# PPP and PPP-AR Kinematic Post-Processed Performance of GPS-Only, Galileo-Only and Multi-GNSS
## 2 Materials and Methods  
The PPP with ambiguity resolution processing has the advantage that the processing can be done directly at the user level without any reference station around. The requirement, however, is that precise and consistent satellite products (derived from a global network of stations) must be available. In this study products are taken from the CNES-CLS AC: satellite orbit file (.sp3), clock file (.clk) and the Wide-Lane satellite biases (.wsb) (given either in the header of the clock files or taken from the CNES/CLS portal. Files Wide_lane_GAL_satellite_bais.wsb and Wide_lane_GPS_satellite_bais.wsb are available at [[ftp://ftpsedr.cls.fr/pub/igsac/](ftp://ftpsedr.cls.fr/pub/igsac/)]([ftp://ftpsedr.cls.fr/pub/igsac/](ftp://ftpsedr.cls.fr/pub/igsac/))). The used zero-difference ambiguity fixing method [8,13] equation model bellow is given for the pseudorange (code) and carrier phase measurements for two frequencies:  
\[P_{r,i}^{s}=\ \rho_{r}^{s}+c\Delta t+T_{r}^{s}+I_{r}^{s}+b_{i}^{s}+b_{r,i}+E_{r,i}^ {s} \tag{1}\]  
\[P_{r,j}^{s}=\ \rho_{r}^{s}+c\Delta t+T_{r}^{s}+\frac{f_{i}^{2}}{f_{j}^{2}}I_{r}^{s }+b_{j}^{s}+b_{r,j}+E_{r,j}^{s} \tag{2}\]  
\[L_{r,i}^{s}=\lambda_{i}\rho_{r,i}^{s}=\ \rho_{r}^{s}+c\Delta t+T_{r}^{s}-I_{r}^{s}+ \lambda_{i}N_{r,i}^{s}+\lambda_{i}W_{r}^{s}+\beta_{i}^{s}+\beta_{r,i}+\varepsilon _{r,i}^{s} \tag{3}\]  
\[L_{r,j}^{s}=\lambda_{j}\rho_{r,j}^{s}=\ \rho_{r}^{s}+c\Delta t+T_{r}^{s}-\frac{f_{i}^{2} }{f_{j}^{2}}I_{r}^{s}+\lambda_{j}N_{r,j}^{s}+\lambda_{j}W_{r}^{s}+\beta_{j}^{s} +\beta_{r,j}+\varepsilon_{r,j}^{s} \tag{4}\]where:  
* \(P^{s}_{r,i^{\prime}}\), \(P^{s}_{r,j}\) are the code measurement at receiver \(r\) from satellite \(s\) on frequency \(i\) or \(j\) (m)
* \(L^{s}_{r,i^{\prime}}\), \(L^{s}_{r,j}\) are the phase measurement at receiver \(r\) from satellite \(s\) on frequency \(i\) or \(j\) (m)
* \(\rho^{s}_{r}\) is the geometric distance between receiver and satellite (m)
* \(\Delta t\) (\(\Delta t=\delta t_{r}-\delta t^{s}\)) is the clock correction related to the satellite (\(\delta t^{s}\)) and the receiver (\(\delta t_{r}\)) with respect to the synchronization to the GPS time (s)
* \(T^{s}_{r}\) is the troposphere delay (m)
* \(I^{s}_{r}\) is the ionosphere delay (m)
* \(E^{s}_{r,i^{\prime}}\), \(E^{s}_{r,j}\) are the code measurement errors at receiver \(r\) from satellite \(s\) on frequency \(i\) or \(j\) (m) including all sources of code errors: multipath and noise. * \(f_{i^{\prime}}\), \(f_{j}\) are the carrier frequency \(i\) or \(j\) (Hz)
* \(c\) is the speed of light in vacuum (m/s)
* \(\lambda_{i}\), \(\lambda_{j}\) are the nominal wavelength of the carrier frequency \(i\) or \(j\) (m)
* \(\rho^{s}_{r,i^{\prime}}\), \(\rho^{s}_{r,j}\) are the carrier phase measurement at receiver \(r\) from satellite \(s\) on frequency \(i\) or \(j\) (cycles)
* \(N^{s}_{r,i^{\prime}}\), \(N^{s}_{r,j}\) are the integer carrier phase ambiguity at receiver \(r\) from satellite \(s\) on frequency \(i\) or \(j\)
* \(W^{s}_{r}\) is the carrier phase wind up effect (cycles)
* \(b^{s}\), \(b_{r}\) are the code phase biases of satellite and receiver (m)
* \(\rho^{s}\), \(\beta_{r}\) are the carrier phase biases of satellite and receiver (m)
* \(\varepsilon^{s}_{r,i^{\prime}}\), \(\varepsilon^{s}_{r,j}\) are the carrier phase measurement error at receiver \(r\) from satellite \(s\) on frequency \(i\) or \(j\) (m) including all sources of phase errors, remaining uncorrected phase center offset and phase center variation, multipath and noise. In this study the frequencies used for the GPS system are in the band of \(L1\) and \(L2\) (\(f_{L1}=154\times f_{0}\) and \(f_{L2}=120\times f_{0}\)). For the Galileo system are in the band of \(E1\) and \(E5a\) (\(f_{E1}=154\times f_{0}\) and \(f_{E5a}=115\times f_{0}\)), where \(f_{0}=10.23\) MHz. From all these four equations it is possible to form a Melbourne-Wubbena linear combination that has the identity to reduce measurement noise and to cancel out any geometric, ionospheric and clock terms [14; 15]:  
\[