The optimized expansion based low-rank method for ...

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GEOPHYSICS, VOL. 79, NO. 2 (MARCH-APRIL 2014); P. T51–T60, 11 FIGS., 2 TABLES. 10.1190/GEO2013-0174.1

The optimized expansion based low-rank method for wavefield extrapolation

Zedong Wu1 and Tariq Alkhalifah1

a result, we obtain lower rank representations compared with the standard low-rank method within reasonable accuracy and thus cheaper extrapolations. Additional bounds set on the range of propagated wavenumbers to adhere to the physical wave limits yield unconditionally stable extrapolations regardless of the time step. An application on the BP model provided superior results compared to those obtained using the decomposition approach. For transversely isotopic media, because we used the pure P-wave dispersion relation, we obtained solutions that were free of the shear wave artifacts, and the algorithm does not require that η > 0. In addition, the required rank for the optimization approach to obtain high accuracy in anisotropic media was lower than that obtained by the decomposition approach, and thus, it was more efficient. A reverse time migration result for the BP tilted transverse isotropy model using this method as a wave propagator demonstrated the ability of the algorithm.

ABSTRACT Spectral methods are fast becoming an indispensable tool for wavefield extrapolation, especially in anisotropic media because it tends to be dispersion and artifact free as well as highly accurate when solving the wave equation. However, for inhomogeneous media, we face difficulties in dealing with the mixed space-wavenumber domain extrapolation operator efficiently. To solve this problem, we evaluated an optimized expansion method that can approximate this operator with a low-rank variable separation representation. The rank defines the number of inverse Fourier transforms for each time extrapolation step, and thus, the lower the rank, the faster the extrapolation. The method uses optimization instead of matrix decomposition to find the optimal wavenumbers and velocities needed to approximate the full operator with its explicit low-rank representation. As

domain operator. These challenges are somewhat addressed with smart variable separation methods (Etgen and Brandsberg-Dahl, 2009; Zhang and Zhang, 2009; Du et al., 2010; Fomel et al., 2010). Despite the effectiveness of these methods, they are not cheap because they require several multidimensional inverse Fourier transforms equal to the rank of the mixed space-wavenumber domain operator. Although the mixed space-wavenumber domain operator is inherently low rank even for complex media, the exposion to the components of the low-rank behavior, short of implementing a computationally exhaustive singular-value decomposition, is hard. Another related method is the Fourier finitedifference method proposed recently by Song and Fomel (2011) and Song et al. (2013). This method uses only one pair of multidimensional forward and inverse fast Fourier transforms per time step. However, it is not accurate enough for large Δt. Alkhalifah (2013) analyzed the mixed domain operator and its dependency

INTRODUCTION Wave extrapolation in time is crucial to seismic modeling, imaging, and full-waveform inversion. The conventional approach of the finite-difference approximation is easy to implement and results in relatively efficient solutions (Etgen, 1986). However, such an approach suffers from dispersion and stability problems requiring fine-space sampling and consequently small time extrapolation steps to satisfy the Courant–Friedrichs–Lewy (CFL) condition (Courant et al., 1928). It does not allow for simple separation of wave modes necessary to isolate desired wavefield solutions like the P-wave mode in anisotropic acoustic media. On the other hand, spectral methods for extrapolation have emerged recently to address such limitations. The Fourier implementation, however, poses challenges in handling inhomogeneous media in an efficient way due to the mixed space-wavenumber

Manuscript received by the Editor 9 May 2013; revised manuscript received 3 November 2013; published online 26 February 2014. 1 King Abdullah University of Science and Technology, Physical Sciences and Engineering Division, Thuwal, Saudi Arabia. E-mail: [email protected] .sa; [email protected]. © 2014 Society of Exploration Geophysicists. All rights reserved. T51

