Distance in the space of elasticity tensors: optimal choices of Hookean models

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In my research, I strive to enhance the quantitative description of seismic phenomena and material properties of the Earth. My work belongs to the realm of applied mathematics in the context of geosciences. To describe quantitatively materials that constitute the Earth, I use a mathematical model provided by the concept of a Hookean solid. Properties of such a hypothetical solid, which are properties of mathematical expressions that describe it, serve as analogies for physical properties of actual materials. As I have demonstrated through my research, Hookean solids exhibit eight symmetry classes. This is of paramount importance in my work, since the behaviour of a real material can be approximated by considering a Hookean solid of a particular symmetry. However, since real materials are not Hookean solids, we cannot say that a given material belongs to a particular symmetry class. We can consider only the proximity of this material to a particular class. Two concepts must be invoked to examine such a proximity, which is the focus of my present research. First, we must invoke a concept of "distance" within the abstract space of Hookean solids. Second, to provide a pragmatic methodology for field data, we have to consider distance in the context of experimental errors; we might ask whether or not the data resolution allows us to distinguish between distances to Hookean solids of two distinct symmetries. To evaluate the reliability of answers to such questions, I will use uncertainty analysis, which requires both advanced analytical methods and efficient computational techniques. The pragmatic importance of this research is motivated by recent increase of accuracy in experimental apparatus that inspires further advancements of quantitative formulations. In other words, there is more information in acquired data than can be extracted by standard techniques. In its immediate application, my research is important to geoscientists concerned with both global and exploration issues. It provides information on material properties with estimates of their accuracy to increase confidence in both scientific and industrial decisions based on such information.