# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 3 Systematic Errors
### Propagation Path (Atmosphere)
#### 3.2.1 Ionospheric Advance
The zenith ionospheric advance (\(\delta_{iono}^{z}\)) at L-band frequency is \(-250\) mm, which is about \(4\), \(19\), and \(61\) times larger than at S-band, C-band, and X-band frequencies, respectively [85]. Figure 3 depicts the effect of TECu's spatiotemporal fluctuation when it spans from \(0\) to \(15\)\(TECu\). Note that the zenith advance (the right subplot) is shown in meters. When the variance is \(15\)\(TECu\), the zenith advance is nearly \(4\) m in L-band, \(1\) m in S-band, and less than \(0.25\) m in C- and X-band. Spatiotemporal ionospheric fluctuations can translate into notable height errors (pertaining to DEM generation) or deformation errors (pertaining to deformation mapping). This translation entails a multiplication of various contributing factors, as expounded upon in the work by [PERSON] et al. [83]. For instance, with a perpendicular baseline of \(200\) m, even a modest \(1\)\(TECu\) of spatiotemporal fluctuation in the ionosphere can give rise to substantial errors in the measured topographic heights and surface displacements across different frequency bands. [PERSON] et al. [83] quantified these effects and revealed errors of \(445.92\), \(24.62\), and \(7.72\) m in topographic heights, as well as \(38\), \(2\), and \(0.7\) cm in observed surface deformation at L-, C-, and X-band frequencies, respectively. Figure 3: The impact of the spatiotemporal variation of \(0\) to \(15\) TECu at L-band, S-band, C-band, and X-band frequencies. The left subplot shows how many phase shifts would be generated at the given TEC variation (modified from [83]). The right subplot shows how much zenith advance in meters is caused at the given TEC variation. The sign is negative because an increase in TEC leads to a phase advance. \begin{table}
\begin{tabular}{l l l l l} \hline \hline  & **L-Band** & **S-Band** & **C-Band** & **X-Band** \\ \hline Wavelength (mm) & \(250\) & \(120\) & \(56.6\) & \(31\) \\ Frequency (GHz) & \(1.27\) & \(2.5\) & \(5.41\) & \(9.65\) \\ Phase cycles (\(2\pi\))/\(\Delta TECu\) & \(2.11\) & \(1.07\) & \(0.5\) & \(0.28\) \\ \(\delta_{iono}^{z}\) (mm)/\(\Delta TECu\) & \(-250\) & \(-64.45\) & \(-13.76\) & \(-4.33\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The impact of \(1\)\(TECu\) spatiotemporal variation in the ionosphere at L-band, S-band, C-band, and X-band frequencies (modified from [83]). Although 1 \(TECu\) of variation could bring in devastating contamination for L-band sensors, the damage it causes on C-band sensors is not negligible either. [PERSON]'s study [99] has shown that 75% of image pairs in that study have the \(\mid\Delta\)\(TECs_{\mid}\) within \(\pm\)5 \(TECu\). The range reaches up to 2.5 phase shifts and corresponds to 123.1 \(m\) of topographic height errors even at the C-band frequency. Other studies also have found that fringe patterns caused by ionospheric effects may be seen in some interferograms obtained with C-band Sentinel-1 and Radarsat images [88; 100], especially when the latitudinal range of the study area is large (>50 km) and ionospheric anomalies are present [28; 99]. Consequently, it is advised that in such cases, ionospheric advance should not be overlooked, even for C-band systems, and that ionospheric artifact correction be performed as an essential step during InSAR processing [88].