Long-wave elastic anisotropy in transversely isotropic media

Compressional waves in horizontally layered media exhibit very weak long-wave anisotropy for short offset seismic data within the physically relevant range of parameters. Shear waves have much stronger anisotropic behavior. Our results generalize the analogous results of Krey and Helbig (1956) in several respects: (1) The inequality (c (sub 11) - c (sub 44) )(c (sub 33) - c (sub 44) ) > or = (c (sub 13) + c (sub 44) ) (super 2) derived by Postma (1955) for periodic isotropic, two-layered media is shown to be valid for any homogeneous, transversely isotropic medium; (2) a general perturbation scheme for analyzing the angular dependence of the phase velocity is formulated and readily yields Krey and Helbig's results in limiting cases; and (3) the effects of relaxing the assumption of constant Poisson's ratio sigma are considered. The phase and group velocities for all three modes of elastic wave propagation are illustrated for typical layered media with (1) one-quarter limestone and three-quarters sandstone, (2) half-limestone and half-sandstone, and (3) three-quarters limestone and one-quarter sandstone. It is concluded that anisotropic effects are greatest in areas where the layering is quite thin (10-50 ft), so that the wavelengths of the seismic signal are greater than the layer thickness and the layers are of alternately high- and low-velocity materials.