Viscoacoustic modeling and imaging using low-rank approximation Junzhe Sun 1, Tieyuan Zhu 2, and Sergey Fomel 1, 1 The University of Texas at Austin; 2 Stanford University SUMMARY A constant-q wave equation involving fractional Laplacians was recently introduced for viscoacoustic modeling and imaging. This fractional wave equation suffers from a mixeddomain problem, because it involves the fractional-laplacian operators with a spatially varying power. We propose to apply low-rank approximation to the mixed-domain symbol, which allows for an arbitrarily variable fractional power of the Laplacians. Using the new low-rank scheme, we formulate the framework of the Q-compensated reverse-time migration (RTM) and least-squares RTM (LSRTM) for attenuation compensation. Numerical examples using synthetic data demonstrate the advantage of using low-rank wave extrapolation with a constant-q fractional-laplacian wave equation for seismic modeling, Q-compensated RTM, as well as LSRTM. INTRODUCTION Modeling acoustic wave propagation in attenuating media accounts for the inelastic characteristics of the real Earth (Carcione, 2007). Numerous studies have shown that some hydrocarbon prospecting areas, such as those where a gas accumulation is present, highly attenuate seismic waves (Carcione et al., 2003; Dvorkin and Mavko, 2006). Seismic attenuation can be expressed as the combined effect of energy loss and velocity dispersion. Attenuation effects can be modeled by incorporating the quality factor, Q, in the time-domain wave equation. One of the approaches involves a superposition of mechanical elements (e.g., Maxwell and standard linear solid elements) to characterize Q, and is known as approximate constant-q models (Liu et al., 1976; Emmerich and Korn, 1987; Carcione, 2007; Zhu et al., 2013). The approximate constant-q approach suffers from large computational and memory requirements. Kjartansson (1979) proposed a constant-q model that assumes a linear relationship between the attenuation coefficient and frequency. This model was proved accurate in capturing constant- Q behavior within the seismic frequency band. However, early implementations of this model involved a fractional time derivative (Caputo and Mainardi, 1971), which had to be calculated from the history of the wavefield. This requirement rendered the memory cost unfeasible for practical applications, even when the fractional operator was truncated after a certain time period (Podlubny, 1999; Carcione et al., 2002; Carcione, 2009). To overcome this issue, fractional-laplacian operators have been introduced to approximate the constant-q viscoacoustic wave equation (Chen and Holm, 2004; Carcione, 2010; Treeby and Cox, 2010; Zhu and Harris, 2014). The fractional- Laplacian approach is attractive because it can be conveniently evaluated in the wavenumber domain using fast Fourier transforms without introducing any extra variables (Carcione, 2010). Using this approach, Zhu and Harris (2014) developed a decoupled wave equation that accounts separately for amplitude attenuation and phase dispersion effects, thus allowing correct compensation for both factors by reversing the sign of the attenuation operator and keeping the sign of the dispersion operator unchanged during back propagation. Zhu et al. (2014) further proposed to use the Q-compensated wave equation for reverse-time migration. Zhang et al. (2010) applied an analogous approach derived from normalization transforms. Other strategies for compensating for attenuation during migration include methods based on wave-equation migration (WEM) using a complex dispersion relation (Valenciano and Chemingui, 2012), as well as methods based on Kirchhoff migration and reverse-time migration (RTM), using time-variant Q filters (Cavalca et al., 2013). Dutta et al. (2013) used least-squares RTM (LSRTM) for attenuation compensation based on standard linear solid model, with a simplified stress-strain relation and incorporated only one relaxation mechanism (Robertsson et al., 1994; Blanch et al., 1995). The fractional-laplacian approach was previously implemented either using a pseudo-spectral method, by taking the average of the fractional power of the Laplacian operator as an approximation (Zhu and Harris, 2014), or using a finite-difference approach (Lin et al., 2009). In this paper, we apply a low-rank approximation scheme (Fomel et al., 2013; Sun and Fomel, 2013) to solve the decoupled fractional Laplacians (Zhu and Harris, 2014), with accurately captured spatially varying fractional power. Additionally, we derive the adjoint of the forward modeling operator, which correctly accounts for dispersion but not for amplitude loss. LSRTM based on the proposed operator and its adjoint can be used to recover the true reflectivity image of the attenuating medium through iterations of migration and modeling (Sun et al., 2014). An alternative is to use an operator that compensates for amplitude loss during back propagation of the viscoacoustic data, following the approach of Zhu et al. (2014). We compare the pros and cons of these two approaches using a synthetic constant-q model and show that, while LSRTM requires more computational effort, it also produces results of higher quality and is more stable. THEORY The time-domain viscoacoustic wave equation can be written as (Carcione et al., 2002) 2 2γ p t 2 2γ = c2 ω 2γ 0 2 p, (1) where p(x,t) is the pressure wavefield, γ(x) = arctan(1/q(x))/π is a dimensionless parameter, c 2 (x) = c 2 0 (x)cos2 (πγ(x)/2), 2 is the Laplacian operator, and c 0 (x) is the velocity model defined at the reference frequency ω 0. When Q is finite, the wave equation involves a fractional time derivative. In a homogeneous medium, substituting the plane-wave solution e (iωt i k x) into equation 1, where ω is the angular fre- Page 3997
