## 5 Numerical Simulations and Comparison
### Point Source  
Complex visibilities \(V\) and \(\widetilde{V}\) have been simulated according to (3) and (4) for the same brightness temperature distribution \(T(\mathbf{\xi})\) of a point-source, which is equal to 0 everywhere except in the boresight direction of the antenna array. They are shown with polar plots on Figure 12. Whatever the pair of elementary antennas \(\mathcal{A}_{p}\) and \(\mathcal{A}_{q}\), in such a special case of brightness temperature distribution, the far-field visibilities \(\widetilde{V}(\mathbf{b}_{pq})\) are real-valued, as expected according to (4) as soon as the only contribution to the integral is for \(\mathbf{\xi}=(0,0)\). This is no longer the case with the near-field visibilities \(V(\mathbf{b}_{pq})\) simulated when the antenna array is placed at a distance \(h=100\) m from the point source. Although the instrument is still the same, the phase of these complex visibilities varies in the range \(\pm 130^{\circ}\), as a consequence of the effects of the geometry illustrated in Figure 1. Furthermore, simulations conducted with (3) for different values of \(h\) have shown that this range of phases is larger as the distance between the antenna array and the point-source gets closer. On the contrary, it tends toward zero as this distance increases and reaches the far-field region, as expected and as shown in Figure 13. For the Fraunhofer distance of \(2D^{2}/\lambda=430\) m, it turns out that this range of phases is about \(\pm 30^{\circ}\). Referring back to the far-field approximation, such a phase translates into a baseline error as large as \(\lambda/12\simeq 18\) mm. This is 6 times larger than the 3 mm baseline uncertainty of SMOS, which corresponds to a phase error of the order of \(5^{\circ}\) that is reached here only from a distance \(h\) greater than 2 Km, that is to say, about \(10D^{2}/\lambda\). As a consequence, there is no doubt that inverting the near-field complex visibilities \(V\) with the pseudo inverse \(\widetilde{\mathbf{G}}^{+}\) of the far-field modeling operator will lead to large errors and artifacts not only for \(h=100\) m, which is evidently in the near-field region, but also for any value of \(h\lesssim 10D^{2}/\lambda\), including of course for the Fraunhofer distance \(2D^{2}/\lambda\). Figure 11: Comparison between the cross-power patterns \(\mathcal{F}_{p}(\mathbf{\xi}_{p})\mathcal{F}_{q}^{*}(\mathbf{\xi}_{q})\) and \(\mathcal{F}_{p}(\mathbf{\xi})\mathcal{F}_{q}^{*}(\mathbf{\xi})\) (the far-field approximation) as a function of the distance between the source and the antenna array. For a distance equal to the Fraunhofer distance \(2D^{2}/\lambda\) with \(D=6.75\) m and \(\lambda=21.21\) cm (the longest baseline and the central wavelength of SMOS antenna array), the largest discrepancy between the two quantities is less than 0.1 dB. The previous complex visibilities \(\widetilde{V}\) and \(V\) have been inverted according to (9) and (10). Shown in Figure 14 is the reconstructed point-spread function of SMOS when the antenna array is placed at a distance \(h=100\) m from the source in the near-field region, and the visibilities \(V\) shown in red in Figure 12 are inverted with the pseudo-inverse of the near-field modeling operator \(\mathbf{G}\) or with that of the far-field one \(\widetilde{\mathbf{G}}\). Both have to be compared to the point-spread function obtained when the array is in the far-field region and the corresponding visibilities \(\widetilde{V}\) shown in green in Figure 12 are inverted with \(\widetilde{\mathbf{G}}^{+}\). In every case, a rectangular windowing function \(W\) has been used so that the reconstructed point-spread function is not apodized and the angular resolution, as estimated by the full-width at half-maximum of the main lobe, is the finest one that could be obtained from this antenna array. As soon as the complex visibilities are inverted with the appropriate operator, the one corresponding to the concerned electromagnetic region, the retrieved point-spread functions exhibit the expected imaging qualities.