# BDS Satellite Clock Prediction Considering Periodic Variations
## 3 Results
### BDS Satellite Clock Prediction Analysis
For different constellation satellites, the MEO satellites have the best prediction accuracy, followed by the GEO satellites and the IGSO satellites. Comparing the prediction accuracy of satellites from different manufacturers, it can be found that the prediction accuracy of SECM satellites is higher than CAST satellites. For different prediction lengths, the prediction accuracy is generally reduced with increasing of the prediction length. The 6 h prediction accuracy of the TFAM for all satellites is approximately 0.9 ns. Considering that the time-varying periodic terms of different satellites occur at different times, the overall improvement of TFAM is limited. For the TFAM, it performs better for the satellites with marked periodic variations. In order to evaluate the performance of the improved model for the satellites with marked periodic variations, the BDS-2 IGSO satellite C06, the BDS-2 MEO satellite C14, the BDS-3 SECM satellite C28, and the BDS-3 CAST satellite C32 are chosen to show improved accuracy. Considering that there are little periodic variations of BDS-2 GEO satellites from DOY 140 to 169, in 2021, four satellites from IGSO and MEO constellations are chosen as an example. Figures 11 and 12 give the amplitude spectrums and spectrograms of WHU and GFZ clock offsets, respectively. It can be seen from the amplitude spectrums and spectrograms in Figures 11 and 12, the main periodic terms detected by FFT and STFT are distinct for C06, C14, and C32. For C28, the main periodic term in the amplitude spectrum and spectrogram of Figure 11 is the same, while different in Figure 12, which may be caused by different clock offset continuity of WHU and GFZ. For C06, C14, and C32, the amplitude spectrums and spectrograms of WHU and GFZ clock offsets have slight discrepancy, while the variation trend is basically the same. The spectral peak at the amplitude spectrum of the C06 is 1 cpd, while 1 cpd during DOY 140 to 145 and 2 cpd during DOY 145 to 170 in the corresponding spectrograms. For the C14, the main periodic term at the amplitude spectrum is 1.86 cpd. A periodic variation can be found in the spectrogram of the C14 satellite. From DOY 140 to 145, the main periodic term is 5.94 cpd, while the main periodic term is changed to 3.72 cpd after DOY 145. The main periodic term at the amplitude spectrum and the spectrogram of C32 are 1.86 cpd and 3.72 cpd, respectively. Therefore, for C06, C14, and C32 in Figure 11 and all of the satellites in Figure 12, the main periodic term has marked variations, and the frequency parameter of the TFAM and the SAM is different. Such significant periodic variation could affect prediction accuracy of the clock prediction model. The 30-day prediction errors (24 h prediction) of C06, C14, C28, and C32 based on the SAM and TFAM are given in Figures 13 and 14. Figure 12: Amplitude spectrums and spectrograms of GFZ clock offsets. As shown in Figures 13 and 14, the prediction errors of the TFAM and the SAM increase with prediction time, and the increase trend is similar for C06, C14, and C32 using WHU and GFZ clock offsets. For the satellites with marked periodic variations, the divergence trend of prediction errors is significantly improved using the TFAM than SAM, which indicates that the prediction errors accumulated with time can be improved by considering the time-varying periodic terms. Apparently, the TFAM can enhance the clock prediction accuracy. In order to intuitively demonstrate the prediction accuracy of the SAM and the TFAM, the mean value of the 30-day prediction errors is calculated, and the mean RMS values of the 30-day are remarked in Figure 15.  
As shown in Figure 15, the prediction accuracy is significantly improved using the TFAM compared to the SAM. For WHU clock offsets, the RMS of the C06, C14, and C32 are improved by 26.7%, 16.2%, and 16.3%, respectively. The GFZ clock offset prediction accuracy improvements for C06, C14, C28, and C32 are 29.8%, 16.0%, 21.0%, and 9.0%. The TFAM performs better for the satellites with marked periodic variations.