# Extended Geometry and Probability Model for GNSS+ Constellation Performance Evaluation
## 2 Methods
### GEO and IGSO Satellite Observing Probability  
GEO satellites are relatively static to earth in earth-centered earth fixed (ECEF) system. Therefore, in theory, a GEO satellite will stay at the specific point all the time. For instance, the 3 BDS-3 GEO satellites are respectively located at 80\({}^{\circ}\) E, 110.5\({}^{\circ}\) E, and 140\({}^{\circ}\) E [27]. So, in theory, the probability of a GEO satellite appearing at the designed position, such as 80 \({}^{\circ}\) E, is 1, while at other arbitrary positions is 0. Then GEO satellite observing probability can be written as [15]:  
\[P_{(X,Y,Z)}^{GEO}=\left\{\begin{array}{l}1,(X,Y,Z)\in P\\ 0,\text{\emph{other positions}}\end{array}\right.. \tag{5}\]  
For an IGSO satellite, their satellite ground track is a closed curve. For example, the 3 BDS-3 IGSO satellites, their ground tracks are coincident, while the longitude of the intersection point is at 118\({}^{\circ}\) E, with a phase difference of 120\({}^{\circ}\)[28]. We can calculate a satellite position (\(X\), \(Y\), \(Z\)) in the ECEF system when the true anomaly \(f\) turns a specific angle and the set of these points in one earth self-rotation period is called B in this study. For an IGSO satellite, when \(f\) turns over from 0\({}^{\circ}\) to 360\({}^{\circ}\), all coordinates in the ECEF system and corresponding velocity in one earth self-rotation period can be obtained. Considering both the accuracy and computation efficiency, this specific angle can be set at 0.25\({}^{\circ}\)[15]. Then, refer to the idea of previous GAPM, the probability of IGSO satellites appearing at one point of B can be determined by the corresponding velocity. So, the modified probability definition, \(P_{(X,Y,Z)}^{IGSO}\), of IGSO satellites to be observed at one point of B can be written as:  
\[P_{(X,Y,Z)}^{IGSO}=\left\{\begin{array}{l}\frac{k^{IGSO}}{V_{(X,Y,Z)}^{IGSO} },(X,Y,Z)\in\text{B}\\ 0,\text{{other positions}}\end{array}\right., \tag{6}\]  
where \(V_{(X,Y,Z)}^{IGSO}\) is the velocity at one point of B in the ECEF system; \(k^{IGSO}\) is the probability constant for IGSO satellites which can be calculated by the following equation:  
\[k^{IGSO}=\frac{n^{IGSO}}{\sum\limits_{(X,Y,Z)\in\text{B}}\frac{1}{V_{(X,Y,Z)} ^{IGSO}}}, \tag{7}\]  
where \(n^{IGSO}\) is the total number of IGSO satellites with the same ground track. \(V_{(X,Y,Z)}^{IGSO}\) at one point of B can be calculated as follows. Firstly, a satellite position (\(X\), \(Y\), \(Z\)) in ECEF can be computed with the true anomaly \(f\) as the independent variable. Then, the velocity of X, Y, and \(Z\) components: \(V_{X}\), \(V_{Y}\), and \(V_{Z}\) can be written as:  
\[\left\{\begin{array}{l}V_{X}=\frac{X_{t+M}-X_{t}}{\Delta t}\\ V_{Y}=\frac{Y_{t+M}-Y_{t}}{\Delta t}\\ V_{Z}=\frac{Z_{t+M}-Z_{t}}{\Delta t}\end{array}\right., \tag{8}\]  
where \(t\) is one specific epoch; \(X\), \(Y\), and \(Z\) are the ECEF coordinates of a satellite at the epoch \(t\) or \(t+\Delta t\); \(\Delta t\) means the time elapsed from one point to another very adjacent point. Then, \(V_{(X,Y,Z)}\) at one point is:  
\[V_{(X,Y,Z)}^{IGSO}=\sqrt{V_{X}^{2}+V_{Y}^{2}+V_{Z}^{2}}. \tag{9}\]  
We investigate the observing probability of the three BDS-3 IGSO satellites using the modified observing probability definition of IGSO satellites. From Figure 1, it can be seen that the observing probability is larger at the northern and southern ends of the figure-8-shape track which is closer to the actual situation than the corresponding results presented in [15].