The application of traveltime inverse in linearly inhomogeneous and elliptically anisotropic media

Kudela, Izabela (2019) The application of traveltime inverse in linearly inhomogeneous and elliptically anisotropic media. Masters thesis, Memorial University of Newfoundland.

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Abstract

We use ray methods for seismological investigations that allow us to address a variety of problems by applying several mathematical tools. With the assumption of elliptical anisotropy and linear inhomogeneity, it results in simplifications of the equations; thereby, it allows us to obtain analytic expressions for the trajectories of seismic signals, as well as their traveltimes. Traveltimes are essential for processing and velocity determination from VSP surveys. To solve the inverse problem, we minimize the difference between the traveltimes calculated from the model and traveltimes from the dataset. We ensure that for a single medium, the inverse is stable and gives unique solutions. For the more complex problems with many parameters, we examine the stability of the inversion. Herein, we investigate one particular case of eight symmetry classes that material described by a Hookean solid can possess. We consider a particular case of the trans- versely isotropic (TI) medium. An accurate enough analogy of a seismic performance with a model of a TI Hookean solid is consistent with parallel layers that serve as an analogy for the behaviour of waves in a sedimentary basin. For such medium, we use the Backus average of a stack of isotropic layers that results in a TI medium. We investigate whether we may encounter the issue with numerical considerations of the Backus (1962) product approximation. For isotropic media, despite the stability conditions, it may lead to an issue within the Backus average. In the context of global seismology, we examine whether this issue may occur. Furthermore, we discuss several physical constraints imposed on elasticity parameters of a TI tensor. Using the assumption of the equal level of increasing velocity of P- and S-waves and the same anisotropy from abχ model and TI medium, we obtain two out of five elasticity parameters of a TI tensor. For different elasticity parameters, we show several numerical examples to examine how these restrictions affect a TI tensor with known values of certain elasticity constants that are acquired from the vertical or horizontal measurements.

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