## 3 Robust Multiscale AT InSAR  
As mentioned previously, the Along Track formation relies on precise control of the along-track separation in order to avoid collision between the leader (or chief) and follower. This entails a risk due to the typical drift between satellites in case the orbit control was not active for a period of time. On the other hand, the possible need for two different baseline scales simultaneously multiplies the risk, as there are now three possible collision events if only one more follower satellite were added to the configuration. [PERSON] showed in [32] a description of a relative orbit in terms of Relative Orbital Elements \(\delta\alpha\ =\ (\delta a,\ \delta\lambda,\ \delta\epsilon,\delta i)\), where \(\delta\epsilon=(\delta\epsilon x,\ \delta\epsilon y)\) is the relative eccentricity vector and \(\delta i=(\delta i_{x},\ \delta i_{y})\) is the relative inclination vector. The orbit phase difference is given by \(\delta\lambda\), while the semi-major axes' relative difference is \(\delta a\). A safe formation is guaranteed when \(\delta\epsilon\) and \(\delta i\) are parallel or anti-parallel, for \(\delta a=0\). A strict AT-InSAR formation only has an orbit phase difference \begin{table}
\begin{tabular}{c c c} \hline
**Baseline** & **SAR Type** & **Measurement and Application** \\ \hline \(\Delta\varphi\) (_look angles difference_) & Across-Track & Topography, digital elevation models \\ \(\Delta t\ =\ ms,\ldots,\ s\) & Along-Track & Ocean currents, moving object detection \\ \(\Delta t\ =\ days\) & Differential & Glacier/ice fields, lava flows, hydrology \\ \(\Delta t\ =\ days\), \(\ldots,\ years\) & Differential & Subsidence, seismic events, volcanic activities, \\ \(\Delta t\ =\ ms,\ldots,\ years\) & Coherent Estimator & Sea-surface decorrelation times, land cover classification \\ \hline \end{tabular}
\end{table} 
Table 1: Some baseline types for interferometric SAR [28]. Figure 5: AT-InSAR baseline difference. between satellites, thus it is not possible to guarantee the safe condition (as \(\delta\epsilon=\delta i=\underline{0}\), here parallelism cannot be evaluated). Here a variation of this along track formation is proposed and as follows:  
1. To add a small helix component to each follower relative to the chief, where the across-track component is one order of magnitude smaller than the chief/follower along-track baseline. 2. To scale the chief/follower relative eccentricity and relative inclination vectors in order to generate low-risk \"pipes\" for each satellite. Figure 6 shows a particular case of a parallel relative eccentricity and relative inclination vectors for two followers with respect to the chief. Notice that the difference between the followers also preserves the relative eccentricity and relative inclinations vectors as collinear, therefore achieving a safe condition. This formation can include more followers by adding other scales on the same _axis_, preserving the collinearity between the relative eccentricity and inclination vectors for the given follower. On the other hand, different along-track baselines could be chosen for each of these followers. As the chief orbit is sun-synchronous and frozen, the satellite altitude and relative orbit baselines are guaranteed to repeat for the same latitude; hence the interferogram products generated on each pass have geometric coherence between different passes, enabling to perform differential interferometry by taking a set of images generated by the SAR system for the same zone. Moreover, the multiple along-track channels make it possible to track different velocity ranges for the targets on the Earth's surface, see [29], which can also be compared along the same pass by adding followers with different along-track baselines with the safe configuration previously presented.