# A Short-Term Forecasting Method of Inter-Frequency Clock Bias for GPS Block IIF Satellites
## 2 Methods
### Method of IFCB Estimation  
The IFCB is usually estimated by calculating the difference of two IF combination carrier phase observations. Take for instance the GPS and combining Equation (10), it is recognized that the difference of the IF combinations of the L1/L2 and L1/L5 carrier phases is equal to the IFCB plus an ambiguity term [19]. \[ \begin{array}{rl}DIF(\widetilde{I}_{r,1}^{s,G},\widetilde{I}_{r,2}^{s,G}, \widetilde{I}_{r,3}^{s,G})&=IF(\widetilde{I}_{r,1}^{s,G},\widetilde{I}_{r,2}^ {s,G})-IF(\widetilde{I}_{r,1}^{s,G},\widetilde{I}_{r,3}^{s,G})\\ &=N_{\rm DIF}+IFCB_{r}^{s,G}\end{array}  \tag{12}\]  
where \(DIF(\widetilde{I}_{r,1}^{s,G},\widetilde{I}_{r,2}^{s,G},\widetilde{I}_{r,3}^ {s,G})\) is the difference in the double IF combination carrier phase observations; \(IF(\widetilde{I}_{r,1}^{s,G},\widetilde{I}_{r,2}^{s,G})\) and \(IF(\widetilde{I}_{r,1}^{s,G},\widetilde{I}_{r,3}^{s,G})\) are the IF combination carrier phase observations on the L1/L2 and L1/L5 frequencies, respectively; \(N_{\rm DIF}\) is the phase ambiguity of \(DIF(\widetilde{I}_{r,1}^{s,G},\widetilde{I}_{r,2}^{s,G},\widetilde{I}_{r,3}^ {s,G})\), which includes the constant ambiguity term and a constant phase hardware delay term. It should be noted that the frequency-related error terms in the Equation (12) were precisely corrected. Without the cycle slip occurring, the ambiguity terms and phase hardware delay can be considered constant for two adjacent calendars or a period of time. Therefore, one can extract the IFCB through eliminating these constant terms by using the difference method of two adjacent calendars for the \(DIF(\widetilde{I}_{r,1}^{s,G},\widetilde{I}_{r,2}^{s,G},\widetilde{I}_{r,3}^ {s,G})\). \[\partial IFCB_{r}^{s,G}=(DIF_{i}(\widetilde{I}_{r,1}^{s,G},\widetilde{I}_{r,2 }^{s,G},\widetilde{I}_{r,3}^{s,G})-DIF_{i-1}(\widetilde{I}_{r,1}^{s,G}, \widetilde{I}_{r,2}^{s,G},\widetilde{I}_{r,3}^{s,G}))/\beta_{f_{i}f_{j}} \tag{13}\]  
where the \(\partial IFCB_{r}^{s,G}\) is the changing value of the \(IFCB_{r}^{s,G}\) for time \(t\) and \(t-1\); \(\beta_{f_{i}f_{j}}\) is the frequency-dependent IF combination coefficient (if GPS, \(\beta_{f_{1}f_{3}}=-(f_{3}^{G})^{2}/((f_{1}^{G})^{2}-(f_{3}^{G})^{2})\)). Given that individual stations may be affected by observation noise and unmodeled errors, an iterative least squares estimation is usually required based on a ground observation network composed of multiple stations. In iterative least squares estimation, firstly, the initial value of the \(\partial IFCB\) at epoch \(t\) is calculated using the equal weight method based on the group stations; secondly, the difference between the initial value and the \(\partial IFCB\) obtained by each stations is calculate, and if the residual is greater than the set threshold, the weight of the station is reduced; thirdly, once more, an adjustment is implemented according to the new weight for each station; fourthly, the difference between the new adjustment value of the IFCB and \(\partial IFCB\) obtained by each station is calculated again, and if the residual is greater than the threshold, the weight of the station is reduced again. The second to fourth steps are repeated until there are no stations left that need to reduce the weight, the iteration is stopped and the last iteration of the IFCB can be used to recover each epoch IFCB. After obtaining an accurate \(\partial IFCB_{t}\) value for each calendar, one calendar IFCB is selected as the reference benchmark. Through the method of accumulation calendar by calendar, each calendar IFCB containing the same reference benchmark can be recovered. \[IFCB_{t}=\partial IFCB_{t_{0}}+\sum_{e=t_{0}+1}^{t}\partial IFCB_{(e,e-1)} \tag{14}\]  
where \(\partial IFCB_{t_{0}}\) is the IFCB of the reference calendar; \(IFCB_{t}\) is the final IFCB for correcting the third- or multi-frequency PPP for GNSS users. It is noted that the recovered IFCB of each calendar contains a common benchmark deviation but does not affect the PPP solution, as the bias will be absorbed into the ambiguity parameter.