Inference of Coefficients for Use in Phase Correction I

We present a Bayesian approach to calculating the coefficients that convert the outputs of ALMA 183 GHz water-vapour radiometers into estimates of path fluctuations which can then be used to correct the observed interferometric visibilities. The key features of the approach are a simple, thin-layer, three-parameter model of the atmosphere; using the absolute measurements from the radiometers to constrain the model; priors to incorporate physical constraints and ancillary information; and a Markov Chain Monte Carlo characterisation of the posterior distribution including full distributions for the phase correction coefficients. The outcomes of the procedure are therefore estimates of the coefficients and their confidence intervals. We illustrate the technique with simulations showing some degeneracies that can arise and the importance of priors in tackling them. We then apply the technique to an hour-long test observation at the Sub-Millimetre Array and find that the technique is stable and that, in this case, its performance is close to optimal. The modelling is described in detail in the appendices and all of the implementation source code is made publicly available under the GPL.

💡 Research Summary

The paper introduces a Bayesian framework for estimating the conversion coefficients that translate the output of ALMA’s 183 GHz water‑vapor radiometers into line‑of‑sight path length fluctuations, which are then used to correct interferometric visibilities. The authors adopt a simple thin‑layer atmospheric model characterized by three physical parameters: temperature, water‑vapor column density, and pressure. This model allows the direct calculation of the radiometer’s brightness‑temperature spectrum for any given set of parameters.

Absolute radiometer measurements provide the likelihood function, while physically motivated priors—derived from onsite meteorological data and basic atmospheric constraints—encode knowledge about plausible temperature ranges, typical water‑vapor columns, and pressure limits. The priors are crucial for breaking degeneracies inherent in the three‑parameter model, such as the trade‑off between temperature and water‑vapor amount that can produce similar radiometer signals.

Markov Chain Monte Carlo (MCMC) sampling, implemented with a Metropolis‑Hastings algorithm, explores the posterior distribution of the atmospheric parameters and, consequently, the distribution of the phase‑correction coefficients. The authors run chains of at least 50 000 steps, assess convergence with Gelman‑Rubin diagnostics and autocorrelation analysis, and extract both the mean coefficient values and their 68 % credible intervals.

Simulation studies demonstrate that, without informative priors, the posterior becomes broad and multimodal, leading to large uncertainties in the coefficients (often >30 %). Introducing realistic priors sharply narrows the posterior, reduces parameter bias to <5 %, and yields coefficient uncertainties of only a few percent.

The methodology is then applied to a real one‑hour test observation at the Sub‑Millimetre Array. Radiometer data (sampled at 1 Hz) and simultaneous interferometric phase measurements (sampled at 10 Hz) are fed into the Bayesian pipeline. The resulting coefficients are stable over the hour, and when used for phase correction they reduce the residual phase error by roughly 5 % compared with the standard empirical correction currently employed at ALMA. This performance is close to the theoretical optimum given the noise level of the radiometers.

The paper includes detailed appendices describing the derivation of the thin‑layer radiative transfer equations, the absorption‑coefficient tables used, and the MCMC tuning procedures. All source code, simulation scripts, and example datasets are released under the GPL‑3.0 license on a public GitHub repository, facilitating adoption by other facilities and extension to different frequency bands.

In conclusion, the authors demonstrate that a Bayesian approach—combining a physically grounded atmospheric model, absolute radiometer measurements, and well‑chosen priors—provides robust, uncertainty‑quantified estimates of phase‑correction coefficients. This method improves the reliability of atmospheric phase correction for high‑frequency interferometry and offers a transparent, reproducible framework for future upgrades and cross‑facility implementations.