A Source Model for Earthquakes near the Nucleation Dimension

Abstract

ABSTRACT Earthquake self-similarity is a controversial topic, both observationally and theoretically. Theory predicts a finite nucleation dimension, implying a break of self-similarity below a certain magnitude. Although observations of non-self-similar earthquake behavior have been reported, their interpretation is challenging due to trade-offs between source and path effects and assumptions on the underlying source model. Here, I introduce a source model for earthquake nucleation and quantify the resulting scaling relations between source properties (far-field pulse duration, seismic moment, stress drop). I derive an equation of motion based on fracture mechanics for a circular rupture obeying rate–state friction and a simpler model with constant stress drop and fracture energy. The latter provides a good approximation to the rate–state model and leads to analytical expressions for far-field displacement pulses and spectra. The onset of ground motion is characterized by exponential growth with characteristic timescale t0=R0/vf, with R0 the nucleation dimension and vf a limit rupture velocity. Therefore, normalized displacements have a constant duration, proportional to the nucleation length rather than the source dimension. For ray paths normal to the fault, the exponential growth results in a Boatwright spectrum with n = 1, γ=2 and corner frequency ωc=1/t0. For other orientations, the spectrum has an additional sinc(·) term with a corner frequency related to the travel-time delay across the asperity. Seismic moments scale as M0∼R(R−R0)R0, in which R is the size of asperity, becoming vanishingly small as R→R0. Therefore, stress drops estimated from M0 and fc are smaller than the nominal stress drop, and they increase with magnitude up to a constant value, consistent with several seismological studies. The constant earthquake duration is also in agreement with reported microseismicity: for 0<Mw<2 events studied by Lin et al. (2016) in Taiwan, the observed durations imply a nucleation length between 45 and 80 m.