


 Imaging in shotgeophone space  

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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 recording parameters
and interpretation parameters is
Travel time may be parameterized in space or space.
Differential relations for this
conversion are given by the chain rule for derivatives:
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 space
to space with (9.15) and (9.16)
and then Fourier transformed to space.
The question is then,
what form would the doublesquareroot equation (9.13)
take in terms of the spatial frequencies ?
Define the seismic data field in either coordinate system as

(19) 
This introduces a new mathematical function with the same
physical meaning as but,
like a computer subroutine or function call,
with a different subscript lookup procedure
for than for .
Applying the chain rule for partial differentiation to (9.19) gives
and utilizing (9.15) and (9.16) gives
In Fourier transform space
where
transforms to ,
equations (9.22) and (9.23),
when and are cancelled, become
Equations (9.24)
and (9.25)
are Fourier representations of (9.22) and (9.23).
Substituting (9.24) and (9.25)
into (9.13) achieves the main purpose of this section,
which is to get the doublesquareroot migration equation
into midpointoffset coordinates:

(26) 
Equation (9.26) is the takeoff point
for many kinds of commonmidpoint seismogram analyses.
Some convenient definitions that simplify its appearance are
The new definitions and 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  or space.
Likewise, may be interpreted as the dip of the data as seen
on a seismic section.
The quantity refers to stepout observed on a commonmidpoint gather.
With these definitions (9.26) becomes slightly less cluttered:

(31) 
EXERCISES:
 Adapt equation (9.26) to allow for a difference in velocity
between the shot and the geophone.
 Adapt equation (9.26) to allow for downgoing pressure waves
and upcoming shear waves.



 Imaging in shotgeophone space  

Next: THE MEANING OF THE
Up: SURVEY SINKING WITH THE
Previous: The DSR equation in
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