Frequency power-law attenuation and dispersion in marine sediments

Abstract

Acoustic waves in marine sediments, including the fine-grained muds and clays, often exhibit an attenuation in the form of a frequency power law in which the exponent is close to unity. The frequency dependence of the wave speed, or dispersion, associated with such a power law has long been a subject of debate. Causality arguments, often in the form of the Kramers-Kronig dispersion relations, are usually applied to the problem, but these are not sufficient to characterize the dispersion fully. Following an alternative approach, a wave equation will be introduced which predicts the dispersion when the attenuation is of the frequency power law type. Based on this wave equation, it will be shown that only certain values of the frequency exponent are allowed, otherwise causality is violated, indicating that the associated attenuation and dispersion are not physically realizable. In effect, for these prohibited values of the frequency exponent, the Fourier components in the dispersion and attenuation cannot combine to give a zero response for negative times. Emerging from the theory is an expression for the complex wavenumber that is complete and exact, and from which all the properties of the dispersion and attenuation may be derived.