A theoretical analysis of outgoing isotropic time-harmonic linear elasticity in a half-space with traction-free boundary

Abstract

Linear elastic wave propagation in a half-space is of fundamental interest in geophysics and has many applications in inverse problems, including imaging the Earth’s mantle, sub-seabed mineral exploration, and cancer screening. The work in this thesis seeks to create a rigorous foundation for future work in the field by establishing existence, uniqueness, and regularity results for outgoing linear elastic waves in a homogeneous half-space. We complete a new derivation of the far-field asymptotic behavior of the outgoing Green’s function and induced solutions, and thereby obtain an explicit radiation condition. We build on the work of previous authors to derive asymptotic error bounds that are proven to be uniform in the polar angle. Previous work has only demonstrated bounds that hold pointwise almost everywhere. We show that there is a critical angle at which the error term is larger than at other angles. We provide tools to construct a far-field asymptotic series near this critical angle. We extend the uniqueness proof of Durán, Muga, and Nédélec from the 2-D case to 3-D, and we refine their method. We prove optimal regularity results up to the boundary through spectral analysis of solutions induced by the outgoing Green’s function and general compactly supported forcing. We expect that the results of our asymptotic analysis will inform refinements to some numerical methods, particularly those that utilize the Dirichlet-to-Neumann map, the far-field Green’s function, or fast volume integrals to implement a non-reflecting computational boundary.