基于近似零范数稀疏恢复的迭代解法的三维重力反演（英文）
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摘要
重力反演是应用地球物理中一个经典的方法,可以获取地下空间的密度分布特征。本文研究了一种新的三维重力反演方法,以压缩感知中的稀疏恢复方法为基本原理。建立以零范数为约束的目标函数,采用近似零范数求解方法对目标函数进行迭代求解。以重力正演模型为基础进行反演方法研究,建立包含深度加权函数的零范数的目标函数,采用相应的最优化数学方法进行求解,得到稀疏的反演结果。同时,为了得到具有实际地质意义的反演结果,采用惩罚函数将反演密度值约束在具有实际地球物理和地质意义范围内。采用本文提出的反演方法得到的反演目标异常源具有较为清晰的边界、深度信息和剩余密度分布信息。本文利用理论模型实验来验证反演方法效果,并将得到的反演结果与设计的理论模型对比分析说明三维反演方法的有效性和可靠性。为了进一步说明本文采用的反演方法的可靠性,将其应用到美国德克萨斯州采集的实际重力数据反演,并将反演结果与前人研究结果和该地区的测井资料进行对比分析,盐丘的深度为4.2km与测井得到深度4.4km基本上是一致的,说明研究的反演方法具有实际应用价值。
This research proposes a novel three-dimensional gravity inversion based on sparse recovery in compress sensing. Zero norm is selected as the objective function, which is then iteratively solved by the approximate zero norm solution. The inversion approach mainly employs forward modeling; a depth weight function is introduced into the objective function of the zero norms. Sparse inversion results are obtained by the corresponding optimal mathematical method. To achieve the practical geophysical and geological significance of the results, penalty function is applied to constrain the density values. Results obtained by proposed provide clear boundary depth and density contrast distribution information. The method's accuracy, validity, and reliability are verified by comparing its results with those of synthetic models. To further explain its reliability, a practical gravity data is obtained for a region in Texas, USA is applied. Inversion results for this region are compared with those of previous studies, including a research of logging data in the same area. The depth of salt dome obtained by the inversion method is 4.2 km, which is in good agreement with the 4.4 km value from the logging data. From this, the practicality of the inversion method is also validated.
This research proposes a novel three-dimensional gravity inversion based on sparse recovery in compress sensing. Zero norm is selected as the objective function, which is then iteratively solved by the approximate zero norm solution. The inversion approach mainly employs forward modeling; a depth weight function is introduced into the objective function of the zero norms. Sparse inversion results are obtained by the corresponding optimal mathematical method. To achieve the practical geophysical and geological significance of the results, penalty function is applied to constrain the density values. Results obtained by proposed provide clear boundary depth and density contrast distribution information. The method's accuracy, validity, and reliability are verified by comparing its results with those of synthetic models. To further explain its reliability, a practical gravity data is obtained for a region in Texas, USA is applied. Inversion results for this region are compared with those of previous studies, including a research of logging data in the same area. The depth of salt dome obtained by the inversion method is 4.2 km, which is in good agreement with the 4.4 km value from the logging data. From this, the practicality of the inversion method is also validated.
引文
Abdelrahman,E. M.,1991,A least-squares minimization approach to invert gravity data:Geophysics, 56(1),115-118.
Barbosa,V. C. F.,and Silva,J. B.,1994,Generalized compact gravity inversion:Geophysics,59(1),57-68.
Bertete-Aguirre, H.,Cherkaev,E.,and Oristaglio,M.,2002, Non-smooth gravity problem with total variation penalization functional:Geophysical Journal International,149(2),499-507.
Blakely,R. J.,1996, Potential theory in gravity and magnetic applications:Cambridge University Press.
Bijani,R.,Ponteneto,C.F.,Carlos,D.U.,and Silva Dias,F.J.S., 2015, Three-dimensional gravity inversion using graph theory to delineate the skeleton of homogeneous sources.Geophysics, 80(2), G53-G66.
Candes,E. J.,and Tao,T.,2005,Decoding by linear programming:IEEE Transactions on,Information Theory,51(12), 4203-4215.
