Next: Constant velocity media
Up: REFERENCES
Previous: Solving equation 20
21
Equation (21) can be written:
 
(39) 
As for Equation (31), we use a CrankNicolson scheme for
the second derivative in x. Equation (39) becomes:
 
(40) 
where
 
(41) 
 (42) 
Equation (40) is solved the same way as (35).
B We provide here the algorithm of our mixeddomain implementation
of both and corrections. The notations refer
to a single spatial axis, but they can be easily extended to two
axes. The correction is only applied to the source wavefield, and is only
applied once, at the surface, not for each depth step. The correction
is . Because it generates strong wraparound that translates as noise in the
final image, it is recommended to pad the data with zeros, apply the
correction, then window it to a smaller size for downward continuation.
For each frequency, to downward continue one step with corrected amplitudes (operator) we Fourier transform to the wavenumber domain:

.
In the wavenumber domain, we perform the following operations:

Phase shift (part of splitstep):
where , in
which v_{0} is the reference velocity at that depth.

First order amplitude correction (part of ):
.

Second order amplitude correction (part of ):
.
Three separate Fourier transforms are needed to transform back to the
spatial domain:
In the spatial domain we perform:

First order amplitude correction (part of ):
,

Second order amplitude correction (part of ):
,

Summation of all terms:
,

Split step:
.
The correction for the receiver wavefield is computed the
same way. Once both wavefields have been computed, the
imaging condition (9) is applied for each depth.
In laterally varying velocity the terms in the spatial wavenumber
domain that contain k_{x} do not commute with terms that contain
functions of x, namely v and v_{z}. We took care to apply the terms
in the right order inside the operator. Outside ,things change. Strictly mathematically speaking, does not
commute with the splitstep, and the right order of applying the
operators remains to be studied. As in the case of splitstep (for
which the two terms that compose it do not commute with each other
either), the differences in output may not warrant the cost of the extra
Fourier transforms.
C At depth z and at each point x we compute:
 
(43) 
where A_{0} and A_{x} are the areas of the unitary wavefront and
a particular wavefront respectively. For spherical wavefronts,
In 2D: , hence .
In 3D: , hence .
To be able to compute the perturbed amplitude I_{x} from the unitary
amplitude I_{0}, we need to compute R, the radius of the spherical
wavefront. Two cases where the wavefronts are spherical are media of constant
velocity and constant vertical velocity gradient.
Next: Constant velocity media
Up: REFERENCES
Previous: Solving equation 20
Stanford Exploration Project
7/8/2003