# High-Resolution Spaceborne SAR Geolocation Accuracy Analysis and Error Correction
## 2 Materials and Methods
### Geolocation Procedure  
A geolocation procedure consists of the following two parts: coordinate system transformation and error correction [18]. The transformation from the geographic coordinate system into the SAR image coordinate system is called indirect geolocation, while the reverse process is known as direct geolocation [35; 36]. Given that our research can be applied to target monitoring and data registration, we focus solely on indirect geolocation. We used the range-Doppler model [37] to perform the transformation. This study does not discuss other models, such as the rational polynomial model, which is based on photogrammetric principles with relatively lower accuracy [38]. The only error we correct in our procedure is atmospheric. We omit orbit correction and correction for solid Earth effects, which might be applied in other studies to reduce the complexity of the SAR geolocation. Assuming that one of the geographic coordinates to be geolocated has been converted into the geocentric coordinate \(\overset{\rightarrow}{R}_{T}\), the range vector, \(\overset{\rightarrow}{R}(t)=\overset{\rightarrow}{R}_{T}-\overset{ \rightarrow}{R}_{S}(t)\), can be obtained with the satellite's position, \(\overset{\rightarrow}{R}_{S}(t)\), at the imaging time, \(t\). The fundamental idea behind geolocation is to determine \(t\) and \(|\overset{\rightarrow}{R}(t)|\), correct the tropospheric delay and ionospheric delay for \(|\overset{\rightarrow}{R}(t)|\), calculate the image coordinates (_row_ and _col_), and resample the image (Figure 3). The imaging time, \(t\), is solved by the Doppler equation (Equation (6)), where \(\overset{\rightarrow}{V}_{S}(t)\) represents the velocity of the satellite, \(f_{0}\) denotes the signal frequency, and \(c\) is the speed of light in a vacuum. \[f_{D}=-2\frac{f_{0}}{c}\frac{\overset{\rightarrow}{V}_{S}(t)\cdot\overset{ \rightarrow}{R}(t)}{\left|\overset{\rightarrow}{R}(t)\right|} \tag{6}\]  
Most SAR images are produced in zero-Doppler geometry, \(f_{D}\equiv 0\), so \(\overset{\rightarrow}{V}_{S}(t)\) is orthogonal to \(\overset{\rightarrow}{R}(t)\). We solve for \(t\) using an iterative method (Equation (7)). \[t_{k+1}=t_{k}+\frac{\overset{\rightarrow}{V}_{S}(t_{k})\cdot\overset{ \rightarrow}{R}(t_{k})}{\left|\overset{\rightarrow}{V}_{S}(t_{k})\right|^{2}} \tag{7}\]  
The principle is as follows (Figure 4a): \(\overset{\rightarrow}{V}_{S}\cdot\overset{\rightarrow}{R}\) is the projection of \(|\overset{\rightarrow}{R}|\) in the direction of \(\overset{\rightarrow}{V}_{S}\). If \(\overset{\rightarrow}{V}_{S}\) is orthogonal to \(\overset{\rightarrow}{R}\), the projection should be zero. We need to increase the imaging time, \(t\), to meet the condition, with the increment value being \(\dfrac{\overset{\rightarrow}{V}_{S}\cdot\overset{\rightarrow}{R}}{\left| \overset{\rightarrow}{V}_{S}\right|^{2}}\). Figure 3: The geolocation procedure. Since the satellite keeps moving between the signal-transmitting and -receiving moments, Equation (6) is only an approximation, but studies have shown that it is sufficiently accurate [39]. With \(t_{start}\) representing the azimuth start time of the image, \(dt\) denoting the azimuth line time, \(R_{0}\) representing the nearest range of the image, \(dR\) representing the range pixel spacing, and \(\Delta\tau_{tro}\) and \(\Delta\tau_{ion}\) representing the tropospheric delay and ionospheric delay, the image coordinates \((row\) and \(col)\) can be calculated by Equations (8) and (9). The signs of \(\Delta\tau_{tro}\) and \(\Delta\tau_{ion}\) are positive, so the range coordinate, \(col\), is relatively smaller without atmospheric correction. \[row=\frac{t-t_{start}}{dt} \tag{8}\]  
\[col=\frac{\left|\overrightarrow{R}(t)\right|-R_{0}+\Delta\tau_{tro}+\Delta \tau_{ion}}{dR} \tag{9}\]  
Owing to the difference in the image standards, the azimuth coordinates of the Tianhui-2 images were calculated using Equation (10) [40].