Thomas C. Hanks
1979
<p><span>In this study the tectonic stress along active crustal fault zones is taken to be of the form </span><span class="math-equation-construct" data-equation-construct="true"><span class="math-equation-image" data-equation-image="true"><img class="inlineGraphic" src="http://api.onlinelibrary.wiley.com/asset/v1/doi/10.1029%2FJB084iB05p02235/asset/equation%2Fjgrb2891-math-0001.gif?l=Cgg2pVVsCMwLOww28sbn56fxTbx6GilFBMCsUrZMoRcYP%2B9NZI%2F0t%2B3p7VO59umu6MIzikkO5ymj%0ANDlW1G0%2F%2BQ%3D%3D" alt="inline image" /></span></span><span>, where </span><span class="math-equation-construct" data-equation-construct="true"><span class="math-equation-image" data-equation-image="true"><img class="inlineGraphic" src="http://api.onlinelibrary.wiley.com/asset/v1/doi/10.1029%2FJB084iB05p02235/asset/equation%2Fjgrb2891-math-0002.gif?l=Cgg2pVVsCMwLOww28sbn56fxTbx6GilFBMCsUrZMoRcYP%2B9NZI%2F0t%2B3p7VO59umuiedpqT79PPhK%0AOsNB85uCWA%3D%3D" alt="inline image" /></span></span><span> is the average tectonic stress at depth </span><i>y</i><span> and Δσ</span><span><i>p</i></span><span>(</span><i>x, y</i><span>) 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</span><i><span>p</span></i><span>(</span><i>x, y</i><span>). 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 </span><i>N</i><span> of dimension ≥</span><i>r</i><span> is of the form </span><i>N</i><span> ∼ 1/</span><i>r</i><span>2</span><span> and that the spectral composition of Δσ</span><i><span>p</span></i><span>(</span><i>x, y</i><span>) is of the form </span><span class="math-equation-construct" data-equation-construct="true"><span class="math-equation-image" data-equation-image="true"><img class="inlineGraphic" src="http://api.onlinelibrary.wiley.com/asset/v1/doi/10.1029%2FJB084iB05p02235/asset/equation%2Fjgrb2891-math-0003.gif?l=Cgg2pVVsCMwLOww28sbn56fxTbx6GilFBMCsUrZMoRcYP%2B9NZI%2F0t%2B3p7VO59umusANOiMJ6380O%0AqOQppeWKzQ%3D%3D" alt="inline image" /></span></span><span>, where </span><span class="math-equation-construct" data-equation-construct="true"><span class="math-equation-image" data-equation-image="true"><img class="inlineGraphic" src="http://api.onlinelibrary.wiley.com/asset/v1/doi/10.1029%2FJB084iB05p02235/asset/equation%2Fjgrb2891-math-0004.gif?l=Cgg2pVVsCMwLOww28sbn56fxTbx6GilFBMCsUrZMoRcYP%2B9NZI%2F0t%2B3p7VO59umuGlAzWdjz2K3Q%0AG1y6nb%2F%2FjA%3D%3D" alt="inline image" /></span></span><span> is the two-dimensional Fourier transform of Δσ</span><i><span>p</span></i><span>(</span><i>x, y</i><span>) expressed in radial wave number </span><i>k</i><span>. The γ = 2 model of the far-field shear wave displacement spectrum is consistent with the spectral composition </span><span class="math-equation-construct" data-equation-construct="true"><span class="math-equation-image" data-equation-image="true"><img class="inlineGraphic" src="http://api.onlinelibrary.wiley.com/asset/v1/doi/10.1029%2FJB084iB05p02235/asset/equation%2Fjgrb2891-math-0005.gif?l=Cgg2pVVsCMwLOww28sbn56fxTbx6GilFBMCsUrZMoRcYP%2B9NZI%2F0t%2B3p7VO59umunzw%2FWHTH5Y0K%0AFHvdyv0N%2Fw%3D%3D" alt="inline image" /></span></span><span>, provided that the number of contributions to the spectral representation of the radiated field at frequency ƒ goes as (</span><i>k</i><span>/</span><i>k</i><span>0</span><span>)</span><span>2</span><span>, consistent with the quasi-static frequency of occurrence relation </span><i>N</i><span> ∼ 1/</span><i>r</i><span>2</span><span>;</span><i>k</i><span>0</span><span> 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, </span><i>b</i><span> 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</span><span class="math-equation-construct" data-equation-construct="true"><span class="math-equation-image" data-equation-image="true"><img class="inlineGraphic" src="http://api.onlinelibrary.wiley.com/asset/v1/doi/10.1029%2FJB084iB05p02235/asset/equation%2Fjgrb2891-math-0006.gif?l=Cgg2pVVsCMwLOww28sbn56fxTbx6GilFBMCsUrZMoRcYP%2B9NZI%2F0t%2B3p7VO59umuaobXgGTYfS%2Fs%0AAwMugfb30g%3D%3D" alt="inline image" /></span></span><span>. Should one of these change as a result of premonitory effects leading to failure, as has been specifically proposed for </span><i>b</i><span> 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, </span><i>a</i><span>rms</span><span>, is given by the expression </span><i>a</i><span>rms</span><span> = 0.85[2</span><span>1/2</span><span>(2π)</span><span>2</span><span>/106] (Δσ/ρ</span><i>R</i><span>)(ƒ</span><span>max</span><span>/ƒ</span><span>0</span><span>)</span><span>1/2</span><span>, where Δσ is the earthquake stress drop, ρ is density, </span><i>R</i><span> is hypocentral distance, ƒ</span><span>0</span><span> is the spectral corner frequency, and ƒ</span><span>max</span><span> is determined by </span><i>R</i><span> and specific attenuation 1/</span><i>Q</i><span>. For several reasons, one of which is that it may be estimated in the absence of empirically defined ground motion correlations, </span><i>a</i><span>rms</span><span> holds considerable promise as a measure of high-frequency strong ground motion for engineering purposes.</span></p>
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10.1029/JB084iB05p02235
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AGU Publications
b values and ω<sup>−γ</sup> seismic source models: Implications for tectonic stress variations along active crustal fault zones and the estimation of high-frequency strong ground motion
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