基于不等式约束的井地电阻率法三维非线性共轭梯度反演研究
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摘要
基于共轭梯度的最小构造反演方法是三维电阻率反演的主要方法,是基于线性系统的线性反演,而反问题的不适定性很大程度上与非线性问题的线性化有关,且存在着多解性较强的问题.为了改善以上问题,本文利用目前较为主流的非线性共轭梯度法(NLCG)实现了三维井地电阻率法的反演,将表征模型参数变化范围的不等式约束作为先验信息以惩罚函数法的方式引入到NLCG反演方法中.通过对多种理论模型的反演结果比较,有效的改善了反演结果的精度,在一定程度上降低了反演的多解性.在目标函数中加入构造位置的先验信息作为局部不等式约束的反演算例表明,给出的范围越精确,反演结果就越准确.
Inversion for minimum structure based on Conjugate Gradient is the main method of three-dimensional resistivity inversion. This is a kind of linear inversion relying on linear systems. However, ill-posed problems of inversion are largely related to linearization of nonlinear questions,and the ambiguity is strong in some cases. In order to alleviate the problems above,a prevailing method,Non Linear Conjugate Gradient( NLCG),was adopted to achieve three-dimensional hole-to-surface resistivity inversion. Also penalty function which represented the range of model parameters was introduced into inversion of NLCG as priori information. After comparing the inversion results from a variety of theoretical models, we found that this method can effectively improve the accuracy of the inversion results, and reduce the ambiguity of inversion. Meanwhile,the inversion instances which took structural position as a local inequality constraints when setting the priori information indicated that the more precise the location of structure was,the more accurate the inversion results were.
Inversion for minimum structure based on Conjugate Gradient is the main method of three-dimensional resistivity inversion. This is a kind of linear inversion relying on linear systems. However, ill-posed problems of inversion are largely related to linearization of nonlinear questions,and the ambiguity is strong in some cases. In order to alleviate the problems above,a prevailing method,Non Linear Conjugate Gradient( NLCG),was adopted to achieve three-dimensional hole-to-surface resistivity inversion. Also penalty function which represented the range of model parameters was introduced into inversion of NLCG as priori information. After comparing the inversion results from a variety of theoretical models, we found that this method can effectively improve the accuracy of the inversion results, and reduce the ambiguity of inversion. Meanwhile,the inversion instances which took structural position as a local inequality constraints when setting the priori information indicated that the more precise the location of structure was,the more accurate the inversion results were.
引文
Dai Q W,Hou Z Q.2013.Anomalous analysis of the 2.5-D finite element numerical simulation of borehole-ground electrical method[J].Computing Techniques for Geophysical and Geochemical Exploration(in Chinese),35(4):457-462.
Dong H,Wei W B,Ye G F,et al.2014.Study of three-dimensional magnetotelluric inversion including surface topography based on finite-difference method[J].Chinese J.Geophys.(in Chinese),57(3):939-952,doi:10.6038/cjg20140323.
Hu Z Z,Hu X Y,He Z X.2006.Pseudo-Three-Dimensional magnetotelluric inversion using nonlinear conjugate gradients[J].Chinese J.Geophys.(in Chinese),49(4):1226-1234.
Huang J G,Ruan B Y,Bao G S.2004a.Research on 3-D resistivity least-square inversion using the finite element method[J].Geology and Prospecting(in Chinese),40(4):70-75.
Huang J G,Ruan B Y,Bao G S.2004b.Resistivity inversion on 3-Dsection based on FEM[J].Journal of Central South University(Science and Technology)(in Chinese),35(2):295-299.
Huang Z L,1999.Gravity,Magnetic and Electrical Prospecting of Petroleum(in Chinese)[M].Beijing:Peroleum University Publishing House.
Jia Z Y,Zhang G.2012.Research on Finite-difference Numerical Simulation for 3-D Borehole-to-ground Resistivity Survey[J].Geoscience(in Chinese),26(6):1199-1205.
Ke G P,Huang Q H.2009.3D forward and inversion problems of borehole-to-surface electrical method[J].Acta Scientiarum Naturalium Universitatis Pekinensis(in Chinese),45(2):264-272.
