基于多重网格有限元法的大地电磁二维正演计算研究
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摘要
为提高大地电磁正演计算速度,开展了基于多重网格有限元法的大地电磁二维正演模拟计算研究.将稳定双共轭梯度算法作为多重网格法的细网格松弛迭代算法,插值算子采用完全加权算子,限制算子设计基于网格单元面积率,使多重网格法更适于求解大型复系数方程组.二维均匀半空间模型、低阻体模型和高阻体模型的大地电磁正演模拟结果表明:当计算量较小时(网格剖分数量少),多重网格法在计算效率方面并未有优势,网格剖分数量较大时,多重网格有限元算法在收敛速度方面的优势明显,多重网格有限元法的大地电磁正演精度优于一般数值算法.这为三维多重网格有限元的大地电磁正演研究奠定了基础.
In order to enhance the computation speed, 2 D magnetotelluric modeling is researched based on multigrid finite element method. To solve the large complex coefficient equations, Bicgstab(Biconjugate gradients stabilized method) is used for the relaxed iterative algorithm in fine grid of multigrid. The interpolation is based on full-weighting operator, and the construction of a restriction operator is the area law. Magnetotelluric response of uniformity half space model、low resistivity body model and high resistivity body moel is studied based on multigrid finite element methods. The results showed that the advantage of computation efficiency of multigrid method in much more mesh generation is more obvious than less mesh generation, Otherwise, the computation precision remains to improve in low resistivity body model. This efficient algorithm of multigrid finite element method has setup basis for research of 3 D magnetotelluric modeling of multigrid finite element method.
In order to enhance the computation speed, 2 D magnetotelluric modeling is researched based on multigrid finite element method. To solve the large complex coefficient equations, Bicgstab(Biconjugate gradients stabilized method) is used for the relaxed iterative algorithm in fine grid of multigrid. The interpolation is based on full-weighting operator, and the construction of a restriction operator is the area law. Magnetotelluric response of uniformity half space model、low resistivity body model and high resistivity body moel is studied based on multigrid finite element methods. The results showed that the advantage of computation efficiency of multigrid method in much more mesh generation is more obvious than less mesh generation, Otherwise, the computation precision remains to improve in low resistivity body model. This efficient algorithm of multigrid finite element method has setup basis for research of 3 D magnetotelluric modeling of multigrid finite element method.
引文
Briggs W L,Henson V E,McCormick S F.2011.A multigrid tutorial [M].Siam,20-30.
Chung Y,Son J S,Lee T J,et al.2014.Three dimensional modelling of controlled-source electromagnetic surveys using an edge finite-element method with a direct solver [J].Geophys prospect,62(6):1468-1483,doi:10.1111/1365-2478.12132.
Egbert G D,Kelbert A.2012.Computational recipes for electromagnetic inverse problems [J].Geophysical journal international,189:251-267,doi:10.1111/j.1365-246x.2011.05347.x.
Farquharson C G,Miensopust M P.2011.Three-dimensional finite-element modelling of magnetotelluric data with a divergence correction [J].Journal of applied geophysics,75(4):699-710,doi:10.1016/j.jappgeo.2011.09.025.
Ghaffar F,Badshah N,Islam S,et al.2016.Multigrid method based on transformation-free high-order scheme for solving 2D Helmholtz equation on nonuniform grids [J].Advances in difference equations,19:1-16.doi:10.1186/s13662- 016- 0745-2.
Gmeiner B,K?stler H,Stürmer M,et al.2014.Parallel multigrid on hierarchical hybrid:a performance study on current high performance computing clusters [J].Concurrency & Computation Practice & Experience,26(1):217-240,doi:10.1002/cpe.2968.
Hu X Y,Huo G P,Gao R,et al.2013.The magnetotelluric anisotropic two-dimensional simulation and case analysis [J].Chinese J.Geophys.(in Chinese),56(12):4268- 4277,doi:10.6038/cjg20131229.
Koldan J,Puzyrev V,Puente J,et al.2014.Algebraic multigrid preconditioning within parallel finite-element solvers for 3-D electromagnetic modelling problems in geophysics [J].Geophysical journal international,197(3):1442-1458,doi:10.1093/gji/ggu086.
Kordy M,Wannamaker P,Maris V,et al.2016.3-D magnetotelluric inversion including topography using deformed hexahedral edge finite elements and direct solvers parallelized on SMP computers-Part 1:forward problem and parameter Jacobians [J].Geophysical journal international,204(1):74-93,doi:10.1093/gji/ggv410.
Li S Y,Jiang S Q,Wang Y Y,et al.2016.A wavelet correction method for the seawater turbulence noise in marine controlled-source electromagnetic data [J].Geophysical prospecting for petroleum (in Chinese),55(5):657- 663.
Liu J X,Guo R W,Tong X Z.2011.Boundary truncation of magnetotelluric modeling based on multigrid method [J].Journal of central south university(Science and technology) (in Chinese),42(11):3429-3437.
