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

Abstract

We show that the normal operator of the X-ray transform in ${\mathbb{R}}^d$, $d\ge 2$, 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.

arXiv comment21 pages, 4 figures

References in corpus (9)

• Ten equivalent definitions of the fractional Laplace operator
• Uniqueness and reconstruction for the fractional Calderón problem with a single measurement
• The fractional Calderón problem: low regularity and stability
• Unique Continuation for Fractional Schrödinger Equations with Rough Potential
• On higher order extensions for the fractional Laplacian
• The geodesic ray transform on Riemannian surfaces with conjugate points
• Abel transforms with low regularity with applications to X-ray tomography on spherically symmetric manifolds
• Spectral rigidity for spherically symmetric manifolds with boundary
• Journey to the Center of the Earth

Cited by in corpus (6)

• Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems
• The fractional $p$-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems
• Fractional Calderón problems and Poincaré inequalities on unbounded domains
• Unique continuation for the momentum ray transform
• Weighted Divergent Beam Ray Transform: Reconstruction, Unique continuation and Stability
• Partial data problems in scalar and vector field tomography