On Identifiability and Estimability of Direction Dependent Calibration of Radio Interferometric Arrays

TL;DR

This paper develops a generic statistical framework to assess the calibratability of radio interferometric arrays, analyzing calibration methods, their assumptions, and effects of using subsets of baselines, supported by theoretical and simulation results.

Contribution

It introduces a new signal processing model for direction-dependent effects and spectral smoothness, and provides a mathematical framework for calibratability analysis of radio arrays.

Findings

Calibration methods differ in assumptions and theoretical properties.

Using only a subset of baselines can introduce excess noise and biases.

The proposed model and framework are validated through numerical simulations.

Abstract

Calibration is a key step in the signal processing pipeline of any radio astronomical instrument. The required sky, ionospheric and instrumental models for this step can suffer from various kinds of incompleteness. In this paper we analyze several important calibration methods, ignoring for now the ionosphere. The aim is to use established statistical and signal processing tools to provide a generic method to assess calibratability of an instrument. We show how currently popular calibration techniques differ in their assumptions and also discuss their theoretical commonalities. We also study the effect of only using a sub-set of baselines on the calibration and provide theoretical methods to analyze excess noise and biases that it might introduce. In order to simplify the physical interpretation of the results, we introduce a new signal processing model which is capable of modeling instrumental direction dependent effects and spectral smoothness of the individual receiver gain within a beam-formed station. The statistical properties of this model are then studied by deriving the Cram\'er--Rao bound (CRB). We finally define a mathematical framework for calibratability of an instrument based on the model used which is generic and can be used to study different instruments. These theoretical results are then verified using numerical simulations.