基于四叉树网格的MT二维正演
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摘要
本文提出了一种基于四叉树网格剖分的大地电磁(MT)响应有限差分正演算法。首先基于四叉树数据结构进行模型网格剖分,然后用待定系数法推导了该网格中各类节点的有限差分计算公式,实现了复杂模型大地电磁响应的正演计算。多个模型计算结果表明,该算法与常规的有限差分法和有限单元法的计算精度相当时,在计算效率上有较大提高。
A finite difference algorithm for forward modeling of magnetotelluric(MT)data based on quadtree gridding is proposed in this paper.First the model is gridded based on the quadtree data structure.Then finite difference formulas for different nodes of the mesh are derived with the method of undetermined coefficient,and the magnetotelluric responses are finally obtained by solving difference equations of nodes.The proposed algorithm is tested on several models with reliefs and its validity is proved by the comparison of conventional finite difference and finite element methods.Based on our test results,the proposed algorithm achieves much better calculation efficiency with comparable accuracy.
A finite difference algorithm for forward modeling of magnetotelluric(MT)data based on quadtree gridding is proposed in this paper.First the model is gridded based on the quadtree data structure.Then finite difference formulas for different nodes of the mesh are derived with the method of undetermined coefficient,and the magnetotelluric responses are finally obtained by solving difference equations of nodes.The proposed algorithm is tested on several models with reliefs and its validity is proved by the comparison of conventional finite difference and finite element methods.Based on our test results,the proposed algorithm achieves much better calculation efficiency with comparable accuracy.
引文
[1] 陈理,秦其明,王楠,等.大地电磁测深正演和反演研究综述[J].北京大学学报(自然科学版),2014,50(5):979-984.CHEN Li,QIN Qiming,WANG Nan,et al.Review of the forward modeling and inversion in magnetotelluric sounding field[J].Acta Scientiarum Naturalium Universitatis Pekinensis,2014,50(5):979-984.
[2] Coggon J H.Electro magnetic and electrical modeling by the finite element method[J].Geophysics,1971,36(1):132-155.
[3] William L,Rodi.A technique for improving the accuracy of finite element solution for magnetotelluric data[J].Geophysical Journal International,1976,4(2):483-506.
[4] Rijo L.Modeling of Electric and Electromagnetic Data[D].University of Utah,1977.
[5] Wannamaker P E,Stodt J A,Rijo L.A stable finite element solution for two-dimensional magnetotelluric modeling[J].Geophysics,1987,88(1):277-296.
[6] Franke A,Ralph B,Klaus S.2D finite element mode-ling of plane-wave diffusive time-harmonic electro-magnetic fields using adaptive unstructured grids[C].Proceeding of the 17th Workshop on Electromagnetic Induction in the Earth,Hyderabad,India,2004.
[7] 陈乐寿.有限元法在大地电磁场正演计算中的应用及改进[J].石油物探,1981,20(3):84-103.CHEN Leshou.Application and improvement of finite element method in forward calculation of geo-electromagnetic field[J].Geophysical Prospecting for Petroleum,1981,20(3):84-103.
[8] 陈乐寿,孙必俊.有限元法在大地电磁场二维正演中的改进[J].石油地球物理勘探,1982,17(3):69-76.CHEN Leshou,SUN Bijun.Improvements on the application of finite element method to the two dimensional forward solution in the magnetotelluric method[J].Oil Geophysical Prospecting,1982,17(3):69-76.
[9] 胡建德,王光锷,陈乐寿,等.大地电磁场二维正演计算中若干问题的讨论[J].石油地球物理勘探,1982,17(6):47-55,65.HU Jiande,WANG Guang’e,CHEN Leshou,et al.Discussion on some problems in the calculation of two dimension forward of magnetotelluric field[J].Oil Geophysical Prospecting,1982,17(6):47-55,65.
[10] 胡建德,蔡纲.用三角形二次插值有限元法计算二维大地电磁测深曲线[J].石油地球物理勘探,1984,19(4):358-367.HU Jiande,CAI Gang.Calculating two dimensional geomagnetic sounding curve with triangular quadratic interpolation finite element method[J].Oil Geophysical Prospecting,1984,19(4):358-367.
[11] 徐世浙.地球物理中的有限单元法[M].北京:科学出版社,1994.
