A constant-Q time-domain wave equation using the fractional Laplacian Tieyuan Zhu and Jerry M. Harris, Stanford University Summary

We present a constant-Q time-domain wave equation. It is derived from Kjartansson’s constant-Q constitutive stress-strain relation in combination with the mass and momentum conservation equations. Our wave equation, expressed by a second order temporal derivative and two fractional Laplacian operators, models attenuation and dispersion effects. The temporal derivative is solved by the staggered-grid finite-difference approach. The fractional Laplacian is easily calculated in the spatial frequency domain using say a Fourier pseudo-spectral implementation. The advantage of using our fractional Laplacian formulation over the traditional fractional time derivative approach is the avoidance of storing the time history of variables and thus more economic in computational costs. In numerical simulations, we incorporate PML (perfectly matched layer) absorbing boundaries. Furthermore, we verify the accuracy of numerical results through comparisons with theoretical constant-Q attenuation and dispersion solutions, McDonal’s measurements in the Pierre shale, and results from 2-D viscoacoustic analytical modeling for the homogeneous Pierre shale. We then generalize our rigorous formulation for viscoacoustic waves in homogeneous media to an approximate equation for heterogeneous media. We demonstrate the applicability and efficiency of our constant-Q time-domain approach wave equation in a heterogeneous medium with highly atttenuative formation. Introduction

Attenuation causes reduced energy and distorted phase of seismic waves (McDonal et al., 1958). In particular, high attenuation (i.e. low Q) due to the presence of gas pockets (Dvorkin and Mavko, 2006) results in the poor seismic imaging of structure within and below such gas zones (Xie et al., 2009; Zhou et al., 2011). Accurate attenuation modeling of seismic waves is therefore important for improved imaging. The constant-Q model (Kjartansson, 1979) in the time-domain wave equation is attractive because it is accurate in producing perfectly constant Q behavior at all frequencies in the band. However, the constant-Q model introduces the fractional time derivative, i.e. irrational (or non-integer) power of the time derivative (Caputo and Mainardi, 1971). Recently this fractional calculus has been introduced to model seismic wave propagation (Carcione et al., 2002; Carcione, 2009; 2010; Zhang et al., 2010). Nevertheless, it is notable that the fractional time derivative of a single

variable, e.g., t, depends on all the previous values of this variable. This property requires large memory of stress-strain history from t=0 to present time (Carcione, 2010). Therefore we have to pay the expense of memory resources even though it is possible to truncate the operator (Podlubny, 1999). To overcome this memory expense, Chen and Holm (2004) proposed to use the fractional Laplacian operator to model anomalous attenuation behavior in fractional wave equations. The fractional Laplacian is easily evaluated in the spatial frequency domain. As a result, the fractional Laplacian successfully avoid additional memory required by the fractional time derivative, and it can be efficiently computed by the generalized Fourier pseudo-spectral approach (Carcione, 2010). In this paper, we first derive a time domain wave equation using the fractional Laplacian to model constant-Q behavior. Starting from Kjartansson’s constant-Q constitutive stress-strain relation, we derive the formulation for homogeneous media and then generalize to heterogeneous media. This constant-Q wave equation introduces decoupled attenuation and velocity dispersion operators based on the fractional Laplacian. For numerical simulations, PML (perfectly matched layer) absorbing boundaries are easily incorporated with the first-order constant-Q equations. The spatial derivative is computed by a staggered-grid pseudo-spectral method while a staggered-grid finite-difference approach is used for temporal simulation. Constant-Q wave equation

In geophysics, attenuation (proportional to 1/Q) is considered to be approximately linear with frequency in many bands (Kolsky, 1956; McDonal et al., 1958; Kjartansson, 1979), i.e. Q is constant over these frequency ranges. Kjartansson (1979) explicitly gave a linear description of attenuation that exhibits the exact constant Q characteristic, which uses fractional derivatives that are

based on a relaxation function of the form t!2" , instead of

integer order of t. The relaxation function is given as

!1"2#

(t ) =

M0

$(1" 2# )

t

t0

%&'

