# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 3 Systematic Errors
### Viewing Geometry (Space)  
#### 3.1.1 Topographic Phase  
Topographic phase signals include the topographic phase (an artifact of elevation differences) and flat-earth phase (an artifact of the side-looking radar). The topographic phase allows for reconstruction of the surface topography from the two-dimensional phase field measurements due to the viewing geometry and the accurate spatial coupling of the two imaging orbits [31]. This is the desired phase component for DEM generation. However, it is an undesirable phase component for deformation mapping. Equation (2) shows the calculation for the topographic phase, where \(h\) is topographic height. \(4\pi\) divided by \(\lambda\) is the conversion factor from pseudo range to phase delay, which will always exist in the phase terms. The most influential component is perpendicular baseline (B\({}_{\perp}\)). The reason is closely related to the height sensitivity of the interferometer, namely, height ambiguity, which symbolizes the elevation difference corresponding to a fringe (\(2\pi\)) of change (i.e., height per phase cycle) [28]. The calculation for height ambiguity is given in Equation (3), where \(h_{2\pi}\) symbolizes the height ambiguity [27; 28]. A longer perpendicular baseline leads to less height ambiguity. Reduced height ambiguity indicates that a fringe is more sensitive to elevation change, making it ideal for DEM generation but not for deformation mapping. On the other hand, a shorter perpendicular baseline leads to significant height uncertainty. A high degree of height ambiguity suggests that a fringe is insensitive to elevation change, making it useful for deformation mapping but not DEM generation. \[\phi_{topo}=\frac{4\pi}{\lambda}\frac{B_{\perp}}{Rsin\theta}h \tag{2}\]  
\[h_{2\pi}=\frac{\lambda}{2}\frac{Rsin(\theta)}{B_{\perp}} \tag{3}\]  
This explains why Sentinel-1 is unsuitable for topographic mapping, as its orbital tube was not designed for DEM generation [53; 54]. Sentinel-1 orbit maintenance strategy provides modest orbital InSAR baselines in the order of 150 m [55]; hence, baselines of Sentinel-1 image pairs are often too narrow to achieve a preferred height ambiguity. In principle, the best baselines for topographic mapping are between 150 and 300 m,whereas baselines below 50 m are recommended for deformation mapping [35]. One issue to keep in mind is that while a larger perpendicular baseline is preferred for DEM production in order to provide higher sensitivity of the InSAR phase to height variation, the value should not be more than the critical baseline (\(B_{c}\)) in order to avoid excessive decorrelation noise. Longer perpendicular baselines increase baseline decorrelation [12], particularly in places with moderate to severe slopes (e.g., [56]), as well as volume decorrelation, since the longer baseline increases the sensitivity of the coherence and the InSAR phase to vegetation characteristics. Height ambiguity is an important value for DEM generation as it is used to translate the number of fringes to the measured topographic height. Here, we use the parameters of C-band Sentinel-1 as an example. Suppose the wavelength is 0.0555 m, slant range is 780,000 m, incident angle is 30\({}^{\circ}\), and perpendicular baseline is 150 m. Using Equation (3), we could derive a height ambiguity of 74 m/fringe. If there are four fringes in the interferogram, this indicates that the topographic height is \(74\times 4=296\) m. As a result, a lower height ambiguity provides a more accurate translation from the number of fringes to the measured elevation, but a large ambiguity height (say, above 500 m/fringe) is unlikely to offer meaningful information for areas where the elevation gradient is less than 500 m.  
Table 1 compares the baselines of different satellite constellations, revised from [PERSON] [57]. Range of height ambiguity (m) is calculated using Equation (3) given the parameters of each SAR system. The lowest limit of the Sentinel-1 and TerraSAR-X/TanDEM-X baselines is set to 10 m for calculation purposes.