Attenuation compensation in least-squares reverse time migration using the visco-acoustic wave equa-tionGaurav Dutta∗, Kai Lu, Xin Wang and Gerard T. Schuster, King Abdullah University of Science and Technology

SUMMARY

Attenuation leads to distortion of amplitude and phase of seis-mic waves propagating inside the earth. Conventional acous-tic and least-squares reverse time migration do not accountfor this distortion which leads to defocusing of migration im-ages in highly attenuative geological environments. To ac-count for this distortion, we propose to use the visco-acousticwave equation for least-squares reverse time migration. Nu-merical tests on synthetic data show that least-squares reversetime migration with the visco-acoustic wave equation correctsfor this distortion and produces images with better balancedamplitudes compared to the conventional approach.

INTRODUCTION

Fluids trapped in overburden structures cause strong attenua-tion of P-waves which hamper the resolution of migrated im-ages. This can be attributed to the fact that the real earth isanelastic and this causes distortion of the amplitude and phaseof the propagating seismic waves (Aki and Richards (1980)).If the attenuation is too strong, ignoring it during migrationcan lead to incorrect positioning of reflectors below these lay-ers. Attenuation of P-waves can be quantified by an attenua-tion parameter or a quality factor,QP, which accounts for theamplitude attenuation and the phase shift as a function of thethe frequency content of the propagating waves and the dis-tance travelled. Lower values ofQP imply more energy lossof the wave per cycle or very high attenuation. The values ofQP for rocks like gas-sandstone and shale are very low (QP≈ 15− 30) and this necessitates the need to account forQP

during migration for more accurate and better resolved images.

Reverse time migration (RTM) has become the standard mi-gration algorithm for imaging in complex geological settingslike the Gulf of Mexico. Conventional acoustic RTM uses thetwo-way wave equation for computing the Green’s functions(Baysal et al. (1983), McMechan (1983)) and is more accuratethan the integral based Kirchhoff methods. Dai et al. (2012)extended the idea of least-squares migration (Nemeth et al.(1999)) to least-squares reverse time migration (LSRTM) andshowed that LSRTM can mitigate the artifacts of RTM andcan produce images of better resolution compared to standardRTM. However, RTM and LSRTM do not take into account theattenuation due toQP if the standard acoustic wave equationis used for wavefield extrapolation. In this work, we proposeto use the visco-acoustic wave equation instead of the standardacoustic wave equation for LSRTM and show with numeri-cal tests that LSRTM using the visco-acoustic wave equationproduces images with better balanced amplitudes and accuratepositioning of reflectors compared to acoustic LSRTM whenthe subsurface attenuation is very strong.

LEAST-SQUARES REVERSE TIME MIGRATION WITHTHE VISCO-ACOUSTIC WAVE EQUATION

The stress-strain relation for a visco-acoustic medium is givenby (Christensen (1982), Carcione et al. (1988)),

m∑

k=0

ckdk p

dtk=

m∑

k=0

dkdke

dtk, (1)

wherep denotes the pressure field,e denotes the trace of thestrain tensor matrix,dk/dtk represents thek-th order time deriva-tive andck anddk are coefficients related to the material prop-erties of the medium. The pressure field can be expressed ex-plicitly from equation (1) using Laplace transform methodsas,

p(t) =−MR

∫ t

∞e(τ)

[1−

L∑

l=1

(1− τε l

τσ l

)exp

(− t − τ

τσ l

)]dτ, (2)

whereτσ l andτε l denote material relaxation times for thel-th mechanism,L is the number of relaxation mechanisms andMR is the relaxed modulus of the medium. The equation of mo-mentum conservation can be written as (Carcione et al. (1988)),

− ∂∂xi

(1ρ

∂ p∂xi

)= e+

∂∂xi

(1ρ

fi

), (3)

whereρ and fi represent the density and body forces, respec-tively. Equations (2) and (3) together describe the deformationin a visco-acoustic medium. For numerical modeling, the con-volution term in equation (2) is simplified by introducing amemory variable term,rp, (Robertsson et al. (1994)) and onlyone relaxation mechanism (L = 1) is sufficient for practicalpurposes (Blanch et al. (1995)). Thus, for a 2D visco-acousticmedium, the linearized equations of motion and deformationbecome (Thorbecke and Draganov (2011)),

