Locally solving fractional Laplacian viscoacoustic wave equationusing Hermite distributed approximating functional method

Jie Yao1, Tieyuan Zhu2, Fazle Hussain3, and Donald J. Kouri1

ABSTRACT

Accurate seismic modeling in realistic media serves asthe basis of seismic full-waveform inversion and imaging.Recently, viscoacoustic seismic modeling incorporating at-tenuation effects has been performed by solving a fractionalLaplacian viscoacoustic wave equation. In this equation, at-tenuation, being spatially heterogeneous, is representedpartially by the spatially varying power of the fractional Lap-lacian operator previously approximated by the global Fou-rier method. We have developed a local-spectral approach,based on the Hermite distributed approximating functional(HDAF) method, to implement the fractional Laplacian inthe viscoacoustic wave equation. Our approach combineslocal methods’ simplicity and global methods’ accuracy. Sev-eral numerical examples are developed to evaluate the fea-sibility and accuracy of using the HDAF method for thefrequency-independentQ fractional Laplacian wave equation.

INTRODUCTION

Attenuation and associated dispersion should be taken intoaccount to accurately characterize wave propagation in realisticmedia. Many methods have been proposed to model seismic-attenu-ation effects during wave propagation (Štekl and Pratt, 1998; Car-cione, 2007; Zhu et al., 2013). One of them is using constant-Qwave propagation in the time domain (Kjartansson, 1979; Carcioneet al., 2002). It is accurate in producing desirable constant-Q behav-ior at all frequencies. However, it requires using a fractional deriva-tive to construct the power-law stress-strain relation (Kjartansson,1979; Carcione, 2010). For instance, a constant-Q wave equation

with the fractional time derivative was proposed to simulate acous-tic and elastic waves and computed by the Grünwald-Letnikovapproximation (Carcione et al., 2002; Carcione, 2009). One draw-back of the fractional time derivative approach is that numericalimplementation requires large computational memory to store allthe previous wavefields. Even when the fractional operators aretruncated after certain time steps, the memory requirements arestill too high (Podlubny, 1998; Carcione, 2009). To overcome thememory expense, Chen and Holm (2004) propose to use fractionalLaplacian operators to model anomalous attenuation. It successfullyavoids additional memory required by the fractional time deriva-tives. Based on this idea, various types of wave equations usingfractional Laplacians are derived to model wave propagation withattenuation and velocity dispersion (Carcione, 2010; Treeby andCox, 2010). For example, Zhu and Harris (2014) develop a de-coupled wave equation that accounts separately for amplitude at-tenuation and phase-dispersion effects, which physically satisfiesthe frequency-independent (constant) Q model.The fractional Laplacian was previously implemented mainly us-

ing the generalized Fourier transform approach (Zhu and Carcione,2014; Zhu and Harris, 2014), by using an averaged fractional powerof the Laplacian operator. This simple average of the power variableof the Laplacian operator undoubtedly introduces numerical errorsinto simulations, especially when an attenuation model is stronglyheterogeneous in the subsurface. To better approximate the spatiallyvarying power variable, Sun et al. (2015) first apply a low-rankapproximation scheme to implement the spatially varying fractionalLaplacians. Chen et al. (2016) reformulate viscoacoustic waveequation with highly accurate time discretization and also adopt thelow-rank approximation for solving the fractional Laplacian. Thelow-rank approach can directly approximate the mixed-domainwave extrapolation operator with a separate representation and en-able one to evaluate the variable fractional power of the Laplacianoperator. However, these methods intensively rely on the Fourier

Manuscript received by the Editor 22 May 2016; revised manuscript received 24 August 2016; published online 20 December 2016.1University of Houston, Departments of Mechanical Engineering, Physics, andMathematics, Houston, Texas, USA. E-mail: yjie2@uh.edu; kouri@central.uh.

edu.2Formerly University of Texas at Austin, John A. and Katherine G. Jackson School of Geosciences, Austin, Texas, USA; presently The Pennsylvania State

University, Department of Geosciences and Institute of Natural Gas Research, University Park, Pennsylvania, USA. E-mail: tyzhu@psu.edu.3Texas Tech University, Department of Mechanical Engineering, Lubbock, Texas, USA. E-mail: fazle.hussain@ttu.edu.© 2017 Society of Exploration Geophysicists. All rights reserved.

