High-Frequency Radiation from an Earthquake Fault: A Review and a Hypothesis of Fractal Rupture Front Geometry

Observed high-frequency (HF) radiation from earthquake faults exhibits specific properties that cannot be deduced or extrapolated from low-frequency fault behavior. In particular: (1) HF time functions look like random signals, with smooth mean spectrum and moderately heavy-tailed probability distribution function for amplitudes; (2) well-known directivity of low-frequency radiation related to rupture propagation is strongly reduced at HF, suggesting incoherent (delta-correlated) behavior of the HF radiator, and contradicting the usual picture of a rupture front as a regular, non-fractal moving line; (3) in the spectral domain, HF radiation occupies a certain specific band seen as a plateau on acceleration source spectra $ K(f) = f^{2} \dot{M}_{0} (f) $ . The lower cutoff frequency f _b of K(f) spectra is often located significantly higher than the common spectral corner frequency f _c, or f _a. In many cases, empirical f _b(M ₀) trends are significantly slower as compared to the simple f _b ∝ M ₀ ^?1/3 , testifying the lack of similarity in spectral shapes; (4) evidence is accumulating in support of the reality of the upper cutoff frequency of K(f): fault-controlled f _max, or f _uf. However, its identification is often hampered by such problems as: (a) strong interference between f _uf and site-controlled f _max; (b) possible location of f _uf above the observable spectral range; and (c) substantial deviations of individual source spectra from the ideal spectral shape; (5) intrinsic structure of random-like HF radiation has been shown to bear significant self-similar (fractal) features. A HF signal can be represented as a product of a random HF “carrier signal” with constant mean square amplitude, and a positive modulation function, again random, that represents a signal envelope. It is this modulation function that shows approximately fractal behavior. This kind of behavior was revealed over a broad range of time scales, from 1 to 300 s from teleseismic data and from 0.04 to 30 s from near-fault accelerogram data. To explain in a qualitative way many of these features, it is proposed that rupture propagation can be visualized as occurring, simultaneously, at two different space–time scales. At a macro-scale (i.e. at a low resolution view), one can safely believe in the reality of a singly connected rupture with a front as a smooth line, like a crack tip, that propagates in a locally unilateral way. At a micro-scale, the rupture front is tortuous and disjoint, and can be visualized as a multiply connected fractal “line” or polyline. It propagates, locally, in random directions, and is governed by stochastic regularities, including fractal time structure. The two scales and styles are separated by a certain characteristic time, of the order of (0.07–0.15) × rupture duration. The domain of fractal behavior spans a certain HF frequency range; its boundaries, related to the lower and upper fractal limits, are believed to be manifested as f _b and f _uf.