∂Vx

∂ t=

1ρ

∂P∂x

,

∂Vz

∂ t=

1ρ

∂P∂ z

,

∂P∂ t

=1κ

τετσ

(∂Vx

∂x+

∂Vz

∂ z

)+ rp,

∂ rp

∂ t=− 1

τσ

(rp +

(τετσ

−1

)(1κ

)(∂Vx

∂x+

∂Vz

∂ z

)). (4)

Here,Vx andVz represent the particle velocities in thex andzdirections, respectively, andκ represents the bulk modulus ofthe medium. The relaxation parameters,τσ andτε , are relatedto the quality factor,QP, and the central frequency,fw, of thesource wavelet as (Robertsson et al. (1994)),

τσ =

√1+ 1

Q2P− 1

QP

fw,

τε =1

f 2wτσ

. (5)

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Attenuation LSRTM

Equation 4 can be used for wavefield extrapolation in a visco-acoustic medium during RTM and LSRTM. Figure 1 showsthe effect of attenuation on amplitude and phase for a homo-geneous medium with a background velocity of 3000m/s andfor different values ofQP. The source is excited at the centerof the model and the snapshots are taken att = 0.8 ms. It isevident that lower values ofQP distort the amplitude and phaseof the propagating waves.

For visco-acoustic LSRTM, the Born modeling equation canbe written as,

P(g,s)≈∫

xG(x|s)m(x)G(g|x)W (ω)dx, (6)

whereG(x′|x) is the Fourier domain representation of the band-limited Green’s function for a source excited atx and an ob-servation point atx′ and is numerically computed using equa-tion (4). W (ω) represents the source wavelet for an angu-lar frequencyω, m(x) represents the reflectivity model andP(g,s) represents the born-modeled data whereg ands are thegeophone and source coordinates, respectively. For notationalconvenience,ω has been dropped from the pressure and theGreen’s functions terms. The RTM operator can be derived byapplying the adjoint operation on equation (6) as,

m(x)≈∫

ω

∫

s

∫

gG(x|s)∗P(g,s)G(g|x)∗dgdsdω. (7)

Using a matrix-vector notation, the forward modeling opera-tion in equation (6) can be written as,

d = Lm, (8)

and the adjoint operation or the RTM operation can be writtenas (Claerbout and Green (2008)),

m ≈ LT d. (9)

The reflectivity model,m(x), can be determined using a least-squares method by minimizing the misfit function,ε, as (Nemethet al. (1999), Dai et al. (2012)),

ε =12||Lm−dobs||2. (10)

Equation (10) can be iteratively solved using any gradient basedmethod like steepest descent as,

m(i+1) = m(i)−αg(i),

g(i) = LT (Lm(i)−dobs),

α =(g(i))T g(i)

(Lg(i))T (Lg(i)). (11)

Here,m(i) represents the reflectivity model at thei-th iteration,g(i) andα represent the gradient and the step-length, respec-tively. The gradient is determined by reverse time migrationof the data residuals between the born-modeled data and theobserved data which has been calculated using a full finite dif-ference simulation of the visco-acoustic wave equation givenin equation (4).

Without Qp

X(m)

Z(m

)

100 300 500

100

300

500

Qp=100

X(m)100 300 500

100

300

500

Qp=20

X(m)100 300 500

100

300

500

(a)

100 200 300 400 500−1

−0.5

0

0.5

1Snapshots at z = 250 m when t = 0.8 ms

X(m)

Am

plitu

de

Without QpQp=100Qp=20

(b)

Figure 1: (a) Wavefield Snapshots att = 0.8 ms for a homo-geneous medium showing the effect of attenuation for differ-ent values ofQP. (b) The amplitude and phase distortion dueto very low value ofQP can be seen by comparing the threetraces.

