technical article
first break volume 26, January 2009
Practical issues in reverse time migration:
true amplitude gathers, noise removal and
harmonic source encoding
Yu Zhang1* and James Sun2
Abstract
Reverse time migration (RTM) is the method of choice for imaging complex subsurface structures. In this paper, we show
that slightly modifying the conventional formulation, plus implementing an appropriate imaging condition, yields a true
amplitude version of RTM that provides the correct amplitude-versus-angle relation. We also discuss different ways to
suppress the migration artifacts and show how noise attenuation can be handled naturally in the common reflection angle
domain. Finally, we introduce a harmonic source phase-encoding method to allow a relatively efficient delayed shot or plane
wave reverse time migration. Taken together, these techniques yield a powerful true amplitude migration method that uses
the complete two-way acoustic wave equation to image complex structures.
Introduction
Recently, reverse time migration (RTM) has drawn a lot of
attention in the industry. Unlike one-way wave equation
migration, RTM does not need to deal with the theory of
singular pseudo-differential operators. A straightforward
implementation of RTM correctly handles complex velocities and produces a complete set of acoustic waves (reflections, refractions, diffractions, multiples, and evanescent
waves). The RTM propagator also carries the correct
propagation amplitude and imposes no dip limitations on
the image. In the past, strong migration artifacts and the
intensive computational cost have been two major problems that prevented RTM from being used in production.
In this paper, we first formulate RTM based on inversion
theory and then we address some solutions to suppress the
low frequency migration artifacts. At the end, we propose
harmonic source migration as a way to improve the efficiency of RTM.
True amplitude reverse time prestack
depth migration
We first discuss how to formulate a true amplitude RTM. To
migrate a shot record Q(x, y; xs, ys; t), we have to compute
the wavefield originating at the source location (xs, ys, zs = 0)
and observed at the receiver locations (x, y, z = 0). Because
the source wavefield expands as time increases and the
recorded receiver wavefield is computed backwards in time,
we denote them by pF and pB, respectively, in the following
two-way wave equations:
(1)
and
(2)
where v=v(x, y, z) is the velocity, ƒ(t) is the source signature, and
is the Laplacian operator.
To obtain a common shot image with correct migration
amplitude, we need to apply the deconvolution imaging
condition (Zhang et al., 2005a),
where
is defined as the inverse of the wavefield
i.e., in frequency domain
(3)
,
(4)
This imaging condition is simple to apply in the frequency
domain for one-way wave equation migration. However,
it is difficult to implement in the time domain for RTM. In
practice, the cross-correlation imaging condition,
(5)
1
CGGVeritas, 10300 Town Park Drive, Houston, TX 77072, USA.
CGGVeritas, 9 Serangoon North Avenue 5, Singapore, Singapore 554531.
*Corresponding author, E-mail: yu.zhang@cggveritas.com.
2
© 2009 EAGE www.firstbreak.org
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technical article
first break volume 26, January 2009
is often preferable for reasons of stability. Although this
imaging condition does not appear to be consistent with
true amplitude migration, Zhang et al. (2007a) proved that
the imaging condition specified in Equation (5) is a proper
choice to obtain true amplitude angle-domain common
image gathers from wave equation-based migration provided
that Equation (1) is modified accordingly as
(6)
Equation (6) is different from Equation (1), the conventional
wave equation for the forward wavefield, because the source
at the surface is treated as a boundary condition instead of
a right hand side forcing term in the equation. An integral
is applied to the source wavelet ƒ(t) to guarantee that the
migrated phase and amplitude are correct.
In summary, the following algorithm allows RTM to
output true amplitude angle domain common image gathers
from RTM:
1. Compute forward and backward wavefields pF and pB by
solving the two-way wave equations, Equations (6) and (2).
2. Apply the cross-correlation imaging condition, Equation
(5), during the migration.
3. Use an existing method, e.g., Sava and Fomel (2003), to
output angle domain common image gathers.
The migration output then provides the angle-dependent
reflectivity in the sense of the high frequency approximation.
To show how true amplitude angle domain RTM works, we
apply it to a 2D horizontal reflector model. The input comprises shot records over five horizontal reflectors (Figure 1).
The shot is at the centre of the section and the receivers
cover the surface across an aperture of 15 km on each side
of the shot. The amplitude variation across traveltime and
lateral distance is due only to the geometrical spreading loss.
We firstly migrated the shot records using the conventional
common shot RTM algorithm, Equations (1) and (2), with
the cross-correlation imaging condition, Equation (5). At an
image location, we stacked all the migrated common image
shot gathers to generate the subsurface offset gathers, and
then converted them to the subsurface reflection angle gathers shown in Figure 2a. The peak amplitudes along the five
migrated reflectors are shown in Figure 2b. We can see that
in this RTM implementation, the migrated amplitude-versusangle curves are not correct. The amplitudes at far angles
are overestimated, especially for the shallow reflectors. We
then migrated the shot records using Equations (2) and (6)
with the cross-correlation imaging condition, Equation (5).
