Gutenberg-Richter-like relations in physical systems

TL;DR

The paper addresses how Gutenberg-Richter-like scale-invariant energy release in earthquakes relates to underlying physical dynamics across different exponent values. Using regional catalogs from Japan (JMA) and Southern California (SCEDC), they define avalanche size by energy release linked to seismic moment $M_0$ and analyze the differential energy distribution $P(E) ∼ E^{-β}$, finding $β ≈ 1.67$ and a corresponding observed size exponent $τ ≈ 1.67$. They generate synthetic energy distributions for a wide range of exponent values $τ$ under two normalization schemes (constant activity and constant energy) and identify a robust earthquake-like regime for $1.5 ≤ τ < 2.0$, tightening to $1.58 ≤ τ ≤ 1.76$ when allowing only factors of ten in energy variation. Physically, these results map distinct regimes to fault dynamics, show that large earthquakes dominate the energy budget near the observed exponent, and provide a quantitative framework linking exponents to energy budgets and event cadences in scale-invariant seismicity.

Abstract

We analyze regional earthquake energy statistics from the Southern California and Japan seismic catalogs and find scale-invariant energy distributions characterized by an exponent $τ\simeq 1.67$. To quantify how closely scale-invariant dynamics with different exponent values resemble real earthquakes, we generate synthetic energy distributions over a wide range of $τ$ under conditions of constant activity. Earthquake-like behavior, in a broad sense, is obtained for $1.5 \leqslant τ< 2.0$. When energy variations are further restricted to be within a factor of ten relative to real earthquakes, the admissible range narrows to $1.58 \leqslant τ\leqslant 1.76$. We identify the physical mechanisms governing the dynamics in the different regimes: fault dynamics characterized by a balance between slow energy accumulation and release through scale-free events in the earthquake-like regime; externally supplied energy relative to a slowly driven fault for $τ< 1.5$; and dominance of small events in the energy budget for $τ> 2$

Figures (4)

• Figure 1: Catalogs. Regions analyzed in the JMA (Japan) and SCEDC (Southern California) catalogs in the time-intervals reported in the text. All M6-M9 earthquakes in Japan and all M4-M7 earthquakes in Southern California are shown, with colors and symbol sizes indicating magnitude (see legend). The dates of the Tōhoku earthquake (M9.1) and the four M8 events in Japan are also indicated. The M8 earthquakes in Japan are used as reference events throughout the article.
• Figure 2: Earthquakes statistics. (a) Gutenberg-Richter relations (Eq. \ref{['GR-def']}) for Japan (JMA catalog) and Southern California (SCEDC catalog). The fits correspond to the solid lines. Magnitudes smaller than M2$\equiv$[M2.0, M2.9] in the SCEDC, and than M5 in JMA, are affected by catalog incompleteness. (b) Non-cumulative number of earthquakes per year as a function of the magnitude (Eq. \ref{['GR-def2']}). (c) Non-cumulative energy distribution. They follow power law distributions with $\beta$ values around $1.67$. (d) Cumulative energy release. Notice that the Tōhoku earthquake (2011, M9.1) accounts for more than 63 $\%$ of the total energy release in the period. Californian earthquakes release two orders of magnitude less energy. Four M7+ quakes dominate the total energy release: Landers (1992, M7.3), Hector Mine (1999, M7.1), Baja California (2010, M7.2), and Ridgecrest (2019, M7.1). (e,f) Extrapolation of the relations obtained in panels (b) and (c) for Japan, assuming $b_1=1$ and $\beta=1.67$ respectively.
• Figure 3: Earthquake-like dynamics. (a) Energy distributions for different exponent $\tau$ values ranging from $0.8$ to $2.2$ (dark to bright colors) considering a constant activity and magnitudes $\text{M}\in[\text{M0.0}-\text{M8.9}]$. In the case of $\tau = 1.67$ (in red) it describes the energy distribution of Japan earthquakes. Curves with exponents $\tau$ = 1.0, 1.5 and 2.0 have been highlighted in blue. Orange diamonds correspond to the mean value of the released energy $\langle E\rangle$. (b) The same data in (a) expressed as number of earthquakes / year as a function of the magnitude. (c) Average energy release $\langle E\rangle$ normalized by the average energy release for $\tau = 1.67$, and corresponding magnitude $\langle \text{M} \rangle$ values as a function of the exponent. A gray rectangle indicates the earthquake-like regime, while the white rectangle indicates the range where the differences in $\langle E\rangle$ are less than ten-times the one of real earthquakes. (d) Energy contributions (in %) of extreme magnitude events M8 (in blue) and M0 (in orange) as a function of the exponent value. At $\tau$ =1.67, 67% of the energy is released by the M8 events. However, for $\tau >2$ the energy budget is controlled by the smallest M0 events. (e) Average period of M8 earthquakes as function of $\tau$. (f) $b$ value as function of $b_1$. The black dashed line has a slope equal 1. $b$ starts to deviate from the identity at $b_1<0.5$, corresponding to $\tau<1.33$
• Figure 4: Dynamics under constant energy. (a) Energy distributions for different exponent $\tau$ values ranging from $0.8$ to $2.2$ (dark to bright colors) considering a constant energy release and magnitudes $\text{M}\in[\text{M0.0}-\text{M8.9}]$. In the case of $\tau = 1.67$ (in red) it describes the energy distribution of Japan earthquakes. Curves with exponents $\tau$ = 1.0, 1.5 and 2.0 have been highlighted in blue. Orange diamonds correspond to the mean value of the released energy $\langle E\rangle$. (b) The same data in (a) expressed as number of earthquakes / year as a function of the magnitude. (c) Total number of events / year as a function of $\tau$.