Paul G. Richards, Dean C. Witte, and Goran Ekstrom
Generalized ray theory for seismic waves in structures with planar nonparallel interfaces
Bulletin of the Seismological Society of America (August 1991), 81(4):1309-1331

Abstract:
A discovery by Diebold (1987) can be used to show that there are analytical methods, developed and widely used for layers with plane parallel interfaces, that can often be extended to the case of plane interfaces that are not paraliel. Layer interfaces can have arbitrary dip and strike, permitting the study of waves in truly three-dimensional structures. We re-derive Diebold's result from first principles, showing that the travel time along a ray path is given by a summation of horizontal slowness times horizontal range and vertical slownesses times vertical "layer thicknesses," in a form essentially identical to the standard relationship between these quantities in a medium that is laterally homogeneous. We then show that, because of an underlying lateral invariance in the ray relationships within a stack of wedge-shaped layers, it is simple to develop a generalized ray theory for wave propagation in such a medium. Finally, we show how such a theory can readily be used to synthesize body-wave pulse shapes at teleseismic distances, in a spherically symmetric Earth modified by dipping structures in the vicinity of source and or receiver.