3D weak-dispersion reverse time migration using a stereomodeling operator
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Citation
Li, Jingshuang, Michael Fehler, Dinghui Yang, and Xueyuan
Huang. “3D Weak-Dispersion Reverse Time Migration Using a
Stereo-Modeling Operator.” GEOPHYSICS 80, no. 1 (December
5, 2014): S19–S30. © 2014 Society of Exploration Geophysicists
As Published
http://dx.doi.org/10.1190/GEO2013-0472.1
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Society of Exploration Geophysicists
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Final published version
Accessed
Wed May 25 22:41:17 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/96309
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GEOPHYSICS, VOL. 80, NO. 1 (JANUARY-FEBRUARY 2015); P. S19–S30, 15 FIGS., 2 TABLES.
10.1190/GEO2013-0472.1
3D weak-dispersion reverse time migration using
a stereo-modeling operator
Jingshuang Li1, Michael Fehler2, Dinghui Yang3, and Xueyuan Huang3
efficient stereo-modeling methods that have been found to be
superior to conventional algorithms in suppressing numerical
dispersion and numerical anisotropy. We generalized one
stereo-modeling method, fourth-order nearly analytic central
difference (NACD), from 2D to 3D and applied it to 3D
RTM. The RTM results for the 3D SEG/EAGE phase A classic
data set 1 and the SEG Advanced Modeling project model demonstrated that, even when using a large grid size, the NACD
method can handle very complex velocity models and produced
better images than can be obtained using the fourth-order and
eighth-order Lax-Wendroff correction (LWC) schemes. We also
applied 3D NACD and fourth-order LWC to a field data set and
illustrated significant improvements in terms of structure imaging, horizon/layer continuity and positioning. We also investigated numerical dispersion and found that not only does the
NACD method have superior dispersion characteristics but also
that the angular variation of dispersion is significantly less than
for LWC.
ABSTRACT
Reliable 3D imaging is a required tool for developing models
of complex geologic structures. Reverse time migration (RTM),
as the most powerful depth imaging method, has become the
preferred imaging tool because of its ability to handle complex
velocity models including steeply dipping interfaces and large
velocity contrasts. Finite-difference methods are among the
most popular numerical approaches used for RTM. However,
these methods often encounter a serious issue of numerical
dispersion, which is typically suppressed by reducing the grid
interval of the propagation model, resulting in large computation and memory requirements. In addition, even with small grid
spacing, numerical anisotropy may degrade images or, worse,
provide images that appear to be focused but position events
incorrectly. Recently, stereo-operators have been developed to
approximate the partial differential operator in space. These operators have been used to develop several weak-dispersion and
Reverse time migration (RTM) (Baysal et al., 1983; Whitmore,
1983) has drawn considerable attention as the most powerful depth
imaging method because of its ability to handle complex velocity
models without dip limitations. Even though it was first introduced
about three decades ago, RTM has only recently become widely
used in the industry because computational capability has finally
caught up with the large memory requirement and computational
cost. RTM uses the two-way wave equation and is relevant for challenging geologic environments because it does not suffer from dip
limitation. Three-dimensional RTM is a suitable approach for seismic imaging in the case of strong lateral variations of velocity. However, 3D RTM still remains expensive, in spite of the incredible
increase in computing power.
INTRODUCTION
Two steps are needed to obtain subsurface seismic images that are
of sufficient quality to be used for resource identification and reservoir characterization in complex geologic environments. First, we
need to reliably estimate parameters such as velocity and density to
form a basic model for conducting imaging. Second, we need migration algorithms that reliably account for wave phenomena in
heterogeneous models. Due to the 3D nature of geologic structures,
2D migrated images may not be accurate due to out-of-plane 3D
effects. Producing 3D seismic images is desirable to make seismic
interpretation easier and help facilitate sound business decision
making.
Manuscript received by the Editor 18 December 2013; revised manuscript received 15 August 2014; published online 5 December 2014.
1
Formerly Tsinghua University, Department of Mathematical Sciences, Beijing, China; presently China University of Mining and Technology (Beijing),
School of Science, Beijing, China. E-mail: lijingshuang322@126.com.
2
Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. E-mail: fehler@mit.edu.
3
Tsinghua University, Department of Mathematical Sciences, Beijing, China. E-mail: dhyang@math.tsinghua.edu.cn; x.y.huang.wb@163.com.
© 2014 Society of Exploration Geophysicists. All rights reserved.
S19
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S20
Li et al.
