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2-D theory

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:  
 \begin{displaymath}
{{\partial t} \over {\partial s}} \,=\,
{ {\sin{(\alpha-\gamma)}} \over {v}}\;,\end{displaymath} (1)
 
 \begin{displaymath}
{{\partial t} \over {\partial r}} \,=\, 
{{\sin{(\alpha+\gamma)}} \over {v}}\;,\end{displaymath} (2)
where s,r stand for the source and receiver spatial coordinates, v is the wave velocity, $\alpha$ is the dip angle, and $\gamma$ is the reflection angle (Figure [*]).

 
local
Figure 1
A sketch of reflection rays in an arbitrary-velocity medium.
local
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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:  
 \begin{displaymath}
{{\partial t} \over {\partial z}} \,=\,
{{\cos{(\alpha-\gamma)}} \over {v}} +
{{\cos{(\alpha+\gamma)}} \over {v}}\;.\end{displaymath} (3)
Equation 3 corresponds to the well-known double-square-root equation Claerbout (1985). This equation simply reflects the fact that the traveltime increases with increasing depth of the reflector.

Transforming Equations (1-3) to the midpoint and half-offset coordinates, we obtain  
 \begin{displaymath}
{{\partial t} \over {\partial x}} \,=\, 
{{\partial t} \over...
 ...rtial r}} \,=\, 
{ {2 \sin{\alpha}\,\cos{\gamma}} \over {v}}\;,\end{displaymath} (4)
 
 \begin{displaymath}
{{\partial t} \over {\partial h}} \,=\,
{{\partial t} \over ...
 ...tial s}} \,=\, 
{ {2 \cos{\alpha}\,\sin{\gamma}} \over {v}} \;,\end{displaymath} (5)
 
 \begin{displaymath}
{{\partial t} \over {\partial z}} \,=\,
- {{2 \cos{\alpha} \cos{\gamma}} \over {v}}\;.\end{displaymath} (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 (5-6), we obtain  
 \begin{displaymath}
{{\partial z} \over {\partial h}} \,=\,- {{\partial t} \over...
 ...l h}} /
 {{\partial t} \over {\partial z}} \,=\,- \tan{\gamma},\end{displaymath} (7)
and using Equations (4-6), we obtain  
 \begin{displaymath}
{{\partial z} \over {\partial x}} \,=\,- {{\partial t} \over...
 ...l x}} /
 {{\partial t} \over {\partial z}} \,=\,- \tan{\alpha}.\end{displaymath} (8)

 
sls
Figure 2
Angle decompositions. Panel (a) corresponds to slant-stack angle-decomposition in the z-h plane. Panel (b) corresponds to slant-stack angle-decomposition in the z-x plane. In both cases, a dipping segment maps at a particular angle.

sls
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Sava and Fomel (2003) use Equation (7) to compute angle-domain common image gathers for images obtained by wavefield extrapolation. This formula can be implemented either as a slant-stack in the space domain, or as a radial-trace 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 $\alpha$. As for equation (7) which represents a slant-stack in the x-h plane, equation (8) represents a slant-stack in the x-z plane (Figure [*]).

 
mag
Figure 3
A sketch of a generic reflection experiment and its associated angle decomposition. The left panel corresponds to the x-z location of the reflection point in the image. In the 2-D case, at every point in image, the reflectivity is described by two angles, $\alpha$ and $\gamma$.
mag
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Next: 3-D theory Up: Sava: Image decomposition Previous: Introduction
Stanford Exploration Project
7/8/2003