起伏地形下大地电磁L-BFGS三维反演方法
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摘要
为推进大地电磁三维反演的实用化,本文实现了基于L-BFGS算法的带地形大地电磁三维反演.首先推导了大地电磁法三维反演的Tikhonov正则化目标函数以及Hessian矩阵逆矩阵近似表达式和计算方法,然后设计了一种既能保证空气电阻率固定不变又能保证模型平滑约束的协方差矩阵统一表达式,解决带地形反演问题.在反演算法中采用正则化因子冷却法以及基于Wolf条件的步长搜索策略,提升了反演的稳定性.利用开发的算法对多个带地形地电模型(山峰地形下的单个异常模型、峰-谷地形下的棋盘模型)的合成数据进行了三维反演,并与已有大地电磁三维反演程序(ModEM)进行对比,验证了本文开发的三维反演算法的正确性和可靠性.最后,利用该算法反演了华南某山区大地电磁实测数据,得到该区三维电性结构,揭示了研究区以高阻介质为基底,中间以低阻不整合面和相对低阻介质连续分布,浅部覆盖高阻介质的电性结构特征,进一步验证了本文算法的实用性.
We present an algorithm of three-dimensional(3D)magnetotelluric(MT)inversion under topographic relief using a limited-memory quasi-Newton optimization based on BroydenFletcher-Goldfarb-Shanno(BFGS)formula in this study.Firstly,we establish the approximate expression and calculation of the inverse of Hessian matrix for 3DMT inversion based on iterative minimization of a classical Tikhonov regularized penalty function.A covariance matrix is designed to keep the resistivity of air unchanged and to impose the smoothness of the model while implementing the inversion with topographic relief.In addition,the cooling method for the regularization factor and a strategy to search for the step length based on the Wolf condition are adopted in this algorithm to improve the stability of the inversion.Secondly,we make a 3D inversion of synthetic data which are generated utilizing a series of conductivity structure models with topographic relief and compare the results with that of the well-known 3D MT inversion program(ModEM)to verify the correctness and reliability of the algorithm in this paper.Finally,we use this 3Dalgorithm to invert observed MT data with topographic relief from a mountainous survey area in South China,yielding an electrical structure model.It can be roughly divided into three layers,i.e.a high-resistivity basement,a conductive layer and the unconformity surface in the middle and the sedimentary cover with high-resistivity.The applicability of this algorithm is further verified.
We present an algorithm of three-dimensional(3D)magnetotelluric(MT)inversion under topographic relief using a limited-memory quasi-Newton optimization based on BroydenFletcher-Goldfarb-Shanno(BFGS)formula in this study.Firstly,we establish the approximate expression and calculation of the inverse of Hessian matrix for 3DMT inversion based on iterative minimization of a classical Tikhonov regularized penalty function.A covariance matrix is designed to keep the resistivity of air unchanged and to impose the smoothness of the model while implementing the inversion with topographic relief.In addition,the cooling method for the regularization factor and a strategy to search for the step length based on the Wolf condition are adopted in this algorithm to improve the stability of the inversion.Secondly,we make a 3D inversion of synthetic data which are generated utilizing a series of conductivity structure models with topographic relief and compare the results with that of the well-known 3D MT inversion program(ModEM)to verify the correctness and reliability of the algorithm in this paper.Finally,we use this 3Dalgorithm to invert observed MT data with topographic relief from a mountainous survey area in South China,yielding an electrical structure model.It can be roughly divided into three layers,i.e.a high-resistivity basement,a conductive layer and the unconformity surface in the middle and the sedimentary cover with high-resistivity.The applicability of this algorithm is further verified.
引文
Avde ev D,Avdeeva A.2009.3Dmagnetotelluric inversion using a limited-memory quasi-Newton optimization.Geophysics,74(3):F45-F57.
Avdeev D B.2005.Three-dimensional electromagnetic modelling and inversion from theory to application.Surveys in Geophysics,26(6):767-799.
Bedrosian P A,Feucht D W.2014.Structure and tectonics of the northwestern United States from EarthScope USArray magnetotelluric data.Earth and Planetary Science Letters,402:275-289.
Beka T I,Smirnov M,Birkelund Y,et al.2016.Analysis and 3Dinversion of magnetotelluric crooked profile data from central Svalbard for geothermal application.Tectonophysics,686:98-115.
