Attenuation compensation for time-reversal imaging in VTImedia

Tong Bai1, Tieyuan Zhu2, and Ilya Tsvankin1

1.Center for Wave Phenomena, Department of Geophysics, Colorado School of Mines2.Department of Geoscience and Institute of Natural Gas Research, Pennsylvania State University

ABSTRACTThe time symmetry of the wave equation enables application of time-reversal model-ing in acoustic and elastic media. Time reversal represents a key component not justin reverse-time migration (RTM), but also in source localization using passive seis-mic (e.g., microseismic) data. This symmetry in time, however, is no longer valid inattenuative media, and attenuation is often anisotropic. Here, we employ a viscoelas-tic anisotropic wave equation that decouples the influence of energy dissipation andvelocity dispersion. That equation helps compensate for anisotropic attenuation andrestore the time symmetry by changing the signs of the dissipation-dominated terms intime-reversed propagation, while keeping the dispersion-related terms unchanged. TheQ-compensated time-reversal imaging algorithm is tested on synthetic microseismicdata from 2D transversely isotropic media with a vertical symmetry axis (VTI). Af-ter back-propagating multicomponent data acquired in a vertical borehole, we imagemicroseismic sources using wavefield focusing. Accounting for attenuation anisotropyproduces superior source images compared to those obtained without attenuation com-pensation or with a purely isotropic Q-factor.

Key words: time reversal, attenuation compensation, VTI

1 INTRODUCTION

A fundamental property of wave propagation through elasticor acoustic media is reciprocity, which enables application oftime reversal (TR). By reversing the recorded data in time andthen injecting them back into the medium, TR is supposed tofocus the energy at the excitation location and time, given asufficiently wide acquisition aperture and knowledge of themedium parameters (e.g., velocities). In addition to its applica-tions in passive surveys to locate and describe seismic sources(e.g., McMechan, 1982; Artman et al., 2010; Gajewski andTessmer, 2005; Saenger, 2011), time reversal is a crucial stepin reverse-time migration (RTM). Time-reversed data are in-jected at the receiver locations and interact with the source(forward) wavefield through the imaging condition to producereflecting interfaces (e.g., Baysal et al., 1983; McMechan,1983).

The time-invariance properties of acoustic and elastic(nonattenuative) wave equations are explained by the presenceof only even-order time derivatives. However, seismic wavesalways experience energy dissipation and velocity dispersionbecause subsurface formations are attenuative (e.g., Single-ton, 2008; Schuler et al., 2014). Attenuation-related termsbreak the time symmetry in the commonly used wave equa-

tions based on either the Generalized Standard Linear Solid(GSLS) model (e.g., Bohlen, 2002; Bai and Tsvankin, 2016) orKjartansson’s constant-Q model (e.g., Carcione, 2008). Theseequations include first-order time derivatives or fractional timederivatives produced by the convolutional stress-strain rela-tionship. As discussed by Zhu et al. (2014), time-reversingand back-propagating recorded viscoacoustic data through at-tenuative media generates a distorted source image because ofadditional attenuation during TR modeling. Therefore, to pre-serve the TR symmetry and reconstruct seismic sources, it isnecessary to compensate for the influence of attenuation dur-ing back-propagation (e.g., Fink and Prada, 2001).

Nearly-constant-Q (NCQ) models (e.g., Emmerich andKorn, 1987; Carcione, 1993; Bohlen, 2002; Bai and Tsvankin,2016) are often adopted in simulating wave propagation inviscoacoustic and viscoelastic media. A convolutional kernel,usually called the relaxation function, relates the stress andstrain fields and ensures nearly invariant Q-values within aspecified frequency band, given a sufficient number of relax-ation mechanisms. Based on Kjartansson’s constant-Q model,Carcione (2008) proposed an alternative approach for Q-simulation that involves fractional time derivatives. Despite itsadvantages (accurate constant-Q function and simple parame-terization), numerical implementation of that method is ham-

2 T. Bai, T. Zhu and I. Tsvankin

pered by the need to store all previously computed wavefields,which entails excessive memory requirements.

