Modeling acoustic wave propagation in heterogeneous attenuatingmedia using decoupled fractional Laplacians

Tieyuan Zhu1 and Jerry M. Harris1

ABSTRACT

We evaluated a time-domain wave equation for modelingacoustic wave propagation in attenuating media. The waveequation was derived from Kjartansson’s constant-Q constitu-tive stress-strain relation in combination with the mass and mo-mentum conservation equations. Our wave equation, expressedby a second-order temporal derivative and two fractional Lap-lacian operators, described very nearly constant-Q attenuationand dispersion effects. The advantage of using our formulationof two fractional Laplacians over the traditional fractional timederivative approach was the avoidance of time history memoryvariables and thus it offered more economic computations.In numerical simulations, we formulated the first-order constit-utive equations with the perfectly matched layer absorbingboundaries. The temporal derivative was calculated with a

staggered-grid finite-difference approach. The fractional Lapla-cians are calculated in the spatial frequency domain using a Fou-rier pseudospectral implementation. We validated our numericalresults through comparisons with theoretical constant-Q attenu-ation and dispersion solutions, field measurements from thePierre Shale, and results from 2D viscoacoustic analytical mod-eling for the homogeneous Pierre Shale. We also evaluated dif-ferent formulations to show separated amplitude loss anddispersion effects on wavefields. Furthermore, we generalizedour rigorous formulation for homogeneous media to an approxi-mate equation for viscoacoustic waves in heterogeneous media.We then investigated the accuracy of numerical modeling in at-tenuating media with differentQ-values and its stability in large-contrast heterogeneous media. Finally, we tested the applicabil-ity of our time-domain formulation in a heterogeneous mediumwith high attenuation.

INTRODUCTION

Conventional seismic modeling approaches usually ignore at-tenuation effects on wavefields. In earth media, however, attenua-tion causes reduced energy and a distorted phase of seismic waves,especially when the attenuation is high due to the presence of, forexample, gas pockets (Dvorkin and Mavko, 2006). To accuratelycharacterize wave propagation in real media, attenuation and asso-ciated dispersion effects should be taken into account in seismicwavefield modeling.Over the past three decades, much work has been done to model

acoustic attenuation effects during wave propagation. Basically,these forward-modeling approaches are split into two categories:One category implements attenuation in the frequency domainby allowing the velocity to be complex (Aki and Richards,1980; Liao and McMechan, 1996; Štekl and Pratt, 1998). The otherintroduces Q in a time-domain wave equation. Time-domain ap-

proaches commonly use superposition of mechanical elements(e.g., Maxwell and standard linear solid [SLS] elements) to describeQ behavior, which are known as approximate constant-Q (ACQ)models (Liu et al., 1976; Emmerich and Korn, 1987; Carcione,2007; Zhu et al., 2013). Nevertheless, the memory and computationtime requirements, especially for, say, 3D applications, may limitthese approaches from being widely applied for seismic migrationand inversion. Instead of such ACQ models, use of a constant-Qmodel (Kjartansson, 1979) in the time-domain wave equation is at-tractive because it is accurate in producing desirable constant-Qbehavior at all frequencies in the band. However, rigorous use ofsuch a constant-Qmodel introduces a fractional time derivative, i.e.,noninteger (irrational or fractional) power of the time derivative(Caputo and Mainardi, 1971; Mainardi and Tomirotti, 1997).Solving the fractional wave equation has been quite common in

mathematics, acoustics, and bioengineering (e.g., Mainardi, 2010;

Manuscript received by the Editor 3 July 2013; revised manuscript received 14 December 2013; published online 11 March 2014.1Stanford University, Department of Geophysics, Stanford, California, USA. E-mail: tyzhu@stanford.edu; jerry.harris@stanford.edu.© 2014 Society of Exploration Geophysicists. All rights reserved.

T105

GEOPHYSICS, VOL. 79, NO. 3 (MAY-JUNE 2014); P. T105–T116, 12 FIGS.10.1190/GEO2013-0245.1

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Treeby and Cox, 2010; Caputo et al., 2011). Recently, this fractionalcalculus was introduced to model seismic wave propagation (Car-cione et al., 2002; Carcione, 2009, 2010). Carcione et al. (2002) andCarcione (2009) simulate acoustic and elastic waves using the con-stant-Q wave equation with the fractional time derivative that wascomputed by the Grünwald-Letnikov (GL) approximation (Pod-lubny, 1999). Nevertheless, it is notable that the fractional timederivative of a single variable depends on all the previous valuesof this variable. This property requires a large memory of stress-strain history from time t ¼ 0 (Carcione, 2010). Therefore, innumerical simulations we have the expense of memory resourceseven though in some situations it is possible to truncate the operator(Podlubny, 1999; Carcione et al., 2002).To overcome this memory expense, Lu and Hanyga (2004) in-

troduce a set of secondary internal variables in addition to the pri-mary internal variables to solve fractional differential equations ofarbitrary order. Nevertheless, additional computational costs are stillpaid. Without introducing extra variables, Chen and Holm (2004)propose to use the fractional Laplacian operator to model anoma-lous attenuation behavior in fractional wave equations for homo-geneous media. The fractional Laplacian is easily evaluated inthe spatial frequency domain using the generalized Fourier ap-proach (Carcione, 2010). As a result, the fractional Laplacian suc-cessfully avoids additional memory required by the fractional timederivative. Carcione (2010) gives a homogeneous constant-Q waveequation using the fractional Laplacian. His formulation capturesamplitude loss and velocity dispersion in a single term. Separationof amplitude loss and dispersion operators in the fractional Lapla-cian wave equation may be preferable because separated forms aremore useful in compensating for attenuation loss in inverse problems,e.g., reverse time imaging by only reversing sign of the attenuationoperator and leaving the sign of the dispersion operator unchanged(Treeby et al., 2010; Zhu, 2014; Zhu et al., 2014). Zhang et al. (2010)also derive the approximate constant-Q wave equation with de-coupled amplitude loss and dispersion effects, but their derivationusing the normalization transform for decoupling amplitude lossand dispersion is not clearly described in the abstract.Following Treeby and Cox’s (2010) approach, here, we derive a

