# A Short-Term Forecasting Method of Inter-Frequency Clock Bias for GPS Block IIF Satellites
## 2 Methods
### Trible-Frequency Uncombined PPP
\tag{2}\]  
where superscript \(G\) denotes a satellite system; \(P_{r,j}^{s,G}\) and \(L_{r,j}^{s,G}\) denote the measured code and carrier phase, respectively; \(\rho_{r}^{s,G}\) is the geometric distance; \(c\) is the speed of light; \(dt_{r}^{G}\) and \(dt^{s,G}\) are the clock offsets of the receiver and satellite, respectively; \(\mathrm{M}_{r}^{s,G}\) is the wet mapping function; \(T_{r}^{s,G}\) is the zenith wet delay; \(\gamma_{f}^{G}=\left(f_{1}^{G}/f_{j}^{G}\right)^{2}\) is the frequency-dependent multiplier factor, which is independent of the satellite pseudorandom noise code; \(I_{r,1}^{s,G}\) is the line-of-sight ionospheric delay on the frequency \(f_{1}^{G}\); \(\lambda_{j}^{s,G}\) is the carrier wavelength on the frequency \(f_{j}^{G}\); \(N_{r,j}^{s,G}\) is the integer phase ambiguity on the frequency \(f_{j}^{G}\); \(B_{r,j}^{s,G}\) and \(B_{j}^{s,G}\) are the time-invariance parts of the frequency-dependent receiver and satellite uncalibrated phase delay (UPD), respectively; \(\partial B_{r,j}^{s,G}\) and \(\partial B_{j}^{s,G}\) are the time-varying parts of the frequency-dependent receiver and satellite UPD, respectively; \(b_{r,j}^{s,G}\) is the code hardware delay (CHD) from the receiver antenna to the signal correlator in the receiver; \(b_{j}^{s,G}\) is the CHD from the satellite signal generator to the satellite antenna; \(e_{r,j}^{s,G}\) and \(e_{r,j}^{s,G}\) are the pseudo-range measurement noise and carrier measurement noise, respectively; \(f_{m}^{s,G}\) is the signal frequency of the \(m_{th}\) band of the system \(G\); \(\alpha_{mn}^{G}\) and \(\beta_{mn}^{G}\) are the frequency-dependent factors, respectively; \(\mathrm{DCB}_{P_{m}P_{n}}^{s,G}\) and \(\mathrm{DCB}_{r,P_{m}P_{n}}^{s,G}\) are the frequency-dependent satellite and receiver differential code bias, respectively; \(b_{I_{Fmn}}^{s,G}\) and \(b_{r,I_{Fmn}}^{s,G}\) are the IF combination of the CHD of the satellite and receiver, respectively; \(\partial\mathrm{UPD}_{P_{m}P_{n}}^{s,G}\) and \(\partial\mathrm{UPD}_{P_{r}P_{n}P_{n}}^{G}\) are the differences in the time-varying part of the UPD on the differential frequency carrier phase in the satellite and receiver, respectively; \(\partial B_{I_{Fmn}}^{s,G}\) and \(\partial B_{P_{r}I_{Fmn}}^{G}\) are the time-varying parts of IF combination of the UPD in the satellite and receiver, respectively. Other error items, such as phase center offsets and variations [28], phase windup [29], relativistic effect and tide loading, were assumed to be precisely corrected with their corresponding models. Precise satellite clock products are generated by using the combination of IF observations. The carrier phase hardware delay is considered relatively stable for 24 h or for a period of time, estimated as a constant, ignoring the time-varying part \(\partial B_{j}^{s,G}\). Therefore, the satellite clock offsets \(dt_{I_{12}}^{s,G}\) absorbs the IF combination satellite's CHD \(b_{I_{12}}^{s,G}\) and the time-varying part of IF combination satellite's UPD \(\partial B_{I_{12}}^{s,G}\)[30]. \[dt_{I_{12}}^{s,G}=c\cdot dt^{s,G}+b_{I_{12}}^{s,G}+\partial B_{I_{12}}^{s,G} \tag{3}\]  
Equation (3) is substituted into Equation (1) and then linearized using [PERSON]'s formula. Moving the known items to the left of the equation and merging the same parameter items above, the equations on each frequency can be written in detail in the following form. For pseudo-range observables:  
\[\left\{