Modelling of earthquake rupturing as a stochastic process and estimation of its distribution function from earthquake observations

SUMMARY The effect on earthquake rupturing of heterogeneities in tectonic stress and in material strength along a large fault zone is incorporated in the potential dynamic stress drop, defined as the difference between the tectonic shear stress and the dynamic frictional strength according to a slip‐weakening model. The distribution of the potential dynamic stress drop Δτ d (x) along the strike of the fault plane is modelled as a 1‐D stochastic process. Using a simple dynamic fracture criterion, a relation is established between earthquake rupturing and potential dynamic stress drop, by which any earthquake rupture process can be regarded as a segment of a realization of the process Δτ d (x) where Δτ d (x) > 0. Since dynamic slip varies approximately linearly with dynamic stress drop, it has the same distribution function as Δτ d (x), provided that Δτ d (x) is a Gaussian process. Three independent earthquake observations, i.e. the average stress drop, the Gutenberg‐Richter relation and the surface slip along earthquake faults, are used to estimate the distribution function of Δτ d (x). An analytical solution is derived for the distribution function of Δτ d (x), which shows that, among all known distribution models, only the fractional Brownian motion with index H → 0 (fractal dimension D = 2 in the 1‐D case) can give rise to the observed approximately constant stress drop independent of earthquake size. The probability distribution of the size of zerosets of the fractional Brownian motion shows a power‐law relation with frequency, which resembles the frequency‐seismic‐moment relation. Using an average b value of 1.O for small earthquakes, an index H → 0 of the fractional Brownian motion is obtained. The model predicts that the b value for large earthquakes is smaller than that for small earthquakes along the same fault zone, which is in agreement with observations. The surface slip data of two strike‐slip‐dominated earthquake faults with rupture lengths larger than 100 km are inverted using power spectral analysis. Both data sets display a power‐law relation between the sample power spectrum and the spatial frequency, which implies a fractional Brownian distribution. The estimated index H is close to zero for both earthquake faults. Stress drops, b values, and surface slips all independently suggest that the earthquake rupturing process can be modelled stochastically as a fractional Brownian motion with index H → 0.