## 3 Instantaneous Multi-Constellation Relative Positioning: Methodology and Performance
### Performance Assessment of Precise Positioning with Focus on E14 and E18 Galileo Satellites  
#### 3.2.1 Data Selection & Experiment Design  
As stated previously, the hypothesis of the applicability of E14 and E18 satellites to positioning was verified on the basis of a multi-baseline, single-epoch RTK solution. For this purpose, the mathematical model given in Section 3.1 was applied to dual-frequency observations (L1/L2 & E1/E5a). The distribution of stations and test baselines used in the experiment is presented in Figure 1. We used one station as a simulated rover receiver (OLSZ), as well as three surrounding permanent stations serving as reference sites (IAWA, CHLE, BYCE), being a part of the VRSNet active reference network [60]. As a consequence, the baselines were of similar length and reached ~60 km. The observational data covered three days, 3-5 October 2017, and were sampled at 30 s intervals. All sites were occupied by Trimble NET9 receivers. BYCE, IAWA and CHLE sites were equipped with TRM115000.00 antennas, while at OLSZ station a choke ring model TRM59900.00 was used. Due to the lack of antenna phase centre offset and variation models for the Galileo E5a, E5b, E5 and E6 signals, the correction values corresponding to GPS L2 signals were applied. We believe that this may have a small impact on the positioning results due to the use of Galileo E5a observables. Nevertheless, at this time there is no better solution to this problem. Several processing scenarios were established for the experimental verification of the hypothesis (see Table 4). The first and second scenarios apply a single system: GPS or Galileo. The third scenario takes advantage of a combined model including Galileo satellites on incorrect orbits (E14 & E18). The last strategy is based on a combined solution; however, it excludes these two satellites. By analysing the results of the last two scenarios, we evaluated the impact of Galileo E14 & E18 on the performance of the precise positioning. \begin{table}
\begin{tabular}{c c} \hline \hline \# & **Constellations** \\ \hline
1. & GPS \\
2. & Galileo \\
3. & Combined GPS + Galileo with E14 and E18 \\
4. & Combined GPS + Galileo without E14 and E18 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Processing scenarios. \begin{table}
\begin{tabular}{c c} \hline \hline
**Date and Time of Session** & **3–5 October 2017 00:00–24:00 UTC** \\ \hline \hline Baseline’s lengths & 61–62 km \\ \hline GNSS constellations & GPS \& Galileo \\ \hline Observation types & dual frequency phase and pseudorange \\ \hline Frequencies & L1 \& L2 (GPS), E1 \& E5a (Galileo) \\ \hline Session length \& solution & Instantaneous (single–epoch) relative solution \\ \hline \# of solutions per day & 288 independent single-epoch solution per day (@300 s) \\ \hline \multirow{2}{*}{Topospheric delay mitigation} & a priori values from Modified Hopfield + GMF mapping function + tightly constrained estimation of the residual ZTD \\ \hline \multirow{2}{*}{Ionospheric delay mitigation} & constrained estimation of DD first-order ionospheric delays \\  & (ionosphere float model) \\ \hline Elevation cut-off angle & 10\({}^{\circ}\) \\ \hline Ephemeris & Precise final from CODE (COM) [57] \\ \hline Observation weighting scheme & Elevation-dependent \\ \hline \multirow{2}{*}{Ambiguities handling} & Fixed to integers with M-Lambda [58] and validated with \(\Psi\)-ratio [59] \\ \hline \multirow{2}{*}{A priori standard deviation of undifferenced observations} & 0.2 m and 0.002 m for code and phase (E1/L1), respectively \\  & \(\sigma_{12}=\sigma_{11}\)\(\lambda_{12}/\lambda_{11}\)\(\sigma_{58}=\sigma_{14}\)\(\lambda_{58}/\lambda_{11}\) \\ \hline \multirow{2}{*}{Estimation method} & least squares adjustment with a priori parameter constraining \\  & (single-epoch) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Processing strategy and session summary. The performance assessment was based on several indicators of solution quality related to the ambiguity resolution--coordinate domains as well as observable residuals.