Lu J J,Wu X P,Spitzer K.2010.Algebraic multigrid method for 3D DC resistivity modelling [J].Chinese J.Geophys.(in Chinese),53(3):700-707.doi:10.3969/1.iss.0001-5733.2010.03.025.
Miensopust M P,Queralt P,Jones A G.2013.Magnetotelluric 3-D inverse-a review of two successful workshops on forward and inversion code testing and comparison [J].Geophysical journal international,193(3):1216-1238.
Nam M J,Kim H J,Song Y.2007.3D magnetotelluric modelling including surface topography [J].Geophysical prospecting,55(2):277-287.
Rodi W,Mackie R L.2001.Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion [J].Geophysics,66(1):174-187,doi:10.1190/1.1444893.
Tang J T,Wang F Y,Ren Z Y.2010.3-D direct current resistivity forward modeling by adaptive multigrid finite element method [J].Journal of central south university of technology (in Chinese),17(3):587-592.
Tang W W,Li Y G,Liu J X,et al.2015.Three-dimensional controlled-source electromagnetic forward modeling by edge-based finite element using secondary electric field [J].Geophysical prospecting for petroleum (in Chinese),54(6):665- 673.
Wang R,Wang M Y,Di Q Y.2006.Electromagnetic modeling due to line source in frequency domain using finite element method [J].Chinese J.Geophys.(in Chinese),49(6):1858-1866,doi:10.3321/j.issn:0001-5733.2006.06.035.
Xiao Qi-bin,Zhao Guo-ze.2010.Comparison of finite difference numerical solutions in magnetotelluric modeling [J].Chinese J.Geophys.(in Chinese),53(3):622- 630,doi:10.3969/ j.issn.0001-5733.2010.03.016.
Zuo C J,Wang Z W,Xiang M,et al.2016.The radial pore heterogeneity of volcanic reservoir based on the porodsity analysis of micro-electric imaging logging [J].Geophysical prospecting for petroleum (in Chinese),55(3):449- 454.
Zyserman F I,Santos J E.2000.Parallel finite element algorithm with domain decomposition for three-dimensional magnetotelluric modelling [J].Journal of applied geophysics,44(4):337-351,doi:10.1016/S0926-9851(00)00012- 4.
胡祥云,霍光谱,高锐,等.2013.大地电磁各向异性二维模拟及实例分析[J].地球物理学报,56(12):4268- 4277,doi:10.6038/cjg20131229.
李肃义,蒋善庆,王跃洋,等.2016.海洋可控源电磁数据中海水扰动噪声的小波校正方法研究[J].石油物探,55(5):657- 663.
柳建新,郭荣文,童孝忠,等.2011.基于多重网格法的MT正演模型边界截取[J].中南大学学报(自然科学版),42(11):3429-3437.
鲁晶津,吴小平,Spitzer Klaus.2010.直流电阻率三维正演的代数多重网格方法[J].地球物理学报,53(3):700-707.doi:10.3969/1.iss.0001-5733.2010.03.025.
汤井田,王飞燕,任政勇,等.2010.自适应多重网格有限元法的三维直流电阻率法正演模拟[J].中南大学学报,17(3):587-592.
汤文武,李耀国,柳建新.2015.基于二次电场的可控源电磁法三维矢量有限元正演模拟[J].石油物探,54(6):665- 673.
王若,王妙月,底青云.2006.频率域线源大地电磁法有限元正演模拟[J].地球物理学报,49(6):1858-1866,doi:10.3321/j.issn:0001-5733.2006.06.035.
肖骑彬,赵国泽.2010.大地电磁有限差分数值解对比[J].地球物理学报,53(3):622 - 630,doi:10.3969/j.issn.0001-5733.2010.03.016.
左程吉,王祝文,向旻,等.2016.基于电成像测井孔隙度分析技术的火山岩孔隙径向非均质性研究[J].石油物探,55(3):449- 454.
Chung Y,Son J S,Lee T J,et al.2014.Three dimensional modelling of controlled-source electromagnetic surveys using an edge finite-element method with a direct solver [J].Geophys prospect,62(6):1468-1483,doi:10.1111/1365-2478.12132.
Egbert G D,Kelbert A.2012.Computational recipes for electromagnetic inverse problems [J].Geophysical journal international,189:251-267,doi:10.1111/j.1365-246x.2011.05347.x.
Farquharson C G,Miensopust M P.2011.Three-dimensional finite-element modelling of magnetotelluric data with a divergence correction [J].Journal of applied geophysics,75(4):699-710,doi:10.1016/j.jappgeo.2011.09.025.
Ghaffar F,Badshah N,Islam S,et al.2016.Multigrid method based on transformation-free high-order scheme for solving 2D Helmholtz equation on nonuniform grids [J].Advances in difference equations,19:1-16.doi:10.1186/s13662- 016- 0745-2.
Gmeiner B,K?stler H,Stürmer M,et al.2014.Parallel multigrid on hierarchical hybrid:a performance study on current high performance computing clusters [J].Concurrency & Computation Practice & Experience,26(1):217-240,doi:10.1002/cpe.2968.
