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Scalar and vector tomography for the weighed transport equation
Thompson, Nathan L.
Thompson, Nathan L.
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dissertation
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2018-05
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Electronic dissertations
Electronic dissertations
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Abstract
Beginning with the weighted transport equation,
P u(x, ω) = hω, ∇xu(x, ω)i + µ(x)u(x, ω) = ρ(x, ω)a(x), x ∈ ω ⊂ R
2
,
we examine the properties of acoustic waves that travel below the surface of a solid, such
as the Earth or the Sun. According to the laws of geometric optics, these acoustic signals
travel along ray paths which obey Fermat’s principle of least time. Every material has an
optimal path and travel time and deviations from this travel time reveal features of the
material hidden within. These optimal paths are encoded into our transport equation by
the weight function ρ(x, ω). Using the Fourier series decomposition of u and ρ with respect
to ω, we examine several approximations to an arbitrary weight function and solve for the
perturbations to the expected travel time in both the scalar and vector tomography cases.
Our results show that the solutions to the resulting systems of differential equations are
unique and stable.
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Thesis (Ph.D.)-- Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics and Physics
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Wichita State University
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Copyright 2018 by nathan Lee Thompson
All Rights Reserved