time-domain wave equation using the fractional Laplacian to modelconstant-Q behavior. Starting fromKjartansson’s constant-Q constit-utive stress-strain relation, we present the formulation for homo-geneous media and then generalize to smoothly heterogeneousmedia. This fractional Laplacian based nearly constant-Q (NCQ)wave equation introduces separated amplitude loss and phase veloc-ity dispersion operators based on the fractional Laplacian. Fornumerical simulations, perfectly matched layer (PML) absorbingboundaries are incorporated with the first-order constant-Q conser-vation equations. The spatial derivative is computed by a staggered-grid pseudospectral method while a staggered-grid finite-differenceapproach is used for temporal discretization.The paper is organized as follows: The beginning is a review of

the constant-Q stress-strain relation and a derivation of the frac-tional time derivative wave equation. Next, we provide detailed der-ivations of the fractional NCQ wave equation for homogeneous andheterogeneous media. Following that, we describe numerical ex-periments for a model of the homogeneous Pierre Shale to inves-tigate the accuracy of our modeling by comparison with theoreticalconstant-Q solutions and for a two-layer model of heterogeneousmedium to investigate stability for large-contrast heterogeneous

media. Finally, we test the capability and efficiency of our formu-lation for heterogeneous media.

CONSTANT-Q AND FRACTIONAL LAPLACIAN

Constant-Q stress-strain relation and fractional timederivative

Even though there is evidence for frequency-dependent Q fromfield and laboratory data (Kan et al., 1983; Sams et al., 1997; Batzleet al., 2006), in this paper we focus on a mathematical and com-putational frequency-independent Q model, which is consideredto be a practical approximate Q model for seismology problems(McDonal et al., 1958; Aki and Richards, 1980). Kjartansson(1979) explicitly gives a linear description of attenuation thatexhibits the exact constant-Q characteristic. His formulation usesfractional derivatives that are based on a relaxation function ofthe form t−2γ , instead of integer power of t. The relaxation functionis given as

ψ1−2γðtÞ ¼M0

Γð1 − 2γÞ�tt0

�−2γ

HðtÞ; t > 0; (1)

where M0 ¼ ρ0c20 cos2ðπγ∕2Þ is a bulk modulus, ρ0 is the acoustic

density, Γ is Euler’s gamma function, and t0 is a reference time(t0 ¼ 1∕ω0). The parameter γ ¼ arctanð1∕QÞ∕π is dimensionless,and we know that 0 < γ < 0.5 for any positive value of Q. HðtÞis the Heaviside step function. Note that only three parameters,the phase velocity c0 at a reference frequency ω0 and the Q, areneeded to describe loss mechanisms, rather than several relaxationtimes and the complex modulus in the SLS models (Car-cione, 2007).In terms of the Caputo’s fractional derivative (Caputo, 1967), the

isotropic stress-strain (σ-ε) relation may be deduced in the follow-ing form:

σ ¼ ψ1−2γðtÞ �dεdt

¼�M0

t−2γ0

�∂2γε∂t2γ

; (2)

where the symbol * denotes the convolution operator. The strain εhas the fractional order 2γ of the time derivative. According to equa-tion 2, the behavior of the fractional time derivative ∂2γε∕∂t2γ de-pends on the characteristics of the relaxation function ψ1−2γðtÞ.Figure 1 shows how ψ1−2γðtÞ varies with Q-values (0.1, 1, 10,and 100). When Q is large, the relaxation function ψ1−2γðtÞ decaysslowly and tends toward the step function. Consequently, the frac-tional time derivative ∂2γε∕∂t2γ approaches ε. When Q is infinite(the elastic case), the current stress will be linear with the currentstrain σ ¼ M0ε.When Q tends toward zero (infinite attenuation), the function

ψ1−2γðtÞ approaches a one-sided delta function and thus the frac-tional time derivative ∂2γε∕∂t2γ reduces to ∂ε∕∂t. When Q lies be-tween zero and infinity, the calculation of the fractional timederivative depends on the state of previous time steps; i.e., the cur-rent stress must be calculated from the time history of the strain. Asa result, numerical evaluation of a fractional time derivative operatorrequires large memory and computation time to store and use thetime history (Carcione et al., 2002; Carcione, 2009; Caputo et al.,2011), even if it is possible to truncate its operator based on someshort memory principle (Podlubny, 1999; Carcione et al., 2002).

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The fractional Laplacian

To overcome this memory requirement and the associatednumerical difficulties, the fractional Laplacian operator ð−∇2Þα∕2is proposed by Chen and Holm (2004) to model power law attenu-ation behavior instead of the fractional time derivative (∂α∕∂tα,where α is real). In this case, the fractional Laplacian, calculatedin the spatial domain, avoids storage of the wavefields from pre-vious time steps. When incorporating this with a Fourier pseudo-spectral numerical scheme, the fractional Laplacian can beimplemented without any particular numerical difficulties.The Fourier transform of the fractional Laplacian is written as

(Chen and Holm, 2004)

ð−∇2Þα∕2ϕðr; tÞ→F kαΦðk;ωÞ; 0 < α < 2; (3)

where k is the spatial wavenumber, ω is the angular frequency, andthe symbol F is the Fourier transform operator. The expressionΦðk;ωÞ is the 2D Fourier transform of the space-time functionϕðr; tÞ; i.e.,

Φðk;ωÞ ¼Z

∞

−∞

Z∞

−∞ϕðr; tÞe−iðωt−krÞdrdt: (4)

From equation 3, we see that the inverse Fourier transform F−1 isalso well defined as kαΦðk;ωÞ→F

−1

ð−∇2Þα∕2ϕðr; tÞ. More detaileddescriptions of the fractional Laplacian can be found in Gorenfloand Mainardi (1998), Hanyga (2001), Chen and Holm (2004),and Carcione (2010).

