Microlocal Analysis of a Seismic Linearized Inverse Problem.

Microlocal analysis of a seismic linearized inverse problem. Abstract: The seismic inverse problem is to determine the wavespeed c(x) in the interior of a medium from measurements at the boundary. In this paper we analyze the linearized inverse problem in general acoustic media. The problem is to nd a left inverse of the linearized forward map F, or, equivalently, to nd the inverse of the normal operator F F. It is well known that in the high frequency approximation the linearized forward map is a Fourier integral operator. If the so called traveltime injec-tivity condition is satissed then the normal operator is an invertible pseu-dodiierential operator. In case this condition is violated the normal operator is the sum of an invertible pseudodiierential operator and a nonlocal part. The normal operator is still asymptotically invertible if the nonlocal part is less singular than the pseudodiierential part. Now there are in general two problems. First the nonlocal part may not be a Fourier integral operator. Second it may be as singular as the pseudodiierential part. We show that both these problems can occur, but that in the generic case they are absent. Acknowledgements: I would like to thank professor J.J. Duistermaat and dr. A.P.E. ten Kroode for their support and the many useful discussions.