基于有限体积法的二维大地电磁各向异性数值模拟
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摘要
为了计算带任意地形的各向异性介质中二维大地电磁响应,本文在非结构化网格的基础上,采用有限体积法,开发了二维大地电磁各向异性正演模拟的新算法.首先,从Maxwell方程出发,推导二维各向异性介质中大地电磁场的边值问题;然后,采用三角网格自动生成技术对求解区域进行非结构化网格剖分,进而构建节点中心控制体积单元,利用有限体积方法,得到求解边值问题的大型稀疏线性方程组;最后,利用Pardiso精确地计算了大地电磁响应值.三个各向异性模型的计算结果表明,本文开发的有限体积算法,不仅能够高精度求解带任意地形的大地电磁电导率各向异性问题,而且对于同一模型,该方法的计算消耗和精度都与有限单元法相当.因此,有限体积法是处理电磁法各向异性问题的一种有效方法.
In order to calculate the two-dimensional magnetotelluric response in anisotropic media with arbitrary topography,we develop a finite-volume approach for this problem.Firstly,based on the energy compensation principle and divergence theorem,the energy compensation equations for two-dimensional magnetotelluric problem with anisotropic conductivity structures are derived from Maxwell′s equations.Then,a triangular grid is used to discretize the two-dimensional conductivity model so that arbitrarily complex cases with topography can be greatly dealt with.The node-centered finite-volume algorithm is used to derive the final system of linear equations.PARDISO,a high-performance parallel solver,is chosen to achieve accurate electrical field and magnetic field efficiently.Finally,three models with anisotropic conductivity structures are used to test our proposed approach.The results show that not only can finite-volume method be used to accurately solve magnetotelluric anisotropic problems,but it also can be used to model the complex cases with arbitrarily surface topography by using unstructured grids.
In order to calculate the two-dimensional magnetotelluric response in anisotropic media with arbitrary topography,we develop a finite-volume approach for this problem.Firstly,based on the energy compensation principle and divergence theorem,the energy compensation equations for two-dimensional magnetotelluric problem with anisotropic conductivity structures are derived from Maxwell′s equations.Then,a triangular grid is used to discretize the two-dimensional conductivity model so that arbitrarily complex cases with topography can be greatly dealt with.The node-centered finite-volume algorithm is used to derive the final system of linear equations.PARDISO,a high-performance parallel solver,is chosen to achieve accurate electrical field and magnetic field efficiently.Finally,three models with anisotropic conductivity structures are used to test our proposed approach.The results show that not only can finite-volume method be used to accurately solve magnetotelluric anisotropic problems,but it also can be used to model the complex cases with arbitrarily surface topography by using unstructured grids.
引文
Cagniard L.1953.Basic theory of the magneto-telluric method of geophysical prospecting.Geophysics,18(3):605-635．
Candansayar M E,Tezkan B.2010.Two-dimensional joint inversion of radiomagnetotelluric and direct current resistivity data.Geophysical Prospecting,56(5):737-749．
Caughey D A,Jameson A.1981.Basic advances in the finite-volume method for transonic potential-flow calculations．∥Numerical and Physical Aspects of Aerodynamic Flows.Berlin,Heidelberg:Springer,445-461．
Chen G B,Wang H N,Yao J J,et al.2009.Three-dimensional numerical modeling of marine controlled-source electromagnetic responses in a layered anisotropic seabed using integral equation method.Acta Physica Sinica(in Chinese),58(6):3848-3857．
