基于非结构化有限元的三维井地电阻率法约束反演

doi:10.11720/wtyht.2022.0181

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Abstract

The inversion of electromagnetic detection data is a typical ill-posed problem and is prone to cause a multiplicity of solutions of the inversion results. The ill-posedness is an inherent characteristic of inversion and is difficult to overcome without additional information. An effective way to solve this problem is constrained inversion. In this study, the Gauss-Newton - conjugate gradient (GN-CG) method was used to directly impose constraints on the inversion objective function. Specifically, the dielectric resistivity range was introduced into the inversion objective function as the prior information and constraints using the exterior penalty function method. Compared with the conventional three-dimensional resistivity inversion objective function, the objective function with inequality constraints can suppress the multiplicity of solutions in theory. As revealed by the testing results of various theoretical models, the three-dimensional borehole-to-surface resistivity inversion algorithm based on inequality constraints effectively improves the precision of inversion results, and the way of imposing inequality constraints using the penalty function method is feasible and effective.

Key words： borehole-to-surface resistivity method inversion GN-CG inequality constraint penalty function method

Received: 13 April 2022 Published: 03 January 2023

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	Zhi WANG
	Cheng WANG
	Si-Nan FANG

Cite this article:

Zhi WANG,Cheng WANG,Si-Nan FANG. Constraint inversion of three-dimensional borehole-to-surface resistivity based on unstructured finite element[J]. Geophysical and Geochemical Exploration, 2022, 46(6): 1431-1443.

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https://www.wutanyuhuatan.com/EN/10.11720/wtyht.2022.0181 OR https://www.wutanyuhuatan.com/EN/Y2022/V46/I6/1431

Sphere model structure and mesh division
a—diagram of the sphere model structure;b—diagram of spherical model meshing;c—diagram of source point and measuring point mesh refinement;d—diagram of mesh generation

Comparison of analytical and numerical solutions of underground sphere

Relative error of analytical and numerical solutions

Cuboid model under flat terrain
a—xoy plane view of the model;b—xoz cross section view of the model;c—3D abnormal body model diagram;d—diagram of mesh refinement at measuring point;e—forward mesh;f—inversion mesh

The relationship between the number of iterations and RMS

Ω · m); f—3D view of inversion results with inequality constraints(resistivity is less than 50

Ω · m

)
">

3D inversion results
a—traditional resistivity inversion results:horizontal slice xoy at z=-7 m; b—traditional resistivity inversion results:cross-section slice xoz at y=25 m; c—inversion results with inequality constraints:horizontal slice xoy at =-7 m; d—inversion results with inequality constraints:cross-section slice xoz at y=25 m; e—3D view of traditional resistivity inversion results(resistivity is less than 50

Ω · m

); f—3D view of inversion results with inequality constraints(resistivity is less than 50

Ω · m

)

长方体编号	电阻率/ $(Ω · m)$	尺寸/(m×m×m)
S1	200	$100 × 100 × 40$
S2	100	$100 × 100 × 40$
S3	2000	$350 × 50 × 40$
B1	2000	$350 × 100 × 200$
B2	100	$100 × 150 × 180$

Cuboid parameter

Diagram of abnormal body model location and mesh generation
a—location of 3D abnormal model;b—observation system of surface data;c—cross-section profile yoz; d—diagram of unstructured refinement mesh

Traditional regularized inversion and constrained inversion results of well-surface with pole-pole device
a—traditional resistivity inversion results:cross-section profile yoz at x=400 m; b—traditional resistivity inversion results:cross-section profile yoz at x=600 m;c—constrained inversion: cross-section profile yoz at x=400 m;d—constrained inversion:cross-section profile yoz at x=600 m