Cardarelli,E.,and Fischanger,F.,2006, 2d data modelling by electrical resistivity tomography for complex subsurface geology:Geophysical Prospecting, 54(2), 121-133.
Chen,S. S.,Donoho, D. L.,and Saunders,M. A.,1998,Atomic decomposition by basis pursuit:SIAM J. Sci.Comput., 20(1), 33-61.
Chen,S.,Saunders,M. A.,and Donoho, D. L.,2001,Atomic decomposition by basis pursuit. Siam Review,43(1),129-159.
Chen, G. X.,Chen, S. C.,and Wang,H. C.,2013,Geophysical data sparse reconstruction based on 10-norm minimization:Applied Geophysics, 10(2), 181-190.
Chen,G. X.,and Chen,S. C.,2014, Based on Lp norm sparse constrained regularization method of reconstruction of potential field data:Journal of Zhejiang University(Engineering Science)(4),748-785.
Cheng,X. L.,Zheng,X.,and Han,W. M.,2013, the sparse algorithm to solve the underdetermined linear equations:Applied Mathematics A Journal of Chinese Universities,No. 2, 235-248.
Commer,M.,2011, Three-dimensional gravity modelling and focusing inversion using rectangular meshes:Geophysical Prospecting, 59(5),966-979.
Commer,M.,and Newman,G. A.,2008, New advances in three-dimensional controlled-source electromagnetic inversion:Geophysical Journal International,172(2),513-535.
Donoho, D. L.,2006, High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension:Discrete&Computational Geometry, 35(4), 617-652.
Donoho and David L.,2006, For most large underdetermined systems of linear equations the minimal$lsb 1$-norm solution is also the sparsest solution. Comm:Pure Appl.Math,59, 797-829.
Elad,M.,2007,Optimized projections for compressed sensing. IEEE Transactions on Signal Processing,55(12),5695-5702.
Essa, K. S.,2007, Gravity data interpretation using the s-curves method:Journal of Geophysics and Engineering,4(2),204.
Farquharson, C. G., 2008, Constructing piecewiseconstant models in multidimensional minimum-structure inversions:Geophysics,73(1),K1-K9.
Fregoso, E.,and Gallardo, L. A.,2009, Cross-gradients joint3D inversion with applications to gravity and magnetic data:Geophysics,74(4),L31-L42.
Figueiredo, M. A.,Nowak,R. D.,and Wright,S. J. 2007,Gradient projection for sparse reconstruction:Application to compressed sensing and other inverse problems:IEEE Journal of Selected Topics in Signal processing,1(4),586-597.
Ghalehnoee,M. H.,Ansari,A.,and Ghorbani,A.,2017,Improving compact gravity inversion using new weighting functions:Geophysical Journal International,208(2),546-560.
Gholami,A.,and Siahkoohi,H. R.,2010,Regularization of linear and non-linear geophysical ill-posed problems with joint sparsity constraints:Geophysical Journal International,180(2),871-882.
Guillen,A.,and Menichetti,V., 1984, Gravity and magnetic inversion with minimization of a specific functional:Geophysics, 49(8), 1354-1360.
Kim,H. J.,Song,Y.,&Lee,K. H.,1999,Inequality constraint in least-squares inversion of geophysical data:Earth Planets&Space, 51(4), 255-259.
Last,B. J.,and Kubik,K.,1983, Compact gravity inversion:Geophysics,48(48),713-721.
Lelievre,P. G.,and Oldenburg,D. W.,2009, A comprehensive study of including structural orientation information in geophysical inversions:Geophysical Journal International,178(178), 623-637.
Li,Y.,and Oldenburg,D. W.,1996,3-D inversion of magnetic data:Geophysics,61(2),394-408.
Li,Y., and Oldenburg, D. W., 1998, 3-D inversion of gravity data:Geophysics, 63(1), 109-119.
Li,Y,,Cichocki,A.,and Amari,S. I.,2003,Sparse component analysis for blind source separation with less sensors than sources:Independent Component Analysis,89-94.
Ma,G. Q.,Du,X. J.,and Li,L. L.,2012, Comparison of the tensor local wavenumber method with the conventional local wavenumber method for interpretation of total tensor data of potential fields:Chinese Journal of Geophysics,55(7), 2450-2461.
