## 4 Geometrical Considerations  
For a regularly spaced antenna array leading to a Cartesian sampling of the Fourier space and therefore of the dual space of the brightness temperature distributions in the direction cosines domain, the synthesized field of view is a square of sides \(\lambda/d\). As a consequence, it remains included in the \(2\pi\) sr space in front of the instrument as soon as \(d/\lambda>\sqrt{2}/2\simeq 0.707\). Likewise, for a regularly spaced antenna array leading to a hexagonal sampling, the synthesized field of view is now a hexagon of sides \(2\lambda/3d\). As a consequence, it remains included in the same unit circle as soon as \(d/\lambda>2/3\simeq 0.667\). These two situations are illustrated in Figure 2. As soon as \(d/\lambda\) becomes smaller than these two limits, the vertices of the synthesized fields of view go beyond the unit circle. The corresponding directions \(\xi\) have neither a physical meaning nor a mathematical one, as they correspond to situations where \(\sin\theta\) would be larger than 1. Contrary to what one might quickly think, these directions are not in the \(2\pi\) sr space in the back of the antenna array. Indeed, these back directions fold on the front ones in the direction cosines domain according to the trigonometric identity \(\sin(\pi-\theta)=\sin\theta\) and, of course, they satisfy the condition \(\sin\theta\leq 1\). As a consequence, they cannot be taken into account in the inversion process. However, there is no physical reason not to attempt to inverse the visibilities with the goal of retrieving the brightness temperature distributions only for those directions of the synthesized field of view included in the unit circle. Unfortunately, it will be shown in the next sections that this limited aperture synthesis operation is unstable in the presence of noise or errors. Shown in Figure 3 are the variations of this proportion of directions with the shortest spacing between the antennas. In every case, two regions are well-separated: in the first one this fraction is strictly equal to zero, whereas in the second one it is a growing function of the inverse of the shortest spacing. The frontier  
Figure 3: Variations of the fraction of directions \(\xi\) without physical or mathematical meaning with the shortest spacing \(d\) between the antennas of arrays leading to (**a**) a Cartesian geometry and to (**b**) a hexagonal geometry. Shown in red are the theoretical variations and with blue bullets the values calculated for sampling grids with 256\(\times\)256 directions (or pixels) in the synthesized field of view. In every case, two regions are well-separated: in the first one this fraction is strictly equal to zero, whereas in the second one it is a growing function of the inverse of the shortest spacing. The frontier between these regions is estimated between 0.704\(\lambda\) and 0.705\(\lambda\) in (**a**) and between 0.664\(\lambda\) and 0.665\(\lambda\) in (**b**). Figure 2: Synthesized field of view (red) and limit of the \(2\pi\) sr space (blue) in front of antenna arrays leading to (**a**) a Cartesian geometry here with \(d/\lambda=\sqrt{2}/2\) and to (**b**,**c**) a hexagonal geometry here with \(d/\lambda=2/3\). The components \(\xi_{1}=\sin\theta\cos\phi\), \(\xi_{2}=\sin\theta\sin\phi\), and \(\xi_{3}=\cos\theta\) of the angular position variable \(\xi\) are direction cosines in the reference frame of the arrays, \(\theta\) and \(\phi\) are the traditional polar coordinates (the colatitude and the azimuth, respectively). Shown with dashed blue lines are the sky aliases. Earth (and its aliases) is not shown as its location, and its extent in this direction cosines domain depends on the elevation and on the attitude of the platform. between these regions is between 0.704\(\lambda\) and 0.705\(\lambda\) with the Cartesian geometry, and it is between 0.664\(\lambda\) and 0.665\(\lambda\) with the hexagonal ones.