Reconstruction along a geodesic from sphere data in Finsler geometry and anisotropic elasticity

TL;DR

This paper addresses a geometric inverse problem in Finsler geometry, reconstructing a manifold from sphere data, with applications to seismology and elastic body recovery.

Contribution

It introduces a local solution to the inverse problem of reconstructing a Finsler manifold from sphere data along geodesics, extending previous approaches in geometric inverse problems.

Findings

Successfully reconstructs Finsler manifolds from sphere data locally

Provides a geometric framework for seismic inverse problems

Extends inverse problem solutions to anisotropic elasticity contexts

Abstract

Dix formulated the inverse problem of recovering an elastic body from the measurements of wave fronts of point sources. We geometrize this problem in the context of seismology, leading to the geometrical inverse problem of recovering a Finsler manifold from certain sphere data in a given open subset of the manifold. We solve this problem locally along any geodesic through the measurement set.