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A particularly efficient implementation of offset continuation results
from a log-stretch transform of the time coordinate
Bolondi et al. (1982), followed by a Fourier transform of the
stretched time axis. After these transforms,
equation () from Chapter takes
the form

| |
(125) |

where is the corresponding frequency, *h* is the half-offset,
*y* is the midpoint, and is the transformed
data Fomel (1995b, 2000b). As in other
*F*-*X* methods, equation () can be applied independently
and in parallel on different frequency slices.
Analogously to the case of the plane-wave-destructor filters discussed
in the previous section, we can construct an effective
offset-continuation finite-difference filter by studying first the
problem of wave extrapolation between neighboring offsets. In the
frequency-wavenumber domain, the extrapolation operator is defined in
accordance with equation (), as follows:

| |
(126) |

where , and is the special
function defined in equation (). The wavenumber *k*
corresponds to the midpoint *y* in the original data domain. In the
high-frequency asymptotics, operator () takes the form
| |
(127) |

where functions *F* and are defined in equations ()
and ().
Returning to the original domain, I approximate the continuation
operator with a finite-difference filter of the form

| |
(128) |

which is somewhat analogous to (). The coefficients
of the filters *G*_{1}(*Z*_{y}) and *G*_{2}(*Z*_{y}) are found by fitting the
Taylor series coefficients of the filter response around the zero
wavenumber. In the simplest case of 3-point filters^{}, this procedure uses four Taylor
series coefficients and leads to the following expressions:
| |
(129) |

| (130) |

where
and
Figure compares the phase characteristic of the
finite-difference extrapolators () with the phase
characteristics of the exact operator () and the
asymptotic operator (). The match between different
phases is poor for very low frequencies (left plot in
Figure ) but sufficiently accurate for frequencies in the
typical bandwidth of seismic data (right plot in
Figure ).
Figure compares impulse responses of the inverse DMO
operator constructed by the asymptotic operator with those
constructed by finite-difference offset continuation. Neglecting
subtle phase inaccuracies at large dips, the two images look similar,
which indicates the high accuracy of the proposed finite-difference
scheme.

When applied on the offset-midpoint plane of an individual frequency
slice, the one-dimensional implicit filter ()
transforms to a two-dimensional explicit filter with the
2-D *Z*-transform

| |
(131) |

analogous to filter () for the case of local
plane-wave destruction. Convolution with filter () is
the regularization operator that I propose for regularizing prestack
seismic data.
**arg
**

Figure 30 Phase of the implicit
offset-continuation operators in comparison with the exact solution.
The offset increment is assumed to be equal to the midpoint spacing.
The left plot corresponds to , the right plot to
.

**off-imp
**

Figure 31 Inverse DMO impulse responses
computed by the Fourier method (left) and by finite-difference
offset continuation (right). The offset is 1 km.

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Stanford Exploration Project

12/28/2000