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

在我的研究中，我努力提高对地震现象和地球物质属性的定量描述。我的工作属于地球科学背景下的应用数学领域。为了定量描述构成地球的物质，我使用了胡肯固体概念提供的数学模型。这种假想固体的性质，即描述它的数学表达式的性质，用作实际材料的物理性质的类比。正如我通过我的研究证明的那样，Hookean固体表现出八种对称性。这在我的工作中非常重要，因为真实材料的行为可以通过考虑特定对称性的Hookean固体来近似。然而，由于实际材料不是Hookean固体，我们不能说给定的材料属于特定的对称类。我们只能考虑这种材料与特定类别的接近程度。必须引用两个概念来考察这种接近，这是我目前研究的重点。首先，我们必须在Hookean固体的抽象空间中引用“距离”的概念。其次，为了给现场数据提供一种实用的方法，我们必须在实验误差的背景下考虑距离；我们可能会问，数据分辨率是否允许我们区分到两个不同对称性的Hookean固体的距离。为了评估这些问题答案的可靠性，我将使用不确定度分析，这既需要先进的分析方法，也需要高效的计算技术。这项研究的实际重要性是由于最近实验仪器精度的提高，这激发了定量公式的进一步进步。换句话说，获取的数据中的信息比标准技术可以提取的信息更多。在它的直接应用中，我的研究对关心全球和勘探问题的地球科学家来说很重要。它提供有关材料特性的信息以及对其准确性的估计，以增加对基于此类信息的科学和工业决策的信心。

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.