大地电磁三维正演聚集多重网格算法
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摘要
为了加快大地电磁三维正演的求解速度,本文将一种新型的代数多重网格算法——聚集多重网格(aggregation-based algebraic multigrid,AGMG)算法引入大地电磁三维正演模拟中。首先从准静态条件下的麦克斯韦方程出发,利用交错网格有限体积法进行离散,并采用第一类Dirichlet边界条件形成大型稀疏复线性方程组;然后阐述AGMG算法的粗化策略和套迭代技术,并实施3种不同的AGMG求解算法:1)传统的V循环AGMG算法;2)AGMG预处理共轭梯度(AGMG-CG);3)AGMG预处理广义共轭残差法(AGMG-GCR)。最终实现大地电磁法三维正演模拟。对典型地电模型进行正演模拟,并与已有的大地电磁三维正反演程序(ModEM)进行结果对比,以验证本文算法的准确性。另外,不同剖分网格和极化方式正演模拟结果与准残量最小化(QMR)迭代算法的对比表明,AGMG预处理求解算法(AGMG-CG、AGMG-GCR)不仅能够改善算法的稳定性,而且能够快速有效地求解正演问题;其中AGMG-GCR迭代次数更少,求解速度更快,误差衰减曲线更光滑,在144×152×104网格剖分情况下,相对于现有ModEM程序能够提高十几倍的计算速度,尤其适合大规模大地电磁三维正演问题。
To speed up 3 D magnetotelluric(MT) modelling,we introduce a novel algebraic multigrid-aggregation-based algebraic multigrid method(AGMG)into three dimensional forward modeling of magnetotelluric.We used the finite-volume algorithm based on the Yee's grids to discretize quasi-static Maxwell's equations with Dirichlet boundary conditions,and used the AGMG method to solve the final large sparse linear equation system in electric field.Through the coarsening of AGMG and the aggregating based on N-passes of a pairwise matching algorithm applied to the matrix graph,we proposed three different AGMG algorithms:1)the classic V-cycle AGMG algorithm;2)the AGMG preconditioned conjugate gradient algorithm(AGMG-CG);3)the AGMG pretreated generalized conjugate residual method(AGMG-GCR).We performed 3D MT modelling for typical geo-electric models with different iteration,and analyzed the features of the AGMG techniques through comparing with ModEM algorithm.The results show that the AGMG methods are accurate and robust,the AGMG preconditioner improves the convergence of the classic V-cycle AGMG and Krylov subspace methods greatly.The AGMG-GCR method is the most effective one presented in this paper,which speeds up the modeling by ten times more than the ModEM codes for large-scale grids(144×152×104).The AGMG-GCR is especially suitable for large-scale 3D MT modeling because of its high precision,fast convergence,and robust iteration.
To speed up 3 D magnetotelluric(MT) modelling,we introduce a novel algebraic multigrid-aggregation-based algebraic multigrid method(AGMG)into three dimensional forward modeling of magnetotelluric.We used the finite-volume algorithm based on the Yee's grids to discretize quasi-static Maxwell's equations with Dirichlet boundary conditions,and used the AGMG method to solve the final large sparse linear equation system in electric field.Through the coarsening of AGMG and the aggregating based on N-passes of a pairwise matching algorithm applied to the matrix graph,we proposed three different AGMG algorithms:1)the classic V-cycle AGMG algorithm;2)the AGMG preconditioned conjugate gradient algorithm(AGMG-CG);3)the AGMG pretreated generalized conjugate residual method(AGMG-GCR).We performed 3D MT modelling for typical geo-electric models with different iteration,and analyzed the features of the AGMG techniques through comparing with ModEM algorithm.The results show that the AGMG methods are accurate and robust,the AGMG preconditioner improves the convergence of the classic V-cycle AGMG and Krylov subspace methods greatly.The AGMG-GCR method is the most effective one presented in this paper,which speeds up the modeling by ten times more than the ModEM codes for large-scale grids(144×152×104).The AGMG-GCR is especially suitable for large-scale 3D MT modeling because of its high precision,fast convergence,and robust iteration.
引文
[1]Newman G A.A Review of High-Performance Computational Strategies for Modeling and Imaging of Electromagnetic Induction Data[J].Surveys in Geophysics,2014,35(1):85-100.
[2]Smith R.Electromagnetic Induction Methods in Mining Geophysics from 2008to 2012[J].Surveys in Geophysics,2014,35(1):123-156.