[1]	柳建新, 赵然, 郭振威. 电磁法在金属矿勘查中的研究进展[J]. 地球物理学进展, 2019, 34(1):151-160.
[1]	Liu J X, Zhao R, Guo Z W. Research progress of electromagnetic methods in the exploration of metal deposits[J]. Progress in Geophysics, 2019, 34(1):151-160.
[2]	吕庆田, 张晓培, 汤井田, 等. 金属矿地球物理勘探技术与设备:回顾与进展[J]. 地球物理学报, 2019, 62(10):3629-3664.
[2]	Lyu Q T, Zhang X P, Tang J T, et al. Review on advancement in technology and equipment of geophysical exploration for metallic deposits in China[J]. Chinese Journal of Geophysics, 2019, 62(10):3629-3664.
[3]	吴小平, 刘洋, 王威. 基于非结构网格的电阻率三维带地形反演[J]. 地球物理学报, 2015, 58(8):2706-2717.
[3]	Wu X P, Liu Y, Wang W. 3D resistivity inversion incorporating topography based on unstructured meshes[J]. Chinese Journal of Geophysics, 2015, 58(8):2706-2717.
[4]	王智, 吴爱平, 李刚. 起伏地表条件下的井中激电井地观测正演模拟研究[J]. 石油物探, 2018, 57(6):927-935, 951.
[4]	Wang Z, Wu A P, Li G. Forward modeling of borehole-ground induced polarization method under undulating topography[J]. Geophyscial Prospecting for Petroleum, 2018, 57(6):927-935,951.
[5]	潘和平. 井中激发极化法在矿产资源勘探中的作用[J]. 物探与化探, 2013, 37(4):620-626.
[5]	Pan H P. The Role of Borehole induced polarization/resistivity method in the exploration of mineral resoueces[J]. Geophysics and Geochemical Exploration, 2013, 37(4):620-626.
[6]	汤井田, 张继锋, 冯兵, 等. 井地电阻率法歧离率确定高阻油气藏边界[J]. 地球物理学报, 2007, 50(3):926-931.
[6]	Tang J T, Zhang J F, Feng B, et al. Detemination of borders for resistive oil and gas reservoirs by deviation rate using the hole-to-surface resistivity method[J]. Chinese Journal of Geophysics, 2007, 50(3):926-931.
[7]	黄俊革, 王家林, 阮百尧. 三维高密度电阻率E-SCAN法有限元模拟异常特征研究[J]. 地球物理学报, 2006, 49(4):1206-1214.
[7]	Huang J G, Wang J L, Ruan B Y. A study on FEM modeling of anomalies of 3-D high-density E-SCAN resistivity survey[J]. Chinese Journal of Geophysics, 2006, 49(4):1206-1214.
[8]	Li Y, Spitzer K. Finite element resistivity modelling for three-dimensional structures with arbitrary anisotropy[J]. Physics of the Earth & Planetary Interiors, 2005, 150(1-3):15-27.
[9]	Wu X. A 3-D finite-element algorithm for DC resistivity modelling using the shifted incomplete Cholesky conjugate gradient method[J]. Geophysical Journal International, 2003, 154(3):947-956.
[10]	吴小平, 汪彤彤. 利用共轭梯度算法的电阻率三维有限元正演[J]. 地球物理学报, 2003, 46(3):428-432.
[10]	Wu X P, Wang T T. A 3-D finite-element resistivity forward modeling using conjugate gradient algorithm[J]. Chinese Journal of Geophysics, 2003, 46(3):428-432.
[11]	Li Y, Klaus S. Three-dimensional DC resistivity forward modelling using finite elements in comparison with finite-difference solutions[J]. Geophysical Journal International, 2002, 51(3):924-934.
[12]	黄俊革, 阮百尧, 鲍光淑. 齐次边界条件下三维地电断面电阻率有限元数值模拟法[J]. 桂林工学院学报, 2002, 22(1):11-14.
[12]	Huang J G, Ruan B Y, Bao G S. Fem under quantic-boundary condition for modeling resistivity on 3-D geoelectric section[J]. Journal of Guilin Institute of Technology, 2002, 22(1):11-14.
