On Backus average for generally anisotropic layers

TL;DR

This paper revisits the Backus average for anisotropic layers, providing mathematical insights, stability conditions, and tensorial analysis to deepen understanding of the long-wave approximation in elastic media.

Contribution

It offers a rigorous mathematical examination of the Backus average for anisotropic layers, including stability proofs and tensorial analysis using Kelvin notation.

Findings

Proves stability preservation under the long-wave approximation.

Analyzes the approximation of the average of a product as the product of averages.

Uses tensor notation to examine effects of coordinate rotations.

Abstract

In this paper, following the Backus (1962) approach, we examine expressions for elasticity parameters of a homogeneous generally anisotropic medium that is long-wave-equivalent to a stack of thin generally anisotropic layers. These expressions reduce to the results of Backus (1962) for the case of isotropic and transversely isotropic layers. In the over half-a-century since the publications of Backus (1962) there have been numerous publications applying and extending that formulation. However, neither George Backus nor the authors of the present paper are aware of further examinations of the mathematical underpinnings of the original formulation; hence this paper. We prove that---within the long-wave approximation---if the thin layers obey stability conditions then so does the equivalent medium. We examine---within the Backus-average context---the approximation of the average of a product as the product of averages, which underlies the averaging process. In the presented examination we use the expression of Hooke's law as a tensor equation; in other words, we use Kelvin's---as opposed to Voigt's---notation. In general, the tensorial notation allows us to conveniently examine effects due to rotations of coordinate systems.