Wu and Alkhalifah

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on perturbations in velocity and the role of the time step length. Considering using the optimization method to get the separable decomposition of the mixed domain operator, Chen et al. (2007) and Liu et al. (2009) propose a method using piecewise linear function as the basis function for approximating the eigenfunction. In this paper, we design an algorithm that can reduce the computational cost of the low-rank method based on optimizing the coefficients that form the low-rank representation. We specifically use the minimum-maximum optimization approach to find the wavenumbers and velocities capable of representing the mixed domain operator as accurately as possible for a fixed number of inverse Fourier transforms. The effectiveness of the approach is demonstrated on the complex isotropic and tilted transverse isotropy (TTI) models. This paper is organized as follows. We first present the derivation of the optimized expansion method, and then we discuss how to make this algorithm unconditionally stable regardless of the time step. The algorithm is then tested on complex models and compared with the decomposition approach. We develop the method for anisotropic media and test it on the BP model. We finally apply the optimized expansion to reverse time migration (RTM) of the Sigsbee2a model.

THEORY Wave propagation, even for purposes of modeling and imaging in isotropic acoustic and possibly inhomogeneous media, is governed by the following wave equation:

∂2 p − v2 ðxÞΔp ¼ 0; ∂t2

(1)

where pðx; tÞ is the pressure wavefield described in a given domain with space coordinates given by x ¼ fx; y; zg, time by t, and v is the velocity. The symbol Δ represents the Laplacian operator defined as

Δ¼

∂ ∂ ∂ þ þ : ∂x2 ∂y2 ∂z2 2

2

Wðv; kÞ ≈

i¼M;j¼N X i¼1;j¼1

ai;j Wðvi ; kÞWðv; kj Þ:

(5)

Thus, we formulate the problem of extracting the best values of vi , kj , ai;j to fit Wðv; kÞ as an optimization problem, instead of a matrix decomposition (Fomel et al., 2010). As a result, we solve the following bounded minimization problem:



i¼M;j¼N X


min min max
Wðv; kÞ − ai;j Wðvi ; kÞWðv; kj Þ

; vi ;kj aij ðv;kÞ∈Ω i¼1;j¼1

(6) with the general case of Ω ¼ Ω0 ¼ ½vmin ; vmax
× ½kmin ; kmax
in which vmin , vmax are the minimum and maximum velocities and kmin , kmax are the minimum and maximum wavenumbers, respectively. In the isotropic case,

kmin ¼ 0; kmax

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2 π π π ; ¼ þ þ Δx Δy Δz

(7)

and, as such, they are bounded by the sampling theorem. This will give us the best approximation parameters for an arbitrary source. However, in most geophysical applications, the source is not arbitrary but band limited. Taking the bandwidth of the source into consideration,

Ω ¼ Ω0 ∩ fðv; kÞjωðv; kÞ ¼ vk ≤ 2πf max g;

(8)

2

(2)

For constant velocity, and after applying spatial Fourier trans^ forms, specifically pðk; tÞ ¼ Ffpðx; tÞg, the acoustic equation 1 reduces to

∂2 p^ þ v2 jkj2 p^ ¼ 0; ∂t2

(3)

where k is the wavenumber vector given by components fkx ; ky ; kz g. A second-order time-marching scheme with a multidimensional inverse Fourier transform led to the familiar expression (Soubaras and Zhang, 2008; Wards et al., 2008; Etgen and Brandsberg-Dahl, 2009; Zhang and Zhang, 2009; Song and Alkhalifah, 2013):

pðx; t þ ΔtÞ þ pðx; t − ΔtÞ Z ∞ ^ ¼ pðk; tÞð2 cos ðjkjvΔtÞÞe−ik·x dk: −∞

implementation as we have to evaluate a full 3D integral at each time step. The basic idea of the optimized expansion method is to approximate the two-variable function Wðv; kÞ ¼ 2 cosðvkΔtÞ with

where f max is the maximum frequency of the source. This will make the optimization domain much smaller, which simplifies the fitting process. Practically, we solve equation 6 and find a representation of Wðv; kÞ on a discrete grid of wavenumbers and velocities. Because the function we are optimizing is inherently smooth, the grid can be coarse with respect to wavenumbers and velocities, which will help to improve the efficiency of the optimization. For any given vi ; kj ; aij , we approximate the continuous maximum problem by the maximum on the uniformly sampled grid points s ;n¼N s . The resulting optimization problem on the fðv~ m ; k~ n Þgm¼M m¼1;n¼1 discrete grid is given as



min min max

Wðv~ m ; k~ n Þ vi ;kj aij ðv~ ;k~ Þ∈Ω m

−

n

i¼M;j¼N X i¼1;j¼1

(4)

