Waves and rays in anisotropic and inhomogeneous Hookean solids: Mathematical formulation and empirical evaluation

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A theory is necessary to mediate between observations and interpretations; there is no other method to infer quantitatively properties of the Earth's interior from measurements on its surface. To describe materials of the Earth, it is common to use a Hookean solid, which is a mathematical object. Properties of that object are analogies for properties of physical materials.Analogies require subtle considerations. An important part of my research is to examine uncertainties of conclusions in the context of experimental errors and model limitations. Thus, properties of materials are stated in terms of ranges of probable values, whose extent depends on experimental errors and model limitations.Models are simpler than their physical counterparts, and their role is to exhibit pertinent qualities of their physical counterparts. Also, models must accommodate the limited accuracy of measurements. Hence, I examine the models obtained by averaging many parameters to obtain fewer parameters. Establishing the criteria for an optimal choice of a model with fewer parameters is an important part of my research.A model can be considered empirically adequate if it accounts, within accepted limits, for experimental measurements. This, however, can be achieved by many models. Thus, I strive to ensure that the chosen model is an optimal analogy for the physical reality. To account for seismic measurements by Hookean solids, we need to optimize between the effects of anisotropy and inhomogeneity, since the same result can be obtained by different effects. The optimal model choice constitutes an essential part of my research.Theoretical challenges arise in examining seismic phenomena in layered media. I strive to infer information about the deep subsurface from the surface measurements of waves bounded by the surface and an interface. Since two types of waves can propagate in such a model, there are two independent sets of measurements pertaining to the same model, which increases the reliability of physical interpretation.The pragmatic importance of my research is motivated by increased accuracy of available experimental apparatus. This research is important to geoscientists concerned with global and exploration issues. Also, this research is linked to material sciences, which is an important field in Canada, spanning from strength of materials to medical sciences.