# Modeling DGNSS Pseudo-Range Correction Messages by Utilizing Satellite Repeat Time
## 4 PRC Modeling
### Accuracy Analysis  
PRC modeling was performed by applying sixth-degree polynomial curve fitting, and satellite-dependent coefficients were derived. The accuracy analysis for PRC modeling was performed by comparing predicted PRC values at all epochs for one day during the experiment date with the corresponding real PRC data obtained from the reference station. Estimates were obtained using the coefficients derived for each satellite and the mathematical equations described in [26]. Figure 5 shows means and standard deviations of differences between real and modeled PRC values for every GPS and BDS satellite in our analysis. Except for a couple of cases, every GPS satellite has a bias error less than \(\pm\)1.0 m and a standard deviation within 1.5 m. Overall, the average bias and standard deviation of GPS modeling is \(-\)0.4 m and 1.0 m, respectively (Figure 5a). BDS also shows a similar level of accuracy, such as a \(-\)0.3 m bias and a 1.4 m standard deviation, on average. However, the PRN-C05 GEO satellite, which was always located at an elevation angle of around 9.4\({}^{\circ}\), showed a 1.7 m bias and a 2.7 m standard deviation. It is interesting to find that standard deviations of MEO satellites from PRN C11 through PRN C14 are relatively high, at 1.7 m (Figure 5b). Those MEO satellites make the same orbit in seven days, so changing tropospheric and ionospheric conditions must have impacted modeling accuracy. The day of modeling, 23 April 2016, was a quiet day with the Kp-indices of global geomagnetic activity was smaller than 3.0 except for 21-24 UTC (Kp = 4.7). On 2 April, when the Kp-index was as high as 5.0, PRC residuals for individual satellite are within the range of \(\pm\)1.0 m on average and the standard deviation was less than 1.5 m. Thus,  
Figure 4: Mean RMSE values of all satellites after polynomial curve fitting to GPS and BeiDou. The left vertical bars refer to RMSE, and the symbols connected by dotted lines refer to improvement ratios in RMSE compared to those corresponding to the polynomial one degree lower. The improvement ratio was the highest when the sixth-degree polynomial was used. no significant differences were observed for two dates with different levels of ionospheric activity. However, in order to evaluate the performance of modelling in general, further experiments are needed for the extremely high Kp-index. After modeling with sixth-order polynomial curve fitting, differences between observed and predicted PRC values were found to be relatively large in the low elevation-angle range. Figure 6 shows discrepancies between observed and predicted values as functions of the elevation angle. GPS residuals are shown in Figure 6a, and the differences usually locate within \(\pm\)2.0 m when the elevation angle is 20\({}^{\circ}\) or larger. However, some satellites showed deviations larger than \(\pm\)4.0 m when elevation angles were less than 20\({}^{\circ}\). Figure 6b depicts PRC differences for BDS according to orbit type. The red \(\times\), blue \(\bullet\), and green * in the figure indicate GEO, IGSO, and MEO, respectively. Overall, patterns of residuals are similar to those of the GPS. However, PRC values for GEO were different depending on the date and time, even though GEO satellites had a nearly constant elevation angle, depending on the location along the equator. As can be seen from Figure 6b, the vertical distribution of difference appears at a nearly constant elevation angle. Meanwhile, IGSO and MEO satellites showed differences within \(\pm\)2.0 m, but some satellites deviated more than \(\pm\)4.0 m in the region where elevation angle is less than 20\({}^{\circ}\). Figure 6c,d shows the mean (bar) and the standard deviation (error bar) of the absolute value of the residuals at 5\({}^{\circ}\) intervals of elevation angle. Statistically, the residuals were less than 2.0 m all segments except for GPS at less than 15\({}^{\circ}\) and BDS at less than 20\({}^{\circ}\).