# Improving Short Term Clock Prediction for BDS-2 Real-Time Precise Point Positioning
## 1 Introduction  
The performance of the Global Navigation Satellite System (GNSS) positioning depends on the accuracies of satellite orbit and clock products [1; 2]. Thanks to the great effort of the International GNSS Service, for example, the Multi-GNSS Experiment project, several types of precise orbit and clock products for GNSS satellites including GPS (Global Positioning System), GLONASS, Galileo, BeiDou and more are provided in routine base with different latencies and accuracies to meet various positioning demands [3; 4; 5; 6; 7; 8]. Among these products, the ultra-rapid product is generated for possible real-time applications by predicting orbits and clocks over the next 24 h. However, the precision ofthe predicted clocks cannot satisfy the requirement of real-time precise point positioning (RTPPP) although the orbit uncertainty of about 5-10 cm can be neglected [9]. With the growth of GNSS high precision applications, there are already a number of RTPPP providers, such as GFZ (German Research Centre for Geosciences) and Wuhan University. Unfortunately, not all users can use the correction information due to either cost of the service or the user location is not within the coverage area of the service. Therefore, to improve the accuracy of predicted satellite clocks, especially over short-term, is of considerable interest to make RTPPP reliable in the abovementioned circumstances [10; 11]. Generally, clock prediction is firstly required for generating clock information for broadcast ephemeris. It is an indispensable GNSS function in order to provide standard positioning service and the prediction time could be several hours to a few days varying from system to system. The positioning performance depends heavily on the accuracy of the predicted clocks. Usually, a low-order polynomial model is employed, and the accuracy is about 0.53 ns and 0.64 ns for example for GPS and BeiDou satellite navigation system (BDS), respectively [12]. Higher accuracy is quite challenging for the prediction process, as the intrinsic physical characteristic of clocks is very complex and unstable also due to varying space environments [12; 13; 14]. As a result of these factors, usually clock offsets, including clock bias, frequency bias, and drift, contain strong non-linear and stochastic features which make the short-term precise modeling and prediction rather difficult [13; 15]. Despite such difficulties, numerous studies were carried out, and several models were established for clock prediction, e.g., quadratic polynomial (QP) model, spectrum analysis (SA) model, Kalman filter (KF) and neural network (NN) model [16; 17]. The widely used prediction models include first and second order polynomial models, for example, and the predicted clock in GPS broadcast ephemeris [18; 19] are well studied and discussed [20]. However, it is certificated that the clock error of the quadratic polynomial model is accumulated rapidly with the predicted time increased [16; 21]. Such a result was expected because the stochastic variations contained by satellite clock accumulated over time [20]. To improve the QP model, studies obtained four main periodic variances for GPS satellites based on the fast Fourier transform (FFT) method and analyzed the relationship between its amplitude and orbit period [22]. They show the incorporation of fixed-period harmonics, especially for the GPS satellite clocks, provides a very accurate predicting model. Even though the SA model with periodic terms included is an improvement to quadratic polynomial model, it is pointed out that the periodic function has to be determined reasonably by a rather long clock time series [14; 17]. Meanwhile, [PERSON] et al. achieve an accuracy of 8 to 9 ns by predicting GPS clocks over six hours with a KF approach, but the estimation is not reliable due to occurring frequency jumps [23].