## 5 Numerical Simulations and Comparison
### Extended Source  
Near-field imaging is no exception to field aliasing; as a consequence, unlike what is often found in the literature, it has to be studied and compared to what is observed and very well known when far-field imaging conditions are met. Here again, complex visibilities \(V\) and \(\widetilde{V}\) have been simulated according to (3) and (4) for the same brightness temperature distribution \(T(\xi)\) of an extended source with a constant temperature. Both have been inverted with the appropriate operator: \(\mathbf{G}^{+}\) for \(V\) according to (10) and \(\widetilde{\mathbf{G}}^{+}\) for \(\widetilde{V}\) according to (9). Two cases have been considered, depending on the extent of the source. The first one corresponds to a situation that has been widely simulated with a circular [16] or a square [24] flat target that never extends beyond the field of view synthesized by the antenna array. The brightness temperature distribution \(T(\xi)\) of an extended source set to \(100\) K in the field of view synthesized by the antenna array of Figure 2 and to \(0\) K outside this hexagon is shown in Figure 19, together with the corresponding complex visibilities \(V\) and \(\widetilde{V}\). Figure 18: Reconstructed point-spread function when the array is in the near-field region at a distance \(h=100\) m from the boresight point and the near-field visibilities \(V\) are corrected from their distortion to obtain far-field visibilities, which are inverted with the far-field modeling operator \(\widetilde{\mathbf{G}}^{+}\). The reconstruction varies with the position \(\xi\) of the source in the field of view: from left to right, \(\xi=(-0.5,0)\), \(\xi=(0,0)\) and \(\xi=(0.5,0)\). The color scale is the same for all images, contour levels are drawn for \(-0.10,0.10,0.25\) and \(0.50\). Shown in Figure 20 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 19 according to (9) and (10). 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 here set to 0 K outside the field of view synthesized by the antenna array, field aliasing is not an issue because it is not expected. Indeed, there is no reason to observe it in such conditions where the brightness temperature from the six neighbors to the synthesized field of view does not contribute to the complex visibilities. As a consequence, in the absence of any modeling error or noise, there is no significant difference observed between \(\mathcal{T}=W\star T\) and the two retrieved maps \(\widetilde{\mathcal{T}}_{r}\) and \(\mathcal{T}_{r}\), as expected. In both cases, the floor error as illustrated here by the Root Mean Squared Error (RMSE) [52] in the whole synthesized field of view is much less than 0.01 K, and it does not vary with the distance \(h\) between the antenna array and the source in the boresight direction. Figure 19: Brightness temperature distribution \(T(\mathbf{\xi})\) of an extended source that is set to 100 K in the field of view synthesized by the antenna array of Figure 2 and to 0 K outside this hexagon. 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. Figure 20: Brightness temperature maps \(\widetilde{\mathcal{T}}_{r}\) (far-field, **left**) and \(\mathcal{T}_{r}\) (near-field, **right**) retrieved after inverting the complex visibilities \(\widetilde{V}\) and \(V\) shown in Figure 19. In both cases, a Blackman windowing function \(W\) has been used to dampen the reconstructed maps \(\widetilde{T}_{r}\) and \(T_{r}\).