# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 3 Systematic Errors
### Viewing Geometry (Space)
#### 3.1.2 Flat-Earth Phase  
Flat-Earth phase is part of the topographic phase signal. It is the phase component of the reference Earth surface and is also called the reference plane phase. Since a flat altitude profile instead of the real topography is initially used for interferogram generation, the resulting interferogram needs to be compensated for the ellipsoid or geoid model of the Earth [35].  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Satellite** & **Band** & **Range of Perpendicular Baseline (m)** & **Range of Height Ambiguity (m)** \\ \hline TerraSAR-X/TanDEM-X & X & 10–4000 & 0.7–1000 \\ ERS-1/ERS-2 & C & 75–400 & 20–120 \\ ERS/ENVISAT & C & 1650–2050 & 3–15 \\ Radarsat & C & 100–1400 & 5–300 \\ Sentinel-1 & C & 10–150 & 60–2000 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Range of perpendicular baseline and height ambiguity for different satellite operations [57; 60].  
The \"flat-Earth\" pattern results from the small change of two viewing geometries (or shift in orbital trajectory) between two imaging acquisitions [31; 46]. Being a long-wavelength phase contribution, the pattern appears as dense stripes on interferograms. Normally, the difference in look angle across a swath of 6000 pixels can generate hundreds of fringes [46]. Due to orbital inaccuracies, the stripes occur in both the range and azimuth directions [66]. These dense stripes can severely obscure other phase variation information and add a significant load to later phase filtering and unwrapping. Therefore, flat-Earth phase is often removed during the initial stages of InSAR processing.  
The fringe frequency (\(f_{\phi}\)) is given by Equation (4) [28]. Fringe frequency (cycle/m) is not a fixed value. It is composed of the perpendicular baseline (\(B_{\perp}\)), slant range (\(R\)), wavelength (\(\lambda\)), incidence angle (\(\theta\)), and local terrain slope (\(\alpha\)). The perpendicular baseline and local terrain slope are the most critical components in determining the density of fringe frequency. Recall that the flat-Earth phase results from the shift in orbital trajectory. A greater shift suggests a wider perpendicular baseline and, as a result, a denser flat-Earth fringe frequency.  
\[f_{\phi}=\frac{2B_{\perp}}{\lambda Rtan(\theta-\alpha)} \tag{4}\]  
In addition to the parameters in Equation (4) that can impact the density of the flat-Earth-phase fringes, additional factors such as system altitude error and baseline error play a role in changing the fringe frequency [29; 30; 67]. System altitude error, hereafter altitude error, refers to the differences in system altitude estimation and true system altitude, also known as altitude indetermination. Baseline error refers to the differences in baseline estimation and true baseline, also known as baseline indetermination. The formation of altitude error and baseline error occurs due to the limitations in determining InSAR geometric parameters [68; 69; 70; 71; 72].  
The errors in altitude and baseline not only pose additional errors on the flat-Earth phase but also on the topographic phase component. In the flat-Earth phase, the altitude error superimposes an additive lower-order polynomial fringe (a linear tilt) in the range direction, and a height shift in the topographic phase, which is dependent on local topography [29; 30]. In the flat-Earth phase, the baseline error superimposes an additive lower-order polynomial fringe (a quadratic surface distortion) in the range direction and a height shift based on local topography in the topographic phase [29; 30; 35]. Reducing the impact of altitude error and baseline error can be achieved by either having more precise geometric parameters or applying tie points in the area of interest [30]. A minimum of two tie points is necessary for the correction of altitude error, and at least three are required for the correction of baseline error [30].