# A Short-Term Forecasting Method of Inter-Frequency Clock Bias for GPS Block IIF Satellites
## 2 Methods
### Trible-Frequency Uncombined PPP
\begin{array}{l}\widehat{p}_{r,1}^{s,G}=\mathbf{u}_{r}^{s}\cdot \mathbf{x}+c\cdot dt_{r}^{G}+\mathrm{M}_{r}^{s,G}\cdot T_{r}^{s,G}+\gamma_{1 }^{G}\cdot\mathrm{I}_{r,1}^{s,G}+\\ \widehat{p}_{r,1}^{s,G}+\partial B_{I_{12}}^{s,G}+\varepsilon_{r,1}^{s,G}\\ \widehat{p}_{r,2}^{s,G}=\mathbf{u}_{r}^{s}\cdot\mathbf{x}+c\cdot dt_{r}^{G}+ \mathrm{M}_{r}^{s,G}\cdot T_{r}^{s,G}+\gamma_{2}^{G}\cdot I_{r,1}^{s,G}+\\ \widehat{b}_{r,2}^{s,G}+\partial B_{I_{12}}^{s,G}+\varepsilon_{r,2}^{s,G}\\ \widehat{p}_{r,3}^{s,G}=\mathbf{u}_{r}^{s}\cdot\mathbf{x}+c\cdot dt_{r}^{G}+ \mathrm{M}_{r}^{s,G}\cdot T_{r}^{s,G}+\gamma_{3}^{G}\cdot I_{r,1}^{s,G}+\\ \widehat{p}_{r,3}^{s,G}+\partial B_{I_{12}}^{s,G}+\varepsilon_{r,3}^{s,G}\\ \end{array} \right. \tag{4}\]  
with  
\[\left\{ \begin{array}{l}\widehat{p}_{r,1}^{s,G}=P_{r,j}^{s,G}-\rho_{r}^{s}+dt _{I_{12}}^{s,G}+\beta_{12}\cdot\mathrm{DCB}_{P_{1}P_{2}}^{s,G}\\ \widehat{p}_{r,2}^{s,G}=P_{r,j}^{s,G}-\rho_{r}^{s}+dt_{I_{12}}^{s,G}-\alpha_{1 2}\cdot\mathrm{DCB}_{P_{1}P_{2}}^{s,G}\\ \widehat{p}_{r,3}^{s,G}=P_{r,j}^{s,G}-\rho_{r}^{s}+dt_{I_{12}}^{s,G}-\alpha_{1 2}\cdot\mathrm{DCB}_{P_{1}P_{3}}^{s,G}-\beta_{12}\cdot\mathrm{DCB}_{P_{2}P_{3}} ^{s,G}\end{array} \right. \tag{5}\]For carrier phase observables:  
\[\left\{ \begin{array}{l}\widetilde{l}_{r,1}^{s,G}=\mathbf{u}_{r}^{s}\cdot \mathbf{x}+c\cdot dt_{r}^{G}+\mathbf{M}_{r}^{s,G}\cdot T_{r}^{s,G}-\gamma_{1}^{G} \cdot l_{r,1}^{s,G}+\\ \lambda_{1}^{s,G}N_{r,1}^{s,G}+B_{r,1}^{s,G}-B_{1}^{s,G}+b_{1F_{2}}^{s,G}+ \beta B_{r,1}^{s,G}-\beta_{12}\partial\text{UPD}_{P_{1}P_{2}}^{s,G}+\epsilon_{r,1}^{s,G}\\ \widetilde{l}_{r,2}^{s,G}=\mathbf{u}_{r}^{s}\cdot\mathbf{x}+c\cdot dt_{r}^{G}+ \mathbf{M}_{r}^{s,G}\cdot T_{r}^{s,G}-\gamma_{2}^{G}\cdot l_{r,1}^{s,G}+\\ \lambda_{2}^{s,G}N_{r,2}^{s,G}+B_{r,2}^{s,G}-B_{r,2}^{s,G}+b_{1F_{2}}^{s,G}+ \beta B_{r,2}^{s,G}+\alpha_{12}\partial\text{UPD}_{P_{1}P_{2}}^{s,G}+\epsilon_{ r,2}^{s,G}\\ \widetilde{l}_{r,3}^{s,G}=\mathbf{u}_{r}^{s}\cdot\mathbf{x}+c\cdot dt_{r}^{G}+ \mathbf{M}_{r}^{s,G}\cdot T_{r}^{s,G}-\gamma_{3}^{G}\cdot l_{r,1}^{s,G}+\\ \lambda_{1}^{s,G}N_{r,3}^{s,G}+B_{r,3}^{s,G}-B_{3}^{s,G}+b_{1F_{2}}^{s,G}+ \partial B_{r,3}^{s,G}-\partial B_{3}^{s,G}+\partial B_{1F_{12}}^{s,G}+\epsilon _{r,3}^{s,G}\end{array} \right. \tag{6}\]  
with  
\[\left\{ \begin{array}{l}\widetilde{l}_{r,1}^{s,G}=L_{r,1}^{s,G}-\rho_{s}^{s}+ dt_{r1}^{s,G}\\ \widetilde{l}_{r,2}^{s,G}=L_{r,2}^{s,G}-\rho_{r}^{s}+dt_{r1}^{s,G}\\ \widetilde{l}_{r,3}^{s,G}=L_{r,3}^{s,G}-\rho_{r}^{s}+dt_{r1}^{s,G}\\ \widetilde{l}_{r,3}^{s,G}=L_{r,3}^{s,G}-\rho_{r}^{s}+dt_{r1}^{s,G}\\ \end{array} \right. \tag{7}\]  
where \(\mathbf{u}_{r}^{s}\) is the unit vector of the component from the receiver to the satellite; \(\mathbf{x}\) is the vector of the receiver position increments relative to the priori position. For \(\widetilde{p}_{r,3}^{s,G}\), since there is no difference code bias product between the first and third frequencies, we used two other DCB products for conversion. Similarly, the receiver clocks are expressed in a form similar to the satellite clock offsets:  
\[dt_{r,I\bar{r}_{12}}^{G}=c\cdot dt_{r}^{G}+b_{r,I\bar{r}_{12}}^{G}+\partial B _{r,I\bar{r}_{12}}^{G} \tag{8}\]  
We substituted (8) into (4) and (6) and merged the same parameter items. The pseudo-range and carrier phase observation equations can be expressed as follows:  
\[\left\{\begin{array}{l}\widetilde{p}_{r,1}^{s,G}=\mathbf{u}_{r}^{s}\cdot \mathbf{x}+dt_{r_{1}I\bar{r}_{12}}^{G}+\mathbf{M}_{r}^{s,G}\cdot T_{r}^{s,G}+ \widetilde{l}_{r,1}^{s,G}+\tau_{r,1}^{s,G}+\epsilon_{r,1}^{s,G}\\ \widetilde{p}_{r,2}^{s,G}=\mathbf{u}_{r}^{s}\cdot\mathbf{x}+dt_  
small magnitude (approximately 2 cm) and relatively low weight for the pseudo-range observation.