T59

GEOPHYSICS, VOL. 82, NO. 2 (MARCH-APRIL 2017); P. T59–T67, 12 FIGS., 1 TABLE.10.1190/GEO2016-0269.1

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transform. For example, the low-rank approximation with a rankof four needs three complex-to-complex fast Fourier transforms(FFTs) per time step (Sun et al., 2015).In this paper, we propose a local approach based on the Hermite

distributed approximating functional (HDAF) to implement thefractional Laplacians of Zhu and Harris (2014). HDAF is originallyintroduced by Hoffman et al. (1991) and Hoffman and Kouri (1992)as a computational tool for treating a variety of problems in physicsand chemistry (Wei et al., 1997, 1998a, 1998b; Zhang et al., 1998,1999; Kouri et al., 1999; Pindza and Maré, 2014; Lesage et al.,2015). In the HDAF approach, an approximation to the delta func-tion is constructed by using even regularized Hermite polynomials.Study of HDAF indicates that it can deliver spectral accuracy whenused to solve partial differential equations. In addition, discretiza-tion of the HDAF method has some features similar in spirit to mostfinite-difference schemes (Zhang et al., 1998). Derivatives of a con-tinuous function can be easily obtained by the convolution betweenthe HDAF kernel and the function. Hence, HDAF is a local spectralmethod. It exhibits global method accuracy for differentiation andlocal method flexibility in handling complex geometries and boun-dary conditions (Wei et al., 1998a).The paper is organized as follows: First, we give a brief review

of the fractional Laplacian viscoacoustic wave equation. Then, weprovide details of using HDAF to approximate functions and deriv-atives. Following that, we apply the HDAFmethod to the viscoacous-tic wave equation. Several numerical experiments are performed todemonstrate the accuracy of the proposed method in seismic viscoa-coustic modeling.

VISCOACOUSTIC WAVE EQUATION

Based on the frequency-independent Q model, in which the at-tenuation coefficient is linear with frequency (Kjartansson, 1979),Zhu and Harris (2014) derive a viscoacoustic wave equation withdecoupled fractional Laplacians:

1

c2∂2P∂t2

¼ ηð−∇2Þγþ1Pþ τ∂∂tð−∇2Þγþ1∕2P; (1)

where Pðx; tÞ is the pressure wavefield and the fractional powerγ ¼ arctanð1∕QÞ∕π. For any positive value of Q, we have 0 < γ< 0.5. We also have

c2 ¼ c20 cos2ðπγ∕2Þ; (2)

η ¼ −c2γ0 ω−2γ0 cosðπγÞ; (3)

τ ¼ −c2γ−10 ω−2γ0 sinðπγÞ; (4)

with c0 is the velocity defined at the reference frequency ω0. Notethat the phase velocities c0 and γ are spatially varying variables.The merit of this viscoacoustic wave equation is that the velocitydispersion and amplitude loss phenomena of seismic attenuation canbe separately modeled, which enables seismic imaging to compen-sate for attenuation with less effort (Zhu et al., 2014).

As we can see, when γ is zero, viscoacoustic equation 1 reducesto the classic acoustic-wave equation. The extra computation com-pared with the acoustic case is to calculate the fractional Laplacianoperators. The fractional Laplacian operators are solved by a Fou-rier-based approach without difficulties when γ is independent ofposition; i.e., the medium is homogeneous. This is not always thecase for subsurface geologic media. When heterogeneity in the sub-surface is taken into account, the fractional Laplacian operator withspatially varying γ becomes a mixed wavenumber-space domainproblem when applying the Fourier transform. Although the spa-tially varying γ was approximated by an averaged value in smoothlyheterogeneous media (Zhu and Harris, 2014), it is inaccurate inheterogeneous media. Recent advances in approximating mixed-domain operators by low-rank approximation are possible to obtainbetter approximation to fractional Laplacian operators (e.g., Sunet al., 2015; Chen et al., 2016). This approach is accurate forapproximating such a mixed-domain fractional Laplacian operator,but it intensively uses the Fourier transform. In this paper, we in-troduce a local method, HDAF, to compute the fractional Laplacianoperators.

