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# Theory

The space-domain shot-profile imaging condition including subsurface offset for shot-profile migration Rickett and Sava (2002) is
 (1)
Where I is the migrated image produced by cross-correlating the up-coming, U, and down-going, D, wavefields at every depth and frequency. Both x and h can be areal vectors. * represents complex conjugation. To derive the Fourier-domain equivalent, we will perform a piece-wise 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 kh variable. Using this fact, we arrive at the multi-dimensional (over x and h, which can be two-dimensional themselves) Fourier transform of the general shot-profile imaging condition
 (7)

From this equation, the result that the Fourier-domain equivalent to the conventional space-domain imaging condition for shot-profile migration is again a lagged multiplication of the up-coming and down-going 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

Next: Synthetic tests Up: Artman and Fomel: Fourier Previous: Introduction
Stanford Exploration Project
5/3/2005