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By converting the DSR equation to midpointoffset space
we will be able to identify the familiar zerooffset migration part
along with corrections for offset.
The transformation between (g,s) recording parameters
and (y,h) interpretation parameters is
 
(37) 
 (38) 
Travel time t may be parameterized in (g,s)space or (y,h)space.
Differential relations for this
conversion are given by the chain rule for derivatives:
 
(39) 
 (40) 
Having seen how stepouts transform from shotgeophone space
to midpointoffset space,
let us next see that spatial frequencies transform in much the same way.
Clearly, data could be transformed from (s,g)space
to (y,h)space with (37) and (38)
and then Fourier transformed to ( k_{y} , k_{h} )space.
The question is then,
what form would the doublesquareroot equation (35)
take in terms of the spatial frequencies ( k_{y} , k_{h} )?
Define the seismic data field in either coordinate system as
 
(41) 
This introduces a new mathematical function U' with the same
physical meaning as U but,
like a computer subroutine or function call,
with a different subscript lookup procedure
for (y,h) than for (s,g).
Applying the chain rule for partial differentiation to (41) gives
 
(42) 
 (43) 
and utilizing (37) and (38) gives
 
(44) 
 (45) 
In Fourier transform space
where transforms to i k_{x},
equations (44) and (45),
when i and U = U' are cancelled, become
 
(46) 
 (47) 
Equations (46)
and (47)
are Fourier representations of (44) and (45).
Substituting (46) and (47)
into (35) achieves the main purpose of this section,
which is to get the doublesquareroot migration equation
into midpointoffset coordinates:
 
(48) 
Equation (48) is the takeoff point
for many kinds of commonmidpoint seismogram analyses.
Some convenient definitions that simplify its appearance are
 
(49) 
 (50) 
 (51) 
 (52) 
Chapter showed that the quantity can
be interpreted as the angle of a wave.
Thus the new definitions S and G are the sines
of the takeoff angle and of the arrival angle of a ray.
When these sines are at their limits of they refer
to the steepest possible slopes in (s,t) or (g,t)space.
Likewise, Y may be interpreted as the dip of the data as seen
on a seismic section.
The quantity H refers to stepout observed on a commonmidpoint gather.
With these definitions (48) becomes slightly less cluttered:
 
(53) 
Most presentday beforestack migration procedures
can be interpreted through
equation (53).
Further analysis of it will explain
the limitations of conventional processing procedures
as well as suggest improvements in the procedures.
EXERCISES:

Adapt equation (48) to allow for a difference in velocity
between the shot and the geophone.

Adapt equation (48) to allow for downgoing pressure waves
and upcoming shear waves.
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Stanford Exploration Project
10/31/1997