火山Mogi模型反演的总体最小二乘联合平差方法
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摘要
针对垂直位移与水平位移的Mogi模型,提出采用总体最小二乘联合(total least squares joint,TLS-J)平差方法进行求解。该方法可同时顾及联合平差函数模型中观测向量与系数矩阵的误差项,且采用3种判别函数最小化法确定相对权比,用以权衡垂直位移与水平位移观测数据在联合求解过程中所占的比重。针对平差过程中出现的病态问题,结合L曲线法确定岭参数。通过实际算例,系统研究了总体最小二乘联合平差方法在长白山天池火山Mogi模型反演中的应用。研究结果表明,以判别函数为∑n1i=1|e1i|+∑n2j=1|e2j|的函数最小化能获得合理的压力源参数估值结果和相对权比大小,具有一定的实际参考价值。
In this paper,a total least squares joint(TLS-J)adjustment method is proposed to the inversion of Mogi model with vertical and horizontal observational data.The proposed method considers the errors in both observation vector and coefficient matrix of the functional model of joint adjustment problem.Three forms of the minimum discriminate function methods are adopted to determine the weight scaling factor which are used to weigh the vertical and horizontal observation data.In view of the existing ill-posed problems in the joint adjustment,the L-curve method is adopted to determine the ridge parameter.Through practical examples,the total least squares joint method is systematically applied to the inversion of the Mogi model of Changbaishan Tianchi volcano.The results show that the discriminant function as the minimum can obtain the reasonable value of pressure source parameters and the relative weight ratio,which has a certain referential value to practical applications.
In this paper,a total least squares joint(TLS-J)adjustment method is proposed to the inversion of Mogi model with vertical and horizontal observational data.The proposed method considers the errors in both observation vector and coefficient matrix of the functional model of joint adjustment problem.Three forms of the minimum discriminate function methods are adopted to determine the weight scaling factor which are used to weigh the vertical and horizontal observation data.In view of the existing ill-posed problems in the joint adjustment,the L-curve method is adopted to determine the ridge parameter.Through practical examples,the total least squares joint method is systematically applied to the inversion of the Mogi model of Changbaishan Tianchi volcano.The results show that the discriminant function as the minimum can obtain the reasonable value of pressure source parameters and the relative weight ratio,which has a certain referential value to practical applications.
引文
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[7]Hu Yaxuan,Shi Xingjue,Wang Qingliang,et al.Analysis of Vertical Deformation in Tengchong Volcano Area[J].Journal of Geodesy and Geodynamics,2003,23(2):37-41(胡亚轩,施行觉,王庆良,等.腾冲火山区地表垂直形变分析[J].大地测量与地球动力学,2003,23(2):37-41)
[8]Shi Xingjue,Hu Yaxuan,Mao Zhu,et al.Vertical Deformation Inversion of Magma Activity in Tengchong Volcanic Area[J].Journal of Seismological Research,2005,28(3):256-261(施行觉,胡亚轩,毛竹,等.以垂直形变资料反演腾冲火山区岩浆活动性的初步研究[J].地震研究,2005,28(3):256-261)
