*2.3. SAR Differential Interferometry Background*

The fundamentals of the conventional DInSAR technique have been presented in many publications [29,31,70]. Therefore, only some aspects relevant to this study are briefly described.

In principle, SAR interferometry exploits the information in the interferometric phase, calculated as the phase difference between two coregistered SAR images acquired from slightly different orbit positions (spatial baseline) and different times (temporal baseline). The interferometric phase (*φint*) is the sum of contributions from several factors, and the following equation can express this:

$$
\phi\_{int} = \phi\_{displ} + \phi\_{topo} + \phi\_E + \phi\_{atm} + \phi\_{noise} \tag{1}
$$

where *φdispl* represents the phase due to surface displacement, *φtopo* refers to the phase caused by local topography (or topographic phase), *φE* is the phase produced by a surface of constant elevation on a spherical Earth (curved Earth), also known as the orbital phase, *φatm* denotes the phase components due to the variation of atmospheric conditions between the image acquisitions (the so-called atmospheric phase screen (APS)), and *φnoise* includes all the phase noise contributions that corrupt the interferometric SAR signal.

All other contributors to the interferometric phase must be removed or diminished, to obtain the Earth surface displacement measurement. Using external DEM and precise orbital information, phase contributions caused by topography and the curved Earth can be estimated and removed from the interferometric phase. This is the basic concept of the Differential SAR Interferometry (DInSAR) approach. However, the differential interferometric phase can still contain some "unwanted" phase components. APS is one of the main sources of errors that influence the differential interferometric phase, and it can degrade the accuracy of surface displacement estimates using DInSAR. Topographic and orbital errors can also contribute to the differential interferometric phase.

The accuracy of surface displacement measurements from DInSAR greatly depends on the quality of the differential interferometric phase. The established criterion to measure the quality of the differential interferometric phase is the value of the complex correlation coefficient, the so-called coherence. The coherence (*γ*<sup>ˆ</sup>) is a measure of phase correlation (or phase reliability) between two complex SAR images, M (master image) and S (slave image), and is defined as [71]:

$$\hat{\gamma} = \frac{\frac{\sum\_{i=1}^{N} M\_i \ S\_i \text{\*}}{\sqrt{\sum\_{i=1}^{N} \left| \mathcal{M}\_i \right|^2} \sqrt{\sum\_{i=1}^{N} \left| \mathcal{S}\_i \right|^2}}} \tag{2}$$

where *S*∗ is the complex conjugate of the complex slave image (S), |M| and |S| are the amplitude of complex SAR master and slave image, respectively, and N indicates the spatial set of samples employed in the coherence estimation. The coherence values lie in the range 0 ≤ *γ*ˆ ≤ 1; a value of zero indicates complete incoherence and a differential interferometric phase value with no useful information, whereas a value of one indicates complete coherence and a differential interferometric phase value with no noise. DInSAR is effectively applied only in areas where the differential interferometric phase is characterized by high coherence. The main factors that affect the coherence are temporal and spatial decorrelation and low accuracy of image coregistration.

The temporal decorrelation phenomenon is caused by changes in the physical and geometric properties of the scatterers on the Earth's surface [72]. Some of the main sources of temporal decorrelation are erosion, vegetation growth, cultivation, snow, and nearsurface moisture changes.

Spatial or geometric decorrelation may result from high variations in imaging geometry. Thus, images with a short spatial baseline must be selected for interferometric processing.

A practical way to overcome the conventional DInSAR limitations mentioned above is combining the information from multiple short-interval differential interferograms, to extract common information. The most basic procedure is to compute integer linear combinations of unwrapped differential interferograms [33] or perform their temporal averaging, the so-called differential interferograms stacking (DIS) approach. The main assumption of this method is that the deformation phase is highly correlated, and the error/noise terms (e.g., APS, signal noise, orbital errors, and nonlinear ground displacements) are uncorrelated between independent pairs. The application of this method increases the signal-to-noise ratio (SNR) and improves the reliability of the Earth surface displacement measurements [73].
