# Cross Sensor Simulation of Tomographic SAR Stacks
## 4 Features of the Simulated Stack  
The correct interpretation of the simulated stack of images is not trivial; some features can be understood by exploring the processing chain outlined in Section 3. Again, the real system will be referred to as \"airborne\" whereas the simulated one as \"spaceborne\" although this layout is not mandatory. Also, this paragraph focuses on the impact of the \(kz\) distribution only rather than both \(ky\) and \(kz\) due to the 1D nature of the issues here raised. In general, the tomographic airborne system makes available some wavenumbers of the target whereas the spaceborne system requests others. Each image of the airborne system samples a specific \(kz\) so that a sampled version of the Fourier transform of the vertical reflectivity profile of the target is collected. It follows that the reconstruction of the reflectivity profile, obtained for example through back-projection, is periodic. This periodicity implies that its Fourier transform exists exclusively in correspondence of the original \(kz\) values, it goes to zero for every other value. Should the spaceborne system require a new \(kz\) then some expedient is needed. A possible solution amounts to interpolating the signal in correspondence of the desired \(kz\) value; this, in turn, corresponds to windowing the reconstructed reflectivity profile lying in the \(z\) domain. It follows that the reflectivity profile may be reconstructed for a limited range of elevations avoiding the replicas. This precaution saves computational time when computing the back-projection and, at the same time, allows the generation of SLC images in correspondence of new \(kz\) values. The generation of data in correspondence of very large \(kz\) values (outside from the sampled range) is possible only through extrapolation and hence it should be avoided as it amounts to fabricating details. The aforementioned concepts are strictly valid for regular spatial sampling only; highly irregular sampling of the \(kz\) axis in the airborne leads to artifacts in the simulated SLCs. Windowing the reflectivity profile amounts to a convolution between the signal and the kernel of the window, for instance a \(sinc\). Should two subsequent sampled \(kz\) be too far away, then the interpolated signal in the middle will be weaker; that is, the synthetic SLC corresponding to a \(kz\) far from the sampled ones will be less powerful. Also, due to the high variability of the \(kz\) distribution in airborne surveys this power loss is dependent on the spatial coordinate. The synthetic SLCs may exhibit power fluctuations due to the space varying sampling of the \(kz\) axis. However, the tomographic profile estimated with the synthetic images is consistent with the one resulting from the original one up to resolution losses. Instability in the \(kz\) distribution of the airborne survey leads to fluctuations in the interferometric phase, in the simulated spaceborne stack this latter is exchanged for amplitude fluctuations of the SLC images. This effect is shown in Figure 9.  
Figure 8: Tomographic profiles returned by the back-projection algorithms when applied to (**a**) real airborne data and (**b**) simulated spaceborne stack. The two axes are analogous to the ones shown in Figure 5; the black line shows the ground topography below the forest layer estimated through SAR tomography [25]. Figure 9: As discussed in Section 4, platform deviations lead to fluctuations in the interferometric phase. The straight orbits of the (simulated) spaceborne system removes phase fluctuations by exchanging them for power oscillations. It must be observed that the simulated stack is not affected by the propagation of the signal more than the real stack is. For a spaceborne simulation to be more realistic the effects of the propagation of the wave through troposphere and ionosphere should be included.