附有相对权比的加权总体最小二乘联合平差方法
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摘要
采用不同类数据联合平差时,不仅观测向量含有误差,其对应的系数矩阵也通常受到误差的影响。将加权总体最小二乘方法应用于多类观测数据的联合平差模型,推导相应迭代计算方法,以相对权比权衡各类数据参与联合平差的比重。设计了多种方案,并给出了确定相对权比的判别函数最小化方法。结果表明,验前单位权方差法与总体最小二乘方差分量估计方法具有一定的局限性,当验前信息不准确或者总体最小二乘方差分量估计方法不可估时,判别函数为n_1∑i=1︱e_(1_i)︱ +n_2∑j=1︱e_(2_j)︱|的判别函数最小化法能取得较优的参数估值结果。
In regard to the joint adjustment problem with different types of dataset, the functional model of each type of dataset is affected by random errors, which indicates the observation vector and coefficient matrix are not error-free. In this paper, the weight total least squares(WTLS) method is applied to joint adjustment model. An iterative WTLS method for joint adjustment model is derived, which uses the weight scaling factor to adjust the contribution of each type of dataset. In view of the determination of the weight scaling factor, more schemes are designed, which includes the minimum discrimination function method. The results show that the prior unit weight variance method and the total least squares variance component estimation(TLS-VCE) method have their limitations. When the prior information is inaccurate or the variance components are not estimable while using the TLS-VCE method, the minimum discriminate function method with n_1∑i=1︱e_(1_i)︱ +n_2∑j=1︱e_(2_j)︱|| as its discriminate function can achieve the relative effective results.
In regard to the joint adjustment problem with different types of dataset, the functional model of each type of dataset is affected by random errors, which indicates the observation vector and coefficient matrix are not error-free. In this paper, the weight total least squares(WTLS) method is applied to joint adjustment model. An iterative WTLS method for joint adjustment model is derived, which uses the weight scaling factor to adjust the contribution of each type of dataset. In view of the determination of the weight scaling factor, more schemes are designed, which includes the minimum discrimination function method. The results show that the prior unit weight variance method and the total least squares variance component estimation(TLS-VCE) method have their limitations. When the prior information is inaccurate or the variance components are not estimable while using the TLS-VCE method, the minimum discriminate function method with n_1∑i=1︱e_(1_i)︱ +n_2∑j=1︱e_(2_j)︱|| as its discriminate function can achieve the relative effective results.
引文
[1]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)
[2]Zhang Jun,Du Zhixing,Du Ning,et al.A Constraint Method to Determine Relative Weight Ratio Factors in Joint Inversion[J].Journal of Geodesy and Geodynamics,2015,35(4):600-603(张俊,独知行,杜宁,等.联合反演模型中相对权比的约束反演[J].大地测量与地球动力学,2015,35(4):600-603)
[3]Xu Caijun,Deng Changyong,Zhou Lixuan.Coseismic Slip Distribution Inversion Method Based on Variance Component Estimation[J].Geomatics and Information Science of Wuhan University,2016,41(1):37-44(许才军,邓长勇,周力璇.利用方差分量估计的地震同震滑动分布反演[J].武汉大学学报·信息科学版,2016,41(1):37-44)
[4]Duan Hurong,Zhang Yongzhi.On Relative Weight Ratio Relation Between Characteristics of Fault Slip and Various Type of Inversion Data[J].Journal of Geodesy and Geodynamics,2009,29(6):18-21(段虎荣,张永志.断层滑动特征与多种反演数据的相对权比关系研究[J].大地测量与地球动力学,2009,29(6):18-21)
[5]Du Zhixing,Ou Jikun,Jin Fengxiang,et al.Optimization Inversion of the Relative Weight Ratioλin the Joint Inversion Models[J].Acta Geodaetica et Cartographica Sinica,2003,32(1):15-19(独知行,欧吉坤,靳奉祥,等.联合反演模型中相对权比的优化反演[J].测绘学报,2003,32(1):15-19)
[6]Wang Leyang,Xu Caijun.Comparative Research on Equality Constraint Inversion and Joint Inversion[J].Journal of Geodesy and Geodynamics,2009,29(1):74-78(王乐洋,许才军.等式约束反演与联合反演的对比研究[J].大地测量与地球动力学,2009,29(1):74-78)
