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Another way to find the relation between z and z' starts from
the cascade of two dispersion relations Biondi and Palacharla (1996).
The first one is for 2D prestack downwardcontinuation along
the inline direction,
 
(15) 
which is equivalent to equation (11).
The second one is for 2D zerooffset downward continuation along
the crossline axis,
 
(16) 
which is equivalent to equation (12). Indeed, by taking
the square and dividing by k_{z},
 
(17) 
and by introducing ,
 
(18) 
BThe coplanarity condition presented in Biondi and Palacharla (1996) is:
 
(19) 
Each square root in the previous equation can be substituted with the
expression including the dip angle of the source or the receiver ray
 
(20) 
 (21) 
where and can be expressed in terms of the aperture
angle and the normal dip angle :
 
(22) 
Or, after applying some trigonometric relations
 
(23) 
We now introduce equation (2) as well as
where
is the projection of the angle on the vertical plane
passing through the sourcereceiver axis.
The two angles are linked by the relation
.Equation 23 becomes
 
(24) 
CConsider one image point in the Fourier domain, i.e. at given
k_{mx}, k_{my} and k_{z}.
For one image point, we have all the values of the offset gather.
Each value is referenced in the Fourier domain by the offset wavenumbers
k_{hx} and k_{hy}.
Given a reflection azimuth and an aperture angle ,we want to get the value of the sample associated with , hence
what are the offset wavenumbers associated with .Given k_{mx}, k_{my}, k_{z}, and :

Because of the reflection azimuth, we need to get into the
new coordinates system. k'_{mx}, k'_{my} are computed using
k_{mx}, k_{my} and in equation (4).

For a given and k_{z} we can determine k'_{hx} from equation
(6).

The second component of the offset wavenumber, k'_{hy}, must satisfy
the coplanarity condition (7).

The two components of the offset wavenumber were obtained in the
new system of coordinates.
To return to the original system of coordinates, we use the inverse
of transformations (4).

We have determined which sample of the offset gather should
be associated with the aperture angle .
The entire angle gather at the image point (k_{mx}, k_{my}, k_{z})
and at the reflection azimuth is obtained by looping over
.
Next: About this document ...
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Stanford Exploration Project
7/8/2003