﻿ 基于有限体积法的二维大地电磁各向异性数值模拟
 地球物理学报  2019, Vol. 62 Issue (10): 3912-3922 PDF

1. 中南大学地球科学与信息物理学院, 长沙 410083;
2. 安徽省地质调查院(安徽省地质科学研究所), 合肥 230001;
3. 有色金属成矿预测与地质环境监测教育部重点实验室, 长沙 410083

Two-dimensional magnetotelluric anisotropic forward modeling using finite-volume method
WANG Ning1,2,3, TANG JingTian1,3, REN ZhengYong1,3, XIAO Xiao1,3, HUANG XiangYu1,3
1. School of Geosciences and Info-Physics, Central South University, Changsha 410083, China;
2. Geological Survey of Anhui Province(Anhui Institute of Geological Sciences), Hefei 230001, China;
3. Key Laboratory of Metallogenic Prediction of Non-ferrous Metals and Geological Environment Monitoring, Ministry of Education, Changsha 410083, China
Abstract: 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.
Keywords: Magnetotelluric    Finite-volume method    Anisotropic conductivity    Unstructured grids
0 引言

1 正演理论 1.1 边值问题

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 图 1 二维各向异性电导率示意图 Fig. 1 Illustration of 2D anisotropic conductivity model

σ′为对角矩阵：

TE极化：

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TM极化：

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1.2 节点中心有限体积法

 图 2 vi节点的控制体单元ci(Pt是控制单元与第t个三角形的公共部分) Fig. 2 A sketch of the node-centered volume ci which is supported by vi (Pt is a shared polygon by the t-th triangle and the control volume ci)

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(10) 式写成矩阵的形式是：

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2 正演分析 2.1 层状各向异性模型

 图 3 含有各向异性层的三层模型 Fig. 3 Three-layer model with an anisotropic layer
 图 4 三层层状模型的非结构化网格图 Fig. 4 The triangulation grids for three-layer model
 图 5 三层各向异性模型视电阻率及相位值 (a)视电阻率值; (b)相位值; (c)视电阻率相对误差; (d)相位残差. Fig. 5 The apparent resistivity and phases for three-layer model Panel (a) is for the apparent resistivity value, panel (b) is for the phases, panel (c) shows the relative error of apparent resistivity and panel (d) is for the phase residuals.
2.2 各向异性矩形棱柱体模型

 图 6 二维电导率各向异性模型 Fig. 6 2D anisotropic conductivity model

 图 7 二维电导率各向异性模型的非结构化网格图 Fig. 7 The triangulation grids for 2D anisotropic conductivity model
 图 8 二维各向异性体模型上频率为0.1 Hz时不同方法的视电阻率对比 (a) TE模式视电阻率; (b) TM模式视电阻率; (c)视电阻率相对误差(与Li，2002); (d)视电阻率相对误差(与Ren，2014). Fig. 8 The comparison of the computed apparent resistivity (TE for a, TM for b) and relative error of apparent resistivity (Li (2002) for c, Ren (2014) for d) among our finite-volume solutions and the two finite-element method solutions when f=0.1 Hz
 图 9 二维电导率各向异性模型上不同方法的视电阻率对比 (a)A=0处TE、TM模式视电阻率; (b) B=0.8 km、C=0.8 km处TM模式视电阻率; (c)视电阻率相对误差(与Ren，2014); (d)视电阻率相对误差(与Li，2002). Fig. 9 The comparison of the computed apparent resistivity (TE and TM at A=0 for a, TM at B=0.8 km, C=0.8 km for b) and relative error of apparent resistivity (Ren (2014) for c, Li (2002) for d) among our finite-volume solutions and the two finite-element method solutions

