三维坐标转换的高斯-赫尔默特模型及其抗差解法
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摘要
对三维坐标转换的高斯-赫尔默特(Gauss-Helmert,GH)模型,采用牛顿-高斯(Newton-Gauss)迭代算法构建了该模型的拉格朗日目标函数,推导了其解算方法,并给出了具体的计算步骤。在此基础上,考虑到可能出现的粗差对观测空间与结构空间的综合影响,基于标准化残差构造权因子函数,推导了该模型的抗差解法。仿真实验结果表明,GH模型用于三维坐标转换时不受旋转角度大小和其他附加条件限制,解算结果与现有算法一致,且估计参数的维数大大降低,计算效率有一定程度的提高;所提出的抗差解法效果良好,与现有基于整体最小二乘的三维坐标转换的抗差解法相比,表现出了更好的稳健性。
For three-dimensional coordinate transformation,it's impossible using the Gauss-Markov model to obtain optimal parameter estimation from the functional model with error in its coefficient matrix.On the other hand,errors-in-variables model has difficulty expressing the functional model,and partial errors-in-variables model is complex as well as too much parameters to be estimated for the quasi-observation method.Therefore,Gauss-Helmert model is employed for three-dimensional coordinate transformation.The target function of the proposed model is established based on NewtonGauss iterative algorithm,and the estimated method and its derivation procedure also are presented in this paper.Beyond the above process,we proposed a new robust estimation method for the proposed model,which is based on the normalized residual error and takes the influence of gross error on both observation and structure spaces into consideration.Meanwhile,derivational process of statistical tests and iterative algorithm are presented.The simulation experiment results show that the proposed estimation method has the same accuracy as the traditional method,which has robust with angular dimension and other additional conditions,but less estimation parameters.In addition,the new robust estimation method has effective robustness when comparing with the other existing robust total least square methods for the coordinate transformation.
For three-dimensional coordinate transformation,it's impossible using the Gauss-Markov model to obtain optimal parameter estimation from the functional model with error in its coefficient matrix.On the other hand,errors-in-variables model has difficulty expressing the functional model,and partial errors-in-variables model is complex as well as too much parameters to be estimated for the quasi-observation method.Therefore,Gauss-Helmert model is employed for three-dimensional coordinate transformation.The target function of the proposed model is established based on NewtonGauss iterative algorithm,and the estimated method and its derivation procedure also are presented in this paper.Beyond the above process,we proposed a new robust estimation method for the proposed model,which is based on the normalized residual error and takes the influence of gross error on both observation and structure spaces into consideration.Meanwhile,derivational process of statistical tests and iterative algorithm are presented.The simulation experiment results show that the proposed estimation method has the same accuracy as the traditional method,which has robust with angular dimension and other additional conditions,but less estimation parameters.In addition,the new robust estimation method has effective robustness when comparing with the other existing robust total least square methods for the coordinate transformation.