The above propagators, however, are not suitable for Q-compensation because the dissipation and dispersion opera-tors are coupled, and amplitude compensation is inevitably ac-companied by a distortion of the velocity dispersion. UsingCarcione’s (2008) formula with the fractional time derivatives,Zhu and Harris (2014) derive a decoupled constant-Q acousticwave equation with two separate fractional Laplacian opera-tors accounting for amplitude dissipation and velocity disper-sion. Zhu and Carcione (2014) generalize that approach forviscoelastic (but still isotropic) media. The new propagatorsare implemented by Zhu (2015) and Zhu and Sun (2017) torestore the time symmetry in time-reversal imaging and RTMfor isotropic attenuative models.

Experimental data (Zhu et al., 2006; Best et al., 2007;Zhubayev et al., 2015) confirm the existence of substantial at-tenuation anisotropy in the subsurface. Estimation of attenua-tion anisotropy can provide new physical attributes for reser-voir characterization and lithology discrimination (e.g., Be-hura et al., 2012; Guo and McMechan, 2017). Within theframework of the GSLS model, Bai and Tsvankin (2016)develop a time-domain finite-difference modeling algorithmfor viscoelastic media with VTI symmetry for both veloc-ity and attenuation. Employing that forward propagator, Baiet al. (2017) present a waveform-inversion methodology forVTI media, which can estimate the attenuation parameters re-quired for anisotropic Q-compensation. Zhu (2017) extendsthe constant-Q modeling approach based on the fractionaltime derivatives (Carcione, 2008) to anisotropic viscoelasticmedia, and Zhu and Bai (2018) approximate the derivativeswith fractional Laplacians for efficient numerical implementa-tion.

Here, we apply the propagator developed by Zhu andBai (2018) to Q-compensation in time-reversal imaging foranisotropic media. First, we briefly review the properties ofthe decoupled (in terms of dissipation and dispersion) vis-coelastic anisotropic wave equation. Next, we show how thatequation can be modified to restore the time symmetry forback-propagation in the presence of both attenuation and ve-locity anisotropy. A synthetic test demonstrates the decou-pling of dispersion and dissipation phenomena in modeling ofanisotropic wavefields based on the developed formalism. Fi-nally, we implement anisotropicQ-compensation during back-propagation with the goal of source imaging from microseis-mic data.

2 METHODOLOGY

2.1 Anisotropic viscoelastic modeling based onfractional Laplacians

By using the fractional Laplacian to approximate the fractionaltime derivatives, the stress (σij)-strain (εij) relationship for

attenuative VTI media can be written as (Zhu and Bai, 2018):

σ11 = η11 v2γ1111 (ε11 + ε33) +

(η13 v

2γ1355 − η11 v

2γ1155

)ε33

+ τ11 v2γ

11−1

11 (ε̇11 + ε̇33) +(τ13 v

2γ13

−1

55 − τ11 v2γ

11−1

55

)ε̇33,

(1)

σ33 = η33 v2γ3333 (ε11 + ε33) +

(η13 v

2γ1355 − η33 v

2γ3355

)ε11

+ τ33v2γ

33−1

33 (ε̇11 + ε̇33) +(τ13 v

2γ13

−1

55 − τ33 v2γ

33−1

55

)ε̇11,

(2)and

σ13 = η55 v2γ

5555 2ε13 + τ55 v

2γ55

−1

55 2ε̇13, (3)

where

ηij = C0ij cos2(

πγij

2) (ω0)

−2γij cos(πγij )

(−∇2)γij ,

(4)

τij = C0ij cos2(

πγij

2) (ω0)

−2γij sin(πγij )

(−∇2)γij− 1

2 ,

(5)and

γij =1

πtan−1

(1

Qij

). (6)

