Inversion of the seismic parabolic Radon transform and the seismic hyperbolic Radon transform

Abstract

Reflection seismology is a method of exploration of the hidden structure of the earth subsurface by processing received seismograms. For a long time, this processing utilizes the so-called slant stack transform, or line Radon transform. More recently, two generalizations of the slant stack transform have been introduced to extract new features of the seismic data and to improve their treatment. These transforms are the parabolic and hyperbolic seismic Radon transforms. The first transform maps a given function to its integrals over parabolas with a fixed axis direction, whereas the second one maps a function to its integrals over hyperbolas (more generally also over ellipses and circles) of fixed axis directions. We show how they can be converted to a line Radon transform, and thereby obtain their inversion formulas. Numerical simulations for each transform were performed and commented to illustrate the suggested algorithms.