## 5 Numerical Simulations and Comparison
### Far-Field vs. Near-Field
Here again, this is large enough to affirm that local viewing directions at the level of the antenna's voltage pattern should be taken into account in near-field synthesis imaging. Figure 8: Spatial distribution of the magnitude (**bottom**) and the phase (**top**) of the difference \(\mathcal{F}_{p}(\xi)\mathcal{F}_{q}^{*}(\xi)-\mathcal{F}_{p}(\xi_{p})\mathcal{F }_{q}^{*}(\xi_{q})\) for the three pairs of elementary antennas corresponding to the shortest baseline at the extremities of the three arms: \(p=22\) and \(q=23\) (**left**), \(p=45\) and \(q=46\) (**middle**), and \(p=68\) and \(q=69\) (**right**). Here, the array is placed at a distance \(h=100\) m from a source. The difference could be as high as 0.4 dB and \(4^{\circ}\). Figure 7: Spatial distribution of the fraction \(r^{2}(\xi)/r_{p}(\xi)r_{q}(\xi)\) for the three pairs of elementary antennas corresponding to the shortest baseline at the extremities of the three arms: \(p=22\) and \(q=23\) (**left**), \(p=45\) and \(q=46\) (**middle**), and \(p=68\) and \(q=69\) (**right**). Here, the array is placed at a distance \(h=100\) m from a source. The deviation from 1 could be as high as 4%. Keeping in mind that the objective of this study is to determine when the far-field approximation can be used and when near-field synthesis imaging has to be performed, shown in Figures 9-11 are the variations of the largest value of the previous spatial distributions with the observing distance \(h\) from the phase center of the antenna array. With regards to the difference \(\mathbf{b}_{pq}\cdot\xi-(r_{q}(\xi)-r_{p}(\xi))\), it turns out that for the Franhaufer distance \(2D^{2}/\lambda\simeq 430\) m, the largest value is here about \(\lambda/16\), as expected, according to the common criterion for separating the Fresnel region from the Fraunhofer one. With respect to the fraction \(r^{2}(\xi)/r_{p}(\xi)r_{q}(\xi)\), the largest deviation from the value 1 for the same Franhaufer distance is slightly less than 1%. Finally, regarding the impact of the local viewing directions \(\xi_{p}\) and \(\xi_{q}\) onto the antenna voltage patterns \(\mathcal{F}_{p}\) and \(\mathcal{F}_{q}\), the largest error introduced at the amplitude level is less than 0.1 dB for the same Franhaufer distance. Figure 10: Comparison between the ratio \(r^{2}(\xi)/r_{q}(\xi)r_{p}(\xi)\) and 1 (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 deviation from 1 is less than 1%. Figure 9: Comparison between the difference \(r_{q}(\xi)-r_{p}(\xi)\) and the dot product \(\mathbf{b}_{pq}\cdot\xi\) (the far-field approximation) as a function of the distance between the source and the phase center of 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 about \(\lambda/16\). As a consequence of this preliminary analysis, none of the three terms identified in Section 3 can be seen as a clear indicator of the boundary between the near-field region and the far-field one. Although the approximation of \(r_{q}(\mathbf{\xi})-r_{p}(\mathbf{\xi})\) by \(\mathbf{b}_{pq}\cdot\mathbf{\xi}\) leads to an error very close to that associated with the Fraunhofer distance, according to Equation (9) in [4], the threshold of \(\lambda/16\) is questionable for imaging purposes with an antenna array: so why would a phase error of \(\pi/8\) at complex visibilities level be acceptable? Referring back to the principle of synthesis imaging, the appropriate threshold has to be found at the level of the imaging process in the comparison made between the reconstructed maps \(\widetilde{\mathcal{T}}_{r}\) and \(\mathcal{T}_{r}\) with the ideal one \(\mathcal{T}\).