Seismic data are commonly modeled by a linearization around a smooth background medium in combination with a high frequency approximation. The perturbation of the medium coefficient is assumed to contain the discontinuities. This leads to two inverse problems, first the linearized inverse problem for the perturbation, and second the estimation of the background, which is a priori unknown (velocity estimation). Here we give a reconstruction formula for the linearized problem using the downward continuation approach. The reconstruction is done microlocally, up to an explicitly given pseudodifferential factor that depends on the aperture. Our main result is a characterization of the wave-equation angle transform, derived from downward continuation, that generates the common image point gathers as an invertible Fourier integral operator, microlocally. We show that the common image point gathers obtained with this particular angle transform are free of so called kinematic artifacts, even in the presence of caustics. The assumption is that the rays in the background that are associated with the reflections due to the medium perturbation are nowhere horizontal. Finally, making use of the mentioned angle transform, pseudodifferential annihilators of the data are constructed. These annihilators detect whether the data are contained in the range of the modeling operator, which is the precise criterion in migration velocity analysis to determine whether a background medium is acceptable, even in the presence of caustics.
- Seismic inversion Microlocal analysis Double-square-root equation