For inhomogeneous media, equation 4 gives good results for small time step Δt, in which we replace v with vðxÞ. In this situation, a mixed domain term cosðΔtvðxÞjkjÞ complicates the numerical



~ ~ ai;j Wðvi ; kn ÞWðvm ; kj Þ

:

(9)

Because the mixed domain operator is inherently smooth, a coarse grid fitting is good enough for the approximation of the continuous maximum. For the special case of a piecewise constant velocity,

vðxÞ ¼ vi ;

x ∈ Ωi ;

i ¼ 1::L;

(10)

Optimized expansion method

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and L is small, we only need to solve the continuous maximum norm on the discrete velocities vi .

SOLUTION TO THE OPTIMIZATION PROBLEM AND THE STABILITY ISSUE In equation 9, the variable components aij are much easier to solve because the objective function is linear with respect to aij . The objective function is nonlinear with respect to vi , kj . Thus, we will handle the inversion for these parameters differently. Specifically, we use genetic algorithms (Wall, 1996) to find the optimal variables vi , kj . To do so, we will need to evaluate the objective function for a given vi , kj , which will require knowledge of aij . As a result, we evaluate aij first using a local derivative-free optimization algorithm (Brent, 1972; Powell, 2004) with the initial

guess given by least-squares optimization. We use the derived aij to evaluate for an the optimal value vi , kj . Of course, any initial guess obtained from other methods, such as those from the low-rank method (Fomel et al., 2010), can help reduce the computational cost of the optimization. We will list the functional call of pseudocode in Algorithm 1. ~ ≤ 2; it is Because the original formulation satisfies −2 ≤ Wð~v; kÞ natural to require the set of inverted parameters to satisfy this condition. Thus, the parameter space is bounded by this condition. To enforce this, we alter the scale factor aij in equation 9 to ensure that the bound is honored. The scaling of aij will not influence the accuracy of the algorithm because the values out of bound are inherently unphysical. Despite this bound, stability is not guaranteed. As shown in Figure 1, our continuous wavefield should also satisfy another condition given by ωðv; kÞ ¼ vk ≤ 2πf max, which means that

~ ¼ 0; ∀jkj ~ > 2πf max : ^ kÞ Pðt; vmin

Algorithm 1. Optimized expansion algorithm.

Input: The upper and lower bound of the position. vmin ; vmax ; kmin ; kmax ; M; N Output: vi , kj , aij and Err function GAOPTIMIZATION ðvmin ; vmax ; kmin ; kmax ; M; NÞ while Doesn’t satisfy the exit condition (Wall, 1996) do now by genetic algorithm (Wall, 1996); Get vnow i , kj now now aij ¼ LocalOptimization (vnow i , kj ; M; N);

Err ¼ max jWðv~ m ; k~ n Þ ðv~ m ;k~ n Þ∈Ω

−

i¼M;j¼N X i¼1;j¼1

now ~ ~ m ;know anow i;j Wðvi ; kn ÞWðv j Þj; (11)

end while return Result; end function function LOCALOPTIMIZATION (vi ; kj ; M; N) anow ij ¼ Initialize A (vi ; kj ; M; N); while Doesn’t satisfy the exit condition do Error=LocalDerivativeFreeOptimization (anow ij , vi , kj ; M; N); /*This function will optimize aij with the initial guess anow ij */ Get new anow ij ; end while return anow ij ; end function function INITIALIZEA (vi ; kj ; M; N) /*The initial value from any algorithm can be used here.*/ P anow ðv;kÞ∈Ω jWðv;kÞ ij ¼argmin a Piji¼M;j¼N − i¼1;j¼1 ai;j Wðvi ;kÞWðv;kj Þj2 return anow ij ; end function

