# PPP and PPP-AR Kinematic Post-Processed Performance of GPS-Only, Galileo-Only and Multi-GNSS
## 2 Materials and Methods
\begin{split} MW^{s}_{r}&=\ \lambda_{\textit{avl}}\rho^{s}_{r}=\left( \frac{f_{i}}{f_{i}-f_{j}}L^{s}_{r,i}-\frac{f_{j}}{f_{i}-f_{j}}L^{s}_{r,j} \right)-\left(\frac{f_{i}}{f_{i}+f_{j}}P^{s}_{r,i}+\frac{f_{j}}{f_{i}+f_{j}}P^ {s}_{r,j}\right)\\ &=\ \lambda_{\textit{avl}}(N^{s}_{r,i}-N^{s}_{r,j}-\mu^{s}+\mu_{r}(t)) =\ \lambda_{\textit{avl}}\left(N^{s}_{\textit{avl},r}-\mu^{s}+\mu_{r}(t)\right) \end{split}  \tag{5}\]  
where:  
* \(MW^{s}_{r}\) is the Melbourne-Wubbena linear combination at receiver \(r\) from satellite \(s\) (m)
* \(\lambda_{\textit{avl}}\) (\(\lambda_{\textit{avl}}=c/\left(f_{i}-f_{j}\right)=\ \lambda_{i}\lambda_{j}/\left(\lambda_{j}-\lambda_{i}\right)\)) is the wide-lane (WL) wavelength (m)
* \(N^{s}_{\textit{avl},r}\) (\(N^{s}_{\textit{avl},r}=N^{s}_{r,i}-N^{s}_{r,j}\)) is the WL ambiguity at receiver \(r\) from satellite \(s\)
* \(\mu^{s}\) is the delay coming from the satellite (also known in the bibliography as WL satellite bias (WSB))
* \(\mu_{r}(t)\) is the delay coming from the receiver (also known in the bibliography as WL receiver bias (WRB))  
It has been observed that for the GPS system the \(\mu^{s}\) are stable over long periods of time and can be considered as constant during at least one day [16]. For the Galileo system they are stable for longer periods; up to months [10]. The \(\mu_{r}(t)\) delay is considered to vary over time because it depends on the behavior of each receiver. As it is seen from the Equation (5), the terms \(N^{s}_{\textit{avl},r}\), \(\mu^{s}\) and \(\mu_{r}(t)\) are totally correlated. Normally, the \(\mu^{s}\) is known; i.e., it is calculated and provided from the CNES/CLS AC to the users [6; 8]. The \(\mu_{r}\) (one per epoch of measurements) and the \(N^{s}_{\textit{avl},r}\) (one per satellite pass) can be separated and solvedusing all available equations (Equation (5) corrected by the \(\mu^{s}\)) from all satellites in view, using a Least Squares Estimation (LSE) processing associated with a bootstrap method [17; 18]. The result after this step are the \(N_{wl,r}^{s}\) and \(\mu_{r}(t)\). Once the \(N_{wl,r}^{s}\) is determined, the following step is to form an ionosphere-free linear combination that has the property to cancel out the first order of ionospheric effects. These combinations use the equations in the two frequencies where they apply to them a coefficient: \(\alpha_{i}\) to the frequency \(i\) and \(\alpha_{j}\) to the frequency \(j\) respectively [19]. \[\alpha_{i}=\ \frac{f_{i}^{2}}{f_{i}^{2}-f_{j}^{2}} \tag{6}\]  
\[\alpha_{j}=\ \frac{-f_{i}^{2}}{f_{i}^{2}-f_{j}^{2}} \tag{7}\]  
Using the above coefficients and Equations (6) and (7), the ionosphere-free linear combinations for code and carrier phase become:  
\[P_{r,\text{IF}}^{s}=\ \alpha_{i}P_{r,i}^{s}+\alpha_{j}P_{r,j}^{s}=\rho_{r}^{s}+c \Delta t+T_{r}^{s}+E_{r,\text{IF}}^{s} \tag{8}\]  
\[ \begin{split} L_{r,\text{IF}}^{s}=\ \alpha_{i}L_{r,i}^{s}+ \alpha_{j}L_{r,j}^{s}\\ =\rho_{r}^{s}+c\Delta t+T_{r}^{s}+\frac{c}{f_{i}^{2}-f_{j}}\Big{(} f_{i}N_{r,i}^{s}-f_{j}N_{r,j}^{s}\Big{)}+\frac{c}{f_{i}^{2}-f_{j}^{2}}\Big{(}f_{i}N_{r }^{s}-f_{j}N_{r}^{s}\Big{)}+\varepsilon_{r,\text{IF}}^{s}\\ =\rho_{r}^{s}+c\Delta t+T_{r}^{s}+\lambda_{nl}N_{r,i}^{s}+\frac{ \lambda_{nl}\lambda_{nl}}{\lambda_{j}}N_{wl,r}^{s}+\lambda_{nl}W_{r}^{s}+ \varepsilon_{r,\text{IF}}^{s}\end{split}