# BDS Satellite Clock Prediction Considering Periodic Variations
## 2 Materials and Methods
### Analysis of Periodic Variations in BDS Satellite Clock Offsets

Table 2: The types of BDS satellites ([[http://www.csno-arc.cn/](http://www.csno-arc.cn/)]([http://www.csno-arc.cn/](http://www.csno-arc.cn/)), accessed on 14 August 2021). Figure 2: Amplitude spectrums of 90-day, 180-day, and 270-day for C01, C10, and C25. the residuals, amplitude spectrum, and spectrogram of the GEO satellite C01, the IGSO satellite C07, and the MEO satellite C27. The figure of residuals shows small gaps, meaning that the raw clock offsets of C01, C07 and C27 have good continuity. The spectrograms in Figure 3 give the information of the time domain and the frequency domain simultaneously. The horizontal axis represents the time domain, and the vertical axis represents the frequency domain. The third dimension, which is represented by the intensity of the color, indicates the Power Spectral Density (PSD) of the frequency at that time. From the results of residuals in Figure 3, it can be seen that the amplitude of the residuals varies over time irregularly, and there is no intuitive information about periodic variations. The amplitude spectrums show the holistic periodic terms of 360-day clock offset series. The spectrograms give the variations of the periodic terms through color variations. Two types of variations can be found through the spectrograms. First, the amplitude of the periodic terms varies with time, and the variations are diverse for different periodic terms and different satellites. Second, the frequencies of periodic terms vary over time. Therefore, the main periodic term is time-varying, and the detected results by FFT and STFT are different. For C01, the highest spectral peak in frame 1 is 3 cpd, while 2 cpd in frame 2. For C07, the main periodic term in frame 1 and frame 2 of the spectrogram is 1 cpd, and 2 cpd, respectively. For C27, there are several significant peaks around 1.86 cpd in the amplitude spectrum, and the amplitude of them has little difference. The picture-in-picture of spectrogram shows that these significant periodic terms around 1.86 cpd play the main role at different times. Such frequency fluctuations of the main periodic term are easily ignored by FFT in a short time. Considering that the accuracy of orbits and clocks during eclipse seasons are much lower than that in non-eclipse seasons, we characterize and quantify the periodic variations in WHU precise clock products during the eclipse seasons. A correlation analysis between the PSD and \(\beta\) angle is performed. The PSD of four significant periodic terms from Figure 3  
Figure 3: Residuals (**top**), amplitude spectrum (**middle**), and spectrogram (**bottom**) of C01, C07, and C27, respectively. are extracted and analyzed. The results of BDS-2 GEO satellites C01 and C02, BDS-2 IGSO satellites C06 and C08, BDS-2 MEO satellites C12 and C14, BDS-3 MEO satellites C19 and C21 made by CAST and BDS-3 MEO satellites C25 and C27 made by SECM are shown in Figure 4. The vertical axis on the left represents the PSD of periodic terms, and the vertical axis on the right represents the \(\beta\) angle. When the absolute value of \(\beta\) angle is less than a critical value, the satellite enters the eclipse seasons [44]. For GEO and IGSO satellites, the value is 8.5 deg, while for MEO satellites the value is 14.5 deg [45; 46]. The shaded regions in Figure 4 represent the eclipse seasons. Figure 4: \(Cont.\)  
Figure 4 shows the variations of four significant periodic terms. The temporal variations of PSD can be found for each satellite in eclipse seasons and non-eclipse seasons. For GEO and IGSO satellites, the PSD of four periodic terms vary seasonally. When the eclipse seasons come, the values of PSD are gradually increasing. The 1.86 cpd periodic term in the PSD of MEO satellites varies significantly, and it tracks the eclipse seasons. Other periodic terms show limited correlation with eclipse seasons. The 1 cpd periodic term of GEO and IGSO satellites and 1.86 cpd periodic term of MEO satellites, which are closely related to their orbit period, are significantly affected by the eclipse seasons.