()*

"2#

H (t ), t > 0 (1)

where

M

0

is a bulk modulus, ! is Euler’s Gamma function and

t

0

is a reference time t

0

= 1 !0

. The

DOI http://dx.doi.org/10.1190/segam2013-0664.1© 2013 SEGSEG Houston 2013 Annual Meeting Page 3417

Dow

nloa

ded

08/1

9/13

to 1

28.1

2.18

7.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Constant-Q wave equation in the time domain

parameter ! = 1 " tan#1

1 Q( ) is dimensionless, and we

know 0 < ! < 0.5 for any positive value of Q. H (t ) is the Heaviside step function. In terms of the Caputo’s fractional derivative (Caputo, 1971), the isotropic stress-strain (! - ! ) relation will be deduced in the form

! ="1#2$

(t ) %d&

dt=

M0

t0

#2$

'()

*+,-2$ &

-t2$

(2)

The symbol ! represents the convolution operator. The strain ! has non-integral order

2! of time derivative so

it is called fractional time derivative. Combining equation 2, the stress-strain relation, with the momentum conversation equations, the fractional wave equation for constant density is obtained

!2"2#

p

!t2"2#

= c0

2

$0

"2#

%2

p,

(3)

where p is pressure field and !2 is the Laplacian

operator. 2 ! 2" is the non-integer order of the time

derivative, which is the so-called fractional time derivative. The velocity 0c is specified at a particular frequency. It is evident that equation 3 reduces to the classical acoustic wave equation as

! " 0 or when Q ! " , and the

diffusion equation as ! " 1 2 when Q ! 0 . This

fractional wave equation is well-known as Caputo’s wave equation. Carcione et al. (2002) successfully solved the above wave equation using Grunwald-Letnikow and central-difference approximations for the time discretization and Fourier method to compute the spatial derivative. Recently, Carcione (2009) extended its derivation to build the elastic wave equation with constant Q. In the following, we will replace the fractional time derivative with the fractional Laplacian operator. The dispersion relation for the fractional wave equation 3 can be obtained by substituting the plane wave solution

p = ei (!t " !kr ) , where the complex wavenumber vector is !k ,

the angular frequency ! , and r is the spatial coordinate vector.

!2

c0

2= (i )

2"

!0

#2"

!2" !k

2

,

(4)

Since i2!

= cos("! ) + i sin("! ) , equation 4 is split into parts as follows:

!2

c0

2= !

0

"2#

cos$#!2# !k

2

+ i!0

"2#

sin$#!2# !k

2

,

(5)

In terms of the fractional Laplacian operator (Chen and Holm, 2004), we obtain the constant-Q wave equation

1

c0

2

!2

p

!t2= " #$

2( )% +1

p + &d

dt#$

2( )% +1/ 2

p,

(6)

where the coefficients are

! = "c

0

2#

$0

"2#

cos%# , ! = "c

0

2# "1

$0

"2#

sin%#

(7)

Under the weak attenuation conditions used previously, the corresponding dispersion relation is given by equation 5. It is evident that the first operator in the right side must correspond to the dispersion while the second is the diffusion part in wave equation that represents attenuation effects. Note that equation 6 is essentially identical to equation 3, the fractional time wave equation. When Q is infinite (no attenuation), i.e.

! = 0 , equation 6 exactly becomes the familiar acoustic wave equation. In addition, Kjartansson (1979) proved that the constant-Q dispersion relation is fulfills the Kramers-Krönig relation and therefore causal. Thus the causality of the constant-Q wave equation (6) is guaranteed. In smoothly heterogeneous media,

0c and Q are the

spatially dependent velocity and Q. Using the local principle of pseudo-differential operator (Stein, 1993), we now adapt the fractional wave equation in a homogeneous medium to approximate a heterogeneous medium. This means replacing the constant parameters appearing in the pseudo-differential operator by the variable parameters. The approximate fractional wave equation in smoothly heterogeneous media is then obtained

!2

p(r,t )

!t2

= "#( r )c0( r )

