带有区间不确定性的约束平差算法及应用
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摘要
针对测量数据中带有区间约束先验信息和附加信息,基于Kuhn-Tucker条件,将测量平差问题转化为二次规划问题,该文提出了一种处理参数带有区间不确定性的新算法,给出了算法具体模型和解算步骤,并且通过模拟数值实验和病态测边网数据计算,分析了在处理病态问题时,最小二乘平差的局限性,通过与岭估计和奇异值分解法的结果相比较,说明了参数带有区间不确定性的平差算法的有效性。
Considering the measurement data contains prior information and additional information.Based on Kuhn-Tucker condition used in quadratic programming,this paper propose a new algorithm with interval uncertain,and give its concrete model and solving steps.According to the results of simulated experiment and ill-posed trilateration net data,it shows that the least-squares has limitation when processing morbid problem.Compared with the results of ridge estimation and singular value decomposition(SVD),it shows that the algorithm with interval uncertain of parameter is effective.
Considering the measurement data contains prior information and additional information.Based on Kuhn-Tucker condition used in quadratic programming,this paper propose a new algorithm with interval uncertain,and give its concrete model and solving steps.According to the results of simulated experiment and ill-posed trilateration net data,it shows that the least-squares has limitation when processing morbid problem.Compared with the results of ridge estimation and singular value decomposition(SVD),it shows that the algorithm with interval uncertain of parameter is effective.
引文
[1]宋迎春,金昊,崔先强.带有不确定性的观测数据平差解算方法[J].武汉大学学报(信息科学版),2014,39(7):788-792.(SONG Yingchun,JIN Hao,CUI Xianqiang.Adjustment algorithm about observation data with uncertainty[J].Geomatics and Information Science of Wuhan University,2014,39(7):788-792.)
[2]张勤.近代测量数据处理与应用[M].北京:测绘出版社,2011.(ZHANG Qin.Advanced theory and application of surveying data[M].Beijing:Surveying Press,2011.)
[3]王乐洋,许才军.总体最小二乘研究进展[J].武汉大学学报(信息科学版),2013,38(7):850-856.(WANG Leyang,XU Caijun.Progress in total least squares[J].Geomatics and Information Science of Wuhan University,2013,38(7):850-856.)
[4]曾文宪,方兴,刘经南,等.附有不等式约束的加权整体最小二乘算法[J].测绘学报,2014,43(10):1013-1018.(ZENG Wenxian,FANG Xing,LIU Jingnan,et al.Weighted total least squares algorithm with inequality constraints[J].Acta Geodaetica et Cartogra phica Sinica,2014,43(10):1013-1018.)
[5]朱建军,谢建,陈宇波,等.附不等式约束平差的理论与方法研究[J].测绘工程,2008,17(6):1-5.(ZHU Jianjun,XIE Jian,CHEN Yubo,et al.Research on theory and methods of inequality constrained least squares[J].Engineering of Surveying and Mapping,2008,17(6):1-5.)
[6]宋迎春,左廷英,朱建军.带有线性不等式约束平差模型的算法研究[J].测绘学报,2008,37(4):433-437.(SONG Yingchun,ZUO Tingying,ZHU Jianjun.Research on algorithm of adjustment model with linear inequality constrained parameters[J].Acta Geodaetica et Cartographica Sinica,2008,37(4):433-437.)
[7]朱建军,谢建,陈宇波.不等式约束对平差结果的影响分析[J].测绘学报,2011,40(4):411-415.(ZHU Jianjun,XIE Jian,CHEN Yubo.Analysis of inequality constraints influence to adjustment results[J].Acta Geodaetica et Cartographica Sinica,2011,40(4):411-415.)
[8]朱建军,谢建.附不等式约束平差的一种简单迭代算法[J].测绘学报,2011,40(2):209-212.(ZHU Jianjun,XIE Jian.A simple iterative algorithm for inequality constrained adjustment[J].Acta Geodaetica et Cartographica Sinica,2011,40(2):209-212.)
