Anisotropic Traveltime Tomography

Abstract

Velocity estimation technique using seismic data is often based on time/distance equations which are independent of direction, and even though we now know that many rocks are quite anisotropic, useful results have been obtained over the years from these isotropic estimates. Nevertheless, if velocities are significantly direction-dependent, then the isotropic assumption may lead to serious structural interpretation errors. Additionally, information on angle-dependence may lead to a better understanding of the lithology of the rocks under measurement. VSP and cross-well data may each lack the necessary aperture to estimate more than two velocity parameters for each wave type, and if the data straddle a symmetry axis, then these may be usefully chosen to be the direct velocities (from time-and-distance measurements along the axis) and NMO velocities (from differential time-offset measurements). These sets of two parameters define ellipses, and provide intermediate models for the variation of velocity with angle which can later be assembled and translated into estimates of the elastic moduli of the rocks under scrutiny.If the aperture of the measurements is large enough e.g. we have access to both VSP and cross-well data, we divide the procedure into two independent steps, first fitting best ellipses around one symmetry axis and then fitting another set around the orthogonal axis. These sets of four elliptical parameters are then combined into a new, double elliptical approximation. This approximation keeps the useful properties of elliptical anisotropy, in particular the simple relation between group and phase velocities which simplifies the route from the traveltimes measurements to the elastic constants of the medium.The inversion proposed in this paper is a simple extension of well-known isotropic schemes and it is conceptually identical for all wave types. Examples are shown to illustrate the application of the technique to cross-well synthetic and field P-wave data. The examples demonstrate three important points: (a) When velocity anisotropy is estimated by iterative Paper presented in two parts at the 53rd EAEG meeting, Florence, techniques such as conjugate gradients, early termination of the iterations may produce artificial anisotropy. (b) Different components of the velocity are subject to different type of artifacts because of differences in ray coverage. (c) Even though most rocks do not have elliptical dispersion relations, our elliptical schemes represent a useful intermediate step in the full characterization of the elastic properties.

A generalized two point ellipsoidal anisotropic ray tracing for converted waves

“…Moreover, the unknown parameters of the problem (6) are only the vectors X and Y since all the components of vector Z can be expressed as a function of the components of X and Y . In the next section, we will obtain a more general equation for the travel time function, when the axis of symmetry is not vertical, as Michelena (1993) presents for a 2-D transversely isotropic media.…”

“…Then we extend the 2-D approach used by Michelena (1993) to obtain a 3-D ellipsoidal expression for the group slowness and the travel time function.…”

“…which correspond to the travel time in 2-D homogeneous elliptically anisotropic media with tilted axis of symmetry given by Michelena (1993) for a 2-D transversely isotropic medium. 3.…”

“…Pratt and Chapman (1992) and Chapman and Pratt (1992) present a scheme that traces rays in isotropic media and then project them in anisotropic media. Michelena et al (1993) and Michelena (1993) present a 2-D elliptical approximation of the phase velocity and trace rays in 2-D heterogeneous transversely isotropic media. More recently, Grechka and McMechan (1996) present an optimization technique for tracing rays in transversely isotropic media which consider the Thomsen weak anisotropic parameters (Thomsen 1986).…”

Assessment of Complex Masonry Structures with GPR Compared to Other Non-Destructive Testing Studies

“…In the case of non-homogeneous media, the seismic wave is refracted, because of changes in the wave velocity associated with adjacent cells, and equations are solved in an iterative computational process until convergence of the solution. The computational process, the simultaneous interactive reconstruction technique (SIRT), contemplates ray curvature as a consequence of internal refractions ( [22,23]). The characteristics of each cell are defined in the case that at least one ray path crosses the cell.…”