2M$

1

#( r )$p(r,t ),

(8)

where

M = !( r ) "#2( )

$+ % ( r )

d

dt"#

2( )$ "1/2&

'()*+

(9)

Simulations

DOI http://dx.doi.org/10.1190/segam2013-0664.1© 2013 SEGSEG Houston 2013 Annual Meeting Page 3418

Dow

nloa

ded

08/1

9/13

to 1

28.1

2.18

7.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Constant-Q wave equation in the time domain

To illustrate the utility of the wave equation (equation 8) to correctly model a constant-Q medium and dispersion, we compare in the first example the numerical results with laboratory measured data and Kjartanssan constant-Q theoretical data. The measured data of Pierre shale in Colorado from McDonal et al. (1958) found an approximate constant-Q behavior with attenuation coefficient ! = 0.12f dB/kft ! 0.0453f Neper/km. From Carcione (2009), we know that the quality factor of Pierre shale rock is Q=32 and the P-wave velocity is about 2164 m/s at reference frequency 100 Hz. For a homogeneous lossy medium, we choose a typical shale density of 2.2 g/cm3. The simulations are performed in 2D with a 512 ! 512 mesh, 1.0 m spacing in x and z direction and a time step of 0.23 ms. The Ricker wavelet source with 100 Hz center frequency is located at (256 m, 256 m) in simulation. Figure 1a and 1b illustrate the computed attenuation and dispersion of Pierre shale model. The experimental data from McDonal et al. (1958) are represented by diamond points. The solid lines in figures give the theoretical attenuation and dispersion based on equation 26 and 27. Values extracted from our numerical simulations at two receivers are shown with open circles. The attenuation coefficient ! and the phase velocity are calculated by the

amplitude spectral ratio in dB m and phase change at each frequency (Treeby and Cox, 2010).

! (" ) = #20 log10A

2

A1

$%&

'()

d

(10)

c(! ) =d

t= "

!d

#2" #

1

(11)

where

1, 2A are amplitude spectrums at two receivers,

!

1,2are

phase spectrums, and d is the propagation distance in meters. To keep the simulation free from numerical artifacts (e.g. numerical dispersions), we use two receivers at 20 m and 70 m from source. The geometric spreading for

2D problem is corrected by the factor r .

Figure 1. Attenuation (a) and phase velocity (b) with frequency in Pierre Shale medium with Q=32. The

numerical phase velocity and attenuation coefficients (open circles) are computed using equation 10 and 11. The solid line is computed with constant-Q model using equation 6. The experimental results (diamond point) are from McDonal‘s paper (1958). (In (a) it appears that the Q value used is too small; note how the line and the circles cross the data values at 100 Hz and the slope is slightly too steep) The calculated attenuation and dispersion appear to fit well the theoretical values as well as experimental data. In Figure 1b the calculated phase velocity drops rapidly below 40 Hz and cannot fit this portion of the theoretical results well. The possible reason is that the loss at low frequency is mostly dominated by geometric spreading (Sheriff and Geldart, 1995). Similar observations were made by Wuenschel (1965) and Liu et al. (2005).

Figure 2. Snapshot at 130 ms in (a) homogeneous constant-Q medium and (b) homogeneous lossless medium (Q = 32); The amplitude is smaller in (a) than (b). (c) Comparison between numerical solution (circle line) and constant-Q Green’s analytic solution (solid line). The receiver is at 200 m away from source (256 m, 256 m). Figure 2a and b show the snapshots at 130 ms propagation time in a constant Q acoustic media. Amplitude in Figure 2a is significantly reduced by attenuation. Figure 2c compares analytical and numerical solutions at a distance of 200 m from the source location. The 2D analytical solution is obtained by the constant-Q Green’s function convolving with source function (Carcione et al., 1988). They show the excellent agreement with RMS error of 6.3e-4. The synthetic example is shown in Figure 3 to demonstrate the applicability of the fractional Laplacian model for a

c)

a) b)

DOI http://dx.doi.org/10.1190/segam2013-0664.1© 2013 SEGSEG Houston 2013 Annual Meeting Page 3419