[3]Siripunvaraporn W.Three-Dimensional Magnetotellu-ric Inversion:An Introductory Guide for Developers and Users[J].Surveys in Geophysics,2012,33(1):5-27.
[4]谭捍东,余钦范,Booker John,等.大地电磁法三维交错采样有限差分数值模拟[J].地球物理学报,2003,46(5):706-711.Tan Handong,Yu Qinfan,Booker J,et al.Magnetotelluric Three-Dimensional Modelling Using the Staggered-Grid Fnite Difference Method[J].Chinese Journal of Geophysics,2003,46(5):706-711.
[5]沈金松.用交错网格有限差分法计算三维频率域电磁响应[J].地球物理学报,2003,46(2):281-289.Shen Jinsong.Modelling of 3-D Eectromagnetic Responses in Frequency Domain by Using StraggeredGrid Finite Difference Method[J].Chinese Journal of Geophysics,2003,46(2):281-289.
[6]Mackie R L,Madden T R,Wannamaker P E.ThreeDimensional Magnetotelluric Modeling Using Difference Equations:Theory and Comparisons to Integral Equation Solutions[J].Geophysics,1993,58(2):215-226.
[7]李焱,胡祥云,杨文采,等.大地电磁三维交错网格有限差分数值模拟的并行计算研究[J].地球物理学报,2012,55(12):4036-4043.Li Yan,Hu Xiangyun,Yang Wencai,et al.A Study on Parallel Computation for 3D Magnetotelluric Modeling Using the Staggered-Grid Finite Difference Method[J].Chinese Journal of Geophysics,2012,55(12):4036-4043.
[8]Haber E,Ascher U M.Fast Finite Volume Simulation of 3D Electromagnetic Problems with Highly Discontinuous Coefficients[J].SIAM Journal on Scientific Computing,2000,22(6):1943-1961.
[9]Haber E,Ruthotto L.A Multiscale Finite Volume Method for Maxwell’s Equations at Low Frequencies[J].Geophysical Journal International,2014,199(2):1268-1277.
[10]陈辉,殷长春,邓居智.基于Lorenz规范条件下磁矢势和标势耦合方程的频率域电磁法三维正演[J].地球物理学报,2016,59(8):3087-3097.Chen Hui,Yin Changchun,Deng Juzhi.A FiniteVolume Solution to 3D Frequency-Domain Electromagnetic Modelling Using Lorenz-Gauged Magnetic Vector and Scalar Potentials[J].Chinese Journal of Geophysics,2016,59(8):3087-3097.
[11]Ren Z,Kalscheuer T,Greenhalgh S,et al.A Finite-Element-Based Domain-Decomposition Approach for Plane Wave 3D Electromagnetic Modeling[J].Geophysics,2014,79(6):E255-E268.
[12]黄临平,戴世坤.复杂条件下3D电磁场有限元计算方法[J].地球科学:中国地质大学学报,2002,27(6):775-779.Huang Linping,Dai Shikun.Finite Element Calculation Method of 3D Electromagnetic Field under Complex Condition[J].Earth Science:Journal of China University of Geoscineces,2002,27(6):775-779.
[13]Mitsuhata Y,Uchida T.3D Magnetotelluric Modeling Using the T-Omega Finite-Element Method[J].Geophysics,2004,69(1):108-119.
[14]李俊杰,严家斌,皇祥宇.无单元Galerkin法大地电磁三维正演模拟[J].地质与勘探,2015,51(5):946-952.Li Junjie,Yan Jiabin,Huang Xiangyu.ThreeDimensional Forward Modeling of Magnetotellurics Using the Element-Free Galerkin Method[J].Geology and Exploration,2015,51(5):946-952.
[15]严家斌,皇祥宇.大地电磁三维矢量有限元正演[J].吉林大学学报(地球科学版),2016,46(5):1538-1549.Yan Jiabin,Huang Xiangyu.3DForward Modeling of Magnetotelluric Field by Vector Finite Element Method[J].Journal of Jilin University(Earth Science Edition),2016,46(5):1538-1549.
[16]Kruglyakov M,Geraskin A,Kuvshinov A.Novel Accurate and Scalable 3-D MT Forward Solver Based on a Contracting Integral Equation Method[J].Computers&Geosciences,2016,96:208-217.
[17]Wannamaker P E.Advances in Three-Dimensional Magnetotelluric Modeling Using Integral Equations[J].Geophysics,1991,56(11):1716-1728.
