Geometrical ray theory: Rays and traveltimes in seismic systems (second‐order approximations of the traveltimes)

Abstract

In this paper, I establish general laws that represent global properties for the reflected and transmitted rays in seismic systems. The method used for establishing these laws is based on Hamilton’s original ideas; the laws do not depend on the specific seismic system being used. The reflected and transmitted rays, as well as their traveltimes, are described by the same four 2 × 2 matrices. These matrices are interrelated by three equations that provide the basis for the general laws and also provide relations between reflected and transmitted rays that eventually will be exploitable for downward continuation and migration. The traveltimes of reflected and transmitted rays are summed up in single functions, giving the traveltime for any pair of source and receiver positions in a seismic system and corresponding to Hamilton’s point characteristics in optics. The traveltimes of the gathers commonly used in seismic exploration are special cases of the general point characteristics. The consequences from the derived laws are directly available as a priori knowledge before any actual ray tracing and computation of traveltimes are done in any particular seismic system. The laws are also available as a general foundation for future treatments of (1) downward continuation, (2) amplitudes of reflected and transmitted events, (3) true amplitude migration, (4) examples of nonuniqueness of seismic interpretations, (5) focus phenomena, and (6) higher order approximations.