Chen G B,Wang H N,Yao J J,et al.2010.Three-dimensional modeling of frequency sounding in layered anisotropic earth using integral equation method.Chinese Journal of Computational Physics(in Chinese),27(2):274-280．
Du H K,Ren Z Y,Tang J T.2016.A finite-volume approach for2D magnetotellurics modeling with arbitrary topographies.Studia Geophysica et Geodaetica,60(2):332-347．
Fu S B,Gao K.2017.A fast solver for the Helmholtz equation based on the generalized multiscale finite-element method.Geophysical Journal International,211(2):797-813．
Hu S G,Tang J T,Ren Z Y,et al.2018.Multiple underwater objects localization with magnetic gradiometry.IEEE Geoscience and Remote Sensing Letters,16(2):296-300．
Huo G P,Hu X Y,Huang Y F,et al.2015.Mt modeling for twodimensional anisotropic conductivity structure with topography and examples of comparative analyses.Chinese Journal of Geophysics(in Chinese),58(12):4696-4708,doi:10.6038/cjg20151230．
Jahandari H,Farquharson C G.2014.A finite-volume solution to the geophysical electromagnetic forward problem using unstructured grids.Geophysics,79(6):653-657．
Jahandari H,Farquharson G C.2015.Finite-volume modelling of geophysical electromagnetic data on unstructured grids using potentials.Geophysical Journal International,202(3):1859-1876．
Ji Y J,Huang T Z,Huang W Y,et al.2016.2D anisotropic magnetotelluric numerical simulation using meshfree method under undulating terrain.Chinese Journal of Geophysics(in Chinese),59(12):4483-4493,doi:10.6038/cjg20161211．
Jin J M.2002.The Finite Element Method in Electromagnetics.2nd ed.New York,NY:John Wiley&Sons．
Li G,Xiao X,Tang J T,et al.2017.Near-source noise suppression of AMT by compressive sensing and mathematical morphology filtering.Applied Geophysics,14(4):581-589,doi:10.1007/s11770-017-0645-6．
Li J,Zhang X,Tang J T,et al.2019.Audio magnetotelluric signalnoise identification and separation based on multifractal spectrum and matching pursuit.Fractals,27(1):1940007．
Li Y.2000.Finite element modeling of electromagnetic fields in two-and three-dimensional anisotropic conductivity structures[Ph.D.thesis].G9ttingen:University of G9ttingen．
Li Y G.2002.A finite-element algorithm for electromagnetic induction in two-dimensional anisotropic conductivity structures.Geophysical Journal International,148(3):389-401．
Li Y G,Pek J.2008.Adaptive finite element modelling of twodimensional magnetotelluric fields in general anisotropic media.Geophysical Journal International,175(3):942-954．
Mo D,Jiang Q Y,Li D Q,et al.2017.Controlled-source electromagnetic data processing based on gray system theory and robust estimation.Applied Geophysics,14(4):570-580．
Pek J,Verner T.2007.Finite-difference modelling of magnetotelluric fields in two-dimensional anisotropic media.Geophysical Journal International,128(3):505-521．
Peng R H,Hu X Y,Han B,et al.2016.3Dfrequency-domain CSEM forward modeling based on the mimetic finite-volume method.Chinese Journal of Geophysics(in Chinese),59(10):3927-3939,doi:10.6038/cjg20161036．
Qiang J K,Wang X Y,Tang J T,et al.2014.The geological structures along Huainan-Liyang magnetotelluric profile:constraints from MT data.Acta Petrologica Sinica(in Chinese),30(4):957-965．