Here the dot denotes the time derivative, ω0 is the ref-erence frequency, which should be larger than the dominantfrequency of the source signal, C0

ij are the stiffness coeffi-cients defined at the frequency ω0, Qij is the VTI quality-factor matrix (Zhu and Tsvankin, 2006), v11 =

√C0

11/ρ,v33 =

√C0

33/ρ, and v55 =√C0

55/ρ (v11, v33, v55 are thevelocities of the horizontally traveling P-wave and verticallytraveling P- and S-waves, respectively). To describe the atten-uation of P- and SV-waves in VTI media, it is convenient touse the Thomsen-style attenuation parameters AP0, AS0, εQ ,and δQ instead of the elements Qij (Appendix A).

The terms multiplied with ηij and τij in equations 1-3account for the dispersion and dissipation, respectively. Notethat with γij = 0 in equations 4-5, we obtain ηij = C0

ij

and τij = 0 (ij = 11, 13, 33, 55), and equations 1-3 de-scribe purely elastic (nonattenuative) VTI medium. On theother hand, setting γij = 0 in equation 4 eliminates velocitydispersion, while γij = 0 in equation 5 removes dissipation.

2.2 Viscoelastic time-reversal imaging

To implement time reversal, we replace the time t in equa-tions 1-3 with T − t̂, where T is the total recorded time andt̂ is the time variable for reverse propagation. The new sys-tem described by t̂ does not coincide with the original equa-tions 1-3 because of the presence of the first-order time deriva-tive in the terms controlling the amplitude dissipation. To pre-serve time symmetry, we need to boost the amplitude duringback-propagation, whereas the dispersion relationship shouldremain the same (Zhu, 2014). Hence, we change the sign in

Attenuation compensation for time-reversal imaging in VTI media 3

front of the dissipation-related operators (the terms containingτij) in equations 1-3:

σ11 = η11 v2γ

1111 (ε11 + ε33) +

(η13 v

2γ13

55 − η11 v2γ

1155

)ε33

− τ11 v2γ

11−1

11 (ε̇11 + ε̇33)−(τ13 v

2γ13

−1

55 − τ11 v2γ

11−1

55

)ε̇33,

(7)

σ33 = η33 v2γ

3333 (ε11 + ε33) +

(η13 v

2γ13

55 − η33 v2γ

3355

)ε11

− τ33v2γ33−1

33 (ε̇11 + ε̇33)−(τ13 v

2γ13−1

55 − τ33 v2γ33−1

55

)ε̇11,

(8)and

σ13 = η55 v2γ

5555 2ε13 − τ55 v

2γ55

−1

55 2ε̇13, (9)

If the time t (hidden in the dot) in equations 7-9 is re-placed with T − t̂ , the expressions for σ11, σ33, and σ13

become identical to equations 1-3. This means that the time-invariance of the viscoelastic system can be restored by em-ploying the modified equations 7-9 and amplifying the ampli-tudes during back-propagation.

To avoid instability that may be caused by enhancinghigh-frequency noise in the data, we apply a low-pass Tukeytaper to the dissipation-related terms during time-reversalmodeling. The taper parameters (cutoff frequency and taperratios) are data-dependent.

3 NUMERICAL EXAMPLES

3.1 Decoupling of dissipation and dispersion

To study the decoupled dissipation and dispersion effects, weexcite the wavefield by an explosive source embedded in a ho-mogeneous VTI medium. Because the focus here is on theinfluence of attenuation (rather than velocity) anisotropy, weset the Thomsen velocity parameters ε and δ at the referencefrequency to zero, while the magnitude of the attenuation-anisotropy parameters is relatively large: εQ = −0.6 andδQ = −1.5 (such negative values have been observed in lab-oratory experiments). The top two panels in Figure 1 showthe displacement generated in the reference elastic medium (a)and in the fully attenuative model (i.e., that with both disper-sion and dissipation, b). The bottom panels display the wave-field obtained by including only dissipation (c) or dispersion(d). As expected, the dissipation operator significantly reducesthe amplitudes in Figures 1 (b) and 1 (c). In contrast, the dis-persion operator only delays the wavefronts in Figures 1 (b)and 1 (d), especially near the vertical direction.

