# BDS Satellite Clock Prediction Considering Periodic Variations
## 1 Introduction  
The real-time satellite orbit and clock products are indispensable for real-time precise point positioning (RT-PPP) [1]. The ultra-rapid orbit products with sufficient accuracy from MGEX Analysis Centers (ACs) can be used as real-time orbit correction directly [2]. However, the clock offsets are closely related to the satellite atomic clocks in space which can be influenced by many factors, such as external environmental effects that are not corrected by conventional models [3]. It is difficult to grasp the behavior of the satellite clocks accurately. Therefore, analyzing the characteristics of the clock offsets and improving the accuracy of real-time clock offsets is of great importance to RT-PPP [4, 5]. For BDS satellites, the domestic atomic clocks are equipped to provide the on-board time reference. BDS-2 satellites are equipped with the rubidium (Rb) atomic clocks, while BeiDou-3 satellites are equipped with the new generation Rb atomic clocks and high-quality hydrogen (H) atomic clocks [6, 7]. The BDS real-time clock products can be obtained through the ultra-rapid clock products or the real-time clock corrections. The ultra-rapid clock products are comprised of observation and prediction parts, and the prediction part is available for real-time applications. Limited by the accuracy of clock prediction model,the accuracy of the predicted clock offsets is lower than that of real-time clock corrections calculated based on observation data from global or regional network stations [8]. The real-time clock corrections enable the RT-PPP via internet. Due to a network outage, a disruption in data streaming is inevitable [9], which will interrupt the availability and stability of RT-PPP [10]. In this case, the predicted clock offsets are usually used to maintain the continuity of real-time applications [11]. Therefore, an accurate clock prediction model is of great importance. Currently, some available models, such as a quadratic polynomial model (QPM), spectral analysis model (SAM), grey mode (GM), and autoregressive integrated moving average model (ARIMA) [12; 13; 14; 15], are given for clock prediction. Among them, the QPM is the most basic clock prediction model, modeled with the parameters of the clock offset, clock velocity, and clock drift, and intuitively reflects the physical characteristics of satellite clocks [16; 17]. However, the QPM ignores the periodic noise, which presents in clock offsets [18; 19]. Some studies attribute the periodic noise to the factors such as orbit determination errors, temperature variations, and perturbation errors [20; 21]. The SAM, composed of quadratic polynomial and periodic terms, improves the prediction accuracy of the QPM by compensating the periodic noise, and is widely used to obtain ultra-rapid clock products [22; 23; 24]. To further improve the prediction accuracy of SAM, different numbers of periodic term and different length of fitting period used in SAM are evaluated. The evaluated results show that the optimal number and length vary with the clock type and its in-orbit performance [25]. The remaining nonlinear system errors in SAM residuals are modeled and compensated by the Back Propagation (BP) neural network, and the accuracy of 24 h predicted clock offsets is around 4 ns [26]. For the SAM, the compensation effect of periodic noise directly affects the fitting and prediction accuracy. Therefore, the accurate periodic terms are the prerequisite for the SAM. The FFT is usually used to extract periodic terms from the clock offsets. For good frequency resolution, long-term clock offset series are applied for FFT, such as 60-day, 100-day, and even one year [25; 26; 27]. However, due to space environment changes, temperature variations, and various disturbances, the periodic noise is time-varying [28].