Mallat,S. G.,and Zhang,Z.,1993, Matching pursuits with time-frequency dictionaries:IEEE Transactions on Signal Processing,41(12),3397-3415.
Mohan,N. L.,Anandababu,L.,and Rao,S. V. S.,1986,Gravity interpretation using the mellin transform:Digital Library Home,51(1),114-122.
Mohimani,H.,Babaie-Zadeh,M.,&Jutten,C.,2009, A fast approach for overcomplete sparse decomposition based on smoothed 10, norm:IEEE Press.
Nettleton,L. L.,1976, Gravity and magnetics in oil prospecting. McGraw-Hill Companies.
Oruc,B.,2010, Location and depth estimation of point-dipole and line of dipoles using analytic signals of the magnetic gradient tensor and magnitude of vector components:Journal of Applied Geophysics, 70(1), 27-37.
Portniaguine,O.,and Zhdanov, M. S.,1999,Focusing geophysical inversion images:Geophysics,64(3),874.
Portniaguine,O.,and Zhdanov,M. S.,2002, 3-D magnetic inversion with data compression and image focusing:Geophysics,67(5),1532-1541.
Qi,G.,Lv,Q. T.,Yan,J. Y.,Wu,M. A.,and Liu,Y.,2012,Geologic constrained 3D gravity and magnetic modeling of Nihe Deposit-A case study:Chinese Journal of Geophysics,55(12),4194-4206.
Salem,A., Ravat,D., Mushayandebvu,M. F.,and Ushijima,K.,2004,Linearized least-squares method for interpretation of potential-field data from sources of simple geometry:Geophysics,69(3),783-788.
Shaw, R. K.,and Agarwal,B. N. P., 1997, A generalized concept of resultant gradient to interpret potential field maps:Geophysical Prospecting,45(6),1003-1011.
Silva Dias,F. J. S.,Barbosa, V. C.,and Silva,J. B. C.,2009, 3D gravity inversion through an adaptive-learning procedure:Geophysics,74(3),I9-I21.
Silva Dias,F. J. S.,Barbosa,V. C. F.,and Silva,J. B. C.,2011, Adaptive learning 3d gravity inversion for salt-body imaging:Geophysics, 76(3), 149-157.
Silva, J. B. C.,and Barbosa,V. C. F.,2006, Gravity inversion of basement relief and estimation of density contrast variation with depth:Geophysics, 71, J51-J58.
Tontini,F. C.,Cocchi,L.,and Carmisciano, C.,2009, Rapid3-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).
Tontini,F.,Ronde,C. E. J.,Yoerger,D.,Kinsey,J.,and Tivey,M.,2012, 3-D focused inversion of near-seafloor magnetic data with application to the Brothers volcano hydrothermal system,Southern Pacific Ocean,New Zealand:Journal of Geophysical Research:Solid Earth,117(B10).
Wang,T. H.,Huang,D. N.,Ma,G. Q.,Meng,Z. H.,and Li,Y., 2017,Improved preconditioned conjugate gradient algorithm and application in 3D inversion of gravitygradiometry data:Applied Geophysics, 14(2),301-313.
Zhang,Y.,Wu,Y., Yan, J.,Wang,H.,Rodriguez,J. A. P., and Qiu,Y. 2018, 3d inversion of full gravity gradient tensor data in spherical coordinate system using local northoriented frame:Earth Planets&Space, 70(1), 58.
Zhdanov, M. S., Ellis, R., and Mukherjee, S., 2004, Threedimensional regularized focusing inversion of gravity gradient tensor component data:Geophysics,69(4),925-937.
Barbosa,V. C. F.,and Silva,J. B.,1994,Generalized compact gravity inversion:Geophysics,59(1),57-68.
Bertete-Aguirre, H.,Cherkaev,E.,and Oristaglio,M.,2002, Non-smooth gravity problem with total variation penalization functional:Geophysical Journal International,149(2),499-507.
Blakely,R. J.,1996, Potential theory in gravity and magnetic applications:Cambridge University Press.
