三维坐标转换的总体最小二乘拟合推估
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摘要
最小二乘的Gauss-Markov误差模型在三维坐标转换参数求解中存在缺陷,而求取坐标转换参数的目的是计算非公共点在目标坐标系的坐标,因此提出了一种三维坐标转换的总体最小二乘拟合推估法。考虑了原坐标系中公共点坐标和非公共点坐标的相关性以及非公共点在原坐标系中的坐标误差对模型的影响,对计算坐标转换参数和计算非公共点在目标坐标系中的坐标进行联合处理,推导出三维坐标转换的总体最小二乘拟合推估模型和求解方法。算例表明,利用新方法计算结果相比其他2种方法更加精确,均方根误差降低至0.077 3,相对百分误差降低至1.54%,证明了新方法的有效性和可靠性。
Gauss-markov error model of least squares has some defects in solving 3 d coordinate transformation parameters, and the purpose of obtaining coordinate transformation parameters is to calculate the coordinates of non-common points in the target coordinate system. Considering the common point coordinates in the coordinate system and the common point coordinates of relevance and the common points in the original coordinate system of the coordinate error of the model, the influence of the coordinate transformation parameters and calculates the common point in joint coordinates of target coordinate system, and deduced the three-dimensional coordinate conversion of total least squares fitting to estimate model and solving method. The results show that the new method is more accurate than the other two methods, and the root-mean-square error is reduced to 0.077 3 and the relative percentage error is reduced to 1.54%, which proves the effectiveness and reliability of the new method.
Gauss-markov error model of least squares has some defects in solving 3 d coordinate transformation parameters, and the purpose of obtaining coordinate transformation parameters is to calculate the coordinates of non-common points in the target coordinate system. Considering the common point coordinates in the coordinate system and the common point coordinates of relevance and the common points in the original coordinate system of the coordinate error of the model, the influence of the coordinate transformation parameters and calculates the common point in joint coordinates of target coordinate system, and deduced the three-dimensional coordinate conversion of total least squares fitting to estimate model and solving method. The results show that the new method is more accurate than the other two methods, and the root-mean-square error is reduced to 0.077 3 and the relative percentage error is reduced to 1.54%, which proves the effectiveness and reliability of the new method.
引文
[1] 李博峰,沈云中,李微晓.无缝三维基准转换模型[J].中国科学:地球科学,2012,42(7):1047-1054.
[2] 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.
[3] 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].大地测量与地球动力学,2008,28(5):77-81.
[4] 袁庆,楼立志,陈玮娴.加权总体最小二乘在三维基准转换中的应用[J].测绘学报,2011,40:115-119.
[5] 陆珏,陈义,郑波.加权总体最小二乘方法在ITRF转换中的应用[J].大地测量与地球动力学,2011,31(4):84-89.
[6] 黄令勇,吕志平,任雅奇,等.多元总体最小二乘在三维坐标转换总的应用[J].武汉大学学报(信息科学版),2014,39(7):793-798.
[7] 陆珏,陈义,郑波.三维非线性基准转换的加权和加约束总体最小二乘解算模型[J].大地测量与地球动力学,2012,32(6):■,et al.Deformation analysis with total least squares[J].Natural Hazards and Earth System Science,2006,6:663-669.
[9] FELUS F,BURTCH R.On sysmmetrical three-dimensional datum conversion[J].GPS Solutions,2009,13(1):65-74.
[10] 鲁铁定,宁津生.总体最小二乘平差理论及其应用[M].北京:中国科学技术出版社,2011.
[11] ZHOU Y J,KOU X J,ZHU J J,et al.A newton algorithm for weighted total least-squares solution to a specific errors-in-variables model with correlated measurements[J].Studia Geophysica et Geodaetica,2014,58(3):349- 375.
[12] SHEN Y Z,LI B F,CHEN Y.An iterative solution of weighted total least-squares adjustment[J].Journal of Geodesy,2011,85(4):229-238.
[2] 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.
[3] 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].大地测量与地球动力学,2008,28(5):77-81.
[4] 袁庆,楼立志,陈玮娴.加权总体最小二乘在三维基准转换中的应用[J].测绘学报,2011,40:115-119.
[5] 陆珏,陈义,郑波.加权总体最小二乘方法在ITRF转换中的应用[J].大地测量与地球动力学,2011,31(4):84-89.
[6] 黄令勇,吕志平,任雅奇,等.多元总体最小二乘在三维坐标转换总的应用[J].武汉大学学报(信息科学版),2014,39(7):793-798.
[7] 陆珏,陈义,郑波.三维非线性基准转换的加权和加约束总体最小二乘解算模型[J].大地测量与地球动力学,2012,32(6):■,et al.Deformation analysis with total least squares[J].Natural Hazards and Earth System Science,2006,6:663-669.
[9] FELUS F,BURTCH R.On sysmmetrical three-dimensional datum conversion[J].GPS Solutions,2009,13(1):65-74.
[10] 鲁铁定,宁津生.总体最小二乘平差理论及其应用[M].北京:中国科学技术出版社,2011.
[11] ZHOU Y J,KOU X J,ZHU J J,et al.A newton algorithm for weighted total least-squares solution to a specific errors-in-variables model with correlated measurements[J].Studia Geophysica et Geodaetica,2014,58(3):349- 375.
[12] SHEN Y Z,LI B F,CHEN Y.An iterative solution of weighted total least-squares adjustment[J].Journal of Geodesy,2011,85(4):229-238.