# Along-Track Multistatic Synthetic Aperture Radar Formations of Miniatellites
## 5 System Limitations
### Across-Track Baseline Effects
Figure 9: Impact of an off-track shift of the sensor on the raw data, in the presence of surface topography. (**a**) geometry, notice the wide azimuth antenna aperture represented in orange line; (**b**) topographic phase screen that is superposed to raw data. The green band defines a range resolution cell. that amounts to about 640 m. This is not so hard requirement to keep, although the presence of a moderate slope of 10\({}^{\circ}\) would reduce it to 400 m. In the rough topography shown in Figure 9, the compensation should be performed on a local base that in turn would require to implement the SIMO image processing in a back-projection framework. This would possibly solve the problem, although, being the SIMO recombination dependent on wavelength \(\lambda\), it could be possible, for very large frequency shifts, to get different performance from the combination. However, the worst case is the one shown in Figure 8b, where multiple targets at different height contribute to the same range with different phase. In fact, even if we were able to perform an adaptive SIMO recombination that integrates a local spectral shift filtering [26], we still would get noise from multiple contributions within a single resolution cell. The impact on the random phase combination that would result can be evaluated in terms of coherence between the ideal data \(d_{0}\) and the actually acquired one, \(d_{B}\), and relates coherence to SNR:  
\[\gamma=\frac{E\left[d_{B}d_{0}^{*}\right]}{\sqrt{d_{B}d_{0}}}=\frac{1}{1+SNR^{ -1}}. \tag{28}\]  
The coherence can be evaluated in terms of the statistics of the target's height dispersion. For a uniform dispersion in a box of elevation \(\Delta q\)[24; 27], \(\gamma=\text{sinc}\left(\Delta q/q_{2\pi}\right)\), where the elevation ambiguity is inversely proportional to baseline:  
\[q_{2\pi}=\frac{\lambda r_{0}\sin\theta}{2B_{n}} \tag{29}\]  
Assuming to tolerate a slight signal to noise ratio degradation, we can approximate Equation (28):  
\[\gamma\approx 1-SNR^{-1}\approx 1-\frac{\pi^{2}}{6}\left(\frac{ \Delta q}{q_{2\pi}}\right)\rightarrow \tag{30}\] \[q_{2\pi}\gg\sqrt{1.6\cdot SNR}\cdot\Delta q.\]  
This means that, if we assume a dispersion of target in a box of \(\Delta q=50\) m, like, for rough topography, mostly urban, and we tolerate an SNR no worse than 10 dB, the elevation of ambiguity should be \(q_{2\pi}>4\Delta q=200\) m. In turn, the maximum offset normal to range should not exceed \(B_{n}=40\) m. This is indeed the most stringent requirement to be fulfilled. If the across track baselines of the satellites are supposed to be perfectly known, the unique unknown affecting the cancellation of the \(R\) ambiguities is the height of the target. The phases with which the target contributes to the \(R\) ambiguities are also related to the baselines. In order to approximately evaluate this effect, let us consider the decrease of ambiguity cancellation when the height error \(qh\) is very small, so that it is possible to linearize. Then, instead of the perfect closure of the phases that would allow for annihilate the interfering ambiguity, a noise component is added to the data. We can briefly approximate the impact of the dispersion of the cross-track baselines on the ambiguity suppression AASR. To do so, we will suppose that the azimuth ambiguity has already been canceled and the receivers are exactly at their optimal positions, so that we will consider only the impact of the cross-track baselines. In this case, the residual ambiguity affecting another pixel at the ambiguity distance will be:  
Figure 7:0  
where \(\Delta q\) is the height error, \(N\) is the number of the receiving satellites with returns to be combined to remove the ambiguity, \(x\) is the local reflectivity, and \(Q_{n}\):  
Figure 7:1are the altitudes of ambiguity of the \(n=1,...,N\) satellites, with positive and negative cross track baselines \(B_{n},\theta\) is the incidence angle, \(r\) is the distance. Obviously, if \(\Delta q\) is zero, the ambiguity is canceled.