The angle domain common image gather is shown in Figure
3a and its normalized peak amplitudes along the reflectors
are shown in Figure 3b. It is clear that the amplitudes in the
A
B
Figure 1 Shot record over five horizontal reflectors in a medium with velocity
v, in units of metres per second, given by v=(2000+0.3z), where z is the depth
in metres.
30
Figure 2 (a) An angle domain common image gather from RTM using Equations
(1) and (2). (b) The corresponding curves of amplitude versus angle.
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first break volume 26, January 2009
angle domain recover the reflectivity accurately over a large
angular range, aside from the edge effects.
Compared with the formulations of true amplitude
Kirchhoff migration (Bleistein, 1987; Bleistein et al., 2001)
and one-way wave equation migration (Zhang et al., 2005a,
2007a), implementing a true amplitude RTM algorithm is
much simpler because the propagator itself naturally carries
correct propagation amplitude, if we assume the observed
seismic wave is well approximated by solving the acoustic
wave equation. Therefore, from the amplitude-versus-angle
point of view, RTM is superior to other existing migration
methods and provides a stable and reliable way for seismic
A
technical article
inversion. However, to take advantage of wide azimuth
acquisition, we need to generate five-dimensional commonimage gathers from prestack depth migration, retaining
both reflection angle and azimuth angle information. This
leads to a dramatic increase in both computation and
input/output costs, by two orders of magnitude, when
compared to a stack output. Efficiently outputting 3D angle
gathers from wide azimuth data remains a challenge to the
industry.
Noise removal from true amplitude migration
point of view
It has been observed that the conventional cross-correlation
imaging condition of Equation (5) produces strong low frequency migration artifacts in RTM. Figure 4 shows a direct
application of RTM to the 2004 BP 2D dataset (Billette and
Brandsberg-Dahl, 2005). Migration artifacts appear mainly
at shallow depths but also above strong reflectors, and
severely mask the migrated structures. They are generated
by the cross-correlation of reflections, backscattered waves,
head waves, and diving waves. Figure 5 illustrates how the
migration noise was generated. For any migration algorithm,
if the sum of ts, the traveltime from the source to a subsurface
location, and tr, the traveltime from the same location to the
receiver, is equal to the recorded two-way traveltime t, then
such a location is considered as a possible imaging point, i.e.,
a point where the reflection can occur (Figure 5a). However,
due to the two-way propagation nature of RTM, it also
generates reflections above a hard interface in the velocity
model. The imaging relation
(7)
is satisfied for any point along the reflection raypath (Figure
5b). Therefore, in addition to the real reflection point at
the hard interface, RTM also produces many unreal imaging points which give the low frequency migration artifacts
(Figure 5c).
B
Figure 3 (a) An angle domain common image gather from RTM using Equations
(6) and (2). (b) The corresponding curves of amplitude versus angle.
© 2009 EAGE www.firstbreak.org
Figure 4 Direct output of RTM applied to the 2004 BP 2D data set.
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technical article
A
first break volume 26, January 2009
B
C
Figure 5 (a) For most of the migrations, the imaging relation ts + tr = t is used to generate an imaging point. (b) The propagation in RTM also generates reflections above a hard interface in the velocity model. As a result, the imaging relation ts + tr = t is satisfied for any point (white triangle) along the reflection ray
path. (c) In addition to the real reflection point (black triangle), RTM also produces many unreal imaging points (white triangles) which give the low frequency
migration artifacts.
A
B
Figure 6 (a) Some angle domain common image gathers from the 2004 BP
dataset. No artifacts show up on 0° to 60° gathers. (b) A stacked image for 2004
BP 2D dataset using reflection angles of 0° to 60°.
32
The first published work to suppress the artifacts is
attributable to Baysal et al. (1984), who introduced a nonreflecting wave equation to remove the normal incidence
reflected energy from an interface for post-stack depth
migration. However, this technique is not effective for
prestack depth RTM because the underlying mechanisms
of noise generation are different. Other techniques have
been proposed in the literature, such as a velocity smoothing high-pass filter (Mulder and Plessix, 2003), Poynting
vectors (Yoon et al., 2004), and a directional damping term
at the interface (Fletcher et al., 2005). In practice, we find
they are either difficult to implement properly or have the
drawbacks of distorting the spectrum or amplitude of the
migrated images undesirably. Liu et al. (2007) proposed a
new imaging condition to address the problem: decompose
the wavefields into one-way components and only crosscorrelate the wave components that occur as reflections. In
3D, fully decomposing the wavefield into eight directions
(up-left-forward, up-left-backward, up-right-forward, upright-backward, down-left-forward, down-left-backward,
down-right-forward and down-right-backward) is computationally intensive, while an incomplete decomposition
may inadvertently remove some steeply dipping reflectors
in complex structures.
Here we point out that suppressing migration artifacts
is simple if we output angle gathers. The migration artifacts have the common feature that the source wavefield
correlates to the receiver wavefield propagating in the
opposite direction, which implies that the reflection
angle is 90º. Therefore the artifacts can be removed by
stacking the migrated angle gathers with a far angle mute
(Figure 6).
