Stability of determination of composite moduli from velocity data in planes of symmetry for weak and strong anisotropies

The determination of elastic constants from ultrasonic bulk wave velocity data taken in a plane of symmetry is studied. It is shown that the problem is well-posed, i.e., the solution is unique and stable. The displacement polarization factors are used to define the degree of material anisotropy and represent solutions of the Christoffel equation. Such a representation in a rotated coordinate system helps to derive simple expressions for coefficients describing the sensitivity of ultrasonic waves to different elastic constants. Thus the optimal refraction angles for velocity measurement can be determined simply. The analysis performed shows that the elastic constants reconstructed from velocity data are not affected by the selection of initial guesses used in a nonlinear least-square optimization (the solution is unique). The solution stability is validated by performing reconstruction with different levels of random scatter in the velocity data. The effect of the angle range used for velocity measurements has been analyzed, and it is shown that, depending on the refraction-angle range, reconstruction of the longitudinal elastic constants is dominated either by the quasilongitudinal or by the quasitransverse velocity. The desirable range of refraction angles for determination of elastic constants with high accuracy is determined.