The kernel of RTM is a method for modeling the two-way wavefield by solving the full wave equation. The finite-difference (FD)
method is a popular and easy way to implement RTM due to its
efficiency and flexibility. A large number of FD schemes have been
developed to solve acoustic and elastic wave equations (Alford et al.,
1974; Kelly et al., 1976; Dablain, 1986; Takeuchi and Geller, 2000)
and also applied to anisotropic and viscoelastic problems (Robertsson et al., 1994; Blanch and Robertsson, 1997; Zhang et al., 1999;
Takeuchi and Geller, 2000). Classical FD methods suffer from serious numerical dispersion when too few samples per wavelength are
used (Yang et al., 2006). To minimize or eliminate numerical
dispersion, many researchers have attempted to develop new algorithms by redefining the operators for spatial and temporal differentiation or making some special treatments (Kosloff and Baysal,
1982; Dablain, 1986; Fei and Larner, 1995; Zhang et al., 1999; Mizutani et al., 2000; Zheng and Zhang, 2005). However, as discussed
in our previous work (Yang et al., 2007), the theoretical properties
of these methods or techniques have some disadvantages. For example, although the so-called flux-corrected transport technique can
suppress the numerical dispersion (Fei and Larner, 1995; Zhang
et al., 1999; Yang et al., 2002; Zheng et al., 2006), it is unable
to fully recover the resolution lost by the numerical dispersion when
the grid size is too coarse (Yang et al., 2002). The spatial operator
for the pseudospectral method (Kosloff and Baysal, 1982; Fornberg,
1987) can be exact up to the Nyquist frequency, but it requires the
fast Fourier transform of the wavefield, which needs additional
computation and introduces difficulties of how to handle nonperiodic boundary conditions and implementation in parallel computing
environments (Mizutani et al., 2000). The performance of high-order FD methods such as the high-order FD scheme (e.g., Fornberg,
1990; Igel et al., 1995), high-order compact FD scheme (Lele,
1992), or the so-called Lax-Wendroff correction (LWC) scheme
(Lax and Wendroff, 1964; Dablain, 1986; Robertsson et al.,
1994; Blanch and Robertsson, 1997) are also affected by numerical
dispersion. Using higher order schemes or finer spatial grids are two
ways to suppress numerical dispersion. Unfortunately, the suppression of numerical dispersion is a trade-off with efficiency because
finer spatial grids bring shorter temporal step sizes, resulting in
large storage space requirements and increased computational cost.
Staggered-grid FD methods (Virieux, 1986; Fornberg, 1990; Igel
et al., 1995) can further reduce numerical dispersion, but they still
suffer from it when too few samples per wavelength are used (Sei
and Symes, 1995). Furthermore, staggered-grid FD methods may
suffer from numerical anisotropy and have additional errors for
the anisotropic case (Igel et al., 1995).
To effectively suppress the numerical dispersion caused by the
discretization of the wave equation, a so-called nearly analytic discrete operator was introduced to approximate the partial differential
operators in space (Yang et al., 2003, 2004, 2006). As described by
Yang et al. (2003), it simultaneously uses the wave displacement
and its gradients to determine the high-order spatial derivatives.
The expression of wave displacements is in a nearly analytical form,
which makes the method different from conventional methods.
Combined with different time schemes, several methods have been
developed (Wang et al., 2009; Chen et al., 2010; Yang and Wang,
2010; Ma et al., 2011) based on this operator, and each has been
found to be better than conventional algorithms at suppressing
numerical dispersion and saving computation and memory cost.
The approach is initially proposed by Konddoh (1991) and applied
to solve parabolic equations (Konddoh et al., 1994). We refer to this
class of modeling methods, stereo-modeling methods, for reasons
that will be described below. In this paper, we extend one of the
stereo-modeling methods from 2D to 3D, discuss issues of numerical dispersion, numerical anisotropy, and computational efficiency,
and, finally, demonstrate its applicability to RTM by investigating
several 3D data sets. Finally, we provide specific examples of how
numerical dispersion varies with grid spacing, frequency, and
propagation angle to show the advantages of the stereo-modeling
method compared with conventional FD methods.
METHOD
In this section, we briefly introduce the concept of stereo-modeling method and illustrate its basic formulation. For simplicity, we
limit our discussion to the scalar isotropic case, which is described
by the following wave equation:
∂2 uðx; tÞ
¼ c2 Δuðx; tÞ;
∂t2
(1)
where uðx; tÞ is the wavefield, c is the propagation velocity in the
medium, and the position vector is indicated by x ¼ ðx; y; zÞ. If a
temporal differencing scheme is applied, we can get a half discretized equation. For example, with the truncated Taylor series expansion in the time domain and equation 1, we can get the following
equation to compute the wavefield at time tnþ1 :
unþ1
i;j;k
¼
2uni;j;k
−
un−1
i;j;k
þ
Δt2
∂2 u
∂t2
n
Δt4 ∂4 u n
þ
12 ∂t4 i;j;k
i;j;k
2
n
¼ 2uni;j;k − un−1
i;j;k þ ðcΔtÞ ðΔuÞi;j;k þ
ðcΔtÞ4 2 n
ðΔ uÞi;j;k ;
12
(2)
where Δt denotes the time increment, n is time index, and i, j, and k
are space indexes for x, y, and z directions, respectively. The main
difference between the stereo-modeling and conventional FD methods is the evaluation of the Laplace operator and the high-order spatial derivatives.
The stereo-modeling method borrows its name from an analogy
to stereotomography (Billette and Lambaré, 1998) for ray-based
seismic velocity. As is well known, traveltime tomography uses
only picked traveltimes in velocity inversion. Stereotomography
uses not only picked travel times but also picked gradients of traveltime versus distance, which include local coherence information.