[12] 徐世浙,于涛,李予国,等.电导率分块连续变化的二维MT有限元模拟(Ⅰ)[J].高校地质学报,1995,1(2):65-73.XU Shizhe,YU Tao,LI Yuguo,et al.The finite element method for modeling 2-D MT field on a geoelectrical model with continuous variation of conductivity within each block (Ⅰ)[J].Geological Journal of China Universities,1995,1(2):65-73.
[13] 徐世浙,于涛,李予国,等.电导率分块连续变化的二维MT有限元模拟(Ⅱ)[J].高校地质学报,1996,2(4):448-452.XU Shizhe,YU Tao,LI Yuguo,et al.The finite element method for modeling 2-D MT field on a geoelectrical model with continuous variation of conductivity within each block (Ⅱ)[J].Geological Journal of China Universities,1996,2(4):448-452.
[14] 陈小斌.有限元直接迭代算法[J].物探化探计算技术,1999,21(2):165-171.CHEN Xiaobin.Direct iterative algorithm of the finite element[J].Computing Techniques for Geophysical and Geochemical Exploration,1999,21(2):165-171.
[15] 陈小斌,张翔,胡文宝.有限元直接迭代算法在MT二维正演计算中的应用[J].石油地球物理勘探,2000,35(4):487-496.CHEN Xiaobin,ZHANG Xiang,HU Wenbao.Application of finite-element direct iteration algorithm to MT 2-D forward computation[J].Oil Geophysical Prospecting,2000,35(4):487-496.
[16] Madden T R,Mackie R L.Three-dimensional magnetotelluric modelling and inversion[J].Proceeding of the IEEE,1989,77(2):318-333.
[17] Mackie R L.Three-Dimensional Magnetotlluric Mode-ling and Inversion with Applications to the California Basin and Range Province[D].Massachusetts Institute of Technology,1991.
[18] 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.
[19] Mackie R L,Madden T R.Conjugate direction relaxation solutions for 3D magnetotelluric modeling[J].Geophysics,1993,58(7):1052-1057.
[20] Mackie R L,Smith J T,Madden T R.Three-dimensional electromagnetic modeling using finite diffe-rence equations:The magnetotelluric example[J].Radio Science,1994,29(4):923-945.
[21] 谭捍东,余钦范,Booker J,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):1011-1020.TAN Handong,YU Qinfan,Booker J,et al.Magnetotelluric three-dimensional modeling using the staggered-grid finite difference method[J].Chinese Journal of Geophysics,2003,46(5):1011-1020.
[22] 肖骑彬,赵国泽.大地电磁有限差分数值解对比[J].地球物理学报,2010,53(3):622-630.XIAO Qibin,ZHAO Guozhe.Comparison of finite difference numerical solutions in magnetotelluric mode-lling[J].Chinese Journal of Geophysics,2010,53(3):622-630.
[23] 苏洲,胡文宝.二维大地电磁正演中的无网格算法[J].物探与化探,2012,36(6):1024-1028,1039.SU Zhou,HU Wenbao.Meshless algorithm in two-dimensional electromagnetic forward calculation[J].Geophysical & Geochemical Exploration,2012,36(6):1024-1028,1039.
[24] 李俊杰,严家斌.无网格点插值法大地电磁二维正演数值模拟[J].石油物探,2014,53(5):617-626.LI Junjie,YAN Jiabin.Magnetotelluric two-dimensional forward numerical modeling by meshfree point interpolation method[J].Geophysical Prospecting for Petroleum,2014,53(5):617-626.
[25] 陈小斌.大地电磁多重网格二维反演方法的应用实践[C].中国地球物理学会第二十七届年会论文集,2011.
[26] 杨振威,冯磊,赵宁,等.多重网格法二维Helmholtz方程解算及其在电磁法正演模拟中的应用[J].石油地球物理勘探,2017,52(1):167-172.YANG Zhenwei,FENG Lei,ZHAO Ning,et al.Calculating 2D Helmholtz equation based on multigrid method and application in magnetotelluric modeling[J].Oil Geophysical Prospecting,2017,52(1):167-172.
[27] Doctor L J,Torborg J G.Display techniques for octree-encoded objects[J].IEEE Computer Graphics and Applications,2006,1(3):29-38.
