The elastic ray transform

TL;DR

This paper introduces a new class of tensor tomography problems related to elastic wave travel times, providing kernel characterizations and mathematical analysis using Fourier, Helmholtz, and cohomology methods.

Contribution

It defines and analyzes a novel family of tensor tomography problems at rank 2, with unique kernel characterizations involving degree 2 differential operators.

Findings

Kernel characterized by potential tensors with degree 2 operators

Provides mathematical framework for elastic wave travel time linearization

Uses Fourier analysis, Helmholtz decompositions, and cohomology techniques

Abstract

We introduce and study a new family of tensor tomography problems. At rank 2 it corresponds to linearization of travel time of elastic waves, measured for all polarizations. We provide a kernel characterization for ranks up to 2. The kernels consist of potential tensors, but in an unusual sense: the associated differential operators have degree 2 instead of the familiar 1. The proofs are based on Fourier analysis, Helmholtz decompositions, and cohomology.