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

Abstract

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 Δσ p ( 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 Δσ p ( 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/ r 2 and that the spectral composition of Δσ p ( x, y ) is of the form , where is the two-dimensional Fourier transform of Δσ p ( 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 / k 0 ) 2 , consistent with the quasi-static frequency of occurrence relation N ∼ 1/ r 2 ; k 0 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, a rms , is given by the expression a rms = 0.85[2 1/2 (2π) 2 /106] (Δσ/ρ R )(ƒ max /ƒ 0 ) 1/2 , where Δσ is the earthquake stress drop, ρ is density, R is hypocentral distance, ƒ 0 is the spectral corner frequency, and ƒ max 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, a rms holds considerable promise as a measure of high-frequency strong ground motion for engineering purposes.