding data[J]. Geophysics,1994, 59: 889~901.
Haoping Huang, Douglas C. Fraser. Magnetic permeability and electrical resistivity mapping with a multifrequency airborne EM system [J].Exploration Gesphysics, 1998, 29:249~253.
Haoping Huang, et al. Maping of the resistivity, susceptibility, and permeability of the earth using a helicopter-borne electromagnetic system [J]. Geophysics,2001, 66:148-157.
Haoping Huang, et al. The use of quad-quad resistivity in helicopter electromagnetic mapping [J]. Geophysics, 2002, 67:727-738.
Haoping Huang, et al. Dielectric permeability and resistivity mapping using high-frequency helicopter-borne EM data [J]. Geophysics, 2002, 67:727~738.
Hohmann, G.W. Numerical modeling for electromagnetic methods of geophysics[M]. In: M.N. Nabighian and J.D. Corbess (Editors), Electromagnetic Methods in Applied Geophysics - Theory, 1987, 1: 313~365.
Hyde,C. The electromagnetic response of a sphere in the moment domain:a building block for the construction of arbitrary of 3D shapes[C].SEG International Exploration and 72nd Annual Meeting, 2002.
Jisoo Ryu,H. Frank Morrison,Stanley H.Ward. Electromagnetic fields about a loop source of current [J].Geophysics, 1970, 35(5):862~896.
Jiuping chen, Art Raiche. Inverting AEM data using a damped eigenparameter method [J]. Exploration Geophysics, 1998, 29:128-132.
Klaus-Peter Sengpie et al. Examples of 1-Dinversion of multifrequency HEM data from 3-D resistivity distributions[J]. Exploration Geophysics, 1998, 29:133~141.
Klaus-Peter Sengpiel, Bernhard Siemon. Examples of 1-D inversion of multifrequency HEM data from 3-D resistivity distributions[J].Exploration Geophysics,1998,29:133~141.
Klaus-Peter Sengpiel, Bernhard Siemon. Advanced inversion methods for airborne electromagnetic exploration [J]. Geophysics, 2000, 65:1983-1992.
Kurt I. Sorensen,Niels B. Christensen. The fields from a finite electrical dipole: A new computational approach [J].Geophysics, 1994, 59(6):864~880.
Macnae, J.C., Smith, R., Polzer, B.D., Lamontangue, Y. and Klinkert, P.S. Conductivity-depth imaging of airborne electromagnetic step-response data.
Geophysics, 1991, 56: 102~114. Martyn J. Unsworth, Bryan J. Travis, and Alan D. Chave .Electromagnetic induction by a finite electric dipole source over a 2-D earth [J].Geophysics ,1993 58:198-214.
Morozova, G.M. and Tabarovsky, L.A. Transient electromagnetic field of a buried dipole source[R]. In: Theory and Experience of Electromagnetic Field Applications in Geophysical Prospecting. Institute for Geology and Geophysics,
Novosibirsk , 1978:54~56. N.P. Singh, Toru Mogi. EMLCLLER-a program for computing the EM response of a large loop source over a layered earth model[J].Computers&Geosciences,2003, 29:1301~1307.
Laust B. Pedersen, Mehran Gharibi. Automatic1-D inversion of magnetotelluric data: Finding the simplest possible model that fits the data[J]. Geophysics, 2000,65:773~782.
Lee K H , Morrison F.A Numerical solution of response of a two-dimensional earth to an oscillating magnetic dipole source[J].1976
Oristagrio, M.L. and Hohmann, G.W. Diffusion of electromagnetic fields in a two-dimensional earth: a finite difference approach[J].Geophysics,1984., 49: 870~894
Peaceman, D.W. and Rachford, H.H. The numerical solution of parabolic and elliptic differential equations [J]. Industr. Math. Sot. , 1955, 3:28~41.
Ryu J, Morrison H F, Ward S H. Electromagnetic filed about a loop source of current[J].Geophysics, 1970, 35 (5):862~896.
Sattel, D. Modelling AEM data with Zohdy’s method[C]. SEG International Exploration and 72nd Annual Meeting, 2002.
Stoyer C.H. and Greenfield R.J. Numerical solutions of the response of a 2D earth toan oscillating magnetic dipole source[J],1976.
Strack, K.-M. Exploration with Deep Transient Electromagnetics[M]. Elsevier, Amsterdam : 374, 1992.
Telford, W.M., Geldart, L.P., Sheriff, R.E. Applied Geophysics[M]. Cambridge University Press, Cambridge, 1990.