T53

(12)

In the numerical implementation investigation, we realized that the source of instability is

~ ≠ 0; ^ kÞ Pðt;

for some ωðv; kÞ >

π : Δt

(13)

As such, we set

~ ¼ 0; ∀jkj ~ > Wðv; kÞ

π ; Δtvmax

(14)

and then we use the optimized expansion method to approximate the modified dispersion relation, constrained by the physical bounds, instead of the original relation. Comparing the constraints 12 with 14, the difference in the dispersion relation does not influence the value in the physical wave space when the following condition is imposed:

2πf max π ≤ : Δtvmax vmin

(15)

As a result accuracy will not be hampered when

Δt ≤

vmin 1 ; vmax 2f max

(16)

Figure 1. An example of the v-k domain. The wavefield in the Green zone caused by the numerical error should be filtered to guarantee the stability.

Wu and Alkhalifah

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and thus, the wavefield can be approximated accurately with a reasonably low rank in this case. In the case of homogeneous media, this constrain degenerates to Δt ≤ ð1∕2f max Þ, which is the Nyquist limit given by the sampling theorem. However, it doesn’t mean that the algorithm is not stable if inequality 16 is not satisfied. Our numerical experiment, as we will see later, show that the method is stable regardless of the time step, though large time steps may induce numerical errors. For each model, the optimization is done only once, and thus, it is an overhead cost for the extrapolation. For practical 2D models for a RTM implementation, the time to generate the coefficients in the optimized expansion method is negligible.

EXTENSION TO THE TI CASE

k^ x ¼ kx cos θ þ kz sin θ;

After doing a variable transformation with η~ ¼ ð8η∕1 þ 2ηÞ, we define the following objective function: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ^2 2 ^2 ~ ¼ 2 cos Δt vx kx þ vz kz þ 1 ðv2x k^ 2x þ v2z k^ 2z Þ2 − η~ v2x v2z k^ 2x k^ 2z ; ~ kÞ WðV; 2 2 

ð19Þ ~ ¼ ðvx ; vz ; η; ~ θÞ is the vector of material parameters and where V k~ ¼ ðkx ; kz Þ is the vector of wavenumbers. Similarly to the isotropic case, we will solve the following optimization problem to get the best coefficients:



i¼M;j¼N X
~ ~ ~~ ; k~ Þ

; ~ i ; k~~ n ÞWðV ~ m ; k~ n Þ − min min max

WðV ai;j WðV m j
~ ~ aij ~ ~

We now consider the wave extrapolation in transversely isotropic (TI) media. We use the pure P-wave dispersion relation proposed by (Alkhalifah, 1998, 2000; Fomel, 2004):

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 uv2 k^ þ v2 k^ 2 1 8η 2 2 ^ 2 ^ 2 z z ðv2x k^ 2x þ v2z k^ 2z Þ2 − v v kk : þ ω¼t x x 2 1 þ 2η x z x z 2 (17) where vx is the P-wave phase velocity in the symmetry plane, vz is the P-wave phase velocity in the direction normal to the symmetry plane, η is the anellipticity parameter (Alkhalifah and Tsvankin, 1995) related to Thomsen’s elastic parameters as ð1 þ 2δ∕1 þ 2ϵÞ ¼ ð1∕1 þ 2ηÞ. For tilted TI, k^ x and k^ z are the wavenumbers corresponding to the axis of symmetry and related to the wavenumbers corresponding to our grid as follows:

k^ z ¼ kz cos θ − kx sin θ: (18)

V i ;kj

ðV~ m ;k~ n Þ∈Ω

i¼1;j¼1

ð20Þ ~~ ; k~~ Þ is the chosen positions of approximate the conin which ðV m n ~ i ; k~j Þ is the material parameters position tinuous maximum error, ðV and wavenumbers position we choose for representing the operator, and aij is the coefficients.

