Delayed dynamic triggering of earthquakes: Evidences from a statistical model of seismicity

I study a recently proposed statistical model of earthquake dynamics that incorporates aging as a fundamental ingredient. The model is known to generate earthquake sequences that quantitatively reproduce the spatial and temporal clustering of events observed in actual seismic patterns. The aim of the present work is to investigate if this model can give support to the empirical evidence that earthquakes can be triggered by transient small perturbations, particularly by the passing of seismic waves originated in events occurring in far geographical locations. The effect of seismic waves is incorporated into the model by assuming that they produce instantaneous small modifications in the dynamical state of the system at the time they are applied. This change in the dynamical state has two main effects. On one side, it induces earthquakes that occur right at the application of the perturbation. These are called immediate events. On the other side, after the application of the perturbation there is a delayed effect: the seismic activity increases abruptly after the perturbation, then falls down below the level of background activity, and eventually recovers to the background value. The time scale of these variations depends on the internal dynamics of the system, and is totally independent of the duration of the perturbation. The number of delayed events in excess of the background activity is typically observed to be around a factor of twenty larger than the number of immediate events. The origin of the enhanced activity period following the perturbation is associated to the existence of aging relaxation, and it does not occur if relaxation is absent. These findings give support to the experimental evidence that earthquake can be remotely triggered by small transient perturbations as those produced by seismic waves.

💡 Research Summary

The paper investigates whether a recently introduced statistical model of seismicity, which explicitly incorporates an aging (relaxation) mechanism, can reproduce the empirically observed phenomenon of remote earthquake triggering by transient perturbations such as passing seismic waves. The model represents the crust as a two‑dimensional lattice; each site carries a stress variable σ and an aging variable θ. Stress evolves through external loading, stress transfer from neighboring ruptures, and a gradual reduction of the frictional strength encoded in θ. When σ exceeds a fixed threshold σc a rupture (an “event”) occurs, redistributing stress to adjacent sites and resetting the local stress while partially restoring θ. This simple rule set has already been shown to generate realistic spatial and temporal clustering, obeying the Gutenberg‑Richter law and Omori aftershock decay.

To mimic the effect of a distant seismic wave, the authors apply an instantaneous, spatially uniform perturbation Δσ to all sites at a prescribed time. Two regimes are explored: (i) a large Δσ that pushes many sites over the failure threshold, producing “immediate events” at the moment of perturbation; (ii) a smaller Δσ that does not directly cause failures but nevertheless perturbs the dynamical state. The system’s response is monitored over many Monte‑Carlo realizations to obtain robust statistics.

Three distinct phases emerge after the perturbation. The first is the immediate burst of events occurring exactly at the time of the perturbation; its magnitude scales with Δσ and is present even when the aging term is switched off. The second phase is a delayed surge of seismic activity that peaks after a characteristic lag that depends solely on the internal aging time τaging, not on the duration or amplitude of the perturbation. During this surge the number of events exceeds the background rate by roughly a factor of twenty relative to the immediate burst. The surge originates from the relaxation of θ: as the aging variable recovers, the effective friction drops, allowing many sites that were previously sub‑critical to become unstable under the redistributed stress. The third phase is a “quiescent” interval in which activity falls below the background level before slowly returning to the stationary rate. This post‑surge depletion reflects a temporary exhaustion of releasable elastic energy.

Crucially, when the aging term is removed the delayed surge and the subsequent quiescent dip disappear, leaving only the immediate events. This demonstrates that the delayed response is a direct consequence of the aging relaxation built into the model. The authors also find a non‑linear relationship between Δσ and the total number of delayed events; below a certain perturbation threshold the delayed surge is negligible, indicating a minimum stress perturbation required to activate the aging‑driven mechanism.

The authors discuss the relevance of these findings to real seismicity. Observations of remote triggering often show a lag of minutes to days between the arrival of surface waves from a distant large earthquake and the onset of increased local seismicity. The model reproduces both the lag and the magnitude of the excess activity without invoking any exotic physics, suggesting that the aging of fault zones—i.e., the time‑dependent weakening of frictional contacts—provides a natural pathway for small, transient stress changes to be amplified into a cascade of earthquakes. The delayed surge also offers a statistical explanation for the “after‑shock‑like” sequences that sometimes follow remote triggering events, while the subsequent dip mirrors the observed temporary reduction in background seismicity after a wave passage.

In summary, the study validates that a simple aging‑based statistical model can capture the essential features of both immediate and delayed remote earthquake triggering. The delayed effect is robust, independent of perturbation duration, and scales with the internal relaxation time of the system. These results lend theoretical support to the growing body of empirical evidence for remote triggering and highlight the importance of incorporating time‑dependent fault weakening into probabilistic seismic hazard assessments and forecasting frameworks.