## 4 Positioning Model  
Quantitative evaluation of the empirical, realistic stochastic model of GNSS observation was made based on the comparison of positioning model solution results that adopted different stochastic parameters. The geometry-based double-differenced code and carrier-phase observations positioning model for multi-frequency and Multi-GNSS systems was used for the tests with the a priori known atmospheric delays (i.e., the ionosphere-fixed troposphere-fixed model). The linearized functional and stochastic formulas of positioning models are given by:  
\[\mathrm{E}\{\mathbf{z}\} = \mathbf{H}\mathbf{x} \tag{12}\] \[\mathrm{D}\{\mathbf{z}\} = \mathbf{V} \tag{13}\]  
where \(\mathbf{z}\) is the observed minus computed DD carrier-phase and the code observation vector, \(\mathbf{H}\) is the design matrix for position and ambiguities and \(\mathbf{x}\) is a vector of unknown real-valued receiver position and integer-valued DD ambiguities. The stochastic model is captured by the dispersion with VC matrix \(\mathbf{V}\) of DD observations. As the functional model used is widely described in the literature (e.g., [53; 85; 86]) it is not defined in detail here. It should only be noted that only the intra-system differences were generated for a multi-system solution without combining observations from different GNSS systems. Figure 21: Cross-correlation coefficient matrix for ZB DD measurements for systems: GPS (**a**), GLOMASS (**b**), Galileo (**c**), BeiDou (**d**). The fully populated VC matrix of the undifferenced measurements for one receiver \(a\) at a single epoch, denoted as \(\mathbf{V}_{0,a}\), is composed from matrices describing the variance \(\mathbf{C}\) and cross-correlation \(\mathbf{\Gamma}\) of observations in the form:  
\[\mathbf{V}_{0,a}= \begin{bmatrix}\mathbf{C}_{f_{1},a}^{2}&\mathbf{\Gamma}_{f_{1},f_{2}}\mathbf{C}_{f_{1},a}\mathbf{C}_{f_{2},a}&\dots&\mathbf{\Gamma}_{f_{1},f_ {m}}\mathbf{C}_{f_{1},a}\mathbf{C}_{f_{m},a}\\ \mathbf{\Gamma}_{f_{1},f_{2}}\mathbf{C}_{f_{1},a}\mathbf{C}_{f_{2},a}&\mathbf{C }_{f_{2},a}^{2}&\dots&\mathbf{\Gamma}_{f_{2},f_{m}}\mathbf{C}_{f_{2},a}\mathbf{ C}_{f_{m},a}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{\Gamma}_{f_{1},f_{m}}\mathbf{C}_{f_{1},a}\mathbf{C}_{f_{m},a}&\mathbf{ \Gamma}_{f_{2},f_{m}}\mathbf{C}_{f_{2},a}\mathbf{C}_{f_{m},a}&\dots&\mathbf{C }_{f_{m},a}^{2}\\ \end{bmatrix}  \tag{14}\]  
where \(f=1,2,\dots,m\) denotes the type of signal/frequency/system. The variance and cross-correlation matrices for \(i=1,2,\dots,k\) satellites are computed using empirical stochastic models of variance (Equation (7)), time-correlation (Equation (9)) and cross-correlation (Equation (11)) parameters:  
\[\mathbf{C}_{f,a} =\left(1-\rho_{t,C_{f}}\right)^{-\frac{1}{2}}\cdot\text{diag} \left(\sigma_{a,C_{f}}^{i=1},\sigma_{a,C_{f}}^{i=2},\dots,\sigma_{a,C_{f}}^{i =k}\right) \tag{15}\] \[\mathbf{\Gamma}_{f_{1},f_{2}} =\rho_{f_{1},f_{2}}\mathbf{I}_{k} \tag{16}\]  
where \(\mathbf{I}_{k}\) denotes identity matrix of size \(k\). In this simplified notation, it was assumed that the same satellites were observed for all signals, but in a general case for a different number of satellites, the appropriate dimensions of the matrices should be considered. The time-correlation coefficient \(\rho_{t}\) in the Equation (15) refers to the correlation between the epochs making up the TD combinations used in the variance model. In the VC matrix for a single epoch, the time-correlation between the observations used in the positioning model cannot be taken into account; this parameter was included in the Kalman Filter for the multi-epoch approach. The final formulation of VC matrix for the DD model for \(a\) and \(b\) receivers is given by:  
\[\mathbf{V}=\mathbf{D}(\mathbf{V}_{0,a}+\mathbf{V}_{0,b})\mathbf{D}^{T} \tag{17}\]  
where \(\mathbf{D}\) is a block diagonal of \(m\) single-differencing matrices [87]. The solution of the positioning model presented by Equations (12) and (13) was employed by the Kalman Filter (KF).