[9]张松林,张昆.附加线性不等式约束的条件平差模型未知参数的解算[J].大地测量与地球动力学,2015,35(6):1049-1052.(ZHANG Songlin,ZHANG Kun.Solving the unknown parameters in conditional adjustment models with linear inequality constraints[J].Journal of Geodesy and Geodynamics,2015,35(6):1049-1052.)
[10]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):1344-1348.(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):1344-1348.)
[11]刘斌,李术才,李树忱,等.基于不等式约束的最小二乘法三维电阻率反演及其算法优化[J].地球物理学报,2012,55(1):260-268.(LIU Bin,LI Shucai,LI Shuchen,et al.3Delectrical resistivity inversion with least-squares method based on inequality constraint and its computation efficiency optimization[J].Chinese Journal of Geophysics,2012,55(1):260-268.)
[12]张松林,陈德虎.带不等式约束的间接平差模型的三种解算方法比较[J].大地测量与地球动力学,2013,33(2):41-44.(ZHANG Songlin,CHEN Dehu.Comparison among three algorithms of parameter adjustment model with inequality constraints[J].Journal of Geodesy and Geodynamics,2013,33(2):41-44.)
[13]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):1440-1444.(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):1440-1444.)
[14]宋迎春,谢雪梅,陈晓林.不确定性平差模型的平差准则与解算方法[J].测绘学报,2015,44(2):135-141.(SONG Yingchun,XIE Xuemei,CHEN Xiaolin.Adjustment criterion and algorithm in adjustment model with uncertainty[J].Acta Geodaetica et Cartographica Sinica,2015,44(2):135-141.)
[15]邹渤,宋迎春,唐争气,等.沉降观测AR模型的不确定性平差算法[J].大地测量与地球动力学,2016,36(8):686-688.(ZOU Bo,SONG Yingchun,TANG Zhengqi,et al.Adjustment algorithm on model with uncertain in settlement observation[J].Journal of Geodesy and Geodynamics,2016,36(8):686-688.)
[16]王乐洋,许才军,鲁铁定.病态加权总体最小二乘平差的岭估计解法[J].武汉大学学报(信息科学版),2010,35(11):1346-1350.(WANG Leyang,XU Caijun,LU Tieding.Ridge estimation method in ill-posed weighted total least squares adjustment[J].Geomatics and Information Science of Wuhan University,2010,35(11):1346-1350.)
[17]徐文,陈义,游为.奇异值分解法在病态问题中的应用[J].测绘通报,2016(1):62-63.(XU Wen,CHEN Yi,YOU Wei.Application of SVD method in ill-posed problem[J].Bulletin of Surveying and Mapping,2016(1):62-63.)
[18]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 illposed total least squares problem[J].Acta Geodaetica et Cartographica Sinica,2014,43(6):575-581.)
[2]张勤.近代测量数据处理与应用[M].北京:测绘出版社,2011.(ZHANG Qin.Advanced theory and application of surveying data[M].Beijing:Surveying Press,2011.)
[3]王乐洋,许才军.总体最小二乘研究进展[J].武汉大学学报(信息科学版),2013,38(7):850-856.(WANG Leyang,XU Caijun.Progress in total least squares[J].Geomatics and Information Science of Wuhan University,2013,38(7):850-856.)
[4]曾文宪,方兴,刘经南,等.附有不等式约束的加权整体最小二乘算法[J].测绘学报,2014,43(10):1013-1018.(ZENG Wenxian,FANG Xing,LIU Jingnan,et al.Weighted total least squares algorithm with inequality constraints[J].Acta Geodaetica et Cartogra phica Sinica,2014,43(10):1013-1018.)
[5]朱建军,谢建,陈宇波,等.附不等式约束平差的理论与方法研究[J].测绘工程,2008,17(6):1-5.(ZHU Jianjun,XIE Jian,CHEN Yubo,et al.Research on theory and methods of inequality constrained least squares[J].Engineering of Surveying and Mapping,2008,17(6):1-5.)
