## 2 Materials and Methods
### Data Processing and Comparison Method
#### 2.3.1 Trajectories Computations from GS14 Data  
The aim of the trajectories computations was to produce differential solutions for the whole set of GS14 acquired data and to perform the subsequent comparisons. Each computation was executed by using the RNX2 RTKP CUI tool of RTKLIB [41] and permanent station data (used to produce a post-processed kinematic solution). For this purpose, a separate RNX2 RTKP configuration file was created for each calculated trajectory (see Table 1) by means of a Python function created ad hoc. To perform a fair comparison among the computed trajectories of the single constellations, a reduced operativeness was simulated considering the maximum available number of Galileo satellites (six) for the performed kinematic survey. All the possible results obtained for all the combinations of four, five, and six satellites of the individual GPS and GLONASS constellations were respectively compared with all the possible results obtained for all the combinations of four and five satellites (and with the unique combination of six satellites) of the individual Galileo constellation. The number of potentially produced trajectories can be calculated as the number of satellite combinations with k dimension, without repetitions, taken from the whole set of available satellites. This number is given by the binomial coefficient (Equation (1)):  
\[\left( \begin{array}{c}\mathrm{n}\\ \mathrm{k}\end{array} \right)=\frac{\mathrm{n!}}{\mathrm{k!(n-k)!}} \tag{1}\]  
where n is the number of available satellites (11, 9, and 6, respectively, for GPS, GLONASS, and Galileo constellations), and k is the number of the considered satellites for each produced combination (four, five, and six). RNX2 RTKP has a wide range of parameters (almost 100) that can be modified within the configuration file to achieve different solutions. In our case, for each tested satellite combination and for each individual constellation, we opted for the following RNX2 RTKP main configuration parameters: (1)the kinematic solution for the whole survey (to create a trajectory of points); L1 frequency (single-frequency analysis for the E1, L1, and G1 signals); broadcasted ionospheric model and Saastamoinen tropospheric model; forward Kalman filtering. For each combination, three different cut-off angles (10\({}^{\circ}\), 15\({}^{\circ}\), and 20\({}^{\circ}\)) and three different integer ambiguity resolution strategies were tested (Continuous, Instantaneous, and Fix and Hold). Moreover, the configuration file allows the user to select one or more constellation and to exclude user-defined satellites for the computations. This feature \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**n. of Solutions** & **Galileo** & **GPS** & **GLONASS** & **Total** \\ \hline
4 satellites & 15 \(\times\) 9 & 330 \(\times\) 9 & 126 \(\times\) 9 & 4239 \\
5 satellites & 6 \(\times\) 9 & 462 \(\times\) 9 & 126 \(\times\) 9 & 5346 \\
6 satellites & 1 \(\times\) 9 & 462 \(\times\) 9 & 84 \(\times\) 9 & 4923 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of computed solutions for the simulated reduced operativeness of GPS, GLONASS, and Galileo: \(\binom{n}{k}\times\) 9, with \(\binom{n}{k}\) as the binomial coefficient for the constellation with n total number of satellites and k considered satellites. In “Total” column, the number of total computed solutions. was exploited to simulate the reduced operativeness. All the other configuration parameters were not changed (further details on the algorithms implemented in RTKLIB can be found in [41]). In this way, for each selected satellite combination, nine different solutions were computed. As shown in Section 2.3.3 (Data filtering and comparisons), among these nine solutions for each combination, potentially only one (the best, according to Equation (6) and the previous applied filters) would represent the selected combination. Table 1 reports the number of solutions (and thus the trajectories potentially computed) for the combinations of four, five, and six satellites for each constellation.