Scalar and vector tomography for the weighed transport equation

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.