Analytic Fourier Integral Operators and a Problem from Seismic Inversion

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.