## 2 Overview of InSAR Methodology
### Noise Signals Corrupting SAR Interferograms
Mathematically, under the hypothesis that SAR processes are stationary, jointly stationary and ergodic (in mean), the ensemble value can be substituted with spatial averages, thus resulting in the so-called sample co-variance:  
\[Z_{n}=E[I]=E[\hat{\gamma}_{1}\cdot\hat{\gamma}_{2}^{*}]=\langle\hat{\gamma}_{ 1}\cdot\hat{\gamma}_{2}^{*}\rangle_{n}=\frac{1}{n}\sum_{i=1}^{n}\hat{\gamma}_ {1}\cdot\hat{\gamma}_{2}^{*}=I_{n}\exp[j\varphi_{n}] \tag{18}\]  
where \(n\) represents the number of average neighbouring samples, the symbol \(\langle\cdot\rangle_{n}\) denotes the spatial average operation and \(I_{n}\) and \(\varphi_{n}\) represent the amplitude and the phase of the \(n\)-multi-looked SAR interferogram, respectively. Therefore, the multi-looking procedure leads to a worsening of the spatial resolution of InSAR measurements of factor \(n\). Nevertheless, this does not represent a serious drawback because, as a counterpart, the average operation leads to a reduction of the phase noise standard deviation, as outlined hereinafter. Statistics of the multi-looked interferometric phase have been investigated in several papers [6; 57; 58; 59; 60]. In particular, the InSAR phase has been modelled as characterized by an additive noise [60]:\[\varphi_{n}=\varphi_{n}+\
u_{n} \tag{19}\]  
with \(\varphi_{x}\) being the true phase and \(\
u_{n}\) denotes a zero-mean additive noise the variance of which is independent of \(\varphi_{x}\). The InSAR multi-looked phase is described by the following pdf:  
\[p_{\varphi_{n}}(\varphi_{n})=\frac{\beta\sqrt{n}}{2\sqrt{\pi}}\cdot\frac{ \left(1-\left|k\right|^{2}\right)^{n}}{\left(1-\beta^{2}\right)^{n+0.5}},\ \ k\cos( \varphi_{n})\
eq 0 \tag{20}\]  
where \(\beta=k\cos(\varphi_{n}-\varphi_{x})\) and the term \(k\) denotes the spatial coherence, which represents the amplitude of the cross-correlation factor \(\chi\) between \(\hat{\gamma}_{1}\) and \(\hat{\gamma}_{2}\) defined as follows:  
\[\chi=\frac{\left\langle\hat{\gamma}_{1}\cdot\hat{\gamma}_{2}^{*}\right\rangle_{ n}}{\sqrt{\left\langle\left|\hat{\gamma}_{1}\right|^{2}\right\rangle_{n}\cdot \left\langle\left|\hat{\gamma}_{2}\right|^{2}\right\rangle_{n}}} \tag{21}\]  
Figure 4: Pdf of the interferometric phase for different values of the (**a**) multi-look factor (with a fixed, given value of spatial coherence that is equal to 0.5) and for different values of the (**b**) spatial coherence (with a fixed value \(n=50\)). Figure 4a,b sketches the pdf in Equation (20) for different values of the number of looks (with a given value of the spatial coherence equal to 0.5) and for different values of the spatial coherence (for a fixed value of the number of looks equal to 50), respectively. Plots depicted in Figure 4 are obtained by imposing \(\varphi_{x}=0\). It is straightforward to note that the pdf distribution of the multi-look phase (see Equation (20)) is symmetrical with respect to its mean value (i.e., \(\varphi_{x}\)). This is in agreement with the additive model of Equation (19). It is worth noting that the validity of the model (19) is restricted to the [\(\varphi_{x}-\pi\), \(\varphi_{x}+\pi\)] interval. A closed-form for the standard deviation of the interferometric phase is hard to investigate, unless in the limit of Cramer-Rao, which is valid for high values of the spatial coherence k, where it assumes the asymptotic value [60]:  
\[\sigma_{\varphi_{n}}=\frac{1}{\sqrt{2n}}\cdot\frac{\sqrt{1-k^{2}}}{k} \tag{22}\]  
Figure 5 plots the standard deviation of the multi-looked phase vs. the spatial coherence for different values of \(n\). For example, if we consider a SAR pixel with given spatial coherence \(k=0.35\) and we assume \(n=80\), we obtain the height estimate has a standard deviation of about 1.5 m (by assuming a perpendicular baseline of the interferogram of 200 m). In the calculations, we took into account asymptotical formulations in Equations (12) and (21) and we used the parameters of ASAR/ENVISAT sensor. Note that the obtained value represents a lower bound for the height measurement accuracy.