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Compensating for visco-acoustic effects with reverse time migrationYu Zhang, Po Zhang and Houzhu Zhang, CGGVeritas

Summary

The anelastic effects of the earth cause frequency

dependent energy attenuation and phase distortion. It isdesirable to correct such unwanted effects in a prestackdepth migration. Based on the dispersion relation under the

linear attenuation assumption, we derive a visco-acousticwave equation in time domain and propose to apply it in a

prestack reverse time migration to compensate for theanelastic effects in the seismic data. The equation is defined

by a pseudo-differential operator and the energy increaseswith frequencies when back propagating the seismic data inmigration. With a regularization process, we show that ourmethod provides stable and amplitude balanced reverse

time migration images.

Introduction

It has been well observed that the anelastic effects of theearth cause seismic energy attenuation and waveletdistortion (Aki and Richards, 1980). For example, gas

trapped in the overburden structures can strongly attenuateseismic P-waves. As a result, not only the migratedamplitude below the gas anomaly is dim, but also the

imaging resolution is much reduced due to the highfrequency energy loss and the phase distortion. Therefore itis important to correct these unwanted effects in seismic

processing and make the final image more interpretable.

Early work to compensate for the seismic absorption wasperformed in data domain by an inverse Q-filter (Bickeland Natarajan, 1985; Hargreaves and Calvert, 1991). Sucha method was based on a one-dimensional backward

propagation and cannot correctly handle the real geologicalcomplexity. Since the anelastic attenuation and dispersionhappen during the wave propagation, it is natural to correct

them in a prestack depth migration. However, commonmigration methods usually treat the earth model as alossless acoustic medium and only intend to correct the

amplitude effect due to geometric spreading (Bleistein et al.2001). Mainly two reasons account for this situation. First,it is difficult to accurately estimate the Q factor fromseismic data. Second, the technology of migrating seismic

data using a visco-acoustic equation or an anelastic

equation has not been well established.

Based on the raytracing tomography algorithm, Xin et al

(2008) proposed a generalized inversion method to estimatethe attenuation loss and to obtain a model for Q. Theanalysis is performed on the migrated data by minimizingthe amplitude discrepancies across the common image

gathers. Xie et al (2009) developed a prestack Kirchhoffinverse Q-migration. They use raytracing to compute theabsorption effects along the ray path and applycompensation to each individual frequency band of the

image during the migration.

In the literature, much effort has been made to develop an

inverse Q-migration using one-way wave equationmigration (Dai and West, 1994; Yu et al. 2002). One-waywave equation is formulated in frequency domain so it is

straightforward to take care of the frequency dependentdissipation. Recently, reverse time migration (RTM) basedon directly solving the two-way wave equation provides asuperior way to image complex geologic regions and has

become a standard migration tool for subsalt imaging,

especially in Gulf of Mexico. To incorporate theattenuation correction in RTM, we need to formulate a timedomain wave equation to model the visco-acoustic effects,

which turns out to be a difficult task.

Liu et al. (1976) showed that a viscoelastic rheology withmultiple relaxation mechanisms can explain experimental

observations of wave propagations in the earth. Carcione etal (1988) designed a system of equations of motion andintroduced the memory variables to obviate storing theentire strain history required by the time convolution. In

this abstract, we derive a pseudo-differential equation tomodel viscoaoustic waves based on its dispersion relation

and apply it to reverse time migration to compensate for theanelastic effects in seismic images. In the numerical

section, we first use a simple example to show our equationcan properly compensate for the frequency dependentabsorption and dispersion effects. Then we provide a real

data example to show how our Q-RTM works on Gulf ofMexico seismic imaging projects.

Theory

The wave propagation through a linear visco-acousticmedium can be described by the following dispersionrelation (Kjartansson, 1979)

vi

vk

||)

2tan(

, (1)

where k and are the spatial wavenumber and the

circular temporal frequency, respectively; is defined by

QQ

1)

1(tan

1 1 , (2)

with the quality factorQ varying with the spatial locations.

Unlike the conventional acoustic equation, the velocity v in equation (1) is frequency dependent, i.e.

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Compensate visco-acoustic effects with RTM

0

0vv , (3)

where0v is the wave speed for the given reference

frequency0 .

To derive a visco-acoustic wave equation based on (1)-(3),

we square both sides of equation (1) and assume the

attenuation is small ( 1/1 Q ). After some algebraic

manipulations, we obtain

20

20

22

02

vQ

i

vk

. (4)

To further simplify the dispersion relation (4), we noticethat

1

0

1

0

0

2

0

0 11vQv

k , (5)

or

1

1

0

0kv . (6)

With the help of (5) and (6), equation (4) can be rewrittenas

01

2

0

01

1

0

02

kvkv

Q

i . (7)

The dispersion equation (7) leads to the following partialdifferential equation in time domain

02

2

2

p

tQt

, (8)

where is a pseudo-differential operator in the spacedomain defined by

1

1

0

2

0v , (9)

and2

2

2

2

2

2

zyx

is the 3-D Laplacian operator.

Here we attempt to strictly follow the dispersion relation(1), and the equation (8) we have derived appears morecomplicated than the one given in Zhang et al (2003).

To ease the implementation, we define the operator

Qt

t e2 , (10)

and introduce the normalized wavefield

);,,();,,( tzyxptzyxq t . (11)

Substituting (11) into (8), we remove the term with the first

temporal derivative and get the following wave equation

02

)(

2

12

2

2

qQ

tt

tt, (12)

which is similar to the conventional acoustic equation

0202

2

pv

t. (13)

Equation (13) describes the frequency dependent phasedispersion caused by the anelasticity effects. To recover the

pressure wavefield, we need to compensate for the

amplitude change after the propagation of qis done

);,,();,,( 1 tzyxqtzyxp t . (13)

By introducing the transform (11), we actually decouple theeffects of amplitude attenuation and phase dispersion.

