# Navigation in GEO, HEO, and Lunar Trajectory Using Multi-GNSS Sidelobe Signals
## 3 Results
### Signal Visibility and GDOP  
The above analysis of \(C/N_{0}\) indicates that abundant signal visibility of GEO and HEO spacecraft can be ensured. The satellite was visible once the nadir angle was greater than the Earth block angle and the \(C/N_{0}\) was beyond the receiver sensitivity. GDOP was an important index to measure the effect of the satellite geometry on the positioning and timing solution [39; 40]. It can be calculated by:  
\[GDOP=\sqrt{tr{(H^{\top}H)}^{-1}} \tag{5}\]  
For multi-GNSS constellations, it was assumed that the measurements of all the satellites were statistically independent. Taking dual-GNSS constellation as an example, \(\alpha\) and \(\beta\) is the satellites from different constellations, the matrix \(H\) is expressed as follows:  
\[H=\left[ \begin{array}{cccc}\frac{\Delta x_{1}^{\alpha}}{\rho_{1}}&\frac{ \Delta y_{1}^{\alpha}}{\rho_{1}}&\frac{\Delta z_{1}^{\alpha}}{\rho_{1}}&1&0\\ \frac{\Delta x_{2}^{\alpha}}{\rho_{2}}&\frac{\Delta y_{2}^{\alpha}}{\rho_{2}} &\frac{\Delta z_{2}^{\alpha}}{\rho_{2}}&1&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \frac{\Delta x_{1}^{\alpha}}{\rho_{m}}&\frac{\Delta y_{m}^{\alpha}}{\rho_{m}} &\frac{\Delta z_{m}^{\alpha}}{\rho_{m}}&1&0\\ \frac{\Delta y_{1}^{\alpha}}{\rho_{1}}&\frac{\Delta y_{1}^{\alpha}}{\rho_{1}} &\frac{\Delta z_{1}^{\alpha}}{\rho_{1}}&0&1\\ \frac{\Delta x_{2}^{\alpha}}{\rho_{2}}&\frac{\Delta y_{2}^{\alpha}}{\rho_{2}} &\frac{\Delta z_{2}^{\beta}}{\rho_{2}}&0&1\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \frac{\Delta x_{m}^{\beta}}{\rho_{m}}&\frac{\Delta y_{m}^{\beta}}{\rho_{m}} &\frac{\Delta z_{m}^{\beta}}{\rho_{m}}&0&1\end{array} \right] \tag{6}\]  
where \(\Delta x_{i}=x_{i}-x,\Delta y_{i}=y_{i}-y,\Delta z_{i}=z_{i}-z\); \((x,y,z)\) is the approximate position of the receiver; \((x_{i},y_{i},z_{i})\) is the position of the \(i\)-th satellite; \(\rho_{i}\) is the distance of the i-th satellite with respect to the receiver. Visible satellites of the GEO spacecraft are shown in Figures 8 and 9. Meanwhile, visible satellites of each GNSS constellation were calculated. For the GEO spacecraft, the mean visible satellites of BDS-3, GPS, Galileo, and GLONASS over 24 h were 25.4/21.3/19.4/19.1, respectively. Satellite visibility of BDS-3 from Figure 9 showed that GEO and IGSO satellites of BDS-3 were constantly visible. This was the reason why visible satellites of BDS-3 were more than that of GPS. Mean visible satellites of GREC combination reached 84.8. GDOP of GEO spacecraft is shown in Figure 10. The mean GDOP of BDS-3 is 3.4. With GPS combined, it improved to 2.6, and four systems yield a mean GDOP of 2.0, which is a rather good satellite geometry. This good satellite geometry was simply benefited from the large number of visible satellites, as we have shown in Figure 8. For GEO spacecraft, the distance between the receiver and the GNSS satellites was not far enough to make serious signal attenuation; therefore, a large number of sidelobe signals could be received, leading to a large number of visible satellites. Visible satellites of HEO spacecraft are shown in Figure 11. Visible satellites around apogee were slightly less than that of perigee, where signals from the same side and different side of the Earth were both received. We noticed that there was a sharp decrease of visible satellites at the lowest altitude of HEO, which was marked as the red circle in Figure 11. This was because the nadir angles of the spacecraft with respect to GNSS satellites were relatively small; therefore, signals from the other side of the Earth were mostly blocked. When the HEO spacecraft moved away from perigee, the altitude of the Earth was increased to 12.55 \(\mu\)m, which was the case for the Earth. Figure 8: Visible satellites of GEO. Figure 9: Satellite visibility of GEO spacecraft. (**a**) Satellite visibility of BDS-3. (**b**) Satellite visibility of GPS. (**c**) Satellite visibility of Galileo. (**d**) Satellite visibility of GLONASS. spacecraft increased and more signals from the other side of the Earth were tracked.