引文
[1]Grafarend E W.Nonlinear Analysis of the Three-Dimensional Datum Transformation[J].Journal of Geodesy,2003,77(1):66-76
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[3]Yang Yuanxi,Xu Tianhe.The Combined Method of Datum Transformation Between Different Coordinate Systems[J].Geomatics and Information Science of Wuhan University,2001,26(6):509-513(杨元喜,徐天河.不同坐标系综合变换法[J].武汉大学学报獉信息科学版,2001,26(6):509-513)
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[5]Zeng Wenxian,Tao Benzao.Non-linear Adjustment Model of Three-Dimensional Coordinate Transformation[J].Geomatics and Information Science of Wuhan University,2003,28(5):566-568(曾文宪,陶本藻.三维坐标转换的非线性模型[J].武汉大学学报獉信息科学版,2003,28(5):566-568)
[6]Lu Jue,Chen Yi,Zheng Bo.Total Least Squares to Three-Dimensional Datum Transformation[J].Journal of Geodesy and Geodynamics,2008,28(5):77-81(陆珏,陈义,郑波.总体最小二乘方法在三维坐标转换中的应用[J].大地测量学与地球动力学,2008,28(5):77-81)
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[8]Schaffrin B,Felus Y A.On the Multivariate Total Least-Squares Approach to Empirical Coordinate Transformations[J].Journal of Geodesy,2008,82(6):373-383
[9]Felus F,Burtch R.On Symmetrical Three-Dimensional Datum Conversion[J].GPS Solutions,2009,13(1):65-74
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[11]Xu Peiliang,Liu Jingnan,Shi Chuang.Total Least Squares Adjustment in Partial Errors-in-Variables Models:Algorithm and Statistical Analysis[J].Journal of Geodesy,2012,86(8):661-675
[12]Xu Peiliang,Liu Jingnan.Variance Components in Errors-in-Variables Models:Estimability,Stability and Bias Analysis[J].Journal of Geodesy,2014,88(8):719-734
[13]Wei Erhu,Yin Zhixiang,Li Guangwen,et al.On 3D Coordinate Transformation with Virtual Observation Method[J].Geomatics and Information Science of Wuhan University,2014,39(2):152-156(魏二虎,殷志祥,李广文,等.虚拟观测值法在三维坐标转换中的应用研究[J].武汉大学学报獉信息科学版,2014,39(2):152-156)
[14]Fang Xing,Zeng Wenxian,Liu Jingnan,et al.A General Total Least Squares Algorithm for ThreeDimensional Coordinate Transformations[J].Acta Geodaetica et Cartographica Sinica,2014,43(11):1 139-1 143(方兴,曾文宪,刘经南,等.三维坐标转换的通用整体最小二乘算法[J].测绘学报,2014,43(11):1 139-1 143)
[15]Neitzel F.Generalization of Total Least-Squares on Example of Unweighted and Weighted 2DSimilarity Transformation[J].Journal of Geodesy,2010,84(12):751-762
[16]Fang Xing,Zeng Wenxian,Liu Jingnan,et al.Mixed LS-TLS Estimation Based on Nonlinear GaussHelmert Model[J].Acta Geodaetica et Cartographica Sinica,2016,45(3):291-296(方兴,曾文宪,刘经南,等.基于非线性高斯-赫尔默特模型的混合整体最小二乘估计[J].测绘学报,2016,45(3):291-296)
[17]Chang Guobin.On Least-Squares Solution to 3D Similarity Transformation Problem Under GaussHelmert Model[J].Journal of Geodesy,2015,89(6):573-576
[18]Ou Jikun.A New Method to Detect Gross Errors:Quasi-Accurate Detection Method[J].Chinese Science Bulletin,1999,44(16):1 777-1 781(欧吉坤.一种检测粗差的新方法——拟准测定法[J].科学通报,1999,44(16):1 777-1 781)
[19]Ou Jikun.Quasi-Accurate Detection of Gross Errors(QUAD)[J].Acta Geodaetica et Cartographica Sinica,1999,28(1):15-20(欧吉坤.粗差的拟准检定法(QUAD法)[J].测绘学报,1999,28(1):15-20)
[20]Yang Yuanxi.Robust Estimation of Geodetic Datum Transformation[J].Journal of Geodesy,1999,73(5):268-274
