# Integrated Precise Orbit Determination of Multi-Gnss and Large LEO Constellations
## 4 Result Analysis and Discussion
### Integrated POD with Full LEO Constellation
#### 4.1.2 Integrated POD with Different LEO Orbital Type  
To assess the impact of LEO orbital type, two typical LEO orbits, nearly-polar and sun-synchronous orbits, are adopted in this study. Both of LEO constellations consist of 60 LEO satellites flying at an altitude of 1000 km (see Figure 1a,b). The ground observations from a global network with 65 stations are used. The results are shown in Figure 7. It can be seen that the sun-synchronous-orbiting constellation presents a slightly stronger enhancement to the GNSS orbits than the polar-orbiting constellation. With the sun-synchronous-orbiting data of 60 satellites, the orbit of GNSS satellites can achieve an accuracy similar to that of the 96-polar-orbiting scheme, and the orbit accuracy of BDS GEO for the sun-synchronous scheme is even better than that for the 96-polar-orbiting scheme. As reported by [PERSON] et al. [9], the polar-orbiting constellation outperforms the sun-synchronous- orbiting constellation in terms of the convergence time of PPP. However, the situation is reversed when the LEO constellation is employed to enhance the GNSS precise orbit determination. In order to determine the reason for this, we computed the actual number of LEO satellites used for the GNSS POD and calculated an orbit dilution of precision (ODOP) value for every GNSS satellite using a similar algorithm like the position dilution of precision (PDOP) for ground stations. Assuming one GNSS satellite can be tracked by \(n\) ground stations simultaneously, the ODOP of this GNSS satellite can be calculated as follows: \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Satellites**} & \multicolumn{2}{c}{**120 MGEX**} & \multicolumn{2}{c}{**65 MGEX+60 LEO-P**} & \multicolumn{2}{c}{**65 MGEX+66 LEO-P**} & \multicolumn{2}{c}{**65 MGEX+96 LEO-P**} \\ \cline{2-9}  & **RMS** & **Percentage** & **RMS** & **Percentage** & **RMS** & **Percentage** & **RMS** & **Percentage** \\ \hline
**GPS** & 3.2 & – & 0.8 & 75.0\% & 0.8 & 75.0\% & 0.7 & 78.1\% \\ \hline
**GLONASS** & 3.3 & – & 0.7 & 78.8\% & 0.7 & 78.8\% & 0.7 & 78.8\% \\ \hline
**Galileo** & 4.4 & – & 0.8 & 81.8\% & 0.8 & 81.8\% & 0.7 & 84.1\% \\ \hline
**BDS GEO** & 237.9 & – & 4.9 & 97.9\% & 4.4 & 98.2\% & 3.7 & 98.4\% \\ \hline
**BDS IGSO** & 6.2 & – & 1.8 & 71.0\% & 1.7 & 72.6\% & 1.8 & 71.0\% \\ \hline
**BDS MEO** & 6.0 & – & 0.9 & 85.0\% & 0.9 & 85.0\% & 0.8 & 86.7\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average 3D RMS of GNSS orbit differences and improvement percentages with respect to the ground-based POD using MGEX data (Unit: cm). Figure 7: Average 3D orbit differences RMS of GNSS satellites for the integrated POD with different orbit types of LEO satellites. Values on the top of each bar indicate the corresponding average 3D RMS. P and S denote the nearly-polar and sun-synchronous orbits, respectively. \[H=\left[ \begin{array}{cccc}\frac{x_{1}-x^{s}}{|\vec{r}_{1}|}&\frac{y_{1}-y^{s}}{| \vec{r}_{1}|}&\frac{z_{1}-z^{s}}{|\vec{r}_{1}|}&1\\ \frac{z_{2}-x^{s}}{|\vec{r}_{2}|}&\frac{y_{2}-y^{s}}{|\vec{r}_{2}|}&\frac{z_{2}- z^{s}}{|\vec{r}_{2}|}&1\\ \vdots&\vdots&\vdots&\vdots\\ \frac{x_{2}-x^{s}}{|\vec{r}_{n}|}&\frac{y_{n}-y^{s}}{|\vec{r}_{n}|}&\frac{z_{n }-z^{s}}{|\vec{r}_{n}|}&1\\ \end{array} \right] \tag{8}\]  
\[Q=\left(H^{T}H\right)^{-1} \tag{9}\]  
\[ODOP=\sqrt{Q(1,1)+Q(2,2)+Q(3,3)} \tag{10}\]  
where \((x_{n},y_{n},z_{n})\) and \((x^{s},y^{s},z^{s})\) are the positions of ground station and GNSS satellite respectively. \(\vec{r}_{n}=(x_{n}-x^{s},y_{n}-y^{s},z_{n}-z^{s})\) is the distance vector with the direction from GNSS satellite to ground station. The ODOP value describes the geometric strength of observations for GNSS satellites. The better the geometric strength, the smaller the ODOP value. Figure 8 shows the used LEO numbers and ODOP values for C02 (GEO) and G13 (MEO) respectively. The two 60-LEO schemes contribute an almost comparable number of LEO satellites to the orbit determination for BDS GEO C02.