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

一个理论是必要的调解之间的观察和解释;没有其他方法来定量推断地球内部的性质从其表面上的测量。为了描述地球的物质，通常使用胡克固体，这是一种数学对象。该物体的性质与物理材料的性质类似。**类比需要微妙的考虑。我研究的一个重要部分是在实验误差和模型局限性的背景下研究结论的不确定性。因此，材料的性质是用可能值的范围来表示的，其范围取决于实验误差和模型限制。模型比物理模型简单，它们的作用是展示物理模型的相关品质。此外，模型必须适应测量的有限准确性。因此，我研究了通过平均许多参数以获得更少参数而获得的模型。建立具有较少参数的模型的最优选择标准是我研究的重要部分。如果一个模型在可接受的限度内考虑了实验测量，则可以认为它在经验上是适当的。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物理现实的最佳类比。为了解释胡克固体的地震测量，我们需要在各向异性和不均匀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。最佳模型选择构成了我研究的重要组成部分。*在研究层状介质中的地震现象时出现了理论上的挑战。我努力推断信息的深层地下从表面测量波的表面和界面的边界。由于两种类型的波可以在这样的模型中传播，因此同一模型有两组独立的测量，这增加了物理解释的可靠性。我的研究的实用性的重要性是由可用的实验仪器的准确性增加的动机。这项研究对关注全球和勘探问题的地球科学家很重要。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.