Borner R U.2010.Numerical modelling in geo-electromagnetics:advances and challenges.Surveys in Geophysics,31(2):225-245.
Cagniard L.1953.Basic theory of the magneto-telluric method of geophysical prospecting.Geophysics,18(3):605-635.
Cao X Y,Yin C C,Zhang B,et al.2018.A goal-oriented adaptive finite-element method for 3D MT anisotropic modeling with topography.Chinese Journal of Geophysics(in Chinese),61(6):2618-2628,doi:10.6038/cjg2018L0068.
Chen H,Yin M,Yin C C,et al.2018.Three-dimensional magnetotelluric modelling using aggregation-based algebraic multigrid method.Journal of Jilin University(Earth Science Edition)(in Chinese),48(1):261-270.
Chen P F,Hou Z Z,Fan G H.1998.Three-dimensional topographic responses in MT using finite difference method.Acta Seismologica Sinica(English Edition),11(5):631-635.
Chen X,Yu P,Zhang L L,et al.2011.Adaptive regularized synchronous joint inversion of MT and seismic data.Chinese Journal of Geophysics(in Chinese),54(10):2673-2681,doi:10.3969/j.issn.0001-5733.2011.10.024.
Deng J Z,Chen H,Yin C C,et al.2015.Three-dimensional electrical structures and significance for mineral exploration in the Jiujiang-Ruichang District.Chinese Journal of Geophysics(in Chinese),58(12):4465-4477,doi:10.6038/cjg20151211.
Dong H,Wei W B,Ye G F,et al.2014.Study of Three-dimensional magnetotelluric inversion including surface topography based on Finite-difference method.Chinese Journal of Geophysics(in Chinese),57(3):939-952,doi:10.6038/cjg20140323.
Dong H,Wei W B,Jin S,et al.2016.Extensional extrusion:Insights into south-eastward expansion of Tibetan Plateau from magnetotelluric array data.Earth and Planetary Science Letters,454:78-85.
Egbert G D,Kelbert A.2012.Computational recipes for electromagnetic inverse problems.Geophysical Journal International,189(1):251-267.
Gürer A,Ilkisik O M.1997.The importance of topographic corrections on magnetotelluric response data from rugged regions of anatolia.Geophysical Prospecting,45(1):111-125.
Grayver A V,Kolev T V.2015.Large-scale 3Dgeoelectromagnetic modeling using parallel adaptive high-order finite element method.Geophysics,80(6):E277-E291.
Haber E,Ascher U M.2001.Fast finite volume simulation of 3Delectromagnetic problems with highly discontinuous coefficients.SIAM Journal on Scientific Computing,22(6):1943-1961.
Haber E,Ascher U M,Oldenburg D W.2004.Inversion of 3Delectromagnetic data in frequency and time domain using an inexact all-at-once approach.Geophysics,69(5):1216-1228.
Haber E,Ruthotto L.2014.A multiscale finite volume method for Maxwell's equations at low frequencies.Geophysical Journal International,199(2):1268-1277.
Henson V E,Yang U M.2002.BoomerAMG:A parallel algebraic multigrid solver and preconditioner.Applied Numerical Mathematics,41(1):155-177.
Jahandari H,Farquharson C G.2017.3-D minimum-structure inversion of magnetotelluric data using the finite-element method and tetrahedral grids.Geophysical Journal International,211(2):1189-1205.
Jiracek G R.1990.Near-surface and topographic distortions in electromagnetic induction.Surveys in Geophysics,11(2-3):163-203.
Kordy M,Wannamaker P,Maris V,et al.2016.3-dimensional magnetotelluric inversion including topography using deformed hexahedral edge finite elements and direct solvers parallelized on symmetric multiprocessor computers-Part II:direct dataspace inverse solution.Geophysical Journal International,204(1):94-110.
Liu Y H,Yin C C.2013.3Dinversion for frequency-domain HEMdata.Chinese Journal of Geophysics(in Chinese),56(12):4278-4287,doi:10.6038/cjg20131230.
Liu Y H,Yin C C.2013.Electromagnetic divergence correction for3Danisotropic EM modeling.Journal of Applied Geophysics,96:19-27.
