rtm migration
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8/3/2019 RTM Migration
http://slidepdf.com/reader/full/rtm-migration 1/5
Submission number: 3784
Practical issues of reverse timemigration: true-amplitude gathers,noise removal and harmonic-source
encoding
Yu Zhang, CGGVeritas, Houston
James Sun, CGGVeritas, Singapore
Summary
We analyze the amplitude behavior of reverse-time migration and show that modifying the
initial-value problem into a boundary-value problem for the source wavefield, plus
implementing an appropriate imaging condition, yields a true-amplitude version of RTM. We
also discuss different ways to suppress the migration artifacts. Finally, we introduce a
“harmonic-source” phase-encoding method to allow a relatively efficient 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.
70th
EAGE Conference & Exhibition — Rome, Italy, 9 - 12 June 2008
8/3/2019 RTM Migration
http://slidepdf.com/reader/full/rtm-migration 2/5
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, evanescent waves, etc.). The RTM propagator also carries the correctpropagation amplitude and imposes no dip limitations on the image. In the past, the strong
migration artifacts and the intensive computational cost have been the two major problems
that prevented RTM from being used in production. In this abstract, 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 delayed-shot RTM.
True-amplitude reverse-time prestack depth migration
We first formulate RTM based on the theory of true-amplitude migration. To migrate a shot
record , with the shot at);,;,( t y x y xQ ss )0,,( =sss z y x and receivers at , we
have to compute the wavefields originating at the source location and observed at the receiverlocations. Because the source wavefield expands as time increases and the recorded receiver
wavefield is computed backward in time, we denote them by and respectively in the
following two-way wave equations:
)0,,( = z y x
F p B p
),()();(1 2
2
2
2t f x xt x p
t csF
rrr
−=⎟⎟ ⎠
⎞⎜⎜⎝
⎛ ∇−
∂
∂δ (1)
and
⎪⎩
⎪⎨
⎧
==
=⎟⎟ ⎠
⎞⎜⎜⎝
⎛ ∇−
∂
∂
),;,;,();0,,(
,0);(1 2
2
2
2
t y x y xQt z y x p
t x pt c
ss B
B
r
(2)
where is the velocity, 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., 2005)
),,( z y xcc = )(t f 2∇
∫ −= dt t x pt x p x R F B );();()( 1 rrr
, (3)
where is defined as the inverse of the wavefield);(1t x pF
r−);( t x pF
r
. 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
∫ = dt t x pt x p x R F B );();()(rrr
(4)
is often preferable for reasons of stability. Although this does not appear to be consistent withtrue-amplitude migration, Zhang et al. (2007a) proved that the imaging condition (4) is a
proper choice to obtain true-amplitude angle gathers from wave equation based migration.
However, for this to occur, equation (1) needs to be modified accordingly as
⎪⎩
⎪⎨
⎧
−==
=⎟⎟ ⎠
⎞⎜⎜⎝
⎛ ∇−
∂
∂
∫ .')'()();0,,(
,0);(1
0
2
2
2
2
t
sF
F
dt t f x xt z y x p
t x pt c
rr
r
δ
(5)
This equation is different from the conventional wave equation (1) 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.
In summary, we propose the following algorithm to output true-amplitude angle-domaincommon-image gathers from RTM:
70th
EAGE Conference & Exhibition — Rome, Italy, 9 - 12 June 2008
8/3/2019 RTM Migration
http://slidepdf.com/reader/full/rtm-migration 3/5
1. Compute forward and backward wavefields and by solving the two-way wave
equations (5) and (2);F p B p
2. Apply the cross-correlation imaging condition (4) 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 angle dependent reflectivity in the sense of the highfrequency approximation.
To show how true-amplitude angle-domain RTM works, we apply it to a 2-D horizontal
reflector model in a medium with velocity sm zc / )3.02000( ⋅+= . The input is shot records
over five horizontal reflectors. The shot is in the center of the section and the receivers cover
the surface in an aperture of 15000m on each side. The amplitude variation across travel time
and lateral distance is due only to geometrical spreading loss. We migrated the shot records
using the common-shot RTM algorithm (2) and (5) with the imaging condition (4). At an
image location, we stack all the migrated common-image shot gathers to generate the
subsurface offset gathers, and then convert them to the subsurface reflection angle gathers
shown in Figure 1. The normalized peak amplitudes along the reflectors in the angle domain
are shown in Figure 2. It is clear that the amplitudes in the angle domain recover thereflectivity accurately over a large angular range, aside from edge effects.
Figure 1: A migrated angle domain common-imagegather.
Figure 2: Normalized peak amplitude vs reflectionangle curves along the migrated reflectors.
Noise removal from true-amplitude migration point of view
It has been observed that the conventional cross-correlation imaging condition (4) produces
strong low-frequency migration artifacts in RTM. Figure 3a shows a direct application of
RTM to the 2004 BP 2-D data set (Billette and Brandsberg-Dahl, 2005). The migration
artifacts appear mainly at shallow depths and severely mask the migrated structures. They are
mainly generated by the cross correlation of reflections, backscattering waves, head waves
and diving waves. The non-reflecting wave equation (Baysal et al., 1984) was proposed to
suppress the artifacts in poststack RTM by avoiding the normal-incidence reflected energy
from an interface. However, this technique is not effective for prestack depth RTM because
the underlying mechanisms of how the noise is generated are different. Other techniques have
been proposed in the literature, such as velocity smoothing, high-pass filter (Mulder and
Plessix, 2003), Poynting vectors (Yoon, et al., 2004), directional damping term at the
interface (Flecher et al., 2005) etc. 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 solve the problem:
decompose the wavefields into one-way components and only cross-correlate the wave
components that occur as reflections. In 3-D, fully decomposing the wavefield into different
directions is computationally intensive, therefore only upgoing and downgoing components
are used in practice. This incomplete decomposition removes some steeply dipping reflectors
for complex structures (Figure 3b).