Low-rank viscoacoustic wave equation quency, k is the complex wavenumber vector, and x is the spatial coordinate vector, leads to the dispersion relation ω 2 c 2 = (i)2γ ω 2γ 0 ω 2γ k 2. (2) Zhu and Harris (2014) adopt the following approximate dispersion relation with decoupled Laplacians: ω 2 c 2 = β 1η k 2γ+2 iβ 2 ωτ k 2γ+1, (3) which corresponds to the new constant-q wave equation: 1 2 p c 2 t 2 = 2 p + β 1 {η( 2 ) γ+1 2 }p (4) +β 2 τ t ( 2 ) γ+1/2 p, where η = c 2γ 0 ω 2γ 0 cos(πγ) and τ = c 2γ 1 0 ω 2γ 0 sin(πγ). Setting β 1 = 1 and β 2 = 1 leads to the constant-q wave equation. Note that both c 0, the phase velocity, and γ in the fractional power can be heterogeneous (a function of space location). Solving for ω in equation 3 yields the phase function φ(x,k, t), which can be supplied to the low-rank one-step wave extrapolation (Sun and Fomel, 2013). The phase function is then defined as where: φ 1 (x,k, t) = k x + ip 1 ± p 2 2 t, (5) p 1 = τc 2 k 2γ+1, (6) p 2 = τ 2 c 4 k 4γ+2 4ηc 2 k 2γ+2. (7) In practice, the sign before p 2 is chosen positive because it corresponds to the correct branch of the solution. When back propagating the wavefield, reversing the sign of β 2 will amplify the amplitude, while β 1 will be kept unchanged to counteract the dispersion effects. The attenuation-compensated constant-q wave equation leads to the dispersion relation: ω 2 c 2 = η k 2γ+2 + iωτ k 2γ+1, (8) which defines the following phase function: φ 2 (x,k, t) = k x + ip 1 ± p 2 2 t (9) Both φ 1 and φ 2 involve the fractional power of wavenumber, and depend on both x and k, the Fourier transform pair. The low-rank one-step wave extrapolation operator (Sun and Fomel, 2013) provides a convenient way to utilize the phase function to extrapolate a viscoacoustic wavefield, while allowing γ(x), the fractional power of wavenumber, to vary in space. The one-step mixed-domain operator has the form of Fourier integral operator: p(x,t + t) = Its adjoint form can be expressed as P(k,t) = P(k,t)e iφ(x,k, t) dk. (10) p(x,t + t)e i φ(x,k, t) dk, (11) where φ denotes the complex conjugate of φ. Low-rank decomposition (Fomel et al., 2013) can be used to approximate the wave extrapolation symbol and speed up the computation. Substituting equation 5 into equation 10, we can write the forward extrapolation operator explicitly as p(x,t + t) = = P(k,t)e iφ 1(x,k, t) dk (12) P(k,t)e ik x+(p 1±ip 2 ) t/2 dk. The adjoint operator that can be used for LSRTM acquires the form (Sun et al., 2014): P(k,t) = p(x,t + t)e i φ 1 (x,k, t) dk (13) = p(x,t + t)e ik x+(p 1 ip 2 ) t/2 dk. On the other hand, the structure of φ 2 provides an alternative strategy for time-reversal wave propagation. The backwardextrapolation operator compensating for attenuation can be written as P(k,t) = = p(x,t + t)e i φ 2 (x,k, t) dk (14) p(x,t + t)e ik x+( p 1 ip 2 ) t/2 dk. We can see that operator 14 is no longer the adjoint of operator 12 (since φ 1 and φ 2 are essentially different). Operator 14 has an exponentially growing term that compensates for the amplitude loss in viscoacoustic data, and might be prone to stability issues. In order to avoid amplifying the high-frequency noise during backward wave propagation, Zhu et al. (2014) employed a low-pass filter for the attenuation and dispersion operators in the wavenumber domain. However, this method may compromise accuracy for stability. To improve the stability and accuracy of attenuation compensation, we propose to use operators 12 and 13 in LSRTM to compensate for attenuation through conjugate-gradient iterations. As will be demonstrated in the numerical examples, LSRTM is capable of recovering the true amplitude loss caused by viscoacoustic attenuation. NUMERICAL EXAMPLES Two-layer model Our first example is chosen to investigate the accuracy of the solution of the constant-q wave equation using the new lowrank scheme, in the presence of a sharp velocity/q contrast. We use an isotropic two-layer model with v = 1800 m/s in the top layer and v = 3600 m/s in the bottom layer (Zhu and Harris, 2014). The model is discretized on a 200 200 grid, with a spatial sampling of 8 m along both X and Z directions. An explosive source with a peak frequency of 50 Hz is located at the center. The constant-q model is defined at a reference frequency ω 0 = 1500 Hz. Wavefield snapshots are taken at t = 330 ms. Figure 1(a) shows the acoustic case, in which the Page 3998
