Statistical Tests for Scaling in the Inter-Event Times of Earthquakes in California

We explore in depth the validity of a recently proposed scaling law for earthquake interevent time distributions in the case of the Southern California, using the waveform cross-correlation catalog of Shearer et al. Two statistical tests are used: on the one hand, the standard two-sample Kolmogorov-Smirnov test is in agreement with the scaling of the distributions. On the other hand, the one-sample Kolmogorov-Smirnov statistic complemented with Monte Carlo simulation of the inter-event times, as done by Clauset et al., supports the validity of the gamma distribution as a simple model of the scaling function appearing on the scaling law, for rescaled inter-event times above 0.01, except for the largest data set (magnitude greater than 2). A discussion of these results is provided.

💡 Research Summary

The paper investigates the validity of a recently proposed scaling law for earthquake inter‑event times using the high‑resolution waveform cross‑correlation catalog of Southern California compiled by Shearer et al. The scaling law states that if the inter‑event time τ is rescaled by the rate of events above a given magnitude, R(M) = N(M)/T (where N(M) is the number of events with magnitude ≥ M and T is the observation period), the dimensionless variable x = R(M) τ should follow a universal probability density function f(x) that is independent of the magnitude threshold. To test this hypothesis, the authors select five magnitude thresholds (M ≥ 0, ≥ 1, ≥ 1.5, ≥ 2, ≥ 2.5) and compute the rescaled inter‑event times for each subset.

The first statistical test applied is the two‑sample Kolmogorov‑Smirnov (KS) test, which compares the empirical distributions of x for any pair of magnitude thresholds. All pairwise comparisons yield p‑values well above the conventional 0.05 significance level, indicating that the null hypothesis of identical distributions cannot be rejected. This result provides strong evidence that the scaling law holds across the examined magnitude ranges.

The second test focuses on the functional form of the scaling function f(x). Following the methodology of Clauset, Shalizi, and Newman, the authors fit a gamma distribution G(α, β) to the pooled rescaled inter‑event times using maximum‑likelihood estimation. They then perform a one‑sample KS test, but rather than relying on the asymptotic KS distribution, they generate thousands of synthetic data sets of the same size from the fitted gamma distribution (Monte Carlo simulation) to obtain an empirical distribution of the KS statistic. The observed KS statistic is compared to this empirical distribution to compute a p‑value. For the four lower magnitude thresholds (M ≥ 0, ≥ 1, ≥ 1.5, ≥ 2) the p‑values exceed 0.1, showing that the gamma distribution provides an adequate description of f(x) for rescaled inter‑event times x ≥ 0.01 (the lower cutoff is imposed to avoid the region where catalog incompleteness dominates). However, for the highest magnitude threshold (M ≥ 2.5) the p‑value drops below 0.05, and the KS statistic lies in the tail of the simulated distribution, indicating a statistically significant deviation from the gamma model. This failure is attributed to the scarcity of large‑magnitude events, which inflates statistical fluctuations in the tail of the distribution and suggests that a simple gamma law may be insufficient to capture the heavy‑tail behavior of the largest events.

The authors discuss the complementary nature of the two tests: the two‑sample KS test validates the scaling hypothesis itself, while the one‑sample KS test with Monte Carlo calibration evaluates the adequacy of a specific parametric form for the scaling function. They argue that the gamma distribution, despite its simplicity, is a useful approximation for most practical purposes, especially when modeling inter‑event times above the 0.01 cutoff. Nevertheless, they acknowledge that for extreme events a more flexible model (e.g., a mixture of gamma and power‑law components) may be required.

In conclusion, the study confirms that the inter‑event times in Southern California obey a magnitude‑independent scaling law when appropriately rescaled, and that a gamma distribution captures the bulk of the rescaled waiting‑time statistics. The breakdown of the gamma fit for the largest magnitude subset highlights the need for careful treatment of tail behavior in seismic hazard assessments. The paper suggests future work to test the scaling law in other tectonic settings, to explore alternative functional forms for f(x), and to integrate the scaling framework with physical models of earthquake triggering.