Mansoori I,Oskooi B,Pedersen L B.2015.Magnetotelluric signature of anticlines in Iran′s Sehqanat oil field.Tectonophysics,654:101-112.10.1016/j.tecto.2015.05.004.
Meqbel N M,Egbert G D,Wannamaker P E,et al.2014.Deep electrical resistivity structure of the northwestern U.S.derived from 3-D inversion of USArray magnetotelluric data.Earth and Planetary Science Letters,402:290-304.
Mohan K,Kumar G P,Chaudhary P,et al.2017.Magnetotelluric investigations to identify geothermal source zone near Chabsar hotwater spring site,Ahmedabad,Gujarat,Northwest India.Geothermics,65:198-209.
Nam M J,Kim H J,Song Y,et al.2008.Three-dimensional topography corrections of magnetotelluric data.Geophysical Journal International,174(2):464-474.
Newman G A,Alumbaugh D L.2002.Three-dimensional magnetotelluric inversion using non-linear conjugate gradients.Geophysical Journal International,140(2):410-424.
Newman G A,Boggs P T.2004.Solution accelerators for large-scale three-dimensional electromagnetic inverse problems.Inverse Problems,20(6):S151-S170.
Qin C,Wang X B,Zhao N.2017.Parallel three-dimensional forward modeling and inversion of magnetotelluric based on a secondary field approach.Chinese Journal of Geophysics(in Chinese),60(6):2456-2468,doi:10.6038/cjg20170633.
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.
Rosenkjaer G K,Gasperikova E,Newman G A,et al.2015.Comparison of 3D MT inversions for geothermal exploration:Case studies for Krafla and Hengill geothermal systems in Iceland.Geothermics,57:258-274.
Roux E,Moorkamp M,Jones A G,et al.2011.Joint inversion of long-period magnetotelluric data and surface-wave dispersion curves for anisotropic structure:Application to data from Central Germany.Geophysical Research Letters,38(5):L05304,doi:10.1029/2010GL046358.
Ruan S.2015.Three-dimensional Magnetotelluric Inversion Based on Limited Memory Quasi-Newton Method(Doctorate thesis).Chengdu University of Technology.
Sarvandani M M,Kalateh A N,Unsworth M,et al.2017.Interpretation of magnetotelluric data from the Gachsaran oil field using sharp boundary inversion.Journal of Petroleum Science and Engineering,149:25-39.10.1016/j.petrol.2016.10.019.
Siripunvaraporn W,Egbert G,Lenbury Y,et al.2005.Threedimensional magnetotelluric inversion:data-space method.Physics of the Earth and Planetary Interiors,150(1-3):3-14.
Siripunvaraporn W.2012.Three-dimensional magnetotelluric inversion:an introductory guide for developers and users.Surveys in Geophysics,33(1):5-27.
Tikhonov A N.1950.On determining electrical characteristics of the deep layers of earth′s crust.Doklady Akademii Nauk,73(2):295-297.
Usui Y.2015.3-D inversion of magnetotelluric data using unstructured tetrahedral elements:applicability to data affected by topography.Geophysical Journal International,202(2):828-849.
Vallianatos F.2002.A note on the topographic distortion of magnetotelluric impedance.Annals of Geophysics,45(2):313-320.
Wannamaker P E,Stodt J A,Rijo L.1986.Two-dimensional topographic responses in magnetotellurics modeled using finite elements.Geophysics,51(11):2131-2144.
Wei W B,Jin S,Ye G F,et al.2003.Methods to study electrical conductivity of continental lithosphere.Earth Science Frontiers(in Chinese),10(1):15-23.
Xu S Z,Ruan B Y,Zhou H,et al.1997.Numerical modeling of 3-D terrain effect on MT field.Science in China Series D:Earth Sciences,40(3):269-275.
Yang D K,Oldenburg D W.2016.Survey decomposition:Ascalable framework for 3D controlled-source electromagnetic inversion.Geophysics,81(2):E69-E87.
Yin Y T,Jin S,Wei W B,et al.2017.Lithospheric rheological heterogeneity across an intraplate rift basin(Linfen Basin,North China)constrained from magnetotelluric data:Implications for seismicity and rift evolution.Tectonophysics,717:1-15.10.1016/j.tecto.2017.07.014.
Yoshimura R,Ogawa Y,Yukutake Y,et al.2018.Resistivity characterisation of Hakone volcano,Central Japan,by threedimensional magnetotelluric inversion.Earth,Planets and Space,70:66.
