Camilla Cattania

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. While 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 from 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, gamma=2 and corner frequency omega_c=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, where R is the size of asperity, becoming vanishingly small as R tends to 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 45-80m.