# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 4 Random Errors
### Baseline/Geometric/Spatial Decorrelation (\(\delta_{B}\))
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline \multirow{2}{*}{**Symbol.**} & \multirow{2}{*}{**Meaning**} & \multirow{2}{*}{**ERS-1**} & \multicolumn{3}{c}{**Sentinel-1**} \\ \cline{3-6}  & & & **IW1** & **IW2** & **IW3** \\ \hline \(c\) & Light velocity (m/s) & \(3\times 10^{8}\) & \(3\times 10^{8}\) & \(3\times 10^{8}\) & \(3\times 10^{8}\) \\ \(B_{\perp}\) & Perpendicular Baseline (m) & \(-600\) & \(-100\) & \(-100\) & \(-100\) \\ \(R\) & Slant Range (m) & 858,200 & 799,300 & 845,800 & 901,400 \\ \(\lambda\) & Wavelength (m) & 0.0566 & 0.0555 & 0.0555 & 0.0555 \\ \(\theta\) & Incident Angle (\({}^{\circ}\)) & 23 & 32.9 & 38.3 & 43.1 \\ \(\alpha\) & Local Terrain Slope (\({}^{\circ}\)) & \(-90\) to 90 & \(-90\) to 90 & \(-90\) to 90 & \(-90\) to 90 \\ \(B_{R}\) & Range Single Bandwidth (MHz) & 16 & 56.5 & 48.3 & 42.8 \\ \hline \multirow{2}{*}{**Symbol.**} & \multirow{2}{*}{**Meaning**} & \multicolumn{3}{c}{**TerraSAR-X**} & \multicolumn{3}{c}{**NISAR (Interferometric Mode)**} \\ \cline{3-6}  & & & **Stripmap Mode** & **ScanSAR Mode** & **S-Band** & **L-Band** \\ \hline \(c\) & Light velocity (m/s) & \(3\times 10^{8}\) & \(3\times 10^{8}\) & \(3\times 10^{8}\) & \(3\times 10^{8}\) \\ \(B_{\perp}\) & Perpendicular Baseline (m) & \(-1000\) & \(-600\) & \(-1000\) & \(-300\) \\ \(R\) & Slant Range (m) & 1,200,000 & 500,000 & 1,800,000 & 800,000 \\ \(\lambda\) & Wavelength (m) & 0.0311 & 0.0311 & 0.126 & 0.238 \\ \(\theta\) & Incident Angle (\({}^{\circ}\)) & 38.4 & 45 & 17.1 & 44 \\ \(\alpha\) & Local Terrain Slope (\({}^{\circ}\)) & \(-90\) to 90 & \(-90\) to 90 & \(-90\) to 90 & \(-90\) to 90 \\ \(B_{R}\) & Range Single Bandwidth (MHz) & 150 & 300 & 75 & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Set of parameters used to calculate the frequency shift in Figure 7.  
Figure 7: Frequency shift as a function of the local terrain slope in the case of ERS-1, Sentinel-1 (IW1/IW2/IW3), TerraSAR-X (Stripmap mode and ScanSAR mode), and NISAR (S-band and L-band). The horizontal red dashed lines symbolize the range single bandwidth. The pink background color shows the blind angle area where the frequency shift exceeds the single range bandwidth. The slopes within the blind angle area (pink) will directly lead to a complete loss of coherence (decorrelation). The light yellow, light green, and light gray background colors represent the area of angles where layover, foreshortening, and shadow occurs, respectively. The area without a colored background means the terrain slope is not subject to the influence of geometric decorrelation (blind angle area) nor geometric distortion (layout, foreshortening, and shadow). Frequency shift is directly related to the overlap between two spectra. When the frequency shift (the black curves in Figure 7) equals 0 (indicated by the horizontal black dashed line), the overlap is 100%. However, when the frequency shift matches the range single bandwidth (indicated by horizontal red dashed lines), the overlap is 0. Once the frequency shift exceeds the range single bandwidth, i.e., \(\Delta f>B_{R}\), there is no overlap in the range direction between both the primary and secondary images. This implies a complete loss of correlation between the two images, i.e., \(\gamma_{B}\) = 0. The frequency shift can surge to infinity when the local terrain slope approaches the incident angle (indicated by the vertical black line). Blind angles (pink background) are local slope angles close to the incidence angle that cause the frequency shift to exceed the range single bandwidth. Baseline decorrelation occurs within this local slope range [127]. ERS-1 has the broadest blind angle range (about incidence angle \(\pm\)14\({}^{\circ}\)), owing to its extremely low-range single bandwidth (16 MHz). This suggests that ERS-1 (with the given parameters) is the most vulnerable to geometric decorrelation. Note that the blind angles in subplots (b), (c), (d), (g), and (h) are quite narrow (about incidence angle \(\pm\)0.5\({}^{\circ}\)), making them less noticeable, but they do exist.