病态加权总体最小二乘模型的高斯-牛顿迭代正则化解  

Gauss-Netwon Iterative Regularizated Solution for Ill-Posed Weighted Total Least Squares Model

作　　者：刘云彤 高琼[2] 何宽[2] 秦奋[1] LIU Yuntong;GAO Qiong;HE Kuan;QIN Fen(College of Environment and Planning,Henan University,Kaifeng 475004,China;Yellow River Water Conservancy Technical Institute,Kaifeng 475004,China)

机构地区：[1]河南大学环境与规划学院,河南开封475004 [2]黄河水利职业技术学院,河南开封475004

出　　处：《测绘科学技术学报》2020年第3期239-245,共7页Journal of Geomatics Science and Technology

摘　　要：根据总体最小二乘模型的高斯-牛顿解法,建立了病态加权总体最小二乘模型的平差准则。由拉格朗日乘数法导出了病态加权总体最小二乘模型的高斯-牛顿正则化迭代解,在等权情形下导出了其与一般正则化解的近似差异。最后用两个算例对算法的有效性进行了验证,结果表明最小二乘解和总体最小二乘解受设计阵病态性影响而严重偏离真值,且病态性对总体最小二乘解的影响远大于最小二乘解;高斯-牛顿正则化迭代法同时考虑了设计阵和观测值的误差,并引入正则化参数削弱了设计阵的病态性,其解的精度较最小二乘解和总体最小二乘解大幅度提升。According to the Gauss-Netwon method of nonlinear adjustment model,the adjustment criterion of ill-posed weighted total least squares model is established.The Gauss-Newton regularizated iterative solution of the ill-posed weighted total least squares model is derived by Lagrange multiplier method and the approximate difference between solutions of new algorithm and generalized regularizated solution is derived in the equal weight case.Finally,two examples are adopted to verify the correctness of the algorithm and results show that solutions of least squares method and total least squares method are seriously deviated from the true value due to the ill-posedness of the design matrix.In addition,the ill-posedness has more negative impacts on the solution of total least squares method than that of least squares method.Conversely,the solution of Gauss-Netwon regularizated iterative method is closest to the true value since it considers the error of the design matrix and observation value and introduces the regularizated parameter to weaken the ill-posedness of the model,therefore the accuracy of the solution of new algorithm is much higher than solutions of least squares method and total least squares method.

关 键 词：总体最小二乘 病态模型 拉格朗日乘数法 正则化 高斯-牛顿迭代法 

分 类 号：P207[天文地球—测绘科学与技术]