## 3 Method
### Reduced-Dimension MUSIC Algorithm Based on Ephemeris Information  
To estimate the elevation and azimuth angles of the signal source in each assumed position, this section proposes utilizing the characteristic of perfect reflection, where the incident angle equals the reflection angle. As shown in Figure 4a, if the reflecting surface is vertical, the satellite elevation angle at the receiver is the same as the satellite elevation angle at the reflecting point, due to the satellite's distance from both the reflecting point and the receiver. According to the law of reflection [22], the incident angle equals the reflection angle, which is the elevation angle at the reflection point. On the other hand, from the receiver's azimuth perspective, the reflection surface is parallel to the direction of travel, and the azimuth angle of the reflection point relative to the direction of travel is equal and opposite to the azimuth angle of the satellite relative to \begin{table}
\begin{tabular}{l c c} \hline \hline \multirow{2}{*}{**Complexity**} & \multicolumn{2}{c}{**Number of Operations**} \\ \cline{2-3}  & **Angle Search Step 1\({}^{\circ}\)** & **Angle Search Step 0.1\({}^{\circ}\)** \\ \hline Number of Operations for & & \\ Spectrum Model Establishment & \(2.06\times 10^{5}\) & \(2.06\times 10^{5}\) \\ Number of Operations for Search & \(3.96\times 10^{12}\) & \(3.96\times 10^{16}\) \\ Total Number of Operations & \(3.96\times 10^{12}\) & \(3.96\times 10^{16}\) \\ Computation Time & \(0.7914\) s & \(95.0814\) s \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Complexity of the MUSIC algorithm under different angle search steps. the direction of travel. As shown in Figure 3(b), without prior information, such as 3D maps, it is not possible to quickly infer the azimuth angle using ephemeris data, so the MUSIC algorithm must still be employed to obtain the azimuth angle. \[\theta^{\prime}=\arctan\biggl{(}\frac{D\sin(\theta)}{D\cos(\theta)+2d}\biggr{)} \tag{28}\]  
Assume that the distance between the satellite and the user is \(d=500\) km and that the elevation angle between the user and the satellite is \(\theta=15^{\circ}\). The reflecting surface is located 100 m in front of the user. Moreover, at this time, \(\theta^{\prime}=14.9941^{\circ}\), resulting in an angular error below 0.002\({}^{\circ}\). When the elevation angle is \(\theta=60^{\circ}\), \(\theta^{\prime}=59.9802^{\circ}\), and the angular error remains below 0.002\({}^{\circ}\). Therefore, the user can directly determine the elevation angle information of one reflected signal using satellite ephemeris. In turn, the geometric relationship between the direct and reflected waves, along with the a priori information, is utilized to continue the dimensionality reduction process and reduce the computational complexity of the algorithm. After obtaining \(K\) elevation angle values, it is straightforward to derive \(K\) azimuth angles. This is achieved by substituting the elevation angle estimates into the spectral function and performing another one-dimensional search, thereby completing the task. As shown in Tables 6 and 7 and Figure 5, reducing the angular search step size significantly enhances the algorithm's angular accuracy. When the search step is reduced from \(1^{\circ}\) to \(0.1^{\circ}\), the average error decreases markedly, particularly in mid-range angles such as \(125^{\circ}\) and \(185^{\circ}\). However, further increasing the precision to 0.01\({}^{\circ}\) results in diminishing returns, offering only marginal improvements. It is also worth noting that the algorithm performs variably across different azimuths, with smaller errors observed near \(0^{\circ}\) and \(360^{\circ}\) (e.g., \(5^{\circ}\) and \(305^{\circ}\)), while larger errors occur in the intermediate angular ranges. Figure 4: View to demonstrate the angular relationship between the direct and reflected signal. [PERSON] and blue line represent reflect and direct paths, respectively.