传统大地测量应用中的基准转换往往涉及小角度旋转,可只考虑旋转角的一阶量采用线性化方法求解。现代空间测量技术成果应用的基准转换涉及大角度旋转,通过将旋转矩阵所有元素作为未知数并利用旋转矩阵正交条件采用附约束条件平差法迭代求解。本文以空间三维基准转换为例,采用多元模型的矩阵形式将多点坐标组成矩阵处理,并利用旋转矩阵的正交条件导出了大角度三维基准转换的解析分步解。同时引入两套公共点坐标误差对传统三维基准转换模型扩展,导出了同时顾及两套公共点坐标误差的大角度三维基准转换模型的解析解。试验表明:给出的大角度三维基准转换解析解能在实现与传统迭代解等效转换结果的同时,有效避免复杂耗时的迭代计算,提高计算效果。
The small rotation angles are typically involved in the traditional geodetic datum transformation, for which one can iteratively solve for its linearized model with ignoring its second-smaller terms. However, the big rotation angles are introduced to transform the outcomes from the advanced space surveying techniques. For this transformation model with big rotation angles, all elements of rotation matrix are usually parameterized as unknown parameters and then solved with the constrained adjustment theory by using the orthogonal condition of rotation matrix. With three-dimensional datum transformation with big rotation angles as example, this paper derives the analytical close-form solutions by formularizing the coordinates of multi-points as a matrix and using the orthogonal condition of rotation matrix. Expanding the transformation model with introducing the errors to common points of both datum, we derive out its analytical solutions as well. The results of simulation computations show that the presented three-dimensional datum transformation can realize the comparable transformation result while the new method can outcome the complicated and time-consuming iterations, therefore improving the computation efficiency.
[1] 朱华统, 杨元喜, 吕志平. GPS坐标系统的变换[M]. 北京:测绘出版社, 1994. ZHU Huatong, YANG Yuanxi, LV Zhiping. GPS Coordinate System Transformation[M]. Beijing:Surveying and Mapping Press, 1994.
[2] RAPP R H. Geometric Geodesy Part Ⅱ[M]. Ohio:Ohio State University, 1993.
[3] LI Bofeng, SHEN Yunzhong, LI Weixiao. The Seamless Model for Three-dimensional Datum Transformation[J]. Science China Earth Sciences, 2012, 55(12):2099-2108.
[4] YANG Yuanxi. Robust Estimation of Geodetic Datum Transformation[J]. Journal of Geodesy, 1999, 73(5):268-274.
[5] 沈云中, 胡雷鸣, 李博峰. Bursa模型用于局部区域坐标变换的病态问题及其解法[J]. 测绘学报, 2006, 35(2):95-98.DOI:10.3321/j.issn:1001-1595.2006.02.001. SHEN Yunzhong, HU Leiming, LI Bofeng. Ill-posed Problem in Determination of Coordinate Transformation Parameters with Small Area's Data Based on Bursa Model[J]. Acta Geodaetica et Cartographica Sinica, 2006, 35(2):95-98. DOI:10.3321/j.issn:1001-1595.2006.02.001.
[6] 沈云中, 卫刚. 利用过渡坐标系改进3维坐标变换模型[J].测绘学报, 1998, 27(2):161-165. SHEN Yunzhong, WEI Gang. Improvement of Three Dimensional Coordinate Transformation Model by Use of Interim Coordinate System[J]. Acta Geodaetica et Cartographica Sinica, 1998, 27(2):161-165.
[7] 杨元喜, 徐天河. 不同坐标系综合变换法[J]. 武汉大学学报(信息科学版),2001, 26(6):509-513. 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.
[8] YOU R, HWANG H.Coordinate Transformation between Two Geodetic Datums of Taiwan by Least-squares Collocation[J]. Journal of Surveying Engineering, 2006, 132(2):64-70.
[9] 杨元喜, 张菊清, 张亮. 基于方差分量估计的拟合推估及其在GIS误差纠正的应用[J]. 测绘学报, 2008, 37(2):152-157. YANG Yuanxi,ZHANG Juqing,ZHANG Liang.Variance Component Estimation Based Collocation and Its Application in GIS Error Fitting[J]. Acta Geodaetica et Cartographica Sinica, 2008, 37(2):152-157.
[10] 李博峰, 沈云中, 楼立志. 基于等效残差的方差-协方差分量估计[J]. 测绘学报, 2010, 39(4):349-354. LI Bofeng,SHEN Yunzhong,LOU Lizhi.Variance-covariance Component Estimation Based on the Equivalent Residuals[J]. Acta Geodaetica et Cartographica Sinica, 2010, 39(4):349-354.
[11] 陈义, 陆珏, 郑波. 近景摄影测量中大角度问题的探讨[J]. 测绘学报, 2008, 37(4):458-463. CHEN Yi, LU Jue, ZHENG Bo. Research on Close-range Photogrammetry with Big Rotation Angle[J]. Acta Geodaetica et Cartographica Sinica, 2008, 37(4):458-463.
[12] SHEN Yunzhong, CHEN Yi, ZHENG Dehua. A Quaternion-Based Geodetic Datum Transformation Algorithm[J]. Journal of Geodesy, 2006,80(5):233-239.
[13] KOCH K R. Parameter Estimation and Hypothesis Testing in Linear Models[M]. 2nd ed.Berlin:Springer,1999.
[14] 张尧庭, 方开泰. 多元统计分析引论[M]. 武汉:武汉大学出版社, 2013. ZHANG Yaoting, FANG Kaitai. Multivariate Statistics Analysis[M]. Wuhan:Wuhan University Press, 2013.
[15] SCHAFFRIN B, WIESER A. On Weighted Total Least-squares Adjustment for Linear Regression[J]. Journal of Geodesy,2008, 82(7):415-421.
[16] SHEN Yunzhong, LI Bofeng, CHEN Yi. An Iterative Solution of Weighted Total Least-squares Adjustment[J]. Journal of Geodesy, 2011,85(4):229-238.
[17] FANG Xing. Weighted Total Least Squares:Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy,2013, 87(8):733-749.
[18] 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.
[19] 李微晓, 沈云中, 李博峰. 顾及2套坐标误差的三维坐标变换方法[J]. 同济大学学报(自然科学版), 2011, 39(8):1243-1246. LI Weixiao, SHEN Yunzhong, LI Bofeng. Three-dimensional Coordinate Transformation with Consideration of Coordinate Errors in Two Coordinate Systems[J].Journal of Tongji University(Natural Science), 2011, 39(8):1243-1246.
[20] 李博峰, 沈云中. 基于等效残差积探测粗差的方差-协方差分量估计[J]. 测绘学报, 2011, 40(1):10-14. LI Bofeng, SHEN Yunzhong. Equivalent Residual Product Based Outlier Detection for Variance and Covariance Component Estimation[J]. Acta Geodaetica et Cartographica Sinica, 2011, 40(1):10-14.
