rtm migration

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.

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EAGE Conference & Exhibition — Rome, Italy, 9 - 12 June 2008



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:

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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

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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

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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.

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