W. Bitterlinch, A. W. Kellner. Determination of electric rock parameters from the field stracture of a magnetic dipole [J]. Geophysical Prospecting, 1970, 18(4):516~549.
W.Qian,J. S.Holladay,Z.Dvorak. Automated inversion of broadband multiple transmitter-receiver airborne EM data for the parameters of a small, confined conductor [J]. Exploration Geophysics, 1998, 29:147-151.
Wait, J R. Mutual coupling of loops lying on the ground:Geo-system assessment[J].Geophysics, 1954, 19(2):290-296.
Walter L. Anderson. Fast Hankel transforms using related and lagged convoluteions[J].ACM Trans. on Math. Software, 1982, 8(4):344~368.
Weidelt, P. Electromagnetic induction in three-dimensional structures[J]. Geophysics, 1975, 41: 85~109.
Zhiyi Zhang. 3D resistivity mapping of airborne EM data[J]. Geophysics, 2003,68:1896-1905.
Zhang, Z., Oldenburg, W. Recovering magnetic susceptibility from electromagnetic data over a one-dimensional earth [J].Geophysics, 1997, 130:422~434.
Zhdanov, M.S., et al. Electromagnetic inversion using quasi-linear approximation[J]. Geophysics, 2000, 65:1501-1513.
程信尧.HDY202型补偿式航电仪试制总结报告[R].物化探研究报导,1982.
傅良魁.电法勘探教程[M].北京:地质出版社,1983.
傅良魁.应用地球物理教程-电法、放射性、地热.北京[M]:地质出版社, 1991.
黄浩平,王维中.时间域航空电磁法的反演[J].地球物理学报,1990,33:87~97.
江乐玉,雷宛.地球物理数据处理教程[M].北京:地质出版社, 2006.
韩悦慧.基于神经网络的时间域航空电磁数据CDI成像与反演初探[D].长春:吉林大学,2009.
雷栋,胡祥云等.航空电磁法的发展现状,地质找矿论丛[J],2006 ,21(1)
李金铭,罗延钟.电法勘探新进展[M],北京:地质出版社,1996.
李金铭.地电场与电法勘探[M].北京:地质出版社,2005.
李文杰.频率域航空电磁数据处理技术研究[D].北京:中国地质大学,2008.
刘尔烈,等.有限单元法及程序设计[M].天津:天津大学出版社, 1999
刘桂芬.回线源层状大地航空瞬变电磁场的理论计算[D].长春:吉林大学,2008.
刘国兴.电法勘探原理与方法[M].北京:地质出版社, 2005.
刘崧.谱激电法[M],武汉:中国地质大学出版社, 1998.
罗延钟,张胜业,王卫平.时间域航空电磁法一维正演研究[J].地球物理学报,2003,46(5):719~724.
孟庆敏.频率域航空电磁法层状反演及应用研究[D].北京:中国地质大学,2005.
孟庆敏.江西省上高-高安地区航空物探(电/磁)综合测量(试生产)成果报告[R].廊
坊:物化探研究所研究报告,1995.
满延龙.国外航空电磁法现状和我国现阶段航电的发展[J].物探与化探,1994,18(3):174~178.
牛之琏.时间域电磁法原理[M].长沙:中南工业大学出版社,1992.
欧介甫.胶东地区航空物探(电/磁)在金矿预测中的应用研究[R].廊坊:物化探研究所研究报告,1993.
彭国伦. Fortran 95程序设计[M].北京:中国电力出版社, 2002
朴化荣.电磁测深法原理[M].北京:地质出版社,1990.
钱立新.频率域航空电磁法的一维反演及在航空电阻率填图中的应用[D].廊坊:物化探研究所,1990.
汤井田,何继善.可控源音频大地电磁法及其应用[M].长沙:中南大学出版社,2005.
沈金松,孙文博.2.5维电磁响应的有限元模拟与波数取值研究[J].物探化探计算技术, 2008 ,30(2).
王卫平,王守坦.直升机频率域航空电磁系统在均匀半空间上方的电磁响应特征与
探测深度[J].地球物理学报,2003,24(3):285~288.
王卫平,周锡华,王守坦,等.吊舱式直升机频率域电磁系统性能及应用效果[J].地球物理学进展,2008,23(3).
王卫平,吴成平.频率域航空电磁法软件研究现状与发展趋势[J].工程地球物理学报, 2010,7(3).
王勖成.有限单元法[M].北京:清华大学出版社, 2003
徐世浙.地球物理中的有限单元法[M],北京:科学出版社, 1994 .
薛琴舫.场论[M].北京:地质出版社, 1978.
