During the past decade, the use of AVO (amplitude variation with offset) analysis in petroleum exploration has become increasingly common. Even though the goal of AVO analysis is to observe the anomalous angle-dependent reflectivity behavior of a reflector, the name amplitude variation with offset was chosen because most of the amplitude analysis is done in the common midpoint domain.
To analyze the properties of the reflector correctly, we need to know the angle-dependent reflectivity, which is often called AVA (amplitude variation with angle). When a target reflector is close to horizontal, and velocity does not change much in the lateral direction, angle-dependent reflectivity can be obtained by measuring amplitudes along the offset, followed by ray tracing for each event to determine the corresponding incidence angle to the reflector. If a target reflector has a complex structure and strong lateral velocity variation, however, AVO may not be the same as AVA because the amplitude can be affected not only by the angle-dependent reflectivity but also by wave focusing or defocusing caused by propagation through complex velocities or structures. Resnick et al. (1987) discuss the fact that dips introduce serious problems for AVO analysis. They conclude that performing prestack migration on the data before doing AVO analysis is a necessity.
Most present-day seismic migration schemes determine only the reflection coefficient of the zero offset or averaged reflectivity over a range of reflection angles for each depth point in the subsurface. This is mainly due to a simplified imaging condition.
Recently, more correct methods for estimating angle-dependent
reflectivity, based on migration algorithms, have been developed.
De Bruin et al. 1990
presented a method for estimating angle-dependent reflectivity
using a scalar prestack 
migration scheme.
First, the multicomponent data were decomposed into
PP, PS, SP, and SS modes and each
of them migrated separately.
Tests with synthetic data showed that the method
was able to retrieve the reflectivity function with considerable
accuracy for horizontal layers,
but when dip was present de et al. (1990),
the retrieved function was incorrectly shifted (in angle)
because of the assumption of a horizontally-layered earth
which is implicit in their imaging principle
(integration along constant Snell parameter traces).
Another limitation of their method is that the estimated
reflectivities do not correspond to the local reflectivities
because the decomposition is carried out in the Fourier domain.
Lumley and Beydoun 1991 introduced a different
method for retrieving the reflectivity function,
using an elastic, prestack Kirchhoff migration scheme.
Their imaging principle was not limited by the horizontal
layer assumption and was successfully applied
to synthetic and real data.
However, as a consequence of the Kirchhoff approach,
their method suffers from the restrictions imposed
by the ray approximation associated with the evaluation
of the Green's function.
Cunha Filho 1992 proposed the local
plane-wave-response imaging after prestack reverse-time migration
McMechan (1983) for the angle-dependent reflectivity recovery.
This approach solves the spatial variability of the
reflectivity by applying the imaging locally
in the space domain instead of the imaging in the wavenumber domain.
However, it requires very expensive elastic forward modelling and
reverse-time migration.
Plane wave synthesis imaging is a promising alternative for the angle-dependent reflectivity recovery because it can be applied to any arbitrary structure and implicitly uses the spatially varying plane-wave-response imaging. In addition to that, the PWS imaging provides incidence-angle of the wavefront along the datum plane where the plane wave is synthesized.
This chapter describes the plane wave synthesis imaging as an tool for the angle-dependent reflectivity recovery. First I describe the forward model of a seismic shot record in detail. Then I explain how the plane wave synthesis imaging recovers the angle-dependent reflectivity. Then I show some examples of angle-dependent reflectivity recovery of each imaging method using synthetic data for comparison.