## 5 Numerical Simulations
### Noisy Inversions
This is exactly what is proven in Figure 14 with the singular vectors \(\mathbf{V}_{\ell}\) of \(\widetilde{\mathbf{G}}\) corresponding to the three smallest singular values \(\sigma_{13859}\simeq 8.1\cdot 10^{-6}\), \(\sigma_{13860}\simeq 7.8\cdot 10^{-6}\), and \(\sigma_{13861}\simeq 1.6\cdot 10^{-6}\). In addition to being associated with the smallest singular values, these three singular vectors are particularly excited in the error map of Figure 9a as the separation angles are, respectively, \(\varphi_{13859}\simeq 78^{\circ}\), \(\varphi_{13860}\simeq 103^{\circ}\), and \(\varphi_{13861}\simeq 41^{\circ}\). Indeed, referring back to (12) and (11), the contribution of the corresponding three singular vectors to \(|\widetilde{T}|^{2}\) is about two thirds. More precisely, it is larger than 4% for \(\mathbf{V}_{13859}\) and for \(\mathbf{V}_{13860}\) as well and about 57% for \(\mathbf{V}_{13861}\). When comparing these singular vectors to the error map of Figure 9a, there is no doubt that the cross-shaped phantom seen in this map is a linear combination of them. The origin of this artifact is therefore coming from the modeling matrix \(\mathbf{G}\) itself and from the way it propagates the additive Gaussian noise \(\Delta V\) from the complex visibilities \(V\) to the retrieved  
Figure 12: Separation angles \(\varphi_{\ell}\) between the singular vectors \(\mathbf{V}_{\ell}\) of \(\widetilde{\mathbf{G}}\) and the brightness temperature maps \(\widetilde{T}\) corresponding to the apodized maps \(\widetilde{\mathcal{T}}\) of Figure 8 retrieved without radiometric noise for two values of the shortest spacing of the cross-shaped array shown in Figure 4: (**a**) \(d=0.69\lambda\) and (**b**) \(d=0.71\lambda\). Only the first 13,861 largest singular values shown in green in Figure 5 are reported here inasmuch as the other ones are discarded prior to the inversion of \(\widetilde{\mathbf{G}}\), the corresponding singular vectors do not contribute to \(\widetilde{T}\). Figure 13: Separation angles \(\varphi_{\ell}\) between the singular vectors \(\mathbf{V}_{\ell}\) of \(\widetilde{\mathbf{G}}\) and the brightness temperature maps \(\widetilde{T}\) corresponding to the apodized maps \(\widetilde{\mathcal{T}}\) of Figure 9 retrieved when complex visibilities \(V\) are blurred by an additive Gaussian noise \(\Delta V\) with standard deviation set to 0.1 K for two values of the shortest spacing of the cross-shaped array shown in Figure 7:0 (**a**) \(d=0.69\lambda\) and (**b**) \(d=0.71\lambda\). Only the first 13,861 largest singular values shown in green in Figure 5 are reported here inasmuch as the other ones are discarded prior to the inversion of \(\widetilde{\mathbf{G}}\), the corresponding singular vectors do not contribute to \(\widetilde{T}\). brightness temperature map \(\widetilde{T}\), and therefore to the apodized one \(\widetilde{\mathcal{T}}\), during the inversion process. As a consequence, the cure has to be found in the regularization of \(\mathbf{G}\). Figure 13a strongly suggests revisiting the truncated singular value decomposition of \(\widetilde{\mathbf{G}}\) when \(d=0.69\lambda\) and to discard those small singular values \(\sigma_{\ell}\) and their singular vectors \(\mathbf{V}_{\ell}\) with a separation angle \(\varphi_{\ell}\) far from \(90^{\circ}\), so that the regularized modeling matrix \(\widetilde{\mathbf{G}}\) now reduces to:  
Figure 7:1  
where \(n\geq 1\) is the number of additional singular values that will discarded prior to the inversion.