# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 4 Random Errors
### Baseline/Geometric/Spatial Decorrelation (\(\delta_{B}\))  
The antenna separation between two repeat SAR acquisitions in space is the baseline of the SAR geometry, and the orthogonal vector of which to the look direction is called the perpendicular baseline (see Figure 6 for InSAR imaging geometry and perpendicular baseline). The difference in two viewing geometries provides the opportunity to observe the topography since the backscattered radar wave contains a different ground reflectivity spectrum for each observation [127]. The reflectivity spectrum difference is defined by a shift and stretch of the imaged terrain spectrum; i.e., the identical spectral components in the first spectrum have a frequency shift in the second spectrum [127]. This is the frequency shift (or spectral shift) that causes decorrelation. The longer the perpendicular baseline, corresponding to a larger spectral shift, the lower the correlation between two SAR images. Once the shift exceeds the range single bandwidth (denoted as \(B_{R}\)), the two images will be completely decorrelated because of the zero overlap between two spectrums. With this restraint, the critical baseline (\(B_{c}\), see Equation (16)) is defined as the maximum effective baseline where the overlap and correlation exist. The length of the perpendicular baseline is one major factor that causes a frequency shift in the range direction. However, there are minor factors that could also lead to frequency shifts (e.g., steepness of the terrain slope and the gradient of the surface displacements) [28]. We mentioned that steep terrain slopes are a major contributor to the loss of coherence; here, we demonstrate why: steep slopes increase the spectral shift, resulting in less overlap. To consider the influence of terrain slope (\(\alpha\)) on frequency shift (\(\Delta f\)), [PERSON] [127] has demonstrated that spectral shift is a function of the local angle (Equation (23)). Terrain slope ranges from \(-90^{\circ}\) to \(90^{\circ}\), where the positive angles (\(\alpha=0^{\circ}\)\(\sim\)\(90^{\circ}\)) mean that the slope is facing the sensor, and the negative angles (\(\alpha=-90^{\circ}\)\(\sim\)\(0^{\circ}\)) mean a backslope relative to the sensor. Equation (23) shows that \(\Delta f\) is dependent on perpendicular baseline, local terrain slope, and incident angle. Suppose the perpendicular baseline is fixed (\(-600\) m for ERS-1 as a reproduction of [PERSON] [127], and \(-100\) m for Sentinel-1 as a reasonable value for using Sentinel-1), the relationship between the local terrain slope and the frequency shift can be visualized (Figure 7). The parameters used for the ERS-1 frequency shift calculation are retrieved from Gatelli [127], and the parameters set for Sentinel-1 (IW1, IW2, and IW3), TerraSAR-X (Stripmap mode and ScanSAR mode), and NISAR (S-band and L-band) are selected within their reasonable range (Table 10). With the frequency shift calculated from Equation (23), one could derive the coherence for baseline term by Equation (24). \[\Delta f=-\frac{cB_{\perp}}{R\lambda\tan(\theta_{inc}-\alpha)} \tag{23}\]  
\[\gamma_{B}=1-\left|\frac{\Delta f}{B_{R}}\right| \tag{24}\]