Waveform inversion with attenuation Jianyong Bai*, David Yingst, Robert Bloor, and Jacques Leveille, ION Geophysical

Summary We present the viscoacoustic wave equation and its adjoint to compensate for the attenuation of the seismic data in waveform inversion. The novelty is that these equations enable us to extrapolate wavefields by high-order finite-difference methods in centered grids rather than in staggered grids. This significantly reduces computation cost and memory requirements. We also present the stability requirement for the finite-difference methods. Numerical examples demonstrate attractive attenuative behaviors in amplitude attenuation, phase dispersion and amplitude spectrum. Tests with a modified Marmousi model clearly show some promise in making waveform inversion more realistic for real earth materials. Introduction Waveform inversion was introduced using a gradient-based optimization method (Lailly, 1983; Tarantola, 1984, 1987). It estimates subsurface parameters by iteratively minimizing the cost function of the difference between recorded seismic data and synthetic seismic data. It is attractive in its ability to produce parameter models with high resolution. Related research work has been done in both time and frequency domains for isotropic and anisotropic media (Yingst et al., 2011; Wang et al., 2012). Today it has become a practical tool in the petroleum industry. Currently most applications apply waveform inversion with acoustic or elastic models. This is a good choice for a lot of practical applications. However, real earth attenuates and disperses waves due to the conversion of elastic energy into heat. This anelastic behavior can decrease amplitude and distort the wavelet and thus can have significant effects on waveform inversion. For example, gas clouds can cause strong attenuation on seismic waves. The attenuation effects can cause the amplitude below the gas clouds to be anomaly dim. As a result, they can reduce the resolution of inversion results or can even cause inversion to produce erroneous results. So it is important to compensate for the anelastic behavior to make the final results more reliable. Viscoelasticity provides a powerful tool to model real earth materials (Robertsson et al., 1994). In a viscoelastic model viscoelastic relaxation functions are applied. A superposition of standard linear solids (SLS) in parallel is controlled by a set of relaxation parameters to simulate the attenuation and dispersion effects that the real earth materials have on wave propagation. The relaxation

parameters can be obtained from certain relationships between the quality factor Q and frequency to approximate a specific viscoelastic model (Blanch et al., 1995). In many applications a single SLS suffices to compensate for the anelastic behavior. Finite-difference wavefield extrapolations implemented in staggered grid have shown that viscoelastic wave equations can simulate wave propagation well (Robertsson et al., 1994; Larsen et al., 1998). In this paper we limit our work to viscoacoustic media. We derive and present the viscoacoustic wave equation and its adjoint to compensate for the attenuation of waves in waveform inversion. Compared to the wave equation in acoustic media, the viscoacoustic wave equation and its adjoint involve a time convolution. We implement the equations by high-order finite-difference methods in centered grids. The stability requirement for the finite-difference schemes is given through a Von Neumann stability analysis. Numerical examples demonstrate attractive attenuative behaviors in amplitude attenuation, phase dispersion and amplitude spectrum. Tests with a modified Marmousi model clearly show some promise in making waveform inversion more realistic for real earth materials. We are now processing 3D real dataset and will have results shortly. Theory The waveform inversion is implemented by minimizing the misfit function (Tarantola, 1987)

€

minvJ[v] = d0 − d

2

, (1)

where d0 = d0(xr,t,xs) is the recorded seismic data and d = d(xr,t,xs) is the synthetic seismic data for a velocity model v at source and receiver positions xs and xr, respectively. The velocity model is updated according to the gradient

€

g(x) = −1v 3

∂ 2P∂t 2

Rt∑

x s

∑

, (2)

where v = v(x) is the velocity at position x, P = P(x,t;xs) is the predicted wavefield at time t obtained from a forward modeling operator and R = R(x,t;xs) is the wavefield obtained by applying the adjoint of forward modeling on the residual d0-d. In viscoacoustic media the predicted wavefield can be obtained by the relationship between pressure and particle velocity for a single SLS (Robertsson et al., 1994)

© 2012 SEG DOI http://dx.doi.org/10.1190/segam2012-1305.1SEG Las Vegas 2012 Annual Meeting Page 1

Dow

nloa

ded

11/1

3/12

to 2

04.2

7.21

3.16

1. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

€

∂P∂t

= −∂[κ(1+τe−t /τσ )H(t)]

∂t*∇⋅ v + f

, (3)