A constant-Q wave equation usingthe fractional Laplacian

For homogeneous acoustic media, the linear first-order momen-tum conservation equation can be written as

∂∂tυ ¼ 1

ρ0∇σ; (5)

and the strain-velocity equation is

∂∂tε ¼ ∇ · υ; (6)

where σ is the stress field, ε is the strain field, and υ ¼ ðυx; υy; υzÞ isthe acoustic particle velocity vector.Combining equation 2, the stress-strain relation, with equations 5

and 6, the first-order conservation equations, the fractional waveequation with constant density ρ0 is obtained:

∂2−2γσ∂t2−2γ

¼ c2ω−2γ0 ∇2σ; (7)

where ∇2 is the Laplacian operator and c2 ¼ M0∕ρ0 ¼c20 cos

2ðπγ∕2Þ. The exponent 2 − 2γ is the fractional order of thetime derivative, the so-called fractional time derivative. It is evidentthat equation 7 reduces to the classical acoustic wave equation asγ → 0 (Q → ∞), and the diffusion equation as γ → 1∕2 (Q → 0).This fractional wave equation is also well-known as the Caputo’swave equation. Carcione et al. (2002) successfully solve thefractional time wave equation using Grünwald-Letnikow and

central-difference approximations for the time discretization andFourier method to compute the spatial derivative. Recently, Car-cione (2009) extends its derivation to build the elastic wave equa-tion with constant-Q. In the following, we will replace the fractionaltime derivative with the fractional Laplacian operator.The dispersion relation for fractional wave equation 7 can be ob-

tained by substituting the plane wave solution expðiωt − i ~k · rÞ,where ~k is the complex wavenumber vector, ω is the angular fre-quency, and r is the spatial coordinate vector:

ω2

c2¼ ðiÞ2γω−2γ

0 ω2γ ~k2: (8)

The attenuation and phase dispersion are derived in Appendix A.After some calculations (see Appendix B), we obtain a newdispersion relation as

ω2

c2≈ c2γ0 ω−2γ

0 cosðπγÞ ~k2γþ2 þ ðiωÞc2γ−10 ω−2γ0 sinðπγÞ ~k2γþ1:

(9)

Equation 9 approximates equation 8; thus, we refer to it as a NCQdispersion relation. Figure 2 shows how well the attenuation anddispersion in equation 9 (dashed line) approximate to the exact con-stant-Q attenuation and dispersion (see Appendix A) in equation 8(solid line) using different Q values (2, 10, and 100). The other me-dia parameters used in Figure 2 are listed in the first example of thenumerical experiments section. One can see that the differences in-crease with decreasing Q value, even though they match very wellfor relatively high Q values (low attenuation).In homogeneous media, velocity c0 and Q are independent of

the space variable. We can apply 2D inverse spatial and temporalFourier transform (see the notation in equation 4) to equation 9,resulting in the fractional Laplacian ~k2γþ2 → ð−∇2Þγþ1 and~k2γþ1 → ð−∇2Þγþ1∕2; thus, we obtain the NCQ wave equationfor homogeneous media:

−1 0 1 2 3 4 50

0.5

1

1.5

2

Time (s)

Ψ1−

2γ (

t)

Q = 0.1

Q = 1

Q = 10

Q = 100

Figure 1. The relaxation function ψ1−2γðtÞ varying with the pre-vious time t in the constant-Q model. The curves denote valuesof Q ¼ 0.1 (solid), 1 (dash), and 10 (triangle), and 100 (circle).Note that whenQ is small, the curve approximates a one-sided deltafunction. In contrast, when Q is large, the dependence becomesmore flat (toward a step function).

A nearly constant-Q wave equation T107

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1

c2∂2σ∂t2

¼ ηð−∇2Þγþ1σ þ τ∂∂tð−∇2Þγþ1∕2σ; (10)

where the coefficients are

η ¼ −c2γ0 ω−2γ0 cosðπγÞ; τ ¼ −c2γ−10 ω−2γ

0 sinðπγÞ: (11)

Note that equation 10 essentially serves the same purpose as equa-tion 7, the fractional time wave equation. When Q is infinite (noattenuation), i.e., γ ¼ 0, our fractional Laplacian equation 10 re-duces exactly to the familiar acoustic wave equation. Numerical sta-bility analysis of equation 10 is shown in Appendix C.

Freezing-unfreezing theory for heterogeneous media

In smoothly heterogeneous media c0ðrÞ and QðrÞ are assumed tovary smoothly in space. Using the locality principle (freezing-un-freezing) of the pseudodifferential operator (Stein [1993], p. 230),we now adapt the fractional NCQ wave equation for homogeneousmedia (equation 10) to describe propagation in heterogeneous me-dia. The locality principle allows us to replace the constant param-eters appearing in the pseudodifferential operator in equation 10 bythe variable parameters. The detailed derivation is shown in Appen-dix D. The fractional NCQ wave equation for smoothly hetero-geneous media is then obtained:

1

c2ðrÞ∂2σðr; tÞ

∂t2¼ ρ0ðrÞ

�ηðrÞð−∇2ÞγðrÞ þ τðrÞ ∂

∂tð−∇2ÞγðrÞ−1∕2

�

×∇ ·

�1

ρ0ðrÞ∇σðr; tÞ

�; (12)

where the coefficients vary in space as follows:

ηðrÞ ¼ −c2γðrÞ0 ðrÞω−2γðrÞ0 cos½πγðrÞ�; (13)

τðrÞ ¼ −c2γðrÞ−10 ðrÞω−2γðrÞ0 sin½πγðrÞ�: (14)

To avoid calculating the spatial derivatives of the medium param-eters in the wave equation 12, we solve instead the equivalent first-order conservation equations with the additional of an external bodyforce f:

∂υðr; tÞ∂t

¼ 1

ρ0ðrÞ∇σðr; tÞ þ f; (15)

∂∂tεðr; tÞ ¼ ∇ · υðr; tÞ; (16)

σðr;tÞ¼M0ðrÞ�ηðrÞð−∇2ÞγðrÞþτðrÞ ∂

∂tð−∇2ÞγðrÞ−1∕2

�εðr;tÞ;(17)

where equation 17 replaces equation 2 as the constitutive stress-strain relation; σðr; tÞ is the stress field; υðr; tÞ is the particle veloc-ity vector; εðr; tÞ is the strain field at position r at time t; and cðrÞand ρ0ðrÞ are the spatial acoustic velocity and density, respectively.Again, equation 17 reduces to the lossless constitutive stress-strainrelation when γ → 0 (Q → ∞).

