## 2 Theory of GLONASS Ionospheric-Free PPP Models
### GLONASS PPP Observation Model  
The GLONASS undifferenced pseudorange \(P\) and carrier phase \(L\) observations can generally be written as [13]  
\[\left\{ \begin{array}{l}P_{r,j}^{s}=\rho_{r}^{s}+c\cdot dt_{r}-c\cdot dt^{s}+ MF_{w}(e)\cdot Z_{w}+I_{r,j}^{s}+\left(d_{r,j}^{s}-d_{j}^{s}\right)+\epsilon_{r,j}^{ s}(P)\\ L_{r,j}^{s}=\lambda_{j}^{s}\cdot\phi_{r,j}^{s}\\ =\rho_{r}^{s}+c\cdot dt_{r}-c\cdot dt^{s}+MF_{w}(e)\cdot Z_{w}-I_{r,j}^{s}+ \lambda_{j}^{s}\cdot\left(N_{r,j}^{s}+B_{r,j}^{s}-B_{j}^{s}\right)+\epsilon_{r,j}^{s}(\phi)\end{array} \right. \tag{1}\]  
where the superscript \(s\) refers to the GLONASS satellites ID; the subscripts \(r\) and \(j\) (\(j=1\), \(2\)) denote the receiver and frequency band, respectively; \(L\) and \(P\) are the carrier phase and pseudorange observations, respectively; \(\lambda_{j}^{s}\) is the carrier wavelength on the frequency band \(f_{j}\); \(\rho\) is the geometric distance; \(c\) is the speed of light; \(dt_{r}\) and \(dt^{s}\) denote the clock error of the receiver and satellite, respectively; \(e\) refer to the elevation angle of the satellite \(s\); \(MF_{w}\) is the wet mapping function; \(Z_{w}\) is the zenith wet delay; \(I_{r,j}^{s}\) is the slant ionospheric delay on the frequency; \(d_{j}^{s}\) and \(d_{r,j}^{s}\) refer to the uncalibrated code delay (UCD) at satellite and receiver end, respectively; \(B_{r,j}^{s}\) and \(B_{j}^{s}\) represent the uncalibrated phase delay (UPD) at receiver and satellite end, respectively; \(\epsilon_{r,j}^{s}(P)\) and \(\epsilon_{r,j}^{s}(\phi)\) represent the observation noise; \(N\) donates the integer phase ambiguity. In our study, the dual-frequency ionosphere-free observations are employed to weaken the effect of the first-order ionospheric delay and can be written as  
\[\left\{ \begin{array}{l}P_{IF,r}^{s}=\frac{\ell_{1}^{2}\cdot P_{1}^{s}-\ell_ {2}^{2}\cdot P_{2}^{s}}{f_{1}^{s}-\ell_{2}^{2}}\\ L_{IF,r}^{s}=\frac{\ell_{1}^{2}\cdot L_{1}^{s}-\ell_{2}^{2}\cdot L_{2}^{s}}{f_{ 1}^{s}-\ell_{2}^{2}}\end{array} \right. \tag{2}\]  
where \(P_{IF}\) and \(L_{IF}\) refer to the ionosphere-free pseudorange and carrier-phase observations, respectively. \(P_{i}\) and \(L_{i}\) are pseudorange and carrier-phase observations on the \(i\)th (\(i=1,2\)) frequency. Since the precise \begin{table}
\begin{tabular}{c c c} \hline
**ACs** & **Scheme** & **Source of Handling Scheme** \\ \hline COD & Estimate IFCB for each GLONASS satellite & Prange, Orlac [26] \\ \hline ESA & Estimate IFCB for each GLONASS satellite & [[ftp://igs.org/pub/center/analysis/esa.ac](ftp://igs.org/pub/center/analysis/esa.ac)]([ftp://igs.org/pub/center/analysis/esa.ac](ftp://igs.org/pub/center/analysis/esa.ac)). Line 406 \\ \hline GFZ & Estimate IFCB for each GLONASS & [[ftp://igs.org/pub/resource/pubs/2011](ftp://igs.org/pub/resource/pubs/2011)]([ftp://igs.org/pub/resource/pubs/2011](ftp://igs.org/pub/resource/pubs/2011))\_techreport.pdf. \\  & frequency number & Page 61, line 21 \\ \hline \end{tabular}
\end{table} 
Table 1: Global Navigation Satellite System (GLONASS) inter-frequency code bias (IFCB) handling schemes of three International GNSS Service (IGS) analysis centers (ACs) in GLONASS satellite clock offset estimation. satellite clock products are derived from the \(L_{1}/L_{2}\) ionospheric-free observations. The ionospheric-free UCD in pseudorange observations will be absorbed by the precise satellite clock (\(c\cdot d_{IF_{12}}^{s}\)), which reads  
\[\left\{ \begin{array}{l}\alpha=\frac{f_{2}^{2}}{f_{2}^{2}-f_{2}^{2}}\\ \beta=-\frac{f_{2}^{2}}{f_{2}^{2}-f_{2}^{2}}\\ c\cdot d_{IF_{12}}^{s}=c\cdot d_{IF_{12}}^{s}\\ =c\cdot d_{IF_{12}}^{s}\end{array} \right. \tag{3}\]  
where \(\alpha\) and \(\beta\) are frequency-dependent factors (\(m\), \(n=1,2\); \(m\
eq n\)). Substituting (2) and (3) into (1), the GLONASS observation model can be expressed as  
\[\left\{