## 4 Experimental Analysis
### BDS/LEO Combined POD during a Quiet Period of Solar Activity  
The BDS-LEO combined POD was calculated with the five designed LEO constellations. In total, six schemes were tested: (S1) the BDS POD with the regional network shown in Figure 3; (S2) the Walker 2/1/0 LEO constellation added to S1; (S3) the Walker 6/2/1 LEO constellation added to S1; (S4) the Walker 12/3/1 LEO constellation added to S1; and (S6) the Walker 60/6/4 LEO constellation added to S1. The BDS orbital differences between the estimated orbits and the \"reference\" are shown in Figure 6. The Swarm POD residuals during the quiet period of solar activity were calculated and added to the simulated spaceborne observations. Figure 6 indicates that the average orbital accuracies of the BDS GEO satellites for the along, cross, and radial components with the regional network are 4.5 m, 1.1 m, and 0.80 m, respectively, and that the orbital accuracies of the IGSO and MEO satellites for the three components are 1.1 m, 0.61 m, and 0.44 m. When the Walker 12/3/1 LEO constellation is included, the orbital accuracy is considerably improved, especially for the along component of the GEO satellites, which is improved to 0.28 m, while the cross and radial components reach 0.24 m and 0.1 m, respectively. Meanwhile, the orbital accuracies of the IGSO and MEO satellites are improved to 0.23 m, 0.18 m, and 0.09 m for the three components. However, the BDS orbital accuracy is not continuously improved with the increasing number of LEO satellites from 12 to 24; the accuracy with the Walker 60/6/4 constellation is nearly the same as that derived with the Walker 12/3/1 constellation. Figure 5: Elevation-dependent GF pseudo-range and carrier-phase measurement residuals for Swarm-A **(top)**, GRACE-A (**middle**), and Sentinel-3A (**below**). To further understand the LEO satellite enhancement of the BDS orbit determination, the orbital dilution of precision (ODOP) for the BDS was calculated. The calculation of the ODOP is similar to the position dilution of precision (PDOP) for ground stations. The ODOP in this study indicates the geometric strength of the BDS tracked by LEO satellites and ground stations. The better the geometry structure, the smaller the ODOP. The ODOP can be calculated based on the below algorithms:  
\[A=\left[ \begin{array}{cccc}\frac{x_{1}-x^{s}}{\left|\overrightarrow{r}_{1} \right|}&\frac{y_{1}-y^{s}}{\left|\overrightarrow{r}_{1}\right|}&\frac{z_{1}- z^{s}}{\left|\overrightarrow{r}_{1}\right|}&1\\ \vdots&\vdots&\vdots&\vdots\\ \frac{x_{n}-x^{s}}{\left|\overrightarrow{r}_{n}\right|}&\frac{y_{n}-y^{s}}{ \left|\overrightarrow{r}_{n}\right|}&\frac{z_{n}-z^{s}}{\left|\overrightarrow{ r}_{n}\right|}&1\\ \frac{x_{t_{1}}-x^{s}}{\left|\overrightarrow{r}_{L_{1}}\right|}&\frac{y_{t_ {1}}-y^{s}}{\left|\overrightarrow{r}_{L_{1}}\right|}&\frac{z_{t_{1}}-z^{s}}{ \left|\overrightarrow{r}_{L_{1}}\right|}&1\\ \vdots&\vdots&\vdots&\vdots\\ \frac{x_{t_{n}}-x^{s}}{\left|\overrightarrow{r}_{L_{n}}\right|}&\frac{y_{t_ {n}}-y^{s}}{\left|\overrightarrow{r}_{L_{n}}\right|}&\frac{z_{t_{n}}-z^{s}}{ \left|\overrightarrow{r}_{L_{n}}\right|}&1\\ \end{array} \right] \tag{1}\]  
\[Q=\left(A^{T}A\right)^{-1} \tag{2}\]  
\[ODOP=\sqrt{Q(1,1)+Q(2,2)+Q(3,3)} \tag{3}\]  
where \((x_{n},y_{n},z_{n})\), \((x_{Ln},y_{Ln},z_{Ln})\), and \((x^{s},y^{s},z^{s})\) represent the coordinates in three directions of the ground station, LEO satellite, and GNSS satellite, respectively. \(\overrightarrow{r}_{n}=(x_{n}-x^{s},y_{n}-y^{s},z_{n}-z^{s})\) is the geometry distance vector from the ground station to a GNSS satellite and is similar for \(\overrightarrow{r}_{L_{n}}\). Figure 7 shows the ODOP values of the G05/G10/C02/C03/C06/C11 satellites. The ODOP decreases with increasing numbers of LEO satellites; however, the BDS orbital accuracy does not improve accordingly. The average ODOP value for each BDS satellite decreases moderately as the LEO number increases from 12 to 60 and the orbital error approaches a stable value. In addition, the LEO orbit is unknown; only the ground network is fixed.