Numerical implementation

Modeling wave propagation using the first-order conservationequations has previously been implemented in the finite-differencetime-domain approach (Zhu et al., 2013). It is easy to implement thePML boundary to absorb waves leaving the domain of interest (Be-renger, 1994). The temporal derivative is solved with the staggered-grid finite-difference approach. The first-order spatial derivativesare calculated by the staggered-grid pseudospectral method thatis known to reduce spatial numerical dispersion and nonphysicalringing (Özdenvar and McMechan, 1996; Carcione, 1999). Here,the fractional Laplacian is calculated by taking the spatial Fouriertransform, as defined in equation 3. Thus, in homogeneous media,the computational form of the fractional Laplacians in equation 17is written as

ð−∇2Þγεðr; tÞ ¼ F−1fk2γF½εðr; tÞ�g; (18)

ð−∇2Þγ−1∕2 ∂∂tεðr; tÞ ¼ F−1

�k2γ−1F

�∂∂tεðr; tÞ

��; (19)

where the symbols F and F−1 are the Fourier transform and inverseFourier transform operators, respectively. This result highlights the

advantage of the derived fractional Laplacian op-erators; in the spatial frequency domain, the frac-tional Laplacian is easily calculated. Whereas insmoothly heterogeneous media (γðrÞ is slowlyvarying in space), equations 18 and 19 will beapproximately satisfied. In our implementation,we use the average value of γðrÞ to approximatethe fractional power terms γðrÞ in equations 18and 19.

NUMERICAL EXPERIMENTS

Attenuation and dispersion

To illustrate the utility of the proposed vis-coacoustic wave equation to correctly modelconstant-Q attenuation and dispersion, we com-pare in the first example the numerical results

0 50 100 150 200 250 3000

200

400

600

800

1000

Frequency (Hz)

α (d

B/k

m)

0 50 100 150 200 250 3001.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

Frequency (Hz)

Vel

oci

ty (

km/s

)

Q = 2

Q = 2

Q = 10

Q = 100

Q = 10

Q = 100

a) b)

Figure 2. Comparisons of the approximate dispersion in equation 9 (dashed line) withthe exact constant-Q dispersion (see Appendix A) in equation 8 (solid line). We cal-culate these curves using different values of Q; i.e., Q ¼ 2, 10, and 100. The smallerthe Q values, the larger the errors.

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with laboratory measured data and Kjartanssan constant-Q theoreti-cal data in a homogeneous medium. McDonal et al. (1958) use mea-sured data of Pierre Shale in Colorado to find an approximateconstant-Q behavior with the attenuation coefficient α ¼ 0.12fdB∕kft ≈ 0.0453f neper∕km, where f is the frequency in hertz.From Carcione (2009), the quality factor of Pierre Shale rock isQ ¼32 and the P-wave acoustic velocity is about 2164 m∕s at the refer-ence frequency 100 Hz. For a homogeneous lossy medium, wechoose a typical shale density of 2.2 g∕cm3. The simulations areperformed in 2D with a 512 × 512 grid, 1.0-m spacing in the x-and z-directions, and a time step of 0.23 × 10−3 s. A Ricker waveletsource with 100-Hz center frequencies is located at (256 m, 256 m)in simulations.The attenuation coefficient α and the phase velocity are

calculated by the amplitude spectral ratio in decibels/meter andthe phase change at each frequency (Buttkus, 2000; Treeby andCox, 2010):

αðωÞ ¼ −20 log10

�A2

A1

�∕d; (20)

cðωÞ ¼ ωdϕ1 − ϕ2

; (21)

where A1;2 are the amplitude spectra at two receivers, ϕ1;2 are phasespectra, and d is the propagation distance in meters. To keepthe simulation free from numerical artifacts (e.g., numericaldispersion), we placed two receivers at 20 and 70 m from thesource. The geometric spreading for 2D problem is corrected bythe factor

ffiffiffir

p.

Figure 3 shows the accuracy of calculated attenuation anddispersion for the Pierre Shale model. The experimental data fromMcDonal et al. (1958) are represented by diamond points. The solidlines in Figure 3 give the theoretical attenuation and dispersionbased on equations A-4 and A-5. The open circles denote the cal-culated attenuation and dispersion values extracted from ournumerical simulations at two receivers. They appear to fit well withthe theoretical values as well as the experimental data. In Figure 3b,the calculated phase velocity drops rapidly below 40 Hz and cannotfit this portion of the theoretical results well. The difference is due to

the underestimation of small attenuation values at lower frequenciesby our NCQ model. Similar observations are made by Wuen-schel (1965).To illustrate the effects of separated amplitude loss and phase

dispersion operators on wavefields, we conduct simulations withdifferent partial formulations of equation 10. For example, ifone only considers the phase dispersion effects and ignores the am-plitude loss effects, the dispersion-dominated wave equation will be

1

c2∂2σ∂t2

¼ ηð−∇2Þγþ1σ: (22)

Similarly, the loss-dominated wave equation that contains theamplitude loss effects will be

1

c2∂2σ∂t2

¼ ∇2σ þ τ∂∂tð−∇2Þγþ1∕2σ: (23)

The second term in equation 23 describes attenuation of the con-stant-Qmodel. We assume the reference velocity c0 ¼ 2164 m∕s isdefined at high frequency, say, 1500 Hz. We simulate wavefieldsusing the model with Q ¼ 10. Figure 4 shows four wavefields at130 ms: (a) acoustic (lossless), (b) loss-dominated (equation 23),(c) dispersion-dominated (equation 22), and (d) viscoacoustic(lossy), respectively. Compared to the acoustic wavefield, Figure 4bshows significantly reduced amplitude due to the loss while weakenergy before the reference wavefront indicated by the red dots inFigure 4 is the result of the acausal attenuation term in equation 23.The dispersive acoustic wavefield in Figure 4c has delayed phasebut comparable amplitude; the viscoacoustic wavefield in Figure 4dhas reduced amplitude and delayed phase.

