# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 3 Systematic Errors
### Scatterer Movement (Ground Surface)
#### 3.3.2 Surface Movements
By putting Equation (17) back to Equation (15), we can derive the upper gradient limit as Equation (18). Consequently, to calculate the upper gradient limit for any sensor, information on the range single bandwidth and the wavelength of the sensor are required. \[B_{c}=\frac{B_{R}R\lambda tan(\theta-\alpha)}{c} \tag{16}\]  
\[f_{\phi,c}=\frac{2B_{c}}{R\lambda tan(\theta-\alpha)}=\frac{2B_{R}R\lambda tan (\theta-\alpha)}{R\lambda tan(\theta-\alpha)c}=\frac{2B_{R}}{c} \tag{17}\]  
\[Upper\ gradient\ limit=\frac{2B_{R}}{c}\times\frac{\lambda}{2}=\frac{B_{R} \lambda}{c} \tag{18}\]  
Following Equation (18), the calculations of the upper gradient limits for ERS-1, Sentinel-1, TerraSAR-X, and NISAR are presented in Table 7. The upper gradient limit for ERS-1 is 3 \(\times\) 10\({}^{-3}\), which meets the statements in [PERSON] and [PERSON] [46]. This limit means that the displacement (in the range direction) is only detectable below 30 cm every 100 m. Since the range single bandwidth of Sentinel-1 are all larger than ERS-1, the upper gradient limit is about 2-4 times larger. In other words, Sentinel-1 is capable of measuring 2-4 times larger surface motions than ERS-1. TerraSAR-X has a shorter wavelength (X-band) but a much larger range single bandwidth (150 MHz and 300 MHz), resulting in higher upper gradient limits. NISAR has two distinct wavelengths: S-band, and L-band. Both wavelengths are intended to offer a wide variety of possible range bandwidths (10, 25, 37.5, 75 MHz for S-band; and 5, 20, 40, 80 MHz for L-band). We selected 75 MHz for S-band and 80 MHz for L-band sensors to show the maximum capability of detectable deformation for NISAR. The specification of NISAR could measure about-20-times-larger surface motion than ERS-1. It is evident from this table that radar satellite technology has evolved significantly over the past 30 years. The lower gradient limit is characterized by two bottom lines: cycle-slicing limit and small-gradient limit. The upper gradient limit represents the maximum border of the phase change inside one pixel as one fringe, where the cycle slicing limit specifies the minimal phase change where incoherent noise within a pixel does not overwhelm the displacement information. In general, phase differences of less than one-tenth of a fringe are challenging to identify [46]. The cycle-slicing limit could be defined as Equation (19). Consequently, the cycle slicing limit is about 12.5 mm for L-band, 6 mm for S-band sensors, 2.8 mm for C-band, and 1.6 mm for X-band (Table 8). In other words, surface motion is detectable for C-band sensors if the motion is larger than 2.8 mm. Unlike the upper and lower gradient limits, the cycle-slicing limit is a fixed value for a sensor and does not vary with different spatial scales. Therefore, no matter how large or small the image frame is, the cycle-slicing limit for C-band sensors is around 2.8 mm. Wavelength is the decisive factor for the cycle-slicing limit. Longer wavelengths uplift the limit while shorter wavelengths lower it. Accordingly, the cycle-slicing limit of L-band sensors is about 4-5 times larger than that of C-band, meaning tiny surface displacements are less detectable for L-band. On the other hand, X-band possesses better sensitivity over small surface motions, with a cycle-slicing limit at 1.6 mm. \[Cycle\text{-}slicing\ limit=\frac{1}{10}\times fringe=\frac{1}{10}\times \frac{\lambda}{2}=\frac{\lambda}{20} \tag{19}\]  
The other bottom line is the small-gradient limit, which is formed because of the mixture of all long wavelength error sources. These sources include the flat-Earth phase, the uncertainty of orbital inaccuracies (orbital error), long wavelength atmospheric gradients, and long wavelength displacements [28; 46]. Since the fringe patterns of the long wavelength signals are similar, the phase variation caused by long wavelength displacements can easily be misinterpreted as other longer wavelength errors [125; 126]. For the application of deformation mapping, the similarities cause bias in the estimation of the correct deformation.