On the Influence of Offset in Plane-Wave Migration
Sergio Grion* and Helmut Jakubowicz, Veritas DGC
where
Summary
This paper considers the influence of offset on plane-wave
illumination and migration for seismic data. It is shown
that, for a single offset, complete illumination and
migration can only be achieved using plane waves
encompassing all angles of incidence. By contrast, when a
full range of offsets is used, it is possible in principle to
illuminate and image the subsurface using only two plane
waves. These results are confirmed using synthetic data.
The number of plane waves required for plane-wave
migration has a direct impact on computational cost. In
practice, offset limitations, as well as other considerations,
make it unrealistic to expect successful imaging using as
few as two plane waves. However, the theory does support
imaging with fewer plane waves than might otherwise be
thought necessary. This is illustrated by a marine data
example for which a good migrated image was obtained
using only six plane waves.
ps =
sin α
V
(2)
is the slowness for a plane wave originating at S, and
traveling in the same direction as the raypath from S to
the reflector.
S
X
R
α
θ
Introduction
Plane-wave seismic data offer an attractive route to
accurate prestack wave-equation imaging (Whitmore,
1995). In particular, plane waves can readily be synthesised
from conventional (point source) measurements, and
migrated using standard algorithms. The computational
cost of plane-wave migration is proportional to the number
of plane waves used, and Zhang et al. (2003 and 2005)
have suggested that, by limiting the number of plane
waves, plane-wave imaging can be achieved with less
computation than for equivalent common-shot techniques.
However, Whitmore (1995) shows successful imaging
using far fewer plane waves than recommended by Zhang
et al.. It is therefore of interest to establish the minimum
number of plane waves that can be used for adequate
imaging using plane-wave migration.
Theory
Figure 1 shows the raypath geometry for the reflection
from an interface with a dip θ. The source is at S, and
separated from the receiver at R by an offset X. It can be
shown that, for a constant velocity, V, and total traveltime,
t,
tan θ =
X + V 2 t ps ,
Vt 1 − V 2 ps2
(1)
Figure 1: Raypath geometry for a dipping reflector.
In general, complete imaging requires that all dips are
illuminated. Equation 1 shows this can be achieved by
varying either, or both, the offset, X, and takeoff angle, α.
Thus, for a single offset, full illumination requires plane
waves with a complete range of takeoff angles. In that case,
the number of plane waves required to represent the output
image is dictated by standard sampling requirements, as,
for example, in zero-offset FK migration. By contrast, with
a single plane wave, different offsets can illuminate
different dips. Thus, with multiple offsets, the number of
plane waves required to represent the output image can be
reduced from that for a single offset (assuming the
acquisition geometry allows plane-wave synthesis to be
performed without aliasing). Indeed, with a complete range
of offsets (-Vt<X<Vt), it can be shown that all dips can be
illuminated with just two plane waves. In that case, one
plane wave predominantly illuminates positive dips, while
the other illuminates negative dips.
In addition to ensuring full illumination, it is also desirable
to ensure the illumination is uniform so amplitude
information can be extracted from the migrated image.
However, while the image from any single plane wave
corresponds to the reflectivity as a function of the angle of
incidence (α−θ), stacks of images from different plane
Influence of Offset on Plane-Wave Imaging
waves do not necessarily contain equal contributions from
all angles of incidence. Uniform illumination therefore
requires careful (multichannel) weighting of each planewave contribution prior to summation.
algorithm. Delayed-shot migration (Zhang et al., 2003 and
2005) is equivalent to plane-wave migration for 2-D data,
and, with sufficient plane waves, should be identical to
conventional common-shot migration. This is confirmed by
these results (compare Figures 2b and 2c).
Synthetic Example
Figure 3 shows delayed-shot migrations of the same data as
in Figure 2, but with different combinations of input offsets
and plane waves. In this case, complete imaging requires
illumination at all angles, and can be achieved in a number
of ways. For example, with a single offset, it is essential to
include sufficient plane waves. Thus, for zero offset
(Figures 3a, 3b and 3c), or a 1 km offset (Figures 3d, 3e
and 3f), the image degrades when the number of plane
waves, Np, is less than that required by sampling theory
(101). However, even with sufficient plane waves, the
images are somewhat distorted because the illumination is
not uniform (and actually different for the two offsets). A
second way of ensuring adequate illumination is to use a
large range of offsets (Figures 3g, 3h and 3i). Indeed,
Figure 3i shows that the diffractor can be imaged with just
two plane waves. Furthermore, because the takeoff angles
were here chosen to provide the most uniform illumination,
the results are similar to those for true-amplitude commonshot migration (compare Figures 2a and 3i).
Although the above theory is based on plane waves and
planar reflectors, more complicated situations can be
treated as superpositions of planar reflections and
reflectors. This is exploited in Figures 2 and 3, which show
prestack migrations for a synthetic dataset corresponding to
a quasi-point diffractor immersed in a medium with a
constant velocity of 2000 m/s. The “diffractor” here is
actually a circular density anomaly with a radius of 100 m
located 5 km (halfway) along the model, 500 m below the
recording datum. The receivers are located at fixed
positions from 0 to 10 km at an interval of 20 m, and shots
are available at every receiver location. The source wavelet
has a maximum frequency of 10 Hz, and has been chosen
so that the migrated image closely represents the density
variations.
