Bay Area Microlocal Analysis Seminar: Travel Time Tomography, Boundary Rigidity and Lens Rigidity

Seminar | March 20 | 4-5 p.m. | Stanford University, Room 384I

 Gunther Uhlmann, University of Washington/HKUST

 Department of Mathematics

We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine a Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.

We consider also the partial data case, where you are making measurements on a subset of the boundary.

This is joint work with Plamen Stefanov and Andras Vasy.