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

Abstract

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 fb of K(f) spectra is often located significantly higher than the common spectral corner frequency fc, or fa. In many cases, empirical fb(M0) trends are significantly slower as compared to the simple fb ∝ M0−1/3, 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 fmax, or fuf. However, its identification is often hampered by such problems as: (a) strong interference between fuf and site-controlled fmax; (b) possible location of fuf 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 fb and fuf.