Dow

nloa

ded

08/1

9/13

to 1

28.1

2.18

7.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Constant-Q wave equation in the time domain

heterogeneous medium. The velocity and Q models are shown in Figure 3. The density model is constant of 2.2 g/cm3. Zone #4 in the model is the high attenuation reservoir zone (Q=30). The model is discretized in a mesh with 200 ! 600 grids. The space equally sampled in x and z at 0.5 m. A PML absorbing boundary with 20 grid points at the four boundaries is implemented. The explosive source with a 200 Hz center frequency Ricker wavelet is chosen for simulation. The time step is 0.014 ms. The location of source is located at 90 m horizontal distance and at 110 m depth. The receivers are in the left well of 0.5 m horizontal distance. They are distributed from depth 5 m to 292.5 m with space 2.5 m. Figure 4 shows the acoustic and viscoacoustic seismograms. Apparently, the reflections from last interface are attenuated after propagating back to receivers above interface 4-5. Direct waves also have loss. It shows clearly that the amplitudes of the reflected and direct waves decrease rapidly with the increasing distance. To check its accuracy of the constant-Q wave equation, we also ran the viscoacoustic modeling using the single relaxation mechanism (Zener model), which is considered to be reasonably accurate in practical applications (Blanch et al., 1995; Zhu et al., 2012). Modeling of nearly constant-Q model can be found in the previous study (Zhu et al., 2012). Figure 5 show comparison of traces between constant-Q wave equation and the nearly constant-Q wave equation based on single Zener model. The two data agree with each other very well in details, even later reflections. Conclusion

We have derived a time-domain constant-Q wave equation using the fractional Laplacian instead of the more familiar fractional time derivative. Starting from Kjartansson’s constant-Q model, we first provided the detailed formulation of this constant-Q wave equation in homogeneous media. We then used a localization principle to extend the homogeneous equation to a generalized wave equation for smoothly heterogeneous media. Numerical results of Pierre Shale rock demonstrate that this proposed wave equation describes the expected constant-Q attenuation and dispersion. Meanwhile, we show that numerical solution is highly accurate by comparing to analytical solution in homogeneous Pierre Shale medium. The final application demonstrates the applicability of the constant-Q wave equation for generally heterogeneous medium. Acknowledgments The author TZ would like to thank Stanford Wave Physics (SWP) Laboratory for financial support.

Figure 3. Heterogeneous velocity model (a) and Q model (b) for simulation.

Figure 4. Synthetic seismograms by acoustic (a) and constant-Q acoustic (b). Due to different attenuation area, reflection at interface 3-4 is less attenuated while reflections at interface 4-5 experience significant loss.

Figure 5: Comparison of two seismograms calculated by the proposed constant-Q equation (dotted line) and single Zener solid model (solid line). Amplitude is not scaled. Partial traces are shown in every other 10 trace.

DOI http://dx.doi.org/10.1190/segam2013-0664.1© 2013 SEGSEG Houston 2013 Annual Meeting Page 3420

Dow

nloa

ded

08/1

9/13

to 1

28.1

2.18

7.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

http://dx.doi.org/10.1190/segam2013-0664.1 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

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, http://dx.doi.org/10.1190/1.1443744.

Carcione, J. M., D. Kosloff, and R. Kosloff, 1988, Wave propagation in a linear viscoacoustic medium: Geophysical Journal, 93, 393–401, http://dx.doi.org/10.1111/j.1365-246X.1988.tb02010.x.

Carcione, J. M., F. Cavallini, F. Mainardi, and A. Hanyga, 2002, Time-domain seismic modeling of constant-Q seismic waves using fractional derivatives: Pure and Applied Geophysics, 159, 1719–1736. http://dx.doi.org/10.1007/s00024-002-8705-z.

Carcione, J. M., 2009, Theory and modeling of constant-Q P- and S-waves using fractional time derivatives: Geophysics, 74, no. 1, T1–T11, http://dx.doi.org/10.1190/1.3008548.