0 100 200 3002.05

2.1

2.15

2.2

Frequency (Hz)

Vel

ocity

(km

/s)

0 100 200 3000

50

100

150

Frequency (Hz)

α (d

B/k

m)

a) b)

Figure 3. (a) Attenuation and (b) phase velocity with frequency forPierre Shale with Q ¼ 32. The numerical attenuation coefficientsand phase velocity (circles) are computed using equations 20and 21. The theoretical constant-Q values (solid lines) are computedusing equations A-4 and A-5. The experimental results (diamondpoints) are from McDonal et al. (1958).

Distance (m)

Dis

tan

ce (

m)

−200 −100 0 100 200

−200

−100

0

100

200

Acoustic a)

c)Dispersion dominated

d) Viscoacoustic

Loss dominated b)

Figure 4. Four wavefield parts split by the dashed lines are gener-ated by: acoustic (a), loss dominated (b), dispersion dominated (c),and viscoacoustic (d), respectively. The red dots indicate the refer-ence acoustic wavefront (no attenuation). The simulation configu-ration is the same as that in the first example. The mediumparameters are c0 ¼ 2164 m∕s and Q ¼ 10.

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Accuracy over long distances

To test the accuracy of our numerical results of this NCQ waveequation over long distances (more attenuation), we increase ourmodel size to a 2048 × 3584 grid to capture wavefields over thelonger distance. The mesh spacing in the x- and z-directions is1.0 m, respectively. Other parameters are the same as the previousexperiment. We compare numerical results at receivers located at500, 1500, and 3000 m from the source with the corresponding ana-lytical solutions for a 2D homogeneous Pierre Shale medium(Figure 5). The 2D analytical solution is obtained by convolvingthe constant-QGreen’s function with the source function (Carcione,2007). We evaluated the accuracy of numerical solutions using theroot-mean-square (rms) errors, which is defined by

E ¼Xntj¼1

ðdnj − daj Þ2∕Xntj¼1

ðdaj Þ2; (24)

where nt is the number of time samples of the seismic trace, dnj isthe calculated value of the numerically simulated trace at sample j,and the superscript a denotes the corresponding analytical value.We found that the rms error deceases with offset, 1.3 × 10−3 at500 m (Figure 5a), 3.2 × 10−3 at 1500 m (Figure 5b), and7.1 × 10−3 at 3000 m (Figure 5c). Numerical solutions agree withthe analytical solutions very well at these distances.

0.2 0.24 0.28 0.32 0.36 0.4−1

−0.5

0

0.5

1

1.5

Time (s)

Pre

ssur

e

Analytical solution Numerical solution

0.6 0.65 0.7 0.75 0.8 0.85 0.9

−0.04

−0.02

0

0.02

0.04

0.06

Time (s)

Pre

ssur

e

Analytical solution Numerical solution

1.3 1.35 1.4 1.45 1.5 1.55 1.6

−4

−2

0

2

4

6

× 10–3

Time (s)

Pre

ssur

e

Analytical solution Numerical solution

a)

b)

c)

Figure 5. Comparison of numerical solutions (circles) with theGreen’s function analytical solutions (solid line). The receiversare at (a) 500, (b) 1500, and (c) 3000 m from the source.

0.1 0.15 0.2−10

0

10

Am

plit

ud

e

0.1 0.15 0.2−5

0

5

10

0.1 0.15 0.2−1

0

1

2

Time (s)

Am

plit

ud

e

0.1 0.15 0.2−0.1

0

0.1

Time (s)

RMS = 2.3e – 4

RMS = 4.7e – 3

RMS = 1.3e – 3

RMS = 1.1e – 1

Analytical

Numerical

a) b)

c) d)

Figure 7. Comparison between simulated seismograms (circles)and analytical seismograms (solid line) corresponding to fourQ val-ues: (a) 100, (b) 30, (c) 10, and (d) 4 at point (0.5, 0.7) km from asource (0.5, 0.5) km. They show excellent agreement, and the lastsmallest Q ¼ 4 value shows a little derivation.

Distance (m)

Dis

tan

ce (

m)

−400 −200 0 200 400

−400

−200

0

200

400

Q = 100 a)

c) Q = 10

Q = 30

d) Q = 4

b)

Figure 6. Four snapshot parts corresponding to four Q values:(a) 100, (b) 30, (c) 10, and (d) 4. A point source located at the centerof the model. Snapshots are recorded at 200 ms in homogeneousattenuating media.

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Accuracy in different Q media

This section investigates the accuracy of the wave equation fordifferent Q media. The velocity model is homogeneous as above.The source is located at the center of the model (0.5, 0.5) km. Weplace a receiver at the location of (0.5, 0.7) km. Figure 6 shows

snapshots taken at 200 ms in the NCQ wave equation withQ = (a) 100, (b) 30, (c) 10, and (d) 4. We can see decreased am-plitudes and delayed phases with decreasing Q values. We do notsimulate for Q < 4 because the simulation is unstable at this dis-tance. Figure 7 compares the analytical and numerical solutions

NCQ SLS

SLSNCQ

Distance (m)Distance (m)

Dis

tan

ce (

m)

0 500 1000 15000

500

1000

1500

0 500 1000 15000

500

1000

1500

0

500

1000

1500

Dis

tan

ce (

m)

0

500

1000

1500

a) b) c)

Dis

tan

ce (

m)

0 500 1000 15000

500

1000

1500

0 500 1000 15000 500 1000 1500

0 500 1000 15000

500

1000

1500

d)

h)g)

e)

0 500 1000 1500−1

0

1

Am

plitu

de

0 500 1000 1500−1

0

1

Distance (m)

Am

plit

ud

e0 500 1000 1500

−1

0

1

Am

plitu

de

0 500 1000 1500−1

0

1

Distance (m)

Am

plit

ud

e

0 500 1000 1500−1

0

1

Am

plitu

de

0 500 1000 1500−1

0

1

Distance (m)

Am

plit

ud

e

j)

f )

