# High-Resolution Spaceborne SAR Geolocation Accuracy Analysis and Error Correction
## 3 Results
### Characteristics of the Topospheric Delay  
We calculated the zenith delay using ERA5 and compared the result with the ZPD in the BJFS station (Figure 10). The difference is larger in summer than in winter, possibly because of higher vapor pressure. The maximum difference is about 0.06 m, corresponding to less than 0.1 pixels, so we believe that ERA5's accuracy is sufficient for geolocation. The difference is not correlated with the delay value (Figure 10b). The tropospheric delay In the study area is calculated using the RT method and ranges from 2.6 m to 2.8 m, demonstrating a strong correlation with the ground elevation (Figure 11a,b). The delay in the plain varies on a smaller scale, from 2.75 m to 2.8 m, while in the mountainous region, it is typically 2.6 m to 2.75 m. This is because the signal must traverse a longer distance before it reaches lower altitude areas, and there tends to be more tropospheric perturbations in mountainous regions. Figure 8: Offsets between the reference image and other images in (**a**) azimuth and (**b**) range. The \(x\)-axis represents the imaging time. Figure 9: Tianhui-2 image geolocation results corresponding to Figure 7a with atmospheric correction: (**a**) reference image; (**b**) reference image with the overall shift; (**c**) secondary image; (**d**) secondary image with the overall shift. We calculated the zenith tropospheric delay in the study area using the same tropospheric parameters as the RT method and then mapped it toward the line-of-sight (LOS) direction. The difference between it and the RT result is less than 0.3 cm (Figure 11c). We examined the relationship between the radio refractivity, \(N\), and elevations less than 1200 m (Figure 11d). Assuming N is 300, only 10 m is needed to form a delay of 0.3 cm, as per Equation (4). Considering that the accuracy of the elevation inside ERA5 is usually less than 10 m, the difference between the two methods is of little reference value. This distribution of N supports this deduction (Figure 11d). At the same elevation, the average fluctuation of N is at most five when the elevation is less than 600 and becomes less than one after that, corresponding to a difference of only 0.3 cm. Compared with the other method, the path length accuracy is more important. Therefore, we believe that the two methods perform equally well. The results suggest that the ZDM method generally meets the accuracy requirements for geolocation. Figure 11: **(a)** DEM of the study area; (b) tropospheric delay on 17 January 2019, derived using the RT method; (c) difference between the ZDM result and the RT result; (d) relationship between the radio refractivity, \(N\), and the elevation in the study area. \(N=-3.1\times 10^{-2}h+293\), when the elevation is greater than 500 m.  
Figure 10: Comparison between the zenith delay calculated using the ERA5 and ZPD data: (a) difference changing with the day of the year; (b) statistical results of the original data. We compared the results of the GACOS-based ZDM method and the ERA5-based ZDM method (Figure 12). The GACOS-based result exhibits the same characteristics as the ERA5-based result; that is, the tropospheric delay was about 5 cm larger in the east plain than in the west plain (Figure 12a). The difference exhibits a noticeable texture associated with the terrain (Figure 12b). It tends to be positive in valleys and negative at ridges, which is related to the WRF-processed ERA5's 1 km lower resolution than the 90 m for GACOS. The discrepancy is about \(-1\) cm on the plain, whereas it barely reaches extremes of around \(-12\) cm in mountainous regions. The difference between the two data sources is much larger than between the two methods. However, owing to its advantages in resolution, although we have verified the high precision of ERA5, we still believe that the GACOS results are at least as good. Compared with the total geolocation error in range, we believe that all three methods demonstrate enough tropospheric accuracy.