Qin L J,Yang C F,Chen K.2013a.Quasi-analytic solution of 2-Dmagnetotelluric fields on an axially anisotropic infinite fault.Geophysical Journal International,192(1):67-74．
Qin L J,Yang C F,Chen K.2013b.Analytic solution to the magnetotelluric response over anisotropic medium and its discussion.Science China Earth Sciences,56(9):1607-1615．
Qiu C K,Yin C C,Liu Y H,et al.2018.3Dforward modeling of controlled-source audio-frequency magnetotellurics in arbitrarily anisotropic media.Chinese Journal of Geophysics(in Chinese),61(8):3488-3498,doi:10.6038/cjg2018L0326．
Reddy I K,Rankin D.1975.Magnetotelluric response of laterally inhomogeneous and anisotropic media.Geophysics,40(6):1035-1045．
Ren Z Y,Kalscheuer T,Greenhalgh S,et al.2013.A goal-oriented adaptive finite-element approach for plane wave 3-D electromagnetic modelling.Geophysical Journal International,194(2):700-718．
Ren Z Y.2014.A C++based 2D magnetotellurics and radiomagnetotellurics finite element solver using unstructured grids.http：∥www.complete-mt-solutions.com/mtnet/main/．
Ren Z Y,Chen C J,Tang J T,et al.2017.A new integral equation approach for 3Dmagnetotelluric modeling.Chinese Journal of Geophysics(in Chinese),60(11):4505-4515,doi:doi:10.6038/cjg20171134．
Saraf P D,Negi J G，ˇCerv V.1986.Magnetotelluric response of a laterally inhomogeneous anisotropic inclusion.Physics of the Earth and Planetary Interiors,43(3):196-198．
Shewchuk J R.1996.Triangle:Engineering a 2D quality mesh generator and Delaunay triangulator．∥Workshop on Applied Computational Geometry.Berlin,Heidelberg:Springer-Verlag,203-222．
Sun L E.2015.Electromagnetic modeling of inhomogeneous and anisotropic structures by volume integral equation methods.Waves in Random and Complex Media,25(4):536-548．
Tang J T,Zhou C,Wang X Y,et al.2013.Deep electrical structure and geological significance of Tongling ore district.Tectonophysics,606:78-96．
Tang J T,Li G,Xiao X,et al.2017.Strong noise separation for magnetotelluric data based on a signal reconstruction algorithm of compressive sensing.Chinese Journal of Geophysics(in Chinese),60(9):3642-3654,doi:10.6038/cjg20170928．
Tang J T,Li G,Zhou C,et al.2018.Denoising AMT data based on dictionary learning.Chinese Journal of Geophysics(in Chinese),61(9):3835-3850,doi:10.6038/cjg2018L0376．
Tikhonov A N.1950.On determining electrical characteristics of the deep layers of the earth′s crust.Dolk Acad Nauk SSSR,73(2):295-297．
Trompert R A,Hansen U.1996.The application of a finite volume multigrid method to three-dimensional flow problems in a highly viscous fluid with a variable viscosity.Geophysical and Astrophysical Fluid Dynamics,83(3):261-291．
Wannamaker P E.2005.Anisotropy versus heterogeneity in continental solid earth electromagnetic studies:fundamental response characteristics and implications for physicochemical state.Surveys in Geophysics,26(6):733-765．
Wei W B,Jin S,Ye G F,et al.2006.Conductivity structure of crust and upper mantle beneath the northern Tibetan Plateau:Results of super-wide band magnetotelluric sounding.Chinese Journal of Geophysics(in Chinese),49(4):1215-1225．
Xie D X,Yang S Y.2009.Engineering Electromagnetic Field Numerical Analysis and Synthesis(in Chinese).Beijing:China Machine Press,129-141．
Xu S Z,Zhao S K.1985.Solution of magnetotelluric field equations for a two-dimensional,anisotropic geoelectric section by the
finite element method.Acta Seismologica Sinica(in Chinese),7(1):80-89．
