# Use of Total Least Squares Adjustment in Geodetic Applications
## 4 Discussion and Conclusions  
The problem of finding an optimal solution for the unknowns in a given system of equations combining observations and unknowns can be approached in several ways. In general, the closest approach to the surveyors is adjustment by the least squares method, in which observations, unknowns, and constants are connected in a linearized form via the Gauss-Markov model (1). In this case, the solution can be computed using a system of normal equations with or without taking into account precision or weights (no weights--similar to case when we give all measurements the same weights, e.g., 1) of the measurement. Such treatment is problematic in some cases. In finding a regression line for a set of points in a horizontal plane, only one coordinate component of the points can be treated as an observation, while the other is a constant. In certain cases, however, it makes sense for both coordinate components to be included in the regression model as observations/measurements. According to the mathematical relationship, the \(x\)-coordinate appears on the side of model matrix \(\mathbf{A}\) and, in this way, also \"forces\" the treatment of errors on the side of the unknowns (EIV). A similar situation arises in the computation of the transformation parameters, where we want points in both the initial and target coordinate systems to have the status of observations in the adjustment. As an extension of the standard LS adjustment procedure, in addition to the well-known generalized adjustment model (GLS), an adjustment procedure based on the least squares method was investigated and developed by many researchers, and is referred to in the literature as \"total least squares\" or TLS. The method takes into account the errors on the side of the unknowns or in the matrix of adjustment model \(\mathbf{A}\), also with the possibility of taking into account weights (weighted total least squares --WTLS). In this article, we presented the adjustment according to classical LS and the TLS approach using the computational example of finding the coefficients of the regression line and the computation of the transformation parameters and compared the results. The  
Figure 5: Case 3—vectors of point displacements in a geodetic network (black—free network, magenta—S-transf., LS and TLS_SVD solution, and red—WTLS solution). regression line of the set of points in the plane computed by the TLS method adapts to the given precision of the \(x\)- and \(y\)-coordinates, which is reasonable and can be treated as a better solution than the LS solution. When transforming the coordinates, based on the computed quality statistics, it can be stated that, in the case of adjustment using the WTLS method, the transformation is better adapted to the points in the final (target) coordinate system. Considering the precision of the given coordinates of the points, which are many times better in the final/newer coordinate system than in the original system (especially in the official state/national coordinate system developing over time), such a result is of course more appropriate due to the generally better quality of coordinates in the newer coordinate system. The method was tested in transforming the positions of geodetic points from the old system to the new national coordinate system in Slovenia, which is quite problematic due to the characteristics of the old coordinate systems. The old system, dating back to the middle of the 20 th century, was developed using totally different measuring and computational methods compared to the new system, which utilizes modern, up-to-date GNSS techniques. The transformation between them is very important due to the need to convert a vast amount of spatial data from various databases (cadastre, GIS, and others) to the new coordinate system. For this purpose, the old geodetic points (e.g., churches), which have positions in the old coordinate system and can also be measured in the new system, are crucial for computing transformation parameters in a local area.