附有等式约束病态模型正则化解的单位权中误差无偏估计
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摘要
利用平差参数间合理的等式约束虽能提升病态模型解的精度,但其本质仍是通过引入正则化参数来改善模型的病态性,由于改变了观测方程的结构,所得的估值残差及单位权中误差均有偏。针对这一不足,在病态模型正则化解的无偏单位权方差估计式基础上引入等式约束条件,根据约束正则化解的残差二次型期望公式,导出约束正则化解的无偏单位权中误差估计式,并用数值算例和病态测边网算例验证其正确性。结果表明,本文公式所估的单位权中误差精度优于传统公式所估结果。
The accuracy of solution for ill-posed model can be significantly improved by introducing the reasonable equality constraint between the adjustment parameters. However, the essence is to improve the ill-posedness of model by introducing regularizated parameters, which results in the change of structure of the observation equation. Therefore, the solution of ill-posed model with equality constraints is biased and its residual is amplified accordingly, which results that the unit weight medium error estimated by traditional formula is biased. In this paper, the unbiased estimation of unit weight medium error of constrained regularizated solution is derived according to mathematic expectancy formula of residual matrix in quadratic form, which is based on the unbiased estimation of unit weight variance for regularizated solution. The correctness of the formula is verified by numerical example and ill-posed trilateration network example and results show that the accuracy of unit weight medium error estimated by formula derived by this paper is better than traditional formula.
The accuracy of solution for ill-posed model can be significantly improved by introducing the reasonable equality constraint between the adjustment parameters. However, the essence is to improve the ill-posedness of model by introducing regularizated parameters, which results in the change of structure of the observation equation. Therefore, the solution of ill-posed model with equality constraints is biased and its residual is amplified accordingly, which results that the unit weight medium error estimated by traditional formula is biased. In this paper, the unbiased estimation of unit weight medium error of constrained regularizated solution is derived according to mathematic expectancy formula of residual matrix in quadratic form, which is based on the unbiased estimation of unit weight variance for regularizated solution. The correctness of the formula is verified by numerical example and ill-posed trilateration network example and results show that the accuracy of unit weight medium error estimated by formula derived by this paper is better than traditional formula.
引文
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[2] 李慕清,刘正华,夏传甲,等.GPS单历元模型病态方程解算方法研究[J].大地测量与地球动力学,2013,33(增1):160-162(Li Muqing,Liu Zhenghua,Xia Chuanjia,et al.Analysis of Ill-Posed Equation Resolution Technique for GPS Positioning in Epoch-Wise[J].Journal of Geodesy and Geodynamics,2013,33(S1):160-162)
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[4] 邓凯亮,黄谟涛,暴景阳,等.向下延拓航空重力数据的Tikhonov双参数正则化法[J].测绘学报,2011,40(6):690-696(Deng Kailiang,Huang Motao,Bao Jingyang,et al.Tikhonov Two-Parameter Regulation Algorithm in Downward Continuation of Airborne Gravity Data[J].Acta Geodaetica et Cartographica Sinica,2011,40(6):690-696)
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[6] Save H,Bettadpur S,Tapley B D.Reducing Errors in the GRACE Gravity Solutions Using Regularization[J].Journal of Geodesy,2012,86(9):695-711
[7] Zhou W N.Normalized Full Gradient of Full Tensor Gravity Gradient Based on Adaptive Iterative Tikhonov Regularization Downward Continuation[J].Journal of Applied Geophysics,2015,118:75-83
[8] Schaffrin B,Felus Y A.An Algorithmic Approach to the Total Least-Squares Problem with Linear and Quadratic Constraints[J].Studia Geophysica et Geodaetica,2009,53(1):1-16
