## 5 Numerical Simulations
### Noisy Inversions
As a consequence, the reconstruction operator is no longer \(\widetilde{\mathbf{G}}^{+}\) but  
Figure 7:2  
and the retrieved brightness temperature distribution is now  
Figure 7:3  
Likewise, the condition number of the regularized modeling matrix is no longer \(\kappa(\widetilde{\mathbf{G}})\) but  
Figure 7:4  
If the number \(N_{f}\) of singular values taken into account in the inversion of \(\widetilde{\mathbf{G}}\) is not subjected to discussion because the other ones are equal to zero from the numerical point of view, as illustrated in Figure 5, the choice of a value for \(n\) is not obvious. When comparing Figures 12 and 13 while keeping an eye to the singular vectors \(\mathbf{V}_{\ell}\) such as those shown in Figure 14, it turns out that \(n=16\) is an appropriate value. As a consequence, the previous visibilities \(V\) have been inverted according to (15), with only \(N_{f}-n=13,\!845\) singular values kept in the pseudo inverse of \(\widetilde{\mathbf{G}}_{n}\), which has a condition number close to 320, a value comparable to that of \(\widetilde{\mathbf{G}}\) with \(d=0.71\lambda\). Shown in Figure 15 are the error maps \(\widetilde{\mathcal{T}}_{n}-\mathcal{T}\) obtained with and without a random Gaussian noise \(\Delta V\) with standard deviation 0.1 K. In contrast to Figure 9a, the propagation of \(\Delta V\) during the inversion no longer results in the presence of an undesirable phantom, as expected with the filtering out of the singular vectors corresponding to the \(n\) smallest singular values. In the absence of noise, this severe truncation does not introduce any additional error in the retrieved brightness temperature map, as illustrated by the comparison with Figure 8a. Average maps and standard deviation  
Figure 7:5 Singular vectors \(\mathbf{V}_{\ell}\) of \(\widetilde{\mathbf{G}}\) corresponding to the smallest singular values \(\sigma_{\ell}\) when the shortest spacing of the cross-shaped array shown in Figure 4 is set to \(d=0.69\lambda\). Only the last three singular vectors are shown here for (**a**) \(\ell=N_{f}-2=13,\!859\) with \(\sigma_{13889}\simeq 8.1\cdot 10^{-6}\), (**b**) \(\ell=N_{f}-1=13,\!860\) with \(\sigma_{13860}\simeq 7.8\cdot 10^{-6}\), and (**c**) \(\ell=N_{f}=13,\!861\) with \(\sigma_{13861}\simeq 1.6\cdot 10^{-6}\). Components of \(\mathbf{V}_{\ell}\) are normalized according to property \(\|\mathbf{V}_{\ell}\|=1\) because matrix \(\mathbf{V}\) is unitary. maps of 1000 error maps \(\widetilde{\mathcal{T}}_{n}-\mathcal{T}\) obtained for different random trials of \(\Delta V\) with the same standard deviation 0.1 K are shown in Figure 7:6 Here again, unlike Figures 10a and 11a, the cross-shaped phantom no longer disturbs these brightness temperature distributions. Due to this additional truncation of \(\mathbf{G}\), the brightness temperature maps retrieved with \(\widetilde{\mathbf{G}}_{n}\) for \(d=0.69\lambda\) are almost identical to those retrieved with \(\widetilde{\mathbf{G}}\) for \(d=0.71\lambda\), as illustrated by the comparison of Figures 15 and 16 with Figures 8b, 9b, 10b, and 11b. Referring back to the radiometric sensitivity [41], from a quantitative point of view, the minimum value at boresight is about 4.43 K in the standard deviation map shown in Figure 11b with \(d=0.71\lambda\) and about 4.20 K in the standard deviation map shown in Figure 16b with \(d=0.69\lambda\). With regard to this latter situation, this estimation is about 13.9 K in the standard deviation map shown in Figure 11a, as a consequence of the presence of the cross-shaped phantom. According to Equation (1) in [43], the radiometric sensitivity should be equal to 4.44 K with  
Figure 7:7 Average map (**a**) and standard deviation map (**b**) of 1000 error maps \(\widetilde{\mathcal{T}}_{n}-\mathcal{T}\) obtained in \(X\) polarization with the cross-shaped array shown in Figure 4 with the shortest spacing \(d\) set to \(0.69\lambda\) (similar results are obtained in \(Y\) polarization).