# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 3 Systematic Errors
### Scatterer Movement (Ground Surface)
#### 3.3.2 Surface Movements  
The other cause for a scatterer's movement are small surface motions. This is the desired phase information for deformation mapping and undesirable phase information for topographic mapping. The traditional approach to calculating surface motion is the differential InSAR (DInSAR) technique, first proposed by [PERSON] et al. [12]. Surface movements resolvable via the DInSAR method include (but are not limited to) volcanic/dike injections, land subsidence, glacier flow, and seismic deformation. The deformation phase component can be calculated by Equation (14), where \(\Delta D\) is the motion along the line-of-sight (LOS) direction. The translation from the number of fringes to the measured deformation is directly related to the sensor wavelength, as a phase cycle corresponds to displacement of half of the wavelength. Therefore, a fringe in an L-band interferogram means a deformation of 12.5 cm, in a C-band means a deformation of 2.8 cm, and in an X-band means a deformation of 1.55 cm (Table 6). \[\phi_{defo}=\frac{4\pi}{\lambda}\Delta D \tag{14}\]  
Because these kinds of movements are spatiotemporally correlated, when such incidences happen, this will induce an additional phase shift to the total phase variation if the displacement happens along the LOS direction. However, as the magnitude of the displacements and the spatial scale of the incidences can vary widely, not every surface motion is detectable if it exceeds the limits of the deformation gradient and/or the spatial scale [46]. The detectable area is confined by five boundary lines: (1) upper gradient limit, (2) cycle-slicing limit, and (3) small-gradient limit are the boundary lines for the deformation gradient, while (4) pixel size limit and (5) swath width limit are for enclosing the detectable spatial ranges. The cost of exceeding the boundary lines could be as minor as not being able to detect the surface movements or the possibility of misinterpretation, but also could be as large as causing decorrelation or phase discontinuity. The following paragraphs will elaborate on the five boundary lines in detail. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Symbol** & **Meaning** & **L-Band** & **S-Band** & **C-Band** & **X-Band** \\ \hline \(\lambda\) & Wavelength (cm) & 25 & 12 & 5.6 & 3.1 \\ Deformation (cm/fringe) & 12.5 & 6 & 2.8 & 1.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The translation from number of fringes to the measured deformation. The limit of upper deformation gradient comes from the constraint of phase variations within a pixel. The phase difference of one pixel between two SAR images must not exceed one fringe of a round trip range shift of one wavelength per pixel as a fundamental criterion for interferometry [46]. If the phase variation exceeds this limit, such as through abrupt changes in topography, the pixels will become incoherent. We could define the upper gradient limit using Equation (15), where \(f_{\phi}\) is fringe frequency (cycle/m) given in Equation (4). The expression uses half of wavelength since one fringe corresponds to half of a wavelength's displacement. \[Upper\ gradient\ limit=f_{\phi}\times\frac{\lambda}{2} \tag{15}\]  
To calculate the upper gradient limit, the critical baseline (\(B_{c}\)) should be applied to Equation (4). The critical baseline is given in Equation (16), where \(B_{R}\) is range single bandwidth in Hz. The definition of critical baseline could be understood from two perspectives. First, as seen from the imaging geometry, \(B_{c}\) is defined as the maximum permissible change in look angle between two acquisitions. The coherence necessary for effective InSAR analysis will be sustained as long as this angular value is not exceeded [32]. Second, as seen from the imaging plane, \(B_{c}\) is defined as the maximum phase change (\(2\pi\), or a fringe) within a resolution cell [29; 30]. The fringe frequency calculated with the critical baseline is formed in Equation (17).