# Use of Total Least Squares Adjustment in Geodetic Applications
## 3 Practical Examples
### The Transformation of Coordinates
From the results of the transformation parameters (Tables 4-6), it can be seen that the results are slightly different if the adjustment is performed classically by the method of least squares, taking into account the weights (the WLS), or if the errors in the unknowns (WTLS) are also taken into account. The differences in the transformed coordinates in D96/TM between the WLS and WTLS are of the order of one centimeter. Using the SVD decomposition in the TLS method, we obtain very similar results to the WTLS method. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Method** & **Weights** & \(a\) & \(b\) & \(c\)**[m]** & \(d\)**[m]** & \(\alpha\)**[\({}^{*}\)]** & \(m\)**[ppm]** & \(\hat{\sigma}_{0}\)**[m]** \\ \hline WLS & yes & 0.9999986 & 0.0000132 & \(-\)370.98998 & 486.51088 & 2.72229 & \(-\)1.35 & 0.09911 \\ \hline TLS\_SVD & no & 0.9999942 & 0.0000143 & \(-\)370.98587 & 486.51985 & 2.94140 & \(-\)5.83 & / \\ \hline WTLS & yes & 0.9999948 & 0.0000140 & \(-\)370.98617 & 486.51914 & 2.89402 & \(-\)5.18 & 0.04278 \\ \hline GLS & yes & 0.9999948 & 0.0000140 & \(-\)370.98617 & 486.51914 & 2.89398 & \(-\)5.18 & 0.06051 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Transformation parameters—results. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & **D48/GK** & & & & & & **D96/TM** & & & \\ \hline
**Point** & **y [m]** & \(x\) [m]** & \(p_{y}\) & \(p_{x}\) & \(e\)**[m]** & \(n\)**[m]** & \(\sigma_{e}\) & \(\sigma_{n}\) & \(p_{e}\) & \(p_{n}\) \\ \hline T1 & 461,832.4600 & 99,989.5000 & 1 & 1 & 461,461.4803 & 100,475.9817 & 1.2 & 1.7 & 10 & 5 \\ \hline T2 & 461,849.9300 & 99,989.2200 & 1 & 1 & 461,478.8845 & 100,475.7180 & 1.4 & 2.0 & 8 & 5 \\ \hline T3 & 459,984.0200 & 99,868.1900 & 1 & 1 & 459,613.0262 & 100,354.6710 & 1.5 & 2.0 & 8 & 5 \\ \hline T4 & 459,937.1900 & 101,340.7500 & 1 & 1 & 459,566.2221 & 101,827.2837 & 0.4 & 0.8 & 25 & 10 \\ \hline T5 & 456,202.4000 & 99,557.5400 & 1 & 1 & 455,831.3858 & 100,044.0614 & 0.2 & 0.3 & 40 & 35 \\ \hline T6 & 454,777.1400 & 100,905.6500 & 1 & 1 & 454,406.2259 & 1013,92.2533 & 0.7 & 0.8 & 10 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: List of points for transformation. Comparing the WLS and WTLS results: Which one is better? Some of the parameters computed here indicate a better quality of adjustment by the WTLS method. If the quality criterion is represented by the a posteriori estimate of precision computed from the vector of measurements/observations residuals (corrections for point coordinates), then the value is better when WTLS is used (Table 4). When the a priori value of the standard deviation is set to 0.020 m for the initial coordinates in D48/GK and 0.005 m for the coordinates in D96/TM, the a posteriori value with the WLS is approximately 10 cm, and 4 cm when taking into account errors in variables (WTLS). Also, the mean value of the squares of the deviations of the transformed coordinates (based on the computed parameters) from those in the final/target coordinate system (Tables 5 and 6) is smaller with WTLS. It follows that the values of the computed transformation parameters are determined/computed by the WTLS method in such a way that the transformed coordinates on average fit better to the final \"measured\" coordinates of points.