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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.

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|January 1, 2023
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Summary
This summary is machine-generated.

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.

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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.