Bayesian MCMC Best-Fitting to GONG Observations

• The paper introduces a Bayesian MCMC framework that rigorously fits power spectral density models to GONG observations, enhancing the detection of solar oscillatory features.
• It employs physically motivated spectral and noise modeling, alongside uninformative priors and convergence diagnostics, for robust uncertainty quantification.
• The study discusses computational challenges and compares MCMC-based gold standards with CNN surrogate models that achieve over 10^4× speedup with minimal loss in fidelity.

Bayesian Markov Chain Monte Carlo (MCMC) best-fitting to GONG observations refers to the use of Bayesian inference and MCMC sampling to rigorously fit parametric models of power spectral density (PSD) to time-series data from the Global Oscillation Network Group (GONG) project. This approach combines physically motivated spectral modeling, statistical noise modeling, and robust uncertainty quantification to extract oscillation parameters and assess signal detection in solar time-series, especially for studies of filament oscillations and helioseismic modes (Castelló et al., 22 Jan 2025, Handberg et al., 2010).

1. Statistical Foundations: Bayesian Inference for Power Spectra

Bayesian inference in the context of helio- and asteroseismic PSD analysis is formalized via Bayes’ theorem,

$$p(\theta\,|\,D) = \frac{p(\theta)\,p(D\,|\,\theta)}{p(D)} \propto p(\theta)\,L(D\,|\,\theta)$$

where $\theta$ represents the parameter vector defining the spectral model, $D$ denotes the observed PSD, $p(\theta)$ is the prior, and $L(D\,|\,\theta)$ is the likelihood function.

For the case of GONG H$\alpha$ observations, the likelihood assumes each periodogram bin $D_j = \mathrm{PSD}(\nu_j)$ is an independent exponential draw about the modeled background $S_j$: $$p(D\,|\,A,\alpha,B) = \prod_{j=1}^N \frac{1}{S_j} \exp\left(-\frac{D_j}{S_j}\right)$$ with $$\ln \mathcal{L}(A,\alpha,B) = -\sum_j[\ln S_j + D_j/S_j]$$ (Castelló et al., 22 Jan 2025). This is analogous to the likelihood function used for fitting helioseismic mode power spectra (Handberg et al., 2010).

2. Parametric Spectral Models for GONG Observations

The PSD for each GONG time-series pixel is parameterized as a combination of red and white noise: $S(\nu) = A\,\nu^{-\alpha} + B$ where $A$ is the red-noise amplitude, $\alpha$ the spectral index, $B$ the frequency-independent white-noise floor, and $\nu$ the frequency (in mHz). This model is motivated by both theoretical and empirical considerations, capturing the dominant noise contributions in H$\alpha$ solar data (Castelló et al., 22 Jan 2025).

In broader helioseismic applications, the spectral model $M(\nu;\theta)$ may also incorporate Lorentzian profiles for oscillation modes and additive background terms (sum of Harvey-like components plus photon noise) (Handberg et al., 2010), but the GONG H$\alpha$ application omits explicit mode peaks in the base noise fit, focusing on the detection of periodic excesses.

3. Choice of Priors

Uninformative, uniform priors are adopted for the PSD model parameters in the GONG H$\alpha$ context:

• $A \sim \mathcal{U}(0,1)$
• $\alpha \sim \mathcal{U}(0,10)$
• $B \sim \mathcal{U}(0,1)$ All parameters are normalized with respect to the Astropy-style Lomb–Scargle periodogram output (Castelló et al., 22 Jan 2025).

For helioseismic peak-bagging of resolved oscillation modes, the prior structure can be more elaborate, employing uniform or (modified) Jeffreys priors for scale parameters, physically motivated bounds for linewidths, timescales, inclinations, and informative or hierarchical priors when applicable (Handberg et al., 2010).

