一维导电薄球层状模型的地磁测深C-响应计算
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摘要
对地磁数据的反演是获取地球深部电性结构的1种重要方法,其反演结果的可靠性必须以准确的正演模拟为基础。文中详细介绍了球坐标系中导电薄球层状地球模型的C-响应计算理论,并对典型的地球模型进行了数值模拟。地磁测深的激发源为磁层中的电流体系,其形态由球谐函数P01近似表示。地球内部,导电层中电矢量位满足亥姆霍兹方程;通过各层界面上磁场法向分量和切向分量满足的连续边界条件,由超导地核确定的核幔边界系数向上逐层递推,进而获得地表的边界系数,最终将地下电性结构和地表磁场各分量联系起来。通过磁场分量比值得到与源强度无关而与地球内部导电性相关的地表C-响应,实现地磁测深一维正演计算。一维典型模型的C-响应与前人结果的一致性验证了本文算法的有效性;通过与直接计算的C-响应曲线进行对比,发现由于忽略了地球曲率的影响,利用地磁函数转换方法获得的C-响应在大周期时(>10~6s)与理论响应存在一定的偏差,会造成反演结果的不准确。用文中的数值模拟方法能获得精确的C-响应,进而支撑地磁测深一维反演结果的可靠性。
This paper tries to formulate the C-response of geomagnetic depth sounding( GDS) on an Earth model with finite electrical conductivity. The computation is performed in a spherical coordinate system. The Earth is divided into a series of thin spherical shells. The source is approximated by a single spherical harmonic P01 due to the spatial structure of electrical currents in the magnetosphere.The whole solution space is separated into inner and external parts by the Earth surface. Omitting displacement current,the magnetic field in the external space obeys Laplacian equation,while in the inner part,due to the finite conductivity,the electromagnetic fields obey Helmholtz equation. To connect the magnetic fields in the inner and external space,the continuity condition of magnetic fields is used on the Earth surface. The external magnetic fields are expressed by the inner and external source coefficients,from which a new parameter called C-response is computed from the inner coefficient divided by the external coefficient,thus normalizing the actual source strength. The inner magnetic fields in each layer can be recursively derived by the continuity boundary condition of both normal and tangential components of the magnetic field from the initial boundary condition at core-mantle-boundary. The consistency of our C-responses with that from a typical 1-D global model validates the accuracy of the proposed algorithm. Numerical results also show that the C-response estimated from the geomagnetic transfer function method will deviate exceeding 5% from the actual response at longer periods than about 106 s,which means that ignoring the curvature of the Earth at extreme long periods will make inversion result unreliable. Therefore,an accurate C-response should be computed in order to lay a solid foundation for reliable inversion.
This paper tries to formulate the C-response of geomagnetic depth sounding( GDS) on an Earth model with finite electrical conductivity. The computation is performed in a spherical coordinate system. The Earth is divided into a series of thin spherical shells. The source is approximated by a single spherical harmonic P01 due to the spatial structure of electrical currents in the magnetosphere.The whole solution space is separated into inner and external parts by the Earth surface. Omitting displacement current,the magnetic field in the external space obeys Laplacian equation,while in the inner part,due to the finite conductivity,the electromagnetic fields obey Helmholtz equation. To connect the magnetic fields in the inner and external space,the continuity condition of magnetic fields is used on the Earth surface. The external magnetic fields are expressed by the inner and external source coefficients,from which a new parameter called C-response is computed from the inner coefficient divided by the external coefficient,thus normalizing the actual source strength. The inner magnetic fields in each layer can be recursively derived by the continuity boundary condition of both normal and tangential components of the magnetic field from the initial boundary condition at core-mantle-boundary. The consistency of our C-responses with that from a typical 1-D global model validates the accuracy of the proposed algorithm. Numerical results also show that the C-response estimated from the geomagnetic transfer function method will deviate exceeding 5% from the actual response at longer periods than about 106 s,which means that ignoring the curvature of the Earth at extreme long periods will make inversion result unreliable. Therefore,an accurate C-response should be computed in order to lay a solid foundation for reliable inversion.
