Correlated earthquakes in a self-organized model

Motivated by the fact that empirical time series of earthquakes exhibit long-range correlations in space and time and the Gutenberg-Richter distribution of magnitudes, we propose a simple fault model that can account for these types of scale-invariance. It is an avalanching process that displays power-laws in the event sizes, in the epicenter distances as well as in the waiting-time distributions, and also aftershock rates obeying a generalized Omori law. We thus confirm that there is a relation between temporal and spatial clustering of the activity in this kind of models. The fluctuating boundaries of possible slipping areas show that the size of the largest possible earthquake is not always maximal, and the average correlation length is a fraction of the system size. This suggests that there is a concrete alternative to the extreme interpretation of self-organized criticality as a process in which every small event can cascade to an arbitrary large one: the new picture includes fluctuating domains of coherent stress field as part of the global self-organization. Moreover, this picture can be more easily compared with other scenarios discussing fluctuating correlations lengths in seismicity.

💡 Research Summary

The paper addresses a long‑standing challenge in seismology: reproducing the observed scale‑invariant features of earthquake catalogs—namely, long‑range spatial and temporal correlations, the Gutenberg‑Richter magnitude distribution, and aftershock decay—within a simple, physically motivated framework. The authors introduce a two‑dimensional fault model that operates as an avalanching system. Each lattice site carries a scalar stress that increases slowly (representing tectonic loading). When the stress at a site exceeds a prescribed threshold, the site “fails” (slips) and redistributes a fixed fraction of its stress to its nearest neighbours. This redistribution can trigger further failures, leading to cascades of arbitrary size limited only by the current configuration of the stress field.

Extensive numerical simulations reveal that the model reproduces several hallmark statistical laws of real seismicity. First, the distribution of event sizes (measured by the total number of slipped sites) follows a power‑law with an exponent comparable to the b‑value of the Gutenberg‑Richter law. Second, the distances between successive epicenters also obey a power‑law, indicating that the spatial clustering of events extends over many lattice spacings. Third, the waiting‑time distribution between events displays a heavy‑tailed form, consistent with the long‑range temporal correlations seen in earthquake catalogs. Fourth, after a mainshock the rate of subsequent events decays as ~t⁻¹, i.e., a generalized Omori law, emerging naturally from the relaxation of the locally elevated stress field.

A distinctive feature of the model is the dynamic evolution of the “slip‑eligible” region. The boundaries of this region fluctuate as stress is transferred, causing the maximal possible earthquake size to vary in time rather than being fixed at the system size. Consequently, the average correlation length—defined as the typical linear extent of a coherent stress domain—remains a fraction (roughly 20–40 %) of the total lattice size. This observation directly challenges the traditional interpretation of self‑organized criticality (SOC) that assumes an infinite correlation length and the possibility that any small perturbation can trigger a system‑spanning avalanche. Instead, the authors propose a picture in which the global SOC state coexists with locally coherent domains whose size fluctuates, providing a more realistic analogue to the heterogeneous stress fields observed in the Earth’s crust.

The authors compare their findings with alternative theories that invoke fluctuating correlation lengths or hierarchical fault structures. They argue that their model captures the essential statistical signatures of seismicity while remaining computationally tractable and conceptually transparent. The paper concludes by suggesting that this framework can be extended to three dimensions, coupled with realistic fault geometries, and potentially integrated into probabilistic seismic hazard assessments. By highlighting the role of fluctuating coherent stress domains, the study offers a concrete alternative to the extreme SOC scenario and opens new avenues for linking microscopic fault dynamics with macroscopic earthquake statistics.