Modeling of seismic data in the downward continuation approach

Seismic data are commonly modeled by a high-frequency single scattering approximation. This amounts to a linearization in the medium coefficient about a smooth background. The discontinuities are contained in the medium perturbation. The high-frequency part of the wavefield in the background medium is described by a geometrical optics representation. It can also be described by a one-way wave equation. Based on this we derive a downward continuation operator for seismic data. This operator solves a pseudodifferential evolution equation in depth, the so-called double-square-root equation. We consider the modeling operator based on this equation. If the rays in the background that are associated with the reflections due to the perturbation are nowhere horizontal, the singular part of the data is described by the solution to an inhomogeneous double-square-root equation.