## 2 Instrument operating characteristics
### Instrument performance: SNR and Height Error  
The signal to noise ratio for an extended target (a resolution cell) can be expressed as follows, where the different terms refer respectively to the range loss, the EIRP (Equivalent Isotropic Radiated Power), the effective antenna surface, the range resolution, the azimuth resolution, a global addition of different losses related to the instrument RF and the atmosphere, the reception channel noise, the range compression gain and the observed scene contribution through the backscattering coefficient (see [16-19]):  
\[SNR\ =\frac{P_{r}}{N}=\left(\frac{1}{4\pi R^{2}}\right)^{2}\left(P_{r}G_{r} \right.\left\{\frac{\lambda^{2}}{4\pi}\,G_{r}\right.\left\|\frac{c}{2\,B_{d}\, \sin\left(\theta\right)}\right\|\left(\frac{R\lambda}{L_{ac}}\right)\left( \frac{1}{Loss}\right)\left\|\frac{1}{F_{\omega}\,kT_{0}\,B_{d}}\right.\right\} \left(B_{d}\,T_{\text{\emph{pulse}}}\right)\left(\sigma\,^{0}\right) \tag{8}\]  
Based on published specifications for WSOA ([1, 2]), the derived coherence of the received signals is plotted in Figure 5 (a). Values are larger than 0.9 for more than 90 % of the swath (Note than the centre of the swath benefits from a coherence value close to 0.97; the bell shape stems from an optimization processing described in the following). The performances of the instrument are related to the onboard interferometric phase estimation that is related to the SNR and the desired geometry of the final pixel; the equation providing the expression of the height error can be written as:  
\[\delta h\ =\ \frac{\lambda r_{1}}{2\,\pi B}\frac{\sin\,\left(\theta\right)}{ \cos\,\left(\theta-\alpha\right)}\delta\Phi\ =\ \frac{\lambda r_{1}}{2\,\pi B}\frac{\sin\,\left(\theta\right)}{\cos\,\left( \theta-\alpha\right)}\sqrt{\frac{1}{2\,N}\frac{1-\gamma\,^{2}}{\gamma\,^{2}}} \tag{9}\]  
Figure 5 (b) shows the height errors (expressed in meters) related to a single looking process, that is to say a resolution cell of 13.5*0.6 km in the NR to 13.5*0.1 km in the FR (assuming an orbit height of 1334 km and the antenna beamwidth in the along track direction to be around 0.55\({}^{\circ}\)). The 10 to 25 m  
Figure 5: (a) WSOA signal coherence throughout the swath with and without the onboard correction of the geometric decorrelation (b) Single looking instrumental noise impact on the height budget. order of magnitude is considerably greater than the scientific objectives (4 to 5 centimeters). Even with a stronger coherence, performances would remain very poor. Therefore we cannot avoid an extensive multilooking process. The size of the onboard post-processed pixels has then to be determined to comply with the following conditions: (i) keep it as small as possible to allow additional ground processing procedures to be used for extra averaging in order to lower noise if needed, (ii) make this size large enough to allow for the feasibility of data downloading to Earth terminals, given affordable telemetry rates on usual satellite platforms. Let us also add that because the complex onboard process is irreversible, a small pixel size would be preferable to avoid losing any information from the raw observations. Figure 6 displays a plot of the number of looks and the standard deviation of the height error as a function of the pixel position in the swath. Four pixel dimensions have been tested in the range direction whereas a fixed 16 km scale (corresponding to 13.5 km plus distance moved in 400 ms) is imposed in the azimuth direction. A 1030.05 Hz PRF per antenna is considered. Looks are counted with regard to the onboard incoherent averaging procedures. Ground segment algorithms may adopt an increased number of looks in order to improve the resulting performances: however, ground algorithms are beyond the scope of this performances study. The 4 to 5 cm objective can be reached when the averaging process is applied over 16*14 km pixels (see Figure 6 (b)). If finer data are required, e.g. with 1 km resolution in the range direction, noise is increased by almost a factor of 4.