Linearization and Microlocal Analysis of Reflection Tomography

Abstract. We examine the linearization of the problem of reflection tomography with data taken to be either the scattering relation or travel time along reflected rays. We obtain coordinate invariant formulae for the Fréchet di↵erentials with respect to perturbations of the wavespeed and reflector locator for the relevant nonlinear maps. The di↵erential with respect to the wavespeed is a system of weighted reflected X-ray transforms and motivated by this we also begin a microlocal study of such transforms. Under a strong condition on the ray geometry we show that the normal operator associated to a weighted reflected X-ray is a pseudodi↵erential operator. As part of this we introduce a new approach to the microlocal analysis of X-ray transforms.