APPROXIMATING A FUNCTION AND ITSDERIVATIVES USING HDAF

HDAF can be viewed as an approximate mapping of a certain setof continuous L2 functions to themselves, accurate to a given tol-erance (Zhang et al., 1997). The HDAF is able to provide an ana-lytical representation of a function and its derivatives in terms of adiscrete set of values of the function only. This is central to its suc-cess in various computational applications (Wei et al., 1997).It is known that the Dirac delta function has the following

properties:

uðxÞ ¼Z

δðx − x 0Þuðx 0Þdx 0; (5)

uαðxÞ ¼Z

δαðx − x 0Þuðx 0Þdx 0; (6)

where α ∈ R and uαðxÞ is the αth derivative of function uðxÞ.However, equations 5 and 6 are of little numerical interest to

practical application due to the occurrence of the delta distribution.In the HDAF approach, an approximation to the delta function isdefined as (Hoffman and Kouri, 1995)

δMðx−x0;σÞ¼ 1ffiffiffiffiffi2π

pσe−

ðx−x0Þ22σ2

XM∕2

n¼0

�−1

4

�n 1

n!H2n

�x−x0ffiffiffi2

pσ

�;

(7)

where the right side is a two-parameter delta sequence, σ andM arethe delta sequence (HDAF) parameters, and the ith Hermite poly-nomial is defined by

HiðxÞ ¼ ð−1Þiex2 di

dxie−x

2

; ∀ x ∈ R: (8)

The HDAF is constructed by using even Hermite polynomials(because the delta distribution is even in its argument), and it is do-

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minated by its Gaussian envelope, expð−ðx − x 0Þ2∕2σ2Þ. For anyfixed σ, the HDAF becomes exactly identical to the delta functionin the limit of infinite M. In addition, for a fixed M, the HDAFbecomes identical to the delta function when σ goes to zero (Hoff-man and Kouri, 1995; Zhang et al., 1997). More detailed propertiesof the HDAF are provided in Appendix A.Unlike the delta function, the HDAF expression (equation 7) can

be discretized by quadratures. The continuous approximation to afunction uðxÞ generated by the HDAF delta sequence is

uðxÞ ≈ uDðxÞ ¼Z

δMðx − x 0; σÞuðx 0Þdx 0: (9)

Given a discrete set of functional values on a grid, the HDAFapproximation to the function at any point xj can be obtained by

uðxjÞ ≈ uDðxjÞ ¼ ΔXWi

δMðxj − xi; σÞuðxiÞ; (10)

where Δ is the uniform grid spacing and W is the number of stencilpoints.The approximation of the derivatives of a continuous function

can also be generated using the HDAF

uαðxÞ ≈ uαDðxÞ ¼Z

δαMðx − x 0; σÞuðx 0Þdx 0; (11)

where δαMðx − x 0; σÞ is the αth derivative of δMðx − x 0; σÞ, and it isgiven by the two-parameter delta sequence

δαMðx−x0;σÞ¼ð−1Þαffiffiffiffiffi2π

pσ

XM∕2

n¼0

�−σ2

2

�n 1

n!d2nþα

dx2nþαe−ðx−x0 Þ2

2σ2 : (12)

When α is an integer, the expression of HDAF is explicitly givenby (Hoffman and Kouri, 1995)

δαMðx−x0;σÞ¼−ð−1Þαffiffiffiffiffi

2πp

2α∕2σαþ1e−

ðx−x0Þ22σ2

×XM∕2

n¼0

�−1

4

�n 1

n!H2nþα

�x−x0ffiffiffi2

pσ

�: (13)

For a noninteger α, there are several knownnumerical algorithms to calculate the fractional-order derivatives of HDAF, such as the Grün-wald-Letnikov method (Petráš, 2008; Kumar andRawat, 2012), finite-difference quadrature ap-proach (Huang and Oberman, 2014), and Fourier-transform method (Carcione, 2010). We use theFourier-transform method:

