Inverse problems from geophysics and transport theory, and applications

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

Seismic tomography plays a central role in our understanding of the substructure of the Earth. The analysis of various seismic data produced by natural earthquakes or artificial seismic sources has important applications in practice, such as characterizing fractured bedrock, and searching for oil and gas deposits. The study of seismology also has a close connection with the transport theory in classical mechanics, which models the behavior of a large number of particles. An essential question in transport theory is to recover the hidden properties of the particles and medium from various physical measurements. This arises in a wide range of applications, including medical imaging, optical tomography, remote sensing, seismology and atmospheric science. This project will address both the theoretical foundations and applications of important challenges arising in seismic tomography and transport theory. The project will provide training opportunities for graduate students, especially those from underrepresented groups. This project aims to address the applied analysis of several linear and non-linear inverse problems. It contains two major lines of research. The first topic is on the travel time tomography arising in geophysics, which consists of reconstructing seismic sound speed from the travel time of seismic waves propagating through the Earth. The goal is to study the uniqueness and stability of the travel time tomography in anisotropic elasticity, which is essentially the boundary rigidity problem in Finsler geometry. The investigator will also address the uncertainty quantification of the Bayesian inversion method for travel time tomography as well as carry out numerical experiments. The second topic addresses inverse problems for time-dependent transport equations, which concerns the recovery of time-independent or time-dependent coefficients or sources inside a bounded domain from the boundary measurements of the solution to the transport equation. The investigator will study both the theoretical aspects, including the uniqueness and stability estimates, and the applied aspects, such as the reconstruction methods and numerical implementations. The outcomes of the project are likely to lead to new developments on related research topics and techniques, both inside and outside the mathematical community. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.