NUMERICAL EXAMPLES

The proposed LSRTM algorithm using the visco-acoustic waveequation is tested on a modified Marmousi model shown inFigure 2(a). TheQP distribution, shown in Figure 2(b), is esti-mated from the true velocity model in Figure 2(a) by assigningthe layers with velocities ranging between 2500-3000m/s tohave very strong attenuation or very lowQP. The migration ve-locity model, shown in Figure 2(c), is obtained by smoothingthe true velocity model. TheQP model for migration, shownin Figure 2(d), is estimated from the migration velocity modelusing a similar mapping of velocity andQP values as in thecase of the trueQP model.

A 2-8 time-space domain staggered grid finite-difference methodis used for visco-acoustic and acoustic RTM and LSRTM in allcases. There are 115 shots fired at a source interval of 40m andthere are 230 receivers evenly distributed on the surface ataninterval of 20m. A fixed spread acquisition geometry is chosenas all the receivers are used for each source. A Ricker waveletwith a 20Hz peak frequency is used as the source wavelet. Theobserved data having strong attenuation is generated usingthevelocity and theQP models shown in Figures 2(a) and (b), re-spectively. To see the effect of attenuation, the same data ismigrated first by taking attenuation into account by using thevisco-acoustic wave equation for extrapolation of the sourceand receiver wavefields and then migrated again by ignoringthe effect of attenuation and using the first order acoustic waveequations. Source side illumination is used as the precondi-tioning factor during LSRTM in both cases.

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Attenuation LSRTM

Z (

km)

X (km)

(a) True Velocity Model

(m/s)

2 4

1

2

2000

3000

4000

Z (

km)

X (km)

(b) True Qp Model

2 4

1

2

20

10000Z

(km

)

X (km)

(c) Migration Velocity Model

(m/s)

2 4

1

2

2000

3000

4000Z

(km

)

X (km)

(d) Migration Qp Model

2 4

1

2

20

10000

Figure 2: The (a) true velocity model, and (b) the trueQPmodel used for generating the observed data. (c) Velocitymodel, and (d)QP model used for RTM and LSRTM.

Figures 4(a) and 4(c) compare the RTM images when the datahaving strong attenuation is migrated using the acoustic andthe visco-acoustic wave equations, respectively. The LSRTMimages after 20 iterations are compared in Figures 4(b) and4(d) for both the cases. Zoomed sections of the images in Fig-ures 4(b) and 4(d) are shown in Figure 5. It is evident thatLSRTM using the acoustic and visco-acoustic wave equationsprovide similar results in shallow layers where there is very lit-tle or no attenuation. However, in deeper layers, where the at-tenuation effect is very strong, the LSRTM image obtained byusing the visco-acoustic wave equation shows better resolutionand focusing compared to the LSRTM image obtained usingthe acoustic wave equation. Also, the migrated amplitudes inthe deeper layers are better balanced in the visco-acousticcase.Careful analysis of the images in Figures 5(b) and 5(e) also re-veals that the reflectors in the deeper layers are slightly mispo-sitioned when compared to the true reflectivity model. How-ever, for the visco-acoustic LSRTM images in Figures 5(c) and5(f), all the reflectors are imaged at the right locations. Thiscan be attributed to the fact that strong attenuation not only af-fects the amplitudes but also the phase of the events and thiseffect becomes more prominent for the waves that travel largedistances and through the high attenuative layers. Migrationwithout taking attenuation because ofQP into considerationcannot correct for this distortion. The convergence curvesforacoustic and visco-acoustic LSRTM are also compared in Fig-ure 3. The convergence in this case is better for visco-acousticLSRTM since the correct physics is accounted for during themodeling and the adjoint operations.

CONCLUSIONS

We presented a least-squares reverse time migration methodusing the visco-acoustic wave equation to compensate for thedistortion in amplitude and phase of seismic waves propagat-

ing in highly attenuative layers. Numerical results using theMarmousi model show that conventional LSRTM using theacoustic wave equation cannot correct for the attenuation loss.However, when the visco-acoustic wave equation is used dur-ing LSRTM, the attenuation effect is correctly accounted for.LSRTM using the visco-acoustic wave equation produces im-ages with better balanced amplitudes and accurately positionedreflectors below highly attenuative layers compared to the acous-tic LSRTM images. Similar to acoustic LSRTM, visco-acousticLSRTM is also sensitive to errors in the migration velocitymodel. Estimation ofQP from real data is also difficult. TheQP tomography methods suggested by Liao and McMechan(1996) and Liao and McMechan (1997) is one possible so-lution to get a startingQP model for visco-acoustic LSRTM.More accurate estimation ofQP should be emphasized on, es-pecially during processing, for accurate imaging in gas-sandsto-nes and shales.

