## 2 Materials and Methods
### Multi-Frequency GFIF Model  
Based on Equation (1), the GFIF combination, which is commonly used for multi-frequency signal consistency analysis, can be easily accessed for B1I/B3I/B1C and B1I/B3I/B2a triple-frequency signals:  
\[\left\{ \begin{array}{l}C261^{s}_{r}=\rho_{L261}+\frac{f_{1}^{2}}{f_{1}^{2} -f_{2}^{2}}IFCB_{r,L1}-\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}IFCB_{L1}^{s}+ \epsilon^{s}_{r,C261}\\ L261^{s}_{r}=\rho_{L261}+\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}IFCB_{r,L1}-\frac {f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}IFCB_{L1}^{s}+\frac{f_{1}}{f_{L261}}\\ C265^{s}_{r}=\rho_{L265}+\frac{-f_{2}^{s}}{f_{2}^{2}-f_{2}^{2}}IFCB_{r,L5}-\frac {-f_{2}^{s}}{f_{2}^{2}-f_{2}^{2}}IFCB_{L5}^{s}+\epsilon^{s}_{r,C265}\\ L265^{s}_{r}=\rho_{L265}+\frac{-f_{2}^{s}}{f_{2}^{2}-f_{2}^{2}}IFCB_{r,L5}-\frac {-f_{2}^{s}}{f_{2}^{2}-f_{2}^{2}}IFCB_{L5}^{s}+\frac{f_{1}}{f_{L265}}+\epsilon^ {s}_{r,L265}\\ \end{array} \right. \tag{7}\]  
where C\(*^{s}_{r}\) and L\(*^{s}_{r}\) are the GFIF combinations of code and phase observations; \(\rho_{L*}\) is the GFIF combination of geometric distance; \(B_{L*}\) is the GFIF combination of phase ambiguity in meters; \(\epsilon^{s}_{r,C*}\) and \(\epsilon^{s}_{r,L*}\) denote the GFIF combinations of code and phase residuals. The GFIF combinations of geometric distances can be further written as:  
\[\left\{ \begin{array}{l}\rho_{L261}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}} \rho_{L1}+\frac{-f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}\rho_{L6}-\frac{f_{2}^{2}}{f_ {2}^{2}-f_{2}^{2}}\rho_{L2}-\frac{-f_{2}^{2}}{f_{2}^{2}-f_{2}^{2}}\rho_{L6}\\ \rho_{L265}=\frac{f_{2}^{2}}{f_{2}^{2}-f_{2}^{2}}\rho_{L2}+\frac{-f_{2}^{2}}{f_ {2}^{2}-f_{2}^{2}}\rho_{L5}-\frac{f_{2}^{2}}{f_{2}^{2}-f_{2}^{2}}\rho_{L2}- \frac{-f_{2}^{2}}{f_{2}^{2}-f_{2}^{2}}\rho_{L6}\\ \end{array} \right. \tag{8}\]  
Combining Equations (3) and (8), the GFIF combination of geometric distances can be simplified as:  
\[\left\{ \begin{array}{l}\rho_{L261}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}} \left(\epsilon^{s}_{r}R_{SCF}^{ECF}\mathbf{PCO}^{s}_{L1}-\epsilon^{s}_{r}\mathbf{R}_{ ENU}^{ECF}\mathbf{PCO}_{r,L1}\right)+\\ \frac{-f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}\left(\epsilon^{s}_{r}\mathbf{R}_{SCF}^{ECF} \mathbf{PCO}^{s}_{L6}-\epsilon^{s}_{r}\mathbf{R}_{ENU}^{ECF}\mathbf{PCO}_{r,L6}\right)-\\ \frac{f_{2}^{2}}{f_{2}^{2}-f_{2}^{2}}\left(\epsilon^{s}_{r}\mathbf{R}_{SCF}^{ECF} \mathbf{PCO}^{s}_{L2}-\epsilon^{s}_{r}\mathbf{R}_{ENU}^{ECF}\mathbf{PCO}_{r,L2}\right)-\\ \frac{-f_{2}^{2}}{f_{2}^{2}-f_{2}^{2}}\left(\epsilon^{s}_{r}\mathbf{R}_{SCF}^{ECF} \mathbf{PCO}^{s}_{L6}-\epsilon^{s}_{r}\mathbf{R}_{ENU}^{ECF}\mathbf{PCO}_{r,L6}\right)\\ \rho_{L265}=\frac{f_{2}^{2}}{f_{2}^{2}-f_{2}^{2}}\left(\epsilon^{s}_{r}\mathbf{R}_{SCF}^{ECF} \mathbf{PCO}^{s}_{L2}-\epsilon^{s}_{r}\mathbf{R}_{ENU}^{ECF}\mathbf{PCO}_{r,L2}\right)+\\ \frac{-f_{2}^{2}}{f_{2}^{2}-f_{2}^{2}}\left(\epsilon^{s}_{r}\mathbf{R}_{SCF}^{ECF} \mathbf{PCO}^{s}_{L5}-\epsilon^{s}_{r}\mathbf{R}_{ENU}^{ECF}\mathbf{PCO}_{r,L5}\right)-\\ \frac{-f_{2}^{2}}{f_{2}^{2}-f_{2}^{2}}\left(\epsilon^{s}_{r}\mathbf{R}_{SCF}^{ECF} \mathbf{PCO}^{s}_{L2}-\epsilon^{s}_{r}\mathbf{R}_{ENU}^{ECF}\mathbf{PCO}_{r,L2}\right)-\\ \frac{-f_{2}^{2}}{f_{2}^{2}-f_{2}^{2}}\left(\epsilon^{s}_{r}\mathbf{R}_{SCF}^{ECF} \mathbf{PCO}^{s}_{L6}-\epsilon^{s}_{r}\mathbf{R}_{ENU}^{ECF}\mathbf{PCO}_{r,L6}\right)\\ \end{array} \right. \tag{9}\]  
As indicated by Equation (9), only frequency-dependent terms, i.e. PCO corrections, are remaining in GFIF combinations. Within IGS ACs' routine multi-GNSS data processing, antenna-related corrections are commonly extracted from the jigs14.atx file. However, the receiver antenna calibrations are absent in jigs14.atx for BDS-3 signals. As a result, the corresponding GPS L1/L2 values are used for adjacent BDS-3 frequencies. Unlike the situation at the receiver end, the satellite antenna information for BDS-3 B1I/B3I/B1C signals is released in the jigs14.atx file. The missing values for B2a signals can be filled with the corresponding B2b values considering the frequency proximity.