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

在观察和解释之间，理论是必要的中介；没有其他方法可以通过对地球表面的测量来定量地推断地球内部的特性。为了描述地球的物质，通常使用胡克固体，这是一个数学对象。物体的性质是物理材料性质的类比。******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限制的背景下检查结论的不确定性。因此，材料的性质是根据可能值的范围来陈述的，其范围取决于实验误差和模型限制。******模型比它们的物理对应物更简单，它们的作用是展示它们的物理对应物的相关特性。此外，模型必须适应测量的有限精度。因此，我检查通过平均许多参数获得的模型，以获得更少的参数。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组成部分。******如果一个模型能够在可接受的范围内解释实验测量，则可以认为它在经验上是充分的。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Hookean固体的地震测量，我们需要在各向异性和非均匀性之间进行优化，因为不同的效果可以获得相同的结果。最优模型选择是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挑战。我努力从以表面和界面为界的波的表面测量中推断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，因此有两组独立的测量值属于同一模型，这增加了物理解释的可靠性。******我的研究的实用重要性是由可用的实验设备的准确性提高所激发的。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.