Zhou J M,Zhang Y,Wang H N,et al.2014.Efficient simulation of three-dimensional marine controlled-source electromagnetic response in anisotropic formation by means of coupled potential finite volume method.Acta Physica Sinica(in Chinese),63(15):159101．
陈桂波，汪宏年，姚敬金等．2009．各向异性海底地层海洋可控源电磁响应三维积分方程法数值模拟．物理学报，58(6):3848-3857．
陈桂波，汪宏年，姚敬金等．2010．利用积分方程法的各向异性地层频率测深三维模拟．计算物理，27(2):274-280．
霍光谱，胡祥云，黄一凡等．2015．带地形的大地电磁各向异性二维模拟及实例对比分析．地球物理学报，58(12):4696-4708,doi:10.6038/cjg20151230．
嵇艳鞠，黄廷哲，黄婉玉等．2016．起伏地形下各向异性的2D大地电磁无网格法数值模拟．地球物理学报，59(12):4483-4493,doi:10.6038/cjg20161211．
彭荣华，胡祥云，韩波等．2016．基于拟态有限体积法的频率域可控源三维正演计算．地球物理学报，59(10):3927-3939,doi:10.6038/cjg20161036．
强建科，王显莹，汤井田等．2014．淮南-溧阳大地电磁剖面与地质结构分析．岩石学报，30(4):957-965．
邱长凯，殷长春，刘云鹤等．2018．任意各向异性介质中三维可控源音频大地电磁正演模拟．地球物理学报，2018,61(8):3488-3498,doi:10.6038/cjg2018L0326．
任政勇，陈超健，汤井田等．2017．一种新的三维大地电磁积分方程正演方法．地球物理学报，60(11):4506-4515,doi:10.6038/cjg20171134．
汤井田，李广，肖晓等．2017．基于压缩感知重构算法的大地电磁强干扰分离．地球物理学报，60(9):3642-3654,doi:10.6038/cjg20170928．
汤井田，李广，周聪等．2018．基于字典学习的音频大地电磁数据处理．地球物理学报，61(9):3835-3850,doi:10.6038/cjg2018L0376．
魏文博，金胜，叶高峰等．2006．藏北高原地壳及上地幔导电性结构---超宽频带大地电磁测深研究结果．地球物理学报，49(4):1215-1225．
谢德馨，杨仕友．2009．工程电磁场数值分析与综合．北京：机械工业出版社，129-141．
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，7(1):80-89．
周建美，张烨，汪宏年等．2014．耦合势有限体积法高效模拟各向异性地层中海洋可控源的三维电磁响应．物理学报，63(15):159101.
Candansayar M E,Tezkan B.2010.Two-dimensional joint inversion of radiomagnetotelluric and direct current resistivity data.Geophysical Prospecting,56(5):737-749．
Caughey D A,Jameson A.1981.Basic advances in the finite-volume method for transonic potential-flow calculations．∥Numerical and Physical Aspects of Aerodynamic Flows.Berlin,Heidelberg:Springer,445-461．
Chen G B,Wang H N,Yao J J,et al.2009.Three-dimensional numerical modeling of marine controlled-source electromagnetic responses in a layered anisotropic seabed using integral equation method.Acta Physica Sinica(in Chinese),58(6):3848-3857．
Chen G B,Wang H N,Yao J J,et al.2010.Three-dimensional modeling of frequency sounding in layered anisotropic earth using integral equation method.Chinese Journal of Computational Physics(in Chinese),27(2):274-280．
Du H K,Ren Z Y,Tang J T.2016.A finite-volume approach for2D magnetotellurics modeling with arbitrary topographies.Studia Geophysica et Geodaetica,60(2):332-347．
Fu S B,Gao K.2017.A fast solver for the Helmholtz equation based on the generalized multiscale finite-element method.Geophysical Journal International,211(2):797-813．
Hu S G,Tang J T,Ren Z Y,et al.2018.Multiple underwater objects localization with magnetic gradiometry.IEEE Geoscience and Remote Sensing Letters,16(2):296-300．
Huo G P,Hu X Y,Huang Y F,et al.2015.Mt modeling for twodimensional anisotropic conductivity structure with topography and examples of comparative analyses.Chinese Journal of Geophysics(in Chinese),58(12):4696-4708,doi:10.6038/cjg20151230．
Jahandari H,Farquharson C G.2014.A finite-volume solution to the geophysical electromagnetic forward problem using unstructured grids.Geophysics,79(6):653-657．
Jahandari H,Farquharson G C.2015.Finite-volume modelling of geophysical electromagnetic data on unstructured grids using potentials.Geophysical Journal International,202(3):1859-1876．
