This paper constitutes, in essence, the Ph.D. thesis proposal of the first author, the research is to be carried out under the supervision of the second author. It is proposed to study certain aspects of wave propagation in elastic, linear, homogeneous (inhomogeneous), and anisotropic media. This formidable subject will be broken into three basic interrelated parts; these are: basic concepts, forward modeling, and inversion. Some areas of research interest in these parts are: the inclusion of the rotational term in the constitutive relations; determinations of macroscopic anisotropic descriptions from preferentially oriented microscopic heterogeneities; the elastodynamic Green's function in anisotropic media and/or its approximations; the statistical approach to determine elastic constants (sparse-matrix techniques); the determination of axes of symmetry by use of the Maxwell multipole representation; the use of (x-p) (plane-wave) methods in modeling and inversion, the effects of singularities on the quasi-shear velocity sheet, the extension of point-source decomposition to anisotropic media, and applying the topics above in an inversion scheme. A brief review of work already done and possible areas of further research of the topics above will be given. Not all these points will be brought to fruition, but at this point it would be unwise to limit myself too severely, as I am still surveying this vast subject. Though the area of thesis research is at present concerned mostly with the theoretical aspects of anisotropy, there are many practical aspects of this subject - such as fracture detection, influences on amplitude-verses-offset and general traveltime effects that profoundly effects seismological data - that can also be explored.
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