Berryman shows elegantly that the “inequality (c11c44)·(c33c44)≥(c13c44)2 is true for any horizontally stratified, homogeneous material whose constituent layers are isotropic…” However, the final clause of this sentence“…, i.e., any homogeneous, transversely isotropic material,” is, if taken at face value, misleading. It is clear from the proof in the section “A fundamental inequality” that this statement is only shown to hold for lamellated media with isotropic lamellae, and that Berryman chooses arbitrarily and without any warning the phrase homogeneous, transversely isotropic to stand as a synonym for what Backus (1962) painstakingly describes as “smoothed, transversely isotropic, long-wave equivalent (STILWE).” In view of the fact that even within the context of exploration seismics transverse isotropy can be due to causes other than horizontal stratification with isotropic constituents (e.g., schists can be intrinsically anisotropic, anisotropy might be due to preferential orientation of sandgrains or joints), I believe this choice to be unfortunate. It leads the unsuspecting reader to assume a wider applicability of the fundamental inequality than Berryman really intends to claim, and thus makes it unnecessarily difficult to understand this significant contribution.

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