On commutativity of Backus and Gazis averages

TL;DR

This paper investigates the mathematical relationship between two types of averages used in geophysics, showing they generally do not commute except in special cases, which has implications for modeling anisotropic media.

Contribution

It demonstrates that Backus and Gazis averages do not generally commute, clarifying their mathematical relationship and identifying conditions for their commutativity.

Findings

Backus and Gazis averages do not commute in general.

They commute only under specific symmetry conditions.

The paper provides examples illustrating these special cases.

Abstract

We show that the Backus (1962) equivalent-medium average, which is an average over a spatial variable, and the Gazis et al. (1963) effective-medium average, which is an average over a symmetry group, do not commute, in general. They commute in special cases, which we exemplify.