Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 4

Special Issue in honor of Victor Guillemin

Guest Editors: Yael Karshon, Richard Melrose, Gunther Uhlmann, and Alejandro Uribe

Seismic imaging with generalized Radon transforms: stability of the Bolker condition

Pages: 1985 – 2036

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n4.a11

Authors

Peer Christian Kunstmann (Department of Mathematics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany)

Eric Todd Quinto (Department of Mathematics, Tufts University, Medford, Massachusetts, U.S.A.)

Andreas Rieder (Department of Mathematics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany)

Abstract

Generalized Radon transforms are Fourier integral operators which are used, for instance, as imaging models in geophysical exploration. They appear naturally when linearizing about a known background compression wave speed. In this work we first consider a linearly increasing background velocity in two spatial dimensions. We verify the Bolker condition for the zero-offset scanning geometry and provide meaningful arguments for it to hold even if the common offset is positive. Based on this result we suggest an imaging operator for which we calculate the top order symbol in the zero-offset case to study how it maps singularities. Second, to support the usage of background models obtained from linear regression we present a stability result for the Bolker condition under perturbations of the background velocity and of the offset.

Keywords

generalized Radon transforms, Fourier integral operators, microlocal analysis, seismic imaging

2010 Mathematics Subject Classification

35S30, 44A12, 58J40, 86A22

Full Text (PDF format)

P.C.K. was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Project-ID 258734477-SFB1173.

W.T.Q. was partially funded by the DFG Project-ID 258734477-SFB1173; by U.S. NSF grant DMS 1712207; and by Simons Foundation award 708556.

A.R. was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 258734477-SFB1173.

Received 15 December 2021

Received revised 17 April 2022

Accepted 3 May 2022

Published 20 November 2023