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Determination of the ray surface and recovery of elastic constants of anisotropic elastic media: a direct and inverse approach

Propagation of plane acoustic waves in anisotropic media can be characterized by the slowness surface and ray surface which represent phase velocities and group velocities, respectively. The mathematical expressions for the slowness surface in terms of elastic constants are well established and provide a foundation for acoustic determination of the elastic constants of anisotropic elastic media. However, a direct mathematical formulation of the ray surface has not been fully developed because of the many-to-one correspondence between the ray surface and the slowness surface. Based upon the Stroh formalism for two-dimensional elastodynamic systems, we establish in this paper a direct and analytical formalism, the degeneracy analysis approach (DAA), for the construction of the ray surface and the recovery of elastic constants for anisotropic media. A special emphasis is put on the group velocities along symmetry directions with respect to symmetry planes, for which an explicit expression for the group velocity is given in terms of the elastic constants so that the inverse problem can be solved easily. Particularly, the complication of cuspidal points due to the presence of axial concavity in the slowness surface is treated and it is apparent that the presence of a cuspidal point becomes an advantage for the inverse problem of recovery of elastic constants.