## 5 Numerical Simulations and Comparison
### Point Source
This is exactly what is suggested in [16], where _\"near-field visibilities from an extended source can be partially corrected just by focussing the near-field visibilities to the pixel at boresight.\"_ This is not always true as illustrated here by Figure 5, where, for three pairs of elementary antennas, the difference between \(\mathbf{b}_{pq}\cdot\boldsymbol{\xi}\) and \(r_{q}(\boldsymbol{\xi})-r_{p}(\boldsymbol{\xi})\) can be as large as \(\pm\lambda/3\) between the boresight direction and other viewing directions in the field of view. Thus, for the corresponding baselines, if the phase correction (14) is correctly introduced at boresight, an error as large as \(\pm 2\pi/3\) is not corrected in the complex visibility for the contributions from other directions. The same remark holds for Figure 6, where, for three other pairs of elementary antennas, this phase error is as large as \(\pm 2\pi/9\), which is still not negligible. As a consequence, owing to the role played by the phase of the complex visibilities in imaging synthesis operations, large errors should be expected in the regions, or for the  
Figure 1.0 Structural SIMilarity (SSIM) index between the far-field point-spread function and the reconstructed point-spread functions obtained when the array is placed at a distance of \(h\) from the source and when the corresponding visibilities are inverted with the far-field modeling operator. Attention should focus on the increase towards \(1\) as \(h\) increases. For the Fraunhofer distance \(2D^{2}/\lambda=430\) m, this index is only about \(0.82\). It is larger than \(0.95\) as soon as \(h\) exceeds \(1\) Km \(\simeq 5D^{2}/\lambda\) and larger than \(0.99\) for any \(h\gtrsim 2\) Km \(\simeq 10D^{2}/\lambda\). point sources, far from the boresight direction. This is exactly what is shown in Figure 18, where the brightness temperature distribution of three point sources located at \(\xi=(0,0)\) and at \(\xi=(\pm 0.5,0)\) is observed when the antenna array is in the near-field region at a distance \(h=100\) m from the first source at boresight (the two other sources are therefore at a distance \(h/cos\,30^{\circ}\simeq 115\) m from the phase center). The near field visibilities \(V\) are corrected according to (14) with \(\xi=(0,0)\), and the corresponding \"almost\" far-field visibilities \(\widetilde{V}\) thus obtained are inverted with the far-field modeling operator \(\widetilde{\mathbf{G}}^{+}\). As expected, the reconstructed point-spread function at boresight is identical to the expected one already shown in Figure 1.1 On the contrary, as soon as the point source moves away from the very center of the field of view, the reconstructed point-spread function is still suffering from strong aberrations, although the situation is obviously slightly better than that shown in Figure 15 without the phase correction (14). Accounting for the previous considerations and referring back to Section 4, it is recommended to invert near-field complex visibilities \(V\) with the appropriate imaging operator \(\mathbf{G}^{+}\) rather than trying to convert them in far-field ones \(\widetilde{V}\) to be inverted with \(\widetilde{\mathbf{G}}^{+}\), contrary to what is suggested in [51] where it is claimed that _\"after near-field correction, far-field image retrieval tools can be used without any additional modification.\"_