Constant Q and a Fractal, Stratified Earth

Frequency-dependent measurements of the quality factor Q typically show a constant behaviour for low frequencies and a positive power-law dependence for higher frequencies. In particular, the constant Q pattern is usually explained using intrinsic attenuation models due to anelasticity with either a single or multiple superposed relaxation mechanisms — each with a particular resonance peak. However, in this study, I show using wave localisation theory that a constant Q may also be due to apparent attenuation due to scattering losses. Namely, this phenomenon occurs if the earth displays fractal characteristics. Moreover, if fractal characteristics exist over a limited range of scales only, even an absorption band can be created—in accordance with observations. This indicates that it may be very difficult to distinguish between intrinsic and scattering attenuation on the basis of frequency-dependent measurements of the quality factor only.