Ji Y J,Huang T Z,Huang W Y,et al.2016.2D anisotropic magnetotelluric numerical simulation using meshfree method under undulating terrain.Chinese Journal of Geophysics(in Chinese),59(12):4483-4493,doi:10.6038/cjg20161211．
Jin J M.2002.The Finite Element Method in Electromagnetics.2nd ed.New York,NY:John Wiley&Sons．
Li G,Xiao X,Tang J T,et al.2017.Near-source noise suppression of AMT by compressive sensing and mathematical morphology filtering.Applied Geophysics,14(4):581-589,doi:10.1007/s11770-017-0645-6．
Li J,Zhang X,Tang J T,et al.2019.Audio magnetotelluric signalnoise identification and separation based on multifractal spectrum and matching pursuit.Fractals,27(1):1940007．
Li Y.2000.Finite element modeling of electromagnetic fields in two-and three-dimensional anisotropic conductivity structures[Ph.D.thesis].G9ttingen:University of G9ttingen．
Li Y G.2002.A finite-element algorithm for electromagnetic induction in two-dimensional anisotropic conductivity structures.Geophysical Journal International,148(3):389-401．
Li Y G,Pek J.2008.Adaptive finite element modelling of twodimensional magnetotelluric fields in general anisotropic media.Geophysical Journal International,175(3):942-954．
Mo D,Jiang Q Y,Li D Q,et al.2017.Controlled-source electromagnetic data processing based on gray system theory and robust estimation.Applied Geophysics,14(4):570-580．
Pek J,Verner T.2007.Finite-difference modelling of magnetotelluric fields in two-dimensional anisotropic media.Geophysical Journal International,128(3):505-521．
Peng R H,Hu X Y,Han B,et al.2016.3Dfrequency-domain CSEM forward modeling based on the mimetic finite-volume method.Chinese Journal of Geophysics(in Chinese),59(10):3927-3939,doi:10.6038/cjg20161036．
Qiang J K,Wang X Y,Tang J T,et al.2014.The geological structures along Huainan-Liyang magnetotelluric profile:constraints from MT data.Acta Petrologica Sinica(in Chinese),30(4):957-965．
Qin L J,Yang C F,Chen K.2013a.Quasi-analytic solution of 2-Dmagnetotelluric fields on an axially anisotropic infinite fault.Geophysical Journal International,192(1):67-74．
Qin L J,Yang C F,Chen K.2013b.Analytic solution to the magnetotelluric response over anisotropic medium and its discussion.Science China Earth Sciences,56(9):1607-1615．
Qiu C K,Yin C C,Liu Y H,et al.2018.3Dforward modeling of controlled-source audio-frequency magnetotellurics in arbitrarily anisotropic media.Chinese Journal of Geophysics(in Chinese),61(8):3488-3498,doi:10.6038/cjg2018L0326．
Reddy I K,Rankin D.1975.Magnetotelluric response of laterally inhomogeneous and anisotropic media.Geophysics,40(6):1035-1045．
Ren Z Y,Kalscheuer T,Greenhalgh S,et al.2013.A goal-oriented adaptive finite-element approach for plane wave 3-D electromagnetic modelling.Geophysical Journal International,194(2):700-718．
Ren Z Y.2014.A C++based 2D magnetotellurics and radiomagnetotellurics finite element solver using unstructured grids.http：∥www.complete-mt-solutions.com/mtnet/main/．
Ren Z Y,Chen C J,Tang J T,et al.2017.A new integral equation approach for 3Dmagnetotelluric modeling.Chinese Journal of Geophysics(in Chinese),60(11):4505-4515,doi:doi:10.6038/cjg20171134．
Saraf P D,Negi J G，ˇCerv V.1986.Magnetotelluric response of a laterally inhomogeneous anisotropic inclusion.Physics of the Earth and Planetary Interiors,43(3):196-198．
