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The spacedomain shotprofile imaging condition including subsurface
offset for shotprofile migration Rickett and Sava (2002) is
 
(1) 
Where I is the migrated image produced by crosscorrelating the
upcoming, U, and downgoing, D, wavefields at every depth and
frequency. Both x and h can be areal vectors. ^{*} represents
complex conjugation. To derive the Fourierdomain equivalent, we will
perform a piecewise proof and begin with the Fourier transform D^{*}
to (neglecting Fourier scaling)
 
(2) 
Continue by Fourier transforming the variable x to find
 
(3) 
By reordering variables, the equivalent form
 

 (4) 
is achieved. From here, we can recognize the inner integral is the
Fourier transform of the wavefield U which can be replaced directly
to yield
 
(5) 
With the use of the definition of offset, , we can
replace several of the above arguments with equivalent expressions to
find
 
(6) 
The last integral is recognized as an inverse Fourier transform, this
time over the k_{h} variable. Using this fact, we arrive at the
multidimensional (over x and h, which can be twodimensional
themselves) Fourier transform of the general shotprofile imaging
condition
 
(7) 
From this equation, the result that the Fourierdomain equivalent to
the conventional spacedomain imaging condition for shotprofile
migration is again a lagged multiplication of the upcoming and
downgoing wavefields at each frequency and depth level. Evaluating
the arguments inside the wavefields to produce a component of the
image shows that the wavefields, in the wavenumber domain, will need
to be interpolated by a factor of two to calculate the image space
output. The Table 1 showing example calculations of the
components of the image space looks like
Table 1:
Layout of wavenumber components in Fourierdomain imaging conditions

Next: Synthetic tests
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Stanford Exploration Project
5/3/2005