New Approach to Gutenberg-Richter Scaling

We introduce a new model for an earthquake fault system that is composed of noninteracting simple lattice models with different levels of damage denoted by $q$. The undamaged lattice models ($q=0$) have Gutenberg-Richter scaling with a cumulative exponent $\beta =1/2$, whereas the damaged models do not have well defined scaling. However, if we consider the “fault system” consisting of all models, damaged and undamaged, we get excellent scaling with the exponent depending on the relative frequency with which faults with a particular amount of damage occur in the fault system. This paradigm combines the idea that Gutenberg-Richter scaling is associated with an underlying critical point with the notion that the structure of a fault system also affects the statistical distribution of earthquakes. In addition, it provides a framework in which the variation, from one tectonic region to another, of the scaling exponent, or $b$ value, can be understood.

Figure 1

The frequency-size distribution for a model fault for various values of $q$. For $q=0$ (i.e., no damage) we observe scaling up to the linear system size. As $q$ is increased, the distribution is no longer described by a power law, rather, large events are exponentially suppressed according to Eq. (4). The black line has the slope predicted for the scaling on undamaged faults.

Figure 2

The frequency-size distribution plotted in terms of the scaling variables. We explicitly account for the $q$ dependence in $n_s$ by plotting $(1-q)/q^3{n}_z$ versus $z=q^{2}s$. Having removed the $q$ dependence in these distributions, the data collapse to a single curve (black line) for all values of $q$.

Figure 3

The number of events with magnitude greater than or equal to the minimum magnitude of completeness [24] versus the area of each event [25] occurring from 1980–2008 within a 20 km swatch on either side of the San Jacinto fault (green crosses), Ft. Tejon segment of the San Andreas Fault (blue diamonds), and the Creeping section of the San Andreas Fault (red triangles). To see the data clearly, the various faults have been offset by factors of ten. The fits (solid colored lines) are to Eq. (6) and the solid black line has unit slope. The data are from the Advanced National Seismic System catalog [22].

Figure 4

The cumulative frequency-size distribution for a model fault system for various damage distributions, $D_q\sim q^{-x}$. By varying the exponent $x$, the GR distribution scales with different exponents $\tilde{\tau }$. The solid black lines are the predictions of Eq. (8). The flattening of the distribution at large $s$ is a finite size effect. We have offset the various distributions by scaling the data by factors of 10.

Figure 5

The scaling exponents, $\tilde{\tau }$ and $b$, versus the damage distribution exponent, $x$. The black exes represent the value of $\tilde{\tau }$ determined by fitting the data to a power law, the blue crosses come from fits keeping the leading asymptotic correction to the scaling form, and the red line is the predicted value of $\tilde{\tau }$.