On the stability/sensitivity of recovering velocity fields from boundary measurements
The thesis investigates the stability/sensitivity of the inverse problem of recovering velocity fields in a bounded domain from the boundary measurements. The problem has important applications in geophysics where people are interested in finding the inner structure (the velocity field in the elastic wave models) of earth from measurements on the surface. Two types of measurements are considered. One is the boundary dynamic DirichlettoNeumann map (DDtN) for the wave equation. The other is the restricted Hamiltonian flow induced by the corresponding velocity field at a sufficiently large time and with domain the cosphere bundle of the boundary, or its equivalent form the scattering relation. Relations between these two type of data are explored. Three main results on the stability/sensitivity of the associated inverse problems are obtained: (1). The sensitivity of recovering scattering relations from their associated DDtN maps. (2). The sensitivity of recovering velocity fields from their induced boundary DDtN maps. (3). The stability of recovering velocity fields from their induced Hamiltonian flows.In addition, a stability estimate for the Xray transform in the resence of caustics is established. The Xray transform is introduced by linearizing the operator which maps a velocity field to its corresponding Hamiltonian flow. Microlocal analysis are used to study the Xray transform and conditions on the background velocity field are found to ensure the stability of the inverse transform. The main results suggest that the DDtN map is very insensitive to small perturbations of the velocity field, namely, small perturbations of velocity field can result changes to the DDtN map at the same level of large perturbations. This differs from existing H$\ddot{o}$lder type stability results for the inverse problem in the case when the velocity fields are simple. It gives hint that the methodology of velocity field inversion by DDtN map is inefficient in some sense. On the other hands, the main results recommend the methodology of inversion by Hamiltonian flow (or its equivalence the scattering relation), where the associated inverse problem has Lipschitz type stability.
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 Copyright Status
 In Copyright
 Material Type

Theses
 Authors

Zhang, Hai
 Thesis Advisors

Bao, Gang
 Committee Members

Li, TienYien
Qian, Jianliang
Tang, Moxue
Zhou, Zhengfang
 Date
 2013
 Subjects

Sensitivity theory (Mathematics)
 Program of Study

Mathematics  Doctor of Philosophy
 Degree Level

Doctoral
 Language

English
 Pages
 vi, 83 pages
 ISBN

9781303290152
1303290154
 Permalink
 https://doi.org/doi:10.25335/M55D48