[9]Bifulco I,Raiconi G,Scarpa R.Computer Algebra Software for Least Squares and Total Least Norm Inversion of Geophysical Models[J].Computers&Geosciences,2009,35(7):1 427-1 438
[10]Wang Leyang.Research on Theory and Application of Total Least Squares in Geodetic Inversion[J].Acta Geodaetica et Cartographica Sinica,2012,41(4):629(王乐洋.基于总体最小二乘的大地测量反演理论及应用研究[J].测绘学报,2012,41(4):629
[11]Yu Dongdong.Research on the Ill-Posed Total Least Squares Algorithm and Its Application[D].Nanchang:East China University of Technology,2015(于冬冬.病态总体最小二乘解算方法及应用[D].南昌:东华理工大学,2015)
[12]Yao Yibin,Kong Jian.A New Combined LS Method Considering Random Errors of Design Matrix[J].Geomatics and Information Science of Wuhan University,2014,39(9):1 028-1 032(姚宜斌,孔建.顾及设计矩阵随机误差的最小二乘组合新解法[J].武汉大学学报獉信息科学版,2014,39(9):1 028-1 032)
[13]Schaffrin B,Wieser A.On Weighted Total Least Squares Adjustment for Linear Regression[J].Journal of Geodesy,2008,82(7):415-421
[14]Wang Leyang,Xu Caijun,Lu Tieding.Ridge Estimation Method in Ill-Posed Weighted Total Least Squares Adjustment[J].Geomatics and Information Science of Wuhan University,2010,35(11):1 346-1 350(王乐洋,许才军,鲁铁定.病态加权总体最小二乘平差的岭估计解法[J].武汉大学学报獉信息科学版,2010,35(11):1 346-1 350)
[15]Wang Leyang,Xu Caijun.Total Least Squares Adjustment with Weight Scaling Factor[J].Geomatics and Information Science of Wuhan University,2011,36(8):887-890(王乐洋,许才军.附有相对权比的总体最小二乘平差[J].武汉大学学报獉信息科学版,2011,36(8):887-890)
[16]Wang Leyang,Yu Hang,Chen Xiaoyong.An Algorithm for Partial EIV Model[J].Acta Geodaetica et Cartographica Sinica,2016,45(1):22-29(王乐洋,余航,陈晓勇.Partial EIV模型的解法[J].测绘学报,2016,45(1):22-29)
[17]Wang Leyang,Xu Guangyu.Variance Component Estimation for Partial Errors-in-Variables Models[J].Studia Geophysica et Geodaetica,2016,60(1):35-55
[18]Wang Leyang,Xu Caijun,Wen Yangmao.Fault Parameters of 2008Qinghai Dacaidan Mw6.3Earthquake from STLN Inversion and InSAR Data[J].Acta Geodaetica et Cartographica Sinica,2013,42(2):168-176(王乐洋,许才军,温扬茂.利用STLN和InSAR数据反演2008年青海大柴旦Mw6.3级地震断层参数[J].测绘学报,2013,42(2):168-176)
[19]Hu Chuan,Chen Yi.An Iterative Algorithm for Nonlinear Total Least Squares Adjustment[J].Acta Geodaetica et Cartographica Sinica,2014,43(7):668-674(胡川,陈义.非线性整体最小二乘平差迭代算法[J].测绘学报,2014,43(7):668-674)
[20]Jazaeri S,Amiri-Simkooei A R.Weighted Total Least Squares for Solving Non-linear Problem:GNSS Point Positioning[J].Survey Review,2015,47(343):265-271
[21]Tikhonov A N.Solution of Incorrectly Formulated Problems and the Regularization Method[J].Soviet Mathematics Doklady,1963,4:1 305-1 308
[22]Yu Hang.Research on the Total Least Squares Joint Adjustment and Its Application[D].Nanchang:East China University of Technology,2016(余航.总体最小二乘联合平差方法及其应用研究[D].南昌:东华理工大学,2016)
[23]Wang Leyang,Yu Hang.Total Least Squares Joint Adjustment[J].Geomatics and Information Science of Wuhan University,2016,41(12):1 683-1 689(王乐洋,余航.总体最小二乘联合平差[J].武汉大学学报獉信息科学版,2016,41(12):1 683-1 689)
