The head-wave transform

2025/09/01 by Maarten V. de Hoop, Antti Kykkänen, de Hoop, Maarten V. +2
Mathematics · #44A12 #44A99 #47G10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2509.01623

openalex publication_date 2025/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce and study a new integral ray transform called the head-wave transform. The head-wave transform integrates a function along a piecewise linear (in general geodesic) path consisting of three parts. The geometry of such paths corresponds to ray paths of head-waves propagating in a medium with sharp changes in sound speed. The middle part of the ray paths corresponds to gliding along the so-called gliding surface. As our main results, we prove inversion formulas and null space characterizations under multiple different sets of assumptions on the geometry of the gliding surface and the integrand function.

Citations

• Weighted Divergent Beam Ray Transform: Reconstruction, Unique continuation and Stability
• Tensor tomography using V-line transforms with vertices restricted to a circle
• V-line 2-tensor tomography in the plane
• Broken ray tensor tomography with one reflecting obstacle
• Inversion of Weighted Divergent Beam and Cone Transforms
• The V-line transform with some generalizations and cone differentiation

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