b values and ω^−γ seismic source models: Implications for tectonic stress variations along active crustal fault zones and the estimation of high-frequency strong ground motion

In this study the tectonic stress along active crustal fault zones is taken to be of the form , where  is the average tectonic stress at depth y and Δσp(x, y) is a seismologically observable, essentially random function of both fault plane coordinates; the stress differences arising in the course of crustal faulting are derived from Δσp(x, y). Empirically known frequency of occurrence statistics, moment-magnitude relationships, and the constancy of earthquake stress drops may be used to infer that the number of earthquakes N of dimension ≥r is of the form N ∼ 1/r2 and that the spectral composition of Δσp(x, y) is of the form , where  is the two-dimensional Fourier transform of Δσp(x, y) expressed in radial wave number k. The γ = 2 model of the far-field shear wave displacement spectrum is consistent with the spectral composition , provided that the number of contributions to the spectral representation of the radiated field at frequency ƒ goes as (k/k0)2, consistent with the quasi-static frequency of occurrence relation N ∼ 1/r2;k0 is a reference wave number associated with the reciprocal source dimension. Separately, a variety of seismologic observations suggests that the γ = 2 model is the one generally, although certainly not always, applicable to the high-frequency spectral decay of the far-field radiation of earthquakes. In this framework, then, b values near 1, the general validity of the γ = 2 model, and the constancy of earthquake stress drops independent of size are all related to the average spectral composition of. Should one of these change as a result of premonitory effects leading to failure, as has been specifically proposed for b values, it seems likely that one or all of the other characteristics will change as well from their normative values. Irrespective of these associations, the far-field, high-frequency shear radiation for the γ = 2 model in the presence of anelastic attenuation may be interpreted as band-limited, finite duration white noise in acceleration. Its rms value, arms, is given by the expression arms = 0.85[21/2(2π)2/106] (Δσ/ρR)(ƒmax/ƒ0)1/2, where Δσ is the earthquake stress drop, ρ is density, R is hypocentral distance, ƒ0 is the spectral corner frequency, and ƒmax is determined by R and specific attenuation 1/Q. For several reasons, one of which is that it may be estimated in the absence of empirically defined ground motion correlations, arms holds considerable promise as a measure of high-frequency strong ground motion for engineering purposes.