On closest isotropic tensors and their norms

Abstract

Theoretical seismology, which is the subject of the thesis, could be viewed as a subject of continuum mechanics, whose mathematical structure relies on tensors. For instance, Hooke’s Law, which underlies the theory of elasticity—a branch of continuum mechanics—is a tensorial equation. A generally anisotropic tensor, obtained from physical measurements, can be approximated by another tensor belonging to a particular material-symmetry class. This tensor is referred to as the effective tensor; among all tensors in a particular symmetry class, it is the closest to the given anisotropic tensor. This ‘closeness’ that we refer to, draws upon the notion of a norm. In this thesis, we compare the effective tensors belonging to the isotropic symmetry class obtained using three different norms—the Frobenius-36, the Frobenius-21, and the operator norms. Furthermore, we utilize another method—a ‘L₂ slowness-curve fit’ method—and compare the results herein. Finally, we explore the associated errors and analyze the relationship between the mathematical and physical models.