Abstract
The basic assumption that the self-similarity and the spectral law of the seismic body-wave radiation (e.g., ω-square model) must find their origin in some simple self-similar process during the seismic rupture led us to construct a kinematic, self-similar model of earthquakes. It is first assumed that the amplitude of the slip distribution high-pass filtered at high wavenumber does not depend on the size of the ruptured fault. This leads to the following “k-square” model for the slip spectrum, for k > 1/L: Δ~uL(k)=CΔσμLk2,
where L is the ruptured fault dimension, k the radial wavenumber, Δσ the mean stress drop, μ the rigidity, and C an adimensional constant of the order of 1. The associated stress-drop spectrum, for k > 1/L, is approximated by Δ~σL(k)=ΔσLk.
The rupture front is assumed to propagate on the fault plane with a constant velocity v, and the rise time function is assumed to be scale dependent. The partial slip associated to a given wavelength 1/k is assumed to be completed in a time 1/kv, based on simple dynamical considerations. We therefore considered a simple dislocation model (instantaneous slip at the final value), which indeed correctly reproduces this self-similar characteristic of the slip duration at any scale. For a simple rectangular fault with isochrones propagating in the x direction, the resulting far-field displacement spectrum is related to the slip spectrum as u˜(ω)=FΔ~u(kx=1Cdωv,ky=0),
where the factor F includes radiation pattern and distance effect, and Cd is the classical directivity coefficient 1/[1 − v/c cos (θ)]. The k-square model for the slip thus leads to the ω-square model, with the assumptions above. Independently of the adequacy of these assumptions, which should be tested with dynamic numerical models, such a kinematic model has several important applications. It may indeed be used for generating realistic synthetics at any frequency, including body waves, surface waves, and near-field terms, even for sites close to the fault, which is often of particular importance; it also provides some clues for estimating the weighting factors for the empirical Green's function methods. Finally, the slip spectrum may easily be modified in order to model other power-law decay of the radiation spectra, as well as composite earthquakes.
Self-similar solutions for the kinematic model equation of spiral waves