At horizontal 1000 m

At horizontal 1000 m

At horizontal 1000 m

At horizontal 500 m

At horizontal 500 m

At horizontal 500 m

Figure 8. Snapshots at 224 ms. A two-layer model with velocities of 1800 m∕s in the top layer and 3600 m∕s in the bottom layer, and aQ value of 30 in the top and 100 in the bottom is used. The interface is at a depth of 800 m. First row: homogeneous Q model with two-layervelocity: (a) computed by solving the NCQwave equation, (b) computed by solving the viscoacoustic (SLS) wave equation, and (c) comparisonof traces at horizontal 500 and 1000 m between (a) NCQ (circle line) and (b) SLS (solid line). Second row: homogeneous velocity model withtwo-layerQ: (d) computed by solving the NCQwave equation, (e) computed by solving the SLS wave equation, and (f) comparison of traces athorizontal 500 and 1000 m between (d) NCQ (circle line) and (e) SLS (solid line). Third row: two-layer velocity and Q models: (g) computedby solving the NCQ wave equation, (h) computed by solving the SLS wave equation, and (j) comparison of traces at horizontal 500 m and1000 m between (g) NCQ (circle line) and (h) SLS (solid line). The dotted line indicates the difference.

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corresponding to theQ values in Figure 6. We found that the smallerQ value results in the larger error, which is consistent with the ob-servation in Figure 2. The corresponding rms error values are2.3 × 10−4, 1.3 × 10−3, 4.7 × 10−3, and 1.1 × 10−1, respectively.

Stability in large-contrast heterogeneous media

To demonstrate the stability of simulations in media with largecontrasts, we design a two-layer model with P-wave velocity of1800 m∕s in the top layer and 3600 m∕s in the bottom layerand a Q value of 30 in the top and 100 in the bottom. The interfaceis at a depth of 800 m. The source function is a Ricker wavelet with25-Hz center frequency located at (968 m, 520 m) in the simulation.We discretize model in a 200 × 200 grid with 8.0 m spacings in thex- and z-directions. The time step is 0.8 × 10−3 s. The accuracy of

the solutions is compared with SLS solutions, which are roughlyaccurate to model constant-Q behavior in simulation over a shortdistance, as Zhu et al. (2013) show.We test three scenarios: (1) two-layer velocity model with homo-

geneousQ ¼ 30, (2) two-layerQmodel with homogeneous P-wavevelocity ¼ 1800 m∕s, and (3) two-layer velocity and Q models.Figure 8 shows the snapshots at 280 ms using the proposedNCQ wave equation and the viscoacoustic wave equation basedon the SLS model (Zhu et al., 2013). The first row shows the effectsof velocity discontinuities on the wavefields. The two numericalresults agree with each other, as shown in Figure 8c. The secondrow shows the effects of Q discontinuities on the wavefields.One can see that the weak reflection is due to Q contrast, as shownin the second row of Figure 8. In the third row, we show simulationsusing velocity and Q contrasts. Overall, the solution by NCQ inFigure 8g matches that by the SLS in Figure 8h very well. Thesecomparisons demonstrate that numerical simulations using the pro-posed NCQ wave equation in such large contrast velocity and Qmodels are stable and practically accurate.

SYNTHETIC MODEL EXAMPLE

Figure 9 shows the P-wave velocity and Q models. The densitymodel is constant of 2.2 g∕cm3. Zone #4 in the model represents ahigh attenuation reservoir (Q ¼ 30). The model is discretized in amesh with 200 × 600 grid points. The space is equally sampled in xand z at 0.5 m. A PML absorbing boundary with 20 grid points isimplemented at four sides. An explosive source with a 200-Hzcenter-frequency Ricker wavelet is chosen for simulation. The timestep is 0.014 ms. The source is located at a 90-m horizontal distanceand at 110-m depth. Receivers are in the left well of 0.5 m horizon-tal distance. They are distributed from depths of 5–292.5 m withspace 2.5 m.Figure 10 shows the snapshots at 20 and 45 ms, respectively. To

illustrate the attenuation effects on the wavefield, we show theacoustic cases (Figure 10a and 10b) as reference. The arrow labeled3-4 indicates reflections from the interface of zones #3 and #4.Figure 10c and 10d show the constant-Q viscoacoustic cases.

Figure 10c shows overall reduced reflection am-plitude compared with acoustic reflections. Re-flections from the interface at the bottom ofthe high attenuation zone (arrow 4-5) suffer fromsignificantly reduced energy because the wavepropagates through zone #4. These observationsare more visible in the seismograms shown inFigure 11.To check the efficiency and accuracy of the

NCQ wave equation, we also ran SLS modelingusing a single SLS, which is considered to be rea-sonably accurate in practical applications (Zhuet al., 2013). All simulations were run on an Intelworkstation (Xeon X5650 2.66 GHz). Theacoustic simulation takes 226 s. The runningtime for the proposed NCQ algorithm and theSLS algorithm is 303 and 368 s, respectively.It may appear surprising that the present NCQalgorithm is more computationally efficient thanthe single SLS modeling algorithm that is consid-ered as the most efficient viscoelastic modelingapproach (Zhu et al., 2013). The reason is that the

Distance (m)

Dep

th (

m)

0 20 40 60 80

0

50

100

150

200

250

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5

Distance (m)

Dep

th (

m)

0 20 40 60 80

0

50

100

150

200

25050

100

150

200

250

300b)a)

1

2

3

45

VP(km/s) Q

Figure 9. (a) Heterogeneous velocity and (b) Q model forsimulation.

Dis

tan

ce (

m)

Distance (m)

0 20 40 60 80

0

50

100

150

200

250

Dis

tan

ce (

m)

Distance (m)

0 20 40 60 80

0

50

100

150

200

250

Dis

tan

ce (

m)

Distance (m)

0 20 40 60 80

0

50

100

150

200

250

Dis

tan

ce (

m)

Distance (m)

0 20 40 60 80

0

50

100

150

200

250

3-4

4-5

a) b) c) d)

Figure 10. Snapshots at 20 and 45 ms. (a and b) Acoustic and (c and d) NCQ. Arrow 3-4indicates reflections from the top of the high attenuation reservoir zone #4, and label 4-5represents reflections from the bottom of the high attenuation reservoir.