[28] Yerry M A,Shephard M S.A modified quadtree approach to finite element mesh generation[J].IEEE Computer Graphies and Applications,1983,3(1):39-46.
[29] Yerry M A,Shephard M S.Automatic three-dimen-sional mesh generation by the modified-octree technique[J].International Journal for Numeric Methods in Engineering,1984,20(11):1965-1990.
[30] 刘晓东,周媛媛,华祖林,等.四叉树网格下水动力及物质输运数值模拟研究进展[J].水电能源科学,2014,32(11):111-114.LIU Xiaodong,ZHOU Yangyuan,HUA Zulin,et al.Advance in numerical simulation of shallow water flow and mass transport on quad-tree grids[J].Water Resources and Power,2014,32(11):111-114.
[31] Haber E,Heldmann S.An octree multigrid method for quasi-static Maxwell's equations with highly discontinuous coefficients[J].Journal of Computational Physics,2007,223(2):783-796.
[32] Lior H and Eldad H.A second order discretization of Maxwell's equations in the quasi-static regime on octree grids[J].SIAM Journal on Scientific Computing,2011,33(5):2805-2822.
[33] 张芩,郭薇.基于四叉树的邻域查询技术[J].系统仿真学报,2001,13(增刊2):48-50.ZHANG Qin,GUO Wei.Neighbor-finding technology based on quadtree[J].Acta Simulata Systematica Sinica,2001,13(S2):48-50.
[34] 刘晓东,华祖林,赵玉萍.基于四叉树网格的Godunov型二维水流数值计算模式[J].河海大学学报(自然科学版),2002,30(6):6-10.LIU Xiaodong,HUA Zulin,ZHAO Yuping.Quad-tree meshes based Godunov-type 2-D flow numerical model[J].Journal of Hohai University,2002,30(6):6-10.
[35] 张文生.微分方程数值解:有限差分理论方法与数值计算[M].北京:科学出版社,2015,111-126.
[36] Ward S H.Electromagnetic theory for geophysical applications[J].Mining Geophysics,1967,2(1):13-196.
[37] Wannamaker P E,John A S,Luis R.Two-dimensional topographic responses in magnetotellurics modeled using finite elements[J].Geophysics,1986,51(11):2131-2144.
[38] 徐世浙,王庆乙,王军.用边界单元法模拟二维地形对大地电磁场的影响[J].地球物理学报,1992,35(3):380-388.XU Shizhe,WANG Qingyi,WANG Jun.Modeling 2D terrain effect on MT by the boundary element method[J].Chinese Journal of Geophysics,1992,35(3):380-388.
[39] 刘云,王绪本.大地电磁二维自适应地形有限元正演模拟[J].地震地质,2010,32(3):382-391.LIU Yun,WANG Xuben.FEM using adaptive topo-graphy in 2-D MT forward modeling[J].Seismology and Geology,2010,32(3):382-391.
[40] 陈小斌.MT二维正演计算中地形影响的研究[J].石油物探,2000,39(3):112-120.CHEN Xiaobin.On the research of the influence of terrain to MT 2D forward computation[J].Geophysical Prospecting for Petroleum,2000,39(3):112-120.
[2] Coggon J H.Electro magnetic and electrical modeling by the finite element method[J].Geophysics,1971,36(1):132-155.
[3] William L,Rodi.A technique for improving the accuracy of finite element solution for magnetotelluric data[J].Geophysical Journal International,1976,4(2):483-506.
[4] Rijo L.Modeling of Electric and Electromagnetic Data[D].University of Utah,1977.
[5] Wannamaker P E,Stodt J A,Rijo L.A stable finite element solution for two-dimensional magnetotelluric modeling[J].Geophysics,1987,88(1):277-296.
[6] Franke A,Ralph B,Klaus S.2D finite element mode-ling of plane-wave diffusive time-harmonic electro-magnetic fields using adaptive unstructured grids[C].Proceeding of the 17th Workshop on Electromagnetic Induction in the Earth,Hyderabad,India,2004.
[7] 陈乐寿.有限元法在大地电磁场正演计算中的应用及改进[J].石油物探,1981,20(3):84-103.CHEN Leshou.Application and improvement of finite element method in forward calculation of geo-electromagnetic field[J].Geophysical Prospecting for Petroleum,1981,20(3):84-103.
