Inversion of seismic-type Radon transforms on the plane

TL;DR

This paper investigates the inversion of Radon-type transforms on the plane, specifically those integrating functions over symmetric curves like parabolas and hyperbolas, with applications in seismology.

Contribution

It provides new inversion formulas for Radon transforms over symmetric plane curves, including parabolic and hyperbolic cases, under certain symmetry and vanishing conditions.

Findings

Derived inversion formulas for Radon transforms on symmetric curves

Applicable to transforms used in seismology

Established conditions for invertibility

Abstract

We study integral transforms mapping a function on the Euclidean plane to the family of its integration on plane curves, that is, a function of plane curves. The plane curves we consider in the present paper are given by the graphs of functions with a fixed axis of the independent variable, and are imposed some symmetry with respect to the axes. These transforms contain the parabolic Radon transform and the hyperbolic Radon transform arising from seismology. We prove the inversion formulas for these transforms under some vanishing and symmetry conditions of functions.