# Weighting of Multi-GNSS Observations in Real-Time Precise Point Positioning
## 2 Data and Methods
### Development of Stochastic Model  
#### 2.3.1 Stochastic Model for GPS  
In majority of GNSS software implementations, pseudoranges are treated as 100 times noisier than carrier phases, and the a priori standard deviation of raw pseudoranges \(\sigma_{P}\) is set between 2.0 m to 3.0 m by default. However, [39] use \(\sigma_{P}\) for GPS equal to 0.3 m and set the standard deviation of GPS  
\begin{table}
\begin{tabular}{l l} \hline \hline Observables & ionosphere-free pseudorange and carrier-phase \\ \hline \multirow{2}{*}{Frequencies (RINEX 3.03 notation)} & GPS: L1/L2, GLONASS: G1/G2, Galileo: E1/E5a, BeiDou: \\  & B1/B2 \\ \hline Inter-system weighting & elevation \(e\) dependent weighting: sin(\(e\)) \\ \hline Elevation cut-off angle & 5\({}^{\circ}\) \\ \hline Sampling rate & 30 s \\ \hline \multirow{2}{*}{Troposphere delay modeling} & UNB3m mapping functions [38] and a priori value for \\  & hydrostatic delay, wet delay estimated as 4 mm/\(\sqrt{\text{hour}}\) \\  & random walk process \\ \hline Receiver clock & estimated as white noise, individual clock for each GNSS \\ \hline Satellite orbits and clocks & fixed from real-time CNES stream (mountpoint CLK93) \\ \hline Code and phase biases & absolute, from real-time CNES stream (mountpoint CLK93) \\ \hline Solution type & daily static with float ambiguities \\ \hline \multirow{2}{*}{Correction models} & phase wind-up, relativistic delays, solid earth tides, receiver \\  & antenna phase center offset and variation [37] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of the Precise Point Positioning (PPP) processing strategy.  
raw carrier phase \(\sigma_{L}\) to 3 mm. During the development stage of the WARP software, we have adopted \(\sigma_{P}\) = 0.3 m, which corresponds to 0.9 m for pseudorange ionosphere-free observable. We have also calibrated the least square adjustment of both pseudoranges and carrier phases in real-time processing with an empirical approach to obtain the a posteriori error of unit weight \(m0\) close to 1. We have resulted with \(\sigma_{L}\) equal to 10 mm.  
In order to check how this stochastic model for GPS observation suits to GPS data from our test stations, we analyze the \(m0\) values from different solutions. First, we perform code-based GPS only positioning using \(\sigma_{P}\) from 0.1 m to 0.5 m (Figure 2a). We noticed that results are station dependent, without any clear relation with station hardware configuration. We suspect that this is rather caused by station environment and software configuration e.g., pseudorange tracking mode or pseudorange smoothing algorithm with carrier phases. All of the stations except UCAL, which is equipped with the relatively old antenna, result in \(m0\) smaller than 1 for \(\sigma_{P}\) = 0.3 m. This indicates that \(\sigma_{P}\) can be set even smaller in code-based only solutions if state-of-the art, modern geodetic hardware is used. We found that \(\sigma_{P}\) = 0.235 m results in the station-averaged \(m0\) equal to 1.  
Secondly, for each \(\sigma_{P}\) we perform GPS-only PPP using \(\sigma_{L}\) from 2 to 50 mm (Figure 2b). We noticed that for small \(\sigma_{P}\) it is not possible to obtain \(m0\) = 1. In general and as expected, we found that the larger the \(\sigma_{P}\) is, the smaller \(\sigma_{L}\) should be in order to achieve \(m0\) = 1, and for unrealistically small \(\sigma_{P}\) it is not possible to obtain \(m0\) = 1 at all. We also found that for \(\sigma_{P}\) = 0.3 m and \(\sigma_{L}\) = 10 mm, the average \(m0\) is very close to 1, which justifies the use of this GPS stochastic model in further processing. For \(\sigma_{P}\) = 0.235 m, in order to achieve \(m0\) close to 1 in PPP, \(\sigma_{L}\) should be set to 50 mm.