Analytic Fourier Integral Operators and a Problem from Seismic Inversion

TL;DR

This paper develops new theoretical tools for analyzing the analytic wavefront set using Fourier integral operators, and applies these to seismic inversion to prove injectivity of a linearized operator.

Contribution

It provides a general result on recovering the analytic wavefront set via elliptic analytic FIOs and constructs an explicit analytic parametrix for second order hyperbolic operators.

Findings

Established a relation between wavefront sets and FIOs under Bolker condition.

Constructed an explicit analytic parametrix for hyperbolic operators.

Proved injectivity of a linearized seismic inversion operator.

Abstract

We establish a general result about the recovery of the analytic wavefront set of a distribution from the analytic wavefront set of its transform coming from a classical elliptic analytic Fourier integral operator (FIO) satisfying some conditions including the Bolker condition. Furthermore, we give a simple explicit analytic parametrix in the form of a classical elliptic analytic FIO for general analytic second order hyperbolic differential operators. Finally, we apply these results together with microlocal analytic continuation and a layer stripping argument to a problem from seismic inversion to prove the injectivity of a linearized operator in the analytic setting. It is the use of wave packet techniques that allows us to give the precise relation how the FIOs under consideration transform the analytic wavefront set, whereas the explicit analytic parametrix comes from a reformulation of the parametrix construction of Hadamard and H\"ormander.