Bijani,R.,Ponteneto,C.F.,Carlos,D.U.,and Silva Dias,F.J.S., 2015, Three-dimensional gravity inversion using graph theory to delineate the skeleton of homogeneous sources.Geophysics, 80(2), G53-G66.
Candes,E. J.,and Tao,T.,2005,Decoding by linear programming:IEEE Transactions on,Information Theory,51(12), 4203-4215.
Cardarelli,E.,and Fischanger,F.,2006, 2d data modelling by electrical resistivity tomography for complex subsurface geology:Geophysical Prospecting, 54(2), 121-133.
Chen,S. S.,Donoho, D. L.,and Saunders,M. A.,1998,Atomic decomposition by basis pursuit:SIAM J. Sci.Comput., 20(1), 33-61.
Chen,S.,Saunders,M. A.,and Donoho, D. L.,2001,Atomic decomposition by basis pursuit. Siam Review,43(1),129-159.
Chen, G. X.,Chen, S. C.,and Wang,H. C.,2013,Geophysical data sparse reconstruction based on 10-norm minimization:Applied Geophysics, 10(2), 181-190.
Chen,G. X.,and Chen,S. C.,2014, Based on Lp norm sparse constrained regularization method of reconstruction of potential field data:Journal of Zhejiang University(Engineering Science)(4),748-785.
Cheng,X. L.,Zheng,X.,and Han,W. M.,2013, the sparse algorithm to solve the underdetermined linear equations:Applied Mathematics A Journal of Chinese Universities,No. 2, 235-248.
Commer,M.,2011, Three-dimensional gravity modelling and focusing inversion using rectangular meshes:Geophysical Prospecting, 59(5),966-979.
Commer,M.,and Newman,G. A.,2008, New advances in three-dimensional controlled-source electromagnetic inversion:Geophysical Journal International,172(2),513-535.
Donoho, D. L.,2006, High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension:Discrete&Computational Geometry, 35(4), 617-652.
Donoho and David L.,2006, For most large underdetermined systems of linear equations the minimal$lsb 1$-norm solution is also the sparsest solution. Comm:Pure Appl.Math,59, 797-829.
Elad,M.,2007,Optimized projections for compressed sensing. IEEE Transactions on Signal Processing,55(12),5695-5702.
Essa, K. S.,2007, Gravity data interpretation using the s-curves method:Journal of Geophysics and Engineering,4(2),204.
Farquharson, C. G., 2008, Constructing piecewiseconstant models in multidimensional minimum-structure inversions:Geophysics,73(1),K1-K9.
Fregoso, E.,and Gallardo, L. A.,2009, Cross-gradients joint3D inversion with applications to gravity and magnetic data:Geophysics,74(4),L31-L42.
Figueiredo, M. A.,Nowak,R. D.,and Wright,S. J. 2007,Gradient projection for sparse reconstruction:Application to compressed sensing and other inverse problems:IEEE Journal of Selected Topics in Signal processing,1(4),586-597.
Ghalehnoee,M. H.,Ansari,A.,and Ghorbani,A.,2017,Improving compact gravity inversion using new weighting functions:Geophysical Journal International,208(2),546-560.
Gholami,A.,and Siahkoohi,H. R.,2010,Regularization of linear and non-linear geophysical ill-posed problems with joint sparsity constraints:Geophysical Journal International,180(2),871-882.
Guillen,A.,and Menichetti,V., 1984, Gravity and magnetic inversion with minimization of a specific functional:Geophysics, 49(8), 1354-1360.
Kim,H. J.,Song,Y.,&Lee,K. H.,1999,Inequality constraint in least-squares inversion of geophysical data:Earth Planets&Space, 51(4), 255-259.
Last,B. J.,and Kubik,K.,1983, Compact gravity inversion:Geophysics,48(48),713-721.
Lelievre,P. G.,and Oldenburg,D. W.,2009, A comprehensive study of including structural orientation information in geophysical inversions:Geophysical Journal International,178(178), 623-637.
Li,Y.,and Oldenburg,D. W.,1996,3-D inversion of magnetic data:Geophysics,61(2),394-408.
