# BDS Satellite Clock Prediction Considering Periodic Variations
## 2 Materials and Methods
### Analysis of Periodic Variations in BDS Satellite Clock Offsets
During the eclipse seasons, the actual attitude of the satellite is inconsistent with the nominal attitude. The yaw attitude will deteriorate the accuracy of non-conservative perturbation model and the correction of phase winding and PCO, thus affecting the accuracy of the satellite orbits [44]. Apart from the yaw attitude, the thermal radiation which is complicated to be accurately modeled also affects the orbit accuracy in eclipse seasons [46]. Considering that up to 97% of the radial orbit errors can be absorbed by the clock offsets [47], the PSD variations of the periodic term related to the orbit period can be affected by low orbit accuracy caused by yaw bias, thermal radiation, and other factors. The PSD variations of other periodic terms may be affected by perturbation errors and environment variations. Compared with BDS-2 MEO satellites, the PSD variations of BDS-3 MEO satellites in eclipse seasons are more outstanding. For BDS-3 MEO satellites, the PSD variations of C19 and C21 are higher than those of C25 and C27. It may be affected by the distinct orbit accuracy of CAST satellites and SECM satellites. From Figure 4, it can be concluded that the PSD of periodic terms vary with time, and the variations of each periodic term are inconsistent. The inconsistent variations lead to the main periodic term varying over time. For GEO and IGSO satellites, the main periodic term varies irregularly, while the variations of the main periodic term generally occur around the \(\beta\) angle peak value for MEO satellites. Considering that the International GNSS Monitoring and Assessment System (IGMAS) and the GFZ only add one periodic term to the SAM for the BDS ultra-rapid clock products [26], we focus on the variations of the main periodic term. The frequency variations of the main periodic term for all satellites considered are shown in Figure 5.  
Figure 4: PSD of four significant periodic terms with respect to the \(\beta\) angle. Figure 5 gives the frequency variations of the main periodic term. The horizontal axis and the vertical axis represent the time and the PRN, respectively. The third dimension indicates the frequency of the main periodic term. The left subfigure contains the frequency variations of the main periodic term for 30 satellites. As can be seen from the figure, the result of C11 is obviously anomalous from the others. The right subfigure gives the results of all satellites except C11. From the right subfigure, it can be seen that the main periodic terms of 29 satellites vary over time, and the frequency range is between 0 and 6 cpd. Therefore, the main periodic term obtained by FFT according to long-term clock offset series cannot be used as the main periodic term simply. As different MGEX ACs have different orbit and clock solutions, apart from the WHU, we also analyze periodic variations based on BDS clock products from GFZ. Since GFZ start to release BDS-3 satellite precise clock products on the website of Crustal Dynamics Data Information System (CDDIS) from DOY 166 in 2020, the 360-day clock products (DOY 166, 2020 to 159, 2021) provided by WHU and GFZ are used to analyze the periodic variations. The frequency variations of the main periodic term for WHU and GFZ clock offsets are shown in Figure 6.  
From Figure 6, it can be seen that the frequency variations exist both in WHU and GFZ clock products, and the inconsistent variations may be caused by the different orbit and clock solutions. In addition to C11, special frequencies are also detected in C14 clock offsets. In order to analyze the special results of C11 and C14, the 360-day (DOY 166, 2020 to 159, 2021) spectrograms are given in Figure 7.  
Figure 5: The frequency variations of main periodic term for 30 satellites (**left**), and 29 satellites except C11 (**right**). Figure 2:0 The frequency variations of main periodic term for WHU (**left**) and GFZ (**right**) clock offsets. As shown in Figure 7, C11 has an obvious periodic term around 20 cpd, and the frequency of this periodic term increases with time.