# Use of Total Least Squares Adjustment in Geodetic Applications
## 3 Practical Examples
### The S-Transformation
#### 3.3.2 The Total Least Squares Solution for the S-Transformation
The comparison of the results in Table 8, Table 9 and Table 10 clearly shows that appropriate solutions for point displacements can only be obtained in the case of the appropriately defined geodetic datum. The results show that the solutions for the appropriate definition of geodetic datums are close to simulated/forced movements of points and are also the same (case 1 and 2--Tables 8 and 9) for all computational approaches. What is apparent from the results is that in case 3 (Table 10) of an inappropriately selected geodetic datum defined by the combination of points that are not all suitable for datum definition \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{} & \multicolumn{2}{c}{**Free Network**} & \multicolumn{2}{c}{**S-Transform.**} & \multicolumn{2}{c}{**LS**} & \multicolumn{2}{c}{**TLS_SVD**} & \multicolumn{2}{c}{**WTLS**} \\ \hline
**Point** & \(\mathbf{dx}\)**[m]** & \(\mathbf{dy}\)**[m]** & \(\mathbf{dx}\)**[m]** & \(\mathbf{dy}\)**[m]** & \(\mathbf{dx}\)**[m]** & \(\mathbf{dy}\)**[m]** & \(\mathbf{dx}\)**[m]** & \(\mathbf{dy}\)**[m]** & \(\mathbf{dx}\)**[m]** & \(\mathbf{dy}\)**[m]** \\ \hline
1 & \(-0.00510\) & \(-0.00780\) & \(0.00137\) & \(-0.00412\) & \(0.00137\) & \(-0.00412\) & \(0.00138\) & \(-0.00412\) & \(0.00162\) & \(-0.00584\) \\ \hline
2 & \(-0.00500\) & \(-0.00530\) & \(-0.00431\) & \(0.00057\) & \(-0.00431\) & \(0.00057\) & \(-0.00431\) & \(0.00057\) & \(-0.00436\) & \(0.00046\) \\ \hline
3 & \(0.00280\) & \(0.00600\) & \(-0.00195\) & \(0.00798\) & \(-0.00195\) & \(0.00798\) & \(-0.00195\) & \(0.00798\) & \(-0.00362\) & \(0.00856\) \\ \hline
4 & \(0.00640\) & \(0.00510\) & \(0.00265\) & \(0.00274\) & \(0.00265\) & \(0.00274\) & \(0.00265\) & \(0.00274\) & \(0.00265\) & \(0.00274\) & \(0.00014\) & \(0.00249\) \\ \hline
5 & \(0.00620\) & \(0.00770\) & \(0.00801\) & \(0.00257\) & \(0.00801\) & \(0.00257\) & \(0.00801\) & \(0.00257\) & \(0.00564\) & \(0.00069\) \\ \hline
6 & \(-0.00650\) & \(-0.00910\) & \(0.00213\) & \(-0.0102\) & \(0.00213\) & \(-0.0102\) & \(0.00213\) & \(-0.01020\) & \(0.00160\) & \(-0.01306\) \\ \hline
7 & \(0.00110\) & \(0.00340\) & \(0.00293\) & \(0.00355\) & \(0.00293\) & \(0.00355\) & \(0.00293\) & \(0.00355\) & \(0.00175\) & \(0.00239\) \\ \hline  & \(\mathbf{x}_{\mathit{Sgl}}^{\overline{\mathrm{T}}}\mathbf{x}_{\mathit{Sgl}}\) & \(0.00049\) & & \(0.00032\) & & \(0.00032\) & & \(0.00032\) & & \(0.00036\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Case 3—solutions for predetermined geodetic datum of points 1, 2 and 7. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{} & \multicolumn{2}{c}{**Free Network**} & \multicolumn{2}{c}{**S-Transform. **} & \multicolumn{2}{c}{**LS**} & \multicolumn{2}{c}{**TLS_SVD**} & \multicolumn{2}{c}{**WTLS**} \\ \hline
**Point** & \(\mathbf{dx}\)**[m]** & \(\mathbf{dy}\)**[m]** & \(\mathbf{dx}\)**[m]** & \(\mathbf{dy}\)**[m]** & \(\mathbf{dx}\)**[m]** & \(\mathbf{dy}\)**[m]** & \(\mathbf{dx}\)**[m]** & \(\mathbf{dy}\)**[m]** & \(\mathbf{dx}\)**[m]** & \(\mathbf{dy}\)**[m]** \\ \hline
1 & \(-0.00510\) & \(-0.007(point 7 has moved), the values of point displacements (coordinate changes) and directions (Figure 5) and also the sum of the squares of all coordinate changes \(\mathbf{x}_{gdj}^{\text{T}}\mathbf{x}_{gdj}\) are wrong and not the same for each solution. It can be seen from Table 10 (case 3) that the WTLS solution significantly differs from other solutions, which is not the case when the geodetic datum is appropriately defined (cases 1 and 2). Taking into account this finding, it is possible to find suitable combinations of points that appropriately define the geodetic datum and result in correct values of point displacements. Consequently, the solution for the appropriate combination of geodetic datums can therefore be found iteratively, combining different sets of datum points in matrix E.