[21]Zhou Jiangwen,Huang Youcai,Yang Yuanxi,et al.Robust Least Squares Method[M].Wuhan:Huazhong University of Science and Technology Press,1997(周江文,黄幼才,杨元喜,等.抗差估计最小二乘法[M].武汉:华中理工大学出版社,1997)
[22]Chen Yi,Lu Jue.Peforming 3DSimilarity Transformation by Robust Total Least Squares[J].Acta Geodaetica et Cartographica Sinica,2012,41(5):715-722(陈义,陆珏.以三维坐标转换为例解算稳健总体最小二乘方法[J].测绘学报,2012,41(5):715-722)
[23]Lu Jue,Chen Yi,Li Bofeng,et al.Robust Total Least Squares with Reweighting Iteration for ThreeDimensional Similarity Transformation[J].Survey Review,2014,46(334):28-36
[24]Mahboub V,Amirisimkooei A R,Sharifi M A.Iteratively Reweighted Total Least Squares:A Robust Estimation in Errors-in-Variables Models[J].Survey Review,2013,45(329):92-99
[25]Gong Xunqiang,Li Zhilin.A Robust Weighted Total Least Squares Method[J].Acta Geodaetica et Cartographica Sinica,2014,43(9):888-894(龚循强,李志林.稳健加权总体最小二乘法[J].测绘学报,2014,43(9):888-894)
[26]Wang Bin,Li Jiancheng,Gao Jingxiang,et al.Newton-Gauss Algorithm of Robust Weighted Total Least Squares Model[J].Acta Geodaetica et Cartographica Sinica,2015,44(6):602-608(王彬,李建成,高井祥,等.抗差加权整体最小二乘模型的牛顿-高斯算法[J].测绘学报,2015,44(6):602-608)
[27]Yang Yuanxi.Robust Estimation for Correlated Observations Based on Bifactor Equivalent Weights[J].Journal of Geodesy,2002,76(6):353-358
[2]Yao Yibin,Huang Chengmeng,Li Chengchun,et al.A New Algorithm for Solution of Transformation Parameters of Big Rotation Angle’s 3D Coordinate[J].Geomatics and Information Science of Wuhan University,2012,37(3):253-256(姚宜斌,黄承猛,李程春,等.一种适用于大角度三维坐标转换参数求解算法[J].武汉大学学报獉信息科学版,2012,37(3):253-256)
[3]Yang Yuanxi,Xu Tianhe.The Combined Method of Datum Transformation Between Different Coordinate Systems[J].Geomatics and Information Science of Wuhan University,2001,26(6):509-513(杨元喜,徐天河.不同坐标系综合变换法[J].武汉大学学报獉信息科学版,2001,26(6):509-513)
[4]Liu Dajie,Shi Yimin,Guo Jingjun.The Theory and Data Processing of Global Positioning System(GPS)[M].Shanghai:Tongji University Press,1996(刘大杰,施一民,过静珺.全球定位系统(GPS)的原理与数据处理[M].上海:同济大学出版社,1996)
[5]Zeng Wenxian,Tao Benzao.Non-linear Adjustment Model of Three-Dimensional Coordinate Transformation[J].Geomatics and Information Science of Wuhan University,2003,28(5):566-568(曾文宪,陶本藻.三维坐标转换的非线性模型[J].武汉大学学报獉信息科学版,2003,28(5):566-568)
[6]Lu Jue,Chen Yi,Zheng Bo.Total Least Squares to Three-Dimensional Datum Transformation[J].Journal of Geodesy and Geodynamics,2008,28(5):77-81(陆珏,陈义,郑波.总体最小二乘方法在三维坐标转换中的应用[J].大地测量学与地球动力学,2008,28(5):77-81)
[7]Golub H G,Vanloan F C.An Analysis of the Total Least Squares Problem[J].SIAM Journal on Numerical Analysis,1980,17(6):883-893
[8]Schaffrin B,Felus Y A.On the Multivariate Total Least-Squares Approach to Empirical Coordinate Transformations[J].Journal of Geodesy,2008,82(6):373-383
[9]Felus F,Burtch R.On Symmetrical Three-Dimensional Datum Conversion[J].GPS Solutions,2009,13(1):65-74
[10]Shen Yunzhong,Li Bofeng,Chen Yi.An Iterative Solution of Weighted Total Least-Squares Adjustment[J].Journal of Geodesy,2011,85(4):229-238