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
70th
EAGE Conference & Exhibition — Rome, Italy, 9 - 12 June 2008
8/3/2019 RTM Migration
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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 3c). Another simple and popular way to remove the migration
artifacts is to apply the Lapacian filter to the stacked migrated image. It removes the
migration artifacts effectively without hurting steep dips. To see how this technique works,
we recall the well-known relation
, (6)222222 / cos4 vk k k z y x θ ω =++
where θ is the reflection angle and v is the local interval velocity. Equation (6) says that
applying a Laplacian filter to the stacked image is equivalent to applying a weight to
the angle gathers. According to (6), to correctly utilize this technique without distorting the
migrated spectrum and amplitude, we have to apply a filter to the input data and rescale
the migration output by a factor. Figure 3d shows the result of applying the above
mentioned technique to BP 2-D dataset.
θ 2cos
2 / 1 ω 2v
Figure 3a: Direct image from RTM. Figure 3b: Image of decomposed imaging condition
Figure 3c: 0º-60º stacked image. Figure 3d: Image of Laplacian filter plus
proper pre and post migration processing
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 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). One may consider applying similar techniques to RTM to improve its efficiency.
However, since we perform RTM in the time domain, delayed-shot or plane-wave RTM
requires a long time padding for long sail lines and large values of p’s. This can slow down
rather than speedup the process considerably. To avoid this problem, Zhang et al. (2007b)
introduced a new phase-encoding scheme that does not suffer from the long time padding
problem. This new phase-encoding algorithm is theoretically equivalent to the delayed-shot
migration and we called it harmonic-source migration. For a typical production project in
Gulf of Mexico, the speedup ratio of harmonic-source migration versus common-shot
migration could be a factor of 2-3. We have applied this migration to both one-way wave
equation migration (Soubaras, 2006) and RTM. Figures 4 compares the results of a one-way
70th
EAGE Conference & Exhibition — Rome, Italy, 9 - 12 June 2008
8/3/2019 RTM Migration
http://slidepdf.com/reader/full/rtm-migration 5/5
wave equation migration to RTM both using harmonic-source encoding with a deep water
GOM dataset . 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.
ConclusionsWe have analyzed the amplitude behavior of reverse-time migration. We have shown that
modifying the initial-value problem into a boundary-value problem for the source wavefield
plus implementing an appropriate imaging condition yields a true-amplitude version of RTM.
We have also discussed different ways to suppress the low frequency migration artifacts.
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.
Figure 4: Real data from GOM with images from one-way wave equation migration and RTM.
References
Baysal, E., Kosloff, D. D. and Sherwood, J. W. C., [1984], A two-way nonreflecting wave equation:
Geophysics, 49, 132-141.
Billette, F. J. and Brandsberg-Dahl, S., [2005], The 2004 BP velocity benchmark: 67 th
Ann. Mtg.:
EAGE , B035.
Fletcher, R. F., Fowler, P., Kitchenside, P. and Albertin, U., [2005], Suppressing artifacts in prestack
reverse time migration, 75th
Ann. Mtg.: SEG, 2049-2051
Liu, F., Zhang, G., Morton, S. and Leveille, J., [2007]. Reverse-time migration using one-way
wavefield imaging condition, 77th Ann. Mtg.: SEG, 2170-2174.
Mulder, W. A. and Plessix R.-E., [2003] One-way and two-way wave equation migration, 73rd
Ann.
Mtg.: SEG, 881-884.Sava, P. C. and Fomel, S., [2003], Angle-domain common-image gathers by wavefield continuation
methods: Geophysics, 68, 1065-1074.
Soubaras, R. [2006], Modulate-shot migration, 76th Ann .Mtg: SEG, 2426-2429.
Whitmore, N. D., [1995], An imaging hierarchy for common angle plane wave seismograms: Ph.D.
thesis, University of Tulsa.
Yoon, K., Marfurt, K. J. and Starr, W., [2004], Challenges in reverse-time migration: 74th Ann. Mtg.:
SEG, 1057-1060.
Zhang, Y., Sun, J. and Gray, S., [2007b], Reverse-time migration: amplitude and implementation
issues, 77th Ann. Mtg.: SEG, 2145-2149.
Zhang, Y., Xu, S., Bleistein, N. and Zhang, G., [2007a], True amplitude angle domain common imagegathers from one-way wave equation migrations: Geophysics, 72, S49-58.
Zhang, Y., Zhang, G. and Bleistein, N., [2005], Theory of true amplitude one-way wave equations and trueamplitude common-shot migration: Geophysics, 70, E1-10.
70th
EAGE Conference & Exhibition — Rome, Italy, 9 - 12 June 2008
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