## 3 Methods
### Functional Model  
In the case of Code Division Multiple Access (CDMA), the linearized phase and code observations for a receiver \(r\), satellite \(s\) of a GNSS \(A\), and signal \(j\) at epoch \(i\) can be formulated according to, e.g., [17] with  
\[ \begin{split}\mathrm{E}\Big{(}\Delta\rho_{r,j}^{s,A}(i)\Big{)}={k _{r}^{s,A}}^{T}(i)\cdot\Delta x_{r}(i)+m\Big{(}\varepsilon_{r}^{s,A}\Big{)} \cdot Z_{w}(i)+\mathrm{d}t_{r}(i)+\lambda_{j}\cdot\Big{(}N_{r,j}^{s,A}+\delta_{ r,j}^{A}(i)-\delta_{j}^{s,A}(i)\Big{)}-\mathrm{d}t^{s,A}(i)-\mu_{j}^{A}\cdot{l_{r}^{s,A} }(i)\\ \mathrm{E}\Big{(}\Delta\rho_{r,j}^{s,A}(i)\Big{)}={k_{r}^{s,A}}^{T }(i)\cdot\Delta x_{r}(i)+m\Big{(}\varepsilon_{r}^{s,A}\Big{)}\cdot Z_{w}(i)+ \mathrm{d}t_{r}(i)+d_{r,j}^{A}(i)-\mathrm{d}t^{s,A}(i)-d_{j}^{s,A}(i)+\mu_{j}^ {A}\cdot{l_{r}^{s,A}}(i)\end{split}  \tag{1}\]  
Here, \(\mathrm{E}(\cdot)\) denotes the expected value of the inherently stochastic observations. Note that it is assumed that the observations have been corrected apriori with appropriate models (see, e.g., [18]); thus, the corrected effects are excluded. Furthermore, the observation equations have been modified to include the notation of the GNSS. It is assumed that different GNSS and signals have different receiver codes and phase biases, while the receiver clock is assumed to be common for each GNSS. The observation equations have been linearized with known satellite positions and approximate receiver coordinates. In (1), \(k_{r}^{s,A}\) denotes the line-of-sight vector from receiver \(r\) to satellite \(s\), \(\Delta x_{r}\) denotes the coordinate residuals of receiver \(r\) to the linearization point, \(Z_{w}\) is the wet tropospheric zenith delay, and \(m\) is the wet mapping function converting zenith delay to slant delay with satellite elevation \(\varepsilon_{r}^{s,A}\). The receiver and satellite clock offsets are denoted by \(\mathrm{d}t_{r}\) and \(\mathrm{d}t^{s,A}\), respectively, receiver / satellite code biases are denoted with \(d_{r,j}/d_{j}^{s,A}\), and phase biases with \(\delta_{r,j}/\delta_{j}^{s,A}\). Ambiguities are denoted with \(N_{r,j}^{s,A}\), with the important property \(N_{r,j}^{s,A}\in\mathbb{Z}\), and, finally, the ionospheric slant delays are denoted with \(l_{r}^{s,A}\). The resulting design matrix of the linear system of equations is rank-deficient when formulated regarding all parameters on the right-hand side of the equation. Solving a rank-deficient system of equations can be done by applying the S-System theory [19], which can be understood as imposing restrictions of the form \(\big{(}S^{\perp}\big{)}^{T}\cdot x=0\) on the parameter vector \(x\), eliminating the underlying rank deficiencies in the functional model. The restrictions imposed on the parameter vector that eliminate all rank deficiencies in the functional model are denoted as the S-basis. The rank deficiencies in the design matrix of the GNSS network equations have been explicitly identified in [2] for the single GNSS case. Furthermore, an in-depth description of the CC-R S-basis, which will be used hereafter, is given in [2]. However, the following three cases require adaptations to be made for the multi-GNSS case with different receiver types: (1) For each additional GNSS introduced, one rank deficiency does not occur that otherwise occurs in the single GNSS case. More specifically, this is the rank deficiency between the receiver clock and the receiver biases, as denoted in [2], since this rank deficiency has already been eliminated by the restrictions imposed on the biases of the first GNSS. As a result, one inter-system bias (ISB) has to be estimated for each additional GNSS. (2) An additional rank deficiency occurs for Galileo satellites when different satellite code biases for the same signal, but different tracking modes, are assumed. The reason is that available receivers currently either obtain the observation in pilot-tracking mode or combined-tracking mode. The receivers can then be classified as type A and type B receivers (see Table 1) with no overlapping signals.