[Paper Review] A Bayesian approach to high fidelity interferometric calibration I: mathematical formalism

Peter Sims, Jonathan C. Pober+1 more|arXiv (Cornell University)|Jun 27, 2022

Radio Astronomy Observations and Technology66 references10 citations

TL;DR

This paper introduces BayesCal, a Bayesian framework that mitigates spurious spectral structure in radio interferometric calibration by modeling missing sky components as a statistically constrained prior on spectral smoothness. By analytically marginalizing over unknown flux contributions and incorporating physically motivated priors on instrumental gains, BayesCal suppresses spectral fluctuations in calibration solutions by up to four orders of magnitude compared to standard methods.

ABSTRACT

High fidelity radio interferometric data calibration that minimises spurious spectral structure in the calibrated data is essential in astrophysical applications, such as 21 cm cosmology, which rely on knowledge of the relative spectral smoothness of distinct astrophysical emission components to extract the signal of interest. Existing approaches to radio interferometric calibration have been shown to impart spurious spectral structure to the calibrated data if the sky model used to calibrate the data is incomplete. In this paper, we introduce BayesCal: a novel solution to the sky-model incompleteness problem in interferometric calibration, designed to enable high fidelity data calibration. The BayesCal data model supplements the a priori known component of the forward model of the sky with a statistical model for the missing and uncertain flux contribution to the data, constrained by a prior on the power in the model. We demonstrate how the parameters of this model can be marginalised out analytically, reducing the dimensionality of the parameter space to be sampled from and allowing one to sample directly from the posterior probability distribution of the calibration parameters. Additionally, we show how physically motivated priors derived from theoretical and measurement-based constraints on the spectral smoothness of the instrumental gains can be used to constrain the calibration solutions. In a companion paper, we apply this algorithm to simulated observations with a HERA-like array and demonstrate that it enables up to four orders of magnitude suppression of power in spurious spectral fluctuations relative to standard calibration approaches.

Motivation & Objective

• . The paper addresses the problem of spurious spectral structure in calibrated interferometric data caused by incomplete sky models.
• . It aims to develop a statistically principled method to model missing flux contributions without assuming perfect knowledge of the sky.
• . The objective is to enable high-fidelity calibration by incorporating uncertainty in the sky model and instrumental response through Bayesian inference.
• . It seeks to improve calibration accuracy in 21 cm cosmology experiments where spectral smoothness of foregrounds is critical for signal extraction.
• . The framework is designed to propagate uncertainties correctly through calibration, enabling robust astrophysical parameter estimation.

Proposed method

• . The BayesCal framework introduces a statistical model for the missing and uncertain flux contribution to the data, constrained by a prior on the power in the model.
• . It uses a forward model that combines the a priori known sky component with a non-parametric spectral model for the unknown component.
• . The parameters of the unknown flux model are analytically marginalized out, reducing the dimensionality of the posterior distribution and enabling direct sampling from the calibration parameter posterior.
• . Physically motivated priors on the spectral smoothness of instrumental gains are incorporated using theoretical and measurement-based constraints.
• . The framework supports joint calibration of multiple time integrations by incorporating temporal models and priors on gain evolution.
• . It enables Bayesian model selection to determine optimal spectral parametrization (e.g., power law vs. more complex models) for the gain solutions.

Experimental results

Research questions

• RQ1 . How can spurious spectral structure in interferometric calibration be suppressed when the sky model is incomplete?
• RQ2 . What statistical framework allows for the marginalization of unknown flux contributions without increasing computational cost?
• RQ3 . How can physically motivated priors on instrumental gain spectral smoothness be incorporated into the calibration process?
• RQ4 . Can the framework robustly handle uncertainty in baseline redundancy and instrument model parameters?
• RQ5 . How does the inclusion of temporal priors improve calibration stability across multiple integrations?

Key findings

• . BayesCal enables up to four orders of magnitude suppression of spurious spectral fluctuations in calibration solutions compared to standard calibration methods.
• . The framework analytically marginalizes over unknown flux components, reducing the dimensionality of the parameter space and enabling efficient posterior sampling.
• . Physically motivated priors on gain spectral smoothness disfavor solutions with spurious high-frequency structure while preserving true low-level spectral features.
• . The method correctly propagates non-uniform noise and redundancy effects through to calibrated data, ensuring accurate uncertainty propagation.
• . Bayesian model selection can determine whether a simple spectral model like a power law is sufficient or if a more complex model is required for a given dataset.
• . The framework generalizes to fully sky-based calibration and can account for uncertainties in antenna positions and beam patterns through joint sampling with priors derived from measurement errors.