# 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  
In addition to the solution in Equation (54), we can find the solution for \(\mathbf{t}_{\mathrm{S}}\) from Equation (53) using SVD decomposition. According to Equation (44), we can extend the \(\mathbf{H}\) matrix by vector \(\mathbf{x}_{gdi}\) and take into account the \"weighting\" matrix \(\mathbf{E}\):  
\[\widetilde{\mathbf{H}}=[\mathbf{E}\mathbf{H}|\mathbf{E}\mathbf{x}_{gdi}], \tag{55}\]  
In this case, the solution can be found according to Equations (46)-(49). We must take care that the system is at least uniquely determined. This means that we have at least uniquely determined the geodetic datum; or in numbers, \(\mathrm{rank}(\mathbf{H})\) should not be smaller than \(\mathrm{rank}(\widetilde{\mathbf{H}})\). The solution for vector \(\mathbf{t}_{\mathrm{S}}\) follows the solutions in Equations (44) to (49). To illustrate the computation of different solutions for \(\mathbf{t}_{S}\), we simulated (the Box-Muller approach [20]) measurements of horizontal angles and distances in a predefined 2D geodetic network (Figure 4) for two measurement epochs, \(t_{1}\) and \(t_{2}\). Epoch \(t_{2}\) is compared to \(t_{1}\), a network with forced point movements (Table 7). For each epoch, the network was adjusted as a free network with the same initial values of point coordinates to provide the same geodetic datum for both epochs. Comparing solutions of free network adjustment for \(t_{1}\) and \(t_{2}\) in Table 7, it is obvious that we cannot obtain the right solution for displacements of points compared to forced values. Therefore, we usually use the S-transformation (51) with appropriately defined matrix \(\mathbf{E}\). Model (53) can be solved in the same way as we performed computations for coordinate transformations using the described TLS approaches. We computed ordinary S-transformation, LS, and TLS using SVD and the WTLS solution of model (53). The results for differently defined geodetic datum (cases 1-3 with values 1 in matrix \(\mathbf{E}\) in positions for selected datum points) are presented in Tables 8-10. For each solution, we also computed the sum of the squares of \(dx\) and \(dy\) coordinate differences \(\mathbf{x}_{gdj}^{\mathrm{T}}\mathbf{x}_{gdj}\). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Initial**} & \multicolumn{2}{c}{**Forced**} & \multicolumn{2}{c}{**Epoch \(t_{1}\)**} & \multicolumn{2}{c}{**Epoch \(t_{2}\)**} \\  & \multicolumn{2}{c}{**Values**} & \multicolumn{2}{c}{**Movement**} & & & & & & & & \\ \hline \multirow{2}{*}{**Point**} & \(x\) & \(y\) & \(dx\) & \(dy\) & \(x\) & \(y\) & \(\sigma_{x}\) & \(\sigma_{y}\) & \(x\) & \(y\) & \(\sigma_{x}\) & \(\sigma_{y}\) \\  & [**m**] & [**m**] & [**mm**] & [**mm**] & [**m**] & [**m**] & [**mm**] & [**mm**] & [**m**] & [**m**] & [**mm**] & [**mm**] \\ \hline \multirow{2}{*}{\(1\)} & 1000 & 1000 & / & / & 999.9999 & 999.9984 & 0.5 & 0.6 & 999.9948 & 999.9906 & 0.5 & 0.6 \\ \hline \multirow{2}{*}{\(2\)} & 2000 & 1000 & / & / & 1999.9998 & 1000.0010 & 0.5 & 0.6 & 1999.9948 & 999.9957 & 0.5 & 0.6 \\ \hline \multirow{2}{*}{\(3\)} & 2600 & 1900 & 10 & 10 & 2600.0002 & 1899.9999 & 0.6 & 0.5 & 2600.0030 & 1900.0059 & 0.6 & 0.5 \\ \hline \multirow{2}{*}{\(4\)} & 2200 & 2500 & 15 & 10 & 2199.9997 & 2499.9996 & 0.5 & 0.6 & 2200.0061 & 2500.0047 & 0.5 & 0.6 \\ \hline \multirow{2}{*}{\(5\)} & 1200 & 2600 & 15 & 15 & 1200.0005 & 2600.0002 & 0.5 & 0.6 & 1200.0067 & 2600.0079 & 0.5 & 0.6 \\ \hline \multirow{2}{*}{\(6\)} & 400 & 1600 & / & / & 400.0001 & 1600.0004 & 0.5 & 0.5 & 399.9936 & 1599.9913 & 0.5 & 0.5 \\ \hline \multirow{2}{*}{\(7\)} & 1500 & 1800 & 8 & 10 & 1500.0000 & 1800.0005 & 0.4 & 0.4 & 1500.0011 & 1800.0039 & 0.4 & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: A simulated geodetic network with free network solutions for epochs \(t_{1}\) and \(t_{2}\). Figure 4: Geometry of the geodetic network.