NUMERICAL EXAMPLES A simple isotropic case Our first example corresponds to wave extrapolation in a 2D smoothly varying isotropic velocity field (Fomel et al., 2013). The velocity model is given by the following function:

a)

b)

c)

d)

e)

f)

g)

h)

Figure 2. Wavefield snapshots at time 1.6 s in a smooth velocity model computed using (a) the fourth-order finite-difference method with Δt ¼ 1 ms, (b) the optimized expansion with Δt ¼ 4 ms, (c) the optimized expansion with Δt ¼ 10 ms, and (d) the optimized expansion with Δt ¼ 20 ms. A horizontal slice at the center point extracted from (e) the finite difference (Δt ¼ 1 ms) (f) the optimized expansion (Δt ¼ 4 ms) (g) the optimized expansion (Δt ¼ 10 ms), and (h) the optimized expansion (Δt ¼ 20 ms).

Optimized expansion method

a)

vðx; zÞ ¼ 550 þ 0.00015ðx − 800Þ2 þ 0.0001ðz − 500Þ2 ;

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0 ≤ x ≤ 2560;

0 ≤ z ≤ 2560:

T55

(21)

We use a point-source Ricker wavelet with a peak frequency of 20 Hz and a maximum frequency of 45 Hz, located in the middle of the model, to initiate the wave. The space grid length is 5 m. The minimum and maximum velocities are, respectively, 550 and 1434.31 m∕s. The accuracy limitation of the time step in the optimized expansion algorithm is Δt ≤ ðvmin ∕vmax Þð1∕2f max Þ ≤ 4.3 ms, so we choose the time step of 4 ms. The wavefield snapshot at time 1.6 s with M ¼ N ¼ 4 is shown in Figure 2b. The stability limit of the standard second-orderpfinite-difference scheme in time ffiffiffi is given by r ¼ ðvmax Δt∕ΔxÞ ≤ ð 2∕πÞ ≈ 0.45, which means that Δt ≤ 1.6 ms for this example. So we choose the time step of finite difference as 1 ms, and the snapshot at time 1.6 s is shown in Figure 2a. The optimized expansion (Figure 2f) result is practically dispersion free, especially compared to the result from the

b)

Figure 3. The horizontal slice at time 1.4 s at the center point extracted from the finite-difference (red) and the optimized expansion (blue) methods with a low-frequency source wavelet.

Figure 4. Snapshot of the wavefield at t ¼ 1.1 s for a two-layer model using (a) the finite-difference method with fourth-order spatial accuracy and (b) the optimized expansion method.

Table 1. The coefficients for different M and N. M N

vi

kj

aij

Error

2 2 3 3

1322.48438 726.7667847 1127.60501 1354.07067 634.984972 992.64550 1260.9412 600.38548 1423.1194

0.16678635 0.469311148 0.06113872 0.499017656 0.351825474 0.01424532 0.304415285 0.392405242 0.508893669

0.123047093 −0.532462397 0.4754973964 0.383803878 −2.201012896 −4.305778193 5.527193209 1.825911556 2.904116766 −4.266642832 0.901329702 1.412990333 −1.298141115 42.1917566 −199.146061 215.460185 −60.2108725 −60.2855685 277.936663 −295.607884 79.3599849 −10.8308675 53.5317606 −58.4275661 17.0333908 29.4276112 −132.330216 138.581478 −36.1838141

9e − 2

4 4

2e − 3

2e − 5

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Wu and Alkhalifah

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finite-difference approach (Figure 2e) even with a time step four times that of the finite-difference extrapolation. For the accuracy comparison with the finite-difference method, we use the same

space and time sampling (Δt ¼ 2 ms) and use a Ricker wavelet with a peak frequency of 4 Hz. Figure 3 shows a horizontal slice at time 1.4 s at the center point extracted from the finite-difference (red) and the optimized expansion method (blue). They are consistent with each other because of the dispersion error in finite difference is small at such a low frequency. Though the accuracy of the optimized expansion required a time step of 4.3 ms for this example,

a)

b)

c)

Figure 5. (a) The exact operator 2 cosðΔtvkÞ. (b) The approximate operator. (c) The difference between the two.

