## 2 SFSE Double-Differenced GPS/Galileo Mathematical Model
### Single GPS/Galileo Code and Phase Observation Equations  
GPS satellites broadcast their position and clock information in GPS Time (GPST), while Galileo satellites work in an independent time system, that is, Galileo System Time (GST). Therefore, it is a prerequisite to unify the time system before the GPS/Galileo dual constellations are used in positioning, navigation, and timing (PNT) service. Assume that both GPS and Galileo code and phase observations are formed in the same time system-GPST; hence, a GPS-to-Galileo Time Offset (GGTO) [30] needs to be adopted to convert the transmission epoch of the Galileo satellite to GPST. Subsequently, the SFSE non-differenced code and phase equations for the receiver b and GPS satellite g at frequency j can be expressed as [24]. \[ \begin{split}& P_{b,j}^{S}=\rho_{b}^{S}-c\cdot\delta t_{b}+c\cdot \delta t^{S}+d_{b,j}^{G}-d_{j}^{S}+I_{b,j}^{S}+T_{b,j}^{S}+\varepsilon_{b,j}^{ S}\\ &\lambda_{j}\cdot\phi_{b,j}^{S}=\rho_{b}^{S}-c\cdot\delta t_{b}+c \cdot\delta t^{S}+\lambda_{j}\left(\delta_{b,j}^{G}-\delta_{j}^{S}+N_{b,j}^{S} \right)-I_{b,j}^{S}+T_{b,j}^{S}+\in\in_{b,j}^{S}\end{split}  \tag{1}\]  
where \(P_{b,j}^{S}\) and \(\phi_{b,j}^{S}\) are the GPS code observation (m) and phase observation (cycle), respectively; \(\rho_{b}^{S}\) denotes the real range (m) between the receiver and the GPS satellite; \(\lambda_{j}\) is the wavelength (m) of the frequency j; \(\delta t_{b}\) and \(\delta t^{S}\) are the receiver clock error (s) and GPS satellite clock error (s), respectively; \(c\) is the velocity of light (m/s); \(d_{b,j}^{C}\) and \(\delta_{b,j}^{C}\) are the hardware GPS code delay (m) and phase delay (cycle) in the receiver, respectively, and the superscript \"G\" represents \"GPS\"; \(d_{,j}^{S}\) and \(\delta_{,j}^{S}\) are the hardware code delay (m) and phase delay (cycle) in the GPS satellite; \(N_{b,j}^{S}\) denotes the unknown integer ambiguities (cycle); \(I_{b,j}^{S}\) and \(T_{b,j}^{S}\) denote the ionospheric delay (m) and tropospheric delay (m), respectively; and, \(\varepsilon_{b,j}^{S}\) and \(\in_{b,j}^{S}\) are the other unmodeled errors (m) of the code and phase equations, including multipath error, random noise, etc. Because the frequency bands of the Galileo system overlap with those of GPS in L1/E1 and L5/E5a, the frequency of Galileo is still denoted by the subscript \"j\". To distinguish from the GPS satellite, the Galileo satellite is expressed by the superscript \"e\". Subsequently, the SFSE non-differenced code and phase equations for Galileo satellites at frequency j can read as:  
\[ \begin{split}& P_{b,j}^{e}=\rho_{b}^{e}-c\cdot\delta t_{b}+c\cdot \delta t^{e}+d_{b,j}^{E}-d_{j}^{e}+I_{b,j}^{e}+T_{b,j}^{e}-GGTO+\varepsilon_{b, j}^{e}\\ &\lambda_{j}\cdot\phi_{b,j}^{e}=\rho_{b}^{e}-c\cdot\delta t_{b}+ c\cdot\delta t^{e}+\lambda_{j}\left(\delta_{b,j}^{E}-\delta_{j}^{e}+N_{b,j}^{ e}\right)-I_{b,j}^{e}+T_{b,j}^{e}-GGTO+\varepsilon_{b,j}^{S}\end{split}  \tag{2}\]  
where GGTO denotes the time offset between GPST and GST. It is clearly revealed in (2) that the Galileo code and phase non-differenced observation equations are very similar to those of GPS, with the only difference being the GGTO. To eliminate the satellite-specific biases (satellite clock error, ionospheric delay, tropospheric delay, and satellite code and phase delay) and receiver-specific biases (receiver clock error, receiver code, and phase delay) and to simplify the observation equations, the double-differenced technique relative to the observations of a base receiver (denoted using subscript b, the rover receiver is denoted using subscript r) and a pivot receiver (denoted using superscript 1G for GPS and 1E for Galileo, the other satellites are denoted while using superscript g and e, respectively) is usually applied in equation construction. Moreover, the system time deviation item GGTO in Galileo observation equations is also offset.