Next: ACKNOWLEDGEMENT
Up: NOTES FROM TIEMAN's SEMINAR
Previous: The midpoint gather
Fortunately, we can convert this commonmidpoint transform (9) into
an equivalent commonsource transform (6).
Let us make two additional Fourier transforms over spatial dimensions
of s and y for the spatial frequencies k_{s} and k_{y}:
 

 (10) 
and
 

 (11) 
To place the second integral (11) in the form of the
first (10), we should change the variables of integration
from h and y to h and s. (The Jacobian of this transformation
is .) Substituting y=s+h/2 we get
 

 
 
 (12) 
Thus, a twodimensional stretch of the midpointgather transform
becomes equivalent to the sourcegather transform.
For a given dip over offset in a midpoint gather
p_{y}, we can identify a dip over midpoint
 
(13) 
The adjustment of p_{s} = p_{y}  k_{y}/2f_{y} subtracts
half of this midpoint dip from the offset dip.
With a careful application of the chain rule, and carefully
distinguishing partial derivatives, we could arrive at the
same result
 
(14) 
Next: ACKNOWLEDGEMENT
Up: NOTES FROM TIEMAN's SEMINAR
Previous: The midpoint gather
Stanford Exploration Project
11/12/1997