Correlation Calibration: A Hybrid Calibration Technique for Radio Interferometric Arrays

TL;DR

CorrCal introduces a covariance-based, hybrid calibration method for drift-scan radio interferometers that integrates sky-based and redundant calibration ideas into a sparse, scalable framework. By modeling the data covariance as a sum of noise, diffuse, and point-source terms and solving for per-antenna gains through likelihood maximization, CorrCal achieves unbiased calibration with robustness to sky and instrument-modeling errors. The approach exploits sparsity via a sparse two-level covariance, enabling efficient likelihood and gradient computations with a practical runtime on large arrays, and is validated against a broad set of sky-modeling and nonredundancy scenarios. The method has direct relevance for current and next-generation 21 cm experiments, offering a computationally efficient path to precise calibration in the presence of complex foregrounds and instrument effects.

Abstract

Calibrating out per-antenna signal chain effects is an essential step in analyzing radio interferometric data. For drift-scanning arrays, robustly calibrating the data is especially challenging due to the lack of the ability to track a calibration source. Consequently, calibration strategies for drift-scanning arrays are limited by our knowledge of the radio sky at large, as well as the direction-dependent instrument response. In the context of 21 cm cosmology, where small calibration errors can conspire to overwhelm the cosmological signal, it is therefore crucially important to develop calibration strategies that are capable of accurately calibrating the data in the presence of sky or instrument modeling errors. In this paper we present CorrCal, a covariance-based calibration strategy for redundant radio interferometric arrays. CorrCal is a hybrid calibration strategy that leverages the strengths of traditional sky-based calibration and redundant calibration in a computationally efficient framework that is fairly insensitive to modeling errors. We find that the calibration errors from CorrCal are unbiased and far below typical thermal noise thresholds across a wide range of modeling error scenarios. We show that CorrCal is computationally efficient: our implementation is capable of evaluating the likelihood and its gradient in less than a second for 1,000-element class arrays using just a single laptop core. Given CorrCal's computational efficiency and robustness to modeling errors, we anticipate that it will serve as a useful tool in the analysis of radio interferometric data from current and next-generation experiments targeting the cosmological 21 cm signal.

Figures (3)

• Figure 1: Schematic showing the product of the beam kernels in the overlap integral that is used to compute the diffuse covariance matrix elements. The dashed lines show the expected position of the first null in the beam kernel, the stars indicate the central $uv$ modes sampled by the different baselines, and the color scale indicates the absolute value of the peak-normalized product of beam kernels at each point in the $uv$-plane. The left panel shows a pair of baselines that are considered non-redundant; the center panel shows a pair of baselines that are partially redundant; and the right panel shows a pair of baselines that are perfectly redundant.
• Figure 2: Results of a benchmark test for the single core execution time (using an Intel i7-8565U processor) of the most computationally intensive tasks in the CorrCal algorithm, shown as a function of array size: black shows the average time to invert the covariance; purple shows the time to invert the covariance and simultaneously accumulate the log-determinant; pink shows the total time for computing the negative log-likelihood; and yellow shows the total time to compute the gradient of the negative log-likelihood. For arrays with less than roughly 100 antennas, the computational cost is dominated by overheads. For larger arrays, the computational cost of all the tasks investigated here roughly scales with the square of the number of antennas, or linearly with the number of baselines. The dashed lines indicate execution times that scale with the square of the number of antennas, normalized to the measured execution time for a 500-element array.
• Figure 3: A representative view of the simulated sky, instrument response, and source model used in the tests in \ref{['sec:VALIDATION']}. The point sources shown in each panel have been smoothed with a $1\hbox{$^\prime$}$ Gaussian kernel. The top panel shows the sky input to the visibility simulations for one of the "bright" fields with the diffuse component scaled down to increase the contrast with the point sources. The middle panel shows the same field, but with the primary beam attenuation applied. The bottom panel provides a representative view of the sources that would be included when modeling the source matrix $\mathbf{\Sigma}$.