δMðx;σÞ→FFTδ̂Mðk;σÞ →×ðikÞα

×ðikÞαδ̂Mðk;σÞ→IFFTδαMðx;σÞ: (14)

When uniformly discretized by the trapezoidalrule (aNewton-Cotes type) quadrature, equation 11gives

uαðxjÞ ≈ uαDðxjÞ ¼ ΔXWi

δαMðxj − xi; σÞuðxiÞ: (15)

Equation 15, together with the expression of the fractional-orderHDAF, provides an algebraic way to approximate fractional deriv-atives of the functions. With proper choice of the parameters (σ,M,andW), HDAF provides arbitrarily high accuracy for estimating thederivatives of the functions. Compared with other differentiationmethods, HDAF is extremely simple. The kernel of the HDAF frac-tional-order differentiator δαM is required to be calculated only oncefor each given α. And arbitrary-order derivatives of a function canbe obtained by the convolution between the fractional-order deriv-atives of the HDAF and the function.To demonstrate the proposed HDAF method for approximating

derivatives Dαu of functions, we test on a Ricker-wavelet signal(Figure 1). The time spacing is ΔT ¼ 1 × 10−3 s. Figure 2 showsthe results for different values of α. Results obtained using pseudo-spectral methods are also displayed for comparison. The HDAFparameters are σ ¼ 3ΔT, M ¼ 48, and W ¼ 51. We evaluated theaccuracy of numerical solutions using the root-mean-square (rms)errors, which are defined by

E ¼Xntj¼1

ðdpj − dhj Þ2∕Xntj¼1

ðdpj Þ2; (16)

where nt is the number of time samples of the signal, dpj is thecalculated value using pseudospectral method at sample j, and dhjdenotes the corresponding value using HDAF method.

0 0.1 0.2 0.3 0.4 0.5−0.5

0

0.5

1

Time (s)

Am

plit

ud

e

Figure 1. A Ricker wavelet signal.

0 0.1 0.2 0.3 0.4 0.5−10

−5

0

5

10

rms = 1.8e−1

Time (s)

Am

plit

ud

e

PseudospectralHDAF

0 0.1 0.2 0.3 0.4 0.5−100

−50

0

50

100

rms = 3.5e−4

Time (s)

Am

plit

ud

e

PseudospectralHDAF

0 0.1 0.2 0.3 0.4 0.5−1.5

−1

−0.5

0

0.5

1

rms = 4.7e−2

Time (s)

Am

plit

ud

e

PseudospectralHDAF

×103

0 0.1 0.2 0.3 0.4 0.5−1.5

−1

−0.5

0

0.5

1x 10

4

rms = 5.3e−4

Time (s)

Am

plit

ud

e PseudospectralHDAF

a)

b)

c)

d)

Figure 2. Comparison between fractional-order derivative results of the Ricker waveletsignal using the pseudospectral and HDAF methods: (a) 0.5th order, (b) first order,(c) 1.5th order, and (d) second order.

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For α ¼ 0.5, 1, 1.5, 2.0, the rms errors are 1.8 × 10−1, 3.5 × 10−4,4.7 × 10−2, and 5.3 × 10−4, respectively. For a given stencil sizeW,the rms errors are almost the same with the other two pairs of HDAFparameters mentioned in Appendix A. The parameters σ and M aredetermined such that the bandwidth of HDAF is greater than themaximum frequency of the signal. A smaller error is observed forinteger-order differentiation (i.e., when α ¼ 1 or 2). We found thatthe rms error is relatively large for α < 1. Differentiation of thefunctions can be regarded as multiplying the function with ðikÞα inthe frequency domain (equation 14). Large-wavelength (low-wave-number) information is more essential for low-order fractionalderivatives. Hence, the corresponding error is bigger when using alocal-differentiation method for α < 1. The accuracy of the approxi-mation can be improved by increasing the number of stencils W.When W increases to 101, the rms errors become 6.5 × 10−2,2.3 × 10−4, 9.5 × 10−3, and 6.44 × 10−5, respectively.

