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

在观察和解释之间调停需要一个理论；没有其他方法可以从地球表面的测量结果中定量地推断地球内部的性质。用胡肯固体来描述地球的物质是很常见的，它是一个数学对象。该物体的性质是对物理材料性质的类比。类比需要细微的考虑。我研究的一个重要部分是在实验误差和模型限制的背景下检查结论的不确定性。因此，材料的性质是用概率值的范围来表示的，其程度取决于实验误差和模型限制。模型比它们的物理对应物简单，它们的作用是展示它们的物理对应物的相关性质。此外，模型还必须适应有限的测量精度。因此，我检查了通过对多个参数进行平均以获得较少参数而获得的模型。为参数较少的模型的最佳选择建立标准是我研究的一个重要部分。如果一个模型在可接受的范围内考虑到了实验测量，则可以认为该模型在经验上是足够的。然而，这可以通过许多模型来实现。因此，我努力确保所选择的模型是对物理现实的最佳类比。为了解释Hookean固体的地震测量，我们需要在各向异性和非均质性的影响之间进行优化，因为不同的影响可以得到相同的结果。最优模型的选择是我研究的重要部分。在研究层状介质中的地震现象时，出现了理论上的挑战。我努力从表面和界面所界定的波的表面测量中推断出有关深层地下的信息。由于两种类型的波可以在这样的模型中传播，同一模型有两组独立的测量数据，这增加了物理解释的可靠性。我的研究的实用重要性是由于现有实验设备的精度提高所致。这项研究对关注全球和勘探问题的地球科学家具有重要意义。此外，这项研究与材料科学有关，这是加拿大的一个重要领域，从材料强度到医学科学。

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.