# An Improved Predicted Model for BDS Ultra-Rapid Satellite Clock Offsets
## 1 Introduction  
The precision of real-time satellite clock products has a direct impact on the capability of the Global Navigation Satellite System (GNSS) Position, Navigation, and Timing (PNT) service [1; 2; 3; 4; 5; 6; 7]. In the case of multi-GNSS PNT, the performances of Chinese BDS satellite clocks are no match for that of GPS and Galileo. However, the accuracy and stability of BDS clock products should be paid more attention [8; 9; 10; 11]. The current BDS real-time clock products include three categories. The first is the real-time broadcast ephemeris, which broadcasts a set of satellite clock coefficients per hour, the user receives the coefficients, and can obtain real-time clock offsets after the quadratic polynomial fitting and forecasting. The precision of BDS broadcast ephemeris clock products is lower than six nanoseconds (ns). The second category are the Beidou ultra-rapid clock offset products, which are based on 24-h clock offset observations for real-time forecasting of 24-h clock offsets. It is updated every six hours. The accuracy of the ultra-rapid products is better than that of the broadcast ephemeris, which is ca. 2\(\sim\)6 ns [12; 13; 14; 15; 16]. However, the ultra-rapid clock offset product still cannot meet the requirements of the real-time navigation and positioning with decimeter precision. The last category is the real-time corrections product of the satellite clock offset, which uses real-time observation flows of local or wide-area BDS stations to estimate the satellite's clock offset in real-time, the interval of the estimated clock offset is generally 5-10 s, and the broadcast cycle is under five minutes. The precision of the estimated clock offset is better than 0.5 ns [17; 18; 19; 20]. However, the real-time clock product relies on high-performance processing equipment and additional stations, and it is also unstable and cannot be obtained by the common user. Generally, as one of the core products of International GNSS Service (IGS), the ultra-rapid clock offset product is an important product in decimeter-level positioning services. At present, there are only two institutes, the German GBM and Chinese ISC, which provide open BDS ultra-rapid clock products. Due to the BDS constellation consisting of Geosynchronous Orbit (GEO), Inclined Geosynchronous Satellite Orbit (IGSO), and Medium Earth Orbit (MEO) satellites, compared with the MEO satellite clocks, the GEO and IGSO satellite clocks have different periodic noises, and all BDS satellite clocks contain evident nonlinear system errors (these errors are discussed in Sections 3 and 4). The ultra-rapid models of GPS, GLONASS, and Galileo, which include stable MEO satellite clocks, are not suitable for BDS [11; 21; 22]. The BDS models of GBM and ISC are similar to that of GPS by a quadratic polynomial plus one period for each satellite [23; 24; 22; 25]. It is necessary to improve and build a more accurate BDS ultra-rapid clock prediction model. In the previous research of real-time clock prediction, GPS clock offset predictions in the IGS ultra-rapid products were modeled by a quadratic polynomial plus one period [14]. However, the forecast result is not significantly improved compared with the broadcast ephemeris accuracy [15]. [PERSON] used the improved model to predict the IGS ultra-rapid clock offset of GPS, but the model does not consider the nonlinear colored noise [9; 26]. [PERSON] et al. proposed a wavelet neural network model based on the single difference sequence to predict GPS clock offsets, but this model does not consider the periodic errors and the frequency shift errors [27]. In order to improve the precision of real-time clock offset prediction, the satellite clock noise type, period characteristics, frequency stability, and the physical model have been researched [9; 10; 13; 15]. However, the previous research was mainly concentrated on the stable GPS clocks rather than the BDS clocks. In the BDS clock offset series, there exist some classified periodic errors and some evident nonlinear colored noise.