[13]	Ren Z, Qiu L, Tang J, et al. 3-D direct current resistivity anisotropic modelling by goal-oriented adaptive finite element methods[J]. Geophysical Journal International, 2018, 212(1):76-87.
[14]	Ren Z, Tang J. A goal-oriented adaptive finite-element approach for multi-electrode resistivity system[J]. Geophysical Journal International, 2014, 199(1):136-145.
[15]	Wei W, Xiaoping W, Spitzer K. Three-dimensional DC anisotropic resistivity modelling using finite elements on unstructured grids[J]. Geophysical Journal International, 2013, 193(2):734-746.
[16]	Ren Z, Jingtian T. 3D direct current resistivity modeling with unstructured mesh by adaptive finite-element method[J]. Geophysics, 2010, 75(1):H7-H17.
[17]	Blome M, Maurer H R, Schmidt K. Advances in three-dimensional geoelectric forward solver techniques[J]. Geophysical Journal International, 2009, 176(3):740-752.
[18]	Rücker C, Günther T, Spitzer K. Three-dimensional modelling and inversion of dc resistivity data incorporating topography-I. Modelling[J]. Geophysical Journal International, 2006, 166(2):495-505.
[19]	Günther T, Rücker C, Spitzer K. Three-dimensional modelling and inversion of DC resistivity data incorporating topography-II. Inversion[J]. Geophysical Journal of the Royal Astronomical Society, 2006, 166(2):506-517.
[20]	Zhou B, Greenhalgh S A. Finite element three-dimensional direct current resistivity modelling; accuracy and efficiency considerations[J]. Geophysical Journal International, 2001, 145(3):679-688.
[21]	Sasaki Y. 3-D resistivity inversion using finite-element method[J]. Geophysics, 1994, 59(12):1839.
[22]	Wu X, Xiao Y, Qi C, et al. Computations of secondary potential for 3-D DC resistivity modelling using an incomplete Choleski conjugate-gradient method[J]. Geophysical Prospecting, 2003, 51(6):567-577.
[23]	吴小平, 徐果明. 利用ICCG迭代技术加快电阻率三维正演计算[J]. 煤田地质与勘探, 1999, 27(3):63-67.
[23]	Wu X P, Xu G M. 3-D resistivity forward calculation accelerated by ICCG iteration technique[J]. Coal Geology and Exploration, 1999, 27(3):63-67.
[24]	吴小平, 徐果明, 李时灿. 利用不完全Cholesky共轭梯度法求解点源三维地电场[J]. 地球物理学报, 1998, 41(6):848-855.
[24]	Wu X P, Xu G M, Li S C. The calculation of three-dimensional geoelectric field of point source by incomplete cholesky conjugate gradient method[J]. Chinese Journal of Geophysics, 1998, 41(6):848-855.
[25]	Zhao S, Yedlin M J. Some refinements on the finite-difference method for 3-D dc resistivity modeling[J]. Geophysics, 1996, 61(5):1301-1307.
[26]	Zhang J, Mackie R L, Madden T R. 3-D resistivity forward modeling and inversion using conjugate gradients[J]. Geophysics, 1995, 60(5):1313-1325.
[27]	Spitzer K. A 3-D finite-difference algorithm for DC resistivity modelling using conjugate gradient methods[J]. Geophysical Journal International, 1995, 123(3):903-914.
