In this thesis, we study the computation of high frequency wave and the related inverse problems based on eikonal equations. Particularly this thesis focuses on the first arrival transmission traveltime tomography, the joint transmission-reflection traveltime tomography and the multi-valued solutions of eikonal equations. We propose Eulerian methods on these subjects, which adopt Cartesian coordinates and yield uniformly distributed solutions.

For the first arrival transmission traveltime tomography, we propose a level set adjoint state method to solve the problems where the underlying slowness is piecewise continuous. The level set method is incorporated to implicitly represent the discontinuity in the slowness. Then this level set adjoint state method is extended to study the...[ Read more ]

In this thesis, we study the computation of high frequency wave and the related inverse problems based on eikonal equations. Particularly this thesis focuses on the first arrival transmission traveltime tomography, the joint transmission-reflection traveltime tomography and the multi-valued solutions of eikonal equations. We propose Eulerian methods on these subjects, which adopt Cartesian coordinates and yield uniformly distributed solutions.

For the first arrival transmission traveltime tomography, we propose a level set adjoint state method to solve the problems where the underlying slowness is piecewise continuous. The level set method is incorporated to implicitly represent the discontinuity in the slowness. Then this level set adjoint state method is extended to study the joint transmission-reflection traveltime tomography, where the reflection part is carefully handled in the algorithm; also we introduce an easily computed quantity which can quantify the reliability of our reconstruction. In addition, we propose an Eulerian labeling method to compute the multi-valued solutions of eikonal equations. Our method is based on the idea of the shock splitting method and we improve the solutions in inhomogeneous medium.