A source model for earthquakes near the nucleation dimension

Abstract

Earthquake self-similarity is a controversial topic, both from an observational and theoretical standpoint. Theory predicts the existence of a finite nucleation dimension, and this characteristic length scale implies a break of self-similarity below a certain magnitude. While observations of non self-similar earthquake behavior have been reported, their interpretation remains debated, since estimating source properties for small earthquakes is challenging due to trade-offs between source and path effects as well as assumptions on the underlying source model, which often assume self-similarity in the first place. Here I introduce a source model that accounts for earthquake nucleation, and I use it to quantify how the nucleation phase affects ground motion. The model consists of an equation of motion for a circular rupture front (derived from fracture mechanics) and the corresponding far-field displacement pulses and spectra. The early phases of far-field ground motion are characterized by exponential growth with characteristic timescale t0=R0/Vr, where R0 is the nucleation dimension and Vr a limit rupture velocity. When displacements are normalized by their peak value, this produces an apparent constant source 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 fc=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 with M0 ~ R(R-R0)R, where R is the size of asperity, and hence become vanishingly small as R->R0. Consequently, stress drops estimated from the scaling between M0 and fc are smaller than the nominal stress drop, and they decrease with decreasing magnitude, consistent with a number of seismological studies. The constant earthquake duration is also in agreement with reported microseismicity, providing an estimate of the nucleation length: for 0 < Mw< 2 events studied by Lin et al. (2016) in Taiwan, a model with a nucleation length between 45-80m provides a good fit to observed durations.