Carcione, J. M., 2010, A generalization of the Fourier pseudospectral method: Geophysics, 75, no. 6, A53–A56, http://dx.doi.org/10.1190/1.3509472.

Caputo, M. and F. Mainardi, 1971, A new dissipation model based on memory mechanism: Pure and Applied Geophysics, 91, 134–147.

Chen, W., and S. Holm, 2004, Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency: The Journal of the Acoustical Society of America, 115, 1424–1430, http://dx.doi.org/10.1121/1.1646399.

Dvorkin , J. P., and G. Mavko, 2006, Modeling attenuation in reservoir and nonreservoir rock: The Leading Edge, 25, 194–197, http://dx.doi.org/10.1190/1.2172312.

Kolsky, H., 1956, The propagation of stress pulses in viscoelastic solids: Philosophical Magazine, 1, 693–710, http://dx.doi.org/10.1080/14786435608238144.

Kjartansson, E., 1979, Constant-Q wave propagation and attenuation: Journal of Geophysical Research, 84, B9, 4737–4748, http://dx.doi.org/10.1029/JB084iB09p04737.

Liu, Y., T. Teng, and Y. Ben-Zion, 2005, Near-surface seismic anisotropy, attenuation and dispersion in the aftershock region of the 1999 Chi-Chi earthquake: Geophysical Journal International, 160, 695–706, http://dx.doi.org/10.1111/j.1365-246X.2005.02512.x.

McDonal, F. J., F. A. Angona, R. L. Mills, R. L. Sengbush, R. G. Van Nostrand, and J. E. White, 1958, Attenuation of shear and compressional waves in Pierre shale : Geophysics, 23, 421–439, http://dx.doi.org/10.1190/1.1438489.

Podlubny, I., 1999, Fractional differential equations: Academic Press.

Sheriff, R. E., and L. P. Geldart, 1995, Exploration seismology: Cambridge University Press.

Stein, E., 1993, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals: Princeton University Press.

DOI http://dx.doi.org/10.1190/segam2013-0664.1© 2013 SEGSEG Houston 2013 Annual Meeting Page 3421

Dow

nloa

ded

08/1

9/13

to 1

28.1

2.18

7.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Treeby, B. E., and B. T. Cox, 2010, Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian: The Journal of the Acoustical Society of America, 127, 2741–2748, http://dx.doi.org/10.1121/1.3377056.

Wuenschel, P. C., 1965, Dispersive body waves — An experimental study: Geophysics, 30, 539–551, http://dx.doi.org/10.1190/1.1439620.

Xie, Y., K. Xin, J. Sun, C. Notfors, A. K. Biswal, and M. K. Balasubramaniam, 2009, 3D prestack depth migration with compensation for frequency dependent absorption and dispersion: 79th Annual International Meeting, SEG, Expanded Abstracts, 2919–2923, http://dx.doi.org/10.1190/1.3255457.

Zhang, Y., P. Zhang, and H. Zhang, 2010, Compensating for viscoacoustic effects in reverse-time migration: 80th Annual International Meeting, SEG, Expanded Abstracts, 3160–3164, http://dx.doi.org/10.1190/1.3513503.

Zhou, J., S. Birdus, B. Hung, K. H. Teng, Y. Xie, D. Chagalov, A. Cheang, D. Wellen, and J. Garrity, 2011, Compensating attenuation due to shallow gas through Q tomography and Q-PSDM, a case study in Brazil: 81st Annual International Meeting, SEG, Expanded Abstracts, 3332–3336, http://dx.doi.org/10.1190/1.3627889.

Zhu, T., J. M. Carcione, and J. M. Harris, 2012, Approximating constant-Q seismic propagation in the time domain: 82nd Annual International Meeting, SEG, Expanded Abstracts, http://dx.doi.org/10.1190/segam2012-1415.1.

DOI http://dx.doi.org/10.1190/segam2013-0664.1© 2013 SEGSEG Houston 2013 Annual Meeting Page 3422

Dow

nloa

ded

08/1

9/13

to 1

28.1

2.18

7.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/