## 2 Observing Equation of Interferometric Arrays  
Whereas classical radiometers measure the power collected by a highly directive antenna, interferometric measurements provided by an aperture synthesis radiometer are obtained by cross-correlating the signals collected by pairs of non-directive antennas pointing in the same direction and sharing the same field of view, as illustrated in Figure 1. As a consequence, if total power radiometers provide direct measurements of the brightness temperature in the main beam direction of the antenna, aperture synthesis ones require inversion of the so-called complex visibilities [6] with the aid of the computer in order to retrieve the brightness temperature distribution in front of the antenna array [12]. The collection of elementary antennas of the interferometric array, as well as their characteristics, play a key role in this inversion process and consequently in the performances of the aperture synthesis radiometer. The relationship between the complex visibilities and the brightness temperature of the scene under observation has been revisited in order to take into account the mutual coupling of close antennas [13]. Without polarimetric considerations [14], the complex visibility \(V_{pq}\) for a pair of elementary antennas \(\mathcal{A}_{p}\) and \(\mathcal{A}_{q}\) is given by:  
\[V(\mathbf{b}_{pq})=\frac{1}{\sqrt{\Omega_{p}\Omega_{q}}}\iint\limits_{\|\xi\| \leq 1}\mathcal{F}_{p}(\xi)\mathcal{F}_{q}^{*}(\xi)\widetilde{r}_{pq}(t_{pq}) \left(T_{b}(\xi)-T_{rec}\right)\mathrm{e}^{-2\mathrm{j}\pi\frac{\mathbf{b}_{pq }\cdot\xi}{\lambda}}\frac{\mathrm{d}\xi}{\sqrt{1-\|\xi\|^{2}}} \tag{1}\]  
where \(\mathbf{b}_{pq}\equiv\mathbf{r}_{q}-\mathbf{r}_{p}\) is the baseline vector associated with the two antennas \(\mathcal{A}_{p}\) and \(\mathcal{A}_{q}\) located in \(\mathbf{r}_{p}\) and in \(\mathbf{r}_{q}\), 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 array, \(\theta\) and \(\phi\) are the traditional spherical coordinates (the colatitude and the azimuth, respectively), \(\mathcal{F}_{p}(\xi)\) and \(\mathcal{F}_{q}(\xi)\) are the voltage patterns of the two antennas (for the readers not familiar with antennas patterns, the difference between amplitude, or voltage, patterns and power patterns can be found in the definition of radiation patterns in [15]) with equivalent solid angles \(\Omega_{p}\) and \(\Omega_{q}\), as defined by Equations (2)-(24) in [16], \(T_{b}(\xi)\) is the brightness temperature distribution of the scene under observation, \(T_{rec}\) is the physical temperature of the receivers (assumed to be the same for all the receivers), \(\widetilde{r}_{pq}(t_{pq})\) is the fringe-washing function that accounts for spatial decorrelation effects, as defined by Equation (3) in [17], with \(t_{pq}=(\mathbf{b}_{pq}\cdot\xi)/c\) the spatial delay and \(\lambda=c/f\) the central wavelength of observation. After discretization of the double integral in modeling Equation (1) over an appropriate sampling grid in the direction cosines domain, the relationship between the complex visibilities \(V\) and the brightness temperature distribution \(T_{b}\) of the scene under observation can be written in the linear algebraic form:  
\[V=\mathbf{G}T, \tag{2}\]  
where \(\mathbf{G}\) is the linear modeling matrix of the instrument. The inverse problem aims at retrieving \(T=T_{b}\) (the constant \(T_{rec}\) is canceled out from the visibilities with the aid of the response to a flat target [18], whatever the method used for the inversion [19]). As this inverse problem is ill-posed [20], it has to be regularized in order to provide a unique and stable solution for \(T\). Among the regularization methods [21], the minimum-norm solution is widely used in imaging radiometry [22]:  
Figure 1: Illustration of the principle of aperture synthesis radiometry.