## 2 Materials and Methods
### Potential Modifications of the ASG-EUPOS's APPS  
Currently, the APPS of the ASG-EUPOS determines point coordinates based on the method for adjusting direct and equally accurate observations with the least-squares method (equally accurate method, EAM). In this method of adjusting, the components of the basis vectors should be treated as observations. Three-dimensional point coordinates (X, Y, Z) are determined with the EAM (including accuracy evaluation after adjustment) following the procedure below (Equations (1)-(7)). Coordinates X, Y, Z of point P are determined from measured baseline vectors to reference stations (Figure 2): \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Service** & \begin{tabular}{c} **Method of Position** \\ **Determination** \\ \end{tabular} & **Assumed Positioning Accuracy** & **Application** \\ \hline \multirow{3}{*}{NAWGEO} & RTK and NRTK with use of & 0.03 m horizontal & \multirow{3}{*}{land surveying, precision agriculture} \\  & dual-frequency GNSS & 0.05 m vertical & \\  & observations & & \\ \hline \multirow{3}{*}{KODGIS} & real-time positioning with & \multirow{3}{*}{down to 0.25–3.00 m in 3D} & construction, spatial information \\  & single-frequency code receivers & & systems, agriculture, tourism and \\ \hline \multirow{3}{*}{POZGEO} & post-processing of dual-frequency & \multirow{3}{*}{\
\begin{tabular}{c} depends on the measurement \\ conditions and duration. It ranges \\ from 0.01 to 0.10 m in 3D \\ \end{tabular} } & \multirow{3}{*}{land surveying} \\  & & & \\ \hline \multirow{3}{*}{POZGEO-D} & & \multirow{3}{*}{
\begin{tabular}{c} provide users with historical GPS or \\ GLONASS observation data from \\ ASC-EUPOS or virtual \\ reference stations \\ \end{tabular} } \\ \cline{1-1} \cline{3-4}  & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Services offered by ASG-EUPOS [17]. \[\left\{ \begin{array}{l}X_{P}^{i}=X_{i}+\Delta X_{Pi}\\ Y_{P}^{i}=Y_{i}+\Delta Y_{Pi}\\ Z_{P}^{i}=Z_{i}+\Delta Z_{Pi}\end{array} \right. \tag{1}\]  
where:  
\(X_{P}^{i}\), \(Y_{P}^{i},Z_{P}^{i}\)--coordinates of determined point P based on baseline vector to the i reference station;  
\(X_{i}\), \(Y_{i}\), \(Z_{i}\)--coordinates of the i reference station;  
\(\Delta X_{Pi}\), \(\Delta Y_{Pi}\), \(\Delta Z_{Pi}\)--components of baseline vector between point P and I reference station. For a GPS vector (\(\Delta X\), \(\Delta Y\), \(\Delta Z\)) between two points P, i (i = 1, 2, 3,..., n; n--the number of tie points), the following observation equations may be written:  
\[\left\{ \begin{array}{l}\Delta X_{Pi}+v_{Pi}^{\Delta X}=X_{P}-X_{i}\\ \Delta Y_{Pi}+v_{Pi}^{\Delta Y}=Y_{P}-Y_{i}\\ \Delta Z_{Pi}+v_{Pi}^{\Delta Z}=Z_{P}-Z_{i}\end{array} \right. \tag{2}\]  
where:  
\(X_{P}\), \(Y_{P}\), \(Z_{P}\)--coordinates of determined point P;  
\(v_{Pi}^{\Delta X}\), \(v_{Pi}^{\Delta Y}\), \(v_{Pi}^{\Delta Z}\)--corrections for components of the baseline vector. As Equation (2) is linear, the correction equations are derived directly (without expansion into a Taylor series):  
\[\left\{ \begin{array}{l}v_{Pi}^{\Delta X}=X_{P}-X_{i}-\Delta X_{Pi}\\ v_{Pi}^{\Delta Y}=Y_{P}-Y_{i}-\Delta Y_{Pi}\\ v_{Pi}^{\Delta Z}=Z_{P}-Z_{i}-\Delta Z_{Pi}\end{array} \right. \tag{3}\]  
In keeping with the least-squares principle, the corrections expressed in Equation (3) should conform (for each coordinate X, Y, Z separately) to:  
\[\sum_{i=1}^{n}\Bigl{(}v_{Pi}^{\Delta X}\Bigr{)}^{2}=min.;\sum_{i=1}^{n}\Bigl{(} v_{Pi}^{\Delta Y}\Bigr{)}^{2}=min.;\sum_{i=1}^{n}\Bigl{(}v_{Pi}^{\Delta Z} \Bigr{)}^{2}=min. \tag{4}\]  
The conditions in Equation (4) are sound assuming that the components of the baseline vector are considered as equally accurate observations. These conditions are satisfied if  
Figure 2: Example of point P positioning from six tie points (i = 1 \(\div\) 6).