## 5 Numerical Simulations
### Noisy Inversions  
Previous complex visibilities \(V\) have been blurred with an additive Gaussian noise \(\Delta V\) with standard deviation 0.1 K. Shown in Figure 9 are the error maps \(\tilde{\mathcal{T}}-\mathcal{T}\) for the same random distribution \(\Delta V\) when the shortest spacing \(d\) is set to \(0.69\lambda\) or to \(0.71\lambda\). As expected, the reconstruction error is no longer equal to 0 in both cases: it is almost randomly distributed in the synthesized FOV, with values increasing from the boresight direction \(\xi=0\) with the colatitude angle \(\theta\). However, when \(d=0.69\lambda\), a cross-shaped phantom is obviously superimposed on this random distribution. Figure 8: Error maps \(\tilde{\mathcal{T}}-\mathcal{T}\) 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 both cases, complex visibilities \(V\) are free from any additional random noise \(\Delta V\). Referring back to Figure 2, Earth aliases are shown with dashed maroon lines. Bias and standard deviation are given in Earth alias-free FOV (dashed maroon line) and in the sky alias-free FOV (dashed blue line). Figure 7: Brightness temperature distributions \(T\) of a typical scene over the ocean in (**a**) \(X\) and (**b**) \(Y\) polarizations at instrument level. Referring back to Figure 2, Earth is here shown with a maroon line from an elevation \(h=685\) km with a tilt angle \(\epsilon=0\) (nadir looking mode, as illustrated in Figure 1). Average maps and standard deviation maps of 1000 error maps \(\widetilde{\mathcal{T}}-\mathcal{T}\) obtained for different random trials of \(\Delta V\) with the same standard deviation 0.1 K are shown in Figures 10 and 11 for the same values of the shortest spacing. When the spacing \(d\) is equal to 0.71\(\lambda\), the average error is almost equal to zero, except at high colatitude angles \(\theta\) in the four corners of the FOV, close to those directions beyond the unit circle. With regards to the radiometric sensitivity [41], the minimum value of the standard deviation map is obtained at boresight. Away from that direction, errors are growing like the inverse of the average antenna radiation pattern times the factor \(\sqrt{1-\|\xi\|^{2}}\), as expected. The same behavior is observed for \(d=0.69\lambda\), except the superimposition of that cross-shaped phantom in the average map as well as in the standard deviation one. Of course, the presence of this cross-shaped phantom is detrimental to any further use of such retrieved maps. The origin of this reconstruction artifact is analyzed hereafter, then a solution is proposed for removing it without altering the performance of the instrument. Figure 10: Average maps of 1000 error maps \(\widetilde{\mathcal{T}}-\mathcal{T}\) 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.  
Figure 9: Error maps \(\widetilde{\mathcal{T}}-\mathcal{T}\) 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 both cases, complex visibilities \(V\) have been blurred by an additive Gaussian noise \(\Delta V\) with standard deviation set to 0.1 K.  
Because the two matrices \(\mathbf{U}\) and \(\mathbf{V}\) are unitary, the corresponding singular vectors \(\mathbf{U}_{\ell}\) and \(\mathbf{V}_{\ell}\) form an orthonormal basis of the underlying real-valued spaces \(\mathbb{R}^{N_{\ell}}\) and \(\mathbb{R}^{N^{2}}\).