# High-Resolution Spaceborne SAR Geolocation Accuracy Analysis and Error Correction
## 2 Materials and Methods  
### Components of Geolocation Errors  
The components of geolocation errors differ in terms of azimuth and range. Typically, the azimuth geolocation accuracy is only affected by the satellite orbit error owing to inaccuracies in satellite positions or mismatched instrumental times [18]. Coregistrating SAR images with digital elevation model (DEM)-simulated amplitude images is a widely used orbit correction method without ground control points [23]. However, it does not apply to high-resolution images and meter-level-accuracy geolocation. We discuss this method in Section 4. In this study, we assumed that there was no significant azimuth geolocation error.  
In addition to the orbit error, the range geolocation accuracy is also affected by the atmospheric error and ground elevation error (Figure 1) [18; 23]. Influenced by the atmosphere, the signal's equivalent velocity, \(v\), is smaller than the velocity of light in a vacuum, \(c\). The propagation time will decrease by \(\left(\frac{1}{v}-\frac{1}{c}\right)\left|ST_{2}\right|\) when calculated using \(c\), called atmospheric delay [18]. For convenience, atmospheric delay can be approximated and expressed in a distance form as \(\Delta s_{atm}=\frac{c-v}{c}|ST_{2}|\). The atmospheric delay comprises the ionospheric delay and the tropospheric delay. The ionospheric delay primarily depends on the ionospheric state and the signal's frequency [22]. In extreme conditions, ionospheric delay is at the decimeter level for X-band SAR, about 1 m for C-band SAR, and may exceed 10 m for L-band SAR [18]. tropospheric delay is mainly influenced by the tropospheric state and the length of the propagation path. Given that the signal's zenith angle, \(\theta\), generally ranges from \(20^{\circ}\) to \(70^{\circ}\), and the zenith tropospheric delay is typically 2 m to 3 m, the total tropospheric delay is about 2 m to 4 m.  
The elevation error, \(\Delta h\), causes a geolocation error, \(\Delta s_{ele}=\Delta lcos\theta\) (Figure 1). Since the accuracies of the currently available DEMs are limited [29], the elevation error tends to have the largest impact on the geolocation accuracy, potentially introducing an error of approximately 10 m. Table 1 lists the components of the range geolocation errors.  
Based on the previous discussion, let \(\Delta\text{s}_{\text{tro}}\) represent the tropospheric delay, \(\Delta\text{s}_{\text{ion}}\) denote the ionospheric delay, and \(\Delta\text{s}_{\text{ele}}\) signify the elevation-related error. We categorized the geolocation error caused by other factors, including the satellite orbit error, into the following residual terms: \(\Delta\text{r}_{\text{rg}}\) in range and \(\Delta\text{r}_{\text{az}}\) in azimuth. Let \(\Delta\text{s}_{\text{rg}}\) and \(\Delta\text{s}_{\text{az}}\) denote the geolocation errors in range and azimuth. We obtain the following:  
\[\Delta\text{s}_{\text{rg}}=\Delta\text{s}_{\text{tro}}+\Delta\text{s}_{\text{ ion}}+\Delta\text{s}_{\text{ele}}+\Delta\text{r}_{\text{rg}} \tag{1}\]  
\[\Delta\text{s}_{\text{az}}=\Delta\text{r}_{\text{az}} \tag{2}\]  
Next, we introduce the tropospheric and ionospheric correction methods and the geolocation procedure.