On the coupled geometrical-mechanical origin of the earthquake b-value in fault networks

Abstract

The Gutenberg-Richter law is a fundamental empirical law in seismology describing earthquake frequency-magnitude distributions, with one of its key parameters, the so-called b-value, quantifying the relative frequency of small versus large events. While the b-value is commonly interpreted as reflecting crustal heterogeneity and regional stress conditions, its underlying physical origin remains poorly understood, particularly the relative roles of geometrical versus mechanical controls. Here, we develop analytical and numerical models to elucidate the origin of the b-value in three-dimensional fault networks subject to mainshock-aftershock sequences. We demonstrate that the b-value emerges from the power-law scaling of fault rupture area together with the scaling of slip magnitude. Our results reveal a two-branch frequency-magnitude distribution, with the regime transition governed by fault criticality and fracture energy dissipation, while the transition magnitude reflects the finite population of faults triggered during the sequence. Our findings provide a physically grounded interpretation of earthquake b-values, establishing a link between fault mechanics and earthquake statistics.

Figures (4)

• Figure 1: Numerical model setup and fault network characteristics.a Model geometry showing a vertical 50 km $\times$ 20 km primary fault surrounded by a network of circular, disk-shaped secondary faults. Probability density functions of b fault diameters and c fault areas, both following power-law scaling. d Fault slip governed by a linear slip-weakening friction law.
• Figure 2: Coseismic stress changes and aftershock patterns triggered by the mainshock.a Coulomb failure stress changes ($\Delta$CFS) in the crustal formation induced by the mainshock (through static and dynamic triggering) and further modified locally by aftershock ruptures. b Slip distribution within the fault network. c Spatial distribution of aftershock moment magnitudes, where each marker represents a fault rupture event, with marker size proportional to moment magnitude $M_{\rm{w}}$, marker center indicating the hypocenter, and marker color reflecting the rupture ratio $\eta=A_{\rm{r}}/A$ (where $A$ is total fault area and $A_{\rm{r}}$ is rupture area). All panels show the case with critical slip distance $d_{\rm{c}}=0.5$ m. Aftershocks represent the cumulative events up to 28 s after mainshock initiation, selected as a representative late-time snapshot when the simulated aftershock sequence has converged.
• Figure 3: Two-branch earthquake frequency--magnitude distributions for different values of critical slip distance $d_{\rm{c}}$.a Cumulative distributions of moment magnitude $M_{\rm{w}}$ for different $d_{\rm{c}}$. Curves and data points are color-coded by the value of $d_{\rm{c}}$, as indicated by the accompanying color map. bb-value and ca-value of the higher-magnitude branch versus $d_{\rm{c}}$. d$b^{\prime}$-value and e$a^{\prime}$-value of the lower-magnitude branch versus $d_{\rm{c}}$. In b--e, numerical estimates are compared with analytical predictions obtained from the measured scaling exponents: b and a derived from $\alpha$ and $\beta$, and $b^{\prime}$ and $a^{\prime}$ derived from $\alpha^{\prime}$ and $\beta^{\prime}$.
• Figure 4: Transitions in rupture regimes in the fault network.a Energy ratio $G_{\rm{c}}/\Delta W_{0}$, which quantifies the proportion of available energy dissipated as fracture energy, plotted against the S-value that characterizes fault stress criticality. Symbols show simulation results for critical slip distance $d_{\rm{c}}=0.005$ m. The dashed curve marks the regime boundary given by the stress--energy formulation, Eq. (\ref{['eq:energy_framework_with_M_0']}). Here, we identify the transition magnitude $M_{\rm{T}}$ from the cumulative frequency--magnitude distribution of this simulation case ($M_{\rm{T}}=4.5$), convert it to the corresponding seismic moment ($M_{0}=7.16\times 10^{15}$ N$\cdot$m), and substitute this $M_{0}$ into Eq. (\ref{['eq:energy_framework_with_M_0']}) to obtain the delineation. b Relationship between fault area $A$ and moment magnitude $M_{\rm{w}}$ for $d_{\rm{c}}=0.005$ m, revealing a transition from fault area-independent behavior in the lower-magnitude branch to a fault area-dependent behavior in the higher-magnitude branch.