杨进.应用地球物理数值模拟的迭代有限元法[M],北京:地质出版社,1996.
张敬华.吉林省四平-长春地区航空物探(电/磁)综合测量成果报告[R].廊坊:物化探研究所研究报告,1992.
雍蔚华.频率域航空电磁法资料的反演解释方法及其应用研究[D].北京:中国地质大学,1993.
周道卿.频率域航空电磁资料实用化处理解释系统研究[D].北京:中国地质大学,2006.
张舒,褚艳利.2009,GPU高性能运算之CUDA[M].北京:水利水电出版社
郑圣谈.高频电磁场模拟及其反演[D].长春:吉林大学,2007
Anderson, W. L. A hybrid fast Hankel transform algorithm for electromagnetic modeling, Geophysics , 1989, 54(2): 263~266.
Ahl, A. Automatic 1D inversion of multifrequency airborne electromagnetic datawith arti- -ficial neural networks: discussion and a case study[J]. GeophysicalProspecting, 2003, v51: 89~97.
Alan D. Chave. Numerical integration of related Hankel transforms by quadrature and continued fraction expansion[J].Geophysics, 1983, 48(12):1671-1686.
Anderson W.L. Numerical integration of related Hankel transforms of order 0 and 1 by adaptive digital filtering[J]. Geophysics, 1979, 44(7):1287~1305.
Anderson W.L. A hybrid fast Hankel transform algorithm for electromagnetic modeling [J] .Geophysics, 1989, 54(2):263~266.
Avdeev, D, B.et al. Three-dimensional frequency-domain modeling of airborne electromagnetic responses [J]. Exploration Geophysics, 1998, v29:111-119.
Commer ,M. and Newman, G.A. New advances in three-dimensional controlled-source electromagnetic inversion[J]. Geophysics ,2008,172(2):513~535.
D.Guptasarma, B.Singh. New digital linear filters for Hankel J0 and J1 transforms[J]. Geophysical prospecting, 1997, 45:745~762.
Dmitriev, V.I. Electromagnetic fields in inhomogeneous media[R]. Computer Center, Moscow University, 1969: 132.
David Fountain, Richard Smith.55 Years of AEM[C].50 Years of Fugro. Fountain, D. Airborne electromagnetic system–50 years of development [J] .Exploration Geophysics, 1998, 29:1-11.
Fraser,D.C. Resistivity mapping with an airborne multi-coil electromagnetic system[J]. Geophysics, 1978, 43:144~172.
Fraser, D.C. A new multi-coil aerial electromagnetic prospecting system [J].Geophysics, 1972, 27:518~537.
Ganquan Xie, Jianhua Li.et al. Oristaglio. 3-D electromagnetic modeling and nonlinear inversion [J]. Geophysics, 2000, 65:804-822.
Gianzero, S., Chemali, R. and Su, S.-M. Induction, resistivity and MWD tools in horizontal wells, Log Analyst, 1990, 3l: 158~171.
Goldman, M.M. and Stayer, C.H. Finite-difference calculations of the transient field of an axially symmetric earth for vertical magnetic dipole excitation[J].
Geophysics, , 1983 48: 953~963.
Goldman, M.M., Tabarovsky, L. and Rabinovich, M. On the influence of three-dimensional structures in the interpretation of transient electromagnetic sounding data[J]. Geophysics,1994, 59: 889~901.
Haoping Huang, Douglas C. Fraser. Magnetic permeability and electrical resistivity mapping with a multifrequency airborne EM system [J].Exploration Gesphysics, 1998, 29:249~253.
Haoping Huang, et al. Maping of the resistivity, susceptibility, and permeability of the earth using a helicopter-borne electromagnetic system [J]. Geophysics,2001, 66:148-157.
Haoping Huang, et al. The use of quad-quad resistivity in helicopter electromagnetic mapping [J]. Geophysics, 2002, 67:727-738.
Haoping Huang, et al. Dielectric permeability and resistivity mapping using high-frequency helicopter-borne EM data [J]. Geophysics, 2002, 67:727~738.
Hohmann, G.W. Numerical modeling for electromagnetic methods of geophysics[M]. In: M.N. Nabighian and J.D. Corbess (Editors), Electromagnetic Methods in Applied Geophysics - Theory, 1987, 1: 313~365.
Hyde,C. The electromagnetic response of a sphere in the moment domain:a building block for the construction of arbitrary of 3D shapes[C].SEG International Exploration and 72nd Annual Meeting, 2002.
