From Group or Phase Velocities to the General Anisotropic Stiffness Tensor
Robert W. Vestrum, R. James Brown, Donald T. Easley, 1996. "From Group or Phase Velocities to the General Anisotropic Stiffness Tensor", Seismic Anisotropy, Erling Fjær, Rune M. Holt, Jaswant S. Rathore
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Two numerical inversions were designed to calculate the 21 independent stiffnesses that define, in general, an anisotropic medium from either group- or phase-velocity data. The accuracy, robustness and computational complexity of the two inversion procedures — group velocity to stiffnesses and phase velocity to stiffnesses — were then compared. The group-velocity inversion overcomes the difficulty of calculating group velocity in a prescribed direction and can calculate group velocities accurately even in directions near shear-wave singularities. Although phase velocities are easier to calculate than group velocities, the group-velocity inversion performed better in laboratory tests because it is easier to acquire measurements of group velocities in many different directions.
The group-velocity inversion for the general anisotropic tensor yielded the best results in its inversion of the 99 velocity data-points from a sphere of phenolic. The smallest velocity error (7.7 m/s) and the least average statistical uncertainty in the stiffnesses (0.04 GPa) came out of this application of the algorithm. The superior performance of this inversion is attributed to the freedom that the experimenter has to make as many measurements in whatever directions as are desirable, without having to cut the sample so that plane waves may be generated.
In contrast to the group-velocity method, the phase-velocity inversion was the simplest, most robust and most accurate method in the theory and numerical testing but did not perform as well when faced with laboratory data. The drawback to the application of this method appears to be the limitation of inverting only plane-wave velocities, which limits the number of measurements that can be made. The model velocities fit the observed velocities with a statistical error of 18 m/s resulting in an average uncertainty in the stiffnesses of 0.16 GPa, statistical errors that are substantially higher than those for the group-velocity inversion. Despite this higher statistical error due to fewer measurements, the errors from the phase-velocity inversion are within the uncertainty estimates for the laboratory measurements.