[7]Wang Leyang,Xu Caijun,Zhang Chaoyu.A Twostep Method to Determine Relative Weight Ratio Factors in Joint Inversion[J].Acta Geodaetica et Cartographica Sinica,2012,41(1):19-24(王乐洋,许才军,张朝玉.一种确定联合反演中相对权比的两步法[J].测绘学报,2012,41(1):19-24)
[8]Zhang Chaoyu,Xu Caijun.Sequential Algorithm of Geodesy Joint Inversion with Auto-match Weight Ratio and Its Application[J].Geomatics and Information Science of Wuhan University,2012,37(10):1 140-1 145(张朝玉,许才军.具有自适应权比的大地测量联合反演序贯算法及其应用[J].武汉大学学报·信息科学版,2012,37(10):1 140-1 145)
[9]Xu Caijun,Ding Kaihua,Cai Jianqiang,et al.Methods of Determining Weight Scaling Factors for GeodeticGeophysical Joint Inversion[J].Journal of GeoDynamics,2009,47(1):39-46
[10]Wang Leyang,Yu Hang.Study on the Applicability of Total Least Squares Method in Surveying Adjustment[J].Journal of Geodesy and Geodynamics,2014,34(3):121-124(王乐洋,余航.总体最小二乘方法的适用性研究[J].大地测量与地球动力学,2014,34(3):121-124)
[11]Wang Leyang,Xu Caijun.Total Least Square Adjustment with Weight Scaling Factor[J].Geomatics and Information Science of Wuhan University,2011,36(8):887-884(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·信息科学版,2011,36(8):887-884)
[12]Wang Leyang,Xu Guangyu,Chen Xiaoyong.Total Least Squares Adjustment of Partial Errors-in-Variables Model with Weight Scaling Factor[J].Geomatics and Information Science of Wuhan University,2017,42(6):857-863(王乐洋,许光煜,陈晓勇.附有相对权比的PEIV模型总体最小二乘平差[J].武汉大学学报·信息科学版,2017,42(6):857-863)
[13]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·信息科学版,2014,39(9):1 028-1 032)
[14]Zhou Yongjun,Fang Xing.A Mixed Weighted Least Squares and Weighted Total Least Squares Adjustment Method and Its Geodetic Applications[J].Survey Review,2016,48(351):421-429
[15]Fang Xing.Weighted Total Least Squares:Necessary and Sufficient Conditions,Fixed and Random Parameters[J].Journal of Geodesy,2013,87(8):733-749
[16]Jazaeri S,Amiri-Simkooei A R,Sharifi M A.Iterative Algorithm for Weighted Total Least Squares Adjustment[J].Survey Review,2014,46(334):19-27
[17]Zeng Wenxian.Effect of the Random Design Matrix on Adjustment of an EIV Model and Its Reliability Theory[D].Wuhan:Wuhan University,2013(曾文宪.系数矩阵误差对EIV模型平差结果的影响研究[D].武汉:武汉大学,2013)
[18]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·信息科学版,2016,41(12):1 683-1 689)
[19]Amiri-Simkooei A R.Application of Least Squares Variance Component Estimation to Errors-in-Variables Models[J].Journal of Geodesy,2013,87(10-12):935-944
[20]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
[21]Wang Leyang,Xu Guangyu.Variance Component Estimation for Partial Errors-in-Variables Models[J].Studia Geophysica et Geodaetica,2016,60(1):35-55
[22]Amiri-Simkooei A R.Non-negative Least-Squares Variance Component Estimation with Application to GPS Time Series[J].Journal of Geodesy,2016,90(5):451-466
[23]Li Bofeng.Variance Component Estimation in the Seamless Affine Transformation Model[J].Acta Geodaetica et Cartographica Sinica,2016,45(1):30-35(李博峰.无缝仿射基准转换模型的方差分量估计[J].测绘学报,2016,45(1):30-35)
[24]Cui Xizhang,Yu Zongchou,Tao Benzao,et al.Generalized Surveying Adjustment[M].2nd ed.Wuhan:Wuhan University Press,2009(崔希璋,於宗俦,陶本藻,等.广义测量平差[M].2版.武汉:武汉大学出版社,2009)
[2]Zhang Jun,Du Zhixing,Du Ning,et al.A Constraint Method to Determine Relative Weight Ratio Factors in Joint Inversion[J].Journal of Geodesy and Geodynamics,2015,35(4):600-603(张俊,独知行,杜宁,等.联合反演模型中相对权比的约束反演[J].大地测量与地球动力学,2015,35(4):600-603)
[3]Xu Caijun,Deng Changyong,Zhou Lixuan.Coseismic Slip Distribution Inversion Method Based on Variance Component Estimation[J].Geomatics and Information Science of Wuhan University,2016,41(1):37-44(许才军,邓长勇,周力璇.利用方差分量估计的地震同震滑动分布反演[J].武汉大学学报·信息科学版,2016,41(1):37-44)
