## 5 Numerical Simulations and Comparison
### Singular Values  
Referring back to (3) and (4), in both cases, the number of complex visibilities is equal to the number of baselines \(\mathbf{b}_{pq}\). For SMOS, it is equal to \(69\times(69-1)=4692\). In addition to these interferometric measurements, 3 out of the 69 receivers are also operating as reference radiometers to provide precise measurements of the average brightness temperature of the observed scene [47]. The total number of measurements is therefore  
Figure 3: Distances \(r_{p}(\xi)\) between a source located in the near-field region at a distance \(h=100\) m in the Nadir direction \(\xi=(0,0)\) of the SMOS array and each of the 69 elementary antennas \(\mathcal{A}_{p}\) (**left**), together with the corresponding local colatitude angles \(\theta_{p}(\xi)\) (**right**). The antenna index \(p\) refers to the numbering of the elementary antennas of MIRAS shown in Figure 2.  
Figure 2: Location and numbering of the 69 elementary antennas of MIRAS in the reference frame of the instrument. The spacing between the elements along the three arms is equal to \(0.875\lambda=18.56\) cm with \(\lambda=21.21\) cm (the central wavelength of the instrument). The longest baselines \(D\) between the antennas at the ends of the three arms are equal to \(21\sqrt{3}\,0.875\lambda=6.75\) m.  
equal to \(4692+3=4695\). However, when accounting for the redundant baselines along the three arms of the Y-shaped array, the number of Fourier frequencies in the corresponding star-shaped frequency coverage \(\mathcal{H}\) is only equal to \(2790+1=2791\), including the zero frequency.  
The distributions of the 4695 singular values of the modeling matrices \(\mathbf{G}\) and \(\widetilde{\mathbf{G}}\) are shown in Figure 4. According to the floating point relative accuracy of Matlab (\(\varepsilon\approx 2\cdot 10^{-16}\)), the 1904 smallest singular values of \(\widetilde{\mathbf{G}}\) could be considered equal to zero [48]. The number of remaining strictly positive singular values is therefore equal to \(4695-1904=2791\), which is also the number of Fourier frequencies in the frequency coverage \(\mathcal{H}\). This property remains true in the distribution of the singular values of the modeling matrix \(\mathbf{G}\); however, the gap between the two groups of singular values is much smaller: here less than one order of magnitude instead of 13 in the far-field conditions.  
As a consequence, when computing the pseudo-inverse with the aid of the truncated singular value decomposition, the impact of the singular values discarded prior to inversion might not be the same and a larger floor error might be observed in the near-field reconstructions compared to that in the far-field ones. Finally, the propagation of random noise is governed by the conditioning of the modeling matrix, which is the ratio between the largest and the smallest singular values kept in the inversion. It turns out that here, this ratio is about 11.2 for \(\widetilde{\mathbf{G}}\) and 12.6 for \(\mathbf{G}\). Although this ratio is an upper bound for the actual amplification factor of random noise, one can expect very similar behaviors of the two matrices.