Figure 6. (a) Portion of BP-2004 synthetic isotropic velocity model. (b) Snapshot of the wavefield at t ¼ 3.2 s for the optimized expansion method with M ¼ N ¼ 2, Δt ¼ 2 ms. (c) The difference between the low rank with the parameter M ¼ N ¼ 4, Δt ¼ 1 ms and the optimized expansion method with M ¼ N ¼ 2, Δt ¼ 2 ms.

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Optimized expansion method exceeding this accuracy limit, unlike in the finite-difference implementation, does not render the method unstable. If we consider the maximum frequency of the source wavelet, the time step bounded by the Nyquist limit is given by Δt ≤ ð1∕2f max Þ ¼ 11 ms. Thus, we choose Δt ¼ 10 ms for the optimized expansion method. The snapshot at time 1.6 s is shown in Figure 2a, and the corresponding to a horizontal slice of the snap shot is plotted in Figure 2g. It shows that even for Δt ¼ 10 ms, the algorithm is stable and the phase is correct for the resulting wavefield, but we end up with relatively low amplitude noise. Figure 2d and 2h shows the result of using Δt ¼ 20 ms, which is beyond even the Nyquist limit. Our algorithm is still stable with this extremely large time step; it is dominated by Nyquist noise. Table 1 shows the resulting coefficients with different M; N and Δt ¼ 4 ms. The last column in Table 1 shows the maximum error defined by equation 11 for different M and N. It indicates that the convergence speed of our method is fast with the increasing of the rank. To display the numerical behavior of our algorithm in the case of a layered medium with high contrast velocity, we use a simple twolayer velocity model. The upper-layer velocity is 1500 m∕s, and the bottom-layer velocity is 4500 m∕s. The mesh parameters are Δt ¼ 2 ms; Δx ¼ Δz ¼ 15 m. The snapshots at time t ¼ 1.1 s computed by the optimized expansion (M ¼ N ¼ 2) and finitedifference method are shown in Figure 4a and 4b.

The 2D BP velocity model Next, we move to isotropic wave extrapolation in a portion of the complex 2D BP velocity model (Billette and Brandsberg-Dahl, 2005), given in Figure 6a, which includes a salt body. The horizontal grid size Δx is 37.5 m, the vertical grid size Δz is 12.5 m, and the time step is 2 ms; the source is given by a Ricker wavelet with a peak frequency of 15 Hz and the maximum frequency 40 Hz. We consider a rank of two in the optimized expansion for this model, which requires two multidimensional inverse Fourier transforms at each time step, and thus we consider M ¼ N ¼ 2. Figure 5a and 5b shows the exact function 2 cosðΔtvkÞ and our approximate function for this example. Figure 5c shows the difference between them, in which the maximum approximate error is 1.2e − 05. The wavefield

a)

b)

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snapshot (shown in Figure 6c) at 3.2 s confirms the ability of our method to handle complex models and sharp velocity variations. We compare our numerical result with that obtained for the low-rank method with Δt ¼ 1 ms and M ¼ N ¼ 4 used as reference. A snapshot of the difference between these two solutions is plotted in Figure 6c.

The transverse isotropy (TI) case The first example for anisotropic media is a simple TTI model with constant η ¼ 0.3 and θ ¼ 45° and smoothly varying P-wave velocities:

vx ¼ 800 ðx − 1000Þðx − 1000Þ þ ðz − 1200Þðz − 1200Þ ; 10; 000 ðz − 1200Þðz − 1200Þ vz ¼ 700 þ : 10; 000 (22) þ