[11]Xu Peiliang,Liu Jingnan,Shi Chuang.Total Least Squares Adjustment in Partial Errors-in-Variables Models:Algorithm and Statistical Analysis[J].Journal of Geodesy,2012,86(8):661-675
[12]Xu Peiliang,Liu Jingnan.Variance Components in Errors-in-Variables Models:Estimability,Stability and Bias Analysis[J].Journal of Geodesy,2014,88(8):719-734
[13]Wei Erhu,Yin Zhixiang,Li Guangwen,et al.On 3D Coordinate Transformation with Virtual Observation Method[J].Geomatics and Information Science of Wuhan University,2014,39(2):152-156(魏二虎,殷志祥,李广文,等.虚拟观测值法在三维坐标转换中的应用研究[J].武汉大学学报獉信息科学版,2014,39(2):152-156)
[14]Fang Xing,Zeng Wenxian,Liu Jingnan,et al.A General Total Least Squares Algorithm for ThreeDimensional Coordinate Transformations[J].Acta Geodaetica et Cartographica Sinica,2014,43(11):1 139-1 143(方兴,曾文宪,刘经南,等.三维坐标转换的通用整体最小二乘算法[J].测绘学报,2014,43(11):1 139-1 143)
[15]Neitzel F.Generalization of Total Least-Squares on Example of Unweighted and Weighted 2DSimilarity Transformation[J].Journal of Geodesy,2010,84(12):751-762
[16]Fang Xing,Zeng Wenxian,Liu Jingnan,et al.Mixed LS-TLS Estimation Based on Nonlinear GaussHelmert Model[J].Acta Geodaetica et Cartographica Sinica,2016,45(3):291-296(方兴,曾文宪,刘经南,等.基于非线性高斯-赫尔默特模型的混合整体最小二乘估计[J].测绘学报,2016,45(3):291-296)
[17]Chang Guobin.On Least-Squares Solution to 3D Similarity Transformation Problem Under GaussHelmert Model[J].Journal of Geodesy,2015,89(6):573-576
[18]Ou Jikun.A New Method to Detect Gross Errors:Quasi-Accurate Detection Method[J].Chinese Science Bulletin,1999,44(16):1 777-1 781(欧吉坤.一种检测粗差的新方法——拟准测定法[J].科学通报,1999,44(16):1 777-1 781)
[19]Ou Jikun.Quasi-Accurate Detection of Gross Errors(QUAD)[J].Acta Geodaetica et Cartographica Sinica,1999,28(1):15-20(欧吉坤.粗差的拟准检定法(QUAD法)[J].测绘学报,1999,28(1):15-20)
[20]Yang Yuanxi.Robust Estimation of Geodetic Datum Transformation[J].Journal of Geodesy,1999,73(5):268-274
[21]Zhou Jiangwen,Huang Youcai,Yang Yuanxi,et al.Robust Least Squares Method[M].Wuhan:Huazhong University of Science and Technology Press,1997(周江文,黄幼才,杨元喜,等.抗差估计最小二乘法[M].武汉:华中理工大学出版社,1997)
[22]Chen Yi,Lu Jue.Peforming 3DSimilarity Transformation by Robust Total Least Squares[J].Acta Geodaetica et Cartographica Sinica,2012,41(5):715-722(陈义,陆珏.以三维坐标转换为例解算稳健总体最小二乘方法[J].测绘学报,2012,41(5):715-722)
[23]Lu Jue,Chen Yi,Li Bofeng,et al.Robust Total Least Squares with Reweighting Iteration for ThreeDimensional Similarity Transformation[J].Survey Review,2014,46(334):28-36
[24]Mahboub V,Amirisimkooei A R,Sharifi M A.Iteratively Reweighted Total Least Squares:A Robust Estimation in Errors-in-Variables Models[J].Survey Review,2013,45(329):92-99
[25]Gong Xunqiang,Li Zhilin.A Robust Weighted Total Least Squares Method[J].Acta Geodaetica et Cartographica Sinica,2014,43(9):888-894(龚循强,李志林.稳健加权总体最小二乘法[J].测绘学报,2014,43(9):888-894)
[26]Wang Bin,Li Jiancheng,Gao Jingxiang,et al.Newton-Gauss Algorithm of Robust Weighted Total Least Squares Model[J].Acta Geodaetica et Cartographica Sinica,2015,44(6):602-608(王彬,李建成,高井祥,等.抗差加权整体最小二乘模型的牛顿-高斯算法[J].测绘学报,2015,44(6):602-608)
[27]Yang Yuanxi.Robust Estimation for Correlated Observations Based on Bifactor Equivalent Weights[J].Journal of Geodesy,2002,76(6):353-358