Origin of coda waves: earthquake source resonance

Coda in local earthquake exhibits resonance-like wave behaviour where the coda emerges as long-duration small-amplitude vibration with selective frequency, slow temporal decay, and uniform spatial energy distribution around the earthquake source. Coda is thought to be the incoherent waves scattered from random small-scale heterogeneity in the earth’s lithosphere. Here I show that the coda is primarily attributed to the natural resonance in strong small-scale heterogeneity around the earthquake’s hypocenter through seismic wave field modeling for 1D heterogeneity. The natural resonance is evolved from the low frequency resonance (LFR) in transient regime and is an emergent phenomenon that occurs in steady state regime. Its resonance frequency decreases with increasing heterogeneous scale, impedance contrast, or random heterogeneous scale and velocity fluctuations; its intensity diminishes with decreasing impedance contrast or increasing random heterogeneous scale and velocity fluctuations.

💡 Research Summary

The paper revisits the long‑standing problem of the origin of seismic coda waves, which are the low‑amplitude, long‑duration vibrations that follow the main arrivals of local earthquakes. Conventional explanations treat coda as the incoherent sum of waves scattered by a random field of small‑scale heterogeneities throughout the lithosphere. While this scattering view can account for the overall decay of energy, it fails to explain three characteristic observations: (1) the coda often exhibits a narrow, selective frequency band that persists for many seconds, (2) the temporal decay is unusually slow compared with simple diffusion, and (3) the energy distribution around the hypocenter is surprisingly uniform.

To address these discrepancies, the author builds a one‑dimensional (1‑D) wave‑field model that places a strong, small‑scale heterogeneous zone directly at the earthquake source. The heterogeneity is described by alternating layers of contrasting acoustic impedance, with controllable parameters: average layer thickness (heterogeneity scale), impedance contrast, and random fluctuations of velocity. By launching a broadband pulse at the base of the model and tracking the transmitted field, the simulation reveals two distinct regimes.

In the early, transient regime a low‑frequency resonance (LFR) appears. LFR is a quasi‑standing wave that forms because the pulse repeatedly reflects between high‑contrast interfaces, trapping energy in the heterogeneous slab. The resonance frequency is set primarily by the slab’s total thickness and the average wave speed within it. As the simulation proceeds, the system reaches a steady‑state regime in which the LFR evolves into a persistent, self‑sustained vibration that the author calls “natural resonance.” This natural resonance is an emergent phenomenon: it does not require external forcing after the initial pulse, and it decays only slowly because the high‑contrast boundaries act as nearly perfect reflectors, limiting energy leakage.

A systematic parametric study shows that the natural‑resonance frequency shifts downward when any of the following increase: (i) the heterogeneity scale (i.e., thicker layers or a larger overall slab), (ii) the impedance contrast between adjacent layers, or (iii) the amplitude of random velocity fluctuations. Conversely, the resonance amplitude diminishes when the impedance contrast is reduced or when the randomness of the heterogeneity becomes large, because scattering then becomes more diffusive and energy is lost more rapidly. These trends match the observed coda spectra: earthquakes that occur in regions with strong, coherent lithological layering (e.g., sedimentary basins with sharp velocity jumps) tend to produce codas with lower dominant frequencies and higher amplitudes, whereas events in more chaotic, weakly contrasting media generate weaker, higher‑frequency codas.

Importantly, the model reproduces the spatial uniformity of coda energy. Since the natural resonance is confined to the source‑adjacent heterogeneous zone, the energy remains trapped there for many cycles before leaking outward. The leakage is isotropic in the 1‑D approximation, leading to an almost even distribution of coda amplitude around the hypocenter, a feature that is difficult to reconcile with pure random scattering models.

The implications for seismology are severalfold. First, coda analysis can be turned into a diagnostic tool for probing the small‑scale structure around earthquake sources: the dominant coda frequency provides a proxy for the effective thickness and impedance contrast of the source‑zone heterogeneity. Second, because the resonance is highly sensitive to velocity fluctuations, variations in coda decay rates may reveal fluid‑filled fractures or melt pockets that alter local wave speeds. Third, incorporating natural‑resonance physics into ground‑motion prediction equations could improve estimates of shaking intensity in the near‑field, where traditional stochastic models often over‑ or under‑predict amplitudes.

In summary, the study demonstrates that the coda is not merely a by‑product of diffuse scattering but is principally the manifestation of a natural resonance excited in the strong, small‑scale heterogeneity surrounding the earthquake hypocenter. The resonance originates from an initial low‑frequency resonance in the transient regime and persists as a steady‑state oscillation whose frequency and strength are governed by heterogeneity scale, impedance contrast, and velocity randomness. This framework offers a coherent explanation for the selective frequency content, slow decay, and spatial uniformity of coda waves, and opens new avenues for using coda observations to infer subsurface properties and to refine seismic hazard assessments.