Seismic rays in media with slight lateral variation in velocity

SUMMARY In this paper, the method of small perturbations is applied to the ray and energy transport equations in an investigation of the effect on the propagation of seismic rays through a layer of fixed thickness of having weak lateral inhomogeneities superimposed on an unperturbed velocity field which varies with depth only. Both plane and spherical geometries are studied. This more fundamental approach highlights inadequacies of the more commonly used Hamiltonian method. In the course of calculations, simple expressions are also obtained for the various components of the propagator matrices for rays in vertically and radially heterogeneous media; these are useful in the study of Gaussian beams and paraxial ray theory in such media and require the calculation of just two integrals along the unperturbed ray path. In particular, these integrals are evaluated analytically for the special case of weak lateral velocity inhomogeneities superimposed on a constant gradient of quadratic slowness. It is found that the perturbation to ray geometry may include a component directed along the ray at the point of observation without violating Fermat's principle and that this may be neglected only when slowness does not vary greatly along the unperturbed ray path. Such perturbations are usually ignored in paraxial ray theory. Expressions are also obtained for the first‐order corrections to traveltime, amplitude, surface slowness and polarization in such media; the approach used in deriving these expressions gives insight into the various complicating factors which need to be considered in inverse modelling problems involving such media.