基于自适应多层快速多极算法的大规模磁法正演模拟
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摘要
提出了一种基于非结构化四面体以及带地形模型的自适应多层快速多极大规模磁法快速正演算法.该算法弥补了传统积分方法采用FFT加速计算时不能采用非结构化网格的缺陷;同时采用自适应快速多极算法突破积分求和方法求解大规模磁法问题耗时长的突出问题.首先,采用非结构化的四面体网格剖分技术能够更好的模拟复杂模型以及带地形模型,实现磁法模型的高精度模拟;其次,采用一种自适应多层快速多极(AMFM)算法实现大规模磁法正演求解.通过将计算区域划分为近区和远区,对近区采用解析计算高精度求解,对远区采用自适应多层快速多极算法进行加速计算,假设有M个观测点,N个四面体源单元,可将计算复杂度由传统积分求和法的O(MN)减少到O(Mlog N).本文设计了组合体模型以及安徽怀宁地区的实际地形模型,模型计算结果体现了采用该方法进行大规模复杂模型三维磁法正演模拟的高效性和准确性.
In this paper,we propose a fast and high-precision approach for large-scale magnetic forward modeling using the adaptive multi-level fast multipole(AMFM) method and an unstructured grid.The algorithm can overcome the shortcomings of the Fast fourier transfom(FFT)method which cannot adopt unstructured grids.In addition,the adaptive multi-level fast multipole(AMFM)method is chosen to solve the problem of high computational cost in traditional integral summation methods.Firstly,the unstructured tetrahedral meshing technique is used to properly approximate the complex model,for instance the terrain model,and realize thehigh-precision simulation of the magnetic model.Secondly,an adaptive multi-level fast multipole(AMFM)method is employed to accelerate large-scale magnetic forward modeling.Then,we divide the integral region into two parts.One is the near part in which we use analytical solution to get high-precision results,and the other is far part in which we choose the the adaptive multilevel fast multipole method to accelerate the calculation.The computational complexity can be reduced from in the traditional integral summation method to,where M and N are the numbers of observation sites and source elements,respectively.A combinational model and a complex model based on the digital elevation model(DEM)of Huaining County in Anhui province are used to test the proposed method.The results indicate that the proposed method is effective and accurate in three-dimensional magnetic forward modeling.
In this paper,we propose a fast and high-precision approach for large-scale magnetic forward modeling using the adaptive multi-level fast multipole(AMFM) method and an unstructured grid.The algorithm can overcome the shortcomings of the Fast fourier transfom(FFT)method which cannot adopt unstructured grids.In addition,the adaptive multi-level fast multipole(AMFM)method is chosen to solve the problem of high computational cost in traditional integral summation methods.Firstly,the unstructured tetrahedral meshing technique is used to properly approximate the complex model,for instance the terrain model,and realize thehigh-precision simulation of the magnetic model.Secondly,an adaptive multi-level fast multipole(AMFM)method is employed to accelerate large-scale magnetic forward modeling.Then,we divide the integral region into two parts.One is the near part in which we use analytical solution to get high-precision results,and the other is far part in which we choose the the adaptive multilevel fast multipole method to accelerate the calculation.The computational complexity can be reduced from in the traditional integral summation method to,where M and N are the numbers of observation sites and source elements,respectively.A combinational model and a complex model based on the digital elevation model(DEM)of Huaining County in Anhui province are used to test the proposed method.The results indicate that the proposed method is effective and accurate in three-dimensional magnetic forward modeling.
引文
Abramowitz M,Stegun I A.1964.Handbook of Mathematical Functions,with Formulas,Graphs,and Mathematical Tables.New York:Courier Corporation.
An Y L,Chen T D,Huang J M.2003.Advance in theory and method for forward and inverse problem of gravity and magnetic prospecting.Earth Science Frontiers(in Chinese),10(1):141-149.
Blakely R J.1996.Potential Theory in Gravity and Magnetic Applications.Cambridge:Cambridge University Press.
Borradaile G J,Henry B.1997.Tectonic applications of magnetic susceptibility and its anisotropy.Earth-Science Reviews,42(1-2):49-93.
Cady J W.1980.Calculation of gravity and magnetic anomalies of finite-length right polygonal prisms.Geophysics,45(10):1507-1512.
Cakir O.2006.The multilevel fast multipole method for forward modelling the multiply scattered seismic surface waves.Geophysical Journal International,167(2):663-678.
