On weighted total least-squares for geodetic transformations

In this contribution, it is proved that the weighted total least-squares (WTLS) approach preserves the structure of the coefficient matrix in errors-in-variables (EIV) model when based on the perfect description of the dispersion matrix. To achieve this goal, first a proper algorithm for WTLS is developed since the quite recent analytical solution for WTLS by Schaffrin and Wieser is restricted to the condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{P}_{\rm A} =\left({P_0 \otimes P_x}\right)}$$\end{document} (where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\otimes}$$\end{document} is used to denote the Kronecker product) for the weight matrix of the coefficient matrix in the EIV model. This situation can be seen in the case of an affine transformation where the univariate approach can be an appropriate alternative to the multivariate WTLS approach, which has been applied to the affine transformation by Schaffrin and Felus, resp. Schaffrin and Wieser with restrictions similar to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{P}_{\rm A} =\left( {P_0 \otimes P_x}\right)}$$\end{document}. In addition, this algorithm for WTLS can be interpreted well in the geodetic literature since it is based on the perfect description of the inverse dispersion matrix (or variance–covariance). By using the algorithm of WTLS, one obtains more realistic results in some applications of transformation where a high precision is needed. Some empirical examples, resp. simulation studies give insight into the efficiency of the procedure.