Fourier-correlation imaging

We investigated whether correlations between the Fourier components at slightly shifted frequencies of the fluctuations of the electric field measured with a one-dimensional antenna array on board a satellite flying over a plane allow one to measure the two-dimensional brightness temperature as a function of position in the plane. We found that the achievable spatial resolution that resulted from just two antennas is on the order of $hχ$⁠, with $χ=c/(Δrω0)$⁠, both in the direction of the flight of the satellite and in the direction perpendicular to it, where $Δr$ is the distance between the antennas, ω₀ is the central frequency, h is the height of the satellite over the plane, and c is the speed of light. Two antennas separated by a distance of about 100 m on a satellite flying with a speed of a few km/s at a height of the order of 1000 km and a central frequency of order GHz allow, therefore, the imaging of the brightness temperature on the surface of Earth with a resolution of the order of 1 km. For a single point source, the relative radiometric resolution is on the order of $χ$⁠, but, for a uniform temperature field in a half plane left or right of the satellite track, it is only on the order of $1/χ3/2$⁠, which indicates that two antennas do not suffice for a precise reconstruction of the temperature field. Several ideas are discussed regarding how the radiometric resolution could be enhanced. In particular, having N antennas all separated by at least a distance on the order of the wave-length allows one to increase the signal-to-noise ratio by a factor of order N but requires averaging over N² temperature profiles obtained from as many pairs of antennas.