Bayesian power spectrum estimation with modelling of systematic effects in delay-fringe rate space

1. Problem Statement

The detection of the redshifted 21cm signal from the Epoch of Reionisation (EoR) is hindered by bright foregrounds (Galactic and extra-galactic) and instrumental systematic effects. While foregrounds are spectrally smooth and typically confined to a "foreground wedge" in delay-fringe rate space, instrumental imperfections—specifically cable reflections and mutual coupling—leak foreground power into the "EoR window" (modes outside the wedge).

• Cable Reflections: Impedance mismatches create attenuated copies of the signal at higher delay modes.
• Mutual Coupling: Electromagnetic interactions between antenna elements create signal copies at low delay modes.
• Current Limitations: Traditional mitigation strategies involve filtering (e.g., fringe-rate filtering) or masking affected modes. These approaches often result in signal loss, introduce spurious correlations in noise statistics, and lack rigorous uncertainty quantification. Furthermore, existing methods often treat systematics as additive noise or require computationally expensive electromagnetic simulations.

2. Methodology

The authors propose a Bayesian forward-modelling approach using a Gibbs Sampler (specifically an extension of the hydra-pspec code) to jointly infer the EoR signal, foregrounds, and systematic effects.

A. Data Model

The visibility data ($d$) is modeled as a multiplicative systematic effect acting on the sky signal ($s = e + f$, where $e$ is the EoR signal and $f$ is the foreground), plus noise ($n$):
$$d = Ys + n$$

• Multiplicative Systematics ($Y$): Instead of modeling systematics as an additive term (which would require modeling a full copy of the complex foreground signal), the authors model them as a multiplicative gain factor.
• Delay-Fringe Rate Parameterization: The gain matrix $Y$ is constructed in the delay-fringe rate space. It is defined as $y = 1 + H b_{sys}$, where:
  • $H$ is a Fourier projection operator selecting a specific, limited set of delay-fringe rate modes known to contain systematics.
  • $b_{sys}$ is a vector of complex amplitudes for these modes.
• Advantage: This parametric approach drastically reduces the number of free parameters compared to additive modeling, as it only requires modeling the coefficients of the specific modes where reflections occur.

B. The Gibbs Sampler

The authors use a Markov Chain Monte Carlo (MCMC) Gibbs sampler to draw samples from the joint posterior distribution $p(e, a_{fg}, b_{sys}, E | \text{data})$. The algorithm iterates through three conditional distributions:

1. Signal Reconstruction (Gaussian Constrained Realisations - GCR): Samples the EoR signal ($e$) and foreground amplitudes ($a_{fg}$) given the current estimate of systematics ($b_{sys}$) and the power spectrum ($E$). This involves solving linear equations (Wiener filtering) with added Gaussian noise terms to generate realizations rather than just Maximum A Posteriori (MAP) estimates.
2. Power Spectrum Sampling: Samples the EoR delay power spectrum ($E$) assuming an inverse Gamma distribution (derived from an inverse Wishart distribution under homogeneity assumptions).
3. Systematics Sampling: Samples the systematics coefficients ($b_{sys}$) given the current signal and foreground estimates, also using a Wiener-filter-like approach with a Gaussian prior.

3. Key Contributions

• Multiplicative Systematics Modeling: Introduced a novel framework to model cable reflections and mutual coupling as multiplicative gains in delay-fringe rate space, significantly reducing the parameter space dimensionality.
• Joint Bayesian Inference: Integrated the systematics model directly into the hydra-pspec Gibbs sampler, allowing for the marginalization of systematics uncertainties rather than filtering them out.
• Uncertainty Quantification: Provided a rigorous statistical framework to propagate uncertainties from systematics modeling into the final EoR power spectrum estimates.
• Simulation Framework: Validated the method on simulated HERA-like data (using pyuvsim) across three distinct test cases representing different locations of systematic contamination in delay-fringe rate space.

4. Results

The method was tested on three simulated cases with different systematic locations:

• Case I: Systematics at low delay modes ($\tau \approx 300-600$ ns) and fringe rate $f \approx 0$ mHz (overlapping with foregrounds).
• Case II: Systematics at high delay modes ($\tau \approx 1000-1300$ ns) and $f \approx 0$ mHz (separated from foregrounds).
• Case III: Systematics at low delay modes but high fringe rate ($f \approx 6$ mHz).

Key Findings:

• EoR Power Spectrum Recovery: The EoR delay power spectrum (DPS) was recovered accurately in all three cases, largely independent of the location of the systematics. The recovered DPS matched the true signal within noise levels, except near the foreground-dominated region.
• Component Recovery:
  • Case II showed the best recovery of systematics parameters and the lowest correlation between foregrounds and systematics.
  • Cases I and III exhibited significant correlations between the foreground parameters and systematics parameters. This led to longer correlation times in the Gibbs chain (low Effective Sample Size) and some bias in the recovered systematics coefficients (offset by several standard deviations).
• Sampling Efficiency: The presence of overlapping structures in delay space (Case I) or shared delay modes with different fringe rates (Case III) caused the sampler to explore the parameter space inefficiently. Case II, where systematics were well-separated in delay, converged much faster.
• Residuals: Residuals between the true and recovered sky visibilities were generally comparable to the noise level, though anti-correlations between EoR and foreground residuals were observed on large scales due to the lack of temporal modeling in the foregrounds.

5. Significance and Future Work

• Robustness: The study demonstrates that Bayesian marginalization of systematics is a viable alternative to aggressive filtering, preserving more of the EoR signal and providing better error bars.
• Limitations: The method currently requires a priori knowledge of which delay-fringe rate modes contain the systematics. In real observational data, identifying these modes precisely is non-trivial.
• Correlation Issues: The paper highlights that strong correlations between foregrounds and systematics can degrade sampling efficiency. Future work suggests:
  • Imposing stronger priors if approximate systematics values are known.
  • Modeling the temporal (LST) correlations of foregrounds explicitly (currently treated as independent time samples).
  • Implementing joint rescaling techniques to break degeneracies in the Gibbs sampler.

In conclusion, this work provides a powerful statistical tool for the next generation of 21cm cosmology experiments (like HERA Phase II and beyond) to handle complex instrumental systematics without sacrificing signal-to-noise ratio through filtering.