Figure 2a shows the image obtained from the synthetic data
using “true-amplitude” phase-shift common-shot migration
(Zhang et al., 2001). This method ensures uniform
illumination within the recording aperture, and is here
taken as a benchmark against which other results can be
compared. Figure 2b shows the results of the same
migration as in 2a, but without corrections for true
amplitude. The omission of the amplitude corrections
reduces the contribution at large dips, and results in an
apparent “squashing” of the diffractor image. Figure 2c
shows the results of delayed-shot migration on the same
data as in Figures 2a and 2b, also using a phase-shift
Figure 3f is of interest because the geometry does not allow
contributions from one of the two plane waves (α=-59o).
This result is therefore the 1 km constant-offset, 59o planewave response of the point diffractor. The image itself
resembles a locally-planar, (bandlimited) reflector. In
general, plane-wave migration can be considered as a linear
superposition
of
similar,
locally-planar,
events.
1 km
1 km
a
b
c
Figure 2: (a) True-amplitude common-shot migration for a point diffractor (see text for details of the model). (b) The same as (a),
but without true-amplitude corrections. (c) Delayed-shot migration of the same data as in (a) and (b), using 101 plane waves
covering all angles.
Np=101
Np=7
Np=2
a
b
c
d
e
f
g
h
i
Zero
Offset
+1 km
Offset
All
Offsets
Figure 3: Delayed-shot migrations of the same data as in Figure 2, but with various combinations of input offsets (fixed across
each row), and numbers of plane waves, Np (fixed in each column). For any single offset (a-c or d-f), all plane waves are required
to image all dips. When all offsets are used (g-i), all dips are imaged, even with only two plane waves (i).
Discussion and Field Example
The above theory and examples show that offset is an
important consideration for plane-wave imaging. However,
the results assume constant velocity, and, in the extreme
case of imaging with only two plane waves, require offsets
up to the length of the raypath for the target reflection.
Furthermore, the analysis focuses on stacks of plane-wave
images, whereas detailed prestack amplitude analysis
requires one plane wave for each angle of incidence. Thus,
the resulting images not only use subsets of the available
data, but, at best, only provide structural information on the
subsurface. It is therefore not normally either realistic, or
desirable, to attempt imaging using only two plane waves.
Influence of Offset on Plane-Wave Imaging
Notwithstanding the above, there is substantial evidence
that structural imaging can be accomplished with planewave migration using relatively few plane waves. Some of
the reasons for this go beyond the analysis presented here
(see, for example, Etgen, 2005, Whitmore, 1995, and
Zhang et al., 2005). In particular, because of velocity
variations, takeoff angles at the surface generally equate to
larger angles at depth, hence sampling requirements at
depth are less stringent than for the near surface. A second
factor is the redundancy inherent in modern, multichannel,
seismic data, since this helps protect against noise. In the
context of this paper, multiple coverage is also significant
because it contributes to offset sampling, and illumination
range of offsets, imaging with relatively limited numbers of
plane waves could have a valuable role in the context of a
more complete imaging strategy.
Acknowledgements
The authors would like to thank their colleagues at Veritas,
particularly Sam Gray, Carl Notfors, Yu Zhang, Leon
Chernis, James Sun and Jerry Young, for many helpful
discussions regarding this work.
References
Etgen, J. T., 2005, How many angles do we really need for
delayed-shot migration?: 75th Meeting, Society of
Exploration Geophysicists, Expanded Abstracts.
The data example shown in Figure 4 provides a direct
demonstration of the power and efficiency of plane-wave
imaging. In this case, the maximum offset is 6 km, and the
targets lie between 3 and 5 km. As a result, even allowing
for the fact that the velocity is not constant, the offset range
is insufficient to expect successful imaging with only two
plane waves. Nevertheless, despite the structural
complexity, the image obtained using 769 plane waves
(Figure 4a) is essentially the same as that obtained using
only 6 plane waves (Figure 4b). Indeed, if anything, the
image obtained with fewer plane waves appears less noisy,
even though it required less computation (by a factor of
128). This is because the plane-waves selected for Figure
4b contained less coherent noise than those included in
Figure 4a.
Whitmore, N. D., 1995, An imaging hierarchy for common
angle plane wave seismograms: Ph.D. thesis, University of
Tulsa.
Zhang, Y., Sun, J., Gray, S. H., Notfors, C. and Bleistein,
N., 2001, Towards accurate amplitudes for one-way
wavefield extrapolation of 3-D common shot records: 71st
Meeting, Society of Exploration Geophysicists, Expanded
Abstracts.
Zhang, Y., Sun, J., Notfors, C., Gray, S. H., Chernis, L. and
Young, J., 2003, Delayed-shot 3D prestack depth
migration: 65th Meeting, European Association of
Geoscientists and Engineers, Expanded Abstracts.
Zhang, Y., Sun, J., Notfors, C., Gray, S. H., Chernis, L. and
Young, J., 2005, Delayed-shot 3D depth migration:
Geophysics, 70, 5, E21-28.
The ability to be selective when choosing plane waves
offers a further practical benefit of plane-wave imaging.
Other advantages include the potential for efficient velocity
analysis and model building. Indeed, while it is unrealistic
to consider imaging with only two plane waves, the theory
and results presented here suggest that, given an adequate
10 km
3 km
4 km
5 km
a
b
Figure 4: Delayed-shot migrations of data from offshore Egypt. (a) Using 769 plane waves, and (b), using only 6 plane waves.