2.3 复杂背斜模型

 图 10 背斜山谷模型 Fig. 10 The anticline model with topography

 图 11 背斜山谷模型的非结构化网格图 Fig. 11 The triangulation grids for the anticline model
 图 12 背斜山谷模型上有限体积法视电阻率和相位拟断面图 (a) TE模式视电阻率; (b) TM模式视电阻率; (c) TE模式相位; (d) TM模式相位. Fig. 12 The pseudo-sections of apparent resistivity and phases computed by the finite-volume method Panel (a) is for the apparent resistivity in TE model, panel (b) is for the apparent resistivity in TM model, panel (c) is for the phase in TE model and panel (d) is for the phase in TM model.
 图 13 背斜山谷模型上，有限体积法和有限单元法计算结果的相对误差分布 (a) TE模式视电阻率相对误差; (b) TM模式视电阻率相对误差; (c) TE模式相位残差; (d) TM模式相位残差. Fig. 13 The pseudo-sections of error computed by the finite-element method and finite-volume method for the anticline model (a) The relative error of apparent resistivity in TE model; (b) The relative error of apparent resistivity in TM model; (c) The residual of phase in TE model and (d) the residual of phase in TM model.

3 结论

(1) 本文提出的基于非结构化网格的大地电磁各向异性有限体积算法，是一种能够适合模拟带任意复杂地形模型的高精度、稳定的算法.

(2) 有限体积法基本思路易于理解，基于守恒原理，利用散度定理可直接将复杂的赫姆霍斯方程转换成线性方程组进行求解，原理简单，易于编程实现.

(3) 对于同一模型，在相同的计算平台和网格条件下，有限体积法与广泛应用的有限单元法的计算精度相似，且消耗相当.因此，有限体积法是处理电磁法各向异性问题的另一种有效方法.

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 附图A1 多边形Pt(i)示意图 Appendix Fig. A1 Illustration of the polygon Pt(i)

，有：

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 附图A2 坐标转换图 Appendix Fig. A2 The coordinate transformation

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，有：

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References
 Cagniard L. 1953. Basic theory of the magneto-telluric method of geophysical prospecting. Geophysics, 18(3): 605-635. DOI:10.1190/1.1437915 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 for 2D magnetotellurics modeling with arbitrary topographies. Studia Geophysica et Geodaetica, 60(2): 332-347. DOI:10.1007/s11200-014-1041-9 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. DOI:10.1093/gji/ggx343 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 two-dimensional 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. DOI:10.1093/gji/ggv257 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 signal-noise identification and separation based on multifractal spectrum and matching pursuit. Fractals, 27(1): 1940007. DOI:10.1142/S0218348X19400073 Li Y. 2000. Finite element modeling of electromagnetic fields in two-and three-dimensional anisotropic conductivity structures [Ph. D. thesis]. Göttingen: University of Göttingen. Li Y G. 2002. A finite-element algorithm for electromagnetic induction in two-dimensional anisotropic conductivity structures. Geophysical Journal International, 148(3): 389-401. DOI:10.1046/j.1365-246x.2002.01570.x Li Y G, Pek J. 2008. Adaptive finite element modelling of two-dimensional magnetotelluric fields in general anisotropic media. Geophysical Journal International, 175(3): 942-954. DOI:10.1111/j.1365-246X.2008.03955.x 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. DOI:10.1007/s11770-017-0646-5 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. 3D frequency-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-D magnetotelluric fields on an axially anisotropic infinite fault. Geophysical Journal International, 192(1): 67-74. DOI:10.1093/gji/ggs018 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. DOI:10.1007/s11430-013-4585-6 Qiu C K, Yin C C, Liu Y H, et al. 2018. 3D forward 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. DOI:10.1190/1.1440579 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. DOI:10.1093/gji/ggt154 Ren Z Y. 2014. A C++ based 2D magnetotellurics and radio-magnetotellurics 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 3D magnetotelluric modeling. Chinese Journal of Geophysics (in Chinese), 60(11): 4505-4515. Saraf P D, Negi J G, erv V. 1986. Magnetotelluric response of a laterally inhomogeneous anisotropic inclusion. Physics of the Earth and Planetary Interiors, 43(3): 196-198. DOI:10.1016/0031-9201(86)90046-4 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. DOI:10.1080/17455030.2015.1058541 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. DOI:10.1016/j.tecto.2013.05.039 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. DOI:10.1007/s10712-005-1832-1 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. 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. DOI:10.3321/j.issn:1000-3290.2009.06.039 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. DOI:10.3969/j.issn.1001-246X.2010.02.017 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, 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. DOI:10.3321/j.issn:0001-5733.2006.04.038 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. DOI:10.7498/aps.63.159101