Travelling times in scattering by obstacles

TL;DR

This paper investigates how to recover obstacle shapes in Euclidean space from measurements of geodesic travel times, establishing conditions for flow conjugacy and providing an algorithm for convex obstacles.

Contribution

It introduces conditions under which obstacle shapes can be uniquely determined from traveling times and offers a constructive algorithm for convex obstacles.

Findings

Obstacles with similar traveling times have conjugate geodesic flows.

A constructive method is provided for convex obstacles in the plane.

The results connect traveling times with the geometric and dynamical properties of obstacles.

Abstract

The paper deals with some problems related to recovering information about an obstacle in an Euclidean space from certain measurements of lengths of generalized geodesics in the exterior of the obstacle. The main result is that if two obstacles satisfy some generic regularity conditions and have (almost) the same traveling times, then the generalized geodesic flows in their exteriors are conjugate on the non-trapping part of their phase spaces with a time preserving conjugacy. In the case of a union of two strictly convex domains in the plane, a constructive algorithm is described to recover the obstacle from traveling times.