The variation of amplitude with angle shown in Figures 1(b) and (c) is in good agreement with the linearized P-wavequality factor QP = 1/(2AP ) derived by Zhu and Tsvankin(2006):

QP (θ) = QP0 (1− δQ sin2 θ cos2 θ − εQ sin4 θ). (10)

For negative εQ and δQ used in Figure 1, the factor QP

(a) (b)

(c) (d)

x(km)

z(km

)

Figure 1. Amplitude snapshots of the wavefield from an explosivesource located at the center of a homogeneous VTI model. The wave-field is computed for: (a) reference elastic medium; (b) fully vis-coelastic medium; (c) dissipation-only medium; (d) dispersion-onlymedium. The parameters are: VP0 = 2 km/s, VS0 = 1 km/s, ε = 0,δ = 0, QP0 = 20, QS0 = 50, εQ = −0.6, δQ = −1.5, and ρ =

2.0 g/cm3. The source excites a Ricker wavelet with a central fre-quency of 100 Hz (the reference angular frequency ω0 = 2000 rad/s).

Figure 2. Linearized P-wave quality factor as a function of the phaseangle (equation 10) for the model from Figure 1 (QP0 = 20, εQ =−0.6, and δQ = −1.5).

increases away from the vertical up to angles close to 65◦ andthen decreases toward 90◦ (Figure 2).

The dispersion-related wavefront delay in Figures 1 (b)and (d) is also anisotropic: the P-wavefront is visibly faster inthe horizontal and oblique directions than in the vertical direc-tion (Figure 3). Similar observations are made by Galvin andGurevich (2015), who study dispersion due to wave-inducedfluid flow in fractured media. In the intermediate frequencyrange where the dispersion is significant, the P-wave velocitydiffers in the directions parallel and perpendicular to alignedfractures, whereas at high frequencies the two velocities co-incide (see Figure 2 in Galvin and Gurevich, 2015). Like-wise, in our model there is no velocity anisotropy at the highreference frequency, for which ε = δ = 0. Note that al-

4 T. Bai, T. Zhu and I. Tsvankin

x (km)z

(km

)

Figure 3. Zoom of Figure 1d, which shows the wavefield in adispersion-only VTI medium. The green dashed line marks theisotropic P-wavefront; the red arrow points to the SV-wave exciteddue to the angle-dependent velocity dispersion.

though we used an explosive source and there is no velocityanisotropy at high frequencies, attenuation anisotropy causesangle-dependent velocity dispersion, which produces a rela-tively weak SV-wave arrival (Figure 3).

3.2 Time-reversal imaging of microseismic sources

To demonstrate the need to apply anisotropicQ-compensationin attenuative VTI media, we conduct time-reversal imagingof synthetic microseismic data for the model in Figure 4. Allthree layers have significant attenuation anisotropy, and lay-ers 2 and 3 have moderate values of the velocity coefficients εand δ typical for shales (Table 1). The wavefields are excitedby three dislocation sources with different magnitudes of themoment tensor. Figure 5 displays the horizontal displacementgenerated in the reference elastic medium (Figure 5(a)) and theactual viscoelastic VTI model (Figure 5(b)). Both energy dis-sipation and dispersion-caused time delay are clearly visiblein Figure 5(b), especially for the later arrivals. Before apply-ing time-reversal imaging, we add band-limited random noiseto the viscoelastic data (Figure 5(c)).