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SLS modeling algorithm requires more computer time in solvingmemory-variable equations at each time step.Figure 12 shows a comparison of simulated traces between our

proposed NCQ wave equation and the SLS wave equation. The twoagree with each other very well in details, even later-arrivingreflections.

DISCUSSION

The fractional Laplacian NCQ wave equation is an alternative tothe fractional time wave equation. Theoretically, the NCQ waveequation is able to model approximately constant-Q behavior whenQ is not very small, a feature that is further verified by numericalsimulations. For smaller Q values, the rms errors of simulationscompared with the analytical solutions become large. We found thatnumerical simulation becomes unstable when Q < 4. We concludethat this NCQ wave equation is highly accurate in Q environments

for most seismic applications. On the other hand, the advantage ofsolving the fractional Laplacian wave equation is computational ef-ficiency in comparison with the fractional time wave equation.Computing the fractional Laplacian is accomplished without usingand storing the previous time wavefields required by the fractionaltime derivative via GL approximation (e.g., Carcione et al., 2002).A synthetic example also indicates that the proposed NCQ waveequation is more efficient than other SLS modeling approaches(Zhu et al., 2013).To obtain the NCQ wave equation for heterogeneous media, we

used the locality principle for pseudodifferential operators, which isvalid for smoothly heterogeneous media. However, it may not holdfor high-contrast heterogeneous media, of which the gradient invelocity and Q is large. Fortunately, from limited numerical tests,we found that numerical simulations are stable for reasonably largecontrast heterogeneous velocity variations and Q variations.We only consider the average γðrÞ to approximate the spatial

varying γðrÞ in the fractional Laplacian terms when using the Fou-rier method. To the best of the authors’ knowledge, numerical meth-ods for spatially varying γðrÞ using the Fourier method are still notreported in the literature. Alternatively, finite-difference approxima-tions for the spatial varying γðrÞ in the fractional Laplacian termsmay be used but are still at an early stage of development (Lin et al.,2009). The finite-difference implementation is untested on practicalproblems. Future research is needed to investigate this topic.Compared to the constant-Q wave equation by Carcione (2010),

our NCQ wave equation with two fractional Laplacian operatorscan model amplitude loss and phase dispersion effects separately.Treeby et al. (2010) report that the inherent separation of loss anddispersion effects in their lossy wave equation is beneficial for at-tenuation compensation in medical imaging. Similarly, in our waveequation, the back-propagation in seismic inversion/migration canbe done by reversing the sign of the loss operator in a way that

0 50 100 150

50

100

150

200

250

Time (ms)

Rec

ever

dep

th (

m)

b)

0 50 100 150

50

100

150

200

250

Time (ms)

Rec

ever

dep

th (

m)

a)

3-4

4-5

Figure 11. Synthetic data calculated by (a) acoustic and (b) NCQ.Due to the different attenuation area, reflections at interface 3-4 areless attenuated and reflections at interface 4-5 experience significantloss.

0.02 0.04 0.06 0.08 0.1

0

50

100

150

200

250

300

Time (s)

Dep

th (

m)

SLS

NCQ

Figure 12. Comparison of two seismograms calculated by the pro-posed NCQ wave equation (red line) and the SLS model (blackline). Partial traces are shown in every other 10 trace.

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naturally compensates for the amplitude loss. The back-propagationhas been successfully demonstrated in applications of Q reverse-time migration (Zhu, 2014; Zhu et al., 2014). It appears that thepresent work has potential applications to seismic inversion/migra-tion with attenuation compensation.

CONCLUSIONS

We have derived a NCQ viscoacoustic wave equation in the timedomain using the fractional Laplacian. The advantage of using ourfractional Laplacian formulation over the traditional fractional timederivative approach is avoidance of storing the time history of var-iables, and thus it is more economic in computational costs. In ad-dition, our viscoacoustic wave equation has two fractionalLaplacian operators to describe separated amplitude loss and phasedispersion effects, respectively. We provided a detailed formulationof this viscoacoustic wave equation for homogeneous media andsmoothly heterogeneous media. We discussed the stability condi-tion of the viscoacoustic wave equation. We also gave the first-orderconservation equations for numerical simulation.Numerical results demonstrate that the proposed viscoacoustic

wave equation models the approximate constant-Q attenuationand dispersion. Modeling with the proposed wave equation is ac-curate in typicalQ environments and stable in large contrast hetero-geneous velocity and Q media. We believe that this fractionalviscoacoustic wave equation is promising for use in accurate wave-field modeling for attenuating media.

ACKNOWLEDGMENTS

We are thankful to the editors, J. Carcione and F. Liu, and threeanonymous reviewers whose comments improved the quality of thepaper. We thank B. Treeby and B. Cox for their open source code(www.k-wave.org) that helped start our work quickly. The work isfinancially supported by the Stanford Wave Physics Laboratory.

APPENDIX A

CONSTANT-Q PHASE VELOCITYAND ATTENUATION

Taking the square root of equation 8 and using the definition ofthe complex wavenumber ~k ¼ ω∕cp − iα ¼ kR þ ikI, we can ob-tain the real wavenumber:

kR ¼ ω

cp¼ ω

c0

�ω

ω0

�−γ

≈ω

c0

�1 − γ ln

ω

ω0

�; (A-1)

for all frequencies of interest that satisfy condition γj lnðω∕ω0Þj≪1

(Kjartansson, 1979). The imaginary wavenumber is

kI ¼ − tan

�πγ

2

�ω

cp: (A-2)

From equation 8, we can obtain the complex velocity of the con-stant-Q model:

vc ¼ω~k¼ c

�iωω0

�γ

: (A-3)

So, the required dispersive phase velocity and attenuation are thesame as the constant-Q model given by Kjartansson (1979); that is,

cp ¼ c0

�ω

ω0

�γ

; (A-4)

α ¼ tan

�πγ

2

�ω

cp; (A-5)

where the velocity c0 is given at a reference frequency ω0. For aconstant-Q, the attenuation coefficient α is linear with frequencyω and the phase velocity cp is slightly dependent on frequency.

