## 3 Methods
### Functional Model
The between-ionospheric slant delays and satellite bias rank deficiencies, as denoted in [2], then occur twice: once for the signals observed by type A receivers and once for type B. The latter additional rank deficiency is eliminated with the additional S-basis restriction on \(d_{\mathrm{GF}}^{s,A}\) (see Table 2). (3) Since receivers can be classified as receiver types A or B, where the tracking of some signals is unique to one receiver type, there is no receiver that observes all existing satellite code biases. Here the pivot receiver, whose clock and biases are used as restrictions in the CC-R S-basis, is assumed to be a receiver of type A. The rank deficiencies caused by the remaining satellitecode biases exclusive to type B receivers, thus not observed by the pivot receiver, are then eliminated with restrictions imposed on the biases of a pivot satellite instead. Each additional rank deficiency requires an additional S-basis restriction. All S-basis restrictions of the CC-R model, including the previously mentioned adaptations, employed on the observation Equation (1) are given in Table 2. Introducing S-basis restrictions into the functional model changes the interpretation of the estimable parameters from their original interpretation. Estimable parameters that differ from their original interpretation in Equation (1) are denoted with a hat 0. The resulting non-rank-deficient observation equations, which are used in this research, are then given with the following:  
\[\mathrm{E}\left\{\Delta\phi_{r,j}^{s,A}\right\}=k_{r}^{s,A}\cdot \Delta x_{r}+m\left(\varepsilon_{r}^{s,A}\right)\cdot T_{r}+d\hat{t}_{r>1}+ \hat{d}_{r>1}^{A>1}-d\hat{t}^{s,A}-\mu_{j}^{A}\cdot\hat{t}_{r}^{s,A}+\lambda_{ j}\cdot\left(\hat{N}_{r>1,j}^{s>1,A}*+\hat{\delta}_{r>1,j}^{A}-\hat{\delta}_{j}^{s,A}\right)\]  
\[\mathrm{\textbf{Code}}\]  
\[\mathrm{\textbf{Receiver Type A or Type B for common signals}}\]  
\[\mathrm{\textbf{Receiver Type B (for GNSS with common signals, for signals exclusive to Type B)}}\]  
\[\mathrm{\textbf{E}\left\{\Delta\rho_{r,j}^{s,A}\right\}=k_{r}^{s,A}\cdot \Delta x_{r}+m\left(\varepsilon_{r}^{s,A}\right)\cdot T_{r}+d\hat{t}_{r>1}+ \hat{d}_{r>1}^{A>1}+\hat{d}_{r>1,j>2}^{A}-d\hat{t}^{s,A}-\hat{d}^{s>1,A}_{j}+ \mu_{j}^{A}\cdot\hat{t}_{r}^{s,A}\]  
When representing the GNSS network as a graph by defining satellites and receivers as nodes and the observations as edges, the presented S-basis restrictions only lead to \begin{table}
\begin{tabular}{c c c} \hline \hline
**S-Basis Restriction** & **Parameter** & **Condition** \\ \hline \multicolumn{3}{c}{CC-R S-basis} \\ Pivot Receiver Clock & \(dt_{r}\) & \(r=1\) \\ Pivot Receiver Code, Phase Biases & \(d_{r,r}^{A},\delta_{r,j}^{A}\) & \(r=1\) \\ Receiver IF Code Bias & \(d_{r,r}^{A}\) & \(r>1\), \(A=1\) \\ Receiver GF Code Bias & \(d_{r,r}^{A,\mathrm{GE}}\) & \(r>1\) \\ Satellite IF Code Bias & \(d_{r,r}^{\mathrm{EH}}\) & For signals exclusive to receiver Type A \\ Satellite GF Code Bias & \(d_{\mathrm{GF}}^{A}\) & For signals exclusive to receiver Type A \\ Pivot Receiver Ambiguities * & \(N_{r,j}^{s,A}\) & \(r=1\) \\ Pivot Satellite Ambiguities * & \(N_{r,j}^{s,A}\) & \(r>1,s=1\) \\ \hline \multicolumn{3}{c}{CC-R S-basis Adaptations} \\ Pivot Satellite Code Bias & \(d_{r,j}^{s,A}\) & \(s=1\) (For satellite code biases not observed by pivot receiver) \\ Satellite GF Code Bias & \(d_{\mathrm{GF}}^{s,A}\) & \(s\
eq 1\) (For GNSS A where no common signal between Receiver Type A and B exists) \\ \hline IF Code Bias \(d_{r,\mathrm{IF}}=\frac{1}{\mu_{2}-\mu_{1}}(\mu_{2}d_{r,1}-\mu_{1}d_{r,2})\), \(d_{\mathrm{IF}}^{\mathrm{IF}}=\frac{1}{\mu_{2}-\mu_{1}}(\mu_{2}d_{r,1}^{\mathrm{ IF}}-\mu_{1}d_{r}^{\mathrm{IF}})\). GF Code Bias \(d_{r,\mathrm{GF}}=\frac{1}{\mu_{2}-\mu_{1}}(d_{r,2}-d_{r,1})\), \(d_{\mathrm{GF}}^{\mathrm{IF}}=\frac{1}{\mu_{2}-\mu_{1}}(d_{r,2}-d_{r,1})\). In practice, this is done with the more flexible Spanning Tree approach, as described later in Section 3.1. \\ \hline \end{tabular}
\end{table}