Linear estimation under the Gauss–Helmert model: geometrical interpretation and general solution

Although the least-squares (LS) adjustment within the Gauss–Markoff model (GMM) and the model with condition equations as dual problem have been geometrically interpreted, no one merged the LS adjustment formulated by the Gauss–Helmert model (GHM) also into the common pattern. We formulate the GHM from the GMM based on a partial orthogonality between their respective coefficient matrices. Then, the LS adjustment within these three models is interpreted geometrically, which implies that the GHM is not the combined model but the intermediate model between the GMM and the model of conditional equations. Meanwhile, the case of a singular dispersion matrix is analyzed once more, but now in the most general way, where the restriction of the Neitzel–Schaffrin rank condition is relaxed. The findings of this part can be summarized as: (1) The LS solution is developed within the general GHM, which yields the unique best linear uniformly unbiased estimation if the parameter coefficient matrix is of full rank; (2) We demonstrated that Baarda’s S-transformation also holds with the most general model setup; (3) In the general case, we proved that the unique S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{S}$$\end{document}-homBLUMBE (best homogeneously linear uniformly minimum bias estimation) can be achieved by S-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{S}^{-1}$$\end{document}-norm minimization within the LS solution set; (4) The S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{S}$$\end{document}-homBLE (best homogeneously linear estimation) type is analyzed for the GHM for the first time, and we find it can function as the intermediate algebraic connection between the LS and the S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{S}$$\end{document}-homBLUMBE solution.