APPENDIX B

APPROXIMATE DISPERSION ANALYSIS

Because i2γ ¼ cosðπγÞ þ i sinðπγÞ, the right-hand side of equa-tion 8 can be split into parts as follows:

ω2

c2¼ ω−2γ

0 cosðπγÞω2γ ~k2 þ iω−2γ0 sinðπγÞω2γ ~k2: (B-1)

Upon multiplication by c2γ0 ∕c2γ0 on the right side, and using theapproximation kR ≈ ω∕c0, where kR is purely real (see equa-tion A-2), the dispersion relation becomes

ω2

c2¼ c2γ0 ω

−2γ0 cosðπγÞk2γR ~k2 þ ic2γ0 ω−2γ

0 sinðπγÞk2γR ~k2:

(B-2)

Because kI∕kR ¼ tanðπγ2Þ ≪ 1 for weak attenuation, expanding ~k2γ

into real and imaginary parts k2γR ≈ ~k2γð1 − i2γkI∕kRÞ then gives

ω2

c2¼ c2γ0 ω−2γ

0 cosðπγÞð1 − i2γkI∕kRÞ ~k2γþ2

þ ic2γ0 ω−2γ0 sinðπγÞ½1 − i2γkI∕kR� ~k2γþ2; (B-3)

¼ c2γ0 ω−2γ0 cosðπγÞ ~k2γþ2f½1þ 2γkI∕kR tanðπγÞ�

þ i tanðπγÞ½1 − 2γkI∕kR∕ tanðπγÞ�g: (B-4)

We also have 2γkI∕kR∕ tanðπγÞ ≪ 1 and 2γkI∕kR tanðπγÞ ≪ 1, andthe dispersion relation is further simplified to

ω2

c2≈ c2γ0 ω

−2γ0 cosðπγÞ ~k2γþ2 þ ðiωÞc2γ−10 ω−2γ

0 sinðπγÞ ~k2γþ1:

(B-5)

APPENDIX C

STABILITY ANALYSIS

To derive the stability criterion of the NCQ wave equation, weapply a generalized Fourier transform on the left side of the waveequation (equation 10) and use the central finite-difference for thesecond-order time derivative. The forward stress field σnþ1 will bewritten as the current stress σn and the previous stress σn−1 fields

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σnþ1 ¼ 2σn − σn−1 þ Δt2c20ηk2γþ2σn

þ Δtc20τk2γþ1ðσn − σn−1Þ: (C-1)

Then, the equation is expressed in a matrix form

�σnþ1

σn

�¼

�2þ aΔt2 þ bΔt −1 − bΔt

1 0

��σn

σn−1

�;

(C-2)

where a ¼ c20ηk2γþ2 and b ¼ c20τk

2γþ1. We analyze the stabilitycondition of the NCQ wave equation using the eigenvalue method,in which the eigenvalues of the matrix must be less than or equal to1 in magnitude for stable simulation (Gazdag, 1981). Solving theeigenvalues λ of matrix A and assuming λ ≤ 1, the stability condi-tion of NCQ wave equation is

Δt ≤ −bþ ffiffiffiffiffiffiffiffiffi

−2ap

a¼ tanðπγÞ

c0kþ

ffiffiffi2

pωγ0

ðc0kÞγþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosðπγÞp :

(C-3)

To guarantee that the solution is stable for all waves, c0 ¼ cmax isthe maximum velocity, k ¼ kNy ¼ π∕Δx is the Nyquist spatialwavenumber, and γ ¼ 1∕π tan−1ð1∕QminÞ. As γ ¼ 0, it becomesthe stability condition of the acoustic wave equation (Gazdag,1981):

Δt ≤ffiffiffi2

pΔx

πcmax

. (C-4)

To our best knowledge, the stability condition in equation C-3 isslightly stricter than equation C-4. In other words, attenuation(1∕Qmin) has a limited effect on the stability condition.

APPENDIX D

CONSTANT-Q WAVE EQUATIONIN HETEROGENEOUS MEDIA

In homogeneous media (c0 and γ are constant with position), theNCQ wave equation (equation 10) is written as

1

c2∂2

∂t2σðr; tÞ ¼

�ηð−∇2Þγ þ τ

∂∂tð−∇2Þγ−1∕2

�∇2σðr; tÞ:

(D-1)

If we write equation D-1 in the first-order mass-momentum con-servation equations 15 and 16, we get the constitutive equation asfollows:

σðr; tÞ ¼ Lεðr; tÞ; (D-2)

L ¼ M0

�ηð−∇2Þγ þ τ

∂∂tð−∇2Þγ−1∕2

�: (D-3)

In smoothly heterogeneous media, we localize the pseudo-differential operator L at local point r0. All parameters related tovelocity c0 and γ are constant at fixed point r0. We have the same

pseudodifferential operator L as equation D-3. It is reasonable tosuppose that the L we want should be well approximated byLðr0Þ when r is near r0; thus, we unfreeze the localized point r0to general spatial r and define the pseudodifferential operator LðrÞ,

LðrÞ ¼ M0ðrÞ�ηðrÞð−∇2ÞγðrÞ þ τðrÞ ∂

∂tð−∇2ÞγðrÞ−1∕2

�:

(D-4)

This generalized pseudodifferential operator from homogeneous toheterogeneous parameters is also called the freezing-unfreezingprinciple that appears in the literature in the context of hyperbolicand elliptic partial differential equations (Stein [1993], p. 230).Plugging equation D-4 into equation D-2, and the mass-

momentum conservation equations 15 and 16, we have the approxi-mate heterogeneous second-order wave equation:

1

c2ðrÞ∂2σðr; tÞ

∂t2¼ ρ0ðrÞ

�ηðrÞð−∇2ÞγðrÞ þ τðrÞ ∂

∂tð−∇2ÞγðrÞ−1∕2

�

×∇ ·

�1

ρ0ðrÞ∇σðr; tÞ

�: (D-5)

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