Lin C H,Tan H D,Shu Q,et al.2012.Three-dimensional conjugate gradient inversion of CSAMT data[J].Chinese J.Geophys.(in Chinese),55(11):3829-3838,doi:10.6038/j.issn.0001-5733.2012.11.030.
Liu B,Li S C,Li S C,et al.2012a.3D electrical resistivity inversion with least-squares method based on inequality constraint and its computation efficiency optimization[J].Chinese J.Geophys.(in Chinese),55(1):260-268,doi:10.6038/j.issn.0001-5733.2012.01.025.
Liu B,Li S C,Nie L C,et al.2012b.Inversion imaging of 3D resistivity detection using adaptive-weighted smooth constraint and PCGalgorithm[J].Chinese Journal of Geotechnical Engineering(in Chinese),34(9):1646-1653.
Liu B,Nie L C,Li S C,et al.2012c.3D electrical resistivity inversion tomography with spatial structural constraint[J].Chinese Journal of Rock Mechanics and Engineering(in Chinese),31(11):2258-2268.
Liu H F.2007.Linear and non-linear inversion methods research on direct current IP measurements(in Chinese)[D].Changsha:Central South University.
Liu H F,Chen D P,Dai Q W,et al.2011.3D FEM modeling of borehole-surface potential with line current source in semiunderground space of continuous variation of conductivity[J].Journal of Guilin University of Technology(in Chinese),31(1):28-38.
Liu J X,Jiang P F,Tong X Z,et al.2009.Application of BICGSTABalgorithm with incomplete LU decomposition preconditioning to twodimensional magnetotelluric forward modeling[J].J.Cent.South Univ.(Sci.Technol.),40(2):484-491.
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Mackie R L,Madden,T R.1993.Three-dimensional magnetotelluric inversion using conjugate gradients[J].Geophysical Journal International,115(1):215-229.
Newman G A,Alumbaugh D L.2000.Three-dimensional magnetotelluric inversion using non-linear conjugate gradients[J].Geophysical Journal International,140(2):410-424.
Qu Y H,Zhang B G,Zhao L F,et al.2007.Study on 3-D resistivity inversion for infinite surface-borehole line current source[J].Progress in Geophysics(in Chinese),22(5):1393-1402,doi:10.3969/j.issn.1004-2903.2007.05.009.
Rodi W,Mackie R L.2001.Nonlinear conjugate gradients algorithm for2-D magnetotelluric inversion[J].Geophysics,66(1):174-187.
Sasaki Y.1994.3-D resistivity inversion using the finite-element method[J].Geophysics,59(12):1839-1850.
Wan X L,Xi D Y,Gao E G,et al.2005.3-D resistivity inversion by the least-squares QR factorization method under improved smoothness constraint condition[J].Chinese J.Geophys.(in Chinese),48(2):439-444.
Wang Z G,He Z X,Wei W B.2006.Study on some problems upon 3Dmodeling of DC borehole-ground method[J].Computing Techniques for Geophysical and Geochemical Exploration(in Chinese),28(4):322-327.
Weng A H,Liu Y H,Jia D Y,et al.2012.Three-dimensional controlled source electromagnetic inversion using non-linear conjugate gradients[J].Chinese J.Geophys.(in Chinese),55(10):3506-3515,doi:10.6038/j.issn.0001-5733.2012.10.034.
Wu X P,Xu G M,1999.Derivation and analysis of partial derivative matrix in resistivity 3-D inversion.[J].Oil Geophysical Prospecting(in Chinese),34(4):363-372.
Wu X P,Xu G M.2000.Study on 3-D resistivity inversion using conjugate gradient method[J].Chinese J.Geophys.(in Chinese),43(3):420-427.
Wu X P,Wang T T.2001.Study on some problems for 3D resistivity inversion using conjugate gradient[J].Seismology and Geology(in Chinese),23(2):321-327.
Wu X P,Wang T T.2002a.Study of stability and reliability of 3-Dresistivity inversion[J].Coal Geology and Exploration(in Chinese),30(1):57-60.