Low-rank viscoacoustic wave equation model has velocity discontinuity but no attenuation (Q = ). For comparison, Figure 1(b) demonstrates the effect of homogeneous attenuation where Q = 30. Both velocity dispersion and amplitude loss can be observed. In Figure 1(c), we set Q = 30 in the top layer and Q = 100 in the bottom layer. The transmitted arrival exhibits less attenuation effect compared with that in Figure 1(b). In Figure 1(d), both velocity and Q remain the same as those in Figure 1(c); however, the fractional power of Laplacians, γ, is taken to be the averaged value, corresponding to the original implementation by Zhu and Harris (2014). To compare the results modeled by the two strategies, a middle trace at x = 800 m is extracted from both wavefield snapshots (Figures 1(c) and 1(d)). Figure 2 shows the two traces, along with their difference. Errors caused by using a constant γ instead of a variable γ can be observed. (a) (c) Figure 1: Viscoacoustic wave propagation in a two-layer model: (a) acoustic modeling with v = 1800 m/s in top layer and v = 3600 m/s in bottom layer; (b) same velocity as (a), homogeneous Q = 30; (c) same velocity as (a), Q = 30 in top layer and Q = 100 in bottom layer; (d) wavefield propagated using averaged fractional power γ using same model as (c). Figure 2: Traces at x = 800 m extracted from wavefield snapshots and their difference. Red, long-dashed line corresponds to variable γ; blue, solid line corresponds to constant γ; black, shot-dashed line is their difference. BP gas-cloud model (b) (d) In the second example, we use a synthetic attenuation model to demonstrate the effect of Q compensation in RTM and LSRTM. Figures 3(a) and 3(b) show a portion of BP 2004 velocity model (Billette and Brandsberg-Dahl, 2004) and the corresponding Q model used by Zhu et al. (2014). The model features a lowvelocity and high-attenuation area in the central-top zone a pattern commonly caused by the presence of a gas chimney. The model is discretized on a 161 398 grid with a spacing of 12.5 m in both horizontal and vertical directions. A total of 16 shots with a spacing of 325 m are used, starting from 12.5 m, and the source is a Ricker wavelet with a peak frequency of 22.5 Hz. Receivers have a spacing of 12.5 m, starting from 0 m and ending at 4962.5 m. Both sources and receivers are located at zero depth. First, acoustic modeling is used to generate acoustic data, and then viscoacoustic modeling is used to generate viscoacoustic data, accounting for seismic attenuation. Figure 3(c) is the reflectivity image generated by applying acoustic RTM to the acoustic data. In the acoustic image, all reflectors, such as the anticline below the gas chimney, can be observed clearly. In contrast, the image generated by viscoacoustic RTM using viscoacoustic data (Figure 3(d)), suffers from a lack of illumination below the gas accumulation. Using the Q-compensated RTM according to equation 14 (Figure 3(e)), the amplitude of structures below the gas accumulation has been recovered to a large extent, with a small mismatch in phase for deeper reflectors. Figure 3(f) demonstrates the result produced by LSRTM using the viscoacoustic kernel. Not only has the amplitude below the high attenuation zone been restored, but the phase is corrected as well. Remarkably, the illumination along the horizontal direction has also become uniform thanks to the least-squares inversion process. The image produced by LSRTM is close to the true amplitude. The least-squares approach is accurate and more stable compared with the Q-compensated RTM, but comes at a cost of multiple conjugate-gradient iterations. CONCLUSIONS We introduce low-rank viscoacoustic wave extrapolation operator and its adjoint based on the constant-q wave equation with decoupled fractional Laplacians. Using a sign reversal, the proposed operator is capable of compensating for attenuation and thus can be applied during reverse-time migration of viscoacoustic data. We also propose an iterative LSRTM strategy to compensate for seismic attenuation during imaging. Both Q-compensated RTM and LSRTM improve the image quality. Comparing the two approaches, we found that the LSRTM is more stable and produces images with better illumination, though requiring additional computational cost. ACKNOWLEDGMENTS We thank TACC (Texas Advanced Computing Center) for providing computational resources. The first author thanks Statoil and other sponsors of the Texas Consortium for Computation Seismology (TCCS) for financial support. The second author thanks the Stanford Wave Physics Lab for financial support. Page 3999
Low-rank viscoacoustic wave equation (a) (c) (e) Figure 3: A portion of BP 2004 velocity/q model and corresponding images. (a) BP gas-cloud velocity model; (b) BP gas-cloud Q model; (c) acoustic RTM image using acoustic data; (d) viscoacoustic RTM image using viscoacoustic data; (e) Q-compensated RTM image using viscoacoustic data; (f) LSRTM image using viscoacoustic data. (b) (d) (f) Page 4000
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