## 2 Materials and Methods
### Validation of the ASG-EUPOS's APPS and the Proposed Modification  
The POZGEO was validated by analysing coordinates from the post-processing of GPS observations from approx. 30 days in three areas of different ASG-EUPOS network densities (Figure 3). The study involved 1 h static GPS observations with 1 s intervals from 318 to 349 DOY 2020 (13 November to 14 December 2020), totalling 767 h of observations. We selected research points with various baseline vector lengths to the six nearest ASG-EUPOS stations. The data were recorded at points in a commercial network of reference stations, TPI NETpro: BROD (B 53\({}^{\circ}\)15\({}^{\prime}\)28.44299\({}^{\prime\prime}\), L 19\({}^{\circ}\)24\({}^{\prime}\)25.82764\({}^{\prime\prime}\) E), OPOC (B 51\({}^{\circ}\)22\({}^{\prime}\)39.36412\({}^{\prime\prime}\) N, L 20\({}^{\circ}\)16\({}^{\prime}\)50.73161\({}^{\prime\prime}\) E), and JASL (B 49\({}^{\circ}\)44\({}^{\prime}\)49.52674\({}^{\prime\prime}\) N, L 21\({}^{\circ}\)28\({}^{\prime}\)16.92604\({}^{\prime\prime}\) E). More information about the TPI NETpro can be found in the paper of [PERSON] and [PERSON] [35]. The data are collected with TPSCR-G5 antennas and Topcon NET-G5 receivers. The APPS employs a general model for calibrating the antenna phase centre developed by the equipment manufacturer, and, in the case of the ASG-EUPOS service, it is possible to enter the type and model of the antenna in the observation file sent to the service. The investigated points are points in the horizontal detailed national control network. Their positions are determined precisely as required by Polish regulations [36]. The OPOC station is situated in a low-density area, while JASL is in a zone of higher network density (Figure 3). In the case of the approach currently employed in the APPS of the ASG-EUPOS, the length of baseline vectors is, for BROD, from 38.2 to 70.5 km (\(\overline{d}=\)52.6 km, \(\sigma_{d}=\)10.9 km), for OPOC, from 41.1 to 89.9 km (\(\overline{d}=\)61.6 km, \(\sigma_{d}=\)18.3 km), and, for JASL, from 23.2 to 60.6 km (\(\overline{d}=\)48.0 km, \(\sigma_{d}=\)13.3 km). If the five nearest stations are used, the mean baseline vector length is reduced, as is the deviation (for JASL, \(\sigma_{d}=\)13.2 km, for OPOC, \(\sigma_{d}=\)13.3 km, and, for BROD, \(\sigma_{d}=\)7.2 km). The sets of three-dimensional coordinates (X, Y, Z) of the research points determined with the APPS and the proposed modified method (UAM) were converted to a local N, E, U system and analysed. Basic accuracy measures, such as the 99% Spherical Accuracy Standard (SAS99), 90% Spherical Accuracy Standard (SAS90), Mean Radial Spherical Error (MRSE), and Twice the Distance Root Mean Squared (2 DRMS), in relation to the reference positions of the points were determined to assess positioning accuracy (Equations (13)-(16)). \[MRSE=\sqrt{\sigma_{N}^{2}+\sigma_{E}^{2}+\sigma_{U}^{2}} \tag{13}\]  
\[SAS90=0.833\left(\sigma_{N}+\sigma_{E}+\sigma_{U}\right) \tag{14}\]  
\[SAS99=1.122\left(\sigma_{N}+\sigma_{E}+\sigma_{U}\right) \tag{15}\]  
Figure 3: Location of research points in the ASG-EUPOS network, visualisation of network density using a kernel density estimator and a 75 km radius (the mean distance required for national EUPOS GNSS networks), and configuration of research points in relation to the six nearest reference stations in the network. \[2 DRMS=2\sqrt{\sigma_{N}^{2}+\sigma_{E}^{2}} \tag{16}\]  
where:  
\(\sigma_{N}\), \(\sigma_{E}\), \(\sigma_{U}\)--standard deviation of coordinate differences with respect to the reference position in the directions of the coordinate axis, calculated from Equation (17). \[\sigma=\sqrt{\frac{\sum(x-\overline{x})^{2}}{(n-1)}} \tag{17}\]  
where:  
\(n\)--the sample size;  
\(\overline{x}\)--the sample mean. In order to characterise the distance between determined points and reference positions, the distance coefficient of variation (\(CV_{d}\)) was determined in accordance with the Equation (18).