# Use of Total Least Squares Adjustment in Geodetic Applications
## 1 Introduction  
In geodesy, the quantities we are interested in cannot usually be measured directly. However, we can compute these unknown quantities indirectly based on measurements. Due to the redundant measurements, the model that connects all measurements and unknown quantities consists of more equations than necessary and is therefore predetermined. The solution for the unknowns is usually obtained by adjusting the measurement residuals using the least squares (LS) method. In adjustment theory, this situation is described as the inverse problem, the goal of which is to determine the best-fitting parameters of the model that minimize the discrepancies between the observed values and those predicted by the model. The computational process involves iteratively adjusting the parameters to reduce the differences between observed and computed values until an optimal solution is achieved. Contrary to the inverse problem, the forward problem involves the prediction or computation of observed values based on known parameters and a mathematical model. We focus on the inverse problem. Due to its non-linearity, the system of equations linking measured values and unknowns is converted into a system of linear equations for measurement residuals, called the Gauss-Markov model [1]. In the next step, it is transformed into a system of normal equations [1, 2], which allows unambiguous determination of the values of the unknowns when the criterion of the least sum of the squares of measurement residuals is satisfied [2]. In this conventional LS adjustment approach, only measurement errors are taken into account. We assume the presence of normally distributed random errors, without gross errors. In geodesy, we have many methods to detect outliers and exclude them from the observations before the adjustment procedure. However, we can look for a solution by assuming that random errors are also present in the model on the side of unknowns orvariables. This approach has been considered by many researchers in the field [3; 4; 5; 6; 7; 8; 9]. It has been referred to as total least squares (TLS) adjustment and has proven to be very useful, especially in the field of regression fitting and transformation computations. In this article, we tested the applicability of the TLS method for two applications which are also common to many other studies [6; 7]. We start with best fitting the regression line to the set of points to practically demonstrate the TLS method explained theoretically with the equations and to clarify the geometrical meaning of the solutions. We used the same set of points as in [7] but our aim, unlike other studies, is to show the difference between different computational procedures. Moreover, the ordinary approaches (LS and LS using Singular Value Decomposition (SVD)) in computing the values of the unknowns under the condition of the minimum sum of the squares of the measurement residuals are also described and practically tested. The scope of the analysis and discussion of the different possibilities for the computation of the unknowns is also to evaluate the quality of each solution and comparison between them. The same concept of testing the TLS approach and other computational methods as well as the analysis of differences between solutions is also applied in the field of coordinate transformation. We test the suitability of the TLS approach for computing the transformation parameters for transitioning between two coordinate systems. The topic is current, and unlike other authors, it is implemented in the specific task of finding the most appropriate solution for the transformation parameters between the old and the new official coordinate system in Slovenia, which has its own peculiarities. Finally, by understanding the computational procedure involved in the task of transforming point coordinates, the TLS approach is also applied to the S-transformation of coordinate solutions from a free network into a datum-dependent solution.