Estimating effective elasticity tensors from Christoffel equations

We consider the problem of obtaining the orientation and elasticity parameters of an effective tensor of particular symmetry that corresponds to measurable traveltime and polarization quantities. These quantities — the wavefront-slowness and polarization vectors — are used in the Christoffel equation, a characteristic equation of the elastodynamic equation that brings seismic concepts to our formulation and relates experimental data to the elasticity tensor. To obtain an effective tensor of particular symmetry, we do not assume its orientation; thus, the regression using the residuals of the Christoffel equation results in a nonlinear optimization problem. We find the absolute extremum and, to avoid numerical instability of a global search, obtain an accurate initial guess using the tensor of given symmetry closest to the generally anisotropic tensor obtained from data by linear regression. The issue is twofold. First, finding the closest tensor of particular symmetry without assuming its orientation is challenging. Second, the closest tensor is not the effective tensor in the sense of regression because the process of finding it carries neither seismic concepts nor statistical information; rather, it relies on an abstract norm in the space of elasticity tensors. To include seismic concepts and statistical information, we distinguish between the closest tensor of particular symmetry and the effective one; the former is the initial guess to search for the latter.