# Improved Short-Term Clock Prediction Method for Real-Time Positioning
## 2 Clock Prediction Methods
### Traditional Model  
Whether in broadcast ephemeris or practical applications, clock correction is usually modeled by a quadratic polynomial model [18], which can be described as  
\[\mathrm{x}(t)\;=\;\mathrm{a}_{0}\;+\;\mathrm{a}_{1}t\;+\;\mathrm{a}_{2}t^{2}\;+\; \varepsilon(t) \tag{2}\]  
where \(\mathrm{x}(t)\) is the clock correction at epoch \(t\); \(\mathrm{a}_{0}\) and \(\mathrm{a}_{1}\) are the clock bias and frequency deviations, respectively; \(\mathrm{a}_{2}\) is the aging rate or frequency drift; and \(\varepsilon(t)\) is the generic random noise process. Given that the quadratic polynomial model has a definite physical meaning, it is suitable for clock correction fitting and forecasting in various situations. Considering that short-term prediction requires only a few hours to make the clock relatively stable during this moment, the frequency correction items of the traditional quadratic polynomial model can be disregarded to reduce Equation (2) to  
\[\mathrm{x}(t)=\mathrm{a}_{0}+\mathrm{a}_{1}+\varepsilon(t) \tag{3}\]  
To verify this viewpoint, the root mean square (RMS) of linear and quadratic polynomial models in 5-h short-term prediction is summarized in Table 3. The unit is nanosecond. The time span of validation clock correction is from day 120 to day 151 in 2016, i.e., a total of 30 days of IGS precision clock products. It must be noted that, since the accuracy of the prediction gradually decreases over time, we only evaluate the accuracy of the last forecast epoch. Furthermore, the prediction in this article means sliding prediction, which is used to simulate real-time application. In other words, the 1-h RMS shown in Table 3 is the statistic results of 30 days of 1-h sliding prediction.  
Table 3 shows that the linear model is more accurate than the quadratic polynomial model regardless of type of satellite and clock. The atomic clock aging does affect the accuracy of G13. The performance of G08 and G24 is much worse than that of the others. Therefore, we discussed short-term prediction of the rubidium clock with the linear model. Considering that the newest Block IIF has better characteristics and would gradually replace other families of satellites, we regarded G25 and G30 as examples of Block IIF to analyze the performance of 2-h linear sliding prediction using 2-h IGS final products. The statistics of 2-h prediction residuals for 30 days are provided in Figure 2.  
Figure 1: The Overlapping Allan DEViation (OADEV) for GPS on-board clocks over 30 days.  
Figure 3 shows pronounced 12-h and 6-h peaks along with minimal 4-h variations. The signal variations are similar to the discovery of [PERSON] et al. [9]. The difference is that the current research focused on the linear prediction residuals, whereas the previous research focused on quadratic polynomial fitting residuals. In fact, regardless of the time span of prediction, the corresponding residual cycles are the same as Figure 1. The difference is that the residuals increase with the extrapolation length of prediction. These special characteristics, which involve much time, can damage the accuracy of short-term prediction; however, few-hour data cannot remove long-term variations. Therefore, we propose a new improved model for short-term prediction.