Super-Resolution Spectral Analysis for Signal Recognition: Theory and Application to Earth’s Free Oscillations

Summary

Enhancing frequency resolution as much as possible from finite-length and noisy observations to identify a larger number of harmonic and quasi-harmonic signals has long been a central objective in data analysis; owing to the intrinsic complexity of the Earth system, this demand is particularly acute in the geosciences. For finite records, the effective resolution of classical Fourier-based spectra is governed by the spectral-window width, commonly characterized by the Rayleigh scale. Although non-Fourier approaches, such as autoregressive (AR) methods and Hankel total least squares (HTLS), are in principle capable of achieving higher frequency resolution, they typically rely on accurate prior information, including the locations of narrow frequency bands and the number of embedded harmonics K, and operate through direct parametric estimation rather than forming an explicit spectrum for visual inspection, thereby limiting their applicability as general blind identification techniques. To address these limitations, we propose a Super-Resolution Spectral Analysis (SRSA) method. This approach constructs a two-dimensional (2D) super-resolution (SR) spectrum ${{\bf I}}( {L,{{{\bf f}}}_c} )$ by exploiting the orthogonality between the multi-scale noise subspaces derived from the singular value decomposition of Hankel matrices (Hankel-SVDs) formed with different window lengths L, and the Vandermonde matrix. Theoretical analysis and numerical simulations demonstrate that the proposed method is robust to moderate perturbations in K and supports reliable signal identification without requiring exact prior knowledge of individual spectral peak locations or the true model order. In addition, SRSA can improve effective frequency discrimination relative to a Fourier benchmark. For a fixed record length T, its resolving capability depends jointly on the sampling frequency ${f}_s$, the intrinsic model order, and the noise level. Under the noise-free simulation setup adopted in this study, numerical experiments show roughly an order-of-magnitude improvement in effective resolution relative to the Fourier benchmark, while the noisy experiments still demonstrate a clear gain in frequency resolution across the tested SNR levels. Using superconducting gravimeter observations following the 2004 Sumatra-Andaman and the 2011 Tohoku earthquakes, we apply the SRSA method to detect the splitting of Earth’s free oscillation eigenmodes. The method improves spectral resolution and weak-signal detectability, successfully resolving multiple low-frequency spheroidal and core-sensitive modes. These results both corroborate previous findings and provide independent constraints on mode splitting, highlighting the broad potential of SRSA as an effective tool for harmonic analysis in geophysical observations.