High-dimensional HDAFs can be obtained using tensor products(Zhang et al., 1999). For example, the 2D HDAF is given as

δMðx; x 0; z; z 0; σÞ ¼ δM1ðx − x 0; σ1ÞδM2

ðz − z 0; σ2Þ: (17)

In general, the parameters M and σ can be different in the x- andz-directions. In this paper, we set M1 ¼ M2 and σ1 ¼ σ2 for sim-plicity. Then, one can write the HDAF approximation for a 2D func-tion uðx; zÞ as

uðx; zÞ ¼ZZ

dx 0dz 0δMðx; x 0; z; z 0; σÞuðx 0; z 0Þ; (18)

and in discretized form

uðxl;znÞ¼ΔxΔz

XW1

i

XW2

j

δMðxl−xi;zn−zj;σÞuðxi;zjÞ; (19)

where W1 and W2 are the stencil lengths along the x- and z-di-rections.Therefore, the fractional 2D Laplacian operators in wave equa-

tion 1 can be approximated as

ð−∇2Þαuðxl;znÞ¼ΔxΔz

XW1

i

XW2

j

δαMðxl−xi;zn−zj;σÞuðxi;zjÞ;

(20)

where δαMðx; z; σÞ is the fractional Laplacian of HDAF, obtainedbased on equations 14 and 17.

NUMERICAL EXAMPLES

Different Q media

First, we investigate the accuracy of the solution of the constant-Q wave equation using the proposed HDAF method for different Qvalues. The velocity model is homogeneous with a reference veloc-ity c0 ¼ 2000 m∕s, defined at high frequency 1500 Hz. Simulationsare performed with a grid of 201 × 201 points, Δh ¼ 10.0 m spac-

ing in the x- and z-directions, and a second-orderfinite-difference scheme for time discretizationwith a time step of 0.25 ms. The source is chosenas a Ricker wavelet with the central frequency of25 Hz located at the center of the model in sim-ulations. Figure 3a–3d shows snapshots taken at400 ms with Q ¼ 100, 30, 10, and 5, respec-tively. The HDAF parameters used throughoutthe paper are σ¼1.65Δh, M¼24, and W¼51.We can see decreased amplitudes and delayedphase with decreasing Q values. Figure 4 com-pares the seismograms recorded 500 m awayfrom the source using pseudospectral and HDAFmethods corresponding to the Q values in Fig-ure 3. The corresponding rms error values are2.6×10−3, 2.6×10−3, 6.5×10−3, and 1.4×10−2,respectively. We found that the smaller Q valueresults in a larger error. The rms errors can bereduced by increasing the stencil length W. Fig-

Distance (km)

Dis

tan

ce (

km)

a) Q = 100 b) Q = 30

c) Q = 10 d) Q = 5

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Figure 3. Four snapshot parts corresponding to four Q values:(a) Q ¼ 100, (b) Q ¼ 30, (c) Q ¼ 10, and (d) Q ¼ 5. A pointsource located at the center of the model. Snapshots are recordedat 0.2 s in homogeneous attenuating media.

0.1 0.2 0.3 0.4 0.5

−5

0

5

10

rms = 2.5e−3

Time (s)

Am

plit

ud

e

PseudospectralHDAF

0.1 0.2 0.3 0.4 0.5

−5

0

5

rms = 2.6e−3

Time (s)

Am

plit

ud

e

PseudospectralHDAF

0.1 0.2 0.3 0.4 0.5

−2

0

2

rms = 6.5e−3

Time (s)

Am

plit

ud

e

PseudospectralHDAF

0.1 0.2 0.3 0.4 0.5

−0.5

0

0.5

rms = 1.4e−2

Time (s)

Am

plit

ud

e PseudospectralHDAF

a)

b)

c)

d)

Figure 4. Comparison between two seismograms calculated by the pseudospectral(solid) and HDAF methods (circle) corresponding to four Q values: (a) Q ¼ 100,(b) Q ¼ 30, (c) Q ¼ 10, and (d) Q ¼ 5 at a point 500 m away from the source.

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ure 5 summarizes the relationship between rms errors and stencilsize W for different Q values.To test the accuracy of our numerical method over long distances,

we increase our model size to a 801 × 801 grid to capture wavefieldsover the longer distance. A larger time step withΔt ¼ 2 ms is used inthis study. Other parameters are the same as the previous experimentwithQ ¼ 10. We compare numerical results using the pseudospectralwith HDAF methods at receivers located at 1500 and 3000 m fromthe source in Figure 6. Numerical solutions with the HDAF methodagree with the pseudospectral method very well at these distances.

