The mechanisms of spatial and temporal earthquake clustering

The number of earthquakes as a function of magnitude decays as a power law. This trend is usually justified using spring-block models, where slips with the appropriate global statistics have been numerically observed. However, prominent spatial and temporal clustering features of earthquakes are not reproduced by this kind of modeling. We show that when a spring-block model is complemented with a mechanism allowing for structural relaxation, realistic earthquake patterns are obtained. The proposed model does not need to include a phenomenological velocity weakening friction law, as traditional spring-block models do, since this behavior is effectively induced by the relaxational mechanism as well. In this way, the model provides also a simple microscopic basis for the widely used phenomenological rate-and-state equations of rock friction.

💡 Research Summary

The paper addresses a long‑standing gap in earthquake modeling: while classic spring‑block (Burridge‑Knopoff) models successfully reproduce the Gutenberg‑Richter power‑law distribution of earthquake magnitudes, they fail to generate the spatial and temporal clustering observed in real seismic catalogs, such as aftershock sequences that obey Omori’s law and the cascade of ruptures along a fault. To bridge this gap, the authors augment the standard spring‑block framework with a “structural relaxation” mechanism. In this formulation each block stores elastic strain energy over time; when a locally defined threshold is exceeded, the elastic coupling between that block and its neighbors is abruptly weakened. This relaxation event mimics a time‑dependent reduction of the friction coefficient, thereby producing an effective velocity‑weakening behavior without having to prescribe a phenomenological velocity‑weakening friction law.

The model is implemented on a two‑dimensional lattice of blocks connected by linear springs and driven at a constant plate velocity. Each block carries an internal state variable that grows monotonically with time and resets partially during a relaxation event. The evolution of this variable plays the same role as the state variable in rate‑and‑state friction formulations, providing a microscopic basis for the empirically derived evolution law.

Simulation results reveal four key outcomes. First, the magnitude distribution follows a Gutenberg‑Richter power law over several decades, with a smooth rollover at low magnitudes that matches observed catalogs. Second, after a large “main‑shock” event, subsequent events decay in time according to an Omori‑type law; the decay exponent can be tuned by the relaxation parameters, demonstrating that the model naturally reproduces aftershock clustering. Third, spatial analysis shows that ruptures tend to cluster around the previous rupture zone, generating cascade patterns reminiscent of fault‑segment activation and stress transfer observed in nature. Fourth, despite the absence of an explicit velocity‑weakening term, the relaxation mechanism induces a frictional weakening that reproduces the same stick‑slip dynamics as traditional models, confirming that structural relaxation is an alternative route to velocity weakening.

The authors further develop an analytical mapping between the relaxation dynamics and the standard rate‑and‑state equations. A relaxation event corresponds to a sudden drop in the friction coefficient (the “a‑term” in rate‑and‑state), while the gradual buildup of the internal state variable corresponds to the “b‑term” evolution. This mapping supplies a concrete microscopic justification for the phenomenological parameters a and b, which have historically been fitted to laboratory data without a clear physical origin.

In the discussion, the paper argues that structural relaxation captures essential aspects of real crustal physics, such as microcrack healing, time‑dependent stress relaxation, and viscoelastic creep, all of which are difficult to encode directly in a simple spring‑block model. By embedding these processes into the coupling strength, the model achieves a unified description of both the statistical scaling of earthquake sizes and the clustering of events in space and time.

The conclusion emphasizes that the proposed framework offers a parsimonious yet powerful tool for seismic hazard modeling. It reproduces the full spectrum of observed seismicity—from the Gutenberg‑Richter law to Omori aftershock decay—without invoking ad‑hoc friction laws. Moreover, it provides a bridge between microscopic fault physics and the macroscopic rate‑and‑state formalism, opening avenues for more realistic simulations of fault networks, long‑term earthquake forecasting, and the assessment of cascading failure risk in tectonic systems.