In this paper we analyze the seismic inverse problem for acoustic media that allow multipathing. The problem is to determine the acoustic wave speed in the interior of the medium from measurements at the boundary. We make two approximations: first, a high-frequency approximation; secondly, we perform a linearization or Born approximation where the wave speed is written as the sum of low-frequency background c(x) and a high-frequency perturbation ฮดc(x). We analyze the problem of reconstructing the high-frequency component given the low-frequency background.
It is well known that under quite general conditions the linearized forward operator F : ฮดc โ data is a Fourier integral operator (FIO). If the so-called travel-time injectivity condition is satisfied, then the normal operator N=F * F is an invertible pseudo-differential operator and (asymptotically) a least-squares inverse exists.
Our main question is whether F is still invertible if the travel-time injectivity condition is violated. In that case the normal operator is the sum of an invertible pseudo-differential operator and a nonmicrolocal part. We first give conditions when this nonmicrolocal part is an FIO. After that we investigate when the nonmicrolocal part is less singular than the pseudo-differential part, in which case F is still invertible. In the usual imaging method the nonmicrolocal terms would lead to artifacts for which we obtain estimates. We give an example where the nonmicrolocal part is as singular as the pseudo-differential part of N. We show also that in general smooth media the nonmicrolocal part is less singular.