## 5 Numerical Simulations and Comparison
### Extended Source
To facilitate the comparison, the color scale is the same for these two images and for the brightness temperature \(T\) shown in Figure 19: there is no significant difference between \(\mathcal{T}=W\star T\) and the two retrieved maps shown here (in both cases, the RMSE is much less than 0.01 K). The second case considered here corresponds to a situation that has not been widely simulated nor reported so far: this is the case where the brightness temperature distribution of the observed scene extends beyond the field of view synthesized by the antenna array. In such a situation, field aliasing has to be observed, depending on the geometry of the antenna array and on the spacing between the elementary antennas. The brightness temperature distribution \(T(\xi)\) of an extended source set to 100 K in the whole space in front of the antenna array of Figure 2 is shown in Figure 21, together with the corresponding complex visibilities \(V\) and \(\widetilde{V}\). Shown in Figure 22 are the brightness temperature maps \(\widetilde{\mathcal{T}}_{r}\) and \(\mathcal{T}_{r}\) retrieved after inverting the complex visibilities \(\widetilde{V}\) and \(V\) of Figure 21 according to (9) and (10). Here again, in every case, a Blackman windowing function \(W\) has been used to damp the reconstructed maps \(\widetilde{T}_{r}\) and \(T_{r}\). As the brightness temperature \(T\) of the observed scene is not set to 0 K outside the field of view synthesized by the antenna array, field aliasing is expected from the six neighbors to that hexagon. This is exactly what can be observed in Figure 22, where only the central part of the hexagonal field of view is free from any aliasing artifact. However, even in the absence of any modeling error or noise, small differences can be observed between \(\mathcal{T}=W\star T\) and the two retrieved maps \(\widetilde{\mathcal{T}}_{r}\) and \(\mathcal{T}_{r}\). More precisely, here, the floor error in that alias-free region of the synthesized field of view is about 0.14 K for the far-field case, whereas it is of the order of 0.53 K for the near-field one. In real situations, this floor error can be efficiently reduced down to a much lower level with the introduction of an artificial scene [53], as close as possible to the observed one, whose visibilities are removed prior to inversion and whose brightness temperature distribution is added back after inversion. However, referring back to Figure 4, such a difference at this raw level of reconstruction was already expected when comparing the singular values of \(\mathbf{G}\) and \(\widetilde{\mathbf{G}}\) and especially those discarded prior to the pseudo-inversion, which are quite different. Finally, shown in Figure 23 are the variations of this floor error with the observing distance \(h\) from the phase center of the antenna array. Although it decreases very rapidly to reach a plateau with a value very similar to that obtained in far-field conditions, it is still very difficult to find a value that makes a clear separation between near-field and far-field regions. With this figure of merit, one can again at most consider a transition zone from the Fresnel region to the Fraunhofer one between \(2D^{2}/\lambda\) and \(10D^{2}/\lambda\). Figure 21: Brightness temperature distribution \(T(\xi)\) of an extended source that is set to 100 K in front of the antenna array of Figure 2. Shown in green are the corresponding complex visibilities \(\widetilde{V}\) when the antenna array is in the far-field region, while those in red are the complex visibilities \(V\) when the phase center of the array is placed at a distance \(h=100\) m from the source in the boresight direction.