Formally determined inverse problems for hyperbolic PDEs

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In geophysics one is interested in recovering the acoustic properties of a three dimensional medium from the medium response when probed by an acoustic wave. The property of the medium is modeled by a function of three variables, loosely called the potential, and the acoustic wave satisfies the wave equation with the zeroth order term coefficient being the potential. The PI proposes studying two specific problems. In the first problem, the medium is probed by a plane wave and the medium response is measured in the same direction as the incoming wave, over a long enough time interval. This data is measured for incoming waves from all possible directions (the back-scattering data) and the goal is to recover the potential. The PI proposes studying the special case when the potential is analytic in the angular variables, by adapting Volterra type methods and using scales of Banach spaces - an idea used by Romanov on a different inverse problem. The PI proposes also using a time domain interpretation of the back-scattering problem which he has already used successfully to obtain new uniqueness results for certain restricted classes of potentials. In the second problem, the medium is excited by a point source placed at the center of the medium, the medium response is measured at the boundary for a long enough time period, and the goal, again, is the recovery of the potential. There are one dimensional versions of this problem, even with interior measurements, which remain unsolved with the chief difficulty being that the response is measured at points away from the source so that Volterra type methods such as layer stripping or downward continuation are not applicable. The PI proposes studying these problems with the help of Carleman estimates using results of Ionescu and Klainerman which give easily verifiable conditions for constructing radial Carleman weights.This work will be of interest to people working in geophysics, oil prospecting, medical imaging, fiber-optics, and in any area where one wishes to determine the properties of the interior of a medium from the response of the medium, measured on the boundary, to an acoustic wave generated at the boundary. The PI will supervise graduate students who will work on these problems towards a PhD and possible careers in the energy industry, the medical devices industry or in academia. Undergraduate students may study discrete versions of these problems in one space dimension. The results of this work will be disseminated through graduate seminars, publications in research journals, and presentations at conferences and introductory workshops for non-specialists.