# A Short-Term Forecasting Method of Inter-Frequency Clock Bias for GPS Block IIF Satellites
## 3 Results
### Time-Varying Characteristics of the IFCB  
In real-time data processing or PPP network solutions, simultaneously receiving real-time data flow and performing IFCB estimation will significantly increase the program's execution burden and calculation time, which is not conducive to real-time multi-frequency PPP and PPP network solutions. Considering that the final precision ephemerides products released by the IGS analysis centers usually lag no more than 7 days, therefore, in order to solve these problems, a short-term prediction model based on the epoch-by-epoch sliding Pearson autocorrelation function was proposed for the IFCB on the GPS Block IIF satellites according to the lag features of the final precision ephemerides product and the short-term variation characteristics of the IFCB described above. Firstly, the autocorrelation coefficient was calculated by the Pearson autocorrelation function [31]. \[r_{i}=\frac{E((X_{d_{1}}-\hat{u})(Y_{d_{1}+i}-\hat{u}))}{\sigma_{X}\cdot\sigma _{Y}} \tag{15}\]  
In Equation (15), \(\hat{u}\) is the average value of IFCB of all days, which is different from the standard autocorrelation function; \(X_{d_{i}}\) is the IFCB time series of the first day, which is the benchmark time series; \(Y_{d_{1}+i}\) is the IFCB time series that sliding \(i\) calendars from the  
Figure 3: Differences in the daily IFCBs relative the first day for the G01 satellite. first epoch, with the same sequence length as \(X_{d_{1}}\); \(r_{i}\) is the autocorrelation coefficient of \(Y_{d_{1}+i}\) and \(X_{d_{1}}\) in the \(i\) epoch; \(\sigma_{X}\) and \(\sigma_{Y}\) are the standard deviations for the different time series, respectively. The autocorrelation coefficients of the IFCB time series for the GPS Block IIF satellites are given in Figure 4. As the lag time increased, the autocorrelation coefficient of the satellite IFCB time series gradually decreased, i.e., the correlation gradually decreased. The autocorrelation coefficient sequences shows two wave peaks within 24 h (the primary and secondary peaks, respectively), indicating that there was a major- and subperiod in the IFCB time series, with approximately one sidereal day and a half-sidereal day, respectively. This result corresponded to the changing features of the IFCB time series in the GPS satellite system. To accurately analyze the changing characteristics of the IFCB period, the epoch moments corresponding to the primary and secondary peaks of the autocorrelation coefficient time series for each GPS Block IIF satellite were extracted as shown in Figure 6.  
In order to verify the validity of the IFCB period extracted by the Pearson autocorrelation function in this paper, a hypothesis test scheme obeying the \(t\)-distribution was set: (1) an IFCB time series of 30 days was divided into two time periods: days 1-15 and days 16-30; (2) the IFCB time series on days 1 and 16 were used as the benchmark values, and the autocorrelation coefficient in the two periods of time could be obtained based on the epoch-by-epoch sliding Pearson autocorrelation function; (3) the epoch moments corresponding to the peaks of the autocorrelation coefficient could be extracted separately, thus obtaining two independent samples; (4) suppose that the average period was equal over the two time periods, namely, \(H_{0}:u_{1}=u_{2}\), and the statistics formula and degrees of freedom were \(t=\frac{|\overline{X}_{1}-\overline{X}_{2}|}{\overline{X}_{1}-\overline{X}_{ 2}}\) and \(n_{1}+n_{2}-2=28\), respectively, where \(\overline{X}_{1}\) and \(\overline{X}_{2}\) are the sample means; \(S_{\overline{X}_{1}-\overline{X}_{2}}\) is the calculated value, and the resulting \(t\) values are shown in Figure 5; (5) according to the Figure 5, the \(t\) values of all satellites were within the acceptance domain and far less than the threshold, namely, it accepts the original hypothesis, where the threshold was 1.701, as determined according to the degrees of freedom and confidence intervals.