A Source Model for Earthquakes near the Nucleation Dimension

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.