## 5 Numerical Simulations and Comparison  
The numerical simulations discussed in this section have been performed with the antenna array of the imaging radiometer MIRAS [38] onboard the SMOS [39] mission. The reader will surely notice that this work and the conclusions drawn from the results presented in this section have evidently no impact on the operational processing of SMOS data as MIRAS is always operating in the far-field region and never in the near-field one. This antenna array has been chosen for feeding these numerical simulations just because it is a concrete and tangible one with available voltage patterns measured for every elementary antenna [40], as well as for establishing some consistency with a previous note [7], as already mentioned in the Section 1. It should also be noted that the study reported in this note is missing real interferometric data from both electromagnetic regions with the same instrument to make the comparison complete. Such a possibility is currently not available, but it will be in the near future, as the French space agency (CNES) and the CESBIO are together developing an L-band antenna array equipped with 32 elementary antennas and with a longest baseline \(D\) of about 1 m intended to be used in a wide range of distances \(h\) (typically from 5 m to 40 m) suspended from a crane over an experienced and controlled field, as was performed few years ago with MIRAS breadboard [41]. Although the main goal of this forthcoming antenna array will not be to compare far-field and near-field imaging conditions but rather to confront synthetic aperture interferometry and digital beam forming [7], the flexibility of the instrument will allow such a comparison. The SMOS mission of the European Space Agency (ESA) is devoted to Soil Moisture and Ocean Salinity remote sensing from space [39]. It was launched in November 2009, and since then, it has been operational and has provided accurate L-band brightness temperature maps [42] with a spatial resolution between \(\sim\)25 km and \(\sim\)60 km, depending on the position in the field of view. These maps are used for retrieving surface soil moisture (SM) [43], as well as sea surface salinity (SSS) [44]. The very principle of the imaging radiometer MIRAS is that of aperture synthesis, a technique initially developed for radio astronomy where the signals collected by each pair of interferometric arrays of elementary antennas are cross-correlated [13]. It is therefore not surprising that the antenna array of MIRAS was inspired by the Very Large Array (VLA) [45] for the geometry of the array and by the Cosmic Background Imager (CBI) [46] for the regular spacing between the antennas. As shown in Figure 2, it is a Y-shaped array equipped with 69 equi-spaced elementary antennas. As a consequence of the longest baseline \(D=6.75\) m and of the central wavelength of the instrument \(\lambda=21.21\) cm, the Fraunhofer distance \(2D^{2}/\lambda\) is here about 430 m [2]. Unless it is explicitly precised, in the remainder of this note, this antenna array will be placed in the near-field region at a distance \(h=100\) m from a source. However, unlike SMOS, which is pointing at the Earth with a constant tilt angle of 32.5\({}^{\circ}\) in the orbital plane [38; 39], here, this offset is reduced to zero so that the number of pixels of the sampling grid in the synthesized field domain belonging to the Earth surface is constant over a wide range of values for \(h\). This is essential for conducting studies where this distance is a key parameter; otherwise, the differences that might be observed could be attributed to this varying number of pixels and not to the modeling equations themselves. Referring back to the geometry of Figure 1 and to illustrate it concretely, shown in Figure 3 are the distances \(r_{p}(\xi)\) from every antenna \(\mathcal{A}_{p}\) of this array to that source in the Nadir direction \(\xi=(0,0)\), as well as the corresponding local angles \(\theta_{p}(\xi)\).