## 2 Overview of InSAR Methodology
### Noise Signals Corrupting SAR Interferograms
Generalized models for the multi-looked interferometric noise have also been proposed in [60]. They are based on the observation that a restriction \(\varphi_{n}\in[\varphi_{x}-\pi,\varphi_{x}+\pi]\) does not apply when the phase is considered in the complex plane instead of the real plane. These models describe the real and imaginary parts of the complex phasor \(\exp[j\varphi_{n}]\) as:  
\[\text{Re}\{\exp[j\varphi_{n}]\}=\text{N}_{c}\cos(\varphi_{x})+\
u_{n1}^{ \prime}\cos(\varphi_{x})-\
u_{n2}^{\prime}\sin(\varphi_{x}) \tag{23}\]  
\[\text{Im}\{\exp[j\varphi_{n}]\}=\text{N}_{c}\sin(\varphi_{x})+\
u_{n1}^{ \prime}\sin(\varphi_{x})+\
u_{n2}^{\prime}\cos(\varphi_{x}) \tag{24}\]  
where \(\text{N}_{c}\) is the expected value of \(\cos(\
u_{n})\) and \(\
u^{\prime}{}_{n1}\) and \(\
u^{\prime}{}_{n2}\) are two zero-mean random variables. \(\text{Re}\{\cdot\}\) and \(\text{Im}\{\cdot\}\) denote the extraction of the real and imaginary parts operations, respectively. The combination of Equations (23) and (24) in the complex plane results in the following noise model for the complex phasor \(\exp[j\varphi_{n}]\):  
\[I=\hat{\gamma}_{1}\cdot\hat{\gamma}_{2}^{*}=|\hat{\gamma}_{1}\cdot\hat{\gamma}_ {2}^{*}|\exp[j\varphi] \tag{17}\]0 Phase standard deviation vs. spatial coherence for different values of the number of independent looks. \[I=\hat{\gamma}_{1}\cdot\hat{\gamma}_{2}^{*}=|\hat{\gamma}_{1}\cdot\hat{\gamma}_ {2}^{*}|\exp[j\varphi] \tag{17}\]1  
It can be demonstrated that, for large values of the multi-look factor \(n\) (and in the case \(k\
eq 0\)):  
\[I=\hat{\gamma}_{1}\cdot\hat{\gamma}_{2}^{*}=|\hat{\gamma}_{1}\cdot\hat{\gamma}_ {2}^{*}|\exp[j\varphi] \tag{17}\]2  
and the standard deviation of the two zero-mean random variables \(\
u^{\prime}{}_{n1}\) and \(\
u^{\prime}{}_{n2}\) are:  
\[I=\hat{\gamma}_{1}\cdot\hat{\gamma}_{2}^{*}=|\hat{\gamma}_{1}\cdot\hat{\gamma}_ {2}^{*}|\exp[j\varphi] \tag{17}\]3  
where \(\alpha\) is a constant value [60]. Interested readers can also find a comprehensive derivation of the noise model for multi-looked multi-dimensional SAR data in [60]. As earlier anticipated, the InSAR height measurement accuracy strictly depends on the InSAR phase measurement accuracy. InSAR phase can be modelled with the pdf function in Equation (20), which depends on the spatial coherence k of the given SAR pixel. Therefore, it is worth to discuss the sources of noise decorrelation that may impact on the value of the measured spatial coherence. It is known in the literature that the cross-correlation factor (see Equation (21)) can be profitably factorized to make it evident the single sources of decorrelation as follows [41]:  
\[I=\hat{\gamma}_{1}\cdot\hat{\gamma}_{2}^{*}=|\hat{\gamma}_{1}\cdot\hat{\gamma}_ {2}^{*}|\exp[j\varphi] \tag{17}\]4  
where \(\chi_{thermal}\) depends on RADAR thermal noise, \(\chi_{spatial}\) depends on geometric relations between the two SAR acquisitions, \(\chi_{Doppler}\) takes account of reduced azimuth spectral overlap of the two interfering SAR images due to their difference of the Doppler Centroid (DC) frequencies, \(\chi_{misreg}\) accounts for possible misregistration errors and, finally, \(\chi_{temp}\) takes into account temporal decorrelation effects, also including the changes over time of the scattering centres location within a volume. A few additional observations on the decorrelation noise sources are now in order. (i) The loss of correlation due to thermal noise is given by:  
\[I=\hat{\gamma}_{1}\cdot\hat{\gamma}_{2}^{*}=|\hat{\gamma}_{1}\cdot\hat{\gamma}_ {2}^{*}|\exp[j\varphi] \tag{17}\]5  
where SNR denotes the signal-to-noise ratio. (ii) The spatial decorrelation arises from the fact that the same ground resolution cell is imaged from slightly different illumination angles, while the master and slave images are collected.