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The splitstep Fourier extrapolation Stoffa and Fokkema (1990)
consists of twostep 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 F_{x} and F_{t} represent the Fourier transform
along the space and the time domains, respectively.
The phases p_{1} and p_{2} quantify the amount of phase shift
in the and in the domain,
and are defined as follows:
 
(56) 
and
 
(57) 
where u_{0} 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 P_{1} is a unitary operator, but P_{2} is a pseudounitary operator
because we substitute for the evanescent component when
p_{2} 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 I_{p} is an idempotent matrix
with some elements that are 1 and others that are zeros.
Therefore, the splitstep Fourier extrapolation is pseudounitary.
Next: Datuming by wavefield depth
Up: Unitarity of the depth
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Stanford Exploration Project
2/5/2001