Inferring the Orientation-distribution Function from Observed Seismic Anisotropy: General Considerations and an Inversion of Surface-wave Dispersion Curves

A general relation linking the elasticity tensor of an anisotropic medium with that of the constituting single crystals and the function describing the orientation distribution of the crystals is derived. By expanding the orientation distribution function (ODF) into tensor spherical harmonics and using canonical components of the elasticity tensors, it is shown that the elastic tensor of the medium is completely determined by a finite number of expansion coefficients, namely those with harmonic degree l h 4. The number of expansion coefficients actually needed to determine the elastic constants of the medium depends on the symmetry of the single crystals. For hexagonal symmetry of the single crystals it is shown that only 8 real numbers are required to fix the 13 elastic constants which are for example needed to determine the azimuthal dependence of surface wave velocities. Thus, inversions of observations of seismic anisotropy are feasible which do not make any a priori assumptions on the orientation of the crystals. As a byproduct of the derivation, a formula is given which allows the easy calculation of the elastic constants of a medium composed of hexagonal crystals obeying an arbitrary ODF. An application of the theoretical results to the inversion of surface wave dispersion curves for an anisotropic 1D-mantle model is presented. For the S-wave velocities the results are similar to those of previous inversions but the new approach also yields P-wave velocities consistent with the assumption of oriented olivine. Moreover it provides a hint of the orientation distribution of the crystals.