Seismic waves in stratified anisotropic media - II. Elastodynamic eigensolutions for some anisotropic systems

The differential system for elastic waves in stratified anisotropic media is, as in the isotropic case, most simply solved by constructing a propagator from the eigenvalues and eigenvectors. In general, the eigensolutions must be found numerically, which may lead to an enormous amount of computation if the ultimate goal is to construct synthetic seismograms. However, if the stiffness tensor for each layer displays a horizontal plane of symmetry, simple analytical solutions exist and the expense of numerical solution can be avoided. We consider monoclinic, orthorhombic, hexagonal (including transversely isotropic), and istropic symmetries and give analytical expressions for the eigensolutions for each of these forms when the alignment is such that a horizontal symmetry plane exists. The analytical solutions are concise enough to make the construction of complete synthetic seismograms (a task which involves the construction and manipulation of 3-D functions) a realistic endeavour. While not applicable to arbitrary anisotropy, these results have widespread application as most of the anisotropies hypothesized in seismology require a horizontal plane of elastic symmetry.