€

∂v∂t

= −1ρ∇P

, (4)

where ρ = ρ(x) is the density, κ = κ(x) is the bulk modulus, v = v(x,t) is the particle velocity vector and f = f(xs,t) is the source term at xs, and H(t) is the Heaviside function. The symbol * stands for a convolution in time domain in equation (3). τσ and τε are respectively stress and strain

relaxation times and τ = τε/τσ - 1. For a reference frequency the relaxation times can be calculated from the quality factor Q according to Blanch et al. (1995). Finite-difference methods in staggered grids are commonly used to extrapolate the wavefield P based on the equations (3) and (4). After differentiating equations (3) and (4) in time, substituting one equation into the other and some algebra, the following equation is obtained:

€

1v 2∂ 2P∂t 2

= (1+τ )ρ∇⋅ ( 1ρ∇P) − r + f

, (5)

where the memory variable r is defined as

€

r =ττσ[e

−tτσ H(t)]*[ρ∇⋅ ( 1

ρ∇P)]

. (6)

r describes the pressure history. In order to accelerate the convolution in equation (6), a recursive convolution method is used

€

∂r∂t

=ττσρ∇⋅ ( 1

ρ∇P) − 1

τσr

. (7)

r is propagated on the same grids as used for the wavefield propagation. The equations (5) and (7) are the forward modeling operator in viscoacoustic media. The adjoint T* of an operator T has the property <Tx, y> = <x, T*y>. After some manipulations we can verify that the adjoint operator solves the following equation

€

1v 2∂ 2P∂t 2 =∇⋅

1ρ∇(1+τ)ρP −∇⋅

1ρ∇ρ˜ r + f

, (8)

where f is the residual d0-d. According to the adjoint-state method the residual is backward propagated by reducing time using the adjoint.

€

˜ r is a memory variable and is responsible for the anelastic behavior.

€

˜ r =ττσ

[etτσ H(−t)]* P

. (9)

We update

€

˜ r according to a recursive convolution method as following

€

∂˜ r ∂t

= −ττσ

P +1τσ

˜ r , (10)

The equations (5), (7), (8) and (10) present all operators for waveform inversion with attenuation compensation. They enable us to extrapolate wavefields using high-order finite-difference methods in centered grids. This can significantly reduce computation time and memory requirements. Finite-difference solutions require determinations of spatial and temporal sampling criteria to avoid grid dispersion. Following the method originally developed by Von Neumann for the stability analysis of finite-difference solutions (Charney et al., 1950), we have

€

Δt ≤ 2vmax a(1+τmax )

, (11)

with

€

a =1Δx 2

wnx

n=−N

N

∑ +1Δy 2

wny +

wnz

Δzn2

n=−N

N

∑n=−N

N

∑ , (12)

where vmax is the maximum velocity in a velocity cube, τmax is the maximum value of a τ cube, Δt is the temporal step size, Δx, Δy and Δz are the spatial sampling intervals along the x, y, and z-axis, respectively, and w’s are finite differencing coefficients in the x, y, and z-directions with (2N)th order accuracy. In order to reduce computation time, the interval Δz can be non-uniform. Examples We first consider a constant density model and a constant velocity model (v = 1500 m/s) with an exploding source at the middle of the surface. A heterogeneous Q model shown in Figure 1(a) is used. In the Q model the attenuation is strong in the lower left corner while it can be ignored in the rest. We use a 15Hz Ricker wavelet to do forward modeling. A 3D snapshot at 1s shown in Figure 1(b) clearly demonstrates amplitude attenuation and phase dispersion caused by attenuation. As expected, we see large attenuation effects in the left part of depth slice and in the lower left part of in-line section. Next a modified Marmousi model is used. The modified model includes a water layer from the surface down to 1000m (Figure 2(a)). A Q model shown in Figure 2(b) is directly mapped from the velocity model. The attenuation in water is very weak (Q = 5000) while below the water layer it is strong since Q ranges from 20 to 60. We generate synthetic data with 59 shots. The shot interval is 200m. Each shot has 241 receivers. The receiver interval is 20m. A shot gather is displayed in Figure 3(a). For the same acquisition geometry a shot gather (Figure 3(b)) is also generated without attenuation. Amplitude spectra (Figure 4) from 2 different windows are obtained from the synthetic data. Obviously seismic attenuation increases with traveltime since the seismic wave takes more oscillations along a longer path. They also indicate that the

Viscoacoustic waveform inversion

© 2012 SEG DOI http://dx.doi.org/10.1190/segam2012-1305.1SEG Las Vegas 2012 Annual Meeting Page 2