[28]	汤井田, 王飞燕, 任政勇. 基于非结构化网格的2.5-D直流电阻率自适应有限元数值模拟[J]. 地球物理学报, 2010, 53(3):708-716.
[28]	Tang J T, Wang F Y, Ren Z Y. 2.5-D DC resistivity modeling by adaptive finite-element method with unstructured triangulation[J]. Chinese Journal of Geophysics, 2010, 53(3):708-716.
[29]	任政勇, 汤井田. 基于局部加密非结构化网格的三维电阻率法有限元数值模拟[J]. 地球物理学报, 2009, 52(10):2627-2634.
[29]	Ren Z Y, Tang J T. Finite element modeling of 3-D DC resistivity using locally refined unstructured meshes[J]. Chinese Journal of Geophysics, 2009, 52(10):2627-2634.
[30]	彭荣华, 胡祥云, 李建慧, 等. 频率域海洋可控源电磁垂直各向异性三维反演[J]. 地球物理学报, 2019, 62(6):2165-2175.
[30]	Peng R H, Hu X Y, Li J H, et al. 3D inversion of frequency-domain marine CSEM data in VTI media[J]. Chinese Journal of Geophysics, 2019, 62(6):2165-2175.
[31]	Gundogdu N Y, Candansayar M E. Three-dimensional regularized inversion of DC resistivity data with different stabilizing functionals[J]. Geophysics, 2018, 83(6):E399-E407.
[32]	郭来功, 戴广龙, 杨本才, 等. 多先验信息约束的三维电阻率反演方法[J]. 石油地球物理勘探, 2018, 53(6):1333-1340.
[32]	Guo L G, Dai G L, Yang B C. et al. 3D resistivity inversion with multiple priori-information constraint[J]. Oil Geophysical Prospecting, 2018, 53(6):1333-1340.
[33]	彭荣华, 胡祥云, 韩波. 基于高斯牛顿法的频率域可控源电磁三维反演研究[J]. 地球物理学报, 2016, 59(9):3470-3481.
[33]	Peng R H, Hu X Y, Han B. 3D inversion of frequency-domain CSEM data based on Gauss-Newton optimization[J]. Chinese Journal of Geophysics, 2016, 59(9):3470-3481.
[34]	Oldenburg D W, Haber E, Shekhtman R. Three dimensional inversion of multisource time domain electromagnetic data[J]. Geophysics, 2013, 78(1):E47-E57.
[35]	Pidlisecky A, Haber E, Knight R J. RESINVM3D: A 3D resistivity inversion package[J]. Geophysics, 2007, 72(2):H1-H10.
[36]	吴小平, 徐果明. 利用共轭梯度法的电阻率三维反演研究[J]. 地球物理学报, 2000, 43(3):420-427.
[36]	Wu X P, Xu G M. Study on 3-D resistivity inversion using conjugate gradient method[J]. Chinese Journal of Geophysics, 2000, 43(3):420-427.
[37]	徐凯军, 李桐林, 张辉, 等. 基于共轭梯度法的垂直有限线源三维电阻率反演[J]. 煤田地质与勘探, 2006, 34(3):68-71.
[37]	Xu K J, Li T L, Zhang H, et al. 3D resistivity inversion of vertical finite line source using conjugate gradients[J]. Coal Geology and Exploration, 2006, 34(3):68-71.
[38]	Cao X Y, Huang X, Yin C C, et al. 3D MT anisotropic inversion based on unstructured finite-element method[J]. Journal of Environmental & Engineering Geophysics, 2021, 26(1):49-60.
[39]	惠哲剑, 殷长春, 刘云鹤, 等. 基于非结构有限元的时间域海洋电磁三维反演[J]. 地球物理学报, 2020, 63(8):3167-3179.
[39]	Hui Z J, Yin C C, Liu Y H, et al. 3D inversion of time-domain marine EM data based on unstructured finite-element method[J]. Chinese Journal of Geophysics, 2020, 63(8):3167-3179.
[40]	余辉, 邓居智, 陈辉, 等. 起伏地形下大地电磁L-BFGS三维反演方法[J]. 地球物理学报, 2019, 62(8):3175-3188.
