# Fourier-Correlation Imaging

## TL;DR

This paper explores how correlating Fourier components from satellite-based electric field measurements can enable two-dimensional imaging of Earth's surface temperature with about one kilometer resolution, highlighting limitations and potential improvements.

## Contribution

It introduces a method using Fourier component correlation for satellite imaging and analyzes the resolution limits with two antennas, proposing ways to enhance radiometric resolution.

## Key findings

- Spatial resolution of about 1 km with two antennas at 100m separation.
- Radiometric resolution improves with more antennas, but requires averaging multiple profiles.
- Two antennas alone are insufficient for precise temperature field reconstruction.

## Abstract

We investigate to what extent correlating the Fourier components at slightly shifted frequencies of the fluctuations of the electric field measured with a one-dimensional antenna array on board of a satellite flying over a plane, allows one to measure the two-dimensional brilliance temperature as function of position in the plane. We find that the achievable spatial resolution resulting from just two antennas is of the order of $h\chi$, with $\chi=c/(\Delta r \omega_0)$, both in the direction of flight of the satellite and in the direction perpendicular to it, where $\Delta r$ is the distance between the antennas, $\omega_0$ the central frequency, $h$ the height of the satellite over the plane, and $c$ the speed of light. Two antennas separated by a distance of about 100m on a satellite flying with a speed of a few km/s at a height of the order of 1000km and a central frequency of order GHz allow therefore the imaging of the brilliance temperature on the surface of Earth with a resolution of the order of one km. For a single point source, the relative radiometric resolution is of order $\sqrt{\chi}$, but for a uniform temperature field in a half plane left or right of the satellite track it is only of order $1/\chi^{3/2}$, indicating that two antennas do not suffice for a precise reconstruction of the temperature field. Several ideas are discussed how the radiometric resolution could be enhanced. In particular, having $N$ antennas all separated by at least a distance of the order of the wave-length, allows one to increase the signal-to-noise ratio by a factor of order $N$, but requires to average over $N^2$ temperature profiles obtained from as many pairs of antennas.