Shewchuk J R.1996.Triangle:Engineering a 2D quality mesh generator and Delaunay triangulator．∥Workshop on Applied Computational Geometry.Berlin,Heidelberg:Springer-Verlag,203-222．
Sun L E.2015.Electromagnetic modeling of inhomogeneous and anisotropic structures by volume integral equation methods.Waves in Random and Complex Media,25(4):536-548．
Tang J T,Zhou C,Wang X Y,et al.2013.Deep electrical structure and geological significance of Tongling ore district.Tectonophysics,606:78-96．
Tang J T,Li G,Xiao X,et al.2017.Strong noise separation for magnetotelluric data based on a signal reconstruction algorithm of compressive sensing.Chinese Journal of Geophysics(in Chinese),60(9):3642-3654,doi:10.6038/cjg20170928．
Tang J T,Li G,Zhou C,et al.2018.Denoising AMT data based on dictionary learning.Chinese Journal of Geophysics(in Chinese),61(9):3835-3850,doi:10.6038/cjg2018L0376．
Tikhonov A N.1950.On determining electrical characteristics of the deep layers of the earth′s crust.Dolk Acad Nauk SSSR,73(2):295-297．
Trompert R A,Hansen U.1996.The application of a finite volume multigrid method to three-dimensional flow problems in a highly viscous fluid with a variable viscosity.Geophysical and Astrophysical Fluid Dynamics,83(3):261-291．
Wannamaker P E.2005.Anisotropy versus heterogeneity in continental solid earth electromagnetic studies:fundamental response characteristics and implications for physicochemical state.Surveys in Geophysics,26(6):733-765．
Wei W B,Jin S,Ye G F,et al.2006.Conductivity structure of crust and upper mantle beneath the northern Tibetan Plateau:Results of super-wide band magnetotelluric sounding.Chinese Journal of Geophysics(in Chinese),49(4):1215-1225．
Xie D X,Yang S Y.2009.Engineering Electromagnetic Field Numerical Analysis and Synthesis(in Chinese).Beijing:China Machine Press,129-141．
Xu S Z,Zhao S K.1985.Solution of magnetotelluric field equations for a two-dimensional,anisotropic geoelectric section by the
finite element method.Acta Seismologica Sinica(in Chinese),7(1):80-89．
Zhou J M,Zhang Y,Wang H N,et al.2014.Efficient simulation of three-dimensional marine controlled-source electromagnetic response in anisotropic formation by means of coupled potential finite volume method.Acta Physica Sinica(in Chinese),63(15):159101．
陈桂波，汪宏年，姚敬金等．2009．各向异性海底地层海洋可控源电磁响应三维积分方程法数值模拟．物理学报，58(6):3848-3857．
陈桂波，汪宏年，姚敬金等．2010．利用积分方程法的各向异性地层频率测深三维模拟．计算物理，27(2):274-280．
霍光谱，胡祥云，黄一凡等．2015．带地形的大地电磁各向异性二维模拟及实例对比分析．地球物理学报，58(12):4696-4708,doi:10.6038/cjg20151230．
嵇艳鞠，黄廷哲，黄婉玉等．2016．起伏地形下各向异性的2D大地电磁无网格法数值模拟．地球物理学报，59(12):4483-4493,doi:10.6038/cjg20161211．
彭荣华，胡祥云，韩波等．2016．基于拟态有限体积法的频率域可控源三维正演计算．地球物理学报，59(10):3927-3939,doi:10.6038/cjg20161036．
强建科，王显莹，汤井田等．2014．淮南-溧阳大地电磁剖面与地质结构分析．岩石学报，30(4):957-965．
邱长凯，殷长春，刘云鹤等．2018．任意各向异性介质中三维可控源音频大地电磁正演模拟．地球物理学报，2018,61(8):3488-3498,doi:10.6038/cjg2018L0326．
任政勇，陈超健，汤井田等．2017．一种新的三维大地电磁积分方程正演方法．地球物理学报，60(11):4506-4515,doi:10.6038/cjg20171134．
汤井田，李广，肖晓等．2017．基于压缩感知重构算法的大地电磁强干扰分离．地球物理学报，60(9):3642-3654,doi:10.6038/cjg20170928．
汤井田，李广，周聪等．2018．基于字典学习的音频大地电磁数据处理．地球物理学报，61(9):3835-3850,doi:10.6038/cjg2018L0376．
魏文博，金胜，叶高峰等．2006．藏北高原地壳及上地幔导电性结构---超宽频带大地电磁测深研究结果．地球物理学报，49(4):1215-1225．
谢德馨，杨仕友．2009．工程电磁场数值分析与综合．北京：机械工业出版社，129-141．
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，7(1):80-89．
周建美，张烨，汪宏年等．2014．耦合势有限体积法高效模拟各向异性地层中海洋可控源的三维电磁响应．物理学报，63(15):159101.
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