## 2 Theory of GLONASS Ionospheric-Free PPP Models
### Handling Schemes of GLONASS IFCsBs  
Generally, GLONASS IFCs vary with each satellite. If improperly modeled, a part of the IFCs will be absorbed by receiver clocks, and the remaining residual biases will degrade the uncertainty of time transfer. To overcome this problematic issue, three handling schemes of IFCBs are proposed [27].  
1. Neglecting IFCBs. When neglecting IFCBs, a part of IFCBs is absorbed by the parameters of the receiver clock offset. The remaining IFCBs will be reflected in the pseudorange residual. The parameter vector of (4) can be conducted as \[\left\{\begin{array}{l}c\cdot d\bar{t}_{r}=c\cdot dt_{r}+d_{r,IF_{12}}^{s,0} \\ \overline{\textbf{N}}_{r,IF_{12}}^{s}=\lambda_{IF_{12}}^{s}\cdot\left(N_{r,IF_ {12}}^{s}+B_{r,IF_{12}}^{s}-B_{IF_{12}}^{s}\right)+d_{IF_{12}}^{s}-\ \ d_{r,IF_{12}}^{s,0}\\ \textbf{X}=\left[\textbf{x},c\cdot d\bar{t}_{r},Z_{w},\overline{\textbf{N}}_{ IF_{12}}\right]^{T}\end{array}\right.\] (5) with **X** the parameter vector of the model; **x** denotes the vector of the receiver position increments.
2. Separating IFCBs by signal frequencies. We set an IFCB parameter to each GLONASS satellite with the same frequency number. The IFCBs are written as follows \[d_{r,j}^{R}=d_{r,j}^{R,0}+K^{i}\cdot\
abla_{r,j}^{i,R}\] (6) where \(d_{r,j}^{R,0}\) refer to the IFCB for the satellite, which frequency number is 0; \(K^{i}\) is the frequency number (\(-7\)-6); \(\
abla_{r,j}^{R}\) indicates the part of IFCB that depend on the frequency number. Note that the necessary zero-mean condition for the improvements of all relevant parameters are automatically applied. \[\left\{\begin{array}{l}\
abla_{r,IF_{12}}^{s}=\alpha\cdot\
abla_{r,1}^{s}+ \beta\cdot\
abla_{r,2}^{i,s}\\ \sum\limits_{i}^{n}\
abla_{r,IF_{12}}^{i,s}=0\end{array}\right.\] (7) with \(n\) the number of the GLONASS frequency. The parameter vector of (4) is conducted as follows: \[\left\{\begin{array}{l}c\cdot d\bar{t}_{r}=c\cdot dt_{r}+d_{r,IF_{12}}^{R,0} \\ \overline{\textbf{N}}_{r,IF_{12}}^{R}=\lambda_{IF_{12}}^{R}\cdot\left(N_{r,IF_ {12}}^{R}+B_{r,IF_{12}}^{R}-B_{IF_{12}}^{R}\right)+d_{IF_{12}}^{R}-d_{r,IF_{12} }^{R}-K^{i}\
abla_{r,IF_{12}}^{i,R}\\ \textbf{X}=\left[\textbf{x},c\cdot d\bar{t}_{r},Z_{w},\overline{\textbf{N}}_{ r,IF_{12}}^{R},K^{i}\
abla_{r,IF_{12}}^{i,R}\right]^{T}\end{array}\right.\] (8)Separating the IFCBs by GLONASS satellites. The IFCBs are modelled as follows:  
\[d_{r,j}^{R}=d_{r,j}^{R,0}+\Omega_{r,j}^{i,R} \tag{9}\]  
The necessary zero-mean condition for the improvements of all relevant parameters are automatically applied.  
\[\left\{\begin{array}{l}\Omega_{r,I\!F_{12}}^{s}=\alpha\cdot\Omega_{r,1}^{s}+ \beta\cdot\Omega_{r,2}^{i,s}\\ \sum\limits_{k}^{n}\Omega_{r,I\!F_{12}}^{k,s}=0\end{array}\right. \tag{10}\]  
where \(n\) again denotes the number of GLONASS satellites; \(k\) is satellite number. Twenty-four constraints exist per day due to 24 GLONASS satellites in each day of 2017. The condition suggests that the IFCBs are not considered to be dependent on a specific GLONASS frequency number. The parameter vector of (4) is conducted as  
\[\left\{\begin{array}{l}c\cdot d\bar{t}_{r}=c\cdot dt_{r}+d_{r,I\!F_{12}}^{R,0 }\\ \overline{\mathbf{N}}_{r,I\!F_{12}}^{R}=\lambda_{I\!F_{12}}^{R}\cdot\left(N_{r,I\!F_{12}}^{R}+B_{r,I\!F_{12}}^{R}-B_{I\!F_{12}}^{R}\right)+d_{I\!F_{12}}^{R} -d_{r,I\!F_{12}}^{R}-\Omega_{r,I\!F_{12}}^{i,R}\\ \mathbf{X}=\left[\mathbf{x},c\cdot d\bar{t}_{r},Z_{w},\overline{\mathbf{N}}_{ r,I\!F_{12}}^{R},\Omega_{r,I\!F_{12}}^{i,R}\right]^{T}\end{array}\right. \tag{11}\]