Head Wave Traveltimes in a Three-Dimensional Multilayered Earth

Traveltimes of head waves propagating within a three-dimensional (3D) multilayered earth are described by straightforward mathematical formulae. The earth model consists of a set of homogeneous and isotropic layers bounded by plane interfaces. Each interface (including the surface) may possess arbitrary strike and dip. In this model, the source-to-receiver raypath of a critically refracted wave consists of a set of straight line segments, not confined to a single plane. Algebraic derivations of the traveltime expressions are greatly simplified by using a novel 3D form of Snell's law of refraction. Various generalizations of the basic traveltime equation extend its applicability to arbitrary source-receiver recording geometries and/or mode-converted waves. Related expressions for the traveltimes of reflected waves and one-way transmitted waves propagating in the same layered earth model are obtained as byproducts of the analysis. The expressions contain a set of unit raypath orientation vectors that depend implicitly on source and receiver coordinates. Hence, the equations cannot be characterized as closed-form in the mathematical sense. However, for critically refracted waves, these vectors can be obtained by a minimal amount of numerical raytracing. The traveltime formulae are useful for a variety of forward modeling and inversion purposes