Elastic wave-mode separation for VTI media

Abstract

The separation of wave modes from isotropic elastic wavefields is typically done using Helmholtz decomposition. However, Helmholtz decomposition using conventional divergence ( ∇ · ) and curl ( ∇ × ) operators in anisotropic media does not give satisfactory results and leaves the different wave modes only partially separated. The separation of anisotropic wavefields requires the use of more sophisticated operators which depend on local material parameters. Dellinger and Etgen (1990) suggest separating wave modes in anisotropic media by projecting the wavefields W onto the directions U in which the P and S modes are polarized: qP = iU(k) · W ̃ = i Ux W̃x + i Uy W̃y + i Uz W̃z. (1) In anisotropic media, polarization vectors U(kx, ky, kz) are different from the wave vector k, and generally are not radial because qP waves in an anisotropic medium are not polarized in the same directions as wave vectors, except in the symmetry planes (kz = 0) and along the symmetry axis (kx = 0). We use the same idea for wave-mode separation, but implement the operator ion the space-domain as opposed to the wavenumber-domain. The equivalent expression to equation 1 in the space domain is