[6]宋迎春,左廷英,朱建军.带有线性不等式约束平差模型的算法研究[J].测绘学报,2008,37(4):433-437.(SONG Yingchun,ZUO Tingying,ZHU Jianjun.Research on algorithm of adjustment model with linear inequality constrained parameters[J].Acta Geodaetica et Cartographica Sinica,2008,37(4):433-437.)
[7]朱建军,谢建,陈宇波.不等式约束对平差结果的影响分析[J].测绘学报,2011,40(4):411-415.(ZHU Jianjun,XIE Jian,CHEN Yubo.Analysis of inequality constraints influence to adjustment results[J].Acta Geodaetica et Cartographica Sinica,2011,40(4):411-415.)
[8]朱建军,谢建.附不等式约束平差的一种简单迭代算法[J].测绘学报,2011,40(2):209-212.(ZHU Jianjun,XIE Jian.A simple iterative algorithm for inequality constrained adjustment[J].Acta Geodaetica et Cartographica Sinica,2011,40(2):209-212.)
[9]张松林,张昆.附加线性不等式约束的条件平差模型未知参数的解算[J].大地测量与地球动力学,2015,35(6):1049-1052.(ZHANG Songlin,ZHANG Kun.Solving the unknown parameters in conditional adjustment models with linear inequality constraints[J].Journal of Geodesy and Geodynamics,2015,35(6):1049-1052.)
[10]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):1344-1348.(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):1344-1348.)
[11]刘斌,李术才,李树忱,等.基于不等式约束的最小二乘法三维电阻率反演及其算法优化[J].地球物理学报,2012,55(1):260-268.(LIU Bin,LI Shucai,LI Shuchen,et al.3Delectrical resistivity inversion with least-squares method based on inequality constraint and its computation efficiency optimization[J].Chinese Journal of Geophysics,2012,55(1):260-268.)
[12]张松林,陈德虎.带不等式约束的间接平差模型的三种解算方法比较[J].大地测量与地球动力学,2013,33(2):41-44.(ZHANG Songlin,CHEN Dehu.Comparison among three algorithms of parameter adjustment model with inequality constraints[J].Journal of Geodesy and Geodynamics,2013,33(2):41-44.)
[13]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):1440-1444.(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):1440-1444.)
[14]宋迎春,谢雪梅,陈晓林.不确定性平差模型的平差准则与解算方法[J].测绘学报,2015,44(2):135-141.(SONG Yingchun,XIE Xuemei,CHEN Xiaolin.Adjustment criterion and algorithm in adjustment model with uncertainty[J].Acta Geodaetica et Cartographica Sinica,2015,44(2):135-141.)
[15]邹渤,宋迎春,唐争气,等.沉降观测AR模型的不确定性平差算法[J].大地测量与地球动力学,2016,36(8):686-688.(ZOU Bo,SONG Yingchun,TANG Zhengqi,et al.Adjustment algorithm on model with uncertain in settlement observation[J].Journal of Geodesy and Geodynamics,2016,36(8):686-688.)
[16]王乐洋,许才军,鲁铁定.病态加权总体最小二乘平差的岭估计解法[J].武汉大学学报(信息科学版),2010,35(11):1346-1350.(WANG Leyang,XU Caijun,LU Tieding.Ridge estimation method in ill-posed weighted total least squares adjustment[J].Geomatics and Information Science of Wuhan University,2010,35(11):1346-1350.)
[17]徐文,陈义,游为.奇异值分解法在病态问题中的应用[J].测绘通报,2016(1):62-63.(XU Wen,CHEN Yi,YOU Wei.Application of SVD method in ill-posed problem[J].Bulletin of Surveying and Mapping,2016(1):62-63.)
[18]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 illposed total least squares problem[J].Acta Geodaetica et Cartographica Sinica,2014,43(6):575-581.)