A Review on Scaling of Earthquake Source Spectra

Abstract

In this review paper, the theoretical and observational studies of scaling of earthquake source displacement spectra (abbreviated as source spectra) are compiled and discussed. The earlier studies, including the kinetic models proposed by several authors [including Haskell (Bull Seismol Soc Am 56:125–140, 1966) and Aki (J Geophys Res 72:1217–1231, 1967), provided the so-called ω−1, ω-square, and ω-cube models. Aki (1967) favored the ω-square model and also assumed constant stress drop, Δσ, and self-similarity of earthquakes. Some observations agree to one of the two models, but others do not. Hence, numerous alternative forms of the three scaling models to interpret the observations have been made by seismologists. For the corner frequencies, fc, the seismic moment Mo scales with fc in the form of Mo ~ f c −3 . For fcP of the P-waves and fcS of the S-waves, fcP > fcS is more reasonable than fcP < fcS. Mo scales as T D 3 , where TD is the duration time of source rupture and inversely related to fc, for both small and large events. The second corner frequency or cut-off frequency, fmax, at higher ω, that exists in the source spectra could be yielded by source, path, and site effects. The mechanisms to cause the patch corner frequency are still open. Analytical and numerical studies of the scaling laws based on the dislocation (e.g., Aki 1967), cracks (e.g., Walter and Brune in J Geophys Res 98(B3):4449–4459, 1993), spring-slider (e.g., Shaw in Geophys Res Lett 20:643–646, 1993), and statistical physics models (e.g., Hanks in J Geophys Res 84:2235–2242, 1997), including self-organized criticality (e.g., Bak et al. in Phys Rev Lett 59:381–384, 1987), show different scaling laws under various physical conditions.