Modelling surface waves in anisotropic structures I. Theory

Abstract

Surface-wave theory in generally-anisotropic laterally-homogeneous media is partially reformulated in order to obtain intuitively-expected extensions of classic body-wave ideas such as Maslov plane-wave summation, the geometrical-ray/WKBJ limit and source-receiver reciprocity. This is done using the ‘reversal’ symmetry of Chapman [Chapman, C.H., 1994. Reflection/transmission coefficient reciprocities in anisotropic media. Geophys. J. Int. 116, 498–501] to generalize the point-source treatment of Kennett [Kennett, B.L N., 1983. Seismic Wave Propagation in Stratified Media. Cambridge Univ. Press, Cambridge, UK] to a stack of anisotropic layers with complex elastic parameters, lateral slowness and frequency. The 2D-integral representation over horizontal slownesses is reduced by residues to a 1D integral over each mode's slowness or dispersion curve at fixed frequency and this may be considered a 1D Maslov summation over local plane waves tangential to the phase front. The residue calculation involves a modified form of the usual variational principle, in which the Lagrangian now contains reversed modes. The 1D slowness integral may be reduced by stationary-phase arguments to the geometrical-ray or WKBJ limit, provided the dispersion-surface curvature does not vanish. This limit satisfies reciprocity, as the reversal symmetry shows. Dimples on the dispersion surface will correspond to folds on the phase front and multiple arrivals. An appendix contains a discussion of orthogonality of the surface-wave modes in relation to the various wave-equation symmetries.