Abstract

Sonic logs are detailed measurements of the in situ seismic velocity along borehole walls. Power spectra of sonic logs typically decay approximately as the reciprocal of spatial frequency f, regardless of the chemical composition, geologic age, and tectonic history of the probed lithologies. Data sequences of this type are fractal or scale-invariant. The origins of this uniform 1/f scaling of seismic structure are not clear, particularly in low-porosity crystalline rocks, but faults, fractures, and cracks are considered to be important. Fault structures also follow fractal scaling laws and have significant effects on seismic velocity. This paper presents a quantitative model that evaluates the role played by faults in determining the scaling laws of seismic velocity fluctuations. The model is based on current knowledge of the structure and scaling properties of brittle faults and of associated regions of microcracking. By approximating the relationship between crack density and velocity variation as linear, this model yields a Brownian power spectrum (∝1/f2) for velocity perturbations across a single fault zone in a medium of otherwise constant velocity. The power spectrum of velocity fluctuations induced by a population of faults is then obtained by superposing the corresponding Brownian power spectra weighted according to the observed frequency-size scaling relationship of brittle faults. The results of this study indicate that the uniform 1/f scaling of velocity fluctuations in crystalline rocks seems to be linked to the correspondingly uniform scaling properties of fault structures.

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