## 1 Introduction
This note is devoted to the comparison between near-field and far-field imaging radiometry in synthetic aperture interferometry from a modeling point of view, from an algebraic point of view, and from an imaging point of view with the aid of numerical simulations. Section 2 gives an overview of the main differences between far-field and near-field synthesis imaging with an antenna array, as illustrated by a pair of elementary antennas with geometrical considerations. The modeling of a synthetic aperture interferometric radiometer is discussed in Section 3, where the integral relation between the brightness temperature distribution \(T\) of an observed scene and the complex visibility \(V\) for a pair of elementary antennas \(\mathcal{A}_{p}\) and \(\mathcal{A}_{q}\) is established for near-field imaging conditions when considering spherical waves as well as for far-field ones within the frame of the approximation with plane waves, as it is assumed in the Van Citter-Zernike theorem. At this level of the modeling equations, three differences between these two regions are clearly identified, and their potential impacts on the synthesis imaging are reported. The regularized reconstruction procedure that will be implemented for data processing is summarized in Section 4. To support the comparison, numerical simulations conducted at microwave frequencies with the Microwave Imaging Radiometer by Aperture Synthesis (MIRAS) onboard the Soil Moisture and Ocean Salinity (SMOS) mission are presented in Section 5. (In order to establish some consistency between this note and a previous one addressing a similar comparison under the same algebraic angle but between two observational paradigms in the far-field region only [7], the same array of elementary antennas have been used, although it is well known that MIRAS does not operate in the near-field region.) Although the two imaging conditions use the same interferometric paradigm and share the same goal, there are several differences that deserve attention while performing the comparison. From an algebraic point of view, this is the case of the corresponding modeling matrices, which are both rank-deficient with the same effective rank but with different distributions of their singular values. The impacts of the three differences identified in Section 3 are illustratedand quantified. Complex visibilities of point sources are simulated and inverted, both in the near-field region and in the far-field one. It is shown that the angular resolution [8] does not depend on the region concerned by synthesis imaging. On the contrary, as soon as near-field visibilities are inverted with the far-field resolving matrix, it turns out that the shape of the angular synthesized point-spread function varies with the distance between the point-source and the phase center of the antenna array. A similar analysis is conducted with extended sources to quantify the so-called reconstruction floor error [9]. Finally, the radiometric sensitivity [10] is addressed, and simulations do not reveal any significant difference in the propagation of random Gaussian noise from the complex visibilities to the retrieved brightness temperature maps. For the sake of transparency and pedagogy, this note is completed with an annex, where the complex cross-correlation between two quasi-monochromatic electromagnetic fields is established for spherical waves. Although this study does not claim to define a boundary between the near-field region and the far-field one for synthetic aperture imaging radiometers, it is shown that for the antenna array of SMOS, the common criterion \(2D^{2}/\lambda\) is under-estimating the extent of the near-field, or conversely over-estimating that of the far-field. Indeed, the safe far zone would probably be around \(10D^{2}/\lambda\) for this antenna array when it is used for synthesis imaging with interferometric data.