[18]王书明,李德山,胡浩.三维/三维构造下大地电磁相位张量数值模拟[J].地球物理学报,2013,56(5):1745-1752.Wang Shuming,Li Deshan,Hu Hao.Numerical Modeling of Magnetotelluric Phase Tensor in the Context of 3D/3D Formation[J].Chinese Journal of Geophysics,2013,56(5):1745-1752.
[19]de Groot-Hedlin C.Finite-Difference Modeling of Magnetotelluric Felds:Error Estimates for Uniform and Nonuniform Grids[J].Geophysics,2006,71(3):G225-G233.
[20]Han N,Nam M J,Kim H J,et al.A Comparison of Accuracy and Computation Time of ThreeDimensional Magnetotelluric Modelling Algorithms[J].Journal of Geophysics and Engineering,2009,6(2):136.
[21]Smith J T.Conservative Mmodeling of 3-D Electromagnetic Fields:Part II:Biconjugate Gradient Solution and an Accelerator[J].Geophysics,1996,61(5):1319-1324.
[22]Siripunvaraporn W,Egbert G,Lenbury Y.Numerical Accuracy of Magnetotelluric Modelling:A Comparison of Finite Difference Approximations[J].Earth Planets Space,2002,54:721-725.
[23]Mackie R L,Madden T R.Conjugate Direction Relaxation Solutions for 3-D Magnetotelluric Modeling[J].Geophysics,1993,58(7):1052-1057.
[24]Weiss C J,Newman G A.Electromagnetic Induction in a Generalized 3D Anisotropic Earth:Part 2:The LIN Preconditioner[J].Geophysics,2003,68(3):922-930.
[25]陈辉,邓居智,谭捍东,等.大地电磁三维交错网格有限差分数值模拟中的散度校正方法研究[J].地球物理学报,2011,54(6):1649-1659.Chen Hui,Deng Juzhi,Tan Handong,et al.Study on Divergence Correction Method in ThreeDimensional Magnetotelluric Modeling with Staggered-Grid Finite Fifference Method[J].Chinese Journal Geophysics,2011,54(6):1649-1659.
[26]Streich R.3D Finite-Difference Frequency-Domain Modeling of Controlled-Source Electromagnetic Data:Direct Solution and Optimization for High Accuracy[J].Geophysics,2009,74(5):95-105.
[27]Puzyrev V,Koric S,Wilkin S.Evaluation of Parallel Direct Sparse Linear Solvers in Electromagnetic Geophysical Problems[J].Computers&Geosciences,2016,89:79-87.
[28]Koldan J,Puzyrev V,de la Puente J,et al.Algebraic Multigrid Preconditioning Within Parallel FiniteElement Solvers for 3-D Electromagnetic Modelling Problems in Geophysics[J].Geophysical Journal International,2014,197(3):1442-1458.
[29]Mulder W A.Geophysical Modelling of 3DElectromagnetic Diffusion with Multigrid[J].Computing and Visualization in Science,2008,11(3):129-138.
[30]Pan K,Tang J.2.5-D and 3-D DC Resistivity Modelling Using an Extrapolation Cascadic Multigrid Method[J].Geophysical Journal International,2014,197(3):1459-1470.
[31]Trottenberg U,Clees T.Multigrid Software for Industrial Applications:From MG00to SAMG[M].Heidelberg:Springer,2009:423-436.
[32]Notay Y.An Aggregation-Based Algebraic Multigrid Method[J].Electronic Transactions on Numerical Analysis,2010,37(6):123-146.
[33]Henson V E,Yang U M.Boomer AMG:A Parallel Algebraic Multigrid Solver and Preconditioner[J].Applied Numerical Mathematics,2002,41(1):155-177.
[34]Saad Y.Iterative Methods for Sparse Linear Systems[M].Philadelphia:SIAM,2003.
[35]Pflaum C.A Multigrid Conjugate Gradient Method[J].Applied Numerical Mathematics,2008,58:1803-1817.
[36]MT 3DInversion Workshop.Dublin Test Model 1(DTM1)[EB/OL].[2016-10-20]http://www.complete-mt-solutions.com/mtnet/workshops/mt3di-nv/2008_Dublin/Dublin/3dmodel.html.
[37]Kelbert A,Meqbel N,Egbert G D,et al.ModEM:A Modular System for Inversion of Electromagnetic Geophysical Data[J].Computers&Geosciences,2014,66:40-53.
[2]Smith R.Electromagnetic Induction Methods in Mining Geophysics from 2008to 2012[J].Surveys in Geophysics,2014,35(1):123-156.