Jisoo Ryu,H. Frank Morrison,Stanley H.Ward. Electromagnetic fields about a loop source of current [J].Geophysics, 1970, 35(5):862~896.
Jiuping chen, Art Raiche. Inverting AEM data using a damped eigenparameter method [J]. Exploration Geophysics, 1998, 29:128-132.
Klaus-Peter Sengpie et al. Examples of 1-Dinversion of multifrequency HEM data from 3-D resistivity distributions[J]. Exploration Geophysics, 1998, 29:133~141.
Klaus-Peter Sengpiel, Bernhard Siemon. Examples of 1-D inversion of multifrequency HEM data from 3-D resistivity distributions[J].Exploration Geophysics,1998,29:133~141.
Klaus-Peter Sengpiel, Bernhard Siemon. Advanced inversion methods for airborne electromagnetic exploration [J]. Geophysics, 2000, 65:1983-1992.
Kurt I. Sorensen,Niels B. Christensen. The fields from a finite electrical dipole: A new computational approach [J].Geophysics, 1994, 59(6):864~880.
Macnae, J.C., Smith, R., Polzer, B.D., Lamontangue, Y. and Klinkert, P.S. Conductivity-depth imaging of airborne electromagnetic step-response data.
Geophysics, 1991, 56: 102~114. Martyn J. Unsworth, Bryan J. Travis, and Alan D. Chave .Electromagnetic induction by a finite electric dipole source over a 2-D earth [J].Geophysics ,1993 58:198-214.
Morozova, G.M. and Tabarovsky, L.A. Transient electromagnetic field of a buried dipole source[R]. In: Theory and Experience of Electromagnetic Field Applications in Geophysical Prospecting. Institute for Geology and Geophysics,
Novosibirsk , 1978:54~56. N.P. Singh, Toru Mogi. EMLCLLER-a program for computing the EM response of a large loop source over a layered earth model[J].Computers&Geosciences,2003, 29:1301~1307.
Laust B. Pedersen, Mehran Gharibi. Automatic1-D inversion of magnetotelluric data: Finding the simplest possible model that fits the data[J]. Geophysics, 2000,65:773~782.
Lee K H , Morrison F.A Numerical solution of response of a two-dimensional earth to an oscillating magnetic dipole source[J].1976
Oristagrio, M.L. and Hohmann, G.W. Diffusion of electromagnetic fields in a two-dimensional earth: a finite difference approach[J].Geophysics,1984., 49: 870~894
Peaceman, D.W. and Rachford, H.H. The numerical solution of parabolic and elliptic differential equations [J]. Industr. Math. Sot. , 1955, 3:28~41.
Ryu J, Morrison H F, Ward S H. Electromagnetic filed about a loop source of current[J].Geophysics, 1970, 35 (5):862~896.
Sattel, D. Modelling AEM data with Zohdy’s method[C]. SEG International Exploration and 72nd Annual Meeting, 2002.
Stoyer C.H. and Greenfield R.J. Numerical solutions of the response of a 2D earth toan oscillating magnetic dipole source[J],1976.
Strack, K.-M. Exploration with Deep Transient Electromagnetics[M]. Elsevier, Amsterdam : 374, 1992.
Telford, W.M., Geldart, L.P., Sheriff, R.E. Applied Geophysics[M]. Cambridge University Press, Cambridge, 1990.
W. Bitterlinch, A. W. Kellner. Determination of electric rock parameters from the field stracture of a magnetic dipole [J]. Geophysical Prospecting, 1970, 18(4):516~549.
W.Qian,J. S.Holladay,Z.Dvorak. Automated inversion of broadband multiple transmitter-receiver airborne EM data for the parameters of a small, confined conductor [J]. Exploration Geophysics, 1998, 29:147-151.
Wait, J R. Mutual coupling of loops lying on the ground:Geo-system assessment[J].Geophysics, 1954, 19(2):290-296.
Walter L. Anderson. Fast Hankel transforms using related and lagged convoluteions[J].ACM Trans. on Math. Software, 1982, 8(4):344~368.
Weidelt, P. Electromagnetic induction in three-dimensional structures[J]. Geophysics, 1975, 41: 85~109.
Zhiyi Zhang. 3D resistivity mapping of airborne EM data[J]. Geophysics, 2003,68:1896-1905.
Zhang, Z., Oldenburg, W. Recovering magnetic susceptibility from electromagnetic data over a one-dimensional earth [J].Geophysics, 1997, 130:422~434.
Zhdanov, M.S., et al. Electromagnetic inversion using quasi-linear approximation[J]. Geophysics, 2000, 65:1501-1513.