The mesh parameters are Δx ¼ Δz ¼ 5 m, Δt ¼ 1 ms, and M ¼ N ¼ 3. Figure 7a shows the wave snapshot at t ¼ 1.5 s. What should be noted is that there is no coupling of qP-waves and qSVwaves because we rely here on the pure P-wave dispersion relation. Figure 7b shows the difference between the standard low-rank method with M ¼ N ¼ 6 at the same scale. Figure 7c shows the difference scaled up 10 times. Finally, we apply the optimized expansion low rank on a complex anisotropic model. The model is a 2007 anisotropic benchmark data set from BP. It exhibits a strong TTI anisotropy (Figure 8a–8d). We use the mesh size Δx ¼ Δz ¼ 12:5 m, Δt ¼ 1 ms, M ¼ 6, N ¼ 8 to model the wavefield. The source is given by a Ricker wavelet with peak frequency of 15 Hz. The snapshot of the wavefield at t ¼ 4.3 s is shown in Figure 8e. For the same model, Fomel et al. (2010) use a rank of 10 to extrapolate the wavefield, which requires at least an additional 50% computational effort.

c)

Figure 7. (a) Wavefield snapshot at t ¼ 1.5 s for the constant gradients P-wave velocity model and constant η and θ with M ¼ N ¼ 3. (b) The difference between (a) and the result produced by stand low-rank method with M ¼ N ¼ 6. (c) The difference at 10× scale.

Wu and Alkhalifah

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Reverse time migration Next, we apply RTM using the optimized expansion method as a wavefield extrapolator on the Sigsbee2a model. The Sigsbee2a model is shown in Figure 9. The data are generated using the finite-difference method with mesh Δx ¼ Δz ¼ 25 ft, Δt ¼ 0.8 ms. The source is a Ricker wavelet with a peak frequency of 24 Hz and a maximum frequency of 50 Hz. We migrate the data on an image grid of Δx ¼ Δz ¼ 50 ft with M ¼ N ¼ 3 and use a time sampling of Δt ¼ 2.4 ms. The source interval is 50 ft, and the receiver interval is 100 ft. For each shot, the receiver range satisfies

−26; 100 ≤ r − s ≤ 26; 100;

(23)

where r and s are the receiver and source coordinate on the surface, respectively. In RTM, we use the random boundary (Clapp, 2009) condition to solve the storage problem of the correlation of the forward and backward wavefield. In this situation, the forward wavefield is recovered by the wavefield at the last time step. The RTM result is shown in Figure 10, which generally shows that the wave extrapolation is accurate and dispersion free. We also apply this method as a wave propagator for RTM of a benchmark data set. This data set was released by BP in 2007 to serve as an anisotropic velocity analysis benchmark data set. Because of the dispersion-free feature, we use a mesh size of Δx ¼ Δz ¼ 25 m,Δt ¼ 1 ms and M ¼ 8; N ¼ 9. Figure 11 shows the RTM result. Even though we use a relatively coarse grid spacing, the resulting image is clean and of good quality.

a)

b)

c)

d)

e)

Figure 8. (a) Velocity along the axis of symmetry. (b) Velocity perpendicular to the axis of symmetry. (c) Anellipticity parameter η. (d) Tilt of the symmetry axis. (e) The snapshot at 4.3 s.

Optimized expansion method

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CONCLUSIONS We developed an optimized expansion method through forming an optimization problem to get the best coefficients to reduce the number of Fourier transformations needed at each time step of a spectral implementation of the wavefield extrapolation. The proposed algorithm uses the source frequency range to define the appropriate wavenumber band for extrapolation. As a result, we achieve unconditional stability, which is supported by the numerical examples. The method yields efficient extrapolations even in complex TTI models, and a RTM application demonstrated the method’s utility.

ACKNOWLEDGMENTS Figure 9. Migration velocity of Sigsbee2a model with Δx ¼ Δz ¼ 50 ft.

We thank KAUST for its support and the SWAG group for the collaborative environment. We also thank the associate editor Y. Zhang, X. Song, and two anonymous reviewers for their fruitful suggestions and comments. The computational examples in this paper use the Madagascar open-source software package http://www .ahay.org/.

REFERENCES

Figure 10. Image generated by RTM with the optimized expansion wavefield extrapolator with Δx ¼ Δz ¼ 50 ft and Δt ¼ 2 ms.

Figure 11. Reverse time migration result for BP TTI data set with Δx ¼ Δz ¼ 25 m.

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