Zhang K,Yan J Y,LüQ T,et al.2017.Three-dimensional nonlinear conjugate gradient parallel inversion with full information of marine magnetotellurics.Journal of Applied Geophysics,139:144-157.
Zhang K,LüQ T,Yan J Y.2015.The lower crust conductor from Nanjing(Ning)-Wuhu(Wu)area in the Middle and Lower Reaches of Yangtze River:Preliminary results from 3Dinversion of mag netotelluric data.Journal of Asian Earth Sciences,101:20-29.
Zhao G Z,Tang J,Zan Y,et al.2005.Relation between electricity structure of the crust and deformation of crustal blocks on the northeastern margin of Qinghai-Tibet Plateau.Science in China Ser.D Earth Sciences,48(10):1613-1626.
曹晓月,殷长春,张博等.2018.面向目标自适应有限元法的带地形三维大地电磁各向异性正演模拟.地球物理学报,61(6):2618-2628,doi:10.6038/cjg2018L0068.
陈辉,尹敏,殷长春等.2018.大地电磁三维正演聚集多重网格算法.吉林大学学报(地球科学版),48(1):261-270.
陈晓,于鹏,张罗磊等.2011.地震与大地电磁测深数据的自适应正则化同步联合反演.地球物理学报,54(10):2673-2681,doi:10.3969/j.issn.0001-5733.2011.10.024.
邓居智,陈辉,殷长春等.2015.九瑞矿集区三维电性结构研究及找矿意义.地球物理学报,58(12):4465-4477,doi:10.6038/cjg20151211.
董浩,魏文博,叶高峰等.2014.基于有限差分正演的带地形三维大地电磁反演方法.地球物理学报,57(3):939-952,doi:10.6038/cjg20140323.
刘云鹤,殷长春.2013.三维频率域航空电磁反演研究.地球物理学报,56(12):4278-4287,doi:10.6038/cjg20131230.
秦策,王绪本,赵宁.2017.基于二次场方法的并行三维大地电磁正反演研究.地球物理学报,60(6):2456-2468,doi:10.6038/cjg20170633.
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Avdeev D B.2005.Three-dimensional electromagnetic modelling and inversion from theory to application.Surveys in Geophysics,26(6):767-799.
Bedrosian P A,Feucht D W.2014.Structure and tectonics of the northwestern United States from EarthScope USArray magnetotelluric data.Earth and Planetary Science Letters,402:275-289.
Beka T I,Smirnov M,Birkelund Y,et al.2016.Analysis and 3Dinversion of magnetotelluric crooked profile data from central Svalbard for geothermal application.Tectonophysics,686:98-115.
Borner R U.2010.Numerical modelling in geo-electromagnetics:advances and challenges.Surveys in Geophysics,31(2):225-245.
Cagniard L.1953.Basic theory of the magneto-telluric method of geophysical prospecting.Geophysics,18(3):605-635.
Cao X Y,Yin C C,Zhang B,et al.2018.A goal-oriented adaptive finite-element method for 3D MT anisotropic modeling with topography.Chinese Journal of Geophysics(in Chinese),61(6):2618-2628,doi:10.6038/cjg2018L0068.
Chen H,Yin M,Yin C C,et al.2018.Three-dimensional magnetotelluric modelling using aggregation-based algebraic multigrid method.Journal of Jilin University(Earth Science Edition)(in Chinese),48(1):261-270.
Chen P F,Hou Z Z,Fan G H.1998.Three-dimensional topographic responses in MT using finite difference method.Acta Seismologica Sinica(English Edition),11(5):631-635.
Chen X,Yu P,Zhang L L,et al.2011.Adaptive regularized synchronous joint inversion of MT and seismic data.Chinese Journal of Geophysics(in Chinese),54(10):2673-2681,doi:10.3969/j.issn.0001-5733.2011.10.024.
Deng J Z,Chen H,Yin C C,et al.2015.Three-dimensional electrical structures and significance for mineral exploration in the Jiujiang-Ruichang District.Chinese Journal of Geophysics(in Chinese),58(12):4465-4477,doi:10.6038/cjg20151211.
