Mode decomposition methods for the analysis of black hole ringdown signals

Abstract: The ringdown phase of a binary black hole merger encodes fundamental information about the remnant through its quasinormal mode (QNM) spectrum. Extracting multiple modes from gravitational-wave data is essential for black-hole spectroscopy but remains challenging due to the short duration of the signal, limited signal-to-noise ratio (SNR), and interference between modes. In this work, we investigate the applicability of Empirical Mode Decomposition (EMD) and Variational Mode Decomposition (VMD) to the analysis of post-merger gravitational-wave signals. Using Monte Carlo simulations of noisy ringdown signals composed of pairs of QNMs, we assessed the ability of these methods to separate the two modes and estimate their frequencies. The instantaneous frequency is obtained via the Hilbert transform (HT) and our new proposed method, named instantaneous frequency and amplitude determination (IFAD). We analyze performance over a wide range of SNR values relevant to current and future gravitational-wave detectors. Our results show that both EMD and VMD can resolve multiple modes, and VMD provides significantly more accurate frequency estimates. We also introduce a modified instantaneous frequency estimator that improves accuracy over the Hilbert transform. The study quantifies the conditions under which two-mode resolution is feasible and highlights the limitations imposed by closely spaced mode frequencies and short signal duration. These results are relevant for current observations and for future high-SNR detections expected from space-based and next-generation ground detectors.

• The paper presents a comprehensive evaluation of adaptive mode decomposition methods for extracting quasinormal modes from black hole ringdown signals.
• It compares empirical mode decomposition with variational mode decomposition and introduces an improved instantaneous frequency estimator (IFAD) to enhance noise robustness.
• The study derives a Fisher-matrix based resolvability criterion to quantify mode separation, validated via Monte Carlo simulations under varying signal-to-noise conditions.

Mode Decomposition Techniques for Black Hole Ringdown Analysis

Introduction

The extraction of multiple quasinormal modes (QNMs) from black hole ringdown signals is a cornerstone of black hole spectroscopy, promising definitive tests of the Kerr hypothesis and robust probes of strong-field gravity. However, practical mode separation is hindered by short signal durations, moderate signal-to-noise ratios (SNR), and spectral interference between proximal modes. The paper "Mode decomposition methods for the analysis of black hole ringdown signals" (2606.13851) provides a comprehensive evaluation of adaptive, minimally model-dependent time-frequency decomposition methods for ringdown analysis, comparing Empirical Mode Decomposition (EMD) and Variational Mode Decomposition (VMD), and proposing an improved instantaneous frequency estimator (IFAD) for noise-robust parameter estimation.

Empirical and Variational Mode Decomposition

Adaptive signal decomposition methodologies, notably EMD and VMD, forgo rigid basis function expansions in favor of extracting intrinsic mode functions (IMFs) directly from data, each corresponding to a narrowband oscillatory mode. EMD operates by recursively sifting the data to isolate locally symmetric oscillations, subject to constraints on zero crossings and envelope symmetry. While EMD has been widely applied to nonstationary astrophysical data, it is susceptible to mode mixing and endpoint artifacts due to spline interpolation.

VMD recasts mode decomposition as a variational optimization problem, solving for a user-set number of band-limited IMFs via Wiener filtering, Lagrange multipliers, and the alternating direction method of multipliers. Each mode's center frequency is iteratively re-estimated to minimize total bandwidth while satisfying a reconstruction constraint. VMD therefore achieves superior noise robustness, convergence guarantees, and sharper spectral separation compared to EMD.

Instantaneous Frequency Estimation (IFAD) Versus Hilbert Transform

Reliable instantaneous frequency extraction from decomposed modes is critical for mode identification and spectroscopy. The Hilbert Transform, the de facto standard, is highly sensitive to edge effects and noise, particularly given the brevity of ringdown signals. The IFAD technique introduced in this work circumvents these issues by interpolating signal extrema and zero crossings for directly estimating phase evolution and instantaneous period, leading to more accurate frequency recovery even in high-noise or short-duration scenarios.

As shown in the evaluation of synthetic signals—including damped sinusoids with frequency drift, standard QNMs, and chirps—IFAD matches ground-truth frequencies and amplitudes with high fidelity, whereas Hilbert-based estimates degrade in boundary regions.

Figure 1: Damped sinusoid signal with frequency drift illustrating the challenge for frequency tracking.