Wu X P,Wang T T.2002b.Progress of study on 3D resistivity inversion methods[J].Progress in Geophysics(in Chinese),17(1):156-162,doi:10.3969/j.issn.1004-2903.2002.01.023.
Wu X P,Wang T T.2003.A 3-D finite-element resistivity forward modeling using conjugate gradient algorithm[J].Chinese J.Geophys.(in Chinese),46(3):428-432.
Wu X P.2004.Rapid 3-D inversion of induced polarization data using conjugate gradient method[J].Coal Geology&Exploration(in Chinese),32(5):62-64.
Wu X P.2005.3-D resistivity inversion under the condition of uneven terrain[J].Chinese J.Geophys.(in Chinese),48(4):932-936.
Xu K J,Li T L.2006.The forward modeling of three-dimensional geoelectric filed of vertical finite line source by finite-difference method[J].Journal of Jilin University(Earth Science Edition)(in Chinese),36(1):137-141.
Xu K J,Li T L,Zhang H,et al.2006.3D resistivity inversion of vertical finite line source using conjugate gradients[J].Coal Geology&Exploration(in Chinese),34(3):68-71.
Yuan Y X,Sun W Y.1997.Theory and Methods of Optimization(in Chinese)[M].Beijing:Science Press.
Zhang J,Mackie R L,Madden T R.1995.3-D resistivity forward modeling and inversion using conjugate gradients[J].Geophysics,60(5):1313-1325.
戴前伟,侯智超,王洪华.2013.井地电位2.5维有限元数值模拟异常分析[J].物探化探计算技术,35(4):457-462.
董浩,魏文博,叶高峰,等.2014.基于有限差分正演的带地形三维大地电磁反演方法[J].地球物理学报,57(3):939-952,doi:10.6038/cjg20140323.
胡祖志,胡祥云,何展翔.2006.大地电磁非线性共轭梯度拟三维反演[J].地球物理学报,49(4):1226-1234.
黄俊革,阮百尧,鲍光淑.2004a.有限单元法三维电阻率最小二乘反演中存在问题的研究[J].地质与勘探,40(4):70-75.
黄俊革,阮百尧,鲍光淑.2004b.基于有限单元法的三维地电断面电阻率反演[J].中南大学学报(自然科学版),35(2):295-299.
黄仲良.1999.石油重·磁·电法勘探[M].北京:石油大学出版社.
贾正元,张刚.2012.三维井地电阻率法有限差分数值模拟研究[J].现代地质,26(6):1199-1205.
柯敢攀,黄清华.2009.井地电法的三维正反演研究[J].北京大学学报(自然科学版),45(2):264-272.
林昌洪,谭捍东,舒晴,等.2012.可控源音频大地电磁三维共轭梯度反演研究[J].地球物理学报,55(11):3829-3838,doi:10.6038/j.issn.0001-5733.2012.11.030.
刘斌,李术才,李树忱,等.2012a.基于不等式约束的最小二乘法三维电阻率反演及其算法优化[J].地球物理学报,55(1):260-268,doi:10.6038/j.issn.0001-5733.2012.01.025.
刘斌,李术才,聂利超,等.2012b.基于自适应加权光滑约束与PCG算法的三维电阻率探测反演成像[J].岩土工程学报,34(9):1646-1653.
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Hu Z Z,Hu X Y,He Z X.2006.Pseudo-Three-Dimensional magnetotelluric inversion using nonlinear conjugate gradients[J].Chinese J.Geophys.(in Chinese),49(4):1226-1234.
Huang J G,Ruan B Y,Bao G S.2004a.Research on 3-D resistivity least-square inversion using the finite element method[J].Geology and Prospecting(in Chinese),40(4):70-75.
Huang J G,Ruan B Y,Bao G S.2004b.Resistivity inversion on 3-Dsection based on FEM[J].Journal of Central South University(Science and Technology)(in Chinese),35(2):295-299.
Huang Z L,1999.Gravity,Magnetic and Electrical Prospecting of Petroleum(in Chinese)[M].Beijing:Peroleum University Publishing House.
