吉林大学学报(地球科学版) ›› 2018, Vol. 48 ›› Issue (1): 261-270.doi: 10.13278/j.cnki.jjuese.20160359
陈辉1,2, 尹敏2, 殷长春1, 邓居智2
Chen Hui1,2, Yin Min2, Yin Changchun1, Deng Juzhi2
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。
中图分类号:
[1] Newman G A. A Review of High-Performance Com-putational 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 Mi-ning Geophysics from 2008 to 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 Straggered-Grid Finite Difference Method[J]. Chinese Journal of Geophysics, 2003, 46(2): 281-289. [6] Mackie R L, Madden T R, Wannamaker P E. Three-Dimensional 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 Finite-Volume 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 Mode-ling 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. Three-Dimensional 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. 3D Forward 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 Three-Dimensional Magnetotelluric Modelling Algorithms[J]. Journal of Geophysics and Engineering, 2009, 6(2): 136. [21] Smith J T. Conservative Mmodeling of 3-D Electro-magnetic Fields: Part Ⅱ: Biconjugate Gradient Solution and an Accelerator[J]. Geophysics, 1996, 61(5): 1319-1324. [22] Siripunvaraporn W, Egbert G, Lenbury Y. Nume-rical 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 Three-Dimensional 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 Finite-Element 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 3D Electro-magnetic 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 MG00 to 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 3D Inversion 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. |
[1] | 李建平, 翁爱华, 李世文, 李大俊, 李斯睿, 杨悦, 唐裕, 张艳辉. 基于球坐标系下有限差分的地磁测深三维正演[J]. 吉林大学学报(地球科学版), 2018, 48(2): 411-419. |
[2] | 殷长春, 卢永超, 刘云鹤, 张博, 齐彦福, 蔡晶. 多重网格准线性近似技术在三维航空电磁正演模拟中的应用[J]. 吉林大学学报(地球科学版), 2018, 48(1): 252-260. |
[3] | 李大俊, 翁爱华, 杨悦, 李斯睿, 李建平, 李世文. 地-井瞬变电磁三维交错网格有限差分正演及响应特性[J]. 吉林大学学报(地球科学版), 2017, 47(5): 1552-1561. |
[4] | 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. |
[5] | 贲放, 刘云鹤, 黄威, 徐驰. 各向异性介质中的浅海海洋可控源电磁响应特征[J]. 吉林大学学报(地球科学版), 2016, 46(2): 581-593. |
[6] | 于向前,赵义平,王明新,刘迪,王文婷,汪馨竹. 音频大地电磁法与核磁共振法结合划分含水层的试验[J]. 吉林大学学报(地球科学版), 2014, 44(1): 350-358. |
[7] | 汤井田,李灏,李晋,强建科,肖晓. top-hat变换与庐枞矿集区大地电磁强干扰分离[J]. 吉林大学学报(地球科学版), 2014, 44(1): 336-343. |
[8] | 张文秀, 周逢道, 林君, 刘长胜, 曹学峰, 陈健, 徐汶东. 分布式电磁探测系统在深部地下水资源勘查中的应用[J]. J4, 2012, 42(4): 1207-1213. |
[9] | 柳建新, 郭荣文, 童孝忠, 刘颖, 刘鹏茂. 大地电磁法正演中多重网格法求解的广义傅里叶谱分析[J]. J4, 2011, 41(5): 1587-1595. |
[10] | 汤井田, 周聪, 张林成. CSAMT电场y方向视电阻率的定义及研究[J]. J4, 2011, 41(2): 552-558. |
[11] | 马瑞杰,李欣. 岩溶裂隙地下水流数学模型求解的有限体积法及应用[J]. J4, 2005, 35(06): 762-0765. |
|