Computation of qP-wave rays, traveltimes and slowness vectors in orthorhombic media

Abstract

The eikonal equation is the equation of the phase slowness surface for isotropic and anisotropic media. In general anisotropic media, there is no simple explicit expression for the phase slowness surface. An approximate expression of the eikonal equation may be obtained in weakly anisotropic media. In orthorhombic media, the approximate eikonal equation of the qP wave is the sum of an ellipsoidal form and a more complicated term. The ellipsoidal form corresponds to what we call ellipsoidal anisotropy. Ray equations written in the Hamiltonian formulation are characteristics of the eikonal equation. Ray perturbation theory may be used to compute changes in ray paths and physical attributes (traveltime, polarization, amplitude) due to changes in the medium with respect to a reference medium. Examples obtained in homogeneous orthorhombic media show that a reference medium with ellipsoidal anisotropy is a better choice to develop the perturbation approach than an isotropic reference medium. Models with strong anisotropy can be considered. The comparison with results obtained by an exact ray program shows a relative traveltime error of less than 0.5 per cent for a model with relatively strong anisotropy. We propose a finite element approach in which the medium is divided into a set of elements with polynomial elastic parameter distributions. Inside each element, using a perturbation approach, analytical expressions for rays and traveltimes are obtained. Ray tracing reduces to connecting these analytical solutions at the vertices of the cells.