## 5 Numerical Simulations and Comparison
### Far-Field vs. Near-Field  
Referring back to Section 3, near-field conditions have three impacts on the modeling equation when comparing expressions (3) and (4). Shown in Figures 5 and 6 are examples of the spatial distribution of the error made when the difference \(r_{q}(\xi)-r_{p}(\xi)\) is approximated by the dot product \(\mathbf{b}_{pq}\cdot\xi\). If the shape of this error is quite simple for pairs of elementary antennas located on the same arm, this is no longer the case when the two antennas are located on two different arms. With regards to the magnitude of this error, it is a not negligible fraction of the wavelength as high as 7.5 cm compared to the 3 mm accuracy required for the baselines coordinates at the central wavelength of SMOS \(\lambda=21.21\) cm. As  
Figure 4: Distributions of the 4695 singular values of the modeling matrices \(\mathbf{G}\) (red) and \(\widetilde{\mathbf{G}}\) (green) of the SMOS array in X polarization when it is placed at a distance \(h=100\) m from a source. Whatever the electromagnetic region concerned by the modeling Equations (5) and (6), two groups of singular values separated by a well-determined gap are observed. In every case, the first group is composed of the 2791 largest singular values, and the second one is composed of the 1904 remaining singular values (\(\varepsilon\approx 2\cdot 10^{-16}\) is the floating point relative accuracy of Matlab). However, this gap between the two groups is much larger in the far-field case (green) than in the near-field one (red). The same behavior is observed in Y polarization. a consequence, such an approximation should have on strong impact on the brightness temperature retrievals. Likewise, shown in Figure 7 are examples of the spatial distribution of the multiplicative real-valued fraction \(r^{2}(\xi)/r_{p}(\xi)r_{q}(\xi)\) of (3). In the far-field approximation (4), this fraction is assumed to be equal to 1. In the near-field case shown here, the deviation from 1 could be as high as 4% for the shortest baselines at the extremities of the arms, but it decreases as soon as the antennas are closer to the phase center of the array. However, this is a not negligible fraction as the brightness temperature seen by the antenna array for these baselines from the corresponding directions is directly affected by this deviation. Although this term does not affect the phase of the complex visibilities, the impact on the magnitude is strong enough to affirm that it should not be neglected in near-field synthesis imaging. Figure 5: Spatial distribution of the difference \(\mathbf{b}_{pq}\,\xi-(r_{q}(\xi)-r_{p}(\xi))\) for three pairs of elementary antennas: \(p=3\) and \(q=23\) (**left**), \(p=26\) and \(q=46\) (**middle**), and \(p=49\) 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 7.5 cm, that is to say, about \(\lambda/3\) with \(\lambda=21.21\) cm (the central wavelength of SMOS antenna array). Figure 6: Spatial distribution of the difference \(\mathbf{b}_{pq}\,\xi-(r_{q}(\xi)-r_{p}(\xi))\) for the three pairs of elementary antennas corresponding to the longest baselines: \(p=23\) and \(q=46\) (**left**), \(p=23\) and \(q=69\) (**middle**), and \(p=46\) 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 2.5 cm, that is to say, about \(\lambda/9\) with \(\lambda=21.21\) cm (the central wavelength of SMOS antenna array). Finally, shown in Figure 8 are examples of the spatial distribution of the magnitude and the phase of the difference \(\mathcal{F}_{p}(\xi)\mathcal{F}_{q}^{*}(\xi)-\mathcal{F}_{p}(\xi_{p})\mathcal{F} _{q}^{*}(\xi_{q})\). The error introduced when ignoring local viewing directions \(\xi_{p}\) and \(\xi_{q}\), assuming \(\xi=\xi_{p}=\xi_{q}\) in that part of the modeling equation involving the voltage pattern of the elementary antennascould be as high as 0.4 dB and \(4^{\circ}\).