Chen C J,Ren Z Y,Pan K J,et al.2018.Exact solutions of the vertical gravitational anomaly for a polyhedral prism with vertical polynomial density contrast of arbitrary orders.Geophysical Journal International,214(3):2115-2132,doi:10.1093/gji/ggy250.
Chen Z X,Meng X H,Liu G F,et al.2012.The GPU-baseds parallel calculation of gravity and magnetic anomalies for 3Darbitrary bodies.Procedia Environmental Sciences,12:628-633.
Davis K,Li Y G.2010.3Djoint inversion of gradient and total-field magnetic data.∥80th Ann.Internat Mtg.,Soc.Expi.Geophys..Expanded Abstracts,1784-1788.
Dolgal A S,Balk P I,Demenev A G,et al.2012.The Finiteelement method application for interpretation of gravity and magnetic data.Vestn Kraunts,1(19):108-127.
Fan J C,Mao X C,Zhao Y,et al.2012.Voxel model and its magnetic forward model of stereoscopic prediction of concealed ore body.The Chinese Journal of Nonferrous Metals(in Chinese),22(1):239-250.
Ferguson A S,Zhang X,Stroink G.1994.A complete linear discretization for calculating the magnetic field using the boundary element method.IEEE Transactions on Biomedical Engineering,41(5):455-460.
Guan Z N.1997.Researches and progresses of magnetic prospecting in China.Acta Geophysica Sinica(in Chinese),40(S1):299-307.
Hwang C,Wang C G,Hsiao Y S.2003.Terrain correction computation using Gaussian quadrature.Computers&Geosciences,29(10):1259-1268.
Jia C S.1999.A method for estimating the magnetic anomaly of an uniformly-magnetized polyhedron.Oil Geophysical Prospecting(in Chinese),34(4):430-464.
Lelièvre P G,Oldenburg D W.2006.Magnetic forward modelling and inversion for high susceptibility.Geophysical Journal International,166(1):76-90.
Lelièvre P G.2003.Forward modelling and inversion of geophysical magnetic data[Master′s thesis].Vancouver:University of British Columbia.
Li Y G,Oldenburg D W.2010.Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method.Geophysical Journal of the Royal Astronomical Society,152(2):251-265.
Liu Y J.2009.Fast Multipole Boundary Element Method:Theory and Applications in Engineering.Cambridge:Cambridge University Press.
May D A,Knepley M G.2011.Optimal,scalable forward models for computing gravity anomalies.Geophysical Journal International,187(1):161-177.
Okabe M.1979.Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies.Geophysics,44(4):730-741.
Ren Z Y,Kalscheuer T,Greenhalgh S,et al.2013a.Boundary element solutions for broad-band 3-D geo-electromagnetic problems accelerated by an adaptive multilevel fast multipole method.Geophysical Journal International,192(2):473-499.
Ren Z Y,Kalscheuer T,Greenhalgh S,et al.2013b.A goal-oriented adaptive finite-element approach for plane wave 3-D electromagnetic modelling.Geophysical Journal International,194(2):700-718.
Ren Z Y,Kalscheuer T,Greenhalgh S,et al.2014.A hybrid boundary element-finite element approach to modeling plane wave 3D electromagnetic induction responses in the Earth.Journal of Computational Physics,258:705-717.
Ren Z Y,Chen C J,Tang J T,et al.2017a.A new integral equation approach for 3D magnetotelluric modeling.Chinese Journal of Geophysics(in Chinese),60(11):4506-4515,doi:10.6038/cjg20171134.
Ren Z Y,Chen C J,Tang J T,et al.2017b.Closed-form formula magnetic gradient tensor for a homogeneous polyhedral magnetic target:A tetrahedral grid example.Geophysics,82(6):WB21-WB28.
Ren Z Y,Chen C J,Pan K J,et al.2017c.Gravity anomalies of arbitrary 3Dpolyhedral bodies with horizontal and vertical mass contrasts.Surveys in Geophysics,38(2):479-502.
Ren Z Y,Tang J T,Kalscheuer T,et al.2017d.Fast 3-D largescale gravity and magnetic modeling using unstructured grids and an adaptive multilevel fast multipole method.Journal of Geophysical Research:Solid Earth,122(1):79-109.
Ren Z Y,Zhong Y J,Chen C J,et al.2018a.Gravity anomalies of arbitrary 3Dpolyhedral bodies with horizontal and vertical mass contrasts up to cubic order.Geophysics,83(1):G1-G13,doi:10.1190/geo2017-0219.1.
