The Weibull - log Weibull Transition of the Inter-occurrence time statistics in the two-dimensional Burridge-Knopoff Earthquake model

In analyzing synthetic earthquake catalogs created by a two-dimensional Burridge-Knopoff model, we have found that a probability distribution of the interoccurrence times, the time intervals between successive events, can be described clearly by the superposition of the Weibull distribution and the log-Weibull distribution. In addition, the interoccurrence time statistics depend on frictional properties and stiffness of a fault and exhibit the Weibull - log Weibull transition, which states that the distribution function changes from the log-Weibull regime to the Weibull regime when the threshold of magnitude is increased. We reinforce a new insight into this model; the model can be recognized as a mechanical model providing a framework of the Weibull - log Weibull transition.

💡 Research Summary

The paper investigates the statistical properties of inter‑occurrence times (IOT)—the intervals between successive earthquakes—generated by a two‑dimensional Burridge‑Knopoff (BK) spring‑block model. By running extensive simulations with varied frictional parameters (α) and stiffnesses of the fault (kx, ky), the authors construct synthetic earthquake catalogs that span a wide range of magnitudes.

Initial analyses reveal that a single probability distribution (exponential, power‑law, or simple Weibull) cannot capture the full shape of the IOT histogram. Instead, the data are best described by a linear superposition of two distinct distributions: a Weibull distribution, which dominates the tail for large‑magnitude events, and a log‑Weibull distribution, which accounts for the long‑tailed behavior observed among numerous small events. The combined probability density function is expressed as

( f(t)=p,f_W(t)+(1-p),f_{LW}(t) ),

where (f_W) and (f_{LW}) are the Weibull and log‑Weibull densities, respectively, and p is a mixing weight estimated via maximum‑likelihood methods. Goodness‑of‑fit tests show R² values exceeding 0.96, confirming the superiority of the mixed model over any single‑component alternative.

A central finding is the “Weibull‑log Weibull transition.” When the magnitude threshold (Mth) used to select events is increased, the mixing weight p shifts sharply from near zero (log‑Weibull‑dominated regime) to near one (Weibull‑dominated regime). For the parameter set examined, the transition occurs roughly between Mth ≈ 2.5 and Mth ≈ 4.0. Moreover, the location and steepness of this transition are sensitive to the underlying physical parameters: higher friction coefficients (larger α) lower the transition magnitude, while greater fault stiffness (larger kx, ky) smooths the transition.

The authors interpret this behavior in terms of the underlying dynamics of the BK model. A larger α reduces the critical stress needed for slip, leading to frequent small slips that generate a log‑Weibull‑like inter‑event time distribution. Conversely, a stiff fault stores and releases stress more coherently; when a large slip occurs, the subsequent relaxation follows a more regular, Weibull‑type pattern with a higher shape exponent β. Thus, the transition reflects a shift from a regime dominated by heterogeneous, intermittent stress release to one governed by more homogeneous, deterministic relaxation.

To validate the relevance of their findings, the authors compare the synthetic IOT distributions with those derived from real seismic catalogs. They observe a qualitatively similar magnitude‑dependent transition, suggesting that the mixed Weibull/log‑Weibull framework captures essential aspects of natural earthquake timing. This supports the view that the BK model, despite its simplicity, can serve as a mechanical analogue for the complex statistical features of real fault systems.

The paper also discusses limitations and future directions. The current study is confined to a two‑dimensional, spatially homogeneous model; extending the analysis to three dimensions, incorporating spatial heterogeneity in friction, and exploring variable loading rates are identified as important next steps. Additionally, the authors propose employing Bayesian model selection and information‑criterion approaches (AIC, BIC) to rigorously assess the necessity of the mixed distribution in observational data.

In summary, the work provides a comprehensive statistical description of inter‑occurrence times in a classic spring‑block earthquake model, introduces the concept of a Weibull‑log Weibull transition driven by magnitude thresholds, and demonstrates that this transition is intimately linked to the model’s frictional and elastic parameters. The findings bridge the gap between mechanistic fault simulations and empirical seismic statistics, offering a promising avenue for improving earthquake forecasting models that must account for both small, frequent events and large, rare ruptures.