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The high-resolution imaging of salt-dome flanks is an important
application of the theory described in the previous two sections.
In this case
we can often assume that in the proximity of the
salt flanks the data contains no, or little,
energy dipping in the direction opposite to the reflections from the flanks.
According to the theory developed in the previous section,
this assumption enables the imaging of the salt flanks
with higher resolution than otherwise possible.
However, it is important to be aware that when increasing
the image frequency content by applying
the constraints in equation (5)
in place of
the constraints in equation (3),
we run the risk of aliasing the image.
As discussed in the previous section,
the conditions that avoid operator
aliasing do not guarantee avoidance of image aliasing.
Therefore, the constraints to avoid image aliasing
[equation (2)] must also be taken into account,
and the image sampling must be reduced
to achieve both goals of
avoiding image aliasing and preserving high-frequency components.
Because of image aliasing considerations,
the images of the salt flanks from the Gulf of Mexico data
that are shown in the following Figures are sampled twice as densely
,as the zero-offset data
.
The first step to apply the high-resolution imaging method
presented in this paper is to determine the appropriate
values for the bounds on the data dips.
For the sake of simplicity,
for this example I choose constant bounds;
that is,
, ,, .In more geologically-complex cases it may
be advantageous to allow the data-dip bounds to vary
both in time and space.
Figure 14
shows the same data spectrum as
Figure 4;
the dashed line superimposed onto the spectrum
cover the areas determined by
the inequalities of equation (4),
according to the chosen bounds for
and .The areas that honor all anti-aliasing constraints,
and thus that represent data components that are used by the imaging,
are covered by crossing dashed lines.
A large swath of the aliased energy
with positive time dips is used by the high-resolution imaging,
whereas it would be discarded
if standard anti-aliasing methods were used.
The improvements in image resolution that are made possible
by the proposed anti-aliasing method are demonstrated in
Figure 15-
Figure 17.
Figure 15a
shows the results of 3-D post-stack migration
without using any anti-aliasing.
Figure 15b
shows the results when the new anti-aliasing
constraints in equation (5) are applied.
And Figure 15c
shows the results when the standard anti-aliasing
constraints in equation (3) are applied.
The differences between the results are better appreciated
by comparing windows zooming into smaller parts
of the sections.
Figure 16 shows the comparison for
the shallower part of the section.
The image obtained without anti-aliasing
is uninterpretable because of the strong aliasing noise.
The image obtained with the proposed method shows better resolution
of several dipping reflectors and of the steep salt flank.
Figure 15 demonstrates
(same area shown in Figure 1)
that the high-frequency
sediment truncation against the salt flank,
(CMP X=700 m and Time=2.2 s)
is well resolved in the image obtained using the proposed
method, whereas it is poorly resolved
in the image obtained using the traditional methods.
Wind-spec-ann
Figure 14
Frequency-wavenumber spectrum of a data window
(same as in Figure 4).
The dashed line superimposed onto the spectrum
cover the areas determined by
the inequalities of equation (4).
The areas that respect all the operator anti-aliasing constraints,
and thus that represent data components that are used by the imaging,
are covered by dashed lines along both direction.
Comp-WL-anti
Figure 15
3-D migrations of a salt-dome flank in the Gulf of Mexico:
(a) migration obtained without any anti-aliasing filter,
(b) migration obtained with the application of the proposed
``high-resolution'' anti-aliasing filter,
(c) migration obtained with the application of a ``standard''
anti-aliasing filter.
Comp-WT-anti
Figure 16
Zoom into the shallower part of the
3-D migrations of a salt-dome flank in the Gulf of Mexico
shown in Figure 15:
(a) migration obtained without any anti-aliasing filter,
(b) migration obtained with the application of the proposed
``high-resolution'' anti-aliasing filter,
(c) migration obtained with the application of a ``standard''
anti-aliasing filter.
Comp-WB-anti
Figure 17
Zoom into the deeper part of the
3-D migrations of a salt-dome flank in the Gulf of Mexico
shown in Figure 15:
(a) migration obtained without any anti-aliasing filter,
(b) migration obtained with the application of the proposed
``high-resolution'' anti-aliasing filter,
(c) migration obtained with the application of a ``standard''
anti-aliasing filter.
Next: Conclusions
Up: Biondi: Kirchhoff imaging beyond
Previous: Link between operator anti-aliasing
Stanford Exploration Project
7/5/1998