Hu X Y,Huo G P,Gao R,et al.2013.The magnetotelluric anisotropic two-dimensional simulation and case analysis [J].Chinese J.Geophys.(in Chinese),56(12):4268- 4277,doi:10.6038/cjg20131229.
Koldan J,Puzyrev V,Puente J,et al.2014.Algebraic multigrid preconditioning within parallel finite-element solvers for 3-D electromagnetic modelling problems in geophysics [J].Geophysical journal international,197(3):1442-1458,doi:10.1093/gji/ggu086.
Kordy M,Wannamaker P,Maris V,et al.2016.3-D magnetotelluric inversion including topography using deformed hexahedral edge finite elements and direct solvers parallelized on SMP computers-Part 1:forward problem and parameter Jacobians [J].Geophysical journal international,204(1):74-93,doi:10.1093/gji/ggv410.
Li S Y,Jiang S Q,Wang Y Y,et al.2016.A wavelet correction method for the seawater turbulence noise in marine controlled-source electromagnetic data [J].Geophysical prospecting for petroleum (in Chinese),55(5):657- 663.
Liu J X,Guo R W,Tong X Z.2011.Boundary truncation of magnetotelluric modeling based on multigrid method [J].Journal of central south university(Science and technology) (in Chinese),42(11):3429-3437.
Lu J J,Wu X P,Spitzer K.2010.Algebraic multigrid method for 3D DC resistivity modelling [J].Chinese J.Geophys.(in Chinese),53(3):700-707.doi:10.3969/1.iss.0001-5733.2010.03.025.
Miensopust M P,Queralt P,Jones A G.2013.Magnetotelluric 3-D inverse-a review of two successful workshops on forward and inversion code testing and comparison [J].Geophysical journal international,193(3):1216-1238.
Nam M J,Kim H J,Song Y.2007.3D magnetotelluric modelling including surface topography [J].Geophysical prospecting,55(2):277-287.
Rodi W,Mackie R L.2001.Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion [J].Geophysics,66(1):174-187,doi:10.1190/1.1444893.
Tang J T,Wang F Y,Ren Z Y.2010.3-D direct current resistivity forward modeling by adaptive multigrid finite element method [J].Journal of central south university of technology (in Chinese),17(3):587-592.
Tang W W,Li Y G,Liu J X,et al.2015.Three-dimensional controlled-source electromagnetic forward modeling by edge-based finite element using secondary electric field [J].Geophysical prospecting for petroleum (in Chinese),54(6):665- 673.
Wang R,Wang M Y,Di Q Y.2006.Electromagnetic modeling due to line source in frequency domain using finite element method [J].Chinese J.Geophys.(in Chinese),49(6):1858-1866,doi:10.3321/j.issn:0001-5733.2006.06.035.
Xiao Qi-bin,Zhao Guo-ze.2010.Comparison of finite difference numerical solutions in magnetotelluric modeling [J].Chinese J.Geophys.(in Chinese),53(3):622- 630,doi:10.3969/ j.issn.0001-5733.2010.03.016.
Zuo C J,Wang Z W,Xiang M,et al.2016.The radial pore heterogeneity of volcanic reservoir based on the porodsity analysis of micro-electric imaging logging [J].Geophysical prospecting for petroleum (in Chinese),55(3):449- 454.
Zyserman F I,Santos J E.2000.Parallel finite element algorithm with domain decomposition for three-dimensional magnetotelluric modelling [J].Journal of applied geophysics,44(4):337-351,doi:10.1016/S0926-9851(00)00012- 4.
胡祥云,霍光谱,高锐,等.2013.大地电磁各向异性二维模拟及实例分析[J].地球物理学报,56(12):4268- 4277,doi:10.6038/cjg20131229.
李肃义,蒋善庆,王跃洋,等.2016.海洋可控源电磁数据中海水扰动噪声的小波校正方法研究[J].石油物探,55(5):657- 663.
柳建新,郭荣文,童孝忠,等.2011.基于多重网格法的MT正演模型边界截取[J].中南大学学报(自然科学版),42(11):3429-3437.
鲁晶津,吴小平,Spitzer Klaus.2010.直流电阻率三维正演的代数多重网格方法[J].地球物理学报,53(3):700-707.doi:10.3969/1.iss.0001-5733.2010.03.025.
汤井田,王飞燕,任政勇,等.2010.自适应多重网格有限元法的三维直流电阻率法正演模拟[J].中南大学学报,17(3):587-592.
汤文武,李耀国,柳建新.2015.基于二次电场的可控源电磁法三维矢量有限元正演模拟[J].石油物探,54(6):665- 673.
王若,王妙月,底青云.2006.频率域线源大地电磁法有限元正演模拟[J].地球物理学报,49(6):1858-1866,doi:10.3321/j.issn:0001-5733.2006.06.035.
肖骑彬,赵国泽.2010.大地电磁有限差分数值解对比[J].地球物理学报,53(3):622 - 630,doi:10.3969/j.issn.0001-5733.2010.03.016.
左程吉,王祝文,向旻,等.2016.基于电成像测井孔隙度分析技术的火山岩孔隙径向非均质性研究[J].石油物探,55(3):449- 454.