Analysis of elastic symmetry from velocity measurements with application to dunite and bronzitite

Abstract

SUMMARY A practical quantitative method is presented for determining an appropriate symmetry and coordinate system in which to express the elastic tensor of rocks. While previous methods have been presented for finding appropriate coordinate system from arbitrary elastic tensors, these methods are either qualitative or purely mathematical and have no mechanism for minimizing the effect of errors which occur when elastic tensors are obtained from real data. In order to find an appropriate coordinate system, a complete elastic tensor must be contracted to one of two symmetric second rank tensors. The principal axes of this tensor are the axes of the coordinate system in which the complete elastic tensor is to be expressed. The traces of the various contracted tensors are scalar invariants which are related to Voight and Reuss moduli. One of these traces also yields a new scalar property, a velocity invariant for anisotropic media. This invariant is the sum of the squares of the three phase velocities in three mutually orthogonal directions and is a quantity whose nature is similar to density. Once an appropriate coordinate system has been found, the highest symmetry that will adequately fit the velocity data can be determined statistically. This method is demonstrated with velocity measurements in dunite and bronzitite made by Babuska (1972). Velocity maxima obtained in this paper differ by 13-21 from those obtained petrographically by Babuska (1972). Further, the symmetry of dunite sample and bronzitite sample can be