## 3 Instrument Modeling
viewing direction \((\theta,\phi)\), the attenuation of the brightness temperature \(T(\theta,\phi)\) by the voltage pattern \(\mathcal{F}\) varies from one elementary antenna to another because the local incoming directions \((\theta_{p},\phi_{p})\) and \((\theta_{q},\phi_{q})\) are no longer equal. In the words of [PERSON] et al. [19]_\"that means for different baselines the source distribution is different, thus the Fourier transform cannot be used for the inversion.\"_ From a modeling point of view, it is as if each antenna had a different diagram from the other one, according to the identities \(\mathcal{F}_{p}(\theta,\phi)\equiv\mathcal{F}(\theta_{p},\phi_{p})\) and \(\mathcal{F}_{q}(\theta,\phi)\equiv\mathcal{F}(\theta_{q},\phi_{q})\). It would therefore not be surprising to see the effect of this disparity of the elementary voltage patterns [20] on the properties of the modeling matrix of the instrument, as well as on the qualities of the inversions. The previous effect cannot be seen in [21] nor in [22] as a _\"simplify visibility function ignoring the antenna voltage pattern\"_ is used. Taking into account the antenna voltage pattern at the modeling level is not a guarantee of being able to capture this effect. This is the case in the recent studies reported in [23; 24], where Equation (4) in the former and Equation (1) in the latter are both suffering from the same assumption that is not justified nor documented by the authors: the local incoming directions \(\xi_{p}\) and \(\xi_{q}\) corresponding to the local spherical angles \((\theta_{p},\phi_{p})\) and \((\theta_{q},\phi_{q})\) are assumed to be identical (and equal to \(\xi\)). This is in contradiction with expression (2), which is often referred to as the near-field complex visibility in the literature, like in Equation (1) of [16], which is very frequently cited in this imaging domain. Moreover, it is clear from Figure 1 that it cannot be the case when one claims to address near-field imaging. What is true with local spherical angles must remain true with direction cosines, as is the case in the present study with expressions (2) and (3). Finally, unlike the assumption made in [17; 23] where the product of the distances between the two antennas and the source element dS is approximated by \(r^{2}\) and therefore the corresponding fraction is neglected in the expression of the complex visibilities, the present note aims at using the complete expression (3) without neglecting any of the three terms listed previously and without introducing any reductive hypothesis, as is often the case when expending the difference \(r_{q}(\xi)-r_{p}(\xi)\) in Taylor series [25] like in [19; 24; 26]. The usefulness of approximations or reductive assumptions in the near-field modeling is questionable, especially since there is no reason not to consider all the terms found in the general expression (3). This is also true for antenna arrays with an aperture so wide [27] that some baselines are in the near-field region while the others are in the far-field one, according to the criteria provided by the Fraunhofer distance [2]. As a consequence, this study focuses only on near-field and far-field synthesis imaging as illustrated by the modeling Equations (3) and (4) and by the corresponding linear operators \(\mathbf{G}\) and \(\widetilde{\mathbf{G}}\) introduced in the next section.