## 3 Results
### LEO-Augmented Four-GNSS  
Similar to four-GNSS constellations, the LEO constellation, serving as a navigation system, should satisfy satellite visibility as well as DOP, which is equally distributed as much as possible on a global scale [39]. In this paper, we simulate six LEO constellation schemes in terms of orbital planes and orbital inclinations to study the distribution of LEO  
Figure 7: DOPs of four-GNSS-NG: (**a**) PDOP; (**b**) TDOP; (**c**) HDOP; (**d**) VDOP; (**e**) NDOP; (**f**) EDOP. satellites. The detailed information of the six Schemes LEO constellations is presented in Table 4. Schemes 1 and 2 LEO constellations adopt polar orbits with altitudes of 1000 km and 600 km, respectively. Schemes 3-5 are designed with an inclined orbit. The altitude of Schemes 3-5 is 1000 km. The inclination angles for Schemes 3-5 are 65\({}^{\circ}\), 60\({}^{\circ}\), and 30\({}^{\circ}\), respectively. For Scheme 6, we designed mixed-orbit LEO constellations and included both the polar orbit and the inclined orbit, and all of the satellites operate in orbit at an altitude of 1000 km. According to the principle of GNSS, availability means that more than four LEO satellites above a specific CEA can be observed on a global scale, for the ground receiver needs at least four satellite signals to determine its position, which can be expressed as follows:  
\[\text{min}(\sum P_{L}(\lambda_{i},\phi_{j}))\geq 4 \tag{17}\]  
According to Equation (17), the results of the minimum number of LEO constellations with global service ability for Schemes 1-5 were calculated, which are listed in Table 5. Because a lower altitude implies narrower coverage for LEO satellites, Schemes 1-3 need at least 126, 220, and 196 LEO satellites, respectively, to ensure global coverage. That is far more than the total number of GNSS satellites. The satellite visibility of LEO constellation Schemes 4 and 5, whose orbit inclinations are smaller, drops rapidly at high latitudes, which results in their coverage being \(|B|\leq 81^{\circ}\) and \(|B|\leq 49^{\circ}\), respectively, with 183 and 202 satellites. According to the number of satellites of each LEO constellation, the observation probabilities of the LEO satellites \(P_{L}(\lambda_{i},\phi_{j})\) for the five schemes are plotted in Figure 8. Due to the increasing number of satellites, the \(P_{L}(\lambda_{i},\phi_{j})\) is much larger than the \(P_{M}(\lambda_{i},\phi_{j})\) of the GNSS. The distribution of polar-orbiting LEO constellations (Schemes 1 and 2) is more even than that of inclined-orbiting LEO constellations at all latitudes. Because a satellite travels north-south over the Earth, and the angular velocity is only in this direction. Comparing to Schemes 3 and 4, the distribution of the Scheme 5 LEO satellites with an inclination angle of 30\({}^{\circ}\) is narrower, which results in higher observation probabilities. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multicolumn{3}{c}{**Orbit Configurations**} \\ \cline{2-4}  & **Orbit Type** & **Angle of Inclination** & **Altitude (km)** \\ \hline
1 & & & 1000 \\
2 & Polar & 90\({}^{\circ}\) & 600 \\ \hline
3 & & 65\({}^{\circ}\) & \\
4 & Inclined & 60\({}^{\circ}\) & 1000 \\
5 & & 30\({}^{\circ}\) & \\ \hline
6 & Polar + Inclined & 90\({}^{\circ}\) + 65\({}^{\circ}\) + 30\({}^{\circ}\) & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Orbital configurations of the designed LEO constellations. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scheme** & **Number** & **Coverage** & \(\kappa\) \\ \hline
1 & 126 & globe & 0.0070 \\
2 & 220 & globe & 0.0133 \\
3 & 196 & globe & 0.0111 \\
4 & 183 & \(|B|\leq 81^{\circ}\) & 0.0104 \\
5 & 202 & \(|B|\leq 49^{\circ}\) & 0.0117 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The minimum number of designed LEO constellations and their coverage areas.