Foreshock and Aftershocks in Simple Earthquake Models

Many models of earthquake faults have been introduced that connect Gutenberg-Richter (GR) scaling to triggering processes. However, natural earthquake fault systems are composed of a variety of different geometries and materials and the associated heterogeneity in physical properties can cause a variety of spatial and temporal behaviors. This raises the question of how the triggering process and the structure interact to produce the observed phenomena. Here we present a simple earthquake fault model based on the Olami-Feder-Christensen and Rundle-Jackson-Brown cellular automata models with long-range interactions that incorporates a fixed percentage of stronger sites, or asperity cells, into the lattice. These asperity cells are significantly stronger than the surrounding lattice sites but eventually rupture when the applied stress reaches their higher threshold stress. The introduction of these spatial heterogeneities results in temporal clustering in the model that mimics that seen in natural fault systems along with GR scaling. In addition, we observe sequences of activity that start with a gradually accelerating number of larger events (foreshocks) prior to a main shock that is followed by a tail of decreasing activity (aftershocks). This work provides further evidence that the spatial and temporal patterns observed in natural seismicity are strongly influenced by the underlying physical properties and are not solely the result of a simple cascade mechanism.

Figure 1

Time series of events (number of failed sites) over $6\times {10}^5$ plate updates for: (a) $\alpha =0.6$, (b) $\alpha =0.4$, and (c) $\alpha =0.2$; (i) 1% of randomly distributed asperity sites (shaded background are times when an asperity site breaks); (ii) homogeneous model with the same conditions as (i). (d) Comparison between frequency distributions $n(s)$, with and without 1% of randomly distributed asperity sites, for three values of $\alpha$. Slope of the linear fit to $(\mathrm{a}\text{−}\mathrm{iii})=2.00$, $(\mathrm{b}\text{−}\mathrm{iii})=1.85$, and $(\mathrm{c}\text{−}\mathrm{iii})=1.65$. (e) Close-up of the box in (d).

Figure 2

(a-i) Number of failed sites at each time step (shaded background as in Fig. 1) for $\alpha =0.2$. Time is binned into coarse-grained units of $\mathrm{\Delta }t=500\mathrm{pu}$. (a-ii) Distance of each event from the largest event in the sequence (main shock, red cross). (a-iii) Distribution of events, $n(s)$, during the period (a-i). Slope for the straight line fit is 1.6. (a-iv) Cumulative number of events greater than the defined threshold versus coarse-grained time. (b-i, ii, iii, iv) as in (a) for stress dissipation of 40% ($\alpha =0.4$). Slope of the linear fit to $(\mathrm{b}\text{−}\mathrm{iii})$ is 1.85.

Figure 3

(a) Seismicity for swarm event, southern Eyjafjarðaráll graben, Aug 20, 2012 through March 25, 2013. Most activity occurred between the graben and the Húsavík-Flatey fault. (b) Spatiotemporal distribution of seismicity associated with the twelve largest events in the sequence in (a). Note that earthquake magnitude is logarithmic, where every unit increase is equivalent to approximately 32 times the energy increase. (c) GR distribution for the longest single sequence in the swarm, $M\ge M_c=2.0$, $M_c$ is minimum magnitude of completeness. (d) Cumulative number of events greater than $M_c$ versus time relative to the main shock (star, $M=4.76$). Data collected by the SIL network was provided by the Icelandic Met Office (en.vedur.is). GMT software [64] was used to create the figures.

Figure 4

The average time period associated with foreshocks and aftershocks as a function of $\alpha$, 1% of randomly distributed asperity sites. FT denotes foreshock time; AT denotes aftershock time; the red data show $\mathrm{FT}/(\mathrm{FT}+\mathrm{AT})$; the blue data show $\mathrm{AT}/(\mathrm{FT}+\mathrm{AT})$; $(\mathrm{FT}+\mathrm{AT})$ represents the total sequence.