Next, we reverse the data in time and inject them backinto the medium to localize the sources. The maximum am-plitude of the horizontal displacement here and in the nextexample is chosen as the imaging condition (optimal choiceof the imaging condition is outside the scope of this paper).Figure 6(a) shows the reference image with accurate sourcelocations obtained by elastic TR of the noise-free elastic data(Figure 5(a)). Next, we apply the same elastic TR algorithmto the noisy viscoelastic data in Figure 5(c). As expected, thesource images are blurry because of the uncompensated influ-ence of attenuation (Figure 6(b)). Then, we compensate for at-tenuation during back-propagation, but under the isotropic Q-assumption. In that case, the time-reversed wavefield is over-compensated due to the negative values of εQ and δQ , andthe source images (especially for the right two sources) are

Table 1. Parameters of the VTI model from Figure 4. The velocityparameters VP0, VS0, ε, and δ correspond to the real parts of thestiffnesses Cij defined at an angular reference frequency of 10000

rad/s.

Layer VP0 VS0 ε δ ρ

(km/s) (km/s) (g/cm3)

1 2.0 1.2 0.1 0.05 2.02 2.5 1.25 0.2 0.15 2.23 2.8 1.5 0.25 0.18 2.4

Layer QP0 QS0 εQ δQ

1 40 30 -0.3 -0.22 20 50 -0.6 -1.23 30 60 -0.4 -0.8

Figure 4. Geometry of a synthetic microseismic survey in a three-layer VTI medium. The model size is 240 m × 320 m, with the gridspacing ∆x = ∆z = 0.4 m; the interval parameters are listed inTable 1. Three dislocation sources (marked by red dots) with nonzeromoment-tensor components M13 = 600 GPa, 800 GPa, and 1200

GPa (from left to right) are initiated at the origin times equal to 18ms, 3 ms, and 30 ms, respectively; the central frequency of the sourcesignal is 250 Hz. The green line at x = 20 m marks the receiver array.

smeared (Figure 6(c)). Finally, taking the actual attenuationanisotropy into account allows us to obtain well-focused im-ages of all three sources, with the quality comparable to thatof the reference result (Figure 6(a)).

Next, we test the Q-compensated TR imaging algorithmon microseismic data simulated for a modified section ofthe BP TI model. The attenuation parameters are generatedby scaling the corresponding velocity parameters: QP0 =1.25QS0 = 10VP0 (in km/s), εQ = −2

√ε, and δQ =

−10 δ (Figure 7). The magenta dots in Figure 7(a) are dislo-cation sources that represent dip-slip faults with the dip closeto 70◦. The sources excite a Ricker wavelet with the centralfrequency ranging from 30 to 40 Hz; the receivers are placedin the vertical monitoring “well” at x = 7.5 km. As in the pre-vious test, we back-propagate the modeled viscoelastic data tofocus the wavefield at the source locations.

Attenuation compensation for time-reversal imaging in VTI media 5

(a) (b) (c)

Figure 5. Horizontal displacement for the model in Figure 4. The data are computed for (a) the reference elastic medium, and (b) the viscoelasticmedium. (c) The data from plot (b) after addition of band-limited random noise with the signal-to-noise ratio close to four.

(a) (b) (c) (d)

Figure 6. (a) Source images of the elastic data from Figure 5(a) obtained by elastic time-reversal (TR). Source images of the noise-contaminatedviscoelastic data from Figure 5(c) obtained using (b) elastic TR (i.e., no Q-compensation), (c) viscoelastic TR with a purely isotropic Q-factor, and(d) viscoelastic TR with the actual anisotropic attenuation. A Tucky taper with a cutoff frequency of 600 Hz and taper ratio of 0.2 is applied tostabilize back-propagation. All images are plotted on the same scale.

The reference source images obtained by elastic TR ofthe purely elastic data are shown in Figure 8(a). The qual-ity of source imaging is lower than in the previous experi-ment because of the interference of the back-propagated wave-fields of the adjacent sources. Still, the TR algorithm focusesthe wavefield at the actual source locations (as indicated bythe red dashed line). Purely elastic TR applied to viscoelas-tic data strongly underestimates the energy at the source lo-cations (Figure 8(b)). In contrast, attenuation compensationbased on the isotropic-Q assumption leads to overestated mag-nitudes and smeared source images (Figure 8(c)). Taking ac-tual attenuation anisotropy into account generates source im-ages (Figure 8(d)) almost indistinguishable from the referenceones (Figure 8(a)).

