论平差模型的关系与几何解释  

Relationship and Geometric Interpretation of Adjustment Models

作　　者：胡宇 姚宜斌[1] 方兴[1] 曾文宪[1] HU Yu;YAO Yibin;FANG Xing;ZENG Wenxian(School of Geodesy and Geomatics,Wuhan University,Wuhan 430079,China)

机构地区：[1]武汉大学测绘学院,湖北武汉430079

出　　处：《武汉大学学报（信息科学版）》2023年第8期1366-1372,共7页Geomatics and Information Science of Wuhan University

基　　金：国家自然科学基金(41774009,42174049)。

摘　　要：高斯马尔可夫模型(间接平差模型)、条件平差模型和高斯赫尔默特模型是3种经典平差模型,通常根据参数个数的不同从模型形式上予以表述,但该方式不能反映模型间的本质关系。首先,以矩阵分析理论为基础,依据系数矩阵的半正交性实现了高斯马尔可夫模型与高斯赫尔默特模型的转换,完善了3种平差函数模型之间的数学转换关系,揭示了模型间内在的数学联系。然后,基于线性空间和投影算子,解释了高斯马尔可夫模型和条件平差模型的最小二乘估计,阐述了两类模型的最小二乘目标函数的等价性,并结合勾股定理和对偶性将两种经典目标函数形式扩展至4种等效目标函数,说明了其在空间意义上的构造原理,揭示了目标函数之间的几何关系和内涵,对于平差模型及其估计的数学本质的理解具有理论和应用价值。Objectives:Gauss-Markov model,the model with condition equations and Gauss-Helmert model are three classical adjustment models.Gauss-Helmert model can be seen as the combined form of the other two models.They are usually distinguished by the number of unknown parameters.However,this classification method cannot reflect the essential relationship between them.In addition,the connec⁃tions between Gauss-Markov model and the conditional model has been intensively investigated,while that of Gauss-Helmert model has been ignored.Methods:Some equivalent objective functions of the adjust⁃ment of the least-square in the context of Gauss-Markov model and the conditional model are proposed based on the projection theory and the dual optimization technique.Based on the theory of matrix analysis,this paper realizes the conversion between Gauss-Markov model and Gauss-Helmert model according to partial orthogonality of coefficient matrices,thus completing the structure of these models and revealing the mathematical relationships.Results:The least-square adjustment within Gauss-Markov model and the model with condition equations are explained geometrically by employing the linear space and projector.The equivalence of the objective functions of these two models are also expounded.Combining the Pythagore⁃an theorem and duality theory,we extend them to four equivalent objective functions and illustrate the con⁃struction principle.The transformation between Gauss-Markov model and Gauss-Helmert model are achieved by the so-called partial-orthogonality.Conclusions:The geometric reasoning of the least-squares adjustment is analyzed once more.Two objection functions are newly proposed,and the geometric relation and connotation between objective functions are revealed.Gauss-Helmert model is not only the combined model in terms of the model appearance,these three adjustment models can actually be mutually trans⁃formed.As a result,the understanding of the mathematical nature of adjustment models and the corresponding estimation i

关 键 词：测量平差 最小二乘估计 正交投影 对偶性 

分 类 号：P207[天文地球—测绘科学与技术]