Scattering in the Presence of Fold Caustics

Abstract

We consider the linearized scattering operator ${\cal F}$ of acoustics which maps singularities in the sound speed in the subsurface of the earth to singularities in the measured pressure at the earth's surface. The source of the acoustic signal is concentrated at a point on the surface. Application ofthe adjoint scattering operator ${\cal F}^*$ to the measured pressure is a standard technique of locating sound speed singularities. The resulting image is equivalent to application of ${\cal F}^*{\cal F}$ to the sound speed singularities. Provided caustics do not occur, ${\cal F}^*{\cal F}$ is a pseudodifferential operator, meaning sound speed singularities do not move when it is applied and one obtains a reliable image. However, if caustics are present, ${\cal F}^*{\cal F}$ is not well understood. Our main result is that for the case of a simple fold caustic (the most common kind), ${\cal F}^*{\cal F}$ is a singular Fourier integral operator, meaning that some sound speed singularities will be propagated to two places: the first location is the original location and the second location (which is an artifact) is given by the graph of a certain function. The strength of the two transported singularities is equal, so conventional techniques designed to construct a local asymptotic inverse for ${\cal F}^*{\cal F}$ (and hence for ${\cal F}$) fail. Therefore, it is necessary to employ an alternative technique that utilizes data from multiple source locations, resulting in artifact smoothing and thus reinstating conventional techniques as tools for imaging.