# Use of Total Least Squares Adjustment in Geodetic Applications
## 3 Practical Examples
### The Transformation of Coordinates  
Considering the transition between the two versions of the coordinate systems is relevant because in everyday official geodetic practice data on points, both coordinate systems are interwoven--cadastral points, legacy points of different levels of geodetic networks, GNSS measurement points, etc. At the state level, or for the data of which the Surveying Authority of the Republic of Slovenia is the administrator, the country-wide triangular transformation model (sectional affine transformations [16]) was officially adopted. In this task, we set out to find the transformation parameters for the transformation in the local area of the part of Ljubljana city based on the points given in the old coordinate system (D48/GK, GK--Gauss-Krueger projection) and determined by the GNSS method in the new coordinate system (D96/TM, TM--transverse Mercator projection). The precision for points in the old official coordinate system (D48/GK) is not listed in Table 3 as we do not have information about them. We estimate that the coordinates were determined with no better than the centimeter precision. This is due to the fact that at the time of determining the coordinates of these points in the old coordinate system (more than 40 years ago), the only measurement method used in geodetic networks for geodetic points of this class was triangulation. The precision of the coordinates of points in D96/TM comes from the results of our measurement and computations of the geodetic network, combining classical terrestrial (measurements of angles and distances) and GNSS surveying methods, and is generally several times more precise than in D48/GK (Table 3). Accordingly, we set the weights of individual points in both coordinate systems (Table 3), with the weights in the initial (D48/GK) coordinate system set to 1. The weights in the new (D96/TM) coordinate systems are presented by the ratio between the precision of points in  
Figure 3: Geometrical representation for a solution of LS (dashed line) in comparison to TLS (dashed–dotted line). the old and new coordinate systems (e.g., for the \(e\) coordinate, \(p_{e}=\sigma_{e}^{old\;cs}/\sigma_{e}^{new\;cs}\)) rounded to the appropriate whole number to emphasize the difference in precision quality. We consider four-parametric similarity transformation in the horizontal plane in the form:  
\[ \begin{bmatrix}e\\ n\end{bmatrix} = \begin{bmatrix}a&b\\ -b&a\end{bmatrix}\begin{bmatrix}y\\ x\end{bmatrix} + \begin{bmatrix}c\\ d\end{bmatrix} , \tag{50}\]  
where \(e\) and \(n\) are the coordinates in new coordinate system, \(y\) and \(x\) are the coordinates in the old coordinate system, \(a\) and \(b\) are transformation parameters including scale and orientation change (\(a=m\cdot\)cos\(\alpha\), \(b=m\cdot\)sin\(\alpha\); \(m\) is scale, \(\alpha\) is rotation) and \(c\) and \(d\) are translation parameters. The transformation parameters are computed in four different ways (Table 4), which are discussed in this paper: according to the WLS, TLS using SVD (TLS_SVD), and with the weighted TLS approach (WTLS). In the computation, we consider reduced coordinates to the center of gravity of the grid in the initial coordinate system (D48/GK). From the results in Table 4, it can be seen that the WTLS method gives quite similar results in the considered case with or without considering the weights (TLS_SVD) in the adjustment and that they are also consistent with the results of the general adjustment model (GLS); only the a posteriori precision estimate and the rotation angle \(\alpha\) are slightly different. We are also interested in the evaluation of the transformation quality from the point of view of the comparison of the transformed coordinates based on the computed parameters (Tables 5 and 6). In this analysis, we restrict ourselves only to the computations that take into account the weights (WTLS and the WLS).