## 4 Practical Applications of GNSS in Railway Surveys and Alignment Results

Table 4: Distribution of mean errors of determining the antenna positions following the alignment over the entire section of the track being measured (n = 507,251) and their percentage for the entire recording. All aligned coordinates of the position of 6 GNSS antennas (3,043,506 records in 507,251 measurement epochs) were plotted on a map of the region where surveys were performed (Figure 8). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Receiver** & **Minimal Value** & **Maximal Value** & **Mean Value** &
\begin{tabular}{c} **Confidence Intervals** \\ **for the Mean (\(\alpha\approx 0.05\))** \\ \end{tabular} & **Variance** & **Standard Deviation** & **Sample Size** \\ \hline
**GNSS \#1** & \(1.592\times 10^{-4}\) & 0.831 & \(3.099\times 10^{-4}\) & \(\left(2.985\times 10^{-4},3.213\times 10^{-4}\right)\) & \(1.712\times 10^{-5}\) & \(4.138\times 10^{-3}\) & 507,216 \\ \hline
**GNSS \#2** & \(4.557\times 10^{-7}\) & 2.068 & \(6.161\times 10^{-4}\) & \(\left(6.004\times 10^{-4},6.318\times 10^{-4}\right)\) & \(3.251\times 10^{-5}\) & \(5.702\times 10^{-3}\) & 507,216 \\ \hline
**GNSS \#3** & \(5.999\times 10^{-7}\) & 1.983 & \(3.297\times 10^{-4}\) & \(\left(3.110\times 10^{-4},3.485\times 10^{-4}\right)\) & \(4.651\times 10^{-5}\) & \(6.820\times 10^{-3}\) & 507,216 \\ \hline
**GNSS \#4** & \(9.948\times 10^{-7}\) & 1.206 & \(3.047\times 10^{-4}\) & \(\left(2.961\times 10^{-4},3.133\times 10^{-4}\right)\) & \(9.743\times 10^{-3}\) & \(3.121\times 10^{-3}\) & 507,216 \\ \hline
**GNSS \#5** & \(8.365\times 10^{-7}\) & 2.908 & \(5.643\times 10^{-4}\) & \(\left(5.395\times 10^{-4},5.880\times 10^{-4}\right)\) & \(8.087\times 10^{-3}\) & \(8.993\times 10^{-3}\) & 507,216 \\ \hline
**GNSS \#6** & \(6.732\times 10^{-7}\) & 2.613 & \(3.611\times 10^{-4}\) & \(\left(3.327\times 10^{-4},3.896\times 10^{-4}\right)\) & \(1.065\times 10^{-4}\) & \(1.002\times 10^{-2}\) & 507,216 \\ \hline
**Total** & \(4.557\times 10^{-7}\) & 2.908 & \(4.143\times 10^{-4}\) & \(\left(4.064\times 10^{-4},4.222\times 10^{-4}\right)\) & \(4.900\times 10^{-5}\) & \(6.993\times 10^{-3}\) & 3,043,296 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results of statistical analyses of the error of determining the antenna position coordinates following the alignment. Figure 6: Distributions of the antennas’ position errors over the test section. Figure 7: Graphical presentation of the mean error distribution over the entire test section. Due to the map scale, the coordinates of GNSS antennas faithfully reproduced the route of the surveying set passage. The figure was prepared using the ArcGIS software by ESRI. After zooming in on sections of the passage route, the positions of individual GNSS receiver antennas on any given section of the route can be seen. For the visualisation provided in Figure 9, orthophotomaps from the GEOPORTAL website [50] were used. The images have a lowered resolution and were taken at a certain angle, which results in minor shifts being visible in the position of receivers in relation to the railway track. Moreover, it can be seen in the figure that the mobile surveying platform moved along the same route in two directions but on different tracks. It should be concluded that the conducted alignment fulfilled its purpose and yielded a very good result for such a large measurement sample implemented in a vast area characterised by various availability of satellites due to the presence of obscuring features (built-up areas, forest areas, an open area). Moreover, the mobile surveying platform moved at different speeds (30-80 km/h), which also affected the continuity of coordinate measurement by GNSS receivers.