Two-layer model

This example is aimed to test the HDAFmethod for viscoacousticwave modeling in the presence of sharp contrast in velocity and Q.We use an isotropic two-layer model with c0 ¼ 1800 m∕s in the toplayer and c0 ¼ 3600 m∕s in the bottom layer (Sun et al., 2015). Theinterface is at a depth of 1040 m. The reference frequency is thesame as above. The model is discretized on a 200 × 200 grid, with

20 50 10010

−4

10−3

10−2

10−1

100

Number of stencil W

rms

erro

r

Q = 100Q = 30Q = 10

Q = 5

Figure 5. The relationship between the rms errors and number ofstencils for different Q values.

0.7 0.8 0.9 1 1.1

−0.2

−0.1

0

0.1

0.2

Time (s)

Am

plit

ud

e

PseudospectralHDAF

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2−0.04

−0.02

0

0.02

Time (s)

Am

plit

ud

e

PseudospectralHDAF

a)

b)

Figure 6. Comparison between two seismograms calculated by thepseudospectral (solid) and HDAF methods (circle) for a long offset.The receivers are at (a) 1500 and (b) 3000 m from the source.

Distance (km)0 0.5 .5

Dep

th (

km)

0

0.5

1

1.5

Distance (km)0 0.51 1 1 1.5

Dep

th (

km)

0

0.5

1

1.5

Distance (km)0 0.5 .5

Dep

th (

km)

0

0.5

1

1.5

Distance (km)

0 0.51 1 1 1.5

Dep

th (

km)

0

0.5

1

1.5

a) b)

c) d)

Figure 7. Snapshots at 330 ms for the two layer model: (a) wave-field for homogeneous Q ¼ 30, (b) wavefield for Q ¼ 30 in the toplayer and Q ¼ 100 in the bottom layer, (c) wavefield using a con-stant averaged fractional power γ in the same model as panel (b),and (d) reference wavefield calculated with pseudospectral methodin the same model as panel (b). Note: All snapshots are displayed inthe same amplitude range.

a)

–0.5

0

0.5

0 0.5 1 1.5

0 0.5 1 1.5

0 0.5 1 1.5

–0.5

0

0.5

Depth (km)

–0.5

0

0.5

b)

c)

Figure 8. Traces at x ¼ 800 m extracted from wavefield snapshotsand their difference: (a) variable γ (solid) using HDAF, homogeneousQ ¼ 30 (dashed), and their difference (dashed dotted); (b) variable γ(solid) using HDAF, average γ (dashed), their difference (dashed dot-ted); and (c) variable γ (solid) using HDAF, variable γ (solid) usingthe pseudospectral method (dashed), and their difference (dasheddotted).

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8 m spacing in both directions. The time step is 1.0 ms. A Rickerwavelet with 25 Hz center frequency is located at the (800, 800 m)position of the model. The snapshots of the wavefield at 330 ms areshown in Figure 7. Figure 7a shows the wavefield with homo-geneous attenuation, where Q ¼ 30. Figure 7b shows the wavefieldusing the HDAF method for the same velocity, but with Q ¼ 30 for

the top layer and Q ¼ 100 in the bottom layer. In Figure 7c, thevelocity and Q remain the same as those in Figure 7b; however,the fractional power of Laplacian γ is the averaged value, which cor-responds to the original implementation in Zhu and Harris (2014).Figure 7d shows the reference wavefield computed by the pseudo-spectral method for a spatially varying Laplacian operator. It is imple-mented by first calculating the fractional Laplacian operator twice inthe wavenumber domain using parameters in the top and bottomlayers, transforming them back to the space domain, and combiningthem in the space domain.To better compare the results, a middle trace at x ¼ 800 m is ex-

tracted from the wavefield snapshots. Figure 8a shows the twotraces from Figure 7a and 7b, along with their difference. In thetop layer, the velocity and Q values are the same for both models.Hence, the first propagating waveforms are identical. The effectof different attenuations can be observed from the other two wave-forms. The transmitted arrival exhibits less attenuation forQ ¼ 100.Figure 8b shows two traces from Figure 7b and 7c. Note the differ-ence between the two traces attributed to error of using an average γwhen implementing fractional Laplacian operators. Figure 8c showstwo traces from Figure 7b and 7d. We find that the results usingHDAF agree with the reference solution very well.

