Abstract
A method is proposed for retrieving source-extent parameters from far-field body-wave data. At low frequency, the normalized P- or S-wave displacement amplitude spectrum can be approximated by |Ω^(r^,ω)| = 1 − τ2(r^)ω2/2 where r^ specifies a point on the focal sphere. For planar dislocation sources, τ2(r^) is linearly related to statistical measures of source dimension, source duration, and directivity. τ2(r^) can be measured as the curvature of |Ω^(r^,ω)| at ω = 0 or the variance of the pulse Ω^(r^,t). The quantity ωc=2τ−1(r^) is contrasted with the traditional corner frequency ω0, defined as the frequency at the intersection of the low- and high-frequency trends of |Ω^(r^,ω)|. For dislocation models without directivity, ωc(P) ≧ ωc(S) for any r^. A mean corner frequency defined by averaging τ2(r^) over the focal sphere, ω¯c=2<τ2(r^)>−1/2, satisfies ωc(P) > ωc(S) for any dislocation source. This behavior is not shared by ω0. It is shown that ω0 is most sensitive to critical times in the rupture history of the source, whereas ωc is determined by the basic parameters of source extent. Evidence is presented that ωc is the corner frequency measured on actual seismograms. Thus, the commonly observed corner frequency shift (P-wave corner greater than the S-wave corner), now viewed as a shift in ωc is simply a result of spatial finiteness and is expected to be a property of any dislocation source. As a result, the shift cannot be used as a criterion for rejecting particular dislocation models.
Long-time asymptotics for evolutionary crystal dislocation models