引文
陈伯舫.1989.中国东南地区深部电导率分布的进一步研究[J].地震研究,12(4):348-352.CHEN Bo-fang.1989.Further study of the electrical conductivity beneath the region of Southeast China[J].Journal of Seismological Research,12(4):348-352(in Chinese).
杜兴信,鲁秀玲.1995.陕西地区单台Z/H地磁测深研究[J].地震地磁观测与研究,16(1):27-34.DU Xing-xin,LU Xiu-ling.1995.Research on the geomagnetic sounding in Shaanxi region by single station Z/H method[J].Seismological and Geomagnetic Observation and Research,16(1):27-34(in Chinese).
范国华,姚同起,顾左文,等.1997.利用磁梯度法研究中国地幔导电率[J].地震学报,19(2):164—173.FAN Guo-hua,YAO Tong-qi,GU Zuo-wen,et al.1997.Research on mantle conductivity of China with gradient method[J].Acta Seismologica Sinica,19(2):164—173(in Chinese).
李墩柱,黄清华,陈小斌.2009.误差对大地电磁测深反演的影响[J].地球物理学报,52(1):268—274.LI Dun-zhu,HUANG Qing-hua,CHEN Xiao-bin.2009.Error effects on magnetotelluric inversion[J].Chinese Journal of Geophysics,52(1):268—274(in Chinese).
李建平,翁爱华,李世文,等.2018.基于球坐标系下有限差分的地磁测深三维正演[J].吉林大学学报(地球科学版),48(2):411—419.LI Jian-ping,WENG Ai-hua,LI Shi-wen,et al.2018.3-D forward modeling of geomagnetic sounding based on finite difference method in spherical coordinates[J].Journal of Jilin University(Earth Science Edition),48(2):411 —419(in Chinese).
罗威,王绪本,覃庆炎.2012.基于球体层状介质模型的大地电磁正演[J].物探化探计算技术,34(4):384—389.LUO Wei,WANG Xu-ben,QIN Qing-yan.2012.MT 1D forward modeling based on sphere layered model[J].Computing Techniques for Geophysical and Geochemical Exploration,34(4):384—389(in Chinese).
覃庆炎,罗威,张伟.2012.地球曲率对长周期大地电磁测深法的影响[J].地震地质,34(3):456—466.doi:10 .3969/j.issn.0253-4967.2012.03.007.QIN Qing-yan,LUO Wei,ZHANG Wei.2012.The influence of the Earth's curvature on the long-period magnetotelluric sounding method[J].Seismology and Geology,34(3):456—466(in Chinese).
汤吉,赵国泽.2005.利用GDS的等效MT响应函数扩展长周期电磁测深研究[C]∥中国地球物理第二十一届年会论文集.北京:中国地球物理学会.TANG Ji,ZHAO Guo-ze.2005.Study on extended long period electromagnetic sounding using GDS equivalent MT response function[C]∥Annual Meeting of China Geophysical Society.Chinese Geophysical Society,Beijing(in Chinese).
滕吉文.2003.地球深部物质和能量交换的动力过程与矿产资源的形成[J].大地构造与成矿学,27(1):3—21.TENG Ji-wen.2003.Dynamic process of substance and energy exchanges in deep Earth and formation of mineral resources[J].Geotectonica et Metallogenia,27(1):3—21(in Chinese).
王桥,黄清华.2016.华北地磁感应矢量时空特征分析[J].地球物理学报,59(1):215—228.WANG Qiao,HUANG Qing-hua.2016.The spatial-temporal characteristics of geomagnetic induction vectors in North China[J].Chinese Journal of Geophysics,59(1):215—228(in Chinese).
