## 2 Instrument operating characteristics
### Interferometric phase  
The interferometric phase measurement quality is strongly related to the coherence value \(\gamma_{\mathrm{T}}\) between the two receiving channels. This coherence is mainly affected by three factors: the instrument signal to noise ratio, the range migration within the swath, and the speckle (its effect is also called geometric decorrelation (see [7]). An onboard process enables to correct the impact of speckle effect (see [1; 2]). Furthermore, accounting for the very small range of incidence angles (less than 5\({}^{\circ}\)), the second term becomes negligible compared to the first term, directly related to the instrument features. Therefore the coherence between signals is mainly driven by the SNR:  
\[\gamma_{N}=\left(1+\frac{1}{SNR}\right)^{-1} \tag{5}\]  
The interferometric phase statistics are distributed according to the Wishart distribution (see [8]), depending on the number of looks and the coherence between the two received signals. Assuming both signals have complex circular Gaussian distributions, the computation of the argument of their product of correlation leads to the following probability density function of the interferometric phase (see [9; 10; 11; 12; 13]):  
\[pdf\ \ ^{(x)}(\varphi\ )=\frac{\Gamma\left(\ N\ +\ \frac{1}{2}\right)\left( \left\|\gamma\right\|^{2}\right)^{N}\left|\varphi\right|\cos\ \left(\ \Phi\ -\ \Phi\ \right)}{2\,\pi}\,F\left(\ N\,1;\frac{1}{2};\left| \gamma\right|^{2}\ \cos\ ^{2}\left(\Phi\ -\ \Phi\ \right)\right)} \tag{6}\]  
where \(\Gamma(.)\) is the Gamma function and F(.) the hypergeometric function (see [14]).  
Figure 3 (a) illustrates the dependence of the pdf on coherence \(\gamma_{\mathrm{T}}\) for a single look and Figure 3 (b) illustrates its dependence on the number of looks for a typical correlation of 0.9 (see next subsection). These plots show that the instrument parameters (related to the SNR) must be selected to guarantee the  
highest possible degree of coherence to avoid the need for extensive multi-look processing, which would result in poorer intrinsic spatial resolution.  
The phase variance can be numerically calculated from (6); above a certain number of looks, it can be shown that it can be well approximated by the Cramer-Rao bound (see [15]), in the case of high coherence values (see Figure 4):  
\[\sigma_{\Phi}^{2}=\frac{1}{2N}\frac{1-\gamma^{2}}{\gamma^{2}} \tag{7}\]  
This approximation strengthens the need for a large degree of coherence. The next subsection will confirm this behaviour when entering the details of the WSOA operating characteristics.  
Therefore, the instrument SNR has to be studied carefully, as the main contributor to the decorrelation between the two channels. The analysis of the derived degree of coherence in the swath will then fix the multilooking strategy to be implemented. Indeed, the objective of such a strategy will  
Figure 4: Statistics of the interferometric phase in multilooking context.  
Figure 3: Interferometric phase densities of probability versus coherence (a) and multi looking (b).  
be to reach the desired standard deviation of the estimated interferometric phase that keeps the height performances within the range of constraints specified by the targeted scientific objectives.