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Recovering obstacles from their traveling times
T Gurfinkel1, L Noakes1, L Stoyanov1
1Department of Mathematics and Statistics, University of Western Australia, Crawley 6009, WA, Australia.
Chaos (Woodbury, N.Y.)
|January 1, 2023
Summary
Researchers extended a method to identify convex obstacles on Riemannian surfaces using travel times. This work advances inverse problems in geometry, enabling obstacle reconstruction in more complex spaces.
Area of Science:
- Mathematics
- Differential Geometry
- Inverse Problems
Background:
- The recovery of strictly convex planar obstacles from travel times was established by Noakes and Stoyanov.
- Inverse problems in geometry involve reconstructing objects from indirect measurements.
Purpose of the Study:
- To extend the obstacle recovery method to Riemannian surfaces.
- To explore the impact of curvature conditions on obstacle reconstruction.
Main Methods:
- Adaptation of the Noakes-Stoyanov construction.
- Analysis of geodesic behavior on Riemannian surfaces.
- Consideration of general curvature conditions.
Main Results:
- A generalized method for recovering strictly convex obstacles on Riemannian surfaces.
- Demonstration of the method's applicability under specific curvature constraints.
- Identification of a condition on non-intersecting geodesics.
Conclusions:
- The study successfully extends obstacle recovery from planar to curved surfaces.
- The findings provide a foundation for inverse problems on manifolds.
- Further research can explore more complex obstacle shapes and surface types.
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