# Extended Geometry and Probability Model for GNSS+ Constellation Performance Evaluation
## 4 Experimental Results and Analyses
### BDS-3+Heo Constellation  
Although the hybrid constellation is applied in BDS and QZSS to improve the visibility in some specific areas, the visibility in polar regions is still unfavorable. This is because that, for the current GNSS constellations, when the station is at higher latitudes, the elevation of satellites that can be observed is lower. If satellites that can cover high latitude regions are included in the GNSS+ constellation, the geometric distribution for such regions will be improved. The HEO satellites, for example, the Russian Molniya satellites, are the kind of satellites that can fulfill this function [19]. These satellites have synchronous 12 h orbits of 500-40,000 km altitude that are inclined at an angle of 63.4\({}^{\circ}\) to the equator which can not only ensure an exceptional coverage of the Northern Hemisphere but also minimizes the impact of orbital perturbations caused by the earth's oblateness. The HEO \begin{table}
\begin{tabular}{c c} \hline \hline
**Orbit Parameter** & **Nominal Value** \\ \hline Semi-major axis & 42,165 km \\ Eccentricity & 0.075 \\ Inclination & 41\({}^{\circ}\) \\ Argument of perigee & 270\({}^{\circ}\) \\ Center of longitude & 139\({}^{\circ}\) East \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Quasi-Zenith orbit (QZO) parameters [29]. Figure 5: The performance improving rates (%) of BDS-3+QZO relative to the BDS-3 constellation in area of 90\({}^{\circ}\) S–90\({}^{\circ}\) N and 50\({}^{\circ}\) E–170\({}^{\circ}\) E. (**a**) Satellites visibility improving rates of BDS-3+QZO relative to BDS-3; (**b**) GDOP improving rates of BDS-3+QZO relative to BDS-3  
satellite's parameters applied in this study are listed in Table 5 and their ground track is continuously repeated as shown in Figure 6. The HEO in the 3D graphic view is displayed in Figure 7.  
Five HEO satellites and BDS-3 satellites are combined to establish the BDS-3+HEO constellation. The primary constellation parameters are listed in Table 6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \multicolumn{4}{c}{**BDS-3+HEO**} \\ \hline Orbit Type & MEO & IGSO & GEO & HEO \\ Nominal Number & 24 & 3 & 3 & 5 \\ Inclination & 55\({}^{\circ}\) & 55\({}^{\circ}\) & 0\({}^{\circ}\) & 63.4\({}^{\circ}\) \\ Altitude (km) & 21,528 & 35,786 & 35,786 & 500–40,000 \\ Period(s) & 46,404 & 86,170.5 & 86,170.5 & 43,061 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of the BDS-3+HEO constellation. Figure 6: Ground track of highly eccentric orbits (HEO) satellites. Figure 7: HEO in the three-dimensional (3D) graphic view. We evaluate the performance of the BDS-3+HEO constellation on a global scale, as illustrated in Figure 8.  
The detailed improvement rates of introducing HEO satellites relative to BDS-3 at 30\({}^{\circ}\) N-90\({}^{\circ}\) N region are summarized in Table 7. As can be seen from Figure 8 and Table 7, the DOPs and satellite visibility are modified favorably after using 5 HEO satellites. In the 30\({}^{\circ}\) N-90\({}^{\circ}\) N region, the satellite visibility is improved on average by 38.76%. In addition, all kinds of DOPs are improved. By introducing five HEO satellites to BDS-3, on average, values of the GDOP, PDOP, HDOP, VDOP, and TDOP are reduced by 15.70%, 16.65%, 10.35%, 16.65%, and 11.52% respectively. However, the correlation level between station height and receiver clock difference is still higher. As shown in Figure 8h, the correlation level in low latitude and the polar region is slightly higher. What is more, as revealed in Figures 8g and 9, after applying HEO satellites, in some regions such as north polar and low latitude regions, positioning precision in longitude is better than in the latitude component. This may be since IGSO and HEO satellites mainly fly along the north-south direction, which would contribute to the DOP value in the longitude component mainly.