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Constant velocity **prestack migration**
in offset-midpoint coordinates (Yilmaz, 1979) can be formulated as:
| |
(1) |

where is the 3-D Fourier transform
of the field *p*(*t*,*y*,*h*,*z*=0) recorded at the surface, using
Claerbout's (1985) sign convention:
The phase is defined in the
dispersion relation as
| |
(2) |

which is also referred to as the double square-root (DSR) equation.
The two integrals in and *k*_{h} in equation (1)
represent the imaging
condition for zero-offset and zero time (*h*=0,*t*=0).
In equation (1), the values of *k*_{z}
have to be real. Imaginary values of *k*_{z} do not
satisfy the downward continuation ordinary differential
equation

| |
(3) |

and have to be excluded. Real values of *k*_{z}
in equation (2) require both of the following
conditions to be satisfied:
By considering the four possible sign cases, given by assigning
*k*_{y} and *k*_{h}
positive and negative values, these two conditions can be reduced
to the condition:
| |
(4) |

Equation (4) will prove to be crucial in
determining the source of artifacts that appear in constant-offset
migrated sections (Figure a).
The prestack DSR algorithm using constant sampling in *k*_{h}
can be summarized as:

`
FFT along all axes
do `*z*
do *k*_{y}
do
do *k*_{h}
if then
endif

Note that in order to perform an FFT along the offset axis, the variable
*k*_{h} is evenly sampled between the
Nyquist negative and positive values.
However, due to
condition (4), for each set of the
loop in *k*_{h} will use either a subset or all of the possible
*k*_{h} sampled values.

** Next:** OFFSET SEPARATION
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** Previous:** Introduction
Stanford Exploration Project

11/16/1997