[40]	Yu H, Deng J Z, Chen H, et al. Three-dimensional magnetotelluric inversion under topographic relief based on the limited-memory quasi-Newton algorithm(L_BFGS)[J]. Chinese Journal of Geophysics, 2019, 62(8):3175-3188.
[41]	邓琰, 汤吉, 阮帅. 三维大地电磁自适应正则化有限内存拟牛顿反演[J]. 地球物理学报, 2019, 62(9):3601-3614.
[41]	Deng Y, Tang J, Ruan S. Adaptive regularized three-dimensional magnetotelluric inversion based on the LBFGS quasi-Newton method[J]. Chinese Journal of Geophysics, 2019, 62(9):3601-3614.
[42]	殷长春, 朱姣, 邱长凯, 等. 航空电磁拟三维模型空间约束反演[J]. 地球物理学报, 2018, 61(6):2537-2547.
[42]	Yin C C, Zhu J, Qiu C K, et al. Spatially constrained inversion for airborne EM data using quasi-3D models[J]. Chinese Journal of Geophysics, 2018, 61(6):2537-2547.
[43]	秦策, 王绪本, 赵宁. 基于二次场方法的并行三维大地电磁正反演研究[J]. 地球物理学报, 2017, 60(6):2456-2468.
[43]	Qin C, Wang X B, Zhao N. Parallel three-dimensional forward modeling and inversion of magnetotelluric based on a secondary field approach[J]. Chinese Journal of Geophysics, 2017, 60(6):2456-2468.
[44]	赵宁, 王绪本, 秦策, 等. 三维频率域可控源电磁反演研究[J]. 地球物理学报, 2016, 59(1):330-341.
[44]	Zhao N, Wang X B, Qin C, et al. 3D frequency-domain CSEM inversion[J]. Chinese Journal of Geophysics, 2016, 59(1):330-341.
[45]	刘云鹤, 殷长春. 三维频率域航空电磁反演研究[J]. 地球物理学报, 2013, 56(12):4278-4287.
[45]	Liu Y H, Yin C C. 3D inversion for frequency-domain HEM data[J]. Chinese Journal of Geophysics, 2013, 56(12):4278-4287.
[46]	Avdeev D, Avdeeva A. 3D Magnetotelluric inversion using a limited-memory quasi-Newton optimization[J]. Geophysics, 2009, 74(3):F45.
[47]	Xiao Y, Wei Z, Wang Z. A limited memory BFGS-type method for large-scale unconstrained optimization[J]. Computers & Mathematics with Applications, 2008, 56(4):1001-1009.
[48]	Eldad, Haber, Douglas, et al. Inversion of time domain three-dimensional electromagnetic data[J]. Geophysical Journal International, 2007, 171(2):B23-B34.
[49]	Avdeeva A, Avdeev D B. A limited memory quasi-Newton inversion for 1D magnetotellurics[J]. Geophysics, 2006, 71(5):G191-G196.
[50]	Haber E. Quasi-Newton methods for large-scale electromagnetic inverse problems[J]. Inverse Problems, 2005, 21(1):305-323.
[51]	Newman G A, Boggs P T. Solution accelerators for large-scale 3D electromagnetic inverse problems[J]. Inverse Problems, 2004, 20(6):S151-S170.
[52]	Nash S G, Nocedal J. A numerical study of the limited memory BFGS method and the truncated-Newton method for large scale optimization[J]. SIAM Journal on Optimization, 1991, 1(3):358-372.
[53]	Nocedal J. Updating quasi-Newton matrices with limited storage[J]. Mathematics of Computation, 1980, 35(151):773-782.
[54]	唐传章, 程见中, 严良俊, 等. 基于边界约束有限内存的拟牛顿CSAMT一维反演及应用[J]. 煤田地质与勘探, 2019, 47(5):193-200.
[54]	Tang C Z, Cheng J Z, Yan L J, et al. LBFGS CSAMT 1D inversion of limited memory based on boundary constraint and its application[J]. Coal Geology and Exploration, 2019, 47(5):193-200.
