# Use of Total Least Squares Adjustment in Geodetic Applications
## 3 Practical Examples
### The S-Transformation  
#### 3.3.1 General information about the S-Transformation  
The S-transformation is a computational method that transforms an arbitrary datum-dependent solution for the vector of unknowns (coordinates of points) in the adjustment of the geodetic network to a selected geodetic datum. The method has been developed and described in detail by many researchers [17; 18]. The basic S-transformation equation that converts the solution for coordinate unknowns from one (\(gdi\)) to another (\(gdj\)) geodetic datum is: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**WTLS**} & \multicolumn{2}{c}{**New Coordinate System**} & \multicolumn{2}{c}{**Differences**} & \multicolumn{2}{c}{**Transformed to D96/TM**} \\ \multicolumn{2}{c}{} & \multicolumn{2}{c}{**(D96/TM)**} & \multicolumn{2}{c}{**} & \multicolumn{2}{c}{**} & \multicolumn{2}{c}{**} \\ \hline
**Point** & \(e\) **[m]** & \(n\) **[m]** & \(de\) **[m]** & \(dn\) **[m]** & \(e\) **[m]** & \(n\) **[m]** \\ \hline T1 & 461,461.4803 & 100,475.9817 & \(-\)0.0247 & 0.0005 & 461,461.4557 & 100,475.9822 \\ \hline T2 & 461,478.8845 & 100,475.7180 & 0.0411 & \(-\)0.0160 & 461,478.9256 & 100,475.7020 \\ \hline T3 & 459,613.0262 & 100,354.6710 & \(-\)0.0027 & 0.0278 & 459,613.0235 & 100,354.6988 \\ \hline T4 & 459,566.2221 & 101,827.2837 & \(-\)0.0077 & \(-\)0.0319 & 459,566.2144 & 101,827.2518 \\ \hline T5 & 455,831.3858 & 100,044.0614 & 0.0330 & 0.0421 & 455,831.4188 & 100,044.1035 \\ \hline T6 & 454,406.2259 & 101,392.2533 & \(-\)0.0408 & \(-\)0.0268 & 454,406.1851 & 101,392.2265 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Differences between transformed and given coordinates in the final coordinate system—using the WTLS approach. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**WLS**} & \multicolumn{2}{c}{**New Coordinate System**} & \multicolumn{2}{c}{**Differences**} & \multicolumn{2}{c}{**Transformed to D96/TM**} \\ \multicolumn{2}{c}{} & \multicolumn{2}{c}{**(D96/TM)**} & \multicolumn{2}{c}{**} & \multicolumn{2}{c}{**} & \multicolumn{2}{c}{**} \\ \hline
**Point** & \(e\) **[m]** & \(n\) **[m]** & \(de\) **[m]** & \(dn\) **[m]** & \(e\) **[m]** & \(n\) **[m]** \\ \hline T1 & 461,461.4803 & 100,475.9817 & \(-\)0.0178 & \(-\)0.0065 & 461,461.4625 & 100,475.9752 \\ \hline T2 & 461,478.8845 & 100,475.7180 & 0.0480 & \(-\)0.0231 & 461,478.9325 & 100,475.6949 \\ \hline T3 & 459,613.0262 & 100,354.6710 & \(-\)0.0028 & 0.0187 & 459,613.0235 & 100,354.6897 \\ \hline T4 & 459,566.2221 & 101,827.2837 & \(-\)0.0092 & \(-\)0.0354 & 459,566.2130 & 101,827.2484 \\ \hline T5 & 455,831.3858 & 100,044.0614 & 0.0187 & 0.0287 & 455,831.4045 & 100,044.0901 \\ \hline T6 & 454,406.2259 & 101,392.2533 & \(-\)0.0617 & \(-\)0.0363 & 454,406.1642 & 101,392.2170 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Differences between transformed and given coordinates in the final coordinate system—using the WLS approach. \[\mathbf{x}_{gdj}=\mathbf{S}_{gdj}\mathbf{x}_{gdi},\] (51) with \[\mathbf{S}_{gdj}=\mathbf{I}-\mathbf{H}(\mathbf{H}^{\mathrm{T}}\mathbf{E}\mathbf{ H})^{-1}\mathbf{H}^{\mathrm{T}}\mathbf{E},\] (52) where \[\mathbf{H}= \begin{bmatrix}&\vdots&\\ 1&0&x_{i}&y_{i}\\ 0&1&-y_{i}&x_{i}\\ &\vdots&\end{bmatrix} \] is the matrix of inner constraints with dimension of \[2m\times(datum\ defect).\] If the datum defect is 3 (the scale is known in the case of measured distances in network), then we, for instance, omit the fourth column. \(\mathbf{E}\) is the matrix of dimensions \(2m\times 2m\) (\(m\)--number of points in network) with off-diagonal elements that are equal to 0; and on the diagonal, there are values of 1 only in places that belong to the individual coordinate component that represents the given quantity to define the geodetic datum _gdi_. Matrix \(\mathbf{E}\) can be considered as a weight matrix and Equation (51) as a weighted S-transformation [19].