Universal mean moment rate profiles of earthquake ruptures

Earthquake phenomenology exhibits a number of power law distributions including the Gutenberg-Richter frequency-size statistics and the Omori law for aftershock decay rates. In search for a basic model that renders correct predictions on long spatiotemporal scales, we discuss results associated with a heterogeneous fault with long-range stress-transfer interactions. To better understand earthquake dynamics we focus on faults with Gutenberg-Richter-like earthquake statistics and develop two universal scaling functions as a stronger test of the theory against observations than mere scaling exponents that have large error bars. Universal shape profiles contain crucial information on the underlying dynamics in a variety of systems. As in magnetic systems, we find that our analysis for earthquakes provides a good overall agreement between theory and observations, but with a potential discrepancy in one particular universal scaling function for moment rates. We primarily use mean field theory for the theoretical analysis, since it has been shown to be in the same universality class as the full three-dimensional version of the model (up to logarithmic corrections). The results point to the existence of deep connections between the physics of avalanches in different systems.

Figure 1

A planar representation of a 3D segmented fault zone by a 2D heterogeneous fault embedded in a 3D solid [19, 20].

Figure 2

Phase diagram of the model (see text and [21] for details).

Figure 3

A collapse of averaged earthquake pulse shapes, $⟨d{m}_0(t\mid M_0)∕dt⟩$, with the size of the moment $M_0$ in N m within 10% of each size given in the legend. In order to obtain each collapsed moment rate shape, five to ten earthquakes were averaged for each value of $M_0$. The collapse was obtained using the mean field scaling relation [9]: $⟨d{m}_0(t\mid M_0)∕dt⟩∕M_0^{1∕2}\sim f(t∕M_0^{1∕2})$ [see text Eq. (16)]. In our mean field theory the universal scaling function is $f_{mf}\left(x\right)=Ax{e}^{-B{x}^2∕2}$ where $x=t∕M_0^{1∕2}$. We plot this functional form (bold curve) with $A=4$ and $B=4.9$. Inset: The raw data and the averaged data (before collapse).

Figure 4

A collapse of averaged earthquake pulse shapes, $⟨d{m}_0(t\mid T)∕dt⟩$ with a duration of $T$ (seconds) within 10% (given in legend), is shown. The collapse was obtained using the mean field scaling relation [38]: $⟨d{m}_0(t\mid T)∕dt⟩\sim g(t∕T)$. In order to obtain each collapsed pulse shape, two to ten earthquakes were averaged for each value of $T$. In our mean field theory the universal scaling function is $g_{mf}\left(x\right)=Ax(1-x)$ with $x=t∕T$. We plot this functional form (bold curve) with $A=80$. Note the apparent asymmetry to the left in the observed data while the theoretical curve is symmetric around its maximum. Inset: The raw data and the averaged data (before collapse).

Figure 5

Rescaled moment rate versus time profiles of seismic events determined with moment scaling (top) and duration scaling (bottom) by [43] (see text). The data shown are obtained from several subduction zones in the indicated geographical locations and the numbers of earthquakes used for each region are given in parentheses.