Jia Z Y,Zhang G.2012.Research on Finite-difference Numerical Simulation for 3-D Borehole-to-ground Resistivity Survey[J].Geoscience(in Chinese),26(6):1199-1205.
Ke G P,Huang Q H.2009.3D forward and inversion problems of borehole-to-surface electrical method[J].Acta Scientiarum Naturalium Universitatis Pekinensis(in Chinese),45(2):264-272.
Lin C H,Tan H D,Shu Q,et al.2012.Three-dimensional conjugate gradient inversion of CSAMT data[J].Chinese J.Geophys.(in Chinese),55(11):3829-3838,doi:10.6038/j.issn.0001-5733.2012.11.030.
Liu B,Li S C,Li S C,et al.2012a.3D electrical resistivity inversion with least-squares method based on inequality constraint and its computation efficiency optimization[J].Chinese J.Geophys.(in Chinese),55(1):260-268,doi:10.6038/j.issn.0001-5733.2012.01.025.
Liu B,Li S C,Nie L C,et al.2012b.Inversion imaging of 3D resistivity detection using adaptive-weighted smooth constraint and PCGalgorithm[J].Chinese Journal of Geotechnical Engineering(in Chinese),34(9):1646-1653.
Liu B,Nie L C,Li S C,et al.2012c.3D electrical resistivity inversion tomography with spatial structural constraint[J].Chinese Journal of Rock Mechanics and Engineering(in Chinese),31(11):2258-2268.
Liu H F.2007.Linear and non-linear inversion methods research on direct current IP measurements(in Chinese)[D].Changsha:Central South University.
Liu H F,Chen D P,Dai Q W,et al.2011.3D FEM modeling of borehole-surface potential with line current source in semiunderground space of continuous variation of conductivity[J].Journal of Guilin University of Technology(in Chinese),31(1):28-38.
Liu J X,Jiang P F,Tong X Z,et al.2009.Application of BICGSTABalgorithm with incomplete LU decomposition preconditioning to twodimensional magnetotelluric forward modeling[J].J.Cent.South Univ.(Sci.Technol.),40(2):484-491.
Liu Y H,Yin C C.2013.3D inversion for frequency-domain HEM data[J].Chinese J.Geophys.(in Chinese),56(12):4278-4287,doi:10.6038/cjg20131230.
Mackie R L,Madden,T R.1993.Three-dimensional magnetotelluric inversion using conjugate gradients[J].Geophysical Journal International,115(1):215-229.
Newman G A,Alumbaugh D L.2000.Three-dimensional magnetotelluric inversion using non-linear conjugate gradients[J].Geophysical Journal International,140(2):410-424.
Qu Y H,Zhang B G,Zhao L F,et al.2007.Study on 3-D resistivity inversion for infinite surface-borehole line current source[J].Progress in Geophysics(in Chinese),22(5):1393-1402,doi:10.3969/j.issn.1004-2903.2007.05.009.
Rodi W,Mackie R L.2001.Nonlinear conjugate gradients algorithm for2-D magnetotelluric inversion[J].Geophysics,66(1):174-187.
Sasaki Y.1994.3-D resistivity inversion using the finite-element method[J].Geophysics,59(12):1839-1850.
Wan X L,Xi D Y,Gao E G,et al.2005.3-D resistivity inversion by the least-squares QR factorization method under improved smoothness constraint condition[J].Chinese J.Geophys.(in Chinese),48(2):439-444.
Wang Z G,He Z X,Wei W B.2006.Study on some problems upon 3Dmodeling of DC borehole-ground method[J].Computing Techniques for Geophysical and Geochemical Exploration(in Chinese),28(4):322-327.
Weng A H,Liu Y H,Jia D Y,et al.2012.Three-dimensional controlled source electromagnetic inversion using non-linear conjugate gradients[J].Chinese J.Geophys.(in Chinese),55(10):3506-3515,doi:10.6038/j.issn.0001-5733.2012.10.034.
