## 3 SFSE Double-Differenced Mathematical Model
### Measurement Function
 \tag{2}\]  
where \(\Delta\mathrm{P}_{\mathrm{rb}}^{\otimes,12}=\mathrm{P}_{\mathrm{r}}^{\otimes,1 }-\mathrm{P}_{\mathrm{r}}^{\otimes,2}-(\mathrm{P}_{\mathrm{b}}^{\otimes,1}- \mathrm{P}_{\mathrm{b}}^{\otimes,2})\) is the DD code observation; \(\Delta\phi_{\mathrm{rb}}^{\otimes,12}=\phi_{\mathrm{r}}^{\otimes,1}-\phi_{ \mathrm{r}}^{\otimes,2}-(\phi_{\mathrm{b}}^{\otimes,1}-\phi_{\mathrm{b}}^{ \otimes,2})\) is the DD phase observation; \(\Delta\rho_{\mathrm{rb}}^{\otimes,12}=]\rho_{\mathrm{r}}^{\otimes,1}-\rho_{ \mathrm{r}}^{\otimes,2}-(\rho_{\mathrm{b}}^{\otimes,1}-\rho_{\mathrm{b}}^{ \otimes,2})\) is the DD receiver-satellite range; \(\Delta\mathrm{N}_{\mathrm{rb}}^{\otimes,12}=\mathrm{N}_{\mathrm{r}}^{\otimes,1 }-\mathrm{N}_{\mathrm{r}}^{\otimes,2}-(\mathrm{N}_{\mathrm{b}}^{\otimes,1}- \mathrm{N}_{\mathrm{b}}^{\otimes,2})\) is the DD ambiguity; \(\Delta\varepsilon_{\mathrm{\phi}}^{\mathrm{B}}\) and \(\Delta\varepsilon_{\mathrm{\phi}}^{\mathrm{J}}\) are, respectively, the DD unmodeled errors of BDS and QZSS; \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Satellite** & **Band** & **Frequency/MHz** & **Wavelength/cm** & **Satellite** & **Satellite** & **Inclination/\({}^{\mathrm{o}}\) & **Orbital Period** & **Number** \\ \hline \multirow{3}{*}{BDS} & B1 & 1561.098 & 19.2 & \multirow{3}{*}{BDS} & GEO & 0 & 23 h 56 min & 5 \\  & B2 & 1207.14 & 24.83 & & IGSO & 55 & 23 h 56 min & 6 \\  & B3 & 1268.52 & 23.63 & & MEO & 55 & 12 h 50 min & 3 \\ \hline \multirow{3}{*}{GPS} & L1 & 1575.42 & 19.03 & \multirow{3}{*}{GPS} & MEO & 55 & 11 h 58 min & 31 \\  & L2 & 1227.6 & 24.42 & & & & \\ \cline{1-1} \cline{5-7}  & L5 & 1176.45 & 25.48 & \multirow{3}{*}{QZSS} & IGSO & 45 & 23 h 56 min & 3 \\ \cline{1-1} \cline{5-7}  & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic states of GPS, BDS, and QZSS. After linearization and simplification, we obtain  
\[y^{B} =\left[ \begin{array}{c}B^{B}x\\ B^{B}x+A^{B}N^{B}\\ B^{I}x\\ B^{I}x+A^{I}N^{I}\\ \end{array} \right]+e^{B} \tag{3}\]  
where \(y\) is the observation vector consisting of code observation and phase observation; \(x\) is the unknown baseline vector with the transition matrix \(B^{\otimes}\); \(N\) is the unknown DD integer ambiguity vector with the transition matrix \(A^{\otimes}\). After further modularization, we can obtain the SFSE DD model of BDS/QZSS:  
\[y^{B/J}=\left[ \begin{array}{ccc}B^{B}&0&0\\ B^{B}&A^{B}&0\\ B^{J}&0&0\\ B^{J}&0&A^{I}\\ \end{array} \right]\times\left[ \begin{array}{c}x\\ N^{B}\\ N^{I}\\ \end{array} \right]+e \tag{4}\]  
QZSS has the same time system as GPS, and the difference between Japan Satellite Navigation Geodetic System (JGS) and WGS84 is negligible. With regard to the DD processing of the GPS/QZSS observation, the tightly combined model (TCM) is used and only one reference satellite need be selected from two constellations. The ISBs between the identical frequency (such as GPS L1 and QZSS L1) of GPS and QZSS remain stable and can be corrected by parameter estimation in advance [22]. Then, Formula (1) turns into  
\[ \begin{array}{l}\Delta\mathrm{P}_{\mathrm{rb}}^{\mathrm{J,12}}=\Delta\rho_{ \mathrm{rb}}^{\mathrm{J,12}}+\Delta\mathrm{E}_{\mathrm{P}}^{\mathrm{J}}\\ \lambda^{\mathrm{J}}\times\Delta\rho_{\mathrm{rb}}^{\mathrm{J,12}}=\Delta\rho _{\mathrm{rb}}^{\mathrm{J,12}}+\lambda^{\mathrm{G}}\times\Delta\mathrm{N}_{ \mathrm{rb}}^{\mathrm{J,12}}+\Delta\mathrm{g}_{\phi}^{\mathrm{J}}\\ \Delta\mathrm{P}_{\mathrm{rb}}^{\mathrm{GJ,12}}=\Delta\rho_{\mathrm{rb}}^{ \mathrm{GJ,12}}+\mathrm{d}_{\mathrm{rb}}^{\mathrm{GJ,12}}+\Delta\mathrm{E}_{ \mathrm{P}}^{\mathrm{GJ}}\\ \lambda^{\mathrm{G}}\times\Delta\rho_{\mathrm{rb}}^{\mathrm{GJ,12}}=\Delta\rho _{\mathrm{rb}}^{\mathrm{GJ,12}}+\lambda^{\mathrm{G}}\times\Delta\mathrm{N}_{ \mathrm{rb}}^{\mathrm{GJ,12}}+\rho_{\mathrm{rb}}^{\mathrm{GJ,12}}+\Delta\mathrm{ e}_{\phi}^{\mathrm{GJ}}\\ \end{array}