## 5 Conclusions  
The inconsistency of the precise satellite clock estimated by different frequencies is defined as the IFCB. Unfortunately, the traditional phase time-variant OSB model has high  
Figure 15: Boxplot of the positioning accuracy for GPS L1 + L2 + L5, BDS-3 BII + B3I + B2a, BDS-3 B1I + B3I + B1C + B2a, and Galileo E1 + E5a + E5b + E5 + E6 multi-frequency PPP models.  
noise and low accuracy and is affected by the receiver time-variant biases. The multi-frequency PPP needs to correct both pseudorange OSB and phase time-variant mixed OSB, which makes the client algorithms more complex. For traditional methods, two bias corrections are required to achieve multi-frequency PPP, which makes the client algorithms more complex. To overcome those flaws and simplify the bias correction process of multi-frequency PPP, the definition of pseudorange and carrier-phase time-variant mixed OSB and a new undifferenced model coupled with satellite clock offsets are presented.  
The new undifferenced mixed OSB model coupled with satellite clock offsets avoids the negative impact of the receiver time-variant biases on the satellite mixed OSB estimation. For instance, the JAVAD TRE-3 receiver OSB for BDS-3 B2a and Galileo E6 exhibits significant periodic variations, which lead to periodic bias in satellite mixed OSB for the ED GFIF model. The STD improvement of the undifferenced OSB is 7.5-60.9%, and the RMS can be improved by 9.4-66.1%. Similarly, the STD improvement of the ED OSB can be improved by 50.0-87.5%, and the RMS can be improved by 60.0-88.9%. Hence, the new undifferenced mixed OSB with characteristics of low noise level and high accuracy is more suitable for MFPP.  
With the mixed OSB correction, the GPS and BDS-3 MFPP performance is obviously improved. With the undifferenced mixed OSB correction, the mean convergence time of GPS, BDS-3, and Galileo PPP can be reduced by several minutes, and the positioning accuracy can be improved by 11.5-20.0% for GPS L1 + L2 + L5 from (0.61, 0.75, 1.30) cm to (0.54, 0.60, 1.11) cm, 15.5-20.5% for BDS-3 B1I + B3I + B2a from (0.75, 0.88, 1.55) cm to (0.61, 0.70, 1.31) cm, 8.3-14.2% for BDS-3 B1I + B3I + B1C + B2a from (0.51, 0.70, 1.20) cm to (0.44, 0.60, 1.10) cm, and 9.0-11.7% for Galileo E1 + E5a + E5b + E5 + E6 from (0.43, 0.68, 1.11) cm to (0.38, 0.60, 1.01) cm compared with the traditional IFCB, respectively.  
The GNSS pseudorange and phase time-variant mixed OSB concept and the new undifferenced model coupled with satellite clock offsets demonstrate that it is reasonable and beneficial for the GNSS field.  
Conceptualization, G.J. and K.S.; methodology, G.J.; software, G.J.; validation, G.J., K.S. and M.F.; formal analysis, G.J.; investigation, G.J.; resources, K.S.; data curation, K.S.; writing--original draft preparation, G.J.; writing--review and editing, M.F.; visualization, Y.Y.; supervision, K.S.; project administration, G.J. and H.H; funding acquisition, K.S. and H.H. All authors have read and agreed to the published version of the manuscript.  
The work is supported by Key Laboratory Fund (No 614201004012209).  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors gratefully acknowledge WHU for providing precise orbit and clock products ([[ftp://igs.gnswhu.cn/pub/](ftp://igs.gnswhu.cn/pub/)]([ftp://igs.gnswhu.cn/pub/](ftp://igs.gnswhu.cn/pub/)) (accessed on 1 October 2024)). Many thanks to the IGS MGEX for providing the observation data ([[ftp://igs.gnswhu.cn/pub/](ftp://igs.gnswhu.cn/pub/)]([ftp://igs.gnswhu.cn/pub/](ftp://igs.gnswhu.cn/pub/)) (accessed on 1 October 2024)).  
The authors declare no conflicts of interest.