# A Study of Landfast Ice with Sentinel-1 Repeat-Pass Interferometry over the Baltic Sea
## 2 Fundamentals of InSAR Theory  
The signal measured by SAR is essentially complex and can be represented in terms of real and imaginary parts or in terms of signal magnitude and phase. The phase of a single SAR image does not have any practical usage, but with two images acquired with slightly different viewing angles (interferometric pair), their phase difference after co-registration can be used to measure topography [16] and terrain displacement. The coherence between two complex SAR images, \(s_{1}\) and \(s_{2}\), is the normalized complex cross-correlation between two complex signals (master and slave images). Coherent signals are used for interferometry [17]. The coherence \(\gamma\) is formulated as [12; 18]:  
\[\gamma=\frac{\langle s_{1}s_{2}^{*}\rangle}{\sqrt{\langle s_{1}s_{1}^{*} \rangle\times\langle s_{2}s_{2}^{*}\rangle}},\;0\leq|\gamma|\leq 1, \tag{1}\]  
where \(s_{1}\) and \(s_{2}\) are two complex SAR image values, and \(\langle\ldots\rangle\) denotes spatial averaging operation. The interferometric SAR coherence is also a complex value and can be represented in terms of the coherence magnitude and InSAR phase [18]. The coherence describes stability of the scattering process and therefore buildings and fixed land features have high levels of coherence, whereas changing and vegetated areas have low values of InSAR coherence. Open sea loses the InSAR coherence completely even within tens of milliseconds [10; 19]. The landfast ice may exhibit both relatively high and low levels of coherence depending on the actual situation and parameters of InSAR measurement. Several factors contribute to the magnitude of coherence:  
\[\gamma=\gamma_{Thermal}\times\gamma_{Process}\times\gamma_{Spatial}\times\gamma_{ Temporal}, \tag{2}\]where \(\gamma_{Thermal}\) is related to the signal to noise ratio (SNR) of the system and caused by thermal noise in receiver; \(\gamma_{Process}\) is processing decorrelation coming from errors in image interpolation, co-registration or spectral filtering; and \(\gamma_{Spatial}\) is signal decorrelation due to the incidence angle differences, proportional to the perpendicular baseline B\({}_{\perp}\) which is dividing the two SAR antennas. As spatial coherence depends on baseline, it is called baseline decorrelation. The \(\gamma_{Temporal}\) in the scattering space is the correlation factor for incoherent changes between satellite acquisitions. It is the most variable contributor while other decorrelation factors are almost stable. This component is the main factor for coherence losses. It can measure decorrelation due to ice movement or changes in scattered positions within a resolution cell [12]. Figure 1 shows how the phase difference between the two SAR images measured by the InSAR technique. The phase of each of the master and slave images shows random appearance. A phase interferogram is created by subtracting the phase of the \"slave\" from that of the \"master\" image. These phase differences become useful when a high coherence image is achieved. Equations (3)-(5) show how these phase differences \(\Delta\varphi\) are calculated:  
\[\varphi_{1}=\frac{4\pi R}{\lambda}, \tag{3}\]  
\[\varphi_{2}=\frac{4\pi\times(R+\Delta R)}{\lambda}, \tag{4}\]  
\[\Delta\varphi=\varphi_{2}-\varphi_{1}=\frac{4\pi\Delta R}{\lambda}, \tag{5}\]  
where \(\varphi_{1,2}\) is the phases of each of master and slave images, \(\lambda\) is the satellite wavelength, and \(R\) is satellite to target distance. The phase difference can be represented as a combination of six sources. Equation (6) shows their contribution to SAR interferometric phase:  
\[\Delta\varphi=\Phi_{flat-earth}+\frac{4\pi B_{T}}{\lambda Rsin(\theta)}\times \Delta h+\frac{4\pi}{\lambda}\Delta\eta+\Phi_{path,\text{ {Atm}}}+\Phi_{noise}+\text{mod}(2\pi), \tag{6}\]  
where \(\theta\) is the incidence angle, \(B_{T}\) is perpendicular baseline, \(\Delta h\) represents the topographic height, and \(\Delta\eta\) represents the relative scatterer displacement projected on the slant range direction [11].