[24]Wang Zhenjie.Research on the Regularization Solution of Ill-Posed Problems in Geodesy[D].Wuhan:Institute of Geodesy and Geophysics,Chinese Academy of Sciences,2003(王振杰.大地测量中不适定问题的正则化解法研究[D].武汉:中国科学院测量与地球物理研究所,2003)
[25]Wang Bin,Gao Jingxiang,Liu Chao,et al.Application of L-curve Method to Equivalent Weighted Robust Ridge Estimation Model[J].Journal of Geodesy and Geodynamics,2012,32(3):97-101(王彬,高井祥,刘超,等.L曲线法在等价权抗差岭估计模型中的应用[J].大地测量与地球动力学,2012,32(3):97-101)
[26]Dieterich J H,Decker R W.Finite Element Modeling of Surface Deformation Associated with Volcanism[J].Journal of Geophysical Research,1975,80(29):4 094-4 102
[2]Hu Yaxuan,Wang Qingliang,Cui Duxin,et al.Application of Mogi Model at Changbaishan Tianchi Volcano[J].Seismology and Geology,2007,29(1):144-151(胡亚轩,王庆良,崔笃信,等.Mogi模型在长白山天池火山区的应用[J].地震地质,2007,29(1):144-151)
[3]Hu Yaxuan,Wang Qingliang,Cui Duxin,et al.Influences on Surface Deformation by the Three Different Stress Models in Volcanic Area[J].Seismology Research of Northeast China,2005,21(3):33-38(胡亚轩,王庆良,崔笃信,等.三种压力源模型对火山区地面变形的影响[J].东北地震研究,2005,21(3):33-38)
[4]Li Ke,Liu Junqing,Pan Xiaodong,et al.Crust Deformation Monitoring and Research in Tianchi Volcano Area,Changbai Mountains from 2000-2007[J].Seismology and Geology,2009,31(4):639-646(李克,刘俊清,盘晓东,等.2000-2007年期间长白山天池火山区地壳变形监测与分析[J].地震地质,2009,31(4):639-646)
[5]Hu Yaxuan,Hao Ming,Wang Xiong,et al.Application of L-curve to the Inversion of Pressure Source Parameters in Volcano Area[J].Journal of Geodesy and Geodynamics,2009,29(2):66-70(胡亚轩,郝明,王雄,等.L曲线法在反演火山区压力源参数中的应用[J].大地测量与地球动力学,2009,29(2):66-70)
[6]Chen Guohu.Deformation Monitoring and Simulation Research of Mt.Changbai Tianchi Volcano[D].Beijing:Institute of Geology,China Earthquake Administration,2007(陈国浒.长白山天池火山形变监测与模拟研究[D].北京:中国地震局地质研究所,2007)
[7]Hu Yaxuan,Shi Xingjue,Wang Qingliang,et al.Analysis of Vertical Deformation in Tengchong Volcano Area[J].Journal of Geodesy and Geodynamics,2003,23(2):37-41(胡亚轩,施行觉,王庆良,等.腾冲火山区地表垂直形变分析[J].大地测量与地球动力学,2003,23(2):37-41)
[8]Shi Xingjue,Hu Yaxuan,Mao Zhu,et al.Vertical Deformation Inversion of Magma Activity in Tengchong Volcanic Area[J].Journal of Seismological Research,2005,28(3):256-261(施行觉,胡亚轩,毛竹,等.以垂直形变资料反演腾冲火山区岩浆活动性的初步研究[J].地震研究,2005,28(3):256-261)
[9]Bifulco I,Raiconi G,Scarpa R.Computer Algebra Software for Least Squares and Total Least Norm Inversion of Geophysical Models[J].Computers&Geosciences,2009,35(7):1 427-1 438
[10]Wang Leyang.Research on Theory and Application of Total Least Squares in Geodetic Inversion[J].Acta Geodaetica et Cartographica Sinica,2012,41(4):629(王乐洋.基于总体最小二乘的大地测量反演理论及应用研究[J].测绘学报,2012,41(4):629
[11]Yu Dongdong.Research on the Ill-Posed Total Least Squares Algorithm and Its Application[D].Nanchang:East China University of Technology,2015(于冬冬.病态总体最小二乘解算方法及应用[D].南昌:东华理工大学,2015)
