## 2 Materials and Methods
### The Multi-GNSS Observation Model of the New Prototype Software Platform
\begin{split}&\Delta\
abla l_{r1r2,IF}^{\text{s1s2}}=\left(\Delta t _{r1r2}^{\text{s1}}\cdot\Phi(t,t_{0})^{\text{s1}}\cdot\omega_{o}^{\text{s1}}- \Delta t_{r1r2}^{\text{s2}}\cdot\Phi(t,t_{0})^{\text{s2}}\cdot\omega_{o}^{ \text{s2}}\right)\\ &+\lambda_{IF}\cdot\Delta\
abla N_{r1r2,IF}^{\text{s1s2}}+\Delta \
abla\partial_{r1r2}^{\text{s1s2}}\cdot\delta_{epr}\\ &-\left(\Delta t_{r1}^{\text{s1s2}}\cdot R_{r1,emp}\cdot r_{r1}- \Delta t_{r2}^{\text{s1s2}}\cdot R_{r2,emp}\cdot r_{r2}\right)\\ &+\left(\Delta m_{r1,trop}^{\text{s1s2}}\cdot\delta_{r1,trop}- \Delta m_{r2,trop}^{\text{s1s2}}\cdot\delta_{r2,trop}\right)+\Delta\
abla \epsilon_{r1r2,IF}^{\text{s1s2}}\\ &\Delta\
abla p_{r1r2,IF}^{\text{s1s2}}=\left(\Delta t_{r1r2}^{ \text{s1}}\cdot\Phi(t,t_{0})^{\text{s1}}\cdot\omega_{o}^{\text{s1}}-\Delta t _{r1r2}^{\text{s2}}\cdot\Phi(t,t_{0})^{\text{s2}}\cdot\omega_{o}^{\text{s2}} \right)\\ &+\Delta\
abla\partial_{r1r2}^{\text{s1s2}}\cdot\delta_{epr}\\ &-\left(\Delta\mu_{r1}^{\text{s1s2}}\cdot R_{r1,emp}\cdot r_{r1}- \Delta\mu_{r2}^{\text{s1s2}}\cdot R_{r2,emp}\cdot r_{r2}\right)\\ &+\left(\Delta m_{r1,trop}^{\text{s1s2}}\cdot\delta_{r1,trop}- \Delta m_{r2,trop}^{\text{s1s2}}\cdot\delta_{r2,trop}\right)+\Delta\
abla \omega_{r1r2,IF}^{\text{s1s2}}\end{split}  \tag{6}\]  
where \(\Delta\
abla l_{r1r2,IF}^{\text{s1s2}}\) is the double-difference IF combined phase of \(l_{r,IF}^{\text{s}}\) between receivers (\(r1,r2\)) and satellites (\(s1,s2\)), and \(\Delta\
abla p_{r1r2,IF}^{\text{s1s2}}\) has an analogous meaning; \(\Delta t_{r1}^{\text{s1s2}}\) is the computed vector \(\mu_{r1}^{\text{s1}}\) minus \(\mu_{r1}^{\text{s2}}\), and \(\Delta\mu_{r2}^{\text{s1s2}}\) has an analogous meaning; \(\Delta\
abla N_{r1r2,IF}^{\text{s1s2}}\) is the double-difference IF combined ambiguity; \(\Delta\
abla\partial_{r1r2}^{\text{s1s2}}\) is the double-difference partial derivative of \(\partial_{r}^{\text{s}}\); \(\Delta m_{r1,trop}^{\text{s1s2}}\) is the computed mapping function given by \(\Delta m_{r1,trop}^{\text{s1}}\) minus \(\Delta m_{r1,trop}^{\text{s2}}\), and \(\Delta m_{r2,trop}^{\text{s1s2}}\) has an analogous meaning; \(\Delta\
abla\epsilon_{r1r2,IF}^{\text{s1s2}}\) is the double-difference value of \(\epsilon_{r,IF}^{\text{s}}\) between receivers (\(r1,r2\)) and satellites (\(s1,s2\)), and \(\Delta\
abla\omega_{r1r2,IF}^{\text{s1s2}}\) has an analogous meaning. Based on the linearized double-difference IF combined observation equation, the five system (GPS/GLONASS/BDS/Galileo/QZSS) ultra-rapid POD observation model using the one-step method is obtained as follows:  
\[ \begin{split}&\Delta\
abla l_{r1r2,IF}^{\text{s1S2}}=\left(\Delta t _{r1r2}^{\text{s1}}\cdot\Phi(t,t_{0})^{\text{s1}}\cdot\omega_{o}^{\text{s1}}- \Delta\mu_{r1r2}^{\text{s2}}\cdot\Phi(t,t_{0})^{\text{s2}}\cdot\omega_{o}^{\text {s2}}\right)\\ &+\lambda_{IF}\cdot\Delta\
abla N_{r1r2,IF}^{\text{s1S2}}+\Delta \
abla\partial_{r1r2}^{\text{s1S2}}\cdot\delta_{epr}\\ &-\left(\Delta\mu_{r1}^{\text{s1S2}}\cdot R_{r1,emp}\cdot r_{r1}- \Delta\mu_{r2}^{\text{s1S2}}\cdot R_{r2,emp}\cdot r_{r2}\right)\\ &+\left(\Delta m_{r1,trop}^{\text{s1S2}}\cdot\delta_{r1,trop}- \Delta m_{r2,trop}^{\text{s1S2}}\cdot\delta_{r2,trop}\right)+\Delta\
abla \epsilon_{r1r2,IF}^{\text{s1S2}}\end{split}  \tag{7}\]\[\Delta\
abla p_{r1r2,1F}^{S1S2}=\left(\Delta\mu_{r1r2}^{S1}\cdot\Phi(t,t_{0})^{S1 }\cdot o_{o}^{S1}-\Delta\mu_{r1r2}^{S2}\cdot\Phi(t,t_{0})^{S2}\cdot o_{o}^{S2}\right) \tag{8}\] \[+\Delta\
abla\partial_{r1r2}^{S1S2}\cdot\delta_{ep}\] \[-\left(\Delta\mu_{r1}^{S1S2}\cdot R_{r1,rep}\cdot r_{r1}-\Delta t _{r2}^{S1S2}\cdot R_{r2,rep}\cdot r_{r2}\right)\] \[+\left(\Delta m_{r1,trop}^{S1S2}\cdot\delta_{r1,trop}-\Delta m_{r2,trop}^{S1S2}\cdot\delta_{r2,trop}\right)+\Delta\
abla\omega_{r1r2,1F}^{S1S2}\]  
where  
\[S=\left(G\text{ or }R_{k}\text{ or }C\text{ or }E\text{ or }J\right), \tag{9}\]  
besides, \(G\), \(R_{k}\), \(C\), \(E\), and \(J\) refer to GPS, GLONASS, BDS, Galileo, and QZSS, respectively; and \(k\) is the frequency factor in the frequency division multiple access (FDMA) mode.