# An Improved Predicted Model for BDS Ultra-Rapid Satellite Clock Offsets
## 6 Discussion  
The real-time prediction precision of on-board satellite atomic clocks is essential for GNSS navigation, positioning, and timing services. At present, few studies have assessed the BDS ultra-rapid clock offset model considering residual characteristics. We analyze the periodicity of the BDS satellite clock offsets and the characteristics of the fitting residual noises. In the end, an adaptive quadratic polynomial prediction model plus neural network and period items is constructed. The orbital duration and on-board statistics of BDS satellites are given (Table 1). The periodic terms of satellites of the on-board clocks are matched using the FFT method (Table 2). The fitting residuals of the BDS clock offsets with quadratic polynomial and periodic terms are shown in Figures 6 and 9. The proposed model improves the forecasting accuracy compared to that of ISU and GBU products (Figures 11-13 and Table 3). As shown in Figures 10 and 11, the accuracies of GBU-P and ISU-P are slightly worse than that of Scheme 1. The improved model with a BP neural network (Table 3) is evidently superior to that of other schemes. For C14 in the GBU product and C06 in the ISU product, their predicting residuals enlarge rapidly with the increases in the length of the predicting times. However, the accuracies of C14 and C06 have been enhanced using the improved model, and their residuals are effectively controlled in the 24 h. Compared with the results of Table 2, the prediction accuracies of Table 3 have more advantages in long-term predictions more than in 3 h.  
In order to further verify the stability and the precision of the proposed model in the long time scale, 30 days of real-time clock predictions are calculated. Based on the ultra-rapid observations of satellite clocks, the real-time predicted precisions of 3 h, 6 h, 12 h, and 24 h in every day are evaluated and compared with that of precise clock observations. The statistics of the average prediction results are calculated and the corresponding RMS are listed in Table 3.  
As shown in Table 3 and Figure 13, the accuracies of ISU-P products are evidently superior to that of GBU-P products, because the ISU clock offset products are integration products of more than ten iGMAS analysis centers in China, which have greater robustness compared to ISU-P products. The average improvements using the proposed model in 3 h, 6 h, 12 h, and 24 h are 23.1%, 21.3%, 20.2%, and 19.8%, respectively. Meanwhile the accuracy improvements of the proposed model are 9.9%, 13.9%, 17.3%, and 21.2%, respectively, compared to GBU products. As shown in Figures 15 and 16 and Table 4, compared with the results of the IGU-P and GBU-P clock product, the RTPPP precisions based on the proposed clock model have improved about 16%, 14%, and 38% in N, E, and U directions, respectively. quadratic polynomial model. The simulation results show that the performance and precision using the improved model are better than that of ISU products and GBU products. Some conclusions can be summarized as follows:  
1. Compared to ISU-P products, the average improvements using the proposed model in 3 h, 6 h, 12 h, and 24 h are 23.1%, 21.3%, 20.2%, and 19.8%, respectively. Meanwhile the accuracy improvements of the proposed model are 9.9%, 13.9%, 17.3%, and 21.2% compared to GBU-P products. 2. The analyzed results of periodic terms show that the main significant periods of GEO and IGSO are 24 h and 12 h, while that of MEO are 12 h and 6 h. Specially, C11 has a significant periodic term of about 1.3 h, which is different from other MEOs. The first and second periodic terms both are evident and should be compensated in the BDS clock offsets prediction. 3. The fitting residuals show that, the accuracies of the old BDS satellites clocks are worse than that of other young satellites. Therefore, evidently, the performance of BDS satellite clocks may degrade during the increase in service years. 4.