# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 3 Systematic Errors
### Scatterer Movement (Ground Surface)
#### 3.3.2 Surface Movements
For the application of topographic mapping, this leads to misinterpretation as well as problems of solving the errors correctly based on their sources. As defined by [PERSON] and [PERSON] [46], the small-gradient limit that will cause misinterpretation is about \(10^{-7}\). The value is unitless and it means long wavelength displacements need to be at least larger than 0.01 mm per 100 m to be distinguished from other long wavelength error sources. The value of \(10^{-7}\) comes from 0.01 mm divided by 100 m.  
The limits of the spatial scale are pixel size limit and swath width limit. The two limits restrict the spatial extent of observable displacement to the dimension as small as a pixel size but could also be as large as the swath width [46]. For example, a surface \begin{table}
\begin{tabular}{l l l l l l} \hline \hline \multirow{2}{*}{**Symbol**} & \multirow{2}{*}{**Meaning**} & \multirow{2}{*}{**ERS-1**} & \multicolumn{2}{c}{**Sentinel-1**} & \\ \cline{3-5}  & & & **IW1** & **IW2** & **IW3** \\ \hline \(c\) & Light velocity (m/s) & \(3\times 10^{8}\) & \(3\times 10^{8}\) & \(3\times 10^{8}\) & \(3\times 10^{8}\) & \(3\times 10^{8}\) \\ \(\lambda\) & Wavelength (m) & 0.0566 & 0.0555 & 0.0555 & 0.0555 \\ \(B_{R}\) & Range Single Bandwidth (Hz) & \(16\times 10^{6}\) & \(56.5\times 10^{6}\) & \(48.3\times 10^{6}\) & \(42.8\times 10^{6}\) \\ Upper Gradient Limit & \(3\times 10^{-3}\) & \(10.5\times 10^{-3}\) & \(8.9\times 10^{-3}\) & \(7.9\times 10^{-3}\) \\ Surface movement (cm)/spatial scale (m) & 30/100 & 105/100 & 89/100 & 79/100 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Calculation for the upper gradient limit (ERS-1, Sentinel-1, TerraSAR-X, and NISAR). \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Symbol** & **Meaning** & **L-Band** & **S-Band** & **C-Band** & **X-Band** \\ \hline \(\lambda\) & Wavelength (mm) & 250 & 120 & 56 & 31 \\ Cycle-Slicing Limit (mm) & 12.5 & 6 & 2.8 & 1.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Range of list of symbols and their corresponding values for each satellite (calculation for cycle-slicing limit). motion that is only 5 m wide in spatial scale is not detectable in a pixel size of 20 m. However, to observe a geophysical phenomenon, the spatial extent of the occurrence needs to be at least 200 m in the real world (corresponding to about 10 pixels). Thus, the pixel size limit is simply the spatial resolution and only depicts the physical constraint of the detectability of a sensor but is not enough for interpreting geophysical phenomena. The pixel size limit is about 20 m for both ERS-1 and Sentinel-1, about 3 m for TerraSAR-X in Stripmap mode, 16 m for TerraSAR-X in ScanSAR mode, and about 5-30 m for NISAR. Contrary to the pixel size limit, the swath width limit is the maximum detectable spatial limit. Displacements that occur at a scale larger than the width of a swath are not observable; thus, the swath width forms the maximum spatial detectability limit. This limit is 100 km for ERS-1 and about 83 km for a subswath of Sentinel-1, 30 km for TerraSAR-X in Stripmap mode, 100 km for TerraSAR-X in ScanSAR mode, and 240 km for NISAR. It is possible to merge three subswatths together for Sentinel-1 imagery, so the limit could be expanded to 250 km if the three subswatths are combined. Figure 5 delineates the five detectability limits of ERS-1, Sentinel-1, TerraSAR-X (Stripmap mode and ScanSAR mode), and NISAR (S-band and L-band), modified from [28; 46]. The light-yellow color (the area enclosed by the polygon) illustrates the detectable area. To be detected via interferometry and be coherent between two images, a signal needs to fall within the detectable area. Overall, since both ERS-1 and Sentinel-1 use C-band sensors, the boundary lines and detectable area are similar in their values and shape, but the upper gradient limit and swath width limit of Sentinel-1 are slightly larger than those of ERS-1.