  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 wave solutions in the background medium admit a geometrical
optics representation. Here we describe the wave propagation in the background
medium by a one-way wave equation. Based on this we derive the
double-square-root equation, which is a first order pseudodifferential
equation, that describes the continuation of seismic data in depth. We consider
the modeling operator, its adjoint and reconstruction 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. We
derive a microlocal reconstruction equation. The main result is a
characterization of the angle transform that generates the common image point
gathers, and a proof that this transform contains no artifacts. Finally,
pseudodifferential annihilators based on the double-square-root equation are
constructed. The double-square-root equation approach is used in seismic data
processing.
