ALBERT

Notes: Ray perturbation theory and the Born approximation have both been used extensively in seismological studies to describe the effects of a slowness perturbation on body and surface wavefields. The relationship between the expressions for the perturbed wavefield calculated using the two methods is investigated here. Using the symplectic symmetry of the ray equations we demonstrate the agreement, in the far field, of the two methods to first order in the slowness perturbation and to leading order in the asymptotic ray series. Thus it is shown that geometrical ray effects, like the traveltime perturbation, ray bending and focusing, are contained within the Born scattering formalism, provided these effects are small. The propagator formalism used to present the results is sufficiently general to include body and surface waves with a smoothly varying inhomogeneous elastic reference medium.

Notes: Reciprocal relationships between the plane-wave reflection/transmission coefficients in anisotropic media are derived directly from the transformed wave equations without use of Betti's theorem. If the eigensolutions are normalized correctly, coefficients with the rǒles of the incident and generated waves reversed are equal, provided the sign of slowness parallel to the interface is also reversed.

Notes: Zeroth-order ray theory is frequently used to calculate synthetic seismograms in media which are both anisotropic and inhomogeneous. One of the principal features of such media is that the polarization vectors of the two quasi-shear (qS) waves are determined by the nature of the anisotropy. Thus, a shear wave entering a region of anisotropy will generally be split into two separate polarizations. Ray theory predicts that these two waves will propagate independently, at different velocities, throughout the anisotropic region. Ray theory solutions also show that in inhomogeneous media, the polarization vectors will rotate along the ray. The rotations of these polarization vectors are strongly influenced by the symmetry and orientation of the anisotropy system, but only weakly depend upon the strength of the anisotropy. In contrast, in isotropic media the polarization of S-waves is determined from the initial conditions and only varies slowly due to the ray curvature. The polarization only changes in the ray direction and at any point does not rotate about the ray.In this paper we show that in the limit of infinitely weak anisotropy, solutions calculated using ray theory in anisotropic media conflict with the known results calculated for a similar isotropic medium. We show this fundamental breakdown in ray theory occurs because coupling between the qS waves is ignored in the zeroth approximation. Thus, the isotropic limit is not equivalent to the high-frequency limit of anisotropic ray theory. The coupling is particularly important in weakly anisotropic media, where the qS velocities are similar, but the same effect is still present in media exhibiting stronger anisotropy. This coupling must be taken into account when calculating waveforms.We show that this coupling may be modelled by treating the ‘error’ terms, produced by substituting a zeroth-order ray theory Green's function into the wave equation, as source terms distributed throughout the medium. For weakly anisotropic media where the qS ray paths are similar, this volume integral may be simplified using perturbation and asymptotic methods and evaluated as a simple integral along the ray path. In the isotropic limit this expression correctly describes the polarization of shear waves along the ray. This integral is easy to compute, requiring only quantities already used in ray tracing and traveltime calculations. A prior knowledge of the location, or even the existence of kiss, intersection, point or other singularities along the ray path, is not required for the method to give accurate results. We present some numerical examples for some simple cases previously investigated by less general or more expensive techniques.

Notes: It is well known that when a seismic wave propagates through an elastic medium with gradients in the parameters which describe it (e.g. slowness and density), energy is scattered from the incident wave generating low-frequency partial reflections. Many approximate solutions to the wave equation, e.g. geometrical ray theory (GRT), Maslov theory and Gaussian beams, do not model these signals. The problem of describing partial reflections in 1-D media has been extensively studied in the seismic literature and considerable progress has been made using iterative techniques based on WKBJ, Airy or Langer type ansätze. In this paper we derive a first-order scattering formalism to describe partial reflections in 3-D media. The correction term describing the scattered energy is developed as a volume integral over terms dependent upon the first spatial derivatives (gradients) of the parameters describing the medium and the solution. The relationship we derive could, in principle, be used as the basis for an iterative scheme but the computational expense, particularly for elastic media, will usually prohibit this approach. The result we obtain is closely related to the usual Born approximation, but differs in that the scattering term is not derived from a perturbation to a background model, but rather from the error in an approximate Green's function. We examine analytically the relationship between the results produced by the new formalism and the usual Born approximation for a medium which has no long-wavelength heterogeneities. We show that in such a case the two methods agree approximately as expected, but that in a media with heterogeneities of all wavelengths the new gradient scattering formalism is superior. We establish analytically the connection between the formalism developed here and the iterative approach based on the WKBJ solution which has been used previously in 1-D media. Numerical examples are shown to illustrate the examples discussed.