## 4 Precise Point Positioning Results  
### CNES Evaluation  
The first evaluation to be done is to demonstrate the benefit of ambiguity fixing. This analysis has been performed on the BRST station located in the north of France with a measurement sampling rate of 30 s for 45 min. Figure 12(a) shows the horizontal positioning errors in float mode (without the biases), in full GNSS mode. Here, the main driver of the error is the noise of the code. The low code noise of the Galileo measurements allows a positioning at a 20 cm level. Using the phase biases with instantaneous widelane ambiguity resolution leads to a horizontal static positioning accuracy of 10 cm, as depicted in Figure 12(b). When the last ambiguity (\(N1\)) is solved, it is possible to obtain sub-centimeter accuracy instantaneously (Figure 12(c)). Note that this last resolution requires very accurate phase biases and clean measurements (open sky stations). Table 5 summarizes all these results for the positioning errors for the three cases in the East and North directions. Figure 13: _Cont._  
In order to assess the better precision of the POST products with respect to the RT ones, a PPP solution is computed with three PPP modes: the PPP-float mode, the PPP-AR mode, and the new PPP-Snapshot mode presented in Section 2.3. The receiver position and velocity are estimated in the Kalman filter along with the receiver clock offsets for each constellation, the receiver code and phase biases for all the frequencies, the carrier phase combinations ambiguities for all the satellites, the ionosphere elongation, and the troposphere delay. A simple linear dynamical model is used in the propagation phase of the Kalman filter to link the receiver position and velocity. Its position and velocity at time \(t_{k}\) are denoted \(p_{k}\) and \(v_{k}\), respectively. This dynamical model reads:  
\[ \begin{split} p_{k+1}&=p_{k}+v_{k}\Delta t+\omega_{ p,k},\\ v_{k+1}&=v_{k}+\omega_{v,k},\end{split}  \tag{36}\]  
where \(\Delta t=t_{k+1}-t_{k}\) is the time interval between two steps of the Kalman filter and \(\omega_{p,k}\) and \(\omega_{v,k}\) are the centered white process noises for the position and velocity, respectively, whose standard deviations \(\sigma_{\omega_{p}}\) and \(\sigma_{\omega_{v}}\) are to be chosen as parameterization of the filter, as well as the a priori standard deviation \(\sigma_{p_{0}}\) and \(\sigma_{v_{0}}\). Two cases are considered for this study:  
1. in the first case, called static (STA), the velocity process noise standard deviation \(\sigma_{\omega_{v}}\) and the velocity a priori standard deviation \(\sigma_{v_{0}}\) are set to zero, in such a way that the velocity is not estimated by the filter;
2. in the second case, called kinematic (KIN), the velocity a priori and process noise standard deviations are not zero, and the position process noise standard deviation is set to zero. For the other estimated parameters, the transition matrix is the identity matrix. Table 6 describes the a priori and process noise standard deviations that have been chosen. The computations have been done for the TLSG station for the day 2023-073 (14 March) with  
Figure 13: Difference in the positioning of the static station BRST with no ambiguity resolution, the widelane ambiguity resolution only, and resolution of all the ambiguities. (**a**) Standard PVT. (**b**) PVT with instantaneous widelane ambiguity resolution. (**c**) PVT with instantaneous ambiguity resolution, all ambiguities. \begin{table}
\begin{tabular}{l c c c} \hline \hline  & **PVT** & **WL Only** & **All Ambiguities** \\ \hline RMS East error [cm] & 21.10 & 10.19 & 0.92 \\ RMS North error [cm] & 24.59 & 13.77 & 0.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results accuracy for three positioning modes for the BRST station, with measures processed at a time interval of 30 s.  
a data sampling rate of 1 Hz.