# An Improved Predicted Model for BDS Ultra-Rapid Satellite Clock Offsets
## 4 Improved Algorithms  
### Periodic Term Detection and Classified Compensation  
The satellites in orbit are affected by the orbital period and the space environment, as well as various disturbances. Therefore, the influence of orbital period and perturbation error should be considered in the satellite clock offset prediction. Considering that the significant period of the satellite clock was usually consistent with the satellite orbital period, the periodic correction with the orbital period is added to the GPS satellite clock prediction by IGS [9; 14]. For the BDS satellite clock, the significant periodic terms not only include the similar orbital period, but other periodic signals that need to be analyzed also exist. In this paper, the Fast Fourier Transform (FFT) algorithm was used to obtain the significant periodic terms of all BDS satellites clocks based on 100 days in 2016 of clock offset observations [4; 8; 10; 36]. Two significant and special periodic terms of BDS different geometric configurations are given, as shown in Figure 4 and Table 2. Additionally, the amplitudes of the significant periodic terms are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**BDS Satellites** & **Main Periodic (h)** & **Amplitude (ns)** & **Secondary Periodic (h)** & **Amplitude (ns)** \\ \hline C01 & 23.9 & 0.19 & 6.00 & 0.12 \\ C02 & 24.0 & 0.22 & 12.0 & 0.13 \\ C03 & 18.8 & 0.24 & 15.8 & 0.14 \\ C04 & 23.9 & 0.25 & 11.8 & 0.10 \\ C05 & 23.9 & 0.30 & 1  
It can be seen from Figure 4 and Table 2 that the different types of BDS satellites are influenced by the different orbital geometric configurations. However, some period terms are not consistent with the orbital periods (see C03, C06 and C13), which are independent of their satellite devices. Usually, in the GNSS clock offset forecasting model, only the first significant period item is selected. It can be seen from Table 2 that the amplitude of the second period is about 0.1 ns, so we omit the third period (about 0.04 ns) and only choose the first periodic terms as the correction of periodic errors. The orbital period is remarkable mainly because of the usage of the precise orbit determination and time synchronization method. Through FFT analysis it can be seen that the periods are different between BDS and other GNSS. Therefore, different geometric configurations should be added with different periods, and the optimal selection of periodic items will have a more significant influence on the prediction accuracy. In addition, from Figure 4, it can be seen that there is a special significant period in the C11 satellite, and the FFT analysis results of the C11 satellite are shown separately in Figure 5.  
From Figure 5, the C11 satellite has a significant periodic term of about 1.3 h, and it is different from the normal period of the MEO orbital geometric configuration. The amplitude of this period is about 0.1 ns. This obvious period will have a large impact on the precision of the prediction model and should be considered. This noticeable period may be caused by the hardware of the atomic clock itself. By using the same observations in Figure 4, through adding one and two significant periodic terms to the quadratic polynomial model, respectively, the fitting residuals can be corrected, and the results are shown in Figure 6.  
Figure 4: BDS satellite periodic terms analysis. The GEO, IGSO, and MEO are classified in the figure. Figure 5: Frequency analysis of the C11 satellite. The main period is 12 h and the second period is 1.3 h.  
From Figure 6, it can be seen that the fitting residuals is more stable after adding two periodic terms, and the fitting accuracies are significantly improved without obvious periodic terms. The results also verify that the existence of the periodic term between different satellites geometric configurations. It can be seen that the fitting precision of GEO and MEO are within 1 ns, and that of IGSO is superior to 1.5 ns.