Gas cloud model

The third example considers a realistic geologic model with aninteresting shallow gas cloud zone just above an anticline structure(Zhu et al., 2014). Figure 9 shows the corresponding velocity and Qmodel. The model has 398 × 161 grid nodes with a spacing of 12.5 min the horizontal and vertical directions. The source is a Ricker wave-let located at 2500 and 25 m with a peak frequency of 15 Hz. Thetime step is 1.0 ms. Receivers are at the same depth as the source, andthey start from 0 to 4975 m with a spacing of 12.5 m. Figure 10ashows the seismograms for acoustic modeling. The seismograms

Distance (km)

Dep

th (

km)

0 1 2 3 4

0

0.5

1

1.5

2

1.5 2 2.5 3 3.5 4 4.5

Distance (km)

Dep

th (

km)

0 1 2 3 4

0

0.5

1

1.5

2

20 40 60 80 100 120

Velocity (km/s)

Q

a)

b)

Figure 9. (a) The gas cloud velocity and (b) Q models.

Distance (km)

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Distance (km)

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Distance (km)

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0.5

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2

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a)

b)

c)

d)

Figure 10. Seismograms: (a) acoustic data, (b) vis-coacoustic data with constant averaged fractionalpower γ, (c) viscoacoustic data with variable γ, and(d) the difference between panels (b and c). Note:All snapshots are displayed in the same amplituderange.

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for viscoacoustic modeling with constant averaged and variable frac-tional power γ are displayed in Figure 10b and 10c, respectively. Fig-ure 10d shows the difference between these two results. We alsoextract the trace at x ¼ 3000 m from three seismograms in Figure 11(direct waves are muted for better display). Figure 11a shows theeffect of phase dispersion and amplitude attenuation for a viscoacous-tic model. When variable γ is used, further phase correction, espe-cially at a high-attenuation region (approximately t ¼ 1.1 s), can beobserved in Figure 11b.

DISCUSSION

The parameters of HDAF provide control over the accuracy ofestimating the derivatives of the functions. The parameters σ andM are determined so that the bandwidth of the HDAF is higher thanthe maximum frequency (or wavenumber) of interest. In general,increasing the size of the stencilW used for HDAF can improve theaccuracy. However, it will require more computational effort. Forinstance, for a 2D model with N × N grid points, the total numberof operations for HDAF method is about N2W2. However, fast con-volution strategies can be used to reduce the computational cost. Forconstant fractional power γ, the operations are at the same level asfor the pseudospectral method. And the computational cost can befurther reduced by using a domain-decomposition convolution

method, i.e., overlap-add, overlap-save methods (Wan et al., 2002).In Table 1, we present the relative central processing unit (CPU)time required for the above testing cases using the pseudospectralmethod with averaged fractional power and HDAF method. Thecalculations were done on a Dell E5430 (Intel Core i7-3540M3.00GHz CPU). For simple models (e.g., homogeneous and two-layer models), no big differences are observed in the CPU times.For spatially varying cases, additional CPU time is required for fre-quently loading different HDAF kernels. The computational cost ofimplementing a complex 3D attenuation will be even higher. Futurework to reduce this computational cost is needed.

CONCLUSION

We propose the HDAF method to compute numerical approxi-mations for the fractional Laplacian, solving the viscoacoustic waveequation. An advantage of the HDAF is that it transforms the Lap-lacian operator into a matrix vector multiplication that involvesbanded matrix representations, similar to local methods (i.e., finitedifference and finite element), while preserving exponential accu-racy of global methods, such as spectral methods. Therefore, HDAFcan obtain the same level of accuracy as spectral methods, but it alsohas sufficient flexibility to handle complicated geometries. Numeri-cal results demonstrate that the proposed method can model wavepropagation in attenuating media with high accuracy. The currentimplementation also shows the drawback of the large computationalcost for the complex attenuation model. This issue will be investi-gated in a future study.

