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We present a newly developed algorithm for three-dimensional (3D) magnetotelluric model based on the vector finite element method. In this paper, unstructured grids which are composed of irregular tetrahedrons are used in our finite element model, which can be refined locally and adaptively according to the complex geometry of computational domain or subsurface structure. For obtaining more accurate solutions, secondary field rather than total field is numerically computed, which makes the errors limited to relatively small secondary field. Traditional node-based finite element method does not satisfy the condition that the normal electrical field is discontinuous at the interface of electric separatrix and the electrical current density is divergence-free throughout the regions without source, which obviously violates Maxwell equations. In order to overcome these drawbacks of node-based finite element, vector finite element method is employed to solve the secondary field-based partial differential equation. Moreover, in this study, the heterogeneous permeability is taken into consideration in our algorithm, as a consequence, which can deal with heterogeneous permeability model. The accuracy of our approach is verified by comparing with previously published numerical simulations of a COMMEMI-3D model. The advantages of our approach are also illustrated by the numerical simulations of model with arbitrary topography and complicated anomalous body. In addition, the simulation results of the irregular anomaly and the complex model of arbitrary terrain are both ideal results. It proves that the effect of 3D terrain is more serious and complex than that of two-dimensional (2D) terrain, and dealing with 3D models or data with 2D algorithm may bring in large errors. In the area where the magnetic permeability is abnormal, the magnetic permeability has an important influence on the numerical simulation results, and the magnetic permeability must be treated as an independent parameter in the magnetotellurics survey.
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Keywords:
- electromagnetic diffusion /
- vector finite element method /
- numerical simulation /
- magnetization /
- magnetotellurics
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Chen G B, Wang H N, Yao J J, Han Z Y 2009 Acta Phys. Sin. 58 3848
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Google Scholar
Tang H D, Yu Q F, John B 2003 Chin. J. Geophys. 46 705
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[5] Streich R 2009 Geophysics 74 F95
Google Scholar
[6] Smith J T 1996 Geophysics 61 1308
Google Scholar
[7] Wang W, Wu X P, Spitzer K 2013 Geophys. J. Int. 193 734
Google Scholar
[8] Coggon J H 1971 Geophysics 36 132
Google Scholar
[9] Fox R C, Hohmann G W, Killpack T J 1980 Geophysics 45 75
Google Scholar
[10] Wannamaker P E, Stodt J A, Rijo L 1987 Geophys. J. Int. 88 277
Google Scholar
[11] 李勇, 吴小平, 林品荣 2013 地球物理学进展 27 2452
Li Y, Wu X P, Lin P R 2013 Prog. Geophys. 27 2452
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Google Scholar
[13] Key K, Weiss C 2006 Geophysics 71 G291
Google Scholar
[14] Franke A, Börner R U, Spitzer K 2007 Geophys. J. Int. 171 71
Google Scholar
[15] Jin J 2002 The Finite Element Method in Electromagnetics (New York: John Wiley & Sons)
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Liu J X, Tong X Z, Guo R W 2012 The Magnetotelluric Sounding Method (Beijing: Science Press) pp48−50
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Xu S Z 1994 FEM in Geophysics (Beijing: Science Press) pp220−224 (in Chinese)
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 背景模型示意图
Figure 1. Schematic diagram for background model.
图 2 四面体单元及其棱边编号示意图
Figure 2. Schematic diagram for tetrahedral element and edge numbers.
图 3 一维层状介质模型
Figure 3. One-dimensional layered medium model.
图 4 基于总场和二次场的二次插值的模拟结果与解析解的对比T为周期) (a) 视电阻率
${\rho _{\rm{s}}}$ ; (b) 相位$\phi $ Figure 4. Comparisons of modelling results of quadratic interpolation and analytical solutions based on total fields and secondary fields: (a) Apparent resistivity
${\rho _{\rm{s}}}$ ; (b) phase$\phi $ .
图 5 COMMEMI-3D1模型, 其中(a)图中虚线为x和y测线
Figure 5. COMMEMI-3D1 model. The dashed lines in (a) are the x and y survey lines.
图 8 COMMEMI-3D2模型
Figure 8. COMMEMI-3D2 model.
图 10 椭球体模型 (a) xy剖面; (b) xz剖面
Figure 10. Ellipsoidal model: (a) xy profile; (b) xz profile.
图 11 椭球体模型模拟结果切片图 (a) XY模式视电阻率; (b) YX模式视电阻率; (c) XY模式相位; (d) YX模式相位
Figure 11. Slices of modelling results for ellipse model: (a) XY-mode apparent resistivities; (b) YX-mode apparent resistivities; (c) XY-mode phase; (d) YX-mode phase.
