## 2 Materials and Methods  
The raw GNSS code and phase observation equations can be expressed as:  
\[\left\{\begin{array}{ll}Ci_{r}^{s}=\rho_{Li}+&(cdt_{r}+d_{r,Ci})-\left(cdt ^{s}+d_{Ci}^{s}\right)+\frac{f_{L}^{2}}{f_{L}^{2}}I_{r,Lj}^{s}+m_{r}^{s}zwd_{r }+\epsilon_{r,Ci}^{s}\\ L_{r}^{s}=\rho_{Li}+&(cdt_{r}+d_{r,Li})-\left(cdt^{s}+d_{Li}^{s}\right)-\frac {f_{L}^{2}}{f_{L}^{2}}I_{r,Lj}^{s}+m_{r}^{s}zwd_{r}\\ &+\lambda_{i}w_{r}^{s}-\lambda_{i}N_{r,Li}^{s}+\epsilon_{r,Li}^{s}\end{array}\right. \tag{1}\]  
where \(C_{r}^{s}\) and \(L_{r}^{s}\) denote the raw code and phase observations in meters, which were corrected for satellite and receiver PCVs (Phase Center Variations), dry part of the tropospheric delay and relativistic effects; \(\rho_{Li}\) denotes the geometric distance between receiver and satellite antenna phase centers; \(cdt_{r}\) and \(cdt^{s}\) denote the receiver and satellite clock offsets; \(dt_{r,Ci}\) and \(dt_{r,Li}\) denote the code and phase hardware delays on the receiver end; \(d_{Ci}^{s}\) and \(d_{LI}^{s}\) denote the code and phase hardware delays on the satellite end; \(f_{i}\) is the phase frequency; \(I_{r,Lj}^{s}\) is the 1 st order ionospheric delay on \(Lj\) frequency; \(m_{r}^{s}\) is the map function of wet tropospheric delay; \(zwd_{r}\) is the zenith wet tropospheric delay; \(\lambda_{i}\) is the phase wavelength in meters; \(w_{r}^{s}\) denotes the phase wind-up effect in cycles; \(N_{r,Li}^{s}\) denotes the phase ambiguity in cycles; \(\epsilon_{r,Ci}^{s}\) and \(\epsilon_{r,Li}^{s}\) denote the code and phase residuals.  
The geometric distance between receiver and satellite antenna phase centers can be expressed as:  
\[\rho_{Li}=\ |(\mathbf{r}^{s}+d\mathbf{r}_{PCO}^{s})-(\mathbf{r}_{r}+d\mathbf{r}_{r,PCO}+d\mathbf{r}_{tides }+d\mathbf{r}_{ARP})| \tag{2}\]  
where \(\mathbf{r}^{s}\) and \(\mathbf{r}_{r}\) denote the coordinates of satellite CoM (Center of Mass) and station MM (Monument Marker) in ECF (Earth-Centered-Fixed) frame; \(d\mathbf{r}_{PCO}^{s}\) and \(d\mathbf{r}_{r,PCO}\) denote the satellite and receiver antenna PCO (Phase Center Offset) corrections; \(d\mathbf{r}_{tides}\) is the station tidal displacements, including solid Earth tides, ocean loading and pole tides; \(d\mathbf{r}_{ARP}\) is the station ARP (Antenna Reference Point) correction.  
Among the coordinate correction terms in Equation (2), only the PCO correction term is frequency-dependent. Additionally, while the station tidal displacements and ARP correction can usually be accurately modeled, the PCO values from either ground calibration or network estimation may be subjected to various errors. As the PCO values are commonly defined in the SCF (Satellite-Centered-Fixed) frame and ENU (East-North-Up) frame for satellite and receiver respectively, Equation (2) can be further linearized as:  
\[\rho_{Li}=\ |\mathbf{r}^{s}-(\mathbf{r}_{r}+d\mathbf{r}_{tides}+d\mathbf{r}_{ARP})|+\mathbf{e}_{r}^{s} \mathbf{R}_{SCF}^{ECF}\mathbf{PCO}_{Li}^{s}-\mathbf{e}_{r}^{s}\mathbf{R}_{ENU}^{ECF}\mathbf{PCO}_{r,Li} \tag{3}\]  
where \(\mathbf{e}_{r}^{s}\) is the line-of-sight unit vector from receiver to satellite; \(\mathbf{R}_{SCF}^{ECF}\) is the rotation matrix from SCF frame to ECF frame; \(\mathbf{R}_{ENU}^{ECF}\) is the rotation matrix from the ENU frame to the ECF frame.