Long-wave elastic anisotropy produced by horizontal layering

TLDR

A horizontally layered inhomogeneous medium, isotropic or transversely isotropic, is studied with properties nearly constant when averaged over a vertical scale l′. The authors derive necessary and sufficient algebraic conditions on the five elastic coefficients of a homogeneous transversely isotropic medium for it to be long‑wave equivalent to a horizontally layered isotropic medium, including the special two‑material case. For wavelengths longer than l′ the layered medium behaves like a homogeneous transversely isotropic medium with averaged density and elastic coefficients, and, except in singular cases, the proportions and coefficients of a two‑layered isotropic medium are uniquely determined, explaining observed crustal P‑wave velocity variations as anisotropy from layering.

Abstract

A horizontally layered inhomogeneous medium, isotropic or transversely isotropic, is considered, whose properties are constant or nearly so when averaged over some vertical height l′. For waves longer than l′ the medium is shown to behave like a homogeneous, or nearly homogeneous, transversely isotropic medium whose density is the average density and whose elastic coefficients are algebraic combinations of averages of algebraic combinations of the elastic coefficients of the original medium. The nearly homogeneous medium is said to be ‘long-wave equivalent’ to the original medium. Conditions on the five elastic coefficients of a homogeneous transversely isotropic medium are derived which are necessary and sufficient for the medium to be ‘long-wave equivalent’ to a horizontally layered isotropic medium. Further conditions are also derived which are necessary and sufficient for the homogeneous medium to be ‘long-wave equivalent’ to a horizontally layered isotropic medium consisting of only two different homogeneous isotropic materials. Except in singular cases, if the latter two-layered medium exists at all, its proportions and elastic coefficients are uniquely determined by the elastic coefficients of the homogeneous transversely isotropic medium. The observed variations in crustal P-wave velocity with depth, obtained from well logs, are shown to be large enough to explain some of the observed crustal anisotropies as due to layering of isotropic material.