[8] 陈乐寿,孙必俊.有限元法在大地电磁场二维正演中的改进[J].石油地球物理勘探,1982,17(3):69-76.CHEN Leshou,SUN Bijun.Improvements on the application of finite element method to the two dimensional forward solution in the magnetotelluric method[J].Oil Geophysical Prospecting,1982,17(3):69-76.
[9] 胡建德,王光锷,陈乐寿,等.大地电磁场二维正演计算中若干问题的讨论[J].石油地球物理勘探,1982,17(6):47-55,65.HU Jiande,WANG Guang’e,CHEN Leshou,et al.Discussion on some problems in the calculation of two dimension forward of magnetotelluric field[J].Oil Geophysical Prospecting,1982,17(6):47-55,65.
[10] 胡建德,蔡纲.用三角形二次插值有限元法计算二维大地电磁测深曲线[J].石油地球物理勘探,1984,19(4):358-367.HU Jiande,CAI Gang.Calculating two dimensional geomagnetic sounding curve with triangular quadratic interpolation finite element method[J].Oil Geophysical Prospecting,1984,19(4):358-367.
[11] 徐世浙.地球物理中的有限单元法[M].北京:科学出版社,1994.
[12] 徐世浙,于涛,李予国,等.电导率分块连续变化的二维MT有限元模拟(Ⅰ)[J].高校地质学报,1995,1(2):65-73.XU Shizhe,YU Tao,LI Yuguo,et al.The finite element method for modeling 2-D MT field on a geoelectrical model with continuous variation of conductivity within each block (Ⅰ)[J].Geological Journal of China Universities,1995,1(2):65-73.
[13] 徐世浙,于涛,李予国,等.电导率分块连续变化的二维MT有限元模拟(Ⅱ)[J].高校地质学报,1996,2(4):448-452.XU Shizhe,YU Tao,LI Yuguo,et al.The finite element method for modeling 2-D MT field on a geoelectrical model with continuous variation of conductivity within each block (Ⅱ)[J].Geological Journal of China Universities,1996,2(4):448-452.
[14] 陈小斌.有限元直接迭代算法[J].物探化探计算技术,1999,21(2):165-171.CHEN Xiaobin.Direct iterative algorithm of the finite element[J].Computing Techniques for Geophysical and Geochemical Exploration,1999,21(2):165-171.
[15] 陈小斌,张翔,胡文宝.有限元直接迭代算法在MT二维正演计算中的应用[J].石油地球物理勘探,2000,35(4):487-496.CHEN Xiaobin,ZHANG Xiang,HU Wenbao.Application of finite-element direct iteration algorithm to MT 2-D forward computation[J].Oil Geophysical Prospecting,2000,35(4):487-496.
[16] Madden T R,Mackie R L.Three-dimensional magnetotelluric modelling and inversion[J].Proceeding of the IEEE,1989,77(2):318-333.
[17] Mackie R L.Three-Dimensional Magnetotlluric Mode-ling and Inversion with Applications to the California Basin and Range Province[D].Massachusetts Institute of Technology,1991.
[18] 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.
[19] Mackie R L,Madden T R.Conjugate direction relaxation solutions for 3D magnetotelluric modeling[J].Geophysics,1993,58(7):1052-1057.
[20] Mackie R L,Smith J T,Madden T R.Three-dimensional electromagnetic modeling using finite diffe-rence equations:The magnetotelluric example[J].Radio Science,1994,29(4):923-945.
[21] 谭捍东,余钦范,Booker J,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):1011-1020.TAN Handong,YU Qinfan,Booker J,et al.Magnetotelluric three-dimensional modeling using the staggered-grid finite difference method[J].Chinese Journal of Geophysics,2003,46(5):1011-1020.
[22] 肖骑彬,赵国泽.大地电磁有限差分数值解对比[J].地球物理学报,2010,53(3):622-630.XIAO Qibin,ZHAO Guozhe.Comparison of finite difference numerical solutions in magnetotelluric mode-lling[J].Chinese Journal of Geophysics,2010,53(3):622-630.
[23] 苏洲,胡文宝.二维大地电磁正演中的无网格算法[J].物探与化探,2012,36(6):1024-1028,1039.SU Zhou,HU Wenbao.Meshless algorithm in two-dimensional electromagnetic forward calculation[J].Geophysical & Geochemical Exploration,2012,36(6):1024-1028,1039.
