三维大地电磁自适应正则化有限内存拟牛顿反演
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摘要
有别于传统基于梯度信息的反演方法在正则化约束中用总梯度逼近海塞逆矩阵的技术,本文将正则化约束问题的数据拟合项和模型光滑项分开考虑,只利用数据拟合函数的梯度信息对数据拟合项的海塞矩阵进行逼近,通过求解类高斯牛顿下降方向方程得到不依赖前几次迭代正则化因子的更精确下降方向,在求解当前迭代下降方向的过程中,通过保证右端项中两个向量的二范数在同一数量级的原则,实现了正则化因子的自动更新.对理论模型的试算表明这种自适应正则化反演方案可以在拟牛顿反演框架下基本达到OCCAM的算法稳定性,反演结果对初始模型依赖性较小,同时又无需在一次迭代中多次搜索最佳正则化因子.本文还基于此算法讨论了大地电磁各参数对于反演结果的影响,由于本文的反演结果能得到充分的正则化约束,因而在此框架下讨论阻抗和倾子在反演中的作用相对更为客观.
In a traditional inversion method based on gradient information,the regularization constraint uses the total gradient to approximate the Hessian inverse matrix.This work proposes a new approach different from this method,in which the data misfit and the model roughness items of the regularization constrained problem are considered separately,and only the gradient information of data fitting function is employed to approximate the Hessian matrix.Then by solving the Gaussian-Newton descent direction equation,a more accurate descent direction is determined without relying on the previous iterations of regularization factors.Consequently an adapting regularization parameter can be obtained aiming at maintaining the same numeric levels for normal numbers of both averaged data misfit gradient and model roughness gradient.A synthetic dataset with a very sparse sites and frequencies is calculated by this approach,yielding very stable and reliable results with different initial model settings.These indicate that our method is as stabilized as OCCAM,little relying on initial models.Meanwhile,since only the data misfit and its gradient need to be computed once almost in every iteration.Based on thisalgorithm,this paper also discusses the influence of the magnetotelluric parameters on the inversion results.As the inversion results in this paper can get sufficient regularization constraints,the role of impedance and tipper in the inversion is relatively more objective.
In a traditional inversion method based on gradient information,the regularization constraint uses the total gradient to approximate the Hessian inverse matrix.This work proposes a new approach different from this method,in which the data misfit and the model roughness items of the regularization constrained problem are considered separately,and only the gradient information of data fitting function is employed to approximate the Hessian matrix.Then by solving the Gaussian-Newton descent direction equation,a more accurate descent direction is determined without relying on the previous iterations of regularization factors.Consequently an adapting regularization parameter can be obtained aiming at maintaining the same numeric levels for normal numbers of both averaged data misfit gradient and model roughness gradient.A synthetic dataset with a very sparse sites and frequencies is calculated by this approach,yielding very stable and reliable results with different initial model settings.These indicate that our method is as stabilized as OCCAM,little relying on initial models.Meanwhile,since only the data misfit and its gradient need to be computed once almost in every iteration.Based on thisalgorithm,this paper also discusses the influence of the magnetotelluric parameters on the inversion results.As the inversion results in this paper can get sufficient regularization constraints,the role of impedance and tipper in the inversion is relatively more objective.