[3]Siripunvaraporn W.Three-Dimensional Magnetotellu-ric Inversion:An Introductory Guide for Developers and Users[J].Surveys in Geophysics,2012,33(1):5-27.
[4]谭捍东,余钦范,Booker John,等.大地电磁法三维交错采样有限差分数值模拟[J].地球物理学报,2003,46(5):706-711.Tan Handong,Yu Qinfan,Booker J,et al.Magnetotelluric Three-Dimensional Modelling Using the Staggered-Grid Fnite Difference Method[J].Chinese Journal of Geophysics,2003,46(5):706-711.
[5]沈金松.用交错网格有限差分法计算三维频率域电磁响应[J].地球物理学报,2003,46(2):281-289.Shen Jinsong.Modelling of 3-D Eectromagnetic Responses in Frequency Domain by Using StraggeredGrid Finite Difference Method[J].Chinese Journal of Geophysics,2003,46(2):281-289.
[6]Mackie R L,Madden T R,Wannamaker P E.ThreeDimensional Magnetotelluric Modeling Using Difference Equations:Theory and Comparisons to Integral Equation Solutions[J].Geophysics,1993,58(2):215-226.
[7]李焱,胡祥云,杨文采,等.大地电磁三维交错网格有限差分数值模拟的并行计算研究[J].地球物理学报,2012,55(12):4036-4043.Li Yan,Hu Xiangyun,Yang Wencai,et al.A Study on Parallel Computation for 3D Magnetotelluric Modeling Using the Staggered-Grid Finite Difference Method[J].Chinese Journal of Geophysics,2012,55(12):4036-4043.
[8]Haber E,Ascher U M.Fast Finite Volume Simulation of 3D Electromagnetic Problems with Highly Discontinuous Coefficients[J].SIAM Journal on Scientific Computing,2000,22(6):1943-1961.
[9]Haber E,Ruthotto L.A Multiscale Finite Volume Method for Maxwell’s Equations at Low Frequencies[J].Geophysical Journal International,2014,199(2):1268-1277.
[10]陈辉,殷长春,邓居智.基于Lorenz规范条件下磁矢势和标势耦合方程的频率域电磁法三维正演[J].地球物理学报,2016,59(8):3087-3097.Chen Hui,Yin Changchun,Deng Juzhi.A FiniteVolume Solution to 3D Frequency-Domain Electromagnetic Modelling Using Lorenz-Gauged Magnetic Vector and Scalar Potentials[J].Chinese Journal of Geophysics,2016,59(8):3087-3097.
[11]Ren Z,Kalscheuer T,Greenhalgh S,et al.A Finite-Element-Based Domain-Decomposition Approach for Plane Wave 3D Electromagnetic Modeling[J].Geophysics,2014,79(6):E255-E268.
[12]黄临平,戴世坤.复杂条件下3D电磁场有限元计算方法[J].地球科学:中国地质大学学报,2002,27(6):775-779.Huang Linping,Dai Shikun.Finite Element Calculation Method of 3D Electromagnetic Field under Complex Condition[J].Earth Science:Journal of China University of Geoscineces,2002,27(6):775-779.
[13]Mitsuhata Y,Uchida T.3D Magnetotelluric Modeling Using the T-Omega Finite-Element Method[J].Geophysics,2004,69(1):108-119.
[14]李俊杰,严家斌,皇祥宇.无单元Galerkin法大地电磁三维正演模拟[J].地质与勘探,2015,51(5):946-952.Li Junjie,Yan Jiabin,Huang Xiangyu.ThreeDimensional Forward Modeling of Magnetotellurics Using the Element-Free Galerkin Method[J].Geology and Exploration,2015,51(5):946-952.
[15]严家斌,皇祥宇.大地电磁三维矢量有限元正演[J].吉林大学学报(地球科学版),2016,46(5):1538-1549.Yan Jiabin,Huang Xiangyu.3DForward Modeling of Magnetotelluric Field by Vector Finite Element Method[J].Journal of Jilin University(Earth Science Edition),2016,46(5):1538-1549.
[16]Kruglyakov M,Geraskin A,Kuvshinov A.Novel Accurate and Scalable 3-D MT Forward Solver Based on a Contracting Integral Equation Method[J].Computers&Geosciences,2016,96:208-217.
[17]Wannamaker P E.Advances in Three-Dimensional Magnetotelluric Modeling Using Integral Equations[J].Geophysics,1991,56(11):1716-1728.
