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Depth extrapolation using split-step Fourier method

The split-step Fourier extrapolation Stoffa and Fokkema (1990) consists of two-step extrapolation, which is a phase shifting in the domain followed by an additional phase shifting in the domain.

Let us consider a simple case that consists of two depth levels of extrapolation. Then the forward operator can be expressed as follows:
 (55)
where Fx and Ft represent the Fourier transform along the space and the time domains, respectively. The phases p1 and p2 quantify the amount of phase shift in the and in the domain, and are defined as follows:
 (56)
and
 (57)
where u0 represents the reference slowness and is a variation of the slowness from the reference slowness. Then the adjoint of the forward operator becomes
 (58)
If we apply the adjoint operator after the forward operator as follows:

 (59)
with and ,where P1 is a unitary operator, but P2 is a pseudo-unitary operator because we substitute for the evanescent component when p2 becomes imaginary. If I group the operators so that they are in the same domain, for example the domain, we get the following:
 (60)
where I is the identity matrix and Ip is an idempotent matrix with some elements that are 1 and others that are zeros. Therefore, the split-step Fourier extrapolation is pseudo-unitary.

Next: Datuming by wavefield depth Up: Unitarity of the depth Previous: Unitarity of the depth
Stanford Exploration Project
2/5/2001