## 5 Numerical Simulations
### Noisy Inversions
As a consequence, the retrieved brightness temperature distribution \(\widetilde{T}\) can be decomposed onto the space spanned by the \(\mathbf{V}_{\ell}\) according to the expansion [42]  
\[\widetilde{T}=\sum_{\ell=1}^{Nf}\langle\widetilde{T}\mid\mathbf{V}_{\ell} \rangle\mathbf{V}_{\ell}. \tag{11}\]  
Of course, the sum is restricted to the singular vectors \(\mathbf{V}_{\ell}\) corresponding to the singular values \(\sigma_{\ell}\) kept in the inversion of \(\widetilde{\mathbf{G}}\), and the inner product \(\langle\widetilde{T}\mid\mathbf{V}_{\ell}\rangle\) is simply the sum of the products between every component of \(\widetilde{T}\) and those of \(\mathbf{V}_{\ell}\) on the grid sampling the FOV. In numerical linear algebra, the error analysis is often initiated by computing the separation angle between the numerical solution of a linear system and every singular vector of the corresponding matrix and especially with those associated to the smallest singular values because their role during the inversion has to be carefully taken into account. The separation angle \(\varphi_{\ell}\) between the solution \(\widetilde{T}\) and the \(\ell\)th singular vector \(\mathbf{V}_{\ell}\) of the modeling matrix \(\widetilde{\mathbf{G}}\) is derived from the well-known definition of the inner product [42]:  
\[\varphi_{\ell}=\arccos\frac{\langle\widetilde{T}\mid\mathbf{V}_{\ell}\rangle} {\|\widetilde{T}\|\|\mathbf{V}_{\ell}\|},\text{ with }1\leq\ell\leq N_{f}. \tag{12}\]  
Referring back to (11), the farther \(\varphi_{\ell}\) to \(90^{\circ}\), the more important the contribution of \(\mathbf{V}_{\ell}\) to the retrieved brightness temperature map \(\widetilde{T}\). Shown in Figures 12 and 13 are the variations of \(\varphi_{\ell}\) with the singular values \(\sigma_{\ell}\) of Figure 5 for the retrieved brightness temperature maps of Figures 8 and 9. In the absence of any additive Gaussian noise \(\Delta V\), whatever the shortest spacing, \(d=0.69\lambda\) or \(d=0.71\lambda\), all the separation angles are confined to the interval \(85^{\circ}\leq\varphi_{\ell}\leq 95^{\circ}\) except \(\varphi_{1}\simeq 146^{\circ}\). As a consequence, apart from \(\mathbf{V}_{1}\), which has the largest contribution to \(\widetilde{T}\), there is no singular vector with a greater contribution than another. This is not surprising, as \(\mathbf{V}_{1}\) contributes to the average value of \(\widetilde{T}\) and is associated with the largest singular value \(\sigma_{1}\simeq 0.05\). As soon as the complex visibilities \(V\) are blurred with an additive Gaussian noise \(\Delta V\), the distribution of the separation angles varies from one shortest spacing to another. When \(d=0.71\lambda\), the situation in Figure 13b is very similar to that in Figure 12b in the absence of noise: this is the reason why no cross-shaped phantom is observed in the retrieved brightness temperature map of Figure 9b, the average map Figure 10b, and in the standard deviation map of Figure 11b. On the contrary, when \(d=0.69\lambda\), there are singular vectors \(\mathbf{V}_{\ell}\) that contribute more to \(\widetilde{T}\) than \(\mathbf{V}_{1}\) as illustrated by their separation angles, which are farther from \(90^{\circ}\) than \(\varphi_{1}\) is. Moreover, these excited singular vectors correspond to the smallest singular values kept in the inversion of \(\widetilde{\mathbf{G}}\). Figure 11: Standard deviation maps of 1000 error maps \(\widetilde{T}-\mathcal{T}\) obtained in \(X\) polarization with the cross-shaped array shown in Figure 4 for two values of the shortest spacing: (a) \(d=0.69\lambda\) and (b) \(d=0.71\lambda\) (similar results are obtained in \(Y\) polarization). In every case, complex visibilities \(V\) have been blurred by an additive Gaussian noise \(\Delta V\) with standard deviation set to 0.1 K.  
Referring back to (8) and to the role played by these singular values, it would not be surprising to see these singular vectors in the error map of Figure 9a.