5. From Group or Phase Velocities to the General Anisotropic Stiffness Tensor

Abstract

Two numerical inversions were designed to calculate the 21 independent stiffnesses that define, in general, an anisotropic medium from either group-or phasevelocity data. The accuracy, robustness and computational complexity of the two inversion procedures-group velocity to stiffnesses and phase velocity to stiffnesseswere then compared. This 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 group velocities are easier to measure.

Kelvin Notation for Stabilizing Elastic-Constant Inversion

“…We have modified Vestrum's original phasevelocity inversion code (Vestrum et al, 1996;Vestrum, 1994) parameterization imposes no assumptions on the orientation of the anisotropy of the sample. It also naturally provides a series of best-fitting stiffnessmatrix results of progressively lower anisotropic symmetry, allowing the likely symmetry system of the sample to be determined from the data, instead of needing to be assumed beforehand.…”

“…We have replaced the 21-constant parameterization in Vestrum's phase-velocity inversion code (Vestrum et al, 1996;Vestrum, 1994) with our 12-dimensional parameterization. By using only 12 parameters, we explicitly excluded any result less symmetric than arbitrarily oriented orthorhombically anisotropic media.…”

“…If the rock's anisotropic symmetry system and orientation are not known, however, a more general anisotropic inversion method is required. Vestrum (1996Vestrum ( , 1994 demonstrated that by using a large number of high-quality P and S-wave velocity measurements, he could successfully invert for a complete set of 21 elastic constants that best fit the velocity measurements in the least-squares sense. His method makes no prior assumptions about the symmetry or orientation of the rock.…”

Estimation of stress-dependent anisotropy from P-wave measurements on a spherical sample

“…In our case, the active area of the piezoelectric element is 3 mm in diameter, which is 1/17 of the diameter of the sample, and the natural frequency of the transmitter is 2.5 MHz. Following Vestrum (1994), we conclude that the measured velocities are the ray velocities.…”

“…Vestrum (1994); Vestrum et al (1996) demonstrated that by using a large number of high-quality P-and S-wave velocity measurements, he could successfully invert for the complete set of 21 elasticity parameters that best fit the velocity measurements in the least-squares sense. His method makes no prior assumptions about the symmetry or orientation of the rock.…”