Imaging in the presence of direction-dependent effects with the MeerKAT radio telescope

Abstract

The radio interferometer measurement equation [1] describes the visibilities measured by a radio interferometer in the following succinct form [2]. For a pair of antennas (i.e. baseline) $p$ and $q$, the measured $2\times 2$ complex visibility matrix $\mathbf{V}_{pq}$ is given by \begin{equation*} \mathbf{V}_{pq}=\iint\limits_{lm}\mathbf{E}_{p}\mathbf{BE}_{q}^{H}\mathrm{e}^{-2\pi \iota\lambda^{-1}(\mathbf{u}_{pq}\cdot\mathbf{l})}\mathrm{d}l\mathrm{d}m+\mathbf{N}_{pq} \tag{1} \end{equation*} where $\mathbf{B}(l, m)$ is a $2\times 2$ matrix describing the (in general, polarized) sky brightness distribution on the tangential plane $l, m, \mathbf{E}_{p}(l, m)$ is a $2\times 2$ Jones matrix describing the propagation effects for antenna $p$ in the direction $l, m, \mathbf{u}_{pq}$ is the baseline vector, $\mathbf{l}=l, m, n$ is the direction cosine vector, $\lambda$ is wavelength, and $\mathbf{N}_{\text{pq}}$ is a $2\times 2$ additive complex normal noise term. Inverting eq. 1 to recover $\mathbf{B}$ from measurements is known as the imaging problem. The problem is ill-posed, particularly so if the $\mathbf{E}$ term is non-trivial (relative to the sensitivity of the telescope given by $\mathbf{N}$); in the latter regime, it is known as the direction-dependent effect (DDE) problem.