# BDS Satellite Clock Prediction Considering Periodic Variations
## 2 Materials and Methods
### Analysis of Periodic Variations in BDS Satellite Clock Offsets  
The atomic clocks are crucial payloads of BDS satellites and can be easily influenced by environment and temperature [16, 17]. Many researchers have confirmed that the periodic noise is significant in BDS satellite clock offsets, which are probably caused by some factors such as the orbit determination errors, temperature variations, and perturbation errors [20, 21]. The FFT is a traditional method used to identify and extract periodic terms from the clock offsets. To improve the frequency resolution of amplitude spectrum, the length of clock offset series used for FFT is usually long, such as 60-day, 100-day, and even one year [25, 26, 27]. There are no uniform criteria for selecting the length of clock offset series in published literature. The FFT based on different lengths of clock offset series may give different periodic terms, especially the main periodic term. Here, the amplitude spectrums of different lengths of clock offset series are given first. The raw BDS satellite clock offset series used here are WHU precise clock products, from day-of-year (DOY) 001 to 270 of the year 2019 ([[https://cddis.nasa.gov/archive/gnss/products/mgex](https://cddis.nasa.gov/archive/gnss/products/mgex)]([https://cddis.nasa.gov/archive/gnss/products/mgex](https://cddis.nasa.gov/archive/gnss/products/mgex)), accessed on 14 August 2021). Considering that the BDS precise clock products are obtained by Orbit Determination and Time Synchronization (ODTS), some orbit determination errors (particularly for the radial component) are absorbed by the clock offsets [31]. Therefore, the orbit determination strategies are essential for the analysis of precise clock offsets [35]. Table 1 gives crucial information of the precise orbit determination strategies with which the used WHU precise clock products are estimated. Solar radiation pressure (SRP) is the dominant non-gravitational perturbation for satellites [36], which can affect the accuracy of the satellite orbit directly. For WHU orbit strategies, the 5-parameter ECOM1 model with an empirical a priori model is used as BDS SRP model [35, 37]. Figure 1: The flowchart of double MAD detection. To analyze the influence of different FFT lengths on spectral analysis, Geosynchronous Orbit (GEO) satellite C01, Inclined Geosynchronous Satellite Orbit (IGSO) satellite C10, and Medium Earth Orbit (MEO) satellite C25 are chosen randomly as an example. First, daily clock offsets are fitted by QPM after preprocessing. Second, the different length residuals of the QPM aligned day by day are utilized for FFT analysis. The amplitude spectrums of the 90-day, 180-day, and 270-day are shown in Figure 2. The horizontal axis represents frequency with the unit cycles per day (cpd), and the vertical axis represents the amplitude with the unit ns. From Figure 2, the periodic terms in clock offsets are detected at multiple frequencies. For C01 and C10, the 1 cpd periodic term approximately equal to their orbit period of 23.9 h. Apart from four significant periodic terms, the amplitude of other periodic terms decreases with n (n is an integer) increases. The periodic terms with low amplitude are ignored in modeling and prediction. For C25, there are only four significant periodic terms in the amplitude spectrums. Among them, the 1.86 cpd periodic term is approximately equal to MEO orbit period of 12.9 h. It can be found that the BDS satellite clock offset periodic terms closely related to the orbit period. Similar periodic terms have been found in other satellites from the same constellation. As shown in Figure 2, it can be seen that the main periodic term obtained by FFT based on different lengths of clock offset series are distinct. For C01, the periodic term of 1 cpd is the most significant in the amplitude spectrum of 90-day, while 3 cpd in the amplitude spectrum of 180-day and 270-day. For C10, the main periodic term in 90-day amplitude spectrum is 2 cpd, while 1 cpd for the 180-day and 270-day amplitude spectrums.