4 CONCLUSIONS

We implemented time-reversal imaging with a viscoelasticVTI wave equation, in which the energy dissipation and ve-locity dispersion are separated. By reversing the signs of thedissipation-related terms while keeping those accounting fordispersion unchanged during back-propagation, we compen-sate for the Q-effect and preserve the time-invariance prop-

erties of the wave equation for attenuative anisotropic me-dia. The presence of attenuation anisotropy causes angle-dependent amplitude and traveltime variation even withoutvelocity anisotropy at the reference frequency. Numericaltests on synthetic viscoelastic microseismic data from a lay-ered VTI model and modified BP TI section validate the Q-compensated time-reversal imaging algorithm. Isotropic Q-compensation generates smeared source images with over-stated magnitudes for typical negative attenuation-anisotropyparameters. In contrast, accounting for attenuation anisotropyleads to superior source images, which are comparable to thereference ones obtained for purely elastic media.

5 ACKNOWLEDGEMENTS

We thank Hemang Shah of BP for generating the TTI model.This work was supported by the Consortium Project on Seis-mic Inverse Methods for Complex Structures at CWP. The re-producible numeric examples in this paper are generated withthe Madagascar open-source software package freely availablefrom http://www.ahay.org.

6 T. Bai, T. Zhu and I. Tsvankin

(a) (b) (c) (d)

Figure 7. Attenuation parameters for a modified section of the BP TI model: (a) QP0, (b) QS0, (c) εQ , and (d) δQ . The model size is 8437.5 m× 11250 m, with grid spacing ∆x = ∆z =18.75 m. The green line on plot (a) marks the receiver array, and the magenta dots denote dislocationsources with the excitation time ranging from 72 to 144 ms and the central frequency from 40 to 50 Hz.

(a) (b) (c) (d)

Figure 8. (a) Source images of the data for the elastic version of the model from Figure 7 obtained by elastic time-reversal (TR). Source imagesof the viscoelastic data obtained using (b) elastic TR (i.e., no Q-compensation), (c) viscoelastic TR with a purely isotropic Q-factor, and (d)viscoelastic TR with the actual anisotropic attenuation. A Tucky taper with a cutoff frequency of 70 Hz and taper ratio of 0.2 is applied to stabilizeback-propagation. All images are plotted on the same scale.

APPENDIX A: THOMSEN-STYLE ATTENUATIONPARAMETERS FOR P- AND SV-WAVES INATTENUATIVE VTI MEDIA

P- and SV-wave propagation in viscoelastic VTI media can bedescribed by the stiffness coefficients Cij (defined at a ref-erence frequency) and the quality factor elements Qij (ij =11, 13, 33, 55). However, for estimating velocity and attenu-ation models, it is more convenient to adopt Thomsen ve-locity parameters and Thomsen-style attenuation parameters(Zhu and Tsvankin, 2006; Tsvankin and Grechka, 2011). TheVTI attenuation parameters for P- and SV-waves are definedas (Zhu and Tsvankin, 2006):

AP0 ≈1

2QP0=

1

2Q33, (A-1)

AS0 ≈1

2QS0=

1

2Q55, (A-2)

εQ =Q33 −Q11

Q11, (A-3)

δQ =1

2AP0

d2APdθ2

∣∣∣∣θ=0◦

, (A-4)

where AP is the P-wave phase attenuation coefficient, θ is thephase angle, AP0 and AS0 are the symmetry (vertical) atten-uation coefficients for P- and SV-waves, εQ is the fractionaldifference between the horizontal and vertical P-wave attenu-ation coefficients, and δQ controls the curvature of AP in thevertical direction.

Attenuation compensation for time-reversal imaging in VTI media 7

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8 T. Bai, T. Zhu and I. Tsvankin