## 2 Instrument operating characteristics
### Geometry of the system  
Both swaths are the results of an off-nadir observation by two passive antennas separated by a mast (see Figure 1 (a)). Compared to conventional SAR interferometry (see [5]), look angles are kept small to get a good level of return power while getting the widest swath possible. The optimal working mode consists in transmitting from one given antenna once every two pulses, while the reception is made by both antennas, to permit further interferometric processing (see Figure 1 (a)). The electronics remains on the central payload to prevent propagation losses all along the interferometric baseline. Then both antennas are tilted by 45 degrees from the mast and double polarization feeds enable both swaths to observe and distinguish signal returns (see Figure 1 (b)).  
The purpose of this subsection is to figure out how the real height and position of an observed target within one given swath is derived from the collected data. The determination of the altimetric measurement requires the precise knowledge of four main parameters: the altitude \(H\) given by the POD (Precise Orbit Determination), the range \(r_{I}\) from the master antenna to one pixel coming from the onboard clock, the roll angle \(\alpha\) and the interferometric phase \(\Phi\), calculated onboard. The roll angle, whose a priori knowledge is given by the PAD (Precise Attitude Determination), happens to be a critical parameter and has then to be accounted for. A dedicated calibration process using crossover data will also have to be implemented to reach optimal performances.  
As illustrated on Figure 2 (a), let us consider a target inside the right hand-side swath. For pulses of wavenumber k, the parameters quoted previously enable us to calculate the height \(h\) above the plane tangential to the surface at nadir:  
\[h=H-r_{\mathrm{i}}\cos\!\left(\arcsin\left(\frac{\Phi}{kB}\!\left[1+\frac{1}{2} \frac{\Phi}{kr_{\mathrm{i}}}\right]-\frac{1}{2}\frac{B}{r_{\mathrm{i}}}\right) +\alpha\right)+\frac{B}{2}\!\cdot\!\sin\left(\alpha\right) \tag{1}\]  
and its position X as:  
\[X=r_{\mathrm{i}}\sin\left(\theta\right) \tag{2}\]  
Figure 2: (a) Geometry of the interferometric measurement (b) Local Frame Transformation.  
Figure 1: (a) Lay-out of the geometry of observation (b) Antenna and feeds configuration arbitrarily for the right swath; on the other side, feeds’ polar are reversed (Emission/Reception H Feed and Reception. V Feed); a V-polarization pulse is emitted by one antenna, and the consecutive pulse, with H-polarization, is emitted by the second antenna; both polarizations are received by the two antennas.  
The height measurement \(h\) and the exact position of the target \(X\) inside the swath, primarily referenced against the tangent plane and reference altitude at nadir, are more conveniently expressed in a local frame: \(h_{real}\) and \(X_{real}\) (see Figure 2 (b)).  
\[h_{real}=\frac{R_{n}+h}{\cos\left(A\right)}-R_{t} \tag{3}\]  
\[A=\arctan\left(\frac{X}{R_{n}+h}\right) \tag{4}\]  
POD performances directly impact the knowledge of \(r_{I}\) and then \(h_{real}\). The precise knowledge of the roll angle comes from the PAD and from a post processing procedure using crossovers estimation (see Section II.3). Last but not least, the quality of the onboard estimation of the interferometric phase is primitive and needs to be further investigated. It should be noted that the phase unwrapping process is not a critical point in our configuration (unlike standard SAR interferometry applications). Indeed, the altitude of ambiguity (see [6]) ranges from 25 m at the NR (Near Range) to 170 m at the FR (Far Range), which is much larger than the few tens of centimeters magnitude of expected observed phenomena.