Three-dimensional traveltime equation for dipping layers

The traveltime equation for homogeneous layers with two-dimensionally dipping interfaces is well known, but it has been proven only for a limited variety of profile geometries. Existing proofs do not necessarily lend themselves to use with other types of profiles, or to extension into three dimensions.When the distance that a ray travels and the elapsed time are decomposed into orthogonal components, and each interaction of the ray and the interface is examined separately, the time contributions for ray travel in the two orthogonal directions can be isolated for ray segments between horizontal planes of reference. The terms in which these contributions are expressed are independent of any part of the structure beyond the ray segment under consideration. Therefore, the total traveltime for a ray can be obtained by simple summation of such contributions. This process represents a new proof of the traveltime equation, but, more importantly, it provides a method for constructing the correct equation for virtually any profile geometry and arrival type, including direct, reflected, and refracted waves as well as converted phases.When the rays and interface normals are treated as three-dimensional vectors, this proof and method can be extended to three dimensions. The result is a general expression describing point-to-point traveltimes for any source-receiver geometry in structures comprising homogeneous layers with interfaces having arbitrary three-dimensional dip. Since this equation accurately describes the incremental traveltime for every portion of any propagating wave, nearly any three-dimensional geometry can be accommodated.