[4]Duan Hurong,Zhang Yongzhi.On Relative Weight Ratio Relation Between Characteristics of Fault Slip and Various Type of Inversion Data[J].Journal of Geodesy and Geodynamics,2009,29(6):18-21(段虎荣,张永志.断层滑动特征与多种反演数据的相对权比关系研究[J].大地测量与地球动力学,2009,29(6):18-21)
[5]Du Zhixing,Ou Jikun,Jin Fengxiang,et al.Optimization Inversion of the Relative Weight Ratioλin the Joint Inversion Models[J].Acta Geodaetica et Cartographica Sinica,2003,32(1):15-19(独知行,欧吉坤,靳奉祥,等.联合反演模型中相对权比的优化反演[J].测绘学报,2003,32(1):15-19)
[6]Wang Leyang,Xu Caijun.Comparative Research on Equality Constraint Inversion and Joint Inversion[J].Journal of Geodesy and Geodynamics,2009,29(1):74-78(王乐洋,许才军.等式约束反演与联合反演的对比研究[J].大地测量与地球动力学,2009,29(1):74-78)
[7]Wang Leyang,Xu Caijun,Zhang Chaoyu.A Twostep Method to Determine Relative Weight Ratio Factors in Joint Inversion[J].Acta Geodaetica et Cartographica Sinica,2012,41(1):19-24(王乐洋,许才军,张朝玉.一种确定联合反演中相对权比的两步法[J].测绘学报,2012,41(1):19-24)
[8]Zhang Chaoyu,Xu Caijun.Sequential Algorithm of Geodesy Joint Inversion with Auto-match Weight Ratio and Its Application[J].Geomatics and Information Science of Wuhan University,2012,37(10):1 140-1 145(张朝玉,许才军.具有自适应权比的大地测量联合反演序贯算法及其应用[J].武汉大学学报·信息科学版,2012,37(10):1 140-1 145)
[9]Xu Caijun,Ding Kaihua,Cai Jianqiang,et al.Methods of Determining Weight Scaling Factors for GeodeticGeophysical Joint Inversion[J].Journal of GeoDynamics,2009,47(1):39-46
[10]Wang Leyang,Yu Hang.Study on the Applicability of Total Least Squares Method in Surveying Adjustment[J].Journal of Geodesy and Geodynamics,2014,34(3):121-124(王乐洋,余航.总体最小二乘方法的适用性研究[J].大地测量与地球动力学,2014,34(3):121-124)
[11]Wang Leyang,Xu Caijun.Total Least Square Adjustment with Weight Scaling Factor[J].Geomatics and Information Science of Wuhan University,2011,36(8):887-884(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·信息科学版,2011,36(8):887-884)
[12]Wang Leyang,Xu Guangyu,Chen Xiaoyong.Total Least Squares Adjustment of Partial Errors-in-Variables Model with Weight Scaling Factor[J].Geomatics and Information Science of Wuhan University,2017,42(6):857-863(王乐洋,许光煜,陈晓勇.附有相对权比的PEIV模型总体最小二乘平差[J].武汉大学学报·信息科学版,2017,42(6):857-863)
[13]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·信息科学版,2014,39(9):1 028-1 032)
[14]Zhou Yongjun,Fang Xing.A Mixed Weighted Least Squares and Weighted Total Least Squares Adjustment Method and Its Geodetic Applications[J].Survey Review,2016,48(351):421-429
[15]Fang Xing.Weighted Total Least Squares:Necessary and Sufficient Conditions,Fixed and Random Parameters[J].Journal of Geodesy,2013,87(8):733-749
[16]Jazaeri S,Amiri-Simkooei A R,Sharifi M A.Iterative Algorithm for Weighted Total Least Squares Adjustment[J].Survey Review,2014,46(334):19-27
[17]Zeng Wenxian.Effect of the Random Design Matrix on Adjustment of an EIV Model and Its Reliability Theory[D].Wuhan:Wuhan University,2013(曾文宪.系数矩阵误差对EIV模型平差结果的影响研究[D].武汉:武汉大学,2013)
[18]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·信息科学版,2016,41(12):1 683-1 689)
[19]Amiri-Simkooei A R.Application of Least Squares Variance Component Estimation to Errors-in-Variables Models[J].Journal of Geodesy,2013,87(10-12):935-944
[20]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
[21]Wang Leyang,Xu Guangyu.Variance Component Estimation for Partial Errors-in-Variables Models[J].Studia Geophysica et Geodaetica,2016,60(1):35-55
[22]Amiri-Simkooei A R.Non-negative Least-Squares Variance Component Estimation with Application to GPS Time Series[J].Journal of Geodesy,2016,90(5):451-466
[23]Li Bofeng.Variance Component Estimation in the Seamless Affine Transformation Model[J].Acta Geodaetica et Cartographica Sinica,2016,45(1):30-35(李博峰.无缝仿射基准转换模型的方差分量估计[J].测绘学报,2016,45(1):30-35)
[24]Cui Xizhang,Yu Zongchou,Tao Benzao,et al.Generalized Surveying Adjustment[M].2nd ed.Wuhan:Wuhan University Press,2009(崔希璋,於宗俦,陶本藻,等.广义测量平差[M].2版.武汉:武汉大学出版社,2009)