## 5 Numerical Simulations and Comparison
### Point Source
As predicted earlier in this subsection when looking at the range of the phase of the near-field visibilities \(V\) reported in Figure 13, large errors and artifacts are observed here at the level of the reconstructed point-spread functions for any value of \(h\lesssim 10D^{2}/\lambda\). Although the Mean Squared Error (MSE) has many attractive features, it is not well suited for quantifying the sameness, or on the contrary the difference, between images [49]. The Structural SIMilarity (SSIM) index [50] is a well-known method for measuring the similarity, or the dissimilarity, between two images. This index can be viewed as a quality measure of one of the images being compared, provided the other image is considered of perfect quality. Here, the reference image is the reconstructed point-spread function shown in Figure 14 when the antenna array is in the far-field region and the visibilities \(\widetilde{V}\) are inverted with the far-field modeling operator \(\widetilde{\mathbf{G}}^{+}\). It is compared to every reconstructed point-spread function, like those shown in Figure 16, obtained when the array is placed at a distance of \(h\) from the source and when the corresponding visibilities \(V\) are nevertheless inverted with the far-field modeling operator \(\widetilde{\mathbf{G}}^{+}\). Shown in Figure 17 are the variations in the SSIM index obtained for this comparison. For the Fraunhofer distance \(2D^{2}/\lambda=430\) m, this index is only about 0.82, but it is larger than 0.99 for any value of \(h\gtrsim 10D^{2}/\lambda\). Figure 16: Reconstructed point-spread function when the array is placed at a distance \(h=2D^{2}/\lambda\), \(3D^{2}/\lambda\), \(4D^{2}/\lambda\), \(5D^{2}/\lambda\), \(10D^{2}/\lambda\), and \(20D^{2}/\lambda\) from a source at boresight \(\xi=(0,0)\), with \(D=6.75\) m and \(\lambda=21.21\) cm (the longest baseline and the central wavelength of SMOS antenna array). In every case, the near-field visibilities \(V\) are inverted with the far-field modeling operator \(\widetilde{\mathbf{G}}^{+}\). The color scale is the same for all images, and contour levels are drawn for \(-0.10,0.10,0.25\) and 0.50. Finally, it is frequently read in the literature (Equation (5) in [16], Equation (9) in [19] or Equation (2) in [51], for example) that near-field complex visibilities \(V\) can be corrected from their distortion to obtain far-field visibilities \(\widetilde{V}\) by simply subtracting the near-field phase and adding the far-field one for every baseline \(\mathbf{b}_{pq}\) according to  
\[V(\mathbf{b}_{pq})\,\mathbf{e} \tag{14}\]  
However, referring back to Equations (3) and (4), only a point source can be concerned by such a correction because these modeling equations are integral relations so that the complex visibility for any antenna pair \(\mathcal{A}_{p}\) and \(\mathcal{A}_{q}\) does not depend on the variable \(\boldsymbol{\xi}\). It should be read between the lines in [16; 19] that \(\boldsymbol{\xi}\) is here nothing but the viewing direction from a single point source; more precisely, the brightness temperature distribution \(T(\boldsymbol{\xi})\) in (3) and (4) is assumed here to be null everywhere except for that particular value of \(\boldsymbol{\xi}\). To be complete with this correction, it is also assumed that the second and the third impact out of the three listed in Section 3 are neglected. This is exactly what should be understood in the words of [16] _\"in the case that only path delay phase errors are taken into account...if we take into account a single punctual source.\"_ Such an approach is therefore useless since observed scenes are not always reduced to a single point source. Moreover, applying this phase correction to an extended source or to a collection of point sources is simply not possible, unless the same phase correction is applied for every viewing direction in the field of view, which would assume that it does not change from one direction to another.