## 3 Results
### Results and Analysis
#### 3.2.2 Atmospheric Stratification Delay  
To assess the feasibility of the proposed satellite position inversion method as an alternative, atmospheric delay values were calculated and compared using satellite positions derived from both the autonomous LOS vector intersection method and the ephemeris data interpolation method. Atmospheric delay is influenced not only by satellite position and slant-range errors but also by the propagation path of electromagnetic waves. Several factors contribute to this delay, including the ionospheric piercing point location, imaging time, VTEC, electron density, pressure, and specific humidity. These parameters undergo continuous dynamic changes during signal propagation. According to the atmospheric delay model, the primary factors affecting the atmospheric delay for a single image are the electron density and specific humidity at the piercing point. Figure 6 shows the distribution of electron density and specific humidity in the zenith direction for different image central points. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline
**Image ID** & **gf1** & **gf2** & **gf3** & **gf4** & **gf5** & **gf6** & **gf7** & **gf8** & **gf9** & **gf10** & **Mean/m** \\ \hline \(\Delta S/\)m & 28.6919 & 5.3345 & 18.3229 & 6.1700 & 63.1564 & 0.8041 & 1.5054 & 25.7691 & 53.8806 & 1.3637 & 20.4998 \\ \(\Delta R/\)m & 0.0477 & 0.0457 & \(-\)0.0297 & \(-\)0.0060 & \(-\)0.0122 & 0.0004 & 0.0027 & \(-\)0.4523 & \(-\)0.0259 & 0.0001 & \(-\)0.0430 \\ \(\Delta\theta/^{\circ}\) & 0.0001 & \(-\)0.0003 & \(-\)0.0010 & 0.0001 & 0.0001 & 0.0000 & 0.0001 & 0.0018 & 0.0000 & 0.0000 & 0.0001 \\ \hline \end{tabular}
\end{table} 
Table 2: Satellite position inversion accuracy. Figure 5: Satellite position inversion accuracy. (a) Satellite position error; (b) slant-range error. Figure 6 illustrates that the distributions of electron density and specific humidity in the zenith direction vary across different image center points. Within the same group of images, the differences in electron density and specific humidity are relatively small, whereas the variations between different groups are more significant. This is likely due to the differences in the geographic locations and imaging times of the images. For a single image, despite the imaging time being largely consistent, the geographic variation in the piercing points introduces uncertainty regarding its impact on atmospheric delay. For large-scale spaceborne SAR imagery, the actual ground footprint of the electromagnetic wave path during the imaging process can extend over several hundred kilometers, while meteorological parameters at varying geographic locations and altitudes are in a state of continuous dynamic change. Therefore, in order to improve the precision of atmospheric delay calculations, the atmospheric layering method is adopted, which stratifies the electromagnetic wave path and analyzes the impact of different layer numbers on the atmospheric delay results. Using gf1 and gf7 satellite imagery, satellite positions are first computed using the autonomous intersection method based on the viewing vectors. The electromagnetic wave propagation path is then segmented  
Tables 3 and 4 present the delay distributions of gf1 and gf7 imagery under different ionospheric stratification conditions. For the ionosphere, the ionospheric delay of gf1 imagery ranges from 0.2246 m to 0.2440 m, with a variation of 0.0194 m, while gf7 imagery ranges from 0.1324 m to 0.1407 m, showing a variation of 0.0083 m. For the neutral atmosphere, the neutral delay for gf1 imagery spans from 2.8262 m to 2.8371 m, with a variation of 0.0109 m, whereas for gf7 imagery, it ranges from 2.4478 m to 2.4588 m, with a variation of 0.0110 m. Figure 7a,b show that when the number of stratifications is less than 20, the ionospheric delay increases progressively, reaching its maximum at 20 stratifications. As the number of stratifications continues to increase beyond this point, the ionospheric delay fluctuates within a certain range.