徐光晶,汤吉,黄清华,等.2015.华北地区上地幔及过渡带电性结构研究[J].地球物理学报,58(2):566—575 .doi:10.6038/cjg20150219.XU Guang-jing,TANG Ji,HUANG Qing-hua,et al.2015.Study on the conductivity structure of the upper mantle and transition zone beneath North China[J].Chinese Journal of Geophysics,58(2):566—575(in Chinese).
徐文耀.2009.地球电磁现象物理学[M].合肥:中国科学技术大学出版社.XU Wen-yao.2009.Physics of Electromagnetic Phenomena of the Earth[M].University of Science and Technology of China Press,Hefei(in Chinese).
张贵宾.1998.地磁梯度测深与位场反演[M].北京:冶金工业出版社.ZHANG Gui-bin.1998.Geomagnetic Gradient Depth Sounding and Potential Field Inversion[M].Metallurgical Industry Press,Beijing(in Chinese).
赵国泽,汤吉,梁竞阁,等.2001.用大地电磁网法在长春等地探测上地幔电导率结构[J].地震地质,23(2):143 —152.doi:10.3969/j.issn.0253-4967.2001.02.003.ZHAO Guo-ze,TANG Ji,LIANG Jing-ge,et al.2001.Measurement of network-MT in two areas of NE China for study of upper mantle conductivity structure of the back-arc region[J].Seismology and Geology,23(2):143—152(in Chinese).
Bai Q,Kohlstedt D L.1992.Substantial hydrogen solubility in olivine and implications for water storage in the mantle[J].Nature,357(6380):672—674.
Banks R J.1969.Geomagnetic variations and the electrical conductivity of the upper mantle[J].Geophysical Journal International,17(5):457—487.
Cagniard L.1953.Basic theory of the magnetotelluric method of geophysical prospecting[J].Geophysics,18(3):605 —635.
Fujii I,Schultz A.2002.The 3D electromagnetic response of the Earth to ring current and auroral oval excitation[J].Geophysical Journal International,151(3):689—709.
Ichiki M,Uyeshima M,Utada H,et al.2001.Upper mantle conductivity structure of the back-arc region beneath northeastern China[J].Geophysical Research Letters,28(19):3773—3776.
Kelbert A,Meqbel N,Egbert G D,et al.2014.Mod EM:A modular system for inversion of electromagnetic geophysical data[J].Computers&Geosciences,66:40—53.
Kelbert A,Schultz A,Egbert G.2009.Global electromagnetic induction constraints on transition-zone water content variations[J].Nature,460(7258):1003—1006.
Kuvshinov A V.2012.Deep electromagnetic studies from land,sea,and space:progress status in the past 10 years[J].Surveys in Geophysics,33(1):169—209.
Olsen N.1998.The electrical conductivity of the mantle beneath Europe derived from C-responses from 3 to 720 hr[J].Geophysical Journal International,133(2):298—308.
Olsen N.1999.Long-period(30 days-1 year)electromagnetic sounding and the electrical conductivity of the lower mantle beneath Europe[J].Geophysical Journal International,138(1):179—187.
Püthe C,Kuvshinov A,Khan A,et al.2015.A new model of Earth's radial conductivity structure derived from over 10yr of satellite and observatory magnetic data[J].Geophysical Journal International,203(3):1864—1872.
Schmucker U.1987.Substitute conductors for electromagnetic response estimates[J].Pure and Applied Geophysics,125 (2-3):341—367.
Schultz A,Larsen J C.1987.On the electrical conductivity of the mid-mantle-I.Calculation of equivalent scalar magnetotelluric response functions[J].Geophysical Journal International,88(3):733—761.
Shimizu H,Koyama T,Baba K,et al.2009.Three—dimensional geomagnetic response functions for global and semiglobal scale induction problems[J].Geophysical Journal International,178(1):123—144.
