## 2 Proposed Method
### Selection of the Available Reflected Signals
\[\varphi_{p}=sin^{-1}\bigg{[}\frac{sin(\phi)sin(\varphi_{LEO})+sin(\theta-\phi )\sin(\varphi_{GNSS})}{sin(\theta)}\bigg{]} \tag{5}\]  
\[\lambda_{p}=\lambda_{LEO}+sin^{-1}\bigg{[}\frac{sin(\phi)cos(\varphi_{GNSS}) sin(\lambda_{GNSS}-\lambda_{LEO})}{cos(\varphi_{p})sin(\theta)}\bigg{]} \tag{6}\]  
where \(\varphi_{LEO}\) and \(\lambda_{LEO}\) are the geodetic latitude and longitude of the LEO, respectively; \(\varphi_{GNSS}\) and \(\lambda_{GNSS}\) are the geodetic latitude and longitude of the GNSS satellite, respectively; \(\varphi_{p}\) and \(\lambda_{p}\) are the geodetic latitude and longitude of the reflected point, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Satellites System**} & \multirow{2}{*}{**H (km)**} & \multicolumn{2}{c}{**Maximum \(\theta\) (\({}^{\circ}\))**} & \multicolumn{2}{c}{**Minimum \(\theta\) (\({}^{\circ}\))**} \\ \cline{3-6}  & & **L1** & **L2** & **L1** & **L2** \\ \hline GPS & 20,200 & 44.01 & 53.55 & 9.95 & 3.15 \\ GLONASS & 19,100 & 43.55 & 53.03 & 9.83 & 3.12 \\ BDS GEO/IGSO & 35,900 & 47.96 & 57.99 & 11.00 & 3.49 \\ BDS MEO & 21,607 & 44.55 & 54.16 & 10.09 & 3.20 \\ GALILEO & 23,222 & 45.10 & 54.78 & 10.24 & 3.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Altitudes and geocentric angles between the LEO and the different GNSS satellites. After computation of the locations of the specular reflection points, we must remove the specular points that are below the minimum antenna gain. We propose a method based on the use of the azimuths and elevations of the vectors between the reflections and the LEO satellites in the body-fixed coordinate system to select the available reflections. We can obtain the azimuth and elevation ranges at both L1 and L2 using the beamwidths and face directions of the antennae. Based on the geometry shown in Figure 4, the elevation angle range can easily be computed to be \(\left(\alpha-\frac{\beta}{2},\alpha+\frac{\beta}{2}\right)\). However, to obtain the azimuth range, which is the angle between the \(x\)-axis vector and the reflection, the vectors OD and OB in the body-fixed coordinate system shown in Figure 3 must be determined. The following characteristics of OD and OB can help us to solve for the values of their unit vectors. Both vectors OD and OB lie perpendicular to the antenna axis \((0,\ cos28^{\circ},-sin28^{\circ})\) and the angle between this axis and the normal vector of the panel \((0,sin28^{\circ},cos28^{\circ})\) is \(\gamma/2\), where \(\gamma\) is the elevation beamwidth. The unit vectors of OB and OD can be set to \((x,y,z)\). The following three equations are therefore derived:  
\[y\cdot cos\alpha-z\cdot sin\alpha=0\]  
\[y\cdot sin\alpha+z\cdot cos\alpha=cos\frac{\gamma}{2} \tag{7}\]  
\[\sqrt{x^{2}+y^{2}+z^{2}}=1\]  
The solutions for \(x,y,z\) are \(x=\pm sin\frac{\gamma}{2}\), \(y=cos\frac{\gamma}{2}\cdot sin\alpha\) and \(\mathrm{z}=\cos\frac{\gamma}{2}\cdot\cos\alpha\). From the solutions to Equation (7) and the geometry, the azimuths of OB and OD in the body-fixed coordinate system are determined to be \(123.07^{\circ}\) and \(56.93^{\circ}\), respectively, at the L1 frequency, while the corresponding angles are \(51.47^{\circ}\) and \(128.53^{\circ}\), respectively, at the L2 frequency. After the elevation and azimuth ranges are computed, the available specular reflection points can then be selected.