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@phdthesis { 11303_10233.2,
author = {Malissiovas, Georgios},
title = {New nonlinear adjustment approaches for applications in geodesy and related fields},
school = {Technische UniversitÃ¤t Berlin},
year = {2019},
type = {Doctoral Thesis},
address = {Berlin},
doi = {10.14279/depositonce-9194.2},
url = {http://dx.doi.org/10.14279/depositonce-9194.2},
keywords = {nonlinear least squares, fitting line, fitting plane, planar transformation, total least squares, singular value decomposition, eigenvalue problem, errors-in-variables, nichtlineare kleinste Quadrate, ausgleichende Gerade, ausgleichende Ebene, Ebene Transformation, Ausgleichungsrechnung},
abstract = {This dissertation deals with a class of nonlinear adjustment problems that has a direct least squares solution for certain weighting cases. In the literature of mathematical statistics these problems are expressed in a nonlinear model called Errors-In-Variables (EIV) and their solution became popular as total least squares (TLS). The TLS solution is direct and involves the use of singular value decomposition (SVD), presented in most cases for adjustment problems with equally weighted and uncorrelated measurements. Additionally, several weighted total least squares (WTLS) algorithms have been published in the last years for deriving iterative solutions, when more general weighting cases have to be taken into account and without linearizing the problem in any step of the solution process.
This research provides firstly a well defined mathematical relationship between TLS and direct least squares solutions. As a by-product, a systematic approach for the direct solution of these adjustments is established, using a consistent and complete mathematical formalization. By transforming the problem to the solution of a quadratic or cubic algebraic equation, which is identical with those resulting from TLS, it will be shown that TLS is an algorithmic approach already known to the geodetic community and not a new method.
A second contribution of this work is the clear overview of weighted least squares solutions for the discussed class of problems, i.e. the WTLS solution in the terminology of the statistical community. It will be shown that for certain weighting cases a direct solution still exists, for which two new solution strategies will be proposed. Further, stochastic models with more general weight matrices are examined, including correlations between the measurements or even singular cofactor matrices. New algorithms are developed and presented, that provide iterative weighted least squares solutions without linearizing the original nonlinear problem.
The aim of this work is the popularization of the TLS approach, by presenting a complete framework for obtaining a (weighted) least squares solution for the investigated class of nonlinear adjustment problems. The proposed approaches and the implemented algorithms can be employed for obtaining direct solutions in engineering tasks for which efficiency is important, while iterative solutions can be derived for stochastic models with more general weights.}
}