Statistical model of earthquake foreshocks

We propose a statistical model of rupture as a mechanism for the occasionally observed marked increase of seismic activity prior to a great earthquake. The physical ingredients of the model are those of geometrical inhomogeneity and viscoelastic creep. We demonstrate that the observed inverse power law for the rate of increase of seismicity before a large earthquake requires no assumptions beyond those of a random featureless distribution of inhomogeneities and a typical power law of creep with exponent m. On the average, the rate of energy release in earthquakes dE/dt before a larg e earthquake that will occur at time ${\mathit{t}}_{\mathit{r}}$ increases with time as (${\mathit{t}}_{\mathit{r}}$-t${)}^{\mathrm{-}\mathrm{\beta }}$, with β decreasing from (t+1) for m=0 to 1 as m increases to infinity; in two dimensions t=1.3 is the percolation-conductance exponent. For large m, which is appropriate for ductile-brittle-fracture laboratory measurements, the value of this exponent is in agreement with observations of the number rate of occurrence of earthquake foreshocks. The exponent β is independent of the amount of initial disorder within a broad interval. The power law is a consequence of the many-body interactions between small cracks formed before an impending large rupture. As a consequence of variations in the initial configuration of inhomogeneities, there are large fluctuations in the rate of energy release dE/dt from system to system, a result also consistent with observations of foreshocks in nature.