4. MCMC Sampling Schemes

MCMC sampling is used to generate posterior samples for $(A, \alpha, B)$ in the GONG analysis. The implementation leverages PyMC, with burn-in and chain sampling of $p(A, \alpha, B | D)$. Specific sampler and configuration (e.g., default NUTS, Metropolis–Hastings, number of chains, steps) are not explicitly detailed in the reference, but typical choices involve 4 chains, $1\,000$–$2\,000$ burn-in steps, and $2\,000$–$5\,000$ post-burn-in draws per chain (Castelló et al., 22 Jan 2025). Convergence diagnostics such as $\hat{R}<1.01$, effective sample size, and trace inspection are implied best practice.

The more general helioseismic pipeline may employ parallel tempering, dynamically tuned proposal widths to target a 25% acceptance rate, and explicit burn-in control. The acceptance probability for Metropolis–Hastings transitions and swap steps (in parallel tempering) are defined explicitly in (Handberg et al., 2010), enabling efficient sampling even in multi-modal, high-dimensional parameter spaces.

5. Posterior Analysis and Confidence Thresholds

After MCMC sampling, marginal and joint posteriors for all spectral parameters are available. For the GONG context, the strategy for establishing detection significance follows:

1. Identify the posterior mode $(A_{\rm mode}, \alpha_{\rm mode}, B_{\rm mode})$.
2. Define the normalized test statistic:

$R_j = \frac{2 D_j}{S_j(\theta_{\rm mode})}$

1. Simulate synthetic periodograms using draws from the MCMC posterior and random deviates $X_j \sim \chi^2_2$:

$D_j^{\rm synth} = S_j(\theta^{\rm synth}) X_j/2$

1. At each $\nu_j$, compute the $(1-\epsilon)$ quantile $R_j^{1-\epsilon}$ of the synthetic $R_j$ ensemble.
2. The $(1-\epsilon)$ confidence threshold curve is then:

$$\mathrm{conf}_j^{1-\epsilon} = \frac{R_j^{1-\epsilon} S_j(\theta_{\rm mode})}{2}$$

This threshold yields statistical control over false positive rates in detecting oscillatory features above the modeled noise, as used to recover known filament oscillation periods (Castelló et al., 22 Jan 2025).

6. Computational Considerations and Scaling

The Bayesian MCMC procedure, though rigorous, is computationally intensive. A single full fit for a PSD takes $\sim$10 seconds per CPU core; at $2048^2$ spatial pixels per solar-disk observation, this results in $\sim$230 days of CPU per day of data (Castelló et al., 22 Jan 2025). This has motivated the exploration of surrogate models—including convolutional neural networks (CNNs)—that emulate the MCMC-based inference and deliver $>10^4\times$ speedup with minimal loss in detection fidelity.

A plausible implication is that MCMC-based Bayesian fitting serves primarily as a gold standard in validating and benchmarking automated detection tools for large-scale spectroscopic solar surveys.

7. Applications and Reproducibility in the GONG Context

The described Bayesian MCMC workflow on GONG H$\alpha$ time series enables systematic, statistically robust detection of periodic solar filament oscillations. Example recoveries include periods such as $75\pm1$ min and $71\pm2.8$ min (2014-01-01), $47.6\pm0.6$ min and $103.3\pm0.4$ min (2014-02-13), aligning with literature-reported results (Castelló et al., 22 Jan 2025). Credible intervals are derived from marginals of the posteriors for relevant parameters (e.g., the spectral index $\alpha$).

The pipeline's reproducibility is determined by clear specification of the spectral model, likelihood, priors, MCMC configuration, and diagnostic procedures. The published work supplies all equations and modeling choices needed for independent re-implementation, modulo standard MCMC configuration decisions.

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References

• "Fast Bayesian spectral analysis using Convolutional Neural Networks: Applications over GONG H$α$ solar data" (Castelló et al., 22 Jan 2025)
• "Bayesian peak-bagging of solar-like oscillators using MCMC: A comprehensive guide" (Handberg et al., 2010)