# A BDGIM-Based Phase-Smoothed Pseudorange Algorithm for BDS-3 High-Precision Time Transfer
## 4 Results and Discussion
### Evaluation of the BDGIM  
The GIM product is used as a reference standard because it is updated every hour and has an inner precision of about 2-8 TECU Total Electron Content Unit (TECU), high accuracy, and continuity. GIM is the most representative global ionospheric VTEC grid product, with stable accuracy and reliability, currently released internationally, and it has high reliability for evaluating the broadcast ionosphere model [29; 30]. The VTEC over the tracking station can be obtained by interpolation with the GIM product. The VTEC difference (dVTEC) between the broadcast ephemeris and the GIM is calculated. The root mean square (\(RMS\)) error and correction rate (\(CR\)) are employed to evaluate the ionospheric model, and they can be written as  
\[RMS=\sqrt{\frac{1}{N}\sum_{k=1}^{N}\left(VTEC_{model}^{k}-VTEC_{GIM}^{k}\right) ^{2}} \tag{16}\]  
\[CR=\frac{1}{N}\sum_{k=1}^{N}\left(1-\frac{\left|VTEC_{model}^{k}-VTEC_{GIM}^{k} \right|}{VTEC_{GIM}^{k}}\right) \tag{17}\]  
where \(N\) denotes the number of epochs involved in the calculation; \(VTEC_{model}^{k}\) is the VTEC calculated with the broadcast ionosphere model; and \(VTEC_{GIM}^{k}\) is the VTEC value calculated from the GIM product.  
Figure 3 shows the time sequence of the dVTEC values of the BDGIM and Klobuchar models relative to the GIM product, which reflects the trend of the accuracy of the different models over time. We have three findings here. First, the BDGIM has the smallest error, and its dVTEC series has the smallest fluctuation range, fluctuating around the zero value, with a peak-to-peak dVTEC of about 10 TECU. The dVTEC series of the Klobuchar model has a relatively larger fluctuation range, with a peak-to-peak dVTEC of approximately 20 TECU. Second, the broadcast ionospheric error varies among the stations. For the BDGIM, the ionospheric errors are basically the same for each station. For the Klobuchar model, the ionospheric model errors vary greatly from station to station. Among all the stations, the ionospheric error of the BRUX station varies very drastically, which may be caused by insufficient BDS-3 observation data [19], while the ionospheric error of the STR1 station varies relatively steadily. This further indicates that the BDGIM outperforms the Klobuchar model. Third, the dVTEC values of both the BDGIM and Klobuchar models are characterized by a daily cyclic variation, which is because the strong sunlight intensity during the day increases the electron content, and the weak sunlight intensity at night results in lower electron content.  
For better analysis and comparison, we calculated the RMS and CR values of the BDGIM and Klobuchar models with the GIM product as a benchmark, as displayed in Figures 4 and 5, respectively. The statistical results indicate that the RMSs of the BDGIM are smaller than those of the Klobuchar model. The RMSs of the BDGIM are all within 4 TECU, with a mean value of 2.6 TECU, while the RMS values of the Klobuchar model are between 4 and 5 TECU, with a mean value of 4.3 TECU. For the ionospheric correction rate, theBDGIM has a better ionospheric correction rate. The CR values of the BDGIM are greater than 60% for all stations, with an average CR of 75.5%. The CRs of the Klobuchar model range from 55% to 75%, with a mean value of 68.2%. Interestingly, we found that the CR values for both the BDGIM and Klobuchar models are greater in the northern hemisphere than those in the southern hemisphere [31]. In summary, the BDGIM outperforms the Klobuchar model. Therefore, the BDGIM was used to conduct time transfer experiments in order to verify our proposed method.  
Figure 4: RMS errors of dVTEC between the broadcast ionosphere model and GIM.  
Figure 3: The dVTEC of BDGIM and Klobuchar relative to GIM.