ACKNOWLEDGMENTS

J. Yao and D. J. Kouri gratefully acknowledge support of thisresearch under R. A. Welch Foundation grant E-0608. T. Zhuwas supported by the Jackson Postdoctoral Fellowship at the Uni-versity of Texas at Austin and startup funding from the Departmentof Geosciences and the Institute of Natural Gas Research at thePennsylvania State University.

APPENDIX A

PROPERTIES OF HDAF

In this appendix, some detailed properties of HDAF are provided.Using equation 8, HDAF can also be written as

δMðx−x0;σÞ¼ 1ffiffiffiffiffi2π

pσ

XM∕2

n¼0

�−σ2

2

�n 1

n!d2n

dx2ne−

ðx−x0Þ22σ2 : (A-1)

The Fourier transform of HDAF is given by Hoffman and Kouri(1995)

δ̂Mðk; σÞ ¼ F ½δMðx − x 0; σÞ� ¼ e−k2σ22

XM∕2

n¼0

ðk2σ2Þn2nn!

: (A-2)

Furthermore, we have

limM→∞

XM∕2

n¼0

ðk2σ2Þn2nn!

¼ ek2σ22 ; (A-3)

0.5 1 1.5 2−2

−1

0

1

2

Time (s)

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ud

e

0.5 1 1.5 2−1

−0.5

0

0.5

1

1.5

Time (s)

Am

plit

ud

e

a)

b)

Figure 11. Comparison of extracted traces between (a) viscoacous-tic modeling with variable γ (solid) and acoustic modeling (dashed)and (b) viscoacoustic modeling with variable γ (solid) and averagedγ (dashed).

Table 1. The relative CPU time (s) for calculations of differentmodels using the pseudospectral method with averagedfractional power and the HDAF method.

Model Homogeneous Two layer Gas cloud

PS 13.9 6.8 37.1

HDAF 15.4 8.1 291.3

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so that

limM→∞

δ̂ðk; σÞ ¼ 1; (A-4)

which means that for any fixed σ, the HDAF becomes identical tothe delta function in the limit of infinite M. In addition, for a fixedM, the HDAF becomes identical to the delta function when σ goesto zero;i.e.,

limσ→0

δMðx − x 0; σÞ ¼ δðx − x 0Þ: (A-5)

Hence, HDAFs provide a controllable two-parameter delta se-quence approximation to the Dirac delta function (Hoffman andKouri, 1995). The frequency response of the HDAF is a smoothedlow-pass window. The half-bandwidth of the HDAF window is(Bodmann et al., 2008)

kc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiM þ 1

p∕σ: (A-6)

In addition to the width of the passband, parameterM controls thesmoothness of HDAF. HDAF is smoother (fewer oscillations) asMdecreases (for more detailed proof of the properties of HDAF, seeBodmann et al., 2008).HDAFs with three different sets of parameters in the (1) coordi-

nate space and (2) frequency space are plotted in Figure A-1. Asexpected, the bandwidths of the HDAF with parameter σ ¼ 3,M ¼8 and σ ¼ 7, M ¼ 48 are the same. The window of the HDAF withσ ¼ 3, M ¼ 48 (dashed line) is broader in Fourier space than theother two HDAFs. The HDAF with M ¼ 8 is smoother than thosewith M ¼ 48. In conclusion, with a proper choice of parameters σ

and M, HDAFs provide an arbitrarily sharp high-frequency cutoffwhile retaining their smoothness.

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−20 −10 0 10 20−1

0

1

2

3

4

X

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ud

e

×10−2

−4 −2 0 2 40

0.2

0.4

0.6

0.8

1

Wavenumber

Am

plit

ud

e

a)

b)

Figure A-1. The HDAF in the (a) space domain and (b) wavenum-ber domain with different sets of parameters: solid, σ ¼ 3, M ¼ 8;dashed dotted, σ ¼ 3, M ¼ 48; and dashed, σ ¼ 7, M ¼ 48.

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