Free oscillations of a slightly anisotropic earth

The effect of anisotropy on the free oscillations of the Earth is examined under the assumptions that anisotropy is general, i.e. there are 21 independent components, and a small perturbation from an isotropic media. It is shown that the deviation of a multiplet frequency δω from an average spherically symmetric earth depends upon 15 functions of θ (co-latitude) and φ (longitude) as opposed to three in the isotropic case shown previously by Woodhouse & Girnius. In the asymptotic limit, δω is an average of a local functional δωlocal (θ, φ, ψ) along a great circle path which contains the source and the receiver, where δωlocal (θ, φ, ψ) is a function of θ, φ and azimuth ψ. Also δωlocal (θ, φ, ψ)is a function of the elastic constants beneath a location (θ, φ). The azimuth dependence of δωlocal (θ, φ, ψ)is 1, cos 2ψ, sin2ψ, cos 4ψ and sin4ψ, which is the same as a flat-layered case. It is shown that formulae equivalent with those of Smith & Dahlen are recovered by an earth flattening transformation. In the non-asymptotic case, in which the angular order l is low, δω shows odd-order azimuthal dependence, i.e. ψ and 3ψ dependence. There exist seven radial kernels in the spherical case as opposed to four in the flat-layered case, although radial resolution remains similar because of similarity of the kernels.