引文
Avdeev D,Avdeeva A.2009.3D magnetotelluric inversion using a limited-memory quasi-Newton optimization.Geophysics,74(3):F45-F57．
Avdeeva A.2008.Three-dimensional magnetotelluric inversion[Ph.D.thesis].Galway:National University of Ireland．
Byrd R H,Nocedal J,Schnabel R B.1994.Representations of quasi-Newton matrices and their use in limited memory methods.Mathematical Programming,63(1-3):129-156．
Chen X B,Zhao G Z,Tang J,et al.2005.An adaptive regularized inversion algorithm for magnetotelluric data.Chinese J.Geophys.(in Chinese),48(4):937-946．
Constable S C,Parker R L,Constable C G.1987.Occam′s inversion:a practical algorithm for generating smooth models from electromagnetic sounding data.Geophysics,52(3):289-300．
Egbert G D,Kelbert A.2012.Computational recipes for electromagnetic inverse problems.Geophysical Journal International,189(1):251-267．
Hansen P C,O′Leary D P.1993.The use of the L-curve in the regularization of discrete ill-posed problems.SIAM Journal on Scientific Computing,14(6):1487-1503．
Kelbert A,Meqbel N,Egbert G D,et al.2014.ModEM:Amodular system for inversion of electromagnetic geophysical data.Computers&Geosciences,66:40-53．
Mackie R L,Madden T R,Wannamaker P E.1993.Three-dimensional magnetotelluric modeling using difference equations;Theory and comparisons to integral equation solutions.Geophysics,58(2):215-226．
MoréJ J,Thuente D J.1994.Line search algorithms with guaranteed sufficient decrease.ACM Transactions on Mathematical Software(TOMS),20(3):286-307．
Newman G A,Alumbaugh D L.2000.Three-dimensional magnetotelluric inversion using non-linear conjugate gradients.Geophysical Journal International,140(2):410-424．
Nocedal J.1980.Updating quasi-Newton matrices with limited storage.Mathematics of Computation,35(151):773-782．
Nocedal J,Wright S J.1999.Numerical optimization.2nd ed.New York:Springer．
Rodi W,Mackie R L.2001.Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion.Geophysics,66(1):174-187．
Siripunvaraporn W,Egbert G,Lenbury Y.2002.Numerical accuracy of magnetotelluric modeling:A comparison of finite difference approximations.Earth,Planets and Space,54(6):721-725．
Siripunvaraporn W,Egbert G,Lenbury Y,et al.2005.Threedimensional magnetotelluric inversion:data-space method.Physics of the Earth and Planetary Interiors,150(1-3):3-14．
Siripunvaraporn W,Egbert G.2009.WSINV3DMT:vertical magnetic field transfer function inversion and parallel implementation.Physics of the Earth and Planetary Interiors,173(3-4):317-329．
Tan H D,Yu Q F,Booker J,et al.2003.Magnetotelluric threedimensional modeling using the staggered-grid finitedifference method.Chinese J.Geophys.(in Chinese),46(5):705-711．
Wu X P,Xu G M.1998.Improvement of OCCAM′s inversion for MT data.Acta Geophysica Sinica(in Chinese),41(4):547-554．
Wu X P,Wang T T.2001.Study on some problems for 3Dresistivity inversion using conjugate gradient.Seismology and Geology(in Chinese),23(2):321-327．
Zhdanov M S.2002.Geophysical Inverse Theory and Regularization Problems,Methods in Geochemistry and Geophysics.Amsterdam:Elsevier．
Zhdanov M S,Gribenko A，ˇCuma M.2007.Regularized focusing inversion of marine CSEM data using minimum verticalsupport stabilizer．∥77th Ann.Internat Mtg.,Soc.Expi.Geophys..Expanded Abstracts,579-583．
陈小斌，赵国泽，汤吉等．2005．大地电磁自适应正则化反演算法．地球物理学报，48(4):937-946．
谭捍东，余钦范，Booker J等．2003．大地电磁法三维交错采样有限差分数值模拟．地球物理学报，46(5):705-711．
吴小平，徐果明．1998．大地电磁数据的OCCAM反演改进．地球物理学报，41(4):547-554．
吴小平，汪彤彤．2001．利用共轭梯度方法的电阻率三维反演的若干问题研究．地震地质，23(2):321-327.