[12]Yao Yibin,Kong Jian.A New Combined LS Method Considering Random Errors of Design Matrix[J].Geomatics and Information Science of Wuhan University,2014,39(9):1 028-1 032(姚宜斌,孔建.顾及设计矩阵随机误差的最小二乘组合新解法[J].武汉大学学报獉信息科学版,2014,39(9):1 028-1 032)
[13]Schaffrin B,Wieser A.On Weighted Total Least Squares Adjustment for Linear Regression[J].Journal of Geodesy,2008,82(7):415-421
[14]Wang Leyang,Xu Caijun,Lu Tieding.Ridge Estimation Method in Ill-Posed Weighted Total Least Squares Adjustment[J].Geomatics and Information Science of Wuhan University,2010,35(11):1 346-1 350(王乐洋,许才军,鲁铁定.病态加权总体最小二乘平差的岭估计解法[J].武汉大学学报獉信息科学版,2010,35(11):1 346-1 350)
[15]Wang Leyang,Xu Caijun.Total Least Squares Adjustment with Weight Scaling Factor[J].Geomatics and Information Science of Wuhan University,2011,36(8):887-890(王乐洋,许才军.附有相对权比的总体最小二乘平差[J].武汉大学学报獉信息科学版,2011,36(8):887-890)
[16]Wang Leyang,Yu Hang,Chen Xiaoyong.An Algorithm for Partial EIV Model[J].Acta Geodaetica et Cartographica Sinica,2016,45(1):22-29(王乐洋,余航,陈晓勇.Partial EIV模型的解法[J].测绘学报,2016,45(1):22-29)
[17]Wang Leyang,Xu Guangyu.Variance Component Estimation for Partial Errors-in-Variables Models[J].Studia Geophysica et Geodaetica,2016,60(1):35-55
[18]Wang Leyang,Xu Caijun,Wen Yangmao.Fault Parameters of 2008Qinghai Dacaidan Mw6.3Earthquake from STLN Inversion and InSAR Data[J].Acta Geodaetica et Cartographica Sinica,2013,42(2):168-176(王乐洋,许才军,温扬茂.利用STLN和InSAR数据反演2008年青海大柴旦Mw6.3级地震断层参数[J].测绘学报,2013,42(2):168-176)
[19]Hu Chuan,Chen Yi.An Iterative Algorithm for Nonlinear Total Least Squares Adjustment[J].Acta Geodaetica et Cartographica Sinica,2014,43(7):668-674(胡川,陈义.非线性整体最小二乘平差迭代算法[J].测绘学报,2014,43(7):668-674)
[20]Jazaeri S,Amiri-Simkooei A R.Weighted Total Least Squares for Solving Non-linear Problem:GNSS Point Positioning[J].Survey Review,2015,47(343):265-271
[21]Tikhonov A N.Solution of Incorrectly Formulated Problems and the Regularization Method[J].Soviet Mathematics Doklady,1963,4:1 305-1 308
[22]Yu Hang.Research on the Total Least Squares Joint Adjustment and Its Application[D].Nanchang:East China University of Technology,2016(余航.总体最小二乘联合平差方法及其应用研究[D].南昌:东华理工大学,2016)
[23]Wang Leyang,Yu Hang.Total Least Squares Joint Adjustment[J].Geomatics and Information Science of Wuhan University,2016,41(12):1 683-1 689(王乐洋,余航.总体最小二乘联合平差[J].武汉大学学报獉信息科学版,2016,41(12):1 683-1 689)
[24]Wang Zhenjie.Research on the Regularization Solution of Ill-Posed Problems in Geodesy[D].Wuhan:Institute of Geodesy and Geophysics,Chinese Academy of Sciences,2003(王振杰.大地测量中不适定问题的正则化解法研究[D].武汉:中国科学院测量与地球物理研究所,2003)
[25]Wang Bin,Gao Jingxiang,Liu Chao,et al.Application of L-curve Method to Equivalent Weighted Robust Ridge Estimation Model[J].Journal of Geodesy and Geodynamics,2012,32(3):97-101(王彬,高井祥,刘超,等.L曲线法在等价权抗差岭估计模型中的应用[J].大地测量与地球动力学,2012,32(3):97-101)
[26]Dieterich J H,Decker R W.Finite Element Modeling of Surface Deformation Associated with Volcanism[J].Journal of Geophysical Research,1975,80(29):4 094-4 102