Unique continuation of the normal operator of the X-ray transform and applications in geophysics

TL;DR

This paper proves a unique continuation property for the normal operator of the X-ray transform in higher dimensions, with applications to partial data tomography and seismology.

Contribution

It establishes a new unique continuation result for the normal operator of the X-ray transform in $\\mathbb{R}^d$, extending previous work to higher dimensions and applications.

Findings

Unique continuation property for the normal operator of the X-ray transform.

Implications for partial data X-ray tomography.

Applications in seismology, including travel time and shear wave tomography.

Abstract

We show that the normal operator of the X-ray transform in $\mathbb{R}^d$, $d\geq 2$, has a unique continuation property in the class of compactly supported distributions. This immediately implies uniqueness for the X-ray tomography problem with partial data and generalizes some earlier results to higher dimensions. Our proof also gives a unique continuation property for certain Riesz potentials in the space of rapidly decreasing distributions. We present applications to local and global seismology. These include linearized travel time tomography with half-local data and global tomography based on shear wave splitting in a weakly anisotropic elastic medium.