[55]	马欢, 郭越, 吴萍萍, 等. 基于MPI并行算法的电阻率法多种装置数据的三维联合反演[J]. 地球物理学报, 2018, 61(12):5052-5065.
[55]	Ma H, Guo Y, Wu P P, et al. 3-D joint inversion of multi-array data set in the resistivity method based on MPI parallel algorithm[J]. Chinese Journal of Geophysics, 2018, 61(12):5052-5065.
[56]	董浩, 魏文博, 叶高峰, 等. 基于有限差分正演的带地形三维大地电磁反演方法[J]. 地球物理学报, 2014, 57(3):939-952.
[56]	Dong H, Wei W B, Ye G F, et al. Study of Three-dimensional magnetotelluric inversion including surface topography based on Finite-difference method[J]. Chinese Journal of Geophysics, 2014, 57(3):939-952.
[57]	张昆, 董浩, 严加永, 等. 一种并行的大地电磁场非线性共轭梯度三维反演方法[J]. 地球物理学报, 2013, 56(11):3922-3931.
[57]	Zhang K, Dong H, Yan J Y, et al. A NLCG inversion method of magnetotellurics with parallel structure[J]. Chinese Journal of Geophysics, 2013, 56(11):3922-3931.
[58]	林昌洪, 谭捍东, 舒晴, 等. 可控源音频大地电磁三维共轭梯度反演研究[J]. 地球物理学报, 2012, 55(11):3829-3838.
[58]	Lin C H, Tan H D, Shu Q, et al. Three-dimensional conjugate gradient inversion of CSAMT data[J]. Chinese Journal of Geophysics, 2012, 55(11):3829-3838.
[59]	胡祖志, 胡祥云, 何展翔. 大地电磁非线性共轭梯度拟三维反演[J]. 地球物理学报, 2006, 49(4):1226-1234.
[59]	Hu Z Z, Hu X Y, He Z X. Pseudo-three-dimensional magnetotelluric using nonlinear conjugate gradients[J]. Chinese Journal of Geophysics, 2006, 49(4):1226-1234.
[60]	Rodi W L, Mackie R L. Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion[J]. Geophysics, 2001, 66(1):174-187.
[61]	Newman G A, Alumbaugh D L. Three-dimensional magnetotelluric inversion using non-linear conjugate gradients[J]. Geophysical Journal International, 2000, 140(2):410-424.
[62]	王智, 潘和平, 骆玉虎, 等. 基于不等式约束的井地电阻率法三维非线性共轭梯度反演研究[J]. 地球物理学进展, 2016, 31(1):360-370.
[62]	Wang Z, Pan H P, Luo Y H, et al. 3-D hole-to-surface resistivity inversion with nonlinear conjugate gradients method under the constraint of inequality[J]. Progress in Geophysics, 2016, 31(1):360-370.
[63]	王智, 潘和平, 吴爱平, 等. 基于不等式约束的井中激电三维反演研究[J]. 石油物探, 2016, 55(3):455-466.
[63]	Wang Z, Pan H P, Wu A P, et al. 3D inversion of borehole induced polarization under the inequality constraint[J]. Geophysical Prospecting for Petroleum, 2016, 55(3):455-466.
[64]	Kim H J, Song Y, Lee K H. Inequality constraint in least-squares inversion of geophysical data[J]. Earth, Planets and Space, 1999, 51(4):255-259.
[65]	Li Y, Oldenburg D W. 3-D Inversion of induced polarization data[J]. Geophysics, 2000, 65(6):1931-1945.
[66]	黄俊革, 阮百尧, 鲍光淑. 基于有限单元法的三维地电断面电阻率反演[J]. 中南大学学报:自然科学版, 2004, 35(2):295-299.
[66]	Huang J G, Ruan B Y, Bao G S. Resistivity inversion on 3-D section based on FEM[J]. Journal of Central South University:Science and Technology, 2004, 35(2):295-299.
[67]	宛新林, 席道瑛, 高尔根, 等. 用改进的光滑约束最小二乘正交分解法实现电阻率三维反演[J]. 地球物理学报, 2005, 48(2):439-444.
