## 4 Results
### The Advantages of the Mixed OSB in MFPPP  
To evaluate the benefits of the mixed OSB on MFPPP, we carried out GPS L1 + L2 + L5, BDS-3 B1I + B3I + B2a, BDS-3 B1I + B3I + B1C + B2a, and Galileo E1 + E5a + E5b + E5 + E6 multi-frequency PPP. First, the PPP results and phase residuals were given. Then, the statistical PPP results of all GNSS stations were statistically analyzed in positioning accuracy and initial convergence time. The observation data for 100 selected stations in a month are used to evaluate GPS, BDS-3, and Galileo MFPPP performance. It is worth noting that the PPP solutions without OSB are the result of only correcting the DCB. Figure 12 depicts the positioning error of the GPS, BDS-3, and Galileo MFPPP. It can be clearly found that GPS, BDS-3, and Galileo MFPPP can be improved in positioning accuracy and convergence time by correcting the satellite pseudorange and phase time-variant mixed OSB. Furthermore, the OSB estimated by Scheme 2 is superior to that of Scheme 1 in stability, accuracy, and noise level, and its PPP performance is relatively more stable and better than that of Scheme 1 in positioning accuracy and initial convergence. Figure 12: Positioning error of the GPS L1 + L2 + L5, BDS-3 B1I + B3I + B2a, BDS-3 B1I + B3I + B1C + B2a, and Galileo E1 + E5a + E5b + E5 + E6 multi-frequency PPP models on DOY 011, 2021. Figure 13 illustrates the GPS, BDS-3, and Galileo MFPPP phase residuals, where different colors identify different satellites. The carrier-phase residuals are analyzed for the 24 h result. Significant systematic errors exist in the GPS, BDS-3, and Galileo MFPPP results without OSB correction. The RMS values of the carrier-phase residuals are obviously reduced with OSB corrections. Figure 14 depicts the boxplot of the convergence time for GPS L1 + L2 + L5, BDS-3 B1I + B3I + B3I + B3I + B1C + B2a, and Galileo E1 + E5a + E5b + E5 + E6 multi-frequency PPP. Moreover, Figure 15 indicates the corresponding boxplot of the positioning accuracy in the north, east, and up components. The corresponding median and mean values are also depicted in the figures. The convergence epoch is defined as the 3D positioning errors kept within 10 cm from the current epoch to the next 20 epochs. The multi-frequency PPP positioning performance in the convergence time and positioning accuracy is significantly improved with the mixed OSB correction. Specifically, the PPP performance of Scheme 2 is superior to Scheme 1 due to the low noise and high accuracy of OSB obtained from Scheme 2. For instance, the GPS L1 + L2 + L5 PPP mean convergence time is reduced from 29.40 min to 24.20 min and then to 20.30 min, the BDS-3 B1I + B3I + B2a PPP mean convergence time is reduced from 34.7 min to 30.50 min and then to 27.30 min, the BDS-3 B1I + B3I + B1C + B2a PPP mean convergence time is reduced from 29.5 min to 25.50 min and then to 21.40 min, and the Galileo E1 + E5a + E5b + E5 + E6 PPP mean convergence time is reduced from 20.90 min to 17.50 min and then to 12.30 min, respectively. The GPS L1 + L2 + L5 PPP positioning accuracy is improved from (0.78, 0.95, 1.53) cm to (0.61, 0.75, 1.30) cm and then to (0.54, 0.60, 1.11) cm in the north, east, and up components. The BDS-3 B1I + B3I + B2a triple-frequency PPP positioning accuracy is improved from (0.83, 1.03, 1.93) cm to (0.75, 0.88, 1.55) cm and then to (0.61, 0.70, 1.31) cm in the north, east, and up components. The BDS-3 B1I + B3I + B1C + B2a PPP positioning accuracy is improved from (0.68 0.94, 1.33) cm to (0.51, 0.70, 1.20) cm and then to (0.44, 0.60, 1.10) cm in the north, east, and up components. The Galileo E1 + E5a + E5b + E5 + E6 PPP positioning accuracy is improved from (0.53 0.93, 1.21) cm to (0.43, 0.68, 1.11) cm and then to (0.38, 0.60, 1.01) cm in the north, east, and up components. In summary, the new undifferenced mixed OSB model coupled with satellite clock offsets avoids the negative impact of the receiver time-variant biases on the satellite OSB estimation.