Earthquake magnitude distribution and aftershocks: A statistical geometry explanation

The emergence of a power-law distribution for the energy released during an earthquake is investigated in several models. Generic features are identified which are based on the self-affine behavior of the stress field prior to an event. This field behaves at large scale as a random trajectory in one dimension of space and a random surface in two dimensions. Using concepts of statistical mechanics and results on the properties of these random objects, several predictions are obtained and verified, in particular the value of the power-law exponent of the earthquake energy distribution (the Gutenberg-Richter law) as well as a mechanism for the existence of aftershocks after a large earthquake (the Omori law).

Figure 1

For the solution of the Coulomb BK model in one dimension with $F_d=0, N=800, v_0={10}^{-6}, k_1=1$, and varying $k_2$ [see color code in the legend of Fig. 1; same color code in Fig. 1 and Fig. 1]: (a) Probability density function (PDF) of the moment of the events $M$. (b) PDF of the event size $N$. (c) Power spectrum density (PSD) of the spatial gradient of the stress $S_i$ before events with $1\le M\le 100$ and $80\le N\le 170$. Straight lines are power-law fits for $K<0.1$; the obtained exponents are $1-2H$ with $H$ the Hurst exponent associated to the large scales of the stress profiles. (d) Exponents $\alpha , \beta , 1+B$, and the prediction $1+(\beta -1)/\alpha$ as a function of $K_2$. Symbols are best fits of the power-law exponents of Figs. 1 and 1. Continuous lines are the predictions obtained from the fBm analogy $[\alpha =1+H, \beta =2-H, 1+B=1+(1-H)/(1+H)]$ with $H$ obtained from the large scales of the gradient of the stress, as in Fig 1.

Figure 2

For the 2D model with $D_1=1, D_2=10, D_3=0.1, N_t={400}^2$: (a) Snapshot of the stress field as a function of position for $s=0$. Values on the color bar on the right. (b) PDF of EQ size $P\left(N\right)$ for red $s=0$, green $s=1$, and blue $s=1.5$. (c) Averaged number of events per unit of time as a function of duration to a main event of size $N\ge 100$. Only events with hypocenter located at a distance smaller than 50 from the main shock are considered. Red: $s=0$; blue: $s=1.5$.

Figure 3

PDF of the distance $d$ between the largest cluster of occupied sites and the other clusters for a random surface with Hurst exponent $H=-1$ in red, $H=0$ in green, and $H=0.5$ in blue. The fraction of occupied sites increases with the line thickness and is 0.3, 0.5, and 0.55. The total number of sites is ${400}^2$.