Model-based discrete relaxation process representation of band-limited power-law attenuation

TL;DR

This paper presents a method to determine discrete relaxation parameters for modeling band-limited power-law attenuation in complex media, using a model-based approach that samples relaxation distributions logarithmically.

Contribution

It introduces an analytical insight that continuous relaxation distributions within a frequency band produce equivalent wave equations, and develops a method to discretize these for power-law attenuation modeling.

Findings

Analytical proof of equivalence for band-limited relaxation distributions

A sampling method for relaxation parameters based on logarithmic frequency intervals

Application to power-law attenuation in biological tissues

Abstract

Frequency-dependent acoustical loss due to a multitude of physical mechanisms is commonly modeled by multiple relaxations. For discrete relaxation distributions, such models correspond with causal wave equations of integer-order temporal derivatives. It has also been shown that certain continuous distributions may give causal wave equations with fractional-order temporal derivatives. This paper demonstrates analytically that if the wave-frequency {\omega} satisfies \Omega_L << {\omega} << \Omega_H, a continuous relaxation distribution populating only {\Omega} belongs to [\Omega_L,\Omega_H] gives the same effective wave equation as for a fully populated distribution. This insight sparks the main contribution: the elaboration of a method to determine discrete relaxation parameters intended for mimicking a desired attenuation behavior for band-limited waves. In particular, power-law attenuation is discussed as motivated by its prevalence in complex media, e.g. biological tissue. A Mittag-Leffler function related distribution of relaxation mechanisms has previously been shown to be related to the fractional Zener wave equation of three power-law attenuation regimes. Because these regimes correspond to power-law regimes in the relaxation distribution, the idea is to sample the distribution's compressibility contributions evenly in logarithmic frequency while appropriately taking the stepsize into account. This work thence claims to provide a model-based approach to determination of discrete relaxation parameters intended to adequately model attenuation power-laws.