[24] 李俊杰,严家斌.无网格点插值法大地电磁二维正演数值模拟[J].石油物探,2014,53(5):617-626.LI Junjie,YAN Jiabin.Magnetotelluric two-dimensional forward numerical modeling by meshfree point interpolation method[J].Geophysical Prospecting for Petroleum,2014,53(5):617-626.
[25] 陈小斌.大地电磁多重网格二维反演方法的应用实践[C].中国地球物理学会第二十七届年会论文集,2011.
[26] 杨振威,冯磊,赵宁,等.多重网格法二维Helmholtz方程解算及其在电磁法正演模拟中的应用[J].石油地球物理勘探,2017,52(1):167-172.YANG Zhenwei,FENG Lei,ZHAO Ning,et al.Calculating 2D Helmholtz equation based on multigrid method and application in magnetotelluric modeling[J].Oil Geophysical Prospecting,2017,52(1):167-172.
[27] Doctor L J,Torborg J G.Display techniques for octree-encoded objects[J].IEEE Computer Graphics and Applications,2006,1(3):29-38.
[28] Yerry M A,Shephard M S.A modified quadtree approach to finite element mesh generation[J].IEEE Computer Graphies and Applications,1983,3(1):39-46.
[29] Yerry M A,Shephard M S.Automatic three-dimen-sional mesh generation by the modified-octree technique[J].International Journal for Numeric Methods in Engineering,1984,20(11):1965-1990.
[30] 刘晓东,周媛媛,华祖林,等.四叉树网格下水动力及物质输运数值模拟研究进展[J].水电能源科学,2014,32(11):111-114.LIU Xiaodong,ZHOU Yangyuan,HUA Zulin,et al.Advance in numerical simulation of shallow water flow and mass transport on quad-tree grids[J].Water Resources and Power,2014,32(11):111-114.
[31] Haber E,Heldmann S.An octree multigrid method for quasi-static Maxwell's equations with highly discontinuous coefficients[J].Journal of Computational Physics,2007,223(2):783-796.
[32] Lior H and Eldad H.A second order discretization of Maxwell's equations in the quasi-static regime on octree grids[J].SIAM Journal on Scientific Computing,2011,33(5):2805-2822.
[33] 张芩,郭薇.基于四叉树的邻域查询技术[J].系统仿真学报,2001,13(增刊2):48-50.ZHANG Qin,GUO Wei.Neighbor-finding technology based on quadtree[J].Acta Simulata Systematica Sinica,2001,13(S2):48-50.
[34] 刘晓东,华祖林,赵玉萍.基于四叉树网格的Godunov型二维水流数值计算模式[J].河海大学学报(自然科学版),2002,30(6):6-10.LIU Xiaodong,HUA Zulin,ZHAO Yuping.Quad-tree meshes based Godunov-type 2-D flow numerical model[J].Journal of Hohai University,2002,30(6):6-10.
[35] 张文生.微分方程数值解:有限差分理论方法与数值计算[M].北京:科学出版社,2015,111-126.
[36] Ward S H.Electromagnetic theory for geophysical applications[J].Mining Geophysics,1967,2(1):13-196.
[37] Wannamaker P E,John A S,Luis R.Two-dimensional topographic responses in magnetotellurics modeled using finite elements[J].Geophysics,1986,51(11):2131-2144.
[38] 徐世浙,王庆乙,王军.用边界单元法模拟二维地形对大地电磁场的影响[J].地球物理学报,1992,35(3):380-388.XU Shizhe,WANG Qingyi,WANG Jun.Modeling 2D terrain effect on MT by the boundary element method[J].Chinese Journal of Geophysics,1992,35(3):380-388.
[39] 刘云,王绪本.大地电磁二维自适应地形有限元正演模拟[J].地震地质,2010,32(3):382-391.LIU Yun,WANG Xuben.FEM using adaptive topo-graphy in 2-D MT forward modeling[J].Seismology and Geology,2010,32(3):382-391.
[40] 陈小斌.MT二维正演计算中地形影响的研究[J].石油物探,2000,39(3):112-120.CHEN Xiaobin.On the research of the influence of terrain to MT 2D forward computation[J].Geophysical Prospecting for Petroleum,2000,39(3):112-120.