Dow

nloa

ded

11/1

3/12

to 2

04.2

7.21

3.16

1. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

seismic attenuation increases with frequency since the seismic wave oscillates more often and dissipates more energy. Both viscoacoustic and acoustic waveform inversions are applied on the synthetic data shown in Figure 3(a). We start the inversions from a velocity model shown in Figure 5(a). In the starting model the velocity below the water layer changes with depth. So the model is far away from the true model. The Q model shown in Figure 2(b) is used for the viscoacoustic inversion. We weight far-offset data more than near-offset data and only use frequencies below 8Hz. The exactly same multi-scale approach is carried out from low to high frequency for the inversions. The final velocity model from the viscoacoustic inversion is displayed in Figure 5(b) and the final velocity model from the acoustic inversion is displayed in Figure 5(c). Both inversions produce results comparable to the true velocity model shown in Figure 2(a). However, compared to the final model from the viscoacoustic inversion, the final model from the acoustic inversion lucks resolution. The geological structure in the lower middle area is complex in the true velocity model. The acoustic inversion gives incorrect solutions in this area while the viscoacoustic inversion still

makes geological sense. This numerical example clearly demonstrates that the seismic attenuation has large effects on waveform inversion. The attenuation effects should be compensated where strong attenuation presents. Conclusions This paper presents the viscoacoustic wave equation and its adjoint to compensate for seismic attenuation in waveform inversion. We implement them by stable finite-difference schemes in centered grids with high-order accuracy. Numerical examples demonstrate attractive attenuative behaviors and thus indicate that the viscoacoustic equation can handle complex Q models well. The tests with the modified Marmousi model clearly show some promise in making waveform inversion more realistic for real earth materials. Acknowledgements We thank our colleagues in ION Geophysical for their helpful discussions, especially Guoquan Chen, Herman Jaramillo, Chao Wang, Helen Delome, Paul Farmer, Adam Gersztenkom, and David Brookes.

Figure 1: (a) A heterogeneous Q model. Red color means weak attenuation (Q = 5000) and cyan color means strong attenuation (Q = 20). (b) A snapshot at 1s obtained from the 3D Q model.

Figure2: (a) A modified Marmousi velocity model includes a water layer from the surface down to 1000m. The water layer is not shown. (b) A Q model is obtained from the velocity model. The attenuation in water is weak (Q = 5000).

Viscoacoustic waveform inversion

© 2012 SEG DOI http://dx.doi.org/10.1190/segam2012-1305.1SEG Las Vegas 2012 Annual Meeting Page 3

Dow

nloa

ded

11/1

3/12

to 2

04.2

7.21

3.16

1. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Figure 3: Shot gathers generated from the modified Marmousi velocity model with attenuation (a) and without attenuation (b).

Figure 4: Amplitude spectra generated from the yellow box (a) and the red box (b) shown in Figure 3(a). The blue curve is obtained from the shot gather without attenuation and the red curve from the shot gather with attenuation.

Figure 5: (a) The starting velocity model for inversions. (b) The final velocity model from viscoacoustic waveform inversion. (c) The final velocity model from acoustic waveform inversion.

Viscoacoustic waveform inversion

© 2012 SEG DOI http://dx.doi.org/10.1190/segam2012-1305.1SEG Las Vegas 2012 Annual Meeting Page 4

Dow

nloa

ded

11/1

3/12

to 2

04.2

7.21

3.16

1. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

http://dx.doi.org/10.1190/segam2012-1305.1 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2012 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

Blanch, J. O., J.O.A. 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.

Charney, J. G., R. Fjørtoft, and J. von Neumann, 1950, Numerical integration of the barotropic vorticity equation: Tellus 2, 237–254.

Lailly, P., 1983, The seismic inverse problem as a sequence of before-stack migration, in J. Bednar, ed., Conference on inverse scattering: Theory and applications: Society for Industrial and Applied Mathematics, 206–220.

Larsen, S., and J. Grieger, 1998, Elastic modeling initiative, Part III: 3D computational modeling: Presented at the 68th Annual International Meeting, SEG, Expanded Abstracts, 68, 1803–1806.

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

Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation: Geophysics, 49, 1259–1266.

Tarantola, A., 1987, Inverse problem theory: Elsevier.

Wang, C., D. Yingst, R. Bloor, and J. Leveille , 2012, Application of VTI waveform inversion with regularization and preconditioning to real 3D data: presented at 74th Annual International Conference and Exhibition, EAGE.

Yingst, D., C. Wang, J. Park, R. Bloor, J. P. Leveille , and P. Farmer, 2011, Application of time domain and single-frequency waveform inversion to real data: Presented at the 73rd Annual International Conference and Exhibition, EAGE.

© 2012 SEG DOI http://dx.doi.org/10.1190/segam2012-1305.1SEG Las Vegas 2012 Annual Meeting Page 5

Dow

nloa

ded

11/1

3/12

to 2

04.2

7.21

3.16

1. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/