Propagation of Singularities and Some Inverse Problems in Wave Propagation
RICE UNIV HOUSTON TX DEPT OF MATHEMATICAL SCIENCES
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We review a number of results relating the propagation of singularities for hyperbolic partial differential equations i.e. the persistence, or non-localization, of wave motion with well-posedness for some inverse problems of reflection type, such as arise for instance in seismology and ultrasonics. By far the most complete information is available for layered problems. We show how a simple but refined propagation-of-singularities theorem, with estimates, yields important functional properties of the model-data relationship for such problems, including regularity in various useful coefficient classes, separation of scales,....We explain the essential role of travel time in the study of these problems, and show how its function may be generalized to multidimensional i.e. non-layered problems.
- Theoretical Mathematics