Li,Y., and Oldenburg, D. W., 1998, 3-D inversion of gravity data:Geophysics, 63(1), 109-119.
Li,Y,,Cichocki,A.,and Amari,S. I.,2003,Sparse component analysis for blind source separation with less sensors than sources:Independent Component Analysis,89-94.
Ma,G. Q.,Du,X. J.,and Li,L. L.,2012, Comparison of the tensor local wavenumber method with the conventional local wavenumber method for interpretation of total tensor data of potential fields:Chinese Journal of Geophysics,55(7), 2450-2461.
Mallat,S. G.,and Zhang,Z.,1993, Matching pursuits with time-frequency dictionaries:IEEE Transactions on Signal Processing,41(12),3397-3415.
Mohan,N. L.,Anandababu,L.,and Rao,S. V. S.,1986,Gravity interpretation using the mellin transform:Digital Library Home,51(1),114-122.
Mohimani,H.,Babaie-Zadeh,M.,&Jutten,C.,2009, A fast approach for overcomplete sparse decomposition based on smoothed 10, norm:IEEE Press.
Nettleton,L. L.,1976, Gravity and magnetics in oil prospecting. McGraw-Hill Companies.
Oruc,B.,2010, Location and depth estimation of point-dipole and line of dipoles using analytic signals of the magnetic gradient tensor and magnitude of vector components:Journal of Applied Geophysics, 70(1), 27-37.
Portniaguine,O.,and Zhdanov, M. S.,1999,Focusing geophysical inversion images:Geophysics,64(3),874.
Portniaguine,O.,and Zhdanov,M. S.,2002, 3-D magnetic inversion with data compression and image focusing:Geophysics,67(5),1532-1541.
Qi,G.,Lv,Q. T.,Yan,J. Y.,Wu,M. A.,and Liu,Y.,2012,Geologic constrained 3D gravity and magnetic modeling of Nihe Deposit-A case study:Chinese Journal of Geophysics,55(12),4194-4206.
Salem,A., Ravat,D., Mushayandebvu,M. F.,and Ushijima,K.,2004,Linearized least-squares method for interpretation of potential-field data from sources of simple geometry:Geophysics,69(3),783-788.
Shaw, R. K.,and Agarwal,B. N. P., 1997, A generalized concept of resultant gradient to interpret potential field maps:Geophysical Prospecting,45(6),1003-1011.
Silva Dias,F. J. S.,Barbosa, V. C.,and Silva,J. B. C.,2009, 3D gravity inversion through an adaptive-learning procedure:Geophysics,74(3),I9-I21.
Silva Dias,F. J. S.,Barbosa,V. C. F.,and Silva,J. B. C.,2011, Adaptive learning 3d gravity inversion for salt-body imaging:Geophysics, 76(3), 149-157.
Silva, J. B. C.,and Barbosa,V. C. F.,2006, Gravity inversion of basement relief and estimation of density contrast variation with depth:Geophysics, 71, J51-J58.
Tontini,F. C.,Cocchi,L.,and Carmisciano, C.,2009, Rapid3-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).
Tontini,F.,Ronde,C. E. J.,Yoerger,D.,Kinsey,J.,and Tivey,M.,2012, 3-D focused inversion of near-seafloor magnetic data with application to the Brothers volcano hydrothermal system,Southern Pacific Ocean,New Zealand:Journal of Geophysical Research:Solid Earth,117(B10).
Wang,T. H.,Huang,D. N.,Ma,G. Q.,Meng,Z. H.,and Li,Y., 2017,Improved preconditioned conjugate gradient algorithm and application in 3D inversion of gravitygradiometry data:Applied Geophysics, 14(2),301-313.
Zhang,Y.,Wu,Y., Yan, J.,Wang,H.,Rodriguez,J. A. P., and Qiu,Y. 2018, 3d inversion of full gravity gradient tensor data in spherical coordinate system using local northoriented frame:Earth Planets&Space, 70(1), 58.
Zhdanov, M. S., Ellis, R., and Mukherjee, S., 2004, Threedimensional regularized focusing inversion of gravity gradient tensor component data:Geophysics,69(4),925-937.
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