Dong H,Wei W B,Ye G F,et al.2014.Study of Three-dimensional magnetotelluric inversion including surface topography based on Finite-difference method.Chinese Journal of Geophysics(in Chinese),57(3):939-952,doi:10.6038/cjg20140323.
Dong H,Wei W B,Jin S,et al.2016.Extensional extrusion:Insights into south-eastward expansion of Tibetan Plateau from magnetotelluric array data.Earth and Planetary Science Letters,454:78-85.
Egbert G D,Kelbert A.2012.Computational recipes for electromagnetic inverse problems.Geophysical Journal International,189(1):251-267.
Gürer A,Ilkisik O M.1997.The importance of topographic corrections on magnetotelluric response data from rugged regions of anatolia.Geophysical Prospecting,45(1):111-125.
Grayver A V,Kolev T V.2015.Large-scale 3Dgeoelectromagnetic modeling using parallel adaptive high-order finite element method.Geophysics,80(6):E277-E291.
Haber E,Ascher U M.2001.Fast finite volume simulation of 3Delectromagnetic problems with highly discontinuous coefficients.SIAM Journal on Scientific Computing,22(6):1943-1961.
Haber E,Ascher U M,Oldenburg D W.2004.Inversion of 3Delectromagnetic data in frequency and time domain using an inexact all-at-once approach.Geophysics,69(5):1216-1228.
Haber E,Ruthotto L.2014.A multiscale finite volume method for Maxwell's equations at low frequencies.Geophysical Journal International,199(2):1268-1277.
Henson V E,Yang U M.2002.BoomerAMG:A parallel algebraic multigrid solver and preconditioner.Applied Numerical Mathematics,41(1):155-177.
Jahandari H,Farquharson C G.2017.3-D minimum-structure inversion of magnetotelluric data using the finite-element method and tetrahedral grids.Geophysical Journal International,211(2):1189-1205.
Jiracek G R.1990.Near-surface and topographic distortions in electromagnetic induction.Surveys in Geophysics,11(2-3):163-203.
Kordy M,Wannamaker P,Maris V,et al.2016.3-dimensional magnetotelluric inversion including topography using deformed hexahedral edge finite elements and direct solvers parallelized on symmetric multiprocessor computers-Part II:direct dataspace inverse solution.Geophysical Journal International,204(1):94-110.
Liu Y H,Yin C C.2013.3Dinversion for frequency-domain HEMdata.Chinese Journal of Geophysics(in Chinese),56(12):4278-4287,doi:10.6038/cjg20131230.
Liu Y H,Yin C C.2013.Electromagnetic divergence correction for3Danisotropic EM modeling.Journal of Applied Geophysics,96:19-27.
Mansoori I,Oskooi B,Pedersen L B.2015.Magnetotelluric signature of anticlines in Iran′s Sehqanat oil field.Tectonophysics,654:101-112.10.1016/j.tecto.2015.05.004.
Meqbel N M,Egbert G D,Wannamaker P E,et al.2014.Deep electrical resistivity structure of the northwestern U.S.derived from 3-D inversion of USArray magnetotelluric data.Earth and Planetary Science Letters,402:290-304.
Mohan K,Kumar G P,Chaudhary P,et al.2017.Magnetotelluric investigations to identify geothermal source zone near Chabsar hotwater spring site,Ahmedabad,Gujarat,Northwest India.Geothermics,65:198-209.
Nam M J,Kim H J,Song Y,et al.2008.Three-dimensional topography corrections of magnetotelluric data.Geophysical Journal International,174(2):464-474.
Newman G A,Alumbaugh D L.2002.Three-dimensional magnetotelluric inversion using non-linear conjugate gradients.Geophysical Journal International,140(2):410-424.
Newman G A,Boggs P T.2004.Solution accelerators for large-scale three-dimensional electromagnetic inverse problems.Inverse Problems,20(6):S151-S170.
Qin C,Wang X B,Zhao N.2017.Parallel three-dimensional forward modeling and inversion of magnetotelluric based on a secondary field approach.Chinese Journal of Geophysics(in Chinese),60(6):2456-2468,doi:10.6038/cjg20170633.
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.
Rosenkjaer G K,Gasperikova E,Newman G A,et al.2015.Comparison of 3D MT inversions for geothermal exploration:Case studies for Krafla and Hengill geothermal systems in Iceland.Geothermics,57:258-274.
