## 2 Materials and Methods
### Satellite Position Inversion  
In accordance with the principles of SAR geometric positioning, the ground point coordinates corresponding to an image point are distributed along concentric circles with a fixed satellite-to-target distance, \(R\)[30], as shown in Figure 1. The discrepancies in the ground coordinates derived from the same image point are primarily influenced by the initial elevation value. By adjusting the initial elevation in the forward RPC model, multiple ground point coordinates located on a specific concentric circle can be obtainedunder various initial elevation assumptions. Based on the mathematical and geometric relationships among these ground points, the radar LOS vector at the imaging moment for the image point can be derived. By calculating the LOS vectors at the leftmost, center, and rightmost positions of the middle row of the image and applying the autonomous intersection method for LOS vectors, the satellite's position can be further determined, as illustrated in Figure 2.  
The explicit steps are as follows:  
1. Construct the RPC model of the image using the RPC parameter file provided with the spaceborne SAR imagery. Iteratively solve the ground coordinates \(P_{g}\) corresponding to the central image point \(m_{0}(r_{0},c_{0})\) by performing a forward transformation of the model using the least squares method. The initial elevation value \(h_{0}\) for the ground point is obtained as an approximate height from the SRTM data in the vicinity of the ground point's location. 2. By varying the initial elevation in the forward transformation of the RPC model, calculate the coordinates of different ground points on a specific concentric circle under various initial elevation values. Set two new initial elevation values \(h_{0}^{\prime}\) and \(h_{0}^{\prime\prime}\), where \(h_{0}^{\prime}=h_{0}+\tau\) (with \(\tau\) approaching 0) and \(h_{0}^{\prime\prime}=h_{0}+v\) (where \(v\) is generally within 1 km). Then, following the procedure in step 1, calculate the corresponding ground coordinates \(P_{g}^{\prime}\) and \(P_{g}^{\prime\prime}\) for these two new initial elevations. 3. To simplify the calculation, standardize the coordinate system. The Earth-Centered, Earth-Fixed (ECEF) coordinates, denoted as \(P_{0}\), \(P_{0}^{\prime}\) and \(P_{0}^{\prime}\), correspond to the geodetic coordinates \(P_{g}\), \(P_{g}^{\prime}\), and \(P_{g}^{\prime\prime}\). The transformation process employs the WGS-84 ellipsoid parameters with the following formulas: \[ \begin{bmatrix}X\\ Y\\ Z\end{bmatrix} = \begin{bmatrix}(N+H)cosB\ cosL\\ (N+H)cosB\ sinL\\ (N(1-e^{2})+H)sinB\end{bmatrix} \] (5)  
Figure 1: Radar LOS vector inversion. Figure 2: Satellite position inversion. In the above formulas, \((X,Y,Z)\) represent the geocentric rectangular coordinates; \(N\) is the radius of curvature in prime vertical, \(N=\frac{a}{\sqrt{1-e^{2}\sin^{2}B}}\); \(a\) is the semimajor axis of ellipsoid \(e\) is the first eccentricity, \(e=\sqrt{\frac{a^{2}-b^{2}}{a^{2}}}\); and \(b\) is the semiminor axis of the ellipsoid. 4. As shown in Figure 1, based on the mathematical and geometric relationships, \(P_{0}P_{0}^{\prime}\) can be approximated as the tangent vector at the point \(P_{0}\), and \(P_{0}P_{0}^{\prime\prime}\) represents the chord of the circle passing through the point \(P_{0}\). Furthermore, by taking the cross product of the normal vector with the tangent vector at the point \(P_{0}\), the resulting vector points from \(P_{0}\) toward the center of the circle, which is the radar LOS vector. Therefore, the calculation formula for the radar LOS vector at the imaging moment of the central point \(m_{0}\) in the SAR image is as follows: \[\alpha=(\overrightarrow{P_{0}^{\prime}P_{1}^{\prime}}\times\overrightarrow{P _{0}^{\prime}P_{2}^{\prime}})\times\overrightarrow{P_{0}^{\prime}P_{1}^{\prime}}\] (6)
5.