# Extended Geometry and Probability Model for GNSS+ Constellation Performance Evaluation
## 2 Methods
### Calculation of DOPs and Satellite Visibility  
A satellite is visible when it is above a specific cutoff elevation angle (CEA). In this paper, to ensure higher accuracy, we apply the 5\({}^{\circ}\) CEA for the following DOP and satellite visibility calculations [15]. We calculated the probability for each satellite that the satellite elevation angle is higher than CEA for each grid element. Then the visible satellite number \(N\) can be written as:  
\[ \begin{array}{l}N=\sum\limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P^{MEO}_{ ij}+\sum\limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P^{LEO}_{ij}+\sum \limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P^{HEO}_{ij}+\\ \sum\limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P^{IGSO}_{ij}+\sum\limits_{i=-8 9}^{90}\sum\limits_{j=-179}^{180}P^{GEO}_{ij}+\sum\limits_{i=-89}^{90}\sum \limits_{j=-179}^{180}P^{QZO}_{ij}\end{array} . \tag{23}\]  
The observation equation of a satellite can be written as [27]:  
\[v=\left(e_{x}e_{y}e_{z}1\right)\left( \begin{array}{c}\delta X\\ \delta Y\\ \delta Z\\ \delta t\end{array} \right)-l, \tag{24}\]  
where \(e_{x}\), \(e_{y}\), and \(e_{z}\) are the unit vectors of station-to-satellite vector; \(v\) is the residual; \(c\delta t\) the receiver clock correction; \(l\) is the vector of observations; \(\delta X\), \(\delta Y\), and \(\delta Z\) are the corrections to a station's approximate coordinates. In EGAPM, observation equations can be established for each grid or point higher than specific CEA and corresponding probability is regarded as its weight. Then, using the  
Figure 2: Distribution of observing probability for three Quasi-Zenith orbit (QZO) satellites. above information, for an arbitrary station on the earth ellipsoid, the coefficient matrix of the normal equations can be written as [16]:  
\[Nbb=\left( \begin{array}{ccccc}\sum\limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P _{ij}e_{x}^{2}|_{D}&\sum\limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P_{ij}e_{ x}e_{y}|_{D}&\sum\limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P_{ij}e_{x}e_{z}|_{D}& \sum\limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P_{ij}e_{x}|_{D}\\ &&\sum\limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P_{ij}e_{y}^{2}|_{D}&\sum \limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P_{ij}e_{y}e_{z}|_{D}&\sum \limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P_{ij}e_{y}|_{D}\\ &&&&\sum\limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P_{ij}e_{x}^{2}|_{D}& \sum\limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P_{ij}e_{z}|_{D}\\ \text{symmetry}&&&&\sum\limits_{i=-89}^{90}\sum\limits_{j=-179}^{180}P_{ij}|_{ D}\\ \end{array} \right) \tag{25}\]  
where \(D\) is the visible area above CEA. The inverse of the normal equation in Equation (25) gives the variance/covariance matrix \(Q\):  
\[Q=Nbb^{-1}. \tag{26}\]  
Then, the geometric dilution of precision, GDOP, by definition, is the square root of the trace of the variance/covariance matrix \(Q\). Cartesian coordinates can be translated into the station's topocentric up (U), north (N), and east (E) coordinate system using the following equations:  
\[\left( \begin{array}{c}\delta X\\ \delta Y\\ \delta Z\\ c\delta t\end{array} \right)=\left( \begin{array}{ccccc}\frac{\partial\left( \begin{array}{ccccc}X&Y&Z\\ \hline U&E&N\end{array}\right)}{\partial\left(\begin{array}{ccccc}U&E&N\\ \hline\end{array}\right)}&0\\ &0&&1\end{array} \right)\left( \begin{array}{c}\delta U\\ \delta E\\ \delta N\\ c\delta t\end{array} \right)=S\left( \begin{array}{c}\delta L\\ \delta E\\ \delta N\\ c\delta t\end{array} \right) \tag{27}\]  
\[\frac{\partial\left( \begin{array}{ccccc}X&Y&Z\\ \hline U&E&N\end{array} \right)}{\partial\left(\begin{array}{ccccc}U&E&N\\ \end{  
So, by definition, the DOPs can be written as:  
\[\left\{