Roux E,Moorkamp M,Jones A G,et al.2011.Joint inversion of long-period magnetotelluric data and surface-wave dispersion curves for anisotropic structure:Application to data from Central Germany.Geophysical Research Letters,38(5):L05304,doi:10.1029/2010GL046358.
Ruan S.2015.Three-dimensional Magnetotelluric Inversion Based on Limited Memory Quasi-Newton Method(Doctorate thesis).Chengdu University of Technology.
Sarvandani M M,Kalateh A N,Unsworth M,et al.2017.Interpretation of magnetotelluric data from the Gachsaran oil field using sharp boundary inversion.Journal of Petroleum Science and Engineering,149:25-39.10.1016/j.petrol.2016.10.019.
Siripunvaraporn W,Egbert G,Lenbury Y,et al.2005.Threedimensional magnetotelluric inversion:data-space method.Physics of the Earth and Planetary Interiors,150(1-3):3-14.
Siripunvaraporn W.2012.Three-dimensional magnetotelluric inversion:an introductory guide for developers and users.Surveys in Geophysics,33(1):5-27.
Tikhonov A N.1950.On determining electrical characteristics of the deep layers of earth′s crust.Doklady Akademii Nauk,73(2):295-297.
Usui Y.2015.3-D inversion of magnetotelluric data using unstructured tetrahedral elements:applicability to data affected by topography.Geophysical Journal International,202(2):828-849.
Vallianatos F.2002.A note on the topographic distortion of magnetotelluric impedance.Annals of Geophysics,45(2):313-320.
Wannamaker P E,Stodt J A,Rijo L.1986.Two-dimensional topographic responses in magnetotellurics modeled using finite elements.Geophysics,51(11):2131-2144.
Wei W B,Jin S,Ye G F,et al.2003.Methods to study electrical conductivity of continental lithosphere.Earth Science Frontiers(in Chinese),10(1):15-23.
Xu S Z,Ruan B Y,Zhou H,et al.1997.Numerical modeling of 3-D terrain effect on MT field.Science in China Series D:Earth Sciences,40(3):269-275.
Yang D K,Oldenburg D W.2016.Survey decomposition:Ascalable framework for 3D controlled-source electromagnetic inversion.Geophysics,81(2):E69-E87.
Yin Y T,Jin S,Wei W B,et al.2017.Lithospheric rheological heterogeneity across an intraplate rift basin(Linfen Basin,North China)constrained from magnetotelluric data:Implications for seismicity and rift evolution.Tectonophysics,717:1-15.10.1016/j.tecto.2017.07.014.
Yoshimura R,Ogawa Y,Yukutake Y,et al.2018.Resistivity characterisation of Hakone volcano,Central Japan,by threedimensional magnetotelluric inversion.Earth,Planets and Space,70:66.
Zhang K,Yan J Y,LüQ T,et al.2017.Three-dimensional nonlinear conjugate gradient parallel inversion with full information of marine magnetotellurics.Journal of Applied Geophysics,139:144-157.
Zhang K,LüQ T,Yan J Y.2015.The lower crust conductor from Nanjing(Ning)-Wuhu(Wu)area in the Middle and Lower Reaches of Yangtze River:Preliminary results from 3Dinversion of mag netotelluric data.Journal of Asian Earth Sciences,101:20-29.
Zhao G Z,Tang J,Zan Y,et al.2005.Relation between electricity structure of the crust and deformation of crustal blocks on the northeastern margin of Qinghai-Tibet Plateau.Science in China Ser.D Earth Sciences,48(10):1613-1626.
曹晓月,殷长春,张博等.2018.面向目标自适应有限元法的带地形三维大地电磁各向异性正演模拟.地球物理学报,61(6):2618-2628,doi:10.6038/cjg2018L0068.
陈辉,尹敏,殷长春等.2018.大地电磁三维正演聚集多重网格算法.吉林大学学报(地球科学版),48(1):261-270.
陈晓,于鹏,张罗磊等.2011.地震与大地电磁测深数据的自适应正则化同步联合反演.地球物理学报,54(10):2673-2681,doi:10.3969/j.issn.0001-5733.2011.10.024.
邓居智,陈辉,殷长春等.2015.九瑞矿集区三维电性结构研究及找矿意义.地球物理学报,58(12):4465-4477,doi:10.6038/cjg20151211.
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