Scaling laws of earthquakes derived by renormalization group method

Size-frequency distribution of earthquakes (Gutenberg–Richter relation) is characterized by the b value, and its average is about 1 in observational works. The high-frequency asymptotes of displacement spectra of seismic waves provide another evidence of scale invariance of seismic faulting. The asymptotes of the observed spectra show ω−θ with θ 2, which was first suggested by Aki [J Geophys Res 1967;72:1217–31]. Matsuba [Chaos, Solitons & Fractals 2002;13:1281–94] applied the renormalization group method to the three-dimensional Burridge–Knopoff (BK) model and obtained the relation between b and θ. b = 0.880 was derived for θ = 2. However, his result does not seem to explain the observed values of b. In this paper we show that a simple modification of block interactions in the formulation for the three-dimensional BK model improves the self-similar solution obtained by the renormalization group method. b = 0.987 is derived for θ = 2 from the improved solution. The introduction of longer-range interactions leads to the change of the universality class of the BK