Recurrent frequency-size distribution of characteristic events

Many complex systems, including sand-pile models, slider-block models, and earthquakes, have been discussed whether they obey the principles of self-organized criticality. Behavior of these systems can be investigated from two different points of view: interoccurrent behavior in a region and recurrent behavior at a given point on a fault or at a given fault. The interoccurrent frequency-size statistics are known to be scale-invariant and obey the power-law Gutenberg-Richter distribution. This paper investigates the recurrent frequency-size behavior of characteristic events at a given point on a fault or at a given fault. For this purpose sequences of creep events at a creeping section of the San Andreas fault are investigated. The applicability of the Brownian passage-time, lognormal, and Weibull distributions to the recurrent frequency-size statistics of slip events is tested and the Weibull distribution is found to be a best-fit distribution. To verify this result the behaviors of the numerical slider-block and sand-pile models are investigated and the applicability of the Weibull distribution is confirmed. Exponents of the best-fit Weibull distributions for the observed creep event sequences and for the slider-block model are found to have close values from 1.6 to 2.2 with the corresponding aperiodicities of the applied distribution from 0.47 to 0.64.

💡 Research Summary

The paper addresses a gap in earthquake statistics by focusing on the “recurrent” behavior of characteristic events—those that repeatedly rupture the same fault segment—rather than the well‑studied “inter‑occurrent” behavior that aggregates all events in a region. While inter‑occurrent size distributions follow the Gutenberg‑Richter power‑law, the statistical form governing the sizes of recurrent events has remained unclear. To investigate this, the author analyzes two complementary data sources: (1) observed slip‑event sequences from creep meters installed on a creeping section of the San Andreas fault, and (2) synthetic event catalogs generated by two classic self‑organized criticality (SOC) models, the slider‑block model and the sand‑pile model.

Observational data
Two USGS creep meters (cwn1 and xhr2) provide continuous measurements from the early 1970s to the early 2000s. Slip events are identified by a “jump‑and‑recovery” criterion: a rapid slip is recorded as a single event only if the creep rate returns to a stationary background before the next jump. This approach yields clean event catalogs containing 70–100 events per series, far more than the handful of earthquakes available from historic fault‑specific catalogs such as Parkfield. Event amplitudes are measured in millimetres of slip, with thresholds of ~0.07 mm for the high‑resolution 10‑minute data and ~0.3 mm for daily averages, ensuring that telemetry noise and rainfall‑induced tilt are negligible.

Candidate probability models
Three distributions are examined: (i) the Brownian passage‑time (BPT) distribution, which models the first‑passage time of a Wiener process with mean μ and coefficient of variation CV; (ii) the log‑normal distribution, obtained by exponentiating a normal variable; and (iii) the Weibull distribution, characterized by a shape parameter β and a scale parameter τ. The Weibull family is particularly flexible: β = 1 reduces to an exponential (memoryless) law, β → ∞ approaches a deterministic (δ‑function) repeat, while intermediate β values produce “stretched‑exponential” behavior.

Goodness‑of‑fit methodology
Two statistical tests are applied to each data set: the Kolmogorov‑Smirnov (K‑S) test, which evaluates the maximum absolute difference D_KS between empirical and theoretical cumulative distribution functions, and the root‑mean‑square error (RMSE), which quantifies the average deviation of predicted probabilities from observed frequencies. The preferred model minimizes D_KS and RMSE while maximizing the K‑S significance level Q_KS.

Results for creep data
Across all four series (two stations, two sampling rates) the Weibull distribution consistently yields the smallest D_KS and RMSE and the largest Q_KS. The fitted Weibull shape parameters β lie between 1.6 and 2.2, corresponding to coefficients of variation (CV) ranging from 0.47 to 0.64. These values indicate that recurrent slip events are more regular than a pure exponential process (β = 1) but still exhibit appreciable variability, contradicting the often‑used assumption of perfectly repeatable characteristic earthquakes (δ‑function).

Synthetic model verification
To test whether the observed Weibull fit is a generic property of SOC‑type systems, the author generates large event catalogs from (a) a slider‑block model (with stiffness driven toward the infinite‑stiffness limit to emulate SOC) and (b) a classic sand‑pile model. Both models produce thousands of events. Applying the same K‑S and RMSE analyses, the Weibull distribution again emerges as the best descriptor, with β≈1.9 (CV≈0.55) for the slider‑block model and β≈2.0 (CV≈0.60) for the sand‑pile model. The close agreement between synthetic and observed β‑values reinforces the conclusion that recurrent event sizes in SOC systems naturally follow a Weibull law.

Interpretation and implications
The findings suggest that the size distribution of characteristic, recurrent earthquakes is not a power‑law nor a perfectly deterministic repeat, but rather a Weibull distribution with a shape exponent modestly greater than one. Physically, this reflects a system where stress accumulation and release are governed by a stochastic process with a characteristic scale (the fault segment size) and a moderate degree of randomness. For seismic hazard assessment, replacing the simplistic δ‑function assumption with a Weibull‑based probabilistic model would capture the observed variability in slip magnitudes, potentially leading to more realistic forecasts of ground‑motion hazards.

Conclusions

1. Recurrent frequency‑size statistics of characteristic events on the San Andreas fault are best described by a Weibull distribution.
2. The Weibull shape parameter β falls in the range 1.6–2.2, yielding coefficients of variation between 0.47 and 0.64.
3. Both observed creep‑event sequences and synthetic SOC models (slider‑block and sand‑pile) support this result, indicating a universal aspect of SOC dynamics.
4. The result challenges the use of a deterministic repeat model for characteristic earthquakes and provides a statistically grounded alternative for seismic risk modeling.