Srivastava S P.1966.Theory of the magnetotelluric method for a spherical conductor[J].Geophysical Journal International,11(4):373—387.
Sun J,Egbert G D.2012.Spherical decomposition of electromagnetic fields generated by quasi-static currents[J].GEM—International Journal on Geomathematics,3(2):279—295.
杜兴信,鲁秀玲.1995.陕西地区单台Z/H地磁测深研究[J].地震地磁观测与研究,16(1):27-34.DU Xing-xin,LU Xiu-ling.1995.Research on the geomagnetic sounding in Shaanxi region by single station Z/H method[J].Seismological and Geomagnetic Observation and Research,16(1):27-34(in Chinese).
范国华,姚同起,顾左文,等.1997.利用磁梯度法研究中国地幔导电率[J].地震学报,19(2):164—173.FAN Guo-hua,YAO Tong-qi,GU Zuo-wen,et al.1997.Research on mantle conductivity of China with gradient method[J].Acta Seismologica Sinica,19(2):164—173(in Chinese).
李墩柱,黄清华,陈小斌.2009.误差对大地电磁测深反演的影响[J].地球物理学报,52(1):268—274.LI Dun-zhu,HUANG Qing-hua,CHEN Xiao-bin.2009.Error effects on magnetotelluric inversion[J].Chinese Journal of Geophysics,52(1):268—274(in Chinese).
李建平,翁爱华,李世文,等.2018.基于球坐标系下有限差分的地磁测深三维正演[J].吉林大学学报(地球科学版),48(2):411—419.LI Jian-ping,WENG Ai-hua,LI Shi-wen,et al.2018.3-D forward modeling of geomagnetic sounding based on finite difference method in spherical coordinates[J].Journal of Jilin University(Earth Science Edition),48(2):411 —419(in Chinese).
罗威,王绪本,覃庆炎.2012.基于球体层状介质模型的大地电磁正演[J].物探化探计算技术,34(4):384—389.LUO Wei,WANG Xu-ben,QIN Qing-yan.2012.MT 1D forward modeling based on sphere layered model[J].Computing Techniques for Geophysical and Geochemical Exploration,34(4):384—389(in Chinese).
覃庆炎,罗威,张伟.2012.地球曲率对长周期大地电磁测深法的影响[J].地震地质,34(3):456—466.doi:10 .3969/j.issn.0253-4967.2012.03.007.QIN Qing-yan,LUO Wei,ZHANG Wei.2012.The influence of the Earth's curvature on the long-period magnetotelluric sounding method[J].Seismology and Geology,34(3):456—466(in Chinese).
汤吉,赵国泽.2005.利用GDS的等效MT响应函数扩展长周期电磁测深研究[C]∥中国地球物理第二十一届年会论文集.北京:中国地球物理学会.TANG Ji,ZHAO Guo-ze.2005.Study on extended long period electromagnetic sounding using GDS equivalent MT response function[C]∥Annual Meeting of China Geophysical Society.Chinese Geophysical Society,Beijing(in Chinese).
滕吉文.2003.地球深部物质和能量交换的动力过程与矿产资源的形成[J].大地构造与成矿学,27(1):3—21.TENG Ji-wen.2003.Dynamic process of substance and energy exchanges in deep Earth and formation of mineral resources[J].Geotectonica et Metallogenia,27(1):3—21(in Chinese).
王桥,黄清华.2016.华北地磁感应矢量时空特征分析[J].地球物理学报,59(1):215—228.WANG Qiao,HUANG Qing-hua.2016.The spatial-temporal characteristics of geomagnetic induction vectors in North China[J].Chinese Journal of Geophysics,59(1):215—228(in Chinese).
徐光晶,汤吉,黄清华,等.2015.华北地区上地幔及过渡带电性结构研究[J].地球物理学报,58(2):566—575 .doi:10.6038/cjg20150219.XU Guang-jing,TANG Ji,HUANG Qing-hua,et al.2015.Study on the conductivity structure of the upper mantle and transition zone beneath North China[J].Chinese Journal of Geophysics,58(2):566—575(in Chinese).