[9] 谢建,朱建军.等式约束对病态问题的影响及约束正则化方法[J].武汉大学学报:信息科学版,2015,40(10):1 344-1 348(Xie Jian,Zhu Jianjun.Influence of Equality Constraints on Ill-Conditioned Problems and Constrained Regularization Method[J].Geomatics and Information Science of Wuhan University,2015,40(10):1 344-1 348)
[10] 马下平.附有约束条件的间接平差扩展模型的参数估计及其精度评定[J].工程勘察,2016,44(5):55-59(Ma Xiaping.Parameter Estimation and Precision Assessment of Extended Indirect Adjustment Model with Constraints[J].Geotechnical Investigation & Surveying,2016,44(5):55-59)
[11] 王乐洋,许才军,汪建军.附有病态约束矩阵的等式约束反演问题研究[J].测绘学报,2009,38(5):397-401(Wang Leyang,Xu Caijun,Wang Jianjun.Research on Equality Constraint Inversion with Ill-Posed Constraint Matrix[J].Acta Geodaetica et Cartographica Sinica,2009,38(5):397-401)
[12] 王乐洋,许才军.附有病态约束反演问题的岭估计法[J].武汉大学学报:信息科学版,2011,36(5):612-616(Wang Leyang,Xu Caijun.Ridge Estimation Method in Inversion Problem with Ill-Posed Constraint[J].Geomatics and Information Science of Wuhan University,2011,36(5):612-616)
[13] 马洋,欧吉坤,袁运斌,等.采用联合平差法处理附有病态等式约束的反演问题[J].武汉大学学报:信息科学版,2011,36(7):816-819(Ma Yang,Ou Jikun,Yuan Yunbin,et al.Solving Equality Constraint Inversion with Ill-Posed Constraint Matrix Using United Method[J].Geomatics and Information Science of Wuhan University,2011,36(7):816-819)
[14] 谢建,朱建军.等式约束病态模型的正则化解及其统计性质[J].武汉大学学报:信息科学版,2013,38(12):1 440-1 444(Xie Jian,Zhu Jianjun.A Regularized Solution and Statistical Properties of Ill-Posed Problem with Equality Constraints[J].Geomatics and Information Science of Wuhan University,2013,38(12):1 440-1 444)
[15] 沈云中,刘大杰.正则化解的单位权方差无偏估计公式[J].武汉大学学报:信息科学版,2002,27(6):604-606(Shen Yunzhong,Liu Dajie.Unbiased Estimation Formula of Unit Weight Mean Square Error in Regularization Solution[J].Geomatics and Information Science of Wuhan University,2002,27(6):604-606)
[16] Xu P L,Shen Y Z,Fukuda Y,et al.Variance Component Estimation in Linear Inverse Ill-Posed Models[J].Journal of Geodesy,2006,80(2):69-81
[17] 张俊,独知行,杜宁,等.补偿最小二乘解的单位权方差的无偏估计[J].大地测量与地球动力学,2014,34(1):153-156(Zhang Jun,Du Zhixing,Du Ning,et al.Unbiased Estimation Formula of Unit Weight Variance in Solution by Penalized Least Square[J].Journal of Geodesy and Geodynamics,2014,34(1):153-156)
[18] 崔希璋.广义测量平差[M].武汉:武汉大学出版社,2009(Cui Xizhang.Generalized Measurement Adjustment[M].Wuhan:Wuhan University Press,2009)
[19] Hansen P C,O’Leary D P.The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems[J].SIAM Journal on Scientific Computing,1993,14(6):1 487-1 503
[20] 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(Lu Tieding,Ning Jinsheng.Total Least Squares Adjustment Theory and Its Application[M].Beijing:Science and Technology of China Press,2011)
[21] 王乐洋,于冬冬.病态总体最小二乘问题的虚拟观测解法[J].测绘学报,2014,43(6):575-581(Wang Leyang,Yu Dongdong.Virtual Observation Method to Ill-Posed Total Least Squares Problem[J].Acta Geodaetica et Cartographica Sinica,2014,43(6):575-581)
[2] 李慕清,刘正华,夏传甲,等.GPS单历元模型病态方程解算方法研究[J].大地测量与地球动力学,2013,33(增1):160-162(Li Muqing,Liu Zhenghua,Xia Chuanjia,et al.Analysis of Ill-Posed Equation Resolution Technique for GPS Positioning in Epoch-Wise[J].Journal of Geodesy and Geodynamics,2013,33(S1):160-162)
[3] 顾勇为,归庆明,张璇,等.大地测量与地球物理中病态性问题的正则化迭代解法[J].测绘学报,2014,43(4):331-336(Gu Yongwei,Gui Qingming,Zhang Xuan ,et al.Iterative Solution of Regularization to Ill-Conditioned Problems in Geodesy and Geophysics[J].Acta Geodaetica et Cartographica Sinica,2014,43(4):331-336)
[4] 邓凯亮,黄谟涛,暴景阳,等.向下延拓航空重力数据的Tikhonov双参数正则化法[J].测绘学报,2011,40(6):690-696(Deng Kailiang,Huang Motao,Bao Jingyang,et al.Tikhonov Two-Parameter Regulation Algorithm in Downward Continuation of Airborne Gravity Data[J].Acta Geodaetica et Cartographica Sinica,2011,40(6):690-696)
