# Precise Point Positioning Using Triple GNSS Constellations in Various Modes
## 2 Un-Differenced Post-Processing PPP Models
\[P_{G_{IF}}=\rho_{G}+c[dt_{rG}-dt^{s}]+c[\alpha d_{P1}-\beta d_{P2}],+c[\alpha d_{ P1}-\beta d_{P2}]^{s}+T_{G}+\varepsilon_{PG_{IF}} \tag{1}\]  
\[P_{E_{IF}}=\rho_{E}+c[dt_{rG}-GGTO-dt^{s}]+c[\alpha d_{E1}-\beta d_{E5a}]_{r}+c[ \alpha d_{E1}-\beta d_{E5a2}]^{s}+T_{E}+\varepsilon_{E_{IF}} \tag{2}\]  
\[P_{B_{IF}}=\rho_{B}+c[dt_{rG}-GB-dt^{s}]+c[\alpha d_{B1}-\beta d_{B2}]_{r}+c[ \alpha d_{B1}-\beta d_{B2}]^{s}+T_{B}+\varepsilon_{B_{IF}} \tag{3}\]  
\[\Phi_{G_{IF}}=\rho_{G}+c[dt_{rG}-dt^{s}]+c[\alpha\delta_{L1}-\beta\delta_{L2}]_{ r}+c[\alpha\delta_{L1}-\beta\delta_{L2}]^{s}+T_{G}+N_{G_{IF}}+\phi_{\Phi_{G_{IF} }}+\phi_{\Phi_{G_{IF}}}^{s}+\varepsilon_{\Phi_{G_{IF}}} \tag{4}\]  
\[\Phi_{E_{IF}}=\rho_{E}+c[dt_{rG}-GGTO-dt^{s}]+c[\alpha\delta_{E1}-\beta\delta_{ E5a}]_{r}+c[\alpha\delta_{E1}-\beta\delta_{E5a}]^{s}+T_{E}+N_{E_{IF}}+\phi_{\Phi_{G_{ IF}}}+\phi_{\Phi_{G_{IF}}}+\varepsilon_{\Phi_{E_{IF}}} \tag{5}\]  
\[\Phi_{B_{IF}}=\rho_{B}+c[dt_{rG}-GB-dt^{s}]+c[\alpha\delta_{B1}-\beta\delta_{B2 }]_{r}+c[\alpha\delta_{B1}-\beta\delta_{B2}]^{s}+T_{B}+N_{B_{IF}}+\phi_{\Phi_{ B_{IF}}}+\phi_{\Phi_{B_{IF}}}^{s}+\varepsilon_{\Phi_{B_{IF}}} \tag{6}\]  
where the subscripts \(G\), \(E\), and \(B\) refer to the GPS, Galileo, and BeiDou satellite systems, respectively; \(P_{G_{IF}}\), \(P_{E_{IF}}\), and \(P_{B_{IF}}\) are the ionosphere-free pseudoranges in meters for GPS, Galileo, and BeiDou systems, respectively; \(\Phi_{G_{IF}}\), \(\Phi_{E_{IF}}\), and \(\Phi_{B_{IF}}\) are the ionosphere-free carrier phase measurements in meters for GPS, Galileo, and BeiDou systems, respectively; \(GGT\)O is the GPS to Galileo time offset; \(GB\) is the GPS to BeiDou time offset; \(\rho\) is the true geometric range from receiver at reception time to satellite at transmission time in meter; \(dt_{r}\), \(dt^{s}\) are the clock errors in seconds for the receiver at signal reception time and the satellite at signal transmission time, respectively; \(d_{P1r}\), \(d_{P2r}\), \(d_{E1r}\), \(d_{E5 ar}\), \(d_{B1r}\), \(d_{B2r}\) are frequency-dependent code hardware delays for the receiver at reception time in seconds; \(d_{P1}\), \(d_{P2}\)\({}^{S}\), \(d_{E1}\)\({}^{S}\), \(d_{E5a}\)\({}^{S}\), \(d_{B1}\)\({}^{S}\), \(d_{B2a}\)\({}^{S}\) are frequency-dependent code hardware delays for the satellite at transmission time in seconds; \(\delta_{L1r}\), \(\delta_{L2r}\), \(\delta_{E1r}\), \(\delta_{E5 ar}\), \(\delta_{B1r}\), \(\delta_{B2r}\) are frequency-dependent carrier-phase hardware delays for the receiver at reception time in seconds; \(\delta_{L1}\)\({}^{S}\), \(\delta_{L2}\)\({}^{S}\), \(\delta_{E1}\)\({}^{S}\), \(\delta_{E5a}\)\({}^{S}\), \(\delta_{B1}\)\({}^{S}\), \(\delta_{B2}\)\({}^{S}\) are frequency-dependent carrier-phase hardware delays for the satellite at transmission time in seconds; \(T\) is the tropospheric delay in meter; \(N_{G_{IF}}\), \(N_{E_{IF}}\), \(N_{B_{IF}}\) are the ionosphere-free linear combinations of the ambiguity parameters for both GPS, Galileo, and BeiDou carrier-phase measurements in meters, respectively; \(\phi_{\Phi_{G_{IF}}}\), \(\phi_{\Phi_{G_{IF}}}^{s}\), \(\phi_{\Phi_{G_{IF}}}\), \(\phi_{\Phi_{E_{IF}}}\), \(\phi_{\Phi_{B_{IF}}}^{s}\), \(\phi_{\Phi_{B_{IF}}}\) are ionosphere-free linear combinations of frequency-dependent initial fractional phase biases in the receiver and satellite channels for both GPS, Galileo, and BeiDou in meters, respectively; \(c\) is the speed of light in vacuum in meter per second; \(\varepsilon_{p_{IF}}\), \(\varepsilon_{E_{IF}}\), \(\varepsilon_{\Phi_{G_{IF}}}\), \(\varepsilon_{\Phi_{E_{IF}}}\), \(\varepsilon_{\Phi_{E_{IF}}}\), \(\varepsilon_{\Phi_{B_{IF}}}\), \(\varepsilon_{\Phi_{B_{IF}}}\) are the ionosphere-free linear combinations of the relevant noise and un-modeled errors in meter; \(\alpha_{G}\), \(\beta_{G}\), \(\alpha_{E}\), \(\beta_{E}\), \(\alpha_{B}\), \(\beta_{B}\) are the ionosphere-free linear combination coefficients for GPS, Galileo, and BeiDou which are given, respectively, by: \(\alpha_{G}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}\), \(\beta_{G}=\frac{f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}\), \(\alpha_{E}=\frac{f_{21}^{2}}{f_{21}^{2}-f_{25a}^{2}}\), \(\beta_{E}=\frac{f_{21}^{2}}{f_{21}^{2}-f_{25a}^{2}}\), \(\alpha_{B}=\frac{f_{21}^{2}}{f_{21}^{2}-f_{25a}^{2}}\), \(\beta_{B}=\frac{f_{22}^{2}}{f_{21}^{2}-f_{22}^{2}}\), where \(f_{1}  
includes the effect of the ionosphere-free linear combination of the satellite hardware delays of L1/L2 P(Y) code, while the Galileo counterpart includes the effect of the ionosphere-free linear combination of the satellite hardware delays of the Galileo E1/E5a pilot code.