# Study of Fast and Reliable Time Transfer Methods Using Low Earth Orbit Enhancement
## 2 Experimental Principles
### LEO-Augmented PPP Model  
In this study, we utilized both GNSS observation data and simulated LEO satellite observation data. The GNSS observation data encompassed pseudorange observations as well as carrier-phase observations. The PPP observation equation is constructed as follows:  
\[P_{f,R}^{G,S}=\rho_{R}^{G,S}+c\cdot\delta T_{R}-c\cdot\delta T^{G,S}+\tau_{ion,f,R }^{G,S}+\tau_{trop}+\tau_{delay_{-}P,f,R}^{G}-\tau_{delay_{-}P,f}^{G,S}+\varepsilon _{P,f,R}^{G,S} \tag{1}\]  
\[\Phi_{f,R}^{G,S}=\rho_{R}^{G,S}+c\cdot\delta T_{R}-c\cdot\delta T^{G,S}+\lambda_ {f}^{G,S}(N_{f,R}^{G,S}+\tau_{delay_{-}\Phi,f,R}^{G,S}-\tau_{delay_{-}\Phi,f}^{G, S})-\tau_{ion,f,R}^{G,S}+\tau_{trop}+\epsilon_{\Phi,f,R}^{G,S} \tag{2}\]  
where the superscript \(G\) indicates the GNSS system type; the superscript \(S\) indicates a specific satellite; the subscript f indicates the carrier frequency of the satellite signal; the subscript R indicates the receiver; \(P_{f,R}^{G,S}\) is the pseudorange observation; \(\rho_{R}^{G,S}\) is the geometric distance between the station and the satellite; \(c\) is the speed of light under vacuum conditions; \(\delta T_{R}\) is the receiver clock offset, which refers to the time deviation between the system time and the receiver clock; \(\delta T^{G,S}\) is the satellite clock offset; \(\tau_{ion,f,R}^{G,S}\) is the ionospheric delay at the f carrier frequency; \(\tau_{trop}\) is the tropospheric delay; \(\tau_{delay_{-}P,f,R}^{G}\) is the hardware delay related to the receiver code, \(\tau_{delay_{-}P,f}^{G,S}\) is the code hardware delay on the satellite side, with the code calibration delay in meters; \(\varepsilon_{P,f,R}^{G,S}\) is the pseudo-distance observation noise; \(\Phi_{f,R}^{G,S}\) is the carrier-phase observation; \(\lambda_{f}^{G,S}\) is the wavelength of the satellite carrier signal; \(N_{f,R}^{G,S}\) is integer ambiguity; \(\tau_{delay_{-}\Phi,f,R}^{G}\) is the hardware delay for the receiver phase, \(\tau_{delay_{-}\Phi,f}^{G,S}\) is the hardware delay for the phase on the satellite end, with phase calibration delays in cycles; and \(\varepsilon_{\Phi,f,R}^{G,S}\) is the carrier-phase observation noise. In this study, to eliminate the first-order ionospheric delay of the pseudorange and carrier-phase observables, the ionosphere-free (_IF_) PPP model was employed as follows:  
\[P_{IF,R}^{G,S}=\frac{{{{({f_{1}^{G,S}})}^{2}}{P_{1,R}^{G,S}}-{{({f_{2}^{G,S}})} ^{2}}{P_{2,R}^{G,S}}}}{{{{({f_{1}^{G,S}})}^{2}}-{{({f_{2}^{G,S}})}^{2}}}} \tag{3}\]  
\[\Phi_{IF,R}^{G,S}=\frac{{{{({f_{1}^{G,S}})}^{2}}{\Phi_{1,R}^{G,S}}-{{({f_{2}^{G,S}})}^{2}}{\Phi_{2,R}^{G,S}}}}{{{{({f_{1}^{G,S}})}^{2}}-{{({f_{2}^{G,S}})}^{2}} }} \tag{4}\]  
where \(f_{1}^{G,S}\) and \(f_{2}^{G,S}\) are the frequencies of different carrier signals. Therefore, the _IF_-PPP model observation equation is as follows:  
\[P_{IF,R}^{G,S}=\rho_{R}^{G,S}+c\cdot\delta T_{R}-c\cdot\delta T^{G,S}+\tau_{ trop}+\tau_{delay_{-}P,IF,R}^{G}-\tau_{delay_{-}P,IF}^{G,S}+\varepsilon_{P,IF,R}^{G,S} \tag{5}\]  
\[\Phi_{IF,R}^{G,S}=\rho_{R}^{G,S}+c\cdot\delta T_{R}-c\cdot\delta T^{G,S}+\lambda _{IF}^{G,S}(N_{IF,R}^{G,S}+\tau_{delay_{-}\Phi,IF,R}^{G,S}-\tau_{delay_{-}\Phi, IF}^{G,S})+\tau_{trop}+\varepsilon_{\Phi,IF,R}^{G,S} \tag{6}\]  
The ionosphere-free combination code pseudorange hardware delay at the satellite end and the receiver end is absorbed by the ambiguity parameters. So, Formulas (5) and (6) are rewritten as Formulas (7) and (8), respectively. \[P_{IF,R}^{G,S}=\rho_{R}^{G,S}+c\cdot\overline{\delta}T_{R}^{G}+\tau_{trop}+ \varepsilon_{P,IF,R}^{G,S} \tag{7}\]  
\[\Phi_{IF,R}^{G,S}=\rho_{R}^{G,S}+c\cdot\overline{\delta}T_{R}^{G}+\lambda_{IF }^{G,S}\overline{N}_{IF,R}^{G,S}+\tau_{trop}+\varepsilon_{\Phi,IF,R}^{G,S} \tag{8}\]The reparameterized receiver clock error and ambiguity are shown in Formula (9):  
\[\left\{