Ren Z Y,Chen H,Tang J T.2018b.Accurate volume integral solutions of direct current resistivity potentials for inhomogeneous conductivities in half space.Journal of Applied Geophysics,151:40-46.
Schmidt P,Clark D,Leslie K,et al.2004.GETMAG-A SQUIDmagnetic tensor gradiometer for mineral and oil exploration.Exploration Geophysics,35(4):297-305.
Schobert D T,Eibert T F.2012.Fast integral equation solution by multilevel Green′s function interpolation combined with multilevel fast multipole method.IEEE Transactions on Antennas and Propagation,60(9):4458-4463.
Si H,TetGen A.2006.A quality tetrahedral mesh generator and three-dimensional delaunay triangulator.Berlin,Germany:Weierstrass Institute for Applied Analysis and Stochastic.
Si H.2015.TetGen,a delaunay-based quality tetrahedral mesh generator.ACM Transactions on Mathematical Software,41(2):1-36.
Tontini F C,Cocchi L,Carmisciano C.2009.Rapid 3-D forward model of potential fields with application to the Palinuro Seamount magnetic anomaly(southern Tyrrhenian Sea,Italy).Journal of Geophysical Research:Solid Earth,114(B2):B02103,doi:10.1029/2008JB005907.
Tontini F C.2012.Rapid interactive modeling of 3D magnetic anomalies.Computers&Geosciences,48:308-315.
Verduzco B,Fairhead J D,Green C M,et al.2004.New insights into magnetic derivatives for structural mapping.The Leading Edge,23(2):116-119,doi:10.1190/1.1651454.
Wang S H.1981.On the theory using the finite element method to solve direct problem of magnetic exploration.Chinese Journal of Geophysics(in Chinese),24(2):206-217.
Won I J,Bevis M.1987.Computing the gravitational and magnetic anomalies due to a polygon:algorithms and fortran subroutines.Geophysics,52(2):232-238.
Xu S Z,Lou Y J.1986.A method of calculating magnetic anomaly of an arbitrary body.Computing Techniques for Geophysical&Geochemical Exploration(in Chinese),18(4):260-275.
Yao C L,Hao T Y,Guan Z N,et al.2003.High-speed computation and efficient storage in 3-D gravity and magnetic inversion.Chinese Journal of Geophysics,46(2):351-361.
Yao C L.2003.High-speed computation and efficient storage in-3Dgravity and magnetic inversion based on genetic algorithms.Chinese Journal of Geophysics(in Chinese),46(2):252-258.
Yoshida K I.2001.Applications of fast multipole method to boundary integral equation method[Ph.D.thesis].Kyoto:Kyoto University.
Zhang Y,Chen C,Du J,et al.2015.Forward modeling method of gravity and magnetic fields and their gradient tensors based on3-D delaunay discretization in cartesian and spherical coordinate systems.∥American Geophysical Union,Fall Meeting 2015.
安玉林,陈玉东,黄金明.2003.重磁勘探正反演理论方法研究的新进展.地学前缘,10(1):141-149.
陈召曦,孟小红,刘国峰等.2012.基于GPU的任意三维复杂形体重磁异常快速计算.物探与化探,36(1):117-121.
樊俊昌,毛先成,赵莹等.2012.隐伏矿体立体预测的体元模型及其磁法正演模型.中国有色金属学报,22(1):239-250.
管志宁.1997.我国磁法勘探的研究与进展.地球物理学报,40(S1):299-307.
贾春生.1999.均匀磁化多面体磁异常计算方法.石油地球物理勘探,34(4):430-464.
任政勇,陈超健,汤井田等.2017.一种新的三维大地电磁积分方程正演方法.地球物理学报,60(11):4506-4515,doi:10.6038/cjg20171134.
王书惠.1981.关于用有限元法作磁法勘探正演计算的理论问题.地球物理学报,24(1):207-217.
徐世浙,楼云菊.1986.计算任意形体磁异常的边界元法.物探化探计算技术,18(4):260-275.
姚长利.2003.重磁遗传算法三维反演中高速计算及有效存储方法技术.地球物理学报,46(2):252-258.
An Y L,Chen T D,Huang J M.2003.Advance in theory and method for forward and inverse problem of gravity and magnetic prospecting.Earth Science Frontiers(in Chinese),10(1):141-149.
Blakely R J.1996.Potential Theory in Gravity and Magnetic Applications.Cambridge:Cambridge University Press.
Borradaile G J,Henry B.1997.Tectonic applications of magnetic susceptibility and its anisotropy.Earth-Science Reviews,42(1-2):49-93.
