Next: 3D theory
Up: Sava: Image decomposition
Previous: Introduction
The derivation in this section follows the one of
Fomel (1996).
Assuming that a reflection event in the extrapolated wavefield is
described by the function t(z,s,r), we find from the Snell's law
the following derivatives:
 
(1) 
 
(2) 
where s,r stand for the source and receiver spatial coordinates,
v is the wave velocity, is the dip angle, and
is the reflection angle (Figure ).
local
Figure 1 A sketch of reflection rays in an
arbitraryvelocity medium.

 
The traveltime derivative with respect to the depth of the
observation surface z has contributions from the two branches
of the reflected ray, as follows:
 
(3) 
Equation 3 corresponds to the wellknown doublesquareroot
equation Claerbout (1985).
This equation simply reflects the fact that the traveltime
increases with increasing depth of the reflector.
Transforming Equations (13)
to the midpoint and halfoffset coordinates, we obtain
 
(4) 
 
(5) 
 
(6) 
At a fixed image location x,
we can transform the derivatives of t(z,x,h) to the derivatives of
z(t,x,h) by applying the implicit function theorem.
Using Equations (56), we obtain
 
(7) 
and using Equations (46), we obtain
 
(8) 
sls
Figure 2 Angle decompositions.
Panel (a) corresponds to slantstack angledecomposition in the zh plane.
Panel (b) corresponds to slantstack angledecomposition in the zx plane.
In both cases, a dipping segment maps at a particular angle.

 
Sava and Fomel (2003) use Equation (7) to
compute angledomain common image gathers for images obtained by
wavefield extrapolation. This formula can be implemented either
as a slantstack in the space domain, or as a radialtrace transform
in the Fourier domain. In both cases, we obtain at every point
in the image the reflection strength as a function of scattering angle,
independent of the reflector dip.
Similarly, we could employ formula (8) for another
decomposition of the migrated image function of the structural dip
. As for equation (7) which represents a slantstack
in the xh plane, equation (8) represents a slantstack in
the xz plane (Figure ).
mag
Figure 3 A sketch of a generic reflection experiment and
its associated angle decomposition. The left panel corresponds to the
xz location of the reflection point in the image. In the 2D case,
at every point in image, the reflectivity is described by two angles,
and . 
 
Next: 3D theory
Up: Sava: Image decomposition
Previous: Introduction
Stanford Exploration Project
7/8/2003