## 2 Materials and Methods
### Data Pre-Processing  
The input data for the least-squares adjustment procedures are planar coordinates of six receivers calculated using projection functions. The first step to achieve this goal is the post-processing of raw satellite observations recorded by six used receivers and synchronous observations from adjacent CORS reference stations of active geodetic networks [30]. In the post-processing calculation, the final standard product 3 (SP3) generated by, e.g., GFZ or IGS [41], and GNSS receiver antennas' parameters (ANTEX) [42] datasets are required to obtain the receiver's coordinates in the WGS84 system [43]. SP3 final orbits  
Figure 2: UTC time series of the GNSS static sessions conducted in Rybina on 20 January 2021 (DOY: 20). contain the most accurate information on satellite positions that are interpolated from 15\({}^{\prime}\) (IGS) or 5\({}^{\prime}\) (GFZ) interval datasets. Satellite measurements carried out in railway conditions are characterized by high frequency due to the need to present the geometry of the route with the shortest possible linear interval. For this purpose, the InnoSatTrack project used receivers measuring with frequencies of 20 Hz (Trimble R10 and Leica GS18) and 100 Hz (Trimble Alloy). The stationary measurement performed in this study used a frequency of 20 Hz to represent the conditions of the railway campaigns. However, one important factor related to the processing of satellite data should be noted. The SP3 product containing the data on satellite positions and satellite clock errors is provided every 15 min (IGS) or 5 min (GFZ). Hence, the estimated positions of receivers and other related geometrical parameters obtained are not fully independent between the epochs. This is a limitation of the study due to the unavailability of other methods for processing raw satellite data. The second step of the initial stage of the calculations is the conversion of the Cartesian ellipsoidal Earth-centered Earth-fixed (ECEF) coordinates into curvilinear latitude and longitude coordinates using the Hirvonen algorithm [44]. Finally, in the third stage, the latitude and longitude are transformed into the Polish coordinate system PL-2000 [45], which is a modified Transverse Mercator projection [46]:  
\[x_{TM}=k_{0}N\Bigg{[}D+\frac{(1-T+C)D^{3}}{6}+\frac{\left(5-18T+T^{2}+72C-58{ \epsilon^{\prime}}^{2}\right)D^{5}}{120}\Bigg{]} \tag{1}\]  
\[y_{TM}=k_{0}\Big{\{}M-M_{0}+N\text{tan}\varphi\Big{[}D^{2}/2+\left(5-T+9C+4C^{ 2}\right)\Big{]}D^{4}/24\,+\Big{(}61-58T+T^{2}+600C-330{\epsilon^{\prime}}^{2 }\Big{)}D^{6}/720\Big{\}}, \tag{2}\]  
where: \(x_{TM},y_{TM}\)--Transverse Mercator planar coordinates, \(k_{0}\)--scaling parameter, \({\epsilon^{\prime}}^{2}=\frac{\epsilon^{2}}{(1-\epsilon^{2})}\) second eccentricity, \(N=a/\left(1-{\epsilon^{2}\text{sin}^{2}\varphi}\right)^{1/2}\)--prime vertical radius, \(T=\text{tan}^{2}\varphi\), \(C={\epsilon^{\prime}}^{2}\text{cos}^{2}\varphi\), \(D=(\lambda-\lambda_{0})\text{cos}\varphi\)--additional computation parameters, \(M=a[(1-{\epsilon^{2}}/{4}-3{\epsilon^{4}}/{64}-5{\epsilon^{6}}/{256})\)\(\varphi-(3{\epsilon^{2}}/8+3{\epsilon^{4}}/32+45{\epsilon^{6}}/{1024})\text{sin} \varphi-(15{\epsilon^{4}}/{256}+45{\epsilon^{6}}/{1024})\text{sin}\varphi-(3{ \epsilon^{6}}/{3072})\text{sin}\varphi]\)--true distance along the central meridian from the equator to the parallel \(\varphi\). The Transverse Mercator projection ensures a low level of projection distortions through the introduction of narrow meridian zones. In the PL-2000 coordinate system, four zones with a width of 3\({}^{\circ}\) are used. The longitude of axial meridians \(\lambda_{0}\) is 15\({}^{\circ}\) E, 18\({}^{\circ}\) E, 21\({}^{\circ}\) E, and 24\({}^{\circ}\) E. To distinguish the longitudinal coordinates in four meridian zones, the value of 500,000 m + (\(\lambda_{0}/3^{\circ}\)) * 1,000,000 m is added to the eastern coordinate [47]. The datum of the PL-2000 system is not WGS84 but GRS80 [48].