# High-Resolution Spaceborne SAR Geolocation Accuracy Analysis and Error Correction
## 2 Materials and Methods
### Tropospheric Correction  
The most common atmospheric correction methods are the RT method and the ZDM method [20]. The RT method models the actual signal propagation process and, thus, possesses superior accuracy [31]. The ZDM method approximates the propagation process and has advantages in computational complexity. Many InSAR studies have demonstrated the effectiveness of the ZDM method for phase correction [32]. We validate its efficacy for geolocation in this study.  
The RT method divides the propagation path into multiple segments, calculates the tropospheric delay corresponding to each segment, and sums them (Figure 2). Since the result is virtually unchanged when segments are more than 20 [18], dividing the path into 20 segments is sufficient to meet the accuracy requirement for geolocation. The bending in the propagation path due to refraction has nearly no impact on the results, so approximating the path as a straight line introduces only negligible errors [22].  
Implementing the RT method requires the pressure, \(P_{i}\) (\(hPa\)); temperature, \(T_{i}\) (\(K\)); vapor partial pressure, \(e_{i}\)(\(hPa\)); and path length, \(h_{i}(m)\), for each segment, as shown in  
\begin{table}
\begin{tabular}{c c c} \hline
**Geolocation Errors** & **Typical Value** & **Factors** \\ \hline Tropospheric delay, \(\Delta\text{s}_{\text{tro}}\) & 2 m–4 m [30] & Atmospheric state, zenith angle \\ Ionospheric delay, \(\Delta\text{s}_{\text{ion}}\) & 0.1 m–20 m [22] & Electronic density, signal frequency \\ Elevation – & 0.01 m–10 m [29] & Elevation accuracy \\ related error, \(\Delta\text{s}_{\text{ele}}\) & & \\ \hline \end{tabular}
\end{table}
Table 1: SAR geolocation errors in the range.  
Figure 2: The signal propagation model upon which the RT method is based. The propagation path in the troposphere is divided into multiple segments.  
Equations (3) and (4) [33; 34], where \(\theta\) is the zenith angle, and \(N\) is the radio refractivity, equaling \(10^{-6}(n-1)\), and \(n\) is the refractivity.  
\[N_{i}=77.6\frac{P_{i}}{T_{i}}+71.6\frac{e_{i}}{T_{i}}+3.75\times 10^{5}\frac{e_{i }}{T_{i}^{2}} \tag{3}\]  
\[\Delta s_{tro}=\frac{10^{-6}}{\cos\theta}{\sum_{i}N_{i}}\cdot h_{i}+\frac{10^{ -6}}{\cos\theta}77.6\times 287.05\times\frac{P_{top}}{9.81} \tag{4}\]  
We added an extra term to the conventional formula in Equation (4) to correct the delay above the tropopause. The vapor partial pressure is virtually zero; hence, the additional delay can be approximated as hydrostatic delay, which can be estimated using the pressure at the tropopause.  
The ZDM method estimates the tropospheric delay in the propagation path using the zenith delay. The distance between the zenith path and the propagation path of the signal increases as the altitude increases. However, the thickness of the troposphere is limited, generally not exceeding 30 km [22]. Therefore, the maximum distance is around 30 km at the tropopause, at which the scales of the tropospheric parameters do not significantly differ. The rationality of replacing the parameters along the propagation path with those along the zenith path is the theoretical foundation of the ZDM method. The mapping function is needed to transform the length of the zenith path into that along the propagation path, which can be \(\sec\theta\) or from related products, such as VMF1. The most significant advantage of the ZDM method is simplifying the computation, allowing the zenith tropospheric delay to be potentially released as a standard product, such as the GACOS and other station observation products.