Tikhonov-regularized weighted total least squares formulation with applications to geodetic problems

Abstract

This contribution presents the Tikhonov regularized weighted total least squares (TRWTLS) solution in an errors-in-variables (EIV) model. The previous attempts had solved this problem based on the hybrid approximation solution (HAPS) within a nonlinear Gauss-Helmert model. The present formulation is a generalized form of the classical nonlinear Gauss-Helmert model, having formulated in an EIV general mixed observation model. It is a follow-up to the previous work throughout the WTLS problems formulated rely on a standard least squares (SLS) theory. Two cases, namely the EIV parametric model and the classical nonlinear mixed model, could be considered special cases of the general mixed observation model. These formulations are conceptually simple; because they are formulated based on the SLS theory, and subsequently, the existing SLS knowledge can directly be applied to the ill-posed mixed EIV model. Two geodetic applications have then adopted to illustrate the developed theory. As a first case, 2D affine transformation parameters (six-parameter affine transformation) for ill-scattered data points are adeptly solved by the TRWTLS method. Second, the circle fitting problem as a nonlinear case is not only tackled for well-scattered data points but also tackled for ill-scattered data points in a nonlinear mixed model. Finally, all results indicate that the Tikhonov regularization provides a stable and reliable solution in an ill-posed WTLS problem, and hence an efficient method applicable to many engineering problems.