While outputting angle gathers is still expensive, a
simple and popular way to remove the migration artifacts
is to apply the Laplacian filter to the stacked migrated
image. It removes the migration artifacts effectively with-
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technical article
first break volume 26, January 2009
the number of shots, thus reducing the project cycle time
and cost. Combining shots by line source synthesis in the
inline direction (delayed shot migration) or in both inline
and crossline directions (plane-wave migration) produces
satisfactory results if enough p-values are used (Whitmore,
1995; Duquet et al., 2001; Liu et al., 2002; Zhang et
al., 2005b; Etgen, 2005). One may consider applying
similar techniques to RTM to improve its efficiency. Taking
delayed shot migration for example, we need to apply a τ-p
transform to the input data to synthesize the line source
response, i.e.,
A
B
Figure 7 (a) A proposed processing flow to remove migration artifacts without
distorting the spectrum or amplitude in the migrated image. (b) The image
after application of a Laplacian filter plus proper pre- and post-migration
processing, as suggested in (a).
For migrations performed in the frequency domain, the
time delays in delayed shot migration can be implemented
as phase shifts, avoiding the need for time padding of the
input traces. Since we perform RTM in the time domain,
delayed shot RTM requires long time padding for long sail
lines and/or large values of p, which can slow down the
computation considerably. To avoid this problem, Zhang
et al. (2007b) introduced a new phase-encoding scheme,
called harmonic source migration, which is theoretically
equivalent to delayed shot migration but does not suffer from the long time padding problem. For harmonic
source migration, the phase-encoding function in the time
domain is
out hurting steep dips. To see how this technique works,
we recall the relation
(8)
where θ is the reflection angle and v is the local interval
velocity. Equation (8) says that applying a Laplacian filter
to the stacked image is equivalent to applying a cos2 θ
weight to the angle gathers. According to Equation (8),
to utilize this technique correctly without distorting the
migrated spectrum and amplitude, we have to apply a
1/w2 filter to the input data and rescale the migration
output by a v2 factor. The proposed processing flow is
summarized in Figure 7a. Figure 7b shows the result of
applying this technique to the 2004 BP 2D dataset. As we
have discussed, such a technique is equivalent to stacking
the common image gathers with an angle domain taper,
although no output in the form of angle domain common
image gathers is required.
Delayed shot, plane-wave, and harmonic
source RTM
For common shot migration, the cost equals the cost of
migrating a single shot times the number of shot migrations. Various approaches have been proposed to reduce
© 2009 EAGE www.firstbreak.org
(9)
(10)
so there is negligible time padding to apply during the
spatial transform. The number of k-values can be determined
by a similar analysis to the number of p-values in delayed
shot migration (Zhang et al., 2005b, 2006). We have applied
this migration to both one-way wave equation migration
(Soubaras, 2006) and RTM. For a typical production project
in Gulf of Mexico, the speed of harmonic source migration
versus common shot migration could be greater by a factor
of 2–3.
Figure 8 compares the results of one-way wave equation migration to RTM, using harmonic source encoding
in both cases, for a deep water dataset from Mississippi
Canyon, Gulf of Mexico. In general, RTM gives better
images of the steeply dipping salt flanks. The sediments
underneath the salt overhangs are extended closer to the
salt flank boundaries. We attribute these improvements
mainly to the high angle or turning wave propagation
absent in the one-way wave equation migration. Figure 9
shows another comparison, for a Gulf of Mexico dataset
in the area of Keathley Canyon, between results from oneway wave equation migration and from RTM. Although
there are no complicated salt bodies and steeply dipping
salt boundaries, thanks to the more accurate propagators,
RTM still gives superior images for the structures in the
subsalt areas.
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technical article
first break volume 26, January 2009
A
A
B
B
Figure 8 A real data example from Mississippi Canyon, Gulf of Mexico, with
images from (a) one-way wave equation migration, and (b) RTM. RTM gives
better images of the steeply dipping salt flanks and sediment termination.
Figure 9 Real data from Keathley Canyon, Gulf of Mexico, with images from
(a) one-way wave equation migration, and (b) RTM. RTM better delineates
the subsalt.
Conclusions
and Sam Gray for their help with this paper and CGGVeritas
US Seismic Imaging for providing the real data examples.
We have shown that by slightly modifying the formulations,
reverse time migration can be calibrated as a stable seismic
inversion technique and provides correct information on
angle dependent reflectivity. We have also discussed different
ways to suppress the low frequency migration artifacts, and
proposed a processing flow to remove migration artifacts
without distorting the migrated amplitude and spectrum.
Finally, we have introduced a harmonic source phase-encoding method which allows a relatively efficient implementation
of delayed shot or plane-wave RTM. Taken together, these
yield a powerful true amplitude migration method that uses
the complete two-way acoustic wave equation to image complex structures.
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Received 24 September 2008; accepted 25 November 2008.
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