## 6 Discussion and Remarks  
The same cross-shaped phantom is observed for any shortest spacing \(d<0.704\lambda\) and for the same reasons it is caused by these singular vectors \(\mathbf{V}_{\ell}\) of \(\widetilde{\mathbf{G}}\) corresponding to the smallest singular values \(\sigma_{\ell}\), which are responsible for an unacceptable propagation of noise from complex visibilities to retrieved brightness temperature maps. In any case, it can be filtered out with the same truncated singular value decomposition of \(\widetilde{\mathbf{G}}\) where the last \(n\) singular vectors that exhibit partial or complete features of this cross-shaped phantom are discarded prior to inversion according to (14), keeping the propagation of noise at an acceptable level. Whatever the shortest spacing \(d\), the appropriate value for \(n\) seems to be linearly correlated with the number of directions \(\boldsymbol{\xi}\) of the synthesized field of view that are not taken into account in the inversion, as illustrated in Figure 3a. More precisely, it turns out that the correlation coefficient between \(n\) and this number of rejected directions is better than 0.99, and the slope of the linear regression between them is very close to 4. This is not surprising, as 4 is the number of vertices of the square-shaped synthesized field of view, which suggests that the key parameter is not the total number of directions but the number of directions per vertex. Each direction in a given vertex is intimately related to another one in the three other vertices, and they are all together responsible for the same suspicious singular vector. Finally, referring back to Figures 2b,c and 3b, a similar phantom is observed with a hexagonal geometry for any shortest spacing \(d<0.664\lambda\): this is the case for the SMOS-HEX project [43]. The singular value decomposition of the modeling matrix \(\mathbf{G}\) strongly depends on the voltage pattern of each elementary antenna of the interferometric array. This is very well illustrated by Figure 1 in [44] with the disparity among the 69 voltage patterns of SMOS that have been measured in an anechoic chamber prior to the launch [45]. As a consequence, the  
Figure 17: Closed views of the normalized synthetic responses of the cross-shaped array shown in Figure 4 when the shortest spacing \(d\) is set to 0.69\(\lambda\). The FWHM value of the main beam, here illustrated by the black circles, is about 0.43\({}^{\circ}\) (**a**) without an apodization window and about 0.61\({}^{\circ}\) (**b**) with a Blackman apodization window. Here, the reconstruction operator is \(\widetilde{\mathbf{G}}^{+}\); identical responses are obtained with \(\widetilde{\mathbf{G}}_{n}^{+}\) when \(n=16\) additional singular values are discarded prior to the inversion. \(n\) singular vectors to be filtered out in the retrieved brightness temperature distribution (15) may vary with these voltage patterns. This is why these patterns have to be measured in their environment [46], as accurately as possible, in order to precisely compute the modeling matrix and its singular value decomposition so that the singular vectors that may propagate noise inappropriately during the inversion process can be clearly identified and filtered out, if necessary. Finally, depending on the extent of the frequency bandwidth of observation, a wide band is often divided into smaller sub-bands for many reasons: reducing the fringe washing effects caused by large spatial decorrelation [47], filtering out the radio frequency interference [48], etc. As a consequence, the fraction \(d/\lambda\) will vary linearly with the frequency from one channel to the other. The situation where few sub-bands (those for the lowest frequencies of the wide band) might suffer from that issue and few others (those for the highest frequencies) might not cannot be excluded.