Wu X P,Xu G M,1999.Derivation and analysis of partial derivative matrix in resistivity 3-D inversion.[J].Oil Geophysical Prospecting(in Chinese),34(4):363-372.
Wu X P,Xu G M.2000.Study on 3-D resistivity inversion using conjugate gradient method[J].Chinese J.Geophys.(in Chinese),43(3):420-427.
Wu X P,Wang T T.2001.Study on some problems for 3D resistivity inversion using conjugate gradient[J].Seismology and Geology(in Chinese),23(2):321-327.
Wu X P,Wang T T.2002a.Study of stability and reliability of 3-Dresistivity inversion[J].Coal Geology and Exploration(in Chinese),30(1):57-60.
Wu X P,Wang T T.2002b.Progress of study on 3D resistivity inversion methods[J].Progress in Geophysics(in Chinese),17(1):156-162,doi:10.3969/j.issn.1004-2903.2002.01.023.
Wu X P,Wang T T.2003.A 3-D finite-element resistivity forward modeling using conjugate gradient algorithm[J].Chinese J.Geophys.(in Chinese),46(3):428-432.
Wu X P.2004.Rapid 3-D inversion of induced polarization data using conjugate gradient method[J].Coal Geology&Exploration(in Chinese),32(5):62-64.
Wu X P.2005.3-D resistivity inversion under the condition of uneven terrain[J].Chinese J.Geophys.(in Chinese),48(4):932-936.
Xu K J,Li T L.2006.The forward modeling of three-dimensional geoelectric filed of vertical finite line source by finite-difference method[J].Journal of Jilin University(Earth Science Edition)(in Chinese),36(1):137-141.
Xu K J,Li T L,Zhang H,et al.2006.3D resistivity inversion of vertical finite line source using conjugate gradients[J].Coal Geology&Exploration(in Chinese),34(3):68-71.
Yuan Y X,Sun W Y.1997.Theory and Methods of Optimization(in Chinese)[M].Beijing:Science Press.
Zhang J,Mackie R L,Madden T R.1995.3-D resistivity forward modeling and inversion using conjugate gradients[J].Geophysics,60(5):1313-1325.
戴前伟,侯智超,王洪华.2013.井地电位2.5维有限元数值模拟异常分析[J].物探化探计算技术,35(4):457-462.
董浩,魏文博,叶高峰,等.2014.基于有限差分正演的带地形三维大地电磁反演方法[J].地球物理学报,57(3):939-952,doi:10.6038/cjg20140323.
胡祖志,胡祥云,何展翔.2006.大地电磁非线性共轭梯度拟三维反演[J].地球物理学报,49(4):1226-1234.
黄俊革,阮百尧,鲍光淑.2004a.有限单元法三维电阻率最小二乘反演中存在问题的研究[J].地质与勘探,40(4):70-75.
黄俊革,阮百尧,鲍光淑.2004b.基于有限单元法的三维地电断面电阻率反演[J].中南大学学报(自然科学版),35(2):295-299.
黄仲良.1999.石油重·磁·电法勘探[M].北京:石油大学出版社.
贾正元,张刚.2012.三维井地电阻率法有限差分数值模拟研究[J].现代地质,26(6):1199-1205.
柯敢攀,黄清华.2009.井地电法的三维正反演研究[J].北京大学学报(自然科学版),45(2):264-272.
林昌洪,谭捍东,舒晴,等.2012.可控源音频大地电磁三维共轭梯度反演研究[J].地球物理学报,55(11):3829-3838,doi:10.6038/j.issn.0001-5733.2012.11.030.
刘斌,李术才,李树忱,等.2012a.基于不等式约束的最小二乘法三维电阻率反演及其算法优化[J].地球物理学报,55(1):260-268,doi:10.6038/j.issn.0001-5733.2012.01.025.
刘斌,李术才,聂利超,等.2012b.基于自适应加权光滑约束与PCG算法的三维电阻率探测反演成像[J].岩土工程学报,34(9):1646-1653.
刘斌,聂利超,李术才,等.2012c.三维电阻率空间结构约束反演成像方法[J].岩石力学与工程学报,31(11):2258-2268.
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