In reflection seismology one places point sources and point receivers
on the Earth's surface. The source generates acoustic waves in the
subsurface, which are reflected where the medium properties vary
discontinuously. The recorded reflections that can be observed in the
data are used to reconstruct these discontinuities.
Seismic data are commonly modeled by a high-frequency single
scattering approximation. This amounts to a linearization in the
medium coefficients about a smooth background. The discontinuities are
contained in the medium perturbation. Both the smooth background and
the perturbation are in general unknown and have to be reconstructed
jointly.
We describe the wave propagation in the background medium by a one-way
wave or single-square-root equation. Based on this we derive a
variation of the double-square-root (DSR) equation, which is a
first-order pseudodifferential equation that describes the
continuation of seismic data in depth. If the rays in the background
associated with the reflections due to the perturbation are nowhere
horizontal, the singular part of the data is described by the solution
of such an equation. Thus, we consider the modeling operator, its
adjoint (imaging) and left inverse based on the DSR equation in the
framework of microlocal analysis. This left inverse is used to
estimate the singular medium perturbation from the data. We then
construct pseudodifferential annihilators from downward continuation
combined with beamforming. The annihilators establish whether the
seismic data are contained in the range of the modeling
operator. This criterion is exploited in the estimation of the smooth
background medium.
We finally establish a system for going between transmission and
reflection collecting only one (transmission or reflection) data
set. The system incorporates the notion of time-reversal mirrors and
acoustic daylight imaging.
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