5 10 15 20

0.2

0.4

0.6

0.8

1Convergence Plots

Number of Iterations

Nor

mal

ized

Dat

a R

esid

ual

Visco−acoustic LSRTMAcoustic LSRTM

Figure 3: Convergence curves for acoustic and visco-acousticLSRTM.

ACKNOWLEDGMENTS

We thank the sponsors of the Center for Subsurface Imag-ing and Modeling (CSIM) consortium for their support andKAUST Supercomputing Laboratory for providing the com-putational resources.

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Attenuation LSRTM

(a) Acoustic RTM ImageZ

(km

)

0 2 4

0

1

2

(b) Acoustic LSRTM Image

0 2 4

0

1

2

(c) Visco−acoustic RTM Image

X (km)

Z (

km)

0 2 4

0

1

2

(d) Visco−acoustic LSRTM Image

X (km)0 2 4

0

1

2

Figure 4: Comparison between images of (a) acoustic RTM, (b)acoustic LSRTM, (c) visco-acoustic RTM, and (d) visco-acousticLSRTM. All the figures have been plotted in the same scale.

(d) True Reflectivity

3 3.5 4

1.8

2.2

(e) Acoustic LSRTM

3 3.5 4

1.8

2.2

(a) True Reflectivity

Z (

km)

1.2 1.6 2

1.8

2.2

(b) Acoustic LSRTM

Z (

km)

1.2 1.6 2

1.8

2.2

(c) Visco−acoustic LSRTM

X (km)

Z (

km)

1.2 1.6 2

1.8

2.2

(f) Visco−acoustic LSRTM

X (km)3 3.5 4

1.8

2.2

Figure 5: Zoomed view of the black (left) and blue (right) boxes in Figure 4. (a), (d) True reflectivity models used only forcomparing the LSRTM images, (b), (e) acoustic LSRTM images,and (c), (f) visco-acoustic LSRTM images. All the figures havebeen plotted in the same scale.

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http://dx.doi.org/10.1190/segam2013-1131.1 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2013 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

Aki, K., and P. G. Richards, 1980, Quantitative seismology: Freeman.

Baysal, E., D. D. Kosloff, and J. W. Sherwood, 1983, Reverse time migration: Geophysics, 48, 1514–1524.

Blanch, J. O., J. O. Robertsson, and W. W. Symes, 1995, Modeling of a constant q: methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique: Geophysics, 60, 176–184.

Carcione, J. M., D. Kosloff, and R. Kosloff, 1988, Wave propagation simulation in a linear viscoacoustic medium: Geophysical Journal International, 93, 393–401.

Christensen, R. M., 1982, Theory of viscoelasticity: An introduction: Academic Press.

Claerbout, J. F., and I. Green, 2008, Basic earth imaging: Citeseer.

Dai, W., P. Fowler, and G. T. Schuster, 2012, Multi-source least-squares reverse time migration: Geophysical Prospecting, 60, 681–695.

Liao, Q., and G. A. McMechan, 1996, Multifrequency viscoacoustic modeling and inversion: Geophysics, 61, 1371–1378.

Liao, Q., and G. A. McMechan, 1997, Tomographic imaging of velocity and Q, with application to crosswell seismic data from the Gypsy pilot site, Oklahoma: Geophysics, 62, 1804–1811.

McMechan, G., 1983, Migration by extrapolation of time-dependent boundary values: Geophysical Prospecting, 31, 413–420.

Nemeth, T., C. Wu, and G. T. Schuster, 1999, Least-squares migration of incomplete reflection data: Geophysics, 64, 208–221.

Robertsson, J. O., J. O. Blanch, and W. W. Symes, 1994, Viscoelastic finite-difference modeling: Geophysics, 59, 1444–1456.

Thorbecke, J. W., and D. Draganov, 2011, Finite-difference modeling experiments for seismic interferometry: Geophysics, 76, no. 6, H1–H18.

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