## 2 Methods
### Satellite Visibility and DOPs  
The original pseudo-range observation equations can be generally described as follows [34]:  
\[\mathbf{v}=\mathbf{A}\mathbf{x}+\mathbf{e}_{n}dt_{r}+\mathbf{\iota} \tag{10}\]  
where \(\mathbf{v}\) denotes the residuals, \(\mathbf{A}\) is the design matrix for the coordinate unknowns \(\mathbf{x}\); \(dt_{r}\) is the receiver clock error, while \(\mathbf{e}_{n}\) is the \(n\)-column vector with all elements being 1, and \(\mathbf{\iota}\) is the observed minus computed (OMC) difference, where the observed part is the pseudo-range observations and where the calculated part is the geometric distance between the satellite and the receiver [35]. Applying the least-squares (LS) criterion to solve Equation (10) yields the coefficient matrix \(\mathbf{G}\) of the normal equations as:  
\[\mathbf{G}=\mathbf{H}^{T}\mathbf{H}=\left[ \begin{array}{cccc}\sum\limits_{s=1}^{n}\left( \frac{\Delta\mathbf{v}_{0}^{s}}{P_{0}^{s}}\right)^{2}&\sum\limits_{s=1}^{n}\frac{ \Delta\mathbf{v}_{0}^{s}\Delta\mathbf{v}_{0}^{s}}{(\phi_{0}^{s})^{2}}&\sum\limits_{s=1 }^{n}\frac{\Delta\mathbf{v}_{0}^{s}\Delta\mathbf{v}_{0}^{s}}{(\phi_{0}^{s})^{2}}&\sum \limits_{s=1}^{n}\frac{\Delta\mathbf{v}_{0}^{s}}{P_{0}^{s}}\\ &\sum\limits_{s=1}^{n}\left(\frac{\Delta\mathbf{v}_{0}^{s}}{P_{0}^{s}}\right)^{2} &\sum\limits_{s=1}^{n}\frac{\Delta\mathbf{v}_{0}^{s}\Delta\mathbf{v}_{0}^{s}}{(\phi_{ 0}^{s})^{2}}&\sum\limits_{s=1}^{n}\frac{\Delta\mathbf{v}_{0}^{s}}{P_{0}^{s}}\\ &&\sum\limits_{s=1}^{n}\left(\frac{\Delta\mathbf{v}_{0}^{s}}{P_{0}^{s}}\right)^{2} &\sum\limits_{s=1}^{n}\frac{\Delta\mathbf{v}_{0}^{s}}{P_{0}^{s}}\\ &&n\end{array} \right] \tag{11}\]  
where \(n\) is the number of visible satellites above the cutoff elevation angle (CEA); \(\mathbf{H}=\left[ \begin{array}{cccc}\mathbf{A}&\mathbf{e}_{n}&\end{array} \right]\) is the geometric observation matrix of \(n\times 4\); \(\rho_{0}^{s}\) is the distance between the station's approximate coordinates \((X_{0},Y_{0},Z_{0})\) and the satellite coordinates \((x^{s},y^{s},z^{s})\); and \(\Delta x^{s}=X_{0}-x^{s}\), \(\Delta y^{s}=Y_{0}-y^{s}\), \(\Delta z^{s}=Z_{0}-z^{s}\). By dividing the whole sky into \(1^{\circ}\times 1^{\circ}\) grids, each element of the \(\mathbf{G}\) can be approximated through the following equation:  
\[ \begin{split}\sum\limits_{s=1}^{n}\left(\frac{\Delta\mathbf{v}_{0}^{s} }{\rho_{0}^{s}}\right)^{2}=\sum\limits_{\lambda_{i}=1}^{360}\sum\limits_{\phi= -89}^{90}P_{M}(\lambda_{i},\phi_{j})(\frac{\Delta\mathbf{v}_{0}^{s}}{\rho_{0}^{s} })^{2}+\sum\limits_{\lambda_{i}=1}^{360}\sum\limits_{\phi=-89}^{90}P_{L}( \lambda_{i},\phi_{j})(\frac{\Delta\mathbf{v}_{0}^{s}}{\rho_{0}^{s}})^{2}+\sum \limits_{\lambda_{i}=1}^{360}\sum\limits_{\phi=-89}^{90}P_{I}(\lambda_{i},\phi _{j})(\frac{\Delta\mathbf{v}_{0}^{s}}{\rho_{0}^{s}})^{2}\\ +\sum\limits_{\lambda_{i}=1}^{360}\sum\limits_{\phi=-89}^{90}P_{Q}(\lambda_{i},\phi_{j})(\frac{\Delta\mathbf{v}_{0}^{s}}{\rho_{0}^{s}})^{2}+\sum\limits_{\lambda_ {i}=1}^{360}\sum\limits_{\phi=-89}^{90}P_{G}(\lambda_{i},\phi_{j})(\frac{\Delta \mathbf{v}_{0}^{s}}{\rho_{0}^{s}})^{2}\\ \end{split}  \tag{12}\]  
and the number of visible satellites \(n\) can be estimated as:  
\[ \begin{split} n=\sum\limits_{\lambda_{i}=1}^{360}\sum\limits_{\phi= -89}^{90}P_{M}(\lambda_{i},\phi_{j})+\sum\limits_{\lambda_{i}=1}^{360}\sum \limits_{\phi=-89}^{90}P_{L}(\lambda_{i},\phi_{j})+\sum\limits_{\lambda_{i}=1}^ {360}\sum\limits_{\phi=-89}^{90}P_{I}(\lambda_{i},\phi_{j})\\ +\sum\limits_{\lambda_{i}=1}^{360}\sum\limits_{\phi=-89}^{90}P_{Q}( \lambda_{i},\phi_{j})+\sum\limits_{\lambda_{i}=1}^{360}\sum\limits_{\phi=-89}^{9 0}P_{G}(\lambda_{i},\phi_{j})\end{split}  \tag{13}\]  
The covariance matrix \(\mathbf{Q}\) can be calculated as follows:  
\[\mathbf{Q}=\mathbf{G}^{-1} \tag{14}\]  
According to the variance and covariance propagation law, \(\mathbf{Q}\) can be translated into \(\mathbf{Q}_{\text{UENT}}\), which represents the covariance matrix expressed in the topocentric coordinate system [31].