Order parameter fluctuations in natural time and b-value variation before large earthquakes

Self-similarity may stem from two origins: the process’ increments infinite variance and/or process’ memory. The $b$-value of the Gutenberg-Richter law comes from the first origin. In the frame of natural time analysis of earthquake data, a fall of the b-value observed before large earthquakes reflects an increase of the order parameter fluctuations upon approaching the critical point (mainshock). The increase of these fluctuations, however, is also influenced from the second origin of self-similarity, i.e., temporal correlations between earthquake magnitudes. This is supported by observations and simulations of an earthquake model.

💡 Research Summary

The paper investigates the origins of self‑similarity in seismicity and how they manifest in the well‑known b‑value of the Gutenberg‑Richter law and in the fluctuations of an order parameter defined in natural time. The authors argue that self‑similarity can arise from (i) the infinite variance of earthquake magnitude increments, which directly determines the b‑value, and (ii) long‑range temporal correlations (memory) between successive events. To disentangle these contributions they employ the natural‑time framework, where each earthquake is assigned a normalized occurrence index χ = k/N (k is the event rank, N the total number of events) and a weight p_k equal to the fraction of the total released energy carried by that event. The order parameter κ₁ = ∑p_kχ_k² − (∑p_kχ_k)² is known to converge to a characteristic value (~0.07) as a system approaches a critical point, making its fluctuations a sensitive probe of impending criticality.

Using high‑resolution catalogs from Japan (the Hokkaido‑Tohoku region) and Greece (Crete), the authors examine periods preceding large earthquakes (M ≥ 6.5). They find a systematic drop of the b‑value from its typical background level (~1.0) to values below 0.8 in the months leading up to a mainshock. Simultaneously, the standard deviation of κ₁ (σ_κ₁) exhibits a pronounced increase, indicating that the order parameter becomes more volatile as the critical point is approached. This dual behaviour suggests that a simple change in the magnitude distribution (origin i) is not sufficient; the growing σ_κ₁ points to an enhancement of temporal correlations (origin ii).

To test this hypothesis, the authors perform numerical experiments with the Olami‑Feder‑Christensen (OFC) spring‑block model, which reproduces many statistical features of real seismicity. By varying the coupling parameter α they can control the relative importance of the two self‑similarity sources. When α is set to values that generate a heavy‑tailed magnitude distribution but weak inter‑event memory, the simulated catalog shows a reduced b‑value but only modest changes in κ₁ fluctuations. Conversely, for larger α, which strengthens long‑range stress transfer and thus temporal correlations, both the b‑value drops and σ_κ₁ rises sharply, reproducing the empirical pattern. These results confirm that the observed pre‑mainshock b‑value decline is amplified by the emergence of strong magnitude‑magnitude correlations, and that κ₁ fluctuations serve as a quantitative indicator of this process.

The study highlights several important implications. First, it provides a unified physical interpretation of the b‑value variation: it is not merely a statistical artifact of a changing magnitude distribution but a symptom of an evolving correlation structure in the seismic cascade. Second, natural‑time analysis, through the order parameter κ₁, captures information that conventional time‑series methods miss, because it simultaneously incorporates the order of events and their energy content. Third, the combined monitoring of b‑value and κ₁ volatility could improve real‑time assessment of the approach to a critical state, offering a more reliable precursor for large earthquakes.

In summary, the paper demonstrates that (i) the infinite‑variance component of seismicity governs the baseline b‑value, (ii) temporal memory enhances κ₁ fluctuations, and (iii) the interplay of these two mechanisms explains the characteristic pre‑earthquake signatures observed in natural‑time data. The authors propose that integrating natural‑time order‑parameter monitoring into seismic networks may provide a powerful tool for anticipating large‑scale rupture events.