图 12 二维纯地形模型
Figure 12. Two-dimensional pure topographical model.
图 13 三维和二维模拟结果对比
Figure 13. Comparisions of three-dimensional and two-dimensional modelling results.
图 14 山峰模型及其非结构化网格剖分(左图点为测线)
Figure 14. Peak model and discretization using unstructured grids (the points in left diagram are survey line).
图 15 三维和二维模拟结果对比 (a), (b)山峰模型; (c), (d)山谷模型
Figure 15. Comparisions of three-dimensional and two-dimensional modelling results: (a), (b) Peak model; (c), (d) valley model.
图 16 不同磁导率模拟结果对比 (a), (b)无电性异常模型; (c), (d)含电性异常模型
Figure 16. Comparisons of modelling results for different magnetic permeability: (a), (b) Model without electrical anomaly; (c), (d) model with electrical anomaly.
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[1] Hohmann G W 1983 Surv. Geophys. 6 27
Google Scholar
[2] Zhdanov M S, Lee S K, Yoshioka K 2006 Geophysics 71 G333
Google Scholar
[3] 陈桂波, 汪宏年, 姚敬金, 韩子夜 2009 58 3848
Google Scholar
Chen G B, Wang H N, Yao J J, Han Z Y 2009 Acta Phys. Sin. 58 3848
Google Scholar
[4] 谭捍东, 余钦范, John B 2003 地球 46 705
Google Scholar
Tang H D, Yu Q F, John B 2003 Chin. J. Geophys. 46 705
Google Scholar
[5] Streich R 2009 Geophysics 74 F95
Google Scholar
[6] Smith J T 1996 Geophysics 61 1308
Google Scholar
[7] Wang W, Wu X P, Spitzer K 2013 Geophys. J. Int. 193 734
Google Scholar
[8] Coggon J H 1971 Geophysics 36 132
Google Scholar
[9] Fox R C, Hohmann G W, Killpack T J 1980 Geophysics 45 75
Google Scholar
[10] Wannamaker P E, Stodt J A, Rijo L 1987 Geophys. J. Int. 88 277
Google Scholar
[11] 李勇, 吴小平, 林品荣 2013 地球物理学进展 27 2452
Li Y, Wu X P, Lin P R 2013 Prog. Geophys. 27 2452
[12] Liu C, Ren Z, Tang J 2008 Appl. Geophys. 5 170
Google Scholar
[13] Key K, Weiss C 2006 Geophysics 71 G291
Google Scholar
[14] Franke A, Börner R U, Spitzer K 2007 Geophys. J. Int. 171 71
Google Scholar
[15] Jin J 2002 The Finite Element Method in Electromagnetics (New York: John Wiley & Sons)
[16] 柳建新, 童孝忠, 郭荣文 2012 大地电磁测深法勘探 (北京: 科学出版社) 第48−50页
Liu J X, Tong X Z, Guo R W 2012 The Magnetotelluric Sounding Method (Beijing: Science Press) pp48−50
[17] Unsworth M J 2010 Surv. Geophys. 31 137
Google Scholar
[18] Rao C K, Jones A G, Moorkamp M 2014 Geophys. J. Int. 198 737
Google Scholar
[19] Juanatey M, Hubert J, Tryggvason A 2013 Geophys. Prospect. 61 200
Google Scholar
[20] Egger A E, Jonathan M G, Glen J M G 2014 Geol. Soc. Am. Bull. 126 523
Google Scholar
[21] Abdelzaher M, Nishijima J, Saibi H 2012 Pure Appl. Geophys. 169 1679
Google Scholar
[22] 徐世浙 1994 地球物理中的有限单元法 (北京: 科学出版社) 第220−224页
Xu S Z 1994 FEM in Geophysics (Beijing: Science Press) pp220−224 (in Chinese)
[23] Farquharson C G, Miensopust M P 2011 J, Appl. Geophys. 75 699
Google Scholar
[24] Zhdanov M S, Varentsov I M, Weaver J T 1997 J. Appl. Geophys. 37 133
Google Scholar
[25] Mitsuhata Y, Uchida T 2004 Geophysics 69 108
Google Scholar
[26] Ren Z, Kalscheuer, Greenhalgh S 2014 Geophysics 79 E255
Google Scholar
[27] Mukherjee S, Everett M E 2011 Geophysics 76 F215
Google Scholar
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