Functional Analysis of Widefield Radio Interferometry

Radio interferometry relies on reconstruction techniques to form images from discrete visibility measurements. As instruments encompass wider fields of view, the nuance of that reconstruction can lead to geometrical errors in the image. As a result, the estimation of the flux density of sources can change. We derive the full functional analysis of widefield radio interferometry to provide a foundation for investigating geometry-based errors. We use this framework to investigate three examples: 1) deconvolution errors resulting from a minor iteration, 2) gridding errors resulting from excluding primary beam sidelobes, and 3) w-term errors resulting from omitting the non-coplanar dimension on a 2 min near-zenith transit. We use their adjoint operator descriptions to simulate each of the three cases for two fields using the Murchison Widefield Array. While the estimated errors on the flux density of reconstructed sources are small, overall errors depend on the field observed. For our two fields, we find fractional flux density errors up to 5% due to deconvolution, 0.1% due to gridding, and 10% due to non-coplanarity. Our operator formalism can be used to investigate many other forms of image reconstruction errors.