[18]王书明,李德山,胡浩.三维/三维构造下大地电磁相位张量数值模拟[J].地球物理学报,2013,56(5):1745-1752.Wang Shuming,Li Deshan,Hu Hao.Numerical Modeling of Magnetotelluric Phase Tensor in the Context of 3D/3D Formation[J].Chinese Journal of Geophysics,2013,56(5):1745-1752.
[19]de Groot-Hedlin C.Finite-Difference Modeling of Magnetotelluric Felds:Error Estimates for Uniform and Nonuniform Grids[J].Geophysics,2006,71(3):G225-G233.
[20]Han N,Nam M J,Kim H J,et al.A Comparison of Accuracy and Computation Time of ThreeDimensional Magnetotelluric Modelling Algorithms[J].Journal of Geophysics and Engineering,2009,6(2):136.
[21]Smith J T.Conservative Mmodeling of 3-D Electromagnetic Fields:Part II:Biconjugate Gradient Solution and an Accelerator[J].Geophysics,1996,61(5):1319-1324.
[22]Siripunvaraporn W,Egbert G,Lenbury Y.Numerical Accuracy of Magnetotelluric Modelling:A Comparison of Finite Difference Approximations[J].Earth Planets Space,2002,54:721-725.
[23]Mackie R L,Madden T R.Conjugate Direction Relaxation Solutions for 3-D Magnetotelluric Modeling[J].Geophysics,1993,58(7):1052-1057.
[24]Weiss C J,Newman G A.Electromagnetic Induction in a Generalized 3D Anisotropic Earth:Part 2:The LIN Preconditioner[J].Geophysics,2003,68(3):922-930.
[25]陈辉,邓居智,谭捍东,等.大地电磁三维交错网格有限差分数值模拟中的散度校正方法研究[J].地球物理学报,2011,54(6):1649-1659.Chen Hui,Deng Juzhi,Tan Handong,et al.Study on Divergence Correction Method in ThreeDimensional Magnetotelluric Modeling with Staggered-Grid Finite Fifference Method[J].Chinese Journal Geophysics,2011,54(6):1649-1659.
[26]Streich R.3D Finite-Difference Frequency-Domain Modeling of Controlled-Source Electromagnetic Data:Direct Solution and Optimization for High Accuracy[J].Geophysics,2009,74(5):95-105.
[27]Puzyrev V,Koric S,Wilkin S.Evaluation of Parallel Direct Sparse Linear Solvers in Electromagnetic Geophysical Problems[J].Computers&Geosciences,2016,89:79-87.
[28]Koldan J,Puzyrev V,de la Puente J,et al.Algebraic Multigrid Preconditioning Within Parallel FiniteElement Solvers for 3-D Electromagnetic Modelling Problems in Geophysics[J].Geophysical Journal International,2014,197(3):1442-1458.
[29]Mulder W A.Geophysical Modelling of 3DElectromagnetic Diffusion with Multigrid[J].Computing and Visualization in Science,2008,11(3):129-138.
[30]Pan K,Tang J.2.5-D and 3-D DC Resistivity Modelling Using an Extrapolation Cascadic Multigrid Method[J].Geophysical Journal International,2014,197(3):1459-1470.
[31]Trottenberg U,Clees T.Multigrid Software for Industrial Applications:From MG00to SAMG[M].Heidelberg:Springer,2009:423-436.
[32]Notay Y.An Aggregation-Based Algebraic Multigrid Method[J].Electronic Transactions on Numerical Analysis,2010,37(6):123-146.
[33]Henson V E,Yang U M.Boomer AMG:A Parallel Algebraic Multigrid Solver and Preconditioner[J].Applied Numerical Mathematics,2002,41(1):155-177.
[34]Saad Y.Iterative Methods for Sparse Linear Systems[M].Philadelphia:SIAM,2003.
[35]Pflaum C.A Multigrid Conjugate Gradient Method[J].Applied Numerical Mathematics,2008,58:1803-1817.
[36]MT 3DInversion Workshop.Dublin Test Model 1(DTM1)[EB/OL].[2016-10-20]http://www.complete-mt-solutions.com/mtnet/workshops/mt3di-nv/2008_Dublin/Dublin/3dmodel.html.
[37]Kelbert A,Meqbel N,Egbert G D,et al.ModEM:A Modular System for Inversion of Electromagnetic Geophysical Data[J].Computers&Geosciences,2014,66:40-53.