We have derived the wave equation to model visco-acoustic propagation. To compensate for the absorptioneffects in the image, we propose the following reverse time

migration algorithm (Zhang and Sun, 2009):

1. Forward propagate the wavefieldFq from the source

,)()();0,,(

,0);(2

)(

0

1

2

21

2

2

t

tsF

Ftt

dssfxxtzyxq

txqQ

tt

(14)

where )(tf is the source wavelet.

2. Backward propagate the wavefieldBq by

reducing time from the recorded seismic data

);,( tyxD

on the surface

).;,();0,,(

,0);(2

2

12

2

2

tyxDtzyxq

txqQt

tB

Btt

(15)

3. Apply the cross-correlation imaging condition

dttxqtxqxR FtBt ));())(;(()(1 (16)

to get the image and compensate for the amplitudeloss.

Numerical experiments and examples

The first example is designed to prove that the waveequation (8) we have derived can correctly handle both theamplitude attenuation and the phase dispersion caused by

the anelasticity. We use a Ricker wavelet with 10hzpeak

frequency as the input at time 0s (Figure 1). Given velocitysmv /10000 at the reference frequency hz100 and

100Q , we use the analytical formula (1) to compute the

wavelet propagation through the anelastic medium at time

5s and show the attenuated amplitude and spectrum inFigure 2 as the dashed curves. Then we back propagate the

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Compensate visco-acoustic effects with RTM

distorted wavelet from time 5s to 0s by numerically solvingequation (8) and depict the results as the solid lines in

Figure 2. It is clear that we almost exactly recover theoriginal wavelet by using equation (8). In the next step, we

test our formulation in an extreme case. We keep all themodel parameters the same, except changing Q to 20 .

After 5s propagation, most of the frequencies aredramatically attenuated (dashed curves in Figure 3). Forexample, at the peak frequency 10hz, the amplitude dropsabout 2560 times. If we directly use equation (8) to back

propagate, the resulting wavelet explodes the numericalnoise at high frequency and appears as noise in the output

with tremendous amplitude. To solve the problem, we haveto stabilize the equation (8) by adding regularization terms.

The solid curves in Figure 3 show a stably recovered resultafter regularization. In this example, we can only faithfullyrecover some low frequency components by an inverse Q

back propagation.

Figure 1: A Ricker wavelet with peak amplitude at 10hz (left) and its spectrum (right).

Figure 2: The distorted wavelet (dashed curves) computed by the analytical formula (1) with 100Q and the recovered wavelet (solid curves)

computed by the numerical solution from (8). Left shows the time domain wavelets. Right shows their frequency spectrums.

`Figure 3: The distorted wavelet (dashed curves) computed by the analytical formula (1) with 20Q and the recovered wavelet (solid curves)

computed by the regularized numerical solution from (8). Left shows the time domain wavelets. For display purpose, the amplitude of the

distorted wavelet is amplifies by 10 times. Right shows their frequency spectrums.

To apply our method in Gulf of Mexico, we need to makeour algorithm handle complex velocity models. Here we

show an example from Alaminos Canyon, where some gashydrate saturated sand has formed in the shallow

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Compensate visco-acoustic effects with RTM

sedimentary layers. It was reported that when the gashydrate is buried under deep water with relatively low

temperature and under high pressure, it behaves like the ice(Ruppel et al., 2008). Therefore, we expect to see a low

velocity zone defined by strong reflectors in the shallowdepth to indicate the gas hydrate pocket and a dim zoneright bellow it (Figure 4). The size and shape of this dimzone vary over the offsets and prohibit us from applying

AVO analysis to deeper targets.

Figure 4: An example of shallow gas hydrate pockets on the near

offset image (the area circled by the solid line) in Alaminos

Canyon. It causes the amplitude attenuation and generates the dim

zone marked by the dashed lines.

To improve the image, a tomographic inversion (Xin et al.2009) is applied to the offset domain common image

gathers to estimate the Q model. We migrate the seismicdata with a conventional RTM and a Q-RTM as define byequations (14)-(16). The comparison is shown in Figure 5.With the Q-RTM, the structures under the anomaly are

more prominent and the migrated amplitude is betterbalanced. A Fourier analysis of the migrated section shows

that in the dim zone, the high frequencies are boosted andthe phases of migrated events are also slightly changed due

to the anelastic compensation embedded in Q-RTM.

Conclusions

Directly compensating the anelastic effects in a prestackdepth migration is an attractive but difficult task. We have

derived a visco-acoustic wave equation and used it in areverse time migration to correct the frequency dependentabsorption and phase distortion. The formulation in timedomain appears more complicated than that in the

frequency domain which adds computational cost. Also,back propagating a visco-acoustic wavefield by reducingtime causes the high frequency amplitude to increase andmay lead to numerical instability. However, the instability

is linear and can be controlled by a regularization process.As we show in this abstract, the method we have developedcan correctly model the anelastic phenomena and can be

used in a reverse time migration to improve the imagequality. Numerical examples in Gulf of Mexico show that astable Q-RTM is achievable for imaging complex

structures and balancing the migrating amplitude below theoverburden.

Acknowledgments

We thank our colleagues in CGGVeritas US Imaging fortheir support to this work, especially Jerry Young, TonyHuang, Yan Huang and Yongping Chen. We thank Yi Xie

and Kefeng Xin in CGGVeritas Singapore Research fortheir helpful discussion. Also we thank Bruce Ver West andSheng Xu for their help with this abstract.

Figure 5: A comparison of conventional RTM (left) and Q-RTM (right) on an Alaminos Canyon project.

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