Generalised ray parameters for vertically inhomogeneous and anisotropic media

Abstract

In this presentation we derive a concise equation for a generalised ray parameter in inhomogeneous and anisotropic media. We illustrate the direct application of this conserved quantity by several examples, involving anisotropic parameters and linear velocity fields. The ray parameter, or raypath parameter, p, is a conserved quantity which plays an important role in both exploration and global seismology. The constancy of the ray parameter contributes to convenient methods of raytracing and imaging. The standard form of p in a homogeneous isotropic medium is the horizontal component of the slowless, expressed in terms of the angle measured from the normal to the interface between media,q, and the velocity, v. This ray parameter, p, is a conserved quantity, ie the first integral of the Euler-Lagrange equation. Increased interest in anisotropic characteristics of sedimentary rocks motivated this work, which provides an exact form of the ray parameter for vertically inhomogeneous and anisotropic media. We use the mathematical tools provided by calculus of variations, in particular the Euler-Lagrange equation with its first integrals, to arrive at this new form of the ray parameter. There always exists a conserved quantity, such as the ray parameter for arbitrarily complex velocity fields. For exploration seismology in sedimentary basins, a relevant form would account for vertical inhomogeneity and anisotropy. In such a case, the ray parameter resembles the standard form mentioned above with an additional term due to the anisotropy. The results allow for convenient modelling and raytracing. They permit certain investigations of the influence of anisotropy on ray trajectories and traveltimes, which play an important role in AVO analysis. Furthermore, presented results could be incorporated in data-processing applications.