## 4 Discussion  
The proposed RPC model that incorporates atmospheric delay is shown to effectively improve the uncontrolled positioning accuracy of the spaceborne SAR imagery under ephemeris-free conditions. The vector-autonomous intersection method accurately inverts the SAR satellite position, serving as an effective substitute for traditional ephemeris data. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Uncorrected**} & \multicolumn{4}{c}{**Corrected System Error**} & \multicolumn{4}{c}{**Corrected Atmospheric Delay Error**} & \multicolumn{4}{c}{**Accuracy Improvement Rate**} \\ \cline{2-13} \multicolumn{13}{c}{**Image**} & \multicolumn{1}{c}{**Image**} & \multicolumn{1}{c}{**Image 2-D**} & \multicolumn{1}{c}{**Sample**} & \multicolumn{1}{c}{**Line**} & \multicolumn{1}{c}{**2-D**} & \multicolumn{1}{c}{**Range**} & \multicolumn{1}{c}{**Azimuth**} & \multicolumn{1}{c}{**2-D**} & \multicolumn{1}{c}{**Sample**} & \multicolumn{1}{c}{**Line**} & \multicolumn{1}{c}{**2-D**} & \multicolumn{1}{c}{**Range**} & \multicolumn{1}{c}{**Azimuth**} & \multicolumn{1}{c}{**2-D**} & \multicolumn{1}{c}{**Range**} & \multicolumn{1}{c}{**Azimuth**} & \multicolumn{1}{c}{**2-D**} & \multicolumn{1}{c}{**Range**} & \multicolumn{1}{c}{**Azimuth**} & \multicolumn{1}{c}{**2-D**} & \multicolumn{1}{c}{**Image-Space**} & \multicolumn{1}{c}{**Object-Space**} \\ \cline{2-13} \multicolumn{13}{c}{**ID**} & **Sample** & **Line** & **2-D** & **Sample** & **Line** & **2-D** & **Range** & **Azimuth** & **2-D** & **Sample** & **Line** & **2-D** & **Range** & **Azimuth** & **2-D** & **Image-Space** & **Object-Space** \\ \hline gf2 & 14.26 & 1.71 & 14.36 & 2.08 & 1.06 & 2.34 & 8.11 &  
Moreover, the RPC model reconstruction method effectively corrects the atmospheric delay errors in the SAR imagery. The satellite position errors obtained through the vector-autonomous intersection method have an average value of approximately 20 m, with slant-range errors less than 0.1 m. Considering the satellite's orbital altitude of several hundred kilometers, these errors represent a negligible fraction, indicating that the method achieves a high level of accuracy in position inversion. The comparative analysis reveals that the atmospheric delay values obtained using the vector-autonomous intersection method are nearly identical to those derived from the ephemeris data interpolation method, with a difference of only 0.0001 m. Additionally, variations in atmospheric delays were observed between the imagery from different study areas, likely due to the differences in temporal spans or geographic locations. Within the same study area, atmospheric delays also differed among images, which may be attributed to differences in the ascending and descending orbit times. Notably, atmospheric delays during descending orbits were found to be potentially smaller than those during ascending orbits. Meanwhile, the atmospheric delay values within the same image group are nearly identical, indicating that small-scale geographic variations have minimal impact on atmospheric correction under similar imaging conditions. This finding confirms that delay deviations derived from atmospheric stratification calculations are negligible. Consequently, it can be inferred that atmospheric delays in forward-looking and backward-looking images within long-strip imagery are consistent. Therefore, the proposed method can be extended to atmospheric correction applications for long-strip imagery. After atmospheric correction, the uncontrolled positioning accuracy of GF-3 imagery improved significantly, with image-space planar accuracy better than 3 pixels and object-space planar accuracy reaching 12 m.  
This study effectively addresses the challenge of atmospheric correction for satellite imagery in the absence of satellite and ephemeris parameters. Similarly to the findings of [PERSON] et al. [53], the pixel positioning accuracy is better than 3 m, but their approach relies on satellite parameters.