Seismic pulse propagation with constant Q and stable probability distributions

TL;DR

This paper models seismic pulse propagation with constant Q using a fractional order evolution equation, linking solutions to stable probability distributions and bridging behaviors between viscous and elastic media.

Contribution

It introduces a fractional order evolution equation for seismic waves with constant Q and relates solutions to stable probability distributions, providing a new mathematical framework.

Findings

Solutions expressed via entire functions of Wright type

Behavior intermediate between viscous fluid and elastic solid

Fundamental solutions linked to stable probability distributions

Abstract

The one-dimensional propagation of seismic waves with constant Q is shown to be governed by an evolution equation of fractional order in time, which interpolates the heat equation and the wave equation. The fundamental solutions for the Cauchy and Signalling problems are expressed in terms of entire functions (of Wright type) in the similarity variable and their behaviours turn out to be intermediate between those for the limiting cases of a perfectly viscous fluid and a perfectly elastic solid. In view of the small dissipation exhibited by the seismic pulses, the nearly elastic limit is considered. Furthermore, the fundamental solutions for the Cauchy and Signalling problems are shown to be related to stable probability distributions with index of stability determined by the order of the fractional time derivative in the evolution equation.