Cady J W.1980.Calculation of gravity and magnetic anomalies of finite-length right polygonal prisms.Geophysics,45(10):1507-1512.
Cakir O.2006.The multilevel fast multipole method for forward modelling the multiply scattered seismic surface waves.Geophysical Journal International,167(2):663-678.
Chen C J,Ren Z Y,Pan K J,et al.2018.Exact solutions of the vertical gravitational anomaly for a polyhedral prism with vertical polynomial density contrast of arbitrary orders.Geophysical Journal International,214(3):2115-2132,doi:10.1093/gji/ggy250.
Chen Z X,Meng X H,Liu G F,et al.2012.The GPU-baseds parallel calculation of gravity and magnetic anomalies for 3Darbitrary bodies.Procedia Environmental Sciences,12:628-633.
Davis K,Li Y G.2010.3Djoint inversion of gradient and total-field magnetic data.∥80th Ann.Internat Mtg.,Soc.Expi.Geophys..Expanded Abstracts,1784-1788.
Dolgal A S,Balk P I,Demenev A G,et al.2012.The Finiteelement method application for interpretation of gravity and magnetic data.Vestn Kraunts,1(19):108-127.
Fan J C,Mao X C,Zhao Y,et al.2012.Voxel model and its magnetic forward model of stereoscopic prediction of concealed ore body.The Chinese Journal of Nonferrous Metals(in Chinese),22(1):239-250.
Ferguson A S,Zhang X,Stroink G.1994.A complete linear discretization for calculating the magnetic field using the boundary element method.IEEE Transactions on Biomedical Engineering,41(5):455-460.
Guan Z N.1997.Researches and progresses of magnetic prospecting in China.Acta Geophysica Sinica(in Chinese),40(S1):299-307.
Hwang C,Wang C G,Hsiao Y S.2003.Terrain correction computation using Gaussian quadrature.Computers&Geosciences,29(10):1259-1268.
Jia C S.1999.A method for estimating the magnetic anomaly of an uniformly-magnetized polyhedron.Oil Geophysical Prospecting(in Chinese),34(4):430-464.
Lelièvre P G,Oldenburg D W.2006.Magnetic forward modelling and inversion for high susceptibility.Geophysical Journal International,166(1):76-90.
Lelièvre P G.2003.Forward modelling and inversion of geophysical magnetic data[Master′s thesis].Vancouver:University of British Columbia.
Li Y G,Oldenburg D W.2010.Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method.Geophysical Journal of the Royal Astronomical Society,152(2):251-265.
Liu Y J.2009.Fast Multipole Boundary Element Method:Theory and Applications in Engineering.Cambridge:Cambridge University Press.
May D A,Knepley M G.2011.Optimal,scalable forward models for computing gravity anomalies.Geophysical Journal International,187(1):161-177.
Okabe M.1979.Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies.Geophysics,44(4):730-741.
Ren Z Y,Kalscheuer T,Greenhalgh S,et al.2013a.Boundary element solutions for broad-band 3-D geo-electromagnetic problems accelerated by an adaptive multilevel fast multipole method.Geophysical Journal International,192(2):473-499.
Ren Z Y,Kalscheuer T,Greenhalgh S,et al.2013b.A goal-oriented adaptive finite-element approach for plane wave 3-D electromagnetic modelling.Geophysical Journal International,194(2):700-718.
Ren Z Y,Kalscheuer T,Greenhalgh S,et al.2014.A hybrid boundary element-finite element approach to modeling plane wave 3D electromagnetic induction responses in the Earth.Journal of Computational Physics,258:705-717.
Ren Z Y,Chen C J,Tang J T,et al.2017a.A new integral equation approach for 3D magnetotelluric modeling.Chinese Journal of Geophysics(in Chinese),60(11):4506-4515,doi:10.6038/cjg20171134.
Ren Z Y,Chen C J,Tang J T,et al.2017b.Closed-form formula magnetic gradient tensor for a homogeneous polyhedral magnetic target:A tetrahedral grid example.Geophysics,82(6):WB21-WB28.
Ren Z Y,Chen C J,Pan K J,et al.2017c.Gravity anomalies of arbitrary 3Dpolyhedral bodies with horizontal and vertical mass contrasts.Surveys in Geophysics,38(2):479-502.
Ren Z Y,Tang J T,Kalscheuer T,et al.2017d.Fast 3-D largescale gravity and magnetic modeling using unstructured grids and an adaptive multilevel fast multipole method.Journal of Geophysical Research:Solid Earth,122(1):79-109.
