## 3 SFSE Double-Differenced Mathematical Model  
### Measurement Function  
Suppose 'r' and 'b' represent the rover station and the base station of the GNSS measurement baseline, respectively; the GNSS constellation is expressed by \(\otimes\) and it is G for GPS, B for BDS, and J for QZSS. Then, the SFSE non-differenced functions for the base station can be expressed as  
\[ \begin{array}{l}\mathrm{P}_{\mathrm{b}}^{\otimes}=\rho_{\mathrm{b}}^{\otimes }-\mathrm{c}\times\delta\mathrm{t}_{\mathrm{b}}+\mathrm{c}\times\delta\mathrm{ t}^{\otimes}+\delta_{\mathrm{ion}}+\delta_{\mathrm{trop}}+\delta_{\mathrm{mul}}+ \varepsilon_{\mathrm{P}}\\ \lambda^{\otimes}\phi_{\mathrm{b}}^{\otimes}=\rho_{\mathrm{b}}^{\otimes}- \mathrm{c}\times\delta\mathrm{t}_{\mathrm{b}}+\mathrm{c}\times\delta\mathrm{ t}^{\otimes}+\lambda^{\otimes}\times\mathrm{N}_{\mathrm{b}}^{\otimes}-\delta_{ \mathrm{ion}}+\delta_{\mathrm{trop}}+\delta_{\mathrm{mul}}+\varepsilon_{ \Phi}\end{array}  \tag{1}\]  
where P and \(\Phi\) represent code observation (m) and phase observation (cycle); \(\rho_{\mathrm{b}}^{\otimes}\) denotes the real receiver-satellite range (m); \(\lambda^{\otimes}\) is the wavelength (m) of the specific frequency of the corresponding system; \(\delta\mathrm{t}_{\mathrm{b}}\) is the receiver clock error (s) and \(\delta\mathrm{t}^{\otimes}\) is satellite clock error (s); c is the velocity of light (m/s); N is the integer ambiguities (cycle); \(\delta_{\mathrm{ion}}\) and \(\delta_{\mathrm{trop}}\) are the ionospheric delay (m) and tropospheric delay (m), respectively; \(\varepsilon_{\mathrm{P}}\) and \(\varepsilon_{\Phi}\) are the other unmodeled errors (m). When the BDS/QZSS dual constellations are used, it is a prerequisite to unify the coordinate and time systems. Therefore, in this paper, the spatiotemporal atom is based on World Geodetic System 1984 (WGS84) and GPS Time (GPST) system. The difference caused by different coordinate systems can be ignored, and the relationship between GPST and BDS Time (BDST) is defined as GPST = BDST + (1356 week, 14 s) [18]. The inter-system biases (ISBs) occur in the combination of BDS/QZSS and would change with the reboots of the receivers [19]. Hence the loosely combined model (LCM) was adopted here and one reference satellite is selected for each satellite system. Then, the common errors existing in each system can be eliminated by subtracting the measurement of the reference satellite from ordinary satellites [5; 20]. In other words, two-times DD processing will be used in the BDS/QZSS combination and the measurement information of two satellites will be consumed [21]. Then Formula (1) turns in to  
\[ \begin{array}{l}\Delta\mathrm{P}_{\mathrm{rb}}^{\mathrm{B},12}=\Delta\rho_ {\mathrm{rb}}^{\mathrm{B},12}+\Delta\varepsilon_{\mathrm{P}}\\ \lambda^{\mathrm{B}}\times\Delta\rho_{\mathrm{rb}}^{\mathrm{B},12}=\Delta\rho_ {\mathrm{rb}}^{\mathrm{B},12}+\lambda^{\mathrm{B}}\times\Delta\mathrm{N}_{ \mathrm{rb}}^{\mathrm{B},12}+\Delta\varepsilon_{\mathrm{\phi}}^{\mathrm{B}}\\ \Delta\mathrm{P}_{\mathrm{rb}}^{\mathrm{J},12}=\Delta\rho_{\mathrm{rb}}^{\mathrm{ J},12}+\Delta\varepsilon_{\mathrm{P}}\\ \lambda^{\mathrm{J}}\times\Delta\rho_{\mathrm{rb}}^{\mathrm{J},12}=\Delta\rho_ {\mathrm{rb}}^{\mathrm{J},12}+\lambda^{\mathrm{B}}\times\Delta\mathrm{N}_{ \mathrm{rb}}^{\mathrm{J},12}+\Delta\varepsilon_{\mathrm{\phi}}^{\mathrm{J}}\end{array}