[5] 许才军,陈庭,张丽琴.地球物理大地测量反演理论与应用[M].武汉:武汉大学出版社,2015(Xu Caijun,Chen Ting,Zhang Liqin.Joint Inversion Theory Geophysics and Geodesy and Its Application[M].Wuhan:Wuhan University Press,2015)
[6] Save H,Bettadpur S,Tapley B D.Reducing Errors in the GRACE Gravity Solutions Using Regularization[J].Journal of Geodesy,2012,86(9):695-711
[7] Zhou W N.Normalized Full Gradient of Full Tensor Gravity Gradient Based on Adaptive Iterative Tikhonov Regularization Downward Continuation[J].Journal of Applied Geophysics,2015,118:75-83
[8] Schaffrin B,Felus Y A.An Algorithmic Approach to the Total Least-Squares Problem with Linear and Quadratic Constraints[J].Studia Geophysica et Geodaetica,2009,53(1):1-16
[9] 谢建,朱建军.等式约束对病态问题的影响及约束正则化方法[J].武汉大学学报:信息科学版,2015,40(10):1 344-1 348(Xie Jian,Zhu Jianjun.Influence of Equality Constraints on Ill-Conditioned Problems and Constrained Regularization Method[J].Geomatics and Information Science of Wuhan University,2015,40(10):1 344-1 348)
[10] 马下平.附有约束条件的间接平差扩展模型的参数估计及其精度评定[J].工程勘察,2016,44(5):55-59(Ma Xiaping.Parameter Estimation and Precision Assessment of Extended Indirect Adjustment Model with Constraints[J].Geotechnical Investigation & Surveying,2016,44(5):55-59)
[11] 王乐洋,许才军,汪建军.附有病态约束矩阵的等式约束反演问题研究[J].测绘学报,2009,38(5):397-401(Wang Leyang,Xu Caijun,Wang Jianjun.Research on Equality Constraint Inversion with Ill-Posed Constraint Matrix[J].Acta Geodaetica et Cartographica Sinica,2009,38(5):397-401)
[12] 王乐洋,许才军.附有病态约束反演问题的岭估计法[J].武汉大学学报:信息科学版,2011,36(5):612-616(Wang Leyang,Xu Caijun.Ridge Estimation Method in Inversion Problem with Ill-Posed Constraint[J].Geomatics and Information Science of Wuhan University,2011,36(5):612-616)
[13] 马洋,欧吉坤,袁运斌,等.采用联合平差法处理附有病态等式约束的反演问题[J].武汉大学学报:信息科学版,2011,36(7):816-819(Ma Yang,Ou Jikun,Yuan Yunbin,et al.Solving Equality Constraint Inversion with Ill-Posed Constraint Matrix Using United Method[J].Geomatics and Information Science of Wuhan University,2011,36(7):816-819)
[14] 谢建,朱建军.等式约束病态模型的正则化解及其统计性质[J].武汉大学学报:信息科学版,2013,38(12):1 440-1 444(Xie Jian,Zhu Jianjun.A Regularized Solution and Statistical Properties of Ill-Posed Problem with Equality Constraints[J].Geomatics and Information Science of Wuhan University,2013,38(12):1 440-1 444)
[15] 沈云中,刘大杰.正则化解的单位权方差无偏估计公式[J].武汉大学学报:信息科学版,2002,27(6):604-606(Shen Yunzhong,Liu Dajie.Unbiased Estimation Formula of Unit Weight Mean Square Error in Regularization Solution[J].Geomatics and Information Science of Wuhan University,2002,27(6):604-606)
[16] Xu P L,Shen Y Z,Fukuda Y,et al.Variance Component Estimation in Linear Inverse Ill-Posed Models[J].Journal of Geodesy,2006,80(2):69-81
[17] 张俊,独知行,杜宁,等.补偿最小二乘解的单位权方差的无偏估计[J].大地测量与地球动力学,2014,34(1):153-156(Zhang Jun,Du Zhixing,Du Ning,et al.Unbiased Estimation Formula of Unit Weight Variance in Solution by Penalized Least Square[J].Journal of Geodesy and Geodynamics,2014,34(1):153-156)
[18] 崔希璋.广义测量平差[M].武汉:武汉大学出版社,2009(Cui Xizhang.Generalized Measurement Adjustment[M].Wuhan:Wuhan University Press,2009)
[19] Hansen P C,O’Leary D P.The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems[J].SIAM Journal on Scientific Computing,1993,14(6):1 487-1 503
[20] 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(Lu Tieding,Ning Jinsheng.Total Least Squares Adjustment Theory and Its Application[M].Beijing:Science and Technology of China Press,2011)
[21] 王乐洋,于冬冬.病态总体最小二乘问题的虚拟观测解法[J].测绘学报,2014,43(6):575-581(Wang Leyang,Yu Dongdong.Virtual Observation Method to Ill-Posed Total Least Squares Problem[J].Acta Geodaetica et Cartographica Sinica,2014,43(6):575-581)