Avdeeva A.2008.Three-dimensional magnetotelluric inversion[Ph.D.thesis].Galway:National University of Ireland．
Byrd R H,Nocedal J,Schnabel R B.1994.Representations of quasi-Newton matrices and their use in limited memory methods.Mathematical Programming,63(1-3):129-156．
Chen X B,Zhao G Z,Tang J,et al.2005.An adaptive regularized inversion algorithm for magnetotelluric data.Chinese J.Geophys.(in Chinese),48(4):937-946．
Constable S C,Parker R L,Constable C G.1987.Occam′s inversion:a practical algorithm for generating smooth models from electromagnetic sounding data.Geophysics,52(3):289-300．
Egbert G D,Kelbert A.2012.Computational recipes for electromagnetic inverse problems.Geophysical Journal International,189(1):251-267．
Hansen P C,O′Leary D P.1993.The use of the L-curve in the regularization of discrete ill-posed problems.SIAM Journal on Scientific Computing,14(6):1487-1503．
Kelbert A,Meqbel N,Egbert G D,et al.2014.ModEM:Amodular system for inversion of electromagnetic geophysical data.Computers&Geosciences,66:40-53．
Mackie R L,Madden T R,Wannamaker P E.1993.Three-dimensional magnetotelluric modeling using difference equations;Theory and comparisons to integral equation solutions.Geophysics,58(2):215-226．
MoréJ J,Thuente D J.1994.Line search algorithms with guaranteed sufficient decrease.ACM Transactions on Mathematical Software(TOMS),20(3):286-307．
Newman G A,Alumbaugh D L.2000.Three-dimensional magnetotelluric inversion using non-linear conjugate gradients.Geophysical Journal International,140(2):410-424．
Nocedal J.1980.Updating quasi-Newton matrices with limited storage.Mathematics of Computation,35(151):773-782．
Nocedal J,Wright S J.1999.Numerical optimization.2nd ed.New York:Springer．
Rodi W,Mackie R L.2001.Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion.Geophysics,66(1):174-187．
Siripunvaraporn W,Egbert G,Lenbury Y.2002.Numerical accuracy of magnetotelluric modeling:A comparison of finite difference approximations.Earth,Planets and Space,54(6):721-725．
Siripunvaraporn W,Egbert G,Lenbury Y,et al.2005.Threedimensional magnetotelluric inversion:data-space method.Physics of the Earth and Planetary Interiors,150(1-3):3-14．
Siripunvaraporn W,Egbert G.2009.WSINV3DMT:vertical magnetic field transfer function inversion and parallel implementation.Physics of the Earth and Planetary Interiors,173(3-4):317-329．
Tan H D,Yu Q F,Booker J,et al.2003.Magnetotelluric threedimensional modeling using the staggered-grid finitedifference method.Chinese J.Geophys.(in Chinese),46(5):705-711．
Wu X P,Xu G M.1998.Improvement of OCCAM′s inversion for MT data.Acta Geophysica Sinica(in Chinese),41(4):547-554．
Wu X P,Wang T T.2001.Study on some problems for 3Dresistivity inversion using conjugate gradient.Seismology and Geology(in Chinese),23(2):321-327．
Zhdanov M S.2002.Geophysical Inverse Theory and Regularization Problems,Methods in Geochemistry and Geophysics.Amsterdam:Elsevier．
Zhdanov M S,Gribenko A，ˇCuma M.2007.Regularized focusing inversion of marine CSEM data using minimum verticalsupport stabilizer．∥77th Ann.Internat Mtg.,Soc.Expi.Geophys..Expanded Abstracts,579-583．
陈小斌，赵国泽，汤吉等．2005．大地电磁自适应正则化反演算法．地球物理学报，48(4):937-946．
谭捍东，余钦范，Booker J等．2003．大地电磁法三维交错采样有限差分数值模拟．地球物理学报，46(5):705-711．
吴小平，徐果明．1998．大地电磁数据的OCCAM反演改进．地球物理学报，41(4):547-554．
吴小平，汪彤彤．2001．利用共轭梯度方法的电阻率三维反演的若干问题研究．地震地质，23(2):321-327.
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