徐文耀.2009.地球电磁现象物理学[M].合肥:中国科学技术大学出版社.XU Wen-yao.2009.Physics of Electromagnetic Phenomena of the Earth[M].University of Science and Technology of China Press,Hefei(in Chinese).
张贵宾.1998.地磁梯度测深与位场反演[M].北京:冶金工业出版社.ZHANG Gui-bin.1998.Geomagnetic Gradient Depth Sounding and Potential Field Inversion[M].Metallurgical Industry Press,Beijing(in Chinese).
赵国泽,汤吉,梁竞阁,等.2001.用大地电磁网法在长春等地探测上地幔电导率结构[J].地震地质,23(2):143 —152.doi:10.3969/j.issn.0253-4967.2001.02.003.ZHAO Guo-ze,TANG Ji,LIANG Jing-ge,et al.2001.Measurement of network-MT in two areas of NE China for study of upper mantle conductivity structure of the back-arc region[J].Seismology and Geology,23(2):143—152(in Chinese).
Bai Q,Kohlstedt D L.1992.Substantial hydrogen solubility in olivine and implications for water storage in the mantle[J].Nature,357(6380):672—674.
Banks R J.1969.Geomagnetic variations and the electrical conductivity of the upper mantle[J].Geophysical Journal International,17(5):457—487.
Cagniard L.1953.Basic theory of the magnetotelluric method of geophysical prospecting[J].Geophysics,18(3):605 —635.
Fujii I,Schultz A.2002.The 3D electromagnetic response of the Earth to ring current and auroral oval excitation[J].Geophysical Journal International,151(3):689—709.
Ichiki M,Uyeshima M,Utada H,et al.2001.Upper mantle conductivity structure of the back-arc region beneath northeastern China[J].Geophysical Research Letters,28(19):3773—3776.
Kelbert A,Meqbel N,Egbert G D,et al.2014.Mod EM:A modular system for inversion of electromagnetic geophysical data[J].Computers&Geosciences,66:40—53.
Kelbert A,Schultz A,Egbert G.2009.Global electromagnetic induction constraints on transition-zone water content variations[J].Nature,460(7258):1003—1006.
Kuvshinov A V.2012.Deep electromagnetic studies from land,sea,and space:progress status in the past 10 years[J].Surveys in Geophysics,33(1):169—209.
Olsen N.1998.The electrical conductivity of the mantle beneath Europe derived from C-responses from 3 to 720 hr[J].Geophysical Journal International,133(2):298—308.
Olsen N.1999.Long-period(30 days-1 year)electromagnetic sounding and the electrical conductivity of the lower mantle beneath Europe[J].Geophysical Journal International,138(1):179—187.
Püthe C,Kuvshinov A,Khan A,et al.2015.A new model of Earth's radial conductivity structure derived from over 10yr of satellite and observatory magnetic data[J].Geophysical Journal International,203(3):1864—1872.
Schmucker U.1987.Substitute conductors for electromagnetic response estimates[J].Pure and Applied Geophysics,125 (2-3):341—367.
Schultz A,Larsen J C.1987.On the electrical conductivity of the mid-mantle-I.Calculation of equivalent scalar magnetotelluric response functions[J].Geophysical Journal International,88(3):733—761.
Shimizu H,Koyama T,Baba K,et al.2009.Three—dimensional geomagnetic response functions for global and semiglobal scale induction problems[J].Geophysical Journal International,178(1):123—144.
Srivastava S P.1966.Theory of the magnetotelluric method for a spherical conductor[J].Geophysical Journal International,11(4):373—387.
Sun J,Egbert G D.2012.Spherical decomposition of electromagnetic fields generated by quasi-static currents[J].GEM—International Journal on Geomathematics,3(2):279—295.