Ren Z Y,Zhong Y J,Chen C J,et al.2018a.Gravity anomalies of arbitrary 3Dpolyhedral bodies with horizontal and vertical mass contrasts up to cubic order.Geophysics,83(1):G1-G13,doi:10.1190/geo2017-0219.1.
Ren Z Y,Chen H,Tang J T.2018b.Accurate volume integral solutions of direct current resistivity potentials for inhomogeneous conductivities in half space.Journal of Applied Geophysics,151:40-46.
Schmidt P,Clark D,Leslie K,et al.2004.GETMAG-A SQUIDmagnetic tensor gradiometer for mineral and oil exploration.Exploration Geophysics,35(4):297-305.
Schobert D T,Eibert T F.2012.Fast integral equation solution by multilevel Green′s function interpolation combined with multilevel fast multipole method.IEEE Transactions on Antennas and Propagation,60(9):4458-4463.
Si H,TetGen A.2006.A quality tetrahedral mesh generator and three-dimensional delaunay triangulator.Berlin,Germany:Weierstrass Institute for Applied Analysis and Stochastic.
Si H.2015.TetGen,a delaunay-based quality tetrahedral mesh generator.ACM Transactions on Mathematical Software,41(2):1-36.
Tontini F C,Cocchi L,Carmisciano C.2009.Rapid 3-D forward model of potential fields with application to the Palinuro Seamount magnetic anomaly(southern Tyrrhenian Sea,Italy).Journal of Geophysical Research:Solid Earth,114(B2):B02103,doi:10.1029/2008JB005907.
Tontini F C.2012.Rapid interactive modeling of 3D magnetic anomalies.Computers&Geosciences,48:308-315.
Verduzco B,Fairhead J D,Green C M,et al.2004.New insights into magnetic derivatives for structural mapping.The Leading Edge,23(2):116-119,doi:10.1190/1.1651454.
Wang S H.1981.On the theory using the finite element method to solve direct problem of magnetic exploration.Chinese Journal of Geophysics(in Chinese),24(2):206-217.
Won I J,Bevis M.1987.Computing the gravitational and magnetic anomalies due to a polygon:algorithms and fortran subroutines.Geophysics,52(2):232-238.
Xu S Z,Lou Y J.1986.A method of calculating magnetic anomaly of an arbitrary body.Computing Techniques for Geophysical&Geochemical Exploration(in Chinese),18(4):260-275.
Yao C L,Hao T Y,Guan Z N,et al.2003.High-speed computation and efficient storage in 3-D gravity and magnetic inversion.Chinese Journal of Geophysics,46(2):351-361.
Yao C L.2003.High-speed computation and efficient storage in-3Dgravity and magnetic inversion based on genetic algorithms.Chinese Journal of Geophysics(in Chinese),46(2):252-258.
Yoshida K I.2001.Applications of fast multipole method to boundary integral equation method[Ph.D.thesis].Kyoto:Kyoto University.
Zhang Y,Chen C,Du J,et al.2015.Forward modeling method of gravity and magnetic fields and their gradient tensors based on3-D delaunay discretization in cartesian and spherical coordinate systems.∥American Geophysical Union,Fall Meeting 2015.
安玉林,陈玉东,黄金明.2003.重磁勘探正反演理论方法研究的新进展.地学前缘,10(1):141-149.
陈召曦,孟小红,刘国峰等.2012.基于GPU的任意三维复杂形体重磁异常快速计算.物探与化探,36(1):117-121.
樊俊昌,毛先成,赵莹等.2012.隐伏矿体立体预测的体元模型及其磁法正演模型.中国有色金属学报,22(1):239-250.
管志宁.1997.我国磁法勘探的研究与进展.地球物理学报,40(S1):299-307.
贾春生.1999.均匀磁化多面体磁异常计算方法.石油地球物理勘探,34(4):430-464.
任政勇,陈超健,汤井田等.2017.一种新的三维大地电磁积分方程正演方法.地球物理学报,60(11):4506-4515,doi:10.6038/cjg20171134.
王书惠.1981.关于用有限元法作磁法勘探正演计算的理论问题.地球物理学报,24(1):207-217.
徐世浙,楼云菊.1986.计算任意形体磁异常的边界元法.物探化探计算技术,18(4):260-275.
姚长利.2003.重磁遗传算法三维反演中高速计算及有效存储方法技术.地球物理学报,46(2):252-258.
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