## 5 Numerical Simulations and Comparison
### Point Source
More precisely, \(\mathbf{G}^{+}V\) in the near-field case and \(\widetilde{\mathbf{G}}^{+}\widetilde{V}\) in the far-field one are perfectly identical, and they evidence the predicted values for many figures of merit (full-width at half-maximum of the main lobe, level of the side lobes, etc.). On the contrary, when inverting near-field visibilities with a far-field  
Figure 12: Complex visibilities of an unresolved point-source when the array is in the far-field region (**left**) and when it is placed at a distance \(h=100\) m from the source (**right**). Attention should focus on the phase of the far-field visibilities \(\widetilde{V}\), which is equal to zero for every baseline \(\mathbf{b}_{pq}\) whereas that of the near-field ones \(V\) varies in the range \(\pm 130^{\circ}\) with an average value of \(0^{\circ}\), depending on the the pair of elementary antennas \(\mathcal{A}_{p}\) and \(\mathcal{A}_{q}\). Figure 13: The range of the phase of the complex visibilities \(V\) of an unresolved point-source as a function of the distance \(h\) between the source and the antenna array. Attention should focus on the decrease towards \(0^{\circ}\) as \(h\) increases. However, for the Fraunhofer distance \(2D^{2}/\lambda=430\) m, this range is still about \(\pm 30^{\circ}\). It remains below \(\pm 5^{\circ}\) as soon as \(h\) exceeds \(2\) Km \(\simeq 10D^{2}/\lambda\). operator, it turns out that \(\widetilde{\mathbf{G}}^{+}V\) is completely different from the expected point-spread function. Clearly, the far-field imaging operator is responsible for large aberrations in the near-field point-spread function. This is exactly what happens when far-field imaging conditions are assumed to be met when they are not. As the energy transported by the electromagnetic field is the same because it does not depend on the distance but only on the brightness temperature of an observed scene, the norm of the retrieved maps is also the same. However, it is clear that the spatial distribution of this energy is distorted so that the shape of the point-spread function has significantly changed. It is no longer that of a main lobe surrounded by six-fold patterns with an alternating succession of positive and negative side lobes, but it is now similar to the shape of the antenna array itself. Finally, as illustrated by Figure 15, this shape also depends on the location of the point source in the field of view synthesized by the instrument. Figure 14: Reconstructed point-spread function when the array is in the far-field region and the visibilities \(\widetilde{V}\) are inverted with the far-field modeling operator \(\widetilde{\mathbf{G}}^{+}\) (**left**), when the array is placed at a distance \(h=100\) m from the source at the boresight \(\boldsymbol{\zeta}=(0,0)\) and the visibilities \(V\) are inverted with the far-field modeling operator \(\widetilde{\mathbf{G}}^{+}\) (**middle**) or with the near-field one \(\mathbf{G}^{+}\) (**right**). The color scale is the same for all images, and contour levels are drawn for \(-0.10,0.10,0.25\) and \(0.50\). Figure 15: 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 inverted with the far-field modeling operator \(\widetilde{\mathbf{G}}^{+}\). The reconstruction varies with the position \(\boldsymbol{\zeta}\) of the source in the field of view: from left to right, \(\boldsymbol{\zeta}=(-0.5,0)\), \(\boldsymbol{\zeta}=(0,0)\) and \(\boldsymbol{\zeta}=(0.5,0)\). 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 16 are reconstructed point-spread functions of SMOS when the antenna array is placed at a distance \(h=2D^{2}/\lambda\), \(3D^{2}/\lambda\), \(4D^{2}/\lambda\), \(5D^{2}/\lambda\), \(10D^{2}/\lambda\) and \(20D^{2}/\lambda\) from a source and when the complex visibilities \(V\) are inverted with the pseudo-inverse of the far-field modeling operator \(\widetilde{\mathbf{G}}\).