[67]	Wan X L, Xi D Y, Gao E G, et al. 3-D resistivity inversion by the least-squares QR factorization method under improved smoothness constraint condition[J]. Chinese Journal of Geophysics, 2005, 48(2):439-444.
[68]	刘斌, 李术才, 聂利超, 等. 基于自适应加权光滑约束与PCG算法的三维电阻率探测反演成像[J]. 岩土工程学报, 2012, 34(9):1646-1653.
[68]	Liu B, Li S C, Nie L C, et al. Inversion imaging of 3D resistivity detection using adaptive-weighted smooth constraint and PCG algorithm[J]. Chinese Journal of Geotechnical Engineering, 2012, 34(9):1646-1653.
[69]	刘斌, 李术才, 李树忱, 等. 基于不等式约束的最小二乘法三维电阻率反演及其算法优化[J]. 地球物理学报, 2012, 55(1):260-268.
[69]	Liu B, Li S C, Li S C, et al. 3D electrical resistivity inversion with least-squares method based on inequality constraint and its computation effciency optimization[J]. Chinese Journal of Geophysics, 2012, 55(1):260-268.
[70]	刘斌, 聂利超, 李术才, 等. 三维电阻率空间结构约束反演成像方法[J]. 岩石力学与工程学报, 2012, 31(11):2258-2268.
[70]	Liu B, Nie L C, Li S C, et al. 3D electrical resistivity inversion tomography with spatial structural constraint[J]. Chinese Journal of Rock Mechanics and Engineering, 2012, 31(11):2258-2268.
[71]	底青云, 薛国强, 殷长春, 等. 中国人工源电磁探测新方法[J]. 中国科学:地球科学, 2020, 50(9):1219-1227.
[71]	Di Q Y, Xue G Q, Yin C C, et al. New methods of controlled-source electromagnetic detection in China[J]. Science China Earth Sciences, 2020, 50(9):1219-1227.
[72]	殷长春, 刘云鹤, 熊彬. 地球物理三维电磁反演方法研究动态[J]. 中国科学:地球科学, 2020, 50(3):432-435.
[72]	Yin C C, Liu Y H, Xiong B. Status and prospect of 3D inversions in EM geophysic[J]. Science China Earth Sciences, 2020, 50(3):432-435.
[73]	Nocedal J, Wright S J, Mikosch T V, et al. Numerical Optimization[M]. Berlin:Springer, 1999.
[74]	徐世浙. 地球物理中的有限单元法[M]. 北京: 科学出版社, 1994.
[74]	Xu S Z. FEM in geophysics[M]. Beijing: Science Press, 1994.
[75]	韩波, 胡祥云, 何展翔, 等. 大地电磁反演方法的数学分类[J]. 石油地球物理勘探, 2012, 47(1):177-188.
[75]	Han B, Hu X Y, He Z X, et al. Mathematical classification of magnetotelluric inversion methods[J]. Oil Geophysical Prospecting, 2012, 47(1):177-188.
[76]	Siripunvaraporn W. Three-dimensional magnetotelluric inversion: An introductory guide for developers and users[J]. Surveys in Geophysics, 2012, 33(1):5-27.
[77]	Geuzaine C, Remacle J F. Gmsh: A three-dimensional finite element mesh generator with built-in pre-and post-processing facilities[J]. International Journal for Numerical Methods in Engineering, 2009, 79(1):1309-1331.
[78]	Ahrens J, Geveci B, Law C. ParaView: An end-user tool for large data visualization[M]. New York: Academic Press, 2005.
[79]	Ellis R G, Oldenburg D W. The pole-pole 3-D Dc-resistivity inverse problem:A conjugategradient approach[J]. Geophysical Journal of the Royal Astronomical Society, 1994, 119(1):187-194.
[80]	Li Y, Oldenburg D W. Inversion of 3-D DC resisitivity data using an approximate inverse mapping[J]. Geophysical Journal International, 1994, 116(4):527-537.