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.

2/5/2001