On Commutativity and Near Commutativity of Translational and Rotational Averages: Analytical Proofs and Numerical Examinations

We show that, in general, the translational average over a spatial variable—discussed by Backus (J. Geophys. Res. 67(11):4427–4440, 1962), and referred to as the equivalent-medium average—and the rotational average over a symmetry group at a point—discussed by Gazis et al. (Acta Crystallogr. 16(9):917–922, 1963), and referred to as the effective-medium average—do not commute. However, they do commute in special cases of particular symmetry classes, which correspond to special relations among the elasticity parameters. We also show that this noncommutativity is a function of the strength of anisotropy. Surprisingly, a perturbation of the elasticity parameters about a point of weak anisotropy results in the commutator of the two types of averaging being of the order of the square of this perturbation. Thus, these averages nearly commute in the case of weak anisotropy, which is of interest in such disciplines as quantitative seismology, where the weak-anisotropy assumption results in empirically adequate models.