## 3 Instrument Modeling  
According to Equation (20) page 573 in [12] and Equation (15.6) page 770 in [13], when neglecting decorrelation effects that can be reduced with spectral sub-banding [14] if narrow-band conditions are not satisfied [15], the complex visibility \(V\) measured with a pair of elementary antennas \(\mathcal{A}_{p}\) and \(\mathcal{A}_{q}\) separated by the baseline vector \(\mathbf{b}_{pq}\), like those in Figure 1, is given by the integral relation established in the Appendix A at the end of this note in spherical coordinates \(\theta\) and \(\phi\):  
\[V(\mathbf{b}_{pq})=\iint\limits_{source}\frac{\mathcal{F}_{p}(\theta_{p}, \phi_{p})}{\sqrt{\Omega_{p}}}\frac{\mathcal{F}_{q}^{*}(\theta_{q},\phi_{q})}{ \sqrt{\Omega_{q}}}T(\theta,\phi)\frac{\mathrm{j}k_{o}\big{(}r_{p}(\theta,\phi )-r_{q}(\theta,\phi)\big{)}}{r_{p}(\theta,\phi)r_{q}(\theta,\phi)}\mathrm{d}S, \tag{1}\]  
where \(\mathcal{F}_{p}\) and \(\mathcal{F}_{q}\) are the voltage patterns of the two elementary antennas with equivalent solid angles \(\Omega_{p}\) and \(\Omega_{q}\), \(T\) is the brightness temperature distribution of an observed scene, \(k_{o}\equiv 2\pi/\lambda_{o}\) is the central angular wavenumber, and \(\lambda_{o}\) is the central wavelength of observation. The angular coordinates \((\theta_{p},\phi_{p})\) and \((\theta_{q},\phi_{q})\) are local spherical angles (the colatitude and the azimuth, respectively) from which the elementary brightness temperature contribution \(T(\theta,\phi)\mathrm{d}S\) is seen by the antennas \(\mathcal{A}_{p}\) and \(\mathcal{A}_{q}\), as illustrated in Figure 1. They are completed by the two distances \(r_{p}\) and \(r_{q}\) between the two antennas and the source element \(\mathrm{d}S\) responsible for that elementary contribution to the complex visibility. On introducing the solid angle \(\mathrm{d}\Omega\equiv\mathrm{d}S/r^{2}\), we therefore have the general expression for \(V\):  
\[V(\mathbf{b}_{pq})=\iint\limits_{source}\frac{\mathcal{F}_{p}(\theta_{p}, \phi_{p})}{\sqrt{\Omega_{p}}}\frac{\mathcal{F}_{q}^{*}(\theta_{q},\phi_{q})}{ \sqrt{\Omega_{q}}}T(\theta,\phi)\frac{r^{2}(\theta,\phi)}{r_{p}(\theta,\phi)r _{q}(\theta,\phi)}\mathrm{d}\Omega, \tag{2}\]  
Figure 1: Geometry of a one-dimensional SAI radiometer with two elementary antennas \(\mathcal{A}_{p}\) and \(\mathcal{A}_{q}\) and with incident wavefronts (in red) from a source: far-field conditions with plane waves (**left**) and near-field ones with spherical waves (**right**). In both cases, \(\mathcal{A}_{p}\) and \(\mathcal{A}_{q}\) are pointing in the same direction; here, the Nadir is illustrated by the main beam of the elementary power patterns (in blue), \(\mathbf{r}_{p}\) and \(\mathbf{r}_{q}\) are location vectors with respect to the source, and the two antennas are separated by the baseline vector \(\mathbf{b}_{pq}\). where \(r(\theta,\phi)\) is the distance from the origin of the reference frame of the antenna array (generally taken at the phase center of the array) to the source element \(\mathrm{d}S\equiv r^{2}\mathrm{d}\Omega\) responsible for the elementary contribution \(T(\theta,\phi)\mathrm{d}S\), as illustrated by Figure 1 in [4], Figure 2 in [16], Figure 3 in [17] and Equation (2) in [16] and Equation (2) in [18]. On introducing the angular position variable \(\xi\) whose components \(\xi_{1}=\sin\theta\cos\phi\) and \(\xi_{2}=\sin\theta\sin\phi\) are direction cosines in the reference frame of the antenna array, the solid angle \(\mathrm{d}\Omega\equiv\mathrm{d}S/r^{2}\equiv\sin\theta\mathrm{d}\phi\) turns out to be equal to \(\mathrm{d}\xi/\sqrt{1-\|\xi\|^{2}}\) when accounting for the Jacobian of the transformation from spherical coordinates \((\theta,\phi)\) to direction cosines \((\xi_{1},\xi_{2})\), so Equation (2) now reads  
\[V(\mathbf{b}_{pq})=\iint\limits_{\|\xi\|\leq 1}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!