## 4 Practical Applications of GNSS in Railway Surveys and Alignment Results
Figure 5: The position of GNSS receivers on the mobile surveying platform. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Station Name** & \(x\) & \(y\) \\ \hline
**Starogard Gdński** & 5981664.913 & 6532343.251 \\
**Konarzyny** & 5965574.243 & 6460849.453 \\
**Czersk** & 5962606.324 & 6498027.094 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Coordinates of the SmartNet network reference stations. parameter is a circle with a radius of \(m_{i}\). Even though the point position error is a conventional parameter and has no deeper probabilistic justification, it is very useful in many comparative analyses; therefore, Table 4 presents the percentage of error sizes, for the entire test section, along with the number of points (n) for which their values are present in the intervals \((0,0.001)\); \(\langle 0.001,\,0.005\rangle\); \(\langle 0.005,\,0.05\rangle\); \(\langle 0.05,\,m^{\max}\rangle\). In analysing Table 4, it can be concluded that the error of determining the aligned coordinates of GNSS antennas does not exceed 0.005 m:  
* for GNSS #1 in 99.59%
* for GNSS #2 in 99.00%
* for GNSS #3 in 99.59%
* for GNSS #4 in 99.63%
* for GNSS #5 in 99.19%
* for GNSS #6 in 99.62%  
The error values in the \(\langle 0.05,\,m^{\max}\rangle\) range appear, on average, only in 0.04%. On the other hand, Figure 6 presents the distributions of the values of errors of determining the individual receivers' position coordinates over the entire test section. Table 5 presents the results of statistical analyses of the series of recorded error values. The table provides minimum values, maximum values, sample mean values, confidence intervals for the mean value, variance, standard deviation and the sample size. The analysis was conducted independently for each of the six GNSS receivers as well as collectively for the entire series of the measured values. The total sample size amounted to 3,043,296. The mean value of the measured error values is in the order of \(10^{-4}\). For the mean value from the sample, confidence intervals with a specified confidence level \(\alpha=0.05\) were determined, which means that the mean value for the entire error population falls within the specified intervals with a probability of 0.95. It was observed that the confidence intervals were very narrow, i.e., the width of the confidence interval was smaller by two orders of magnitude than the mean value of the sample. This proves that the surveys were conducted very precisely, and the collected data set can be regarded as reliable and credible. This is also confirmed by the very low variance, in the order of \(10^{-5}\). Despite the fact that values higher by as much as four orders of magnitude than the mean value are found in the set; these cases can be considered extremely rare and should be regarded as exceptions or anomalies. These exceptionally high values should be isolated and omitted in further research. Error values of more than 5 cm are primarily found in the urbanised area (Tczew, Chojnice), which is confirmed by Figure 7. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{**m [m]**} \\ \cline{2-10}
**Receiver** & \multicolumn{2}{c}{**0–0.001**} & \multicolumn{2}{c}{**0.001–0.005**} & \multicolumn{2}{c}{**0.005–0.05**} & \multicolumn{2}{c}{**0.05–max**} & \multicolumn{2}{c}{**m**} \\ \cline{2-10}  & **n** & **\%** & **n** & **\%** & **n** & **\%** & **n** & **\%** & \\ \hline
**GNSS \#1** & 495,231 & 97.63\% & 9938 & 1.96\% & 1965 & 0.39\% & 116 & 0.02\% & 0.831 \\
**GNSS \#2** & 455,199 & 89.74\% & 46,981 & 9.26\% & 4738 & 0.93\% & 333 & 0.07\% & 2.068 \\
**GNSS \#3** & 496,176 & 97.82\% & 8971 & 1.77\% & 1963 & 0.39\% & 141 & 0.03\% & 1.983 \\
**GNSS \#4** & 492,935 & 97.18\% & 12,438 & 2.45\% & 1707 & 0.34\% & 171 & 0.03\% & 1.206 \\
**GNSS \#5** & 465,407 & 91.75\% & 37,738 & 7.44\% & 3824 & 0.75\% & 282 & 0.06\% & 2.908 \\
**GNSS \#6** & 497,205 & 98.02\% & 8122 & 1.60\% & 1784 & 0.35\% & 140 & 0.03\% & 1.506 \\ \hline \hline \end{tabular}
\end{table}