Resolvability Criteria and Fisher Analysis

A central contribution is the derivation and deployment of a geometric resolvability criterion based on the projected Fisher matrix, encapsulated by the parameter $$\rho^2 = d\theta \cdot \bar{\Gamma} \cdot d\theta^T$$, which accounts for correlations between frequency and damping time across modes. This approach generalizes previous criteria (e.g., by Isi and Farr) to allow for full covariance structure, enabling accurate quantitative predictions for mode resolvability in both Bayesian and frequentist analyses.

Empirical exploration demonstrates that resolvability increases with SNR and mode parameter separation, but exhibits pronounced asymmetry depending on relative damping times—longer-lived subdominant modes yield more observable cycles and facilitate better resolution.

Figure 2: Standard deviations of damping times and frequencies as a function of mode separation, as estimated via the Fisher matrix.

Figure 3: Resolvability parameter $\rho^2$ as a function of frequency and damping time separation, revealing conditions for robust mode distinction.

Comparative analysis establishes that the geometric criterion is more stringent, with the previous $r^2$ statistic systematically underestimating resolvability when off-diagonal Fisher terms are significant.

Figure 4: Comparison between the geometric ($\rho^2$) and previous ($r^2$) resolvability metrics.

Figure 5: Resolvability dependence on SNR, amplitude ratio, and relative phase, demonstrating optimal separation for large phase differences and balanced amplitudes.

QNM Resolution in Kerr Black Holes

Applying the geometric criterion to astrophysically motivated signal models, the study examines decompositions of the fundamental $[2\,2\,0]$ mode and its prominent overtones and angular harmonics as predicted for GW250114-like remnant parameters. These simulations show that resolvability is set primarily by frequency separation, with the $[2\,2\,1]$ overtone remaining marginally resolvable under most scenarios due to its proximity to the fundamental, despite favorable SNR.

Figure 6: The resolvability parameter $\rho^2$ for select QNM pairs as a function of SNR, demonstrating the dominant impact of frequency separation and SNR.

Figure 7: r$^2$ vs. $\rho^2$ for the case of $\rho^2$0 and $\rho^2$1 mode resolution.

Monte Carlo Performance: EMD, VMD, and IFAD

Systematic Monte Carlo studies with noisy two-mode ringdown signals demonstrate that both EMD and VMD can separate constituent QNMs, but VMD delivers lower variance and bias in frequency estimates across SNR spans, particularly as mode parameter proximity increases. Limitations become acute when energy is concentrated in closely spaced, rapidly damped (short $\rho^2$2) modes or when SNR is below the threshold defined by $\rho^2$3.

The IFAD estimator further improves the precision and stability of extracted frequencies compared to Hilbert-based approaches, especially for intermediate and high SNRs, offering substantial benefits for next-generation detectors.

Figure 8: Example VMD extraction and reconstruction: (top) noisy input, (middle/bottom) decomposition results.

Figure 9: EMD frequency extraction performance in the low SNR regime.

Figure 10: VMD frequency extraction performance in the low SNR regime.

Figure 11: VMD+IFAD frequency extraction performance, showing reduced bias and variance across SNRs compared to Hilbert-based estimation.

Implications and Future Directions

The adoption of adaptive decomposition and improved frequency tracking holds significant promise for gravitational-wave spectroscopy. These techniques are largely agnostic to model-specific waveform assumptions, positioning them as complementary or even primary analysis tools as detector sensitivity improves. For current Earth-based interferometers, robust two-mode resolution is achievable only within specific amplitude, phase, and SNR windows, but the pathway to fully multi-mode, high-precision spectroscopy is clearly enabled for planned next-generation observatories.

Key challenges persist, notably for the resolution of weak or near-degenerate overtones, and extensions to scenarios with more than two modes, non-Gaussian noise, or signals with strong environmental couplings (see e.g., [RAJ2026101072] for PINN-based extensions). Incorporating physical priors or hybridizing with machine learning approaches may yield further gains.

Conclusion

This study delivers a careful, foundational evaluation of adaptive mode decomposition for black hole ringdown signals, establishing VMD and refined instantaneous frequency estimation as preferred methods for multi-mode, high-fidelity gravitational-wave spectroscopy. The rigorous quantification of resolvability conditions via a Fisher-matrix-based geometric metric, coupled with exhaustive Monte Carlo validation, sets a new standard for nonparametric ringdown analysis. The techniques surveyed here are poised to facilitate the extraction of rich black hole structure information from next-generation gravitational-wave observations, with ongoing and future work needed to handle even more complex signal scenarios and noise environments.