Joint inversion of traveltimes and gradients greatly increases the resolution of inverted model parameters. Analogously, conventional
FD modeling uses only the wavefield for computing high-order spatial derivatives. Correspondingly, stereo-modeling uses not only the
wavefield but also its gradient or first-order spatial derivative when
constructing high-order spatial derivatives. During wave propagation, the wavefield and its gradient are propagated simultaneously.
Including wave gradient information greatly increases the accuracy
of the representation of the high-order spatial derivatives, thus improving the wave simulation quality by reducing numerical
dispersion and numerical anisotropy while leading to improved efficiency.
To illustrate the main difference between conventional FD and
stereo-modeling schemes, we take the 1D case as an example.
3D weak-dispersion RTM with NACD
The conventional representation of spatial derivatives is given by
(Fornberg, 1988)
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n
X
dm fðxÞ
j
≍
αm fðx Þ m ¼ 1; ···;M;n ¼ m;m þ 1; ···;N;
dxm x¼x̄ j¼0 n;j j
(3)
where M is the order of the highest order derivatives to be approximated and αm
n;j is the coefficient for the mth-order derivative of function fðxÞ at grid point j when n þ 1 points are used. Note that the
number of terms in the sum n depends on the order of the derivative.
Inspection of this formula leads us to conclude that conventional FD
has an expanded computational stencil, which means the number of
grid points used is no less than the order of the derivative to be
calculated. For example, for a fourth-order derivative (m ¼ 4), at
least four grid points (n ≥ 4) are needed, and the higher the order
of the derivative, the more grid points are needed. Furthermore,
when used in a wave equation solver, only grid points along axis
directions are used, which makes a linear computational stencil
structure. Large numerical dispersion will appear unless one makes
a careful selection of grid size, which means that the number of grid
points per wavelength should be large. In contrast to conventional
FD approaches, the representation of high-order spatial derivatives
for the stereo-modeling approximation is formulated as follows
(Yang et al., 2003, 2004, 2006; Tong and Yang, 2013):
L X
dfðxj Þ
dm fðxÞ
m fðx Þþβ m
j
≈
m ¼ 1; ···;4 Lþ1:
α
j
j
dx
dxm x¼x̄ j¼−L j
S21
compact computational stencil. When used in a wave equation
solver, the computational stencil has an areal structure for 2D case
and cubic structure for 3D case as shown in Figure 1. The 2D 9point (3 × 3) and 3D 27-point (3 × 3 × 3) stencils for stereo-modeling (nearly analytic central difference [NACD]) use off-axis
points, which, we will show later, leads to much less numerical
anisotropy and numerical dispersion than when using a conventional linear stencil. An added advantage is that the spatial gradient
information available from stereo-modeling would be useful for angle gather computation in RTM, filtering, some new imaging
conditions (Fleury and Vasconcelos, 2010), and some new acquisition approaches including the direct use of measured wavefield
gradients.
We will now briefly illustrate the theory of 3D NACD. NACD
gets its name because we use a central difference operator and save
fourth-order terms in the time and space domains. The NACD
method is relatively straight forward, and the time discretization
is simpler than that of the other stereo-modeling methods, which
use more complex time discretization, for example, the symplectic
partitioned Runge-Kutta (Ma et al., 2011). The discretization in
space is also different from other stereo-modeling methods, which
are based on interpolation extrapolation. The differences can be
seen, for example, by comparing the formulas in Appendix A with
those by Yang et al. (2007). Because we need to update the gradients as well as the displacements at each time step in stereo-modeling, we introduce the definition U ¼ ðu; ∂u∕∂x; ∂u∕∂y; ∂u∕∂zÞ
and replace u with U in equation 2 to get
(4)
In this formula, function values and their first-order derivatives
are used in the construction of higher order derivatives. The value of
L is independent of m and only depends on the accuracy of the
method.
Larger values of L lead to higher spatial accuracy. We can achieve
4Lth order accuracy in space when we use 2L þ 1 grid points for
the 1D case. For example, we can achieve fourth-order accuracy in
space when L is 1. Thus, to achieve fourth-order accuracy in space
for a 1D case, only three grid points are required for all high-order
derivatives. To achieve higher order accuracy, the number of grid
points needed increases, but it will remain the same for all derivatives. The number of grid points used for different order derivatives
to achieve different orders of accuracy for conventional FD and
stereo-modeling is listed in Table 1. Stereo-modeling thus has a
Figure 1. Schematic diagram of the 3D computational stencil for
(a) NACD and (b) fourth-order LWC.
Table 1. The number of grid points used for different order derivatives to achieve different orders of accuracy for conventional
1D FD and stereo-modeling.
Order of accuracy (LWC)
Order of derivative
Second
Third
Fourth
Order of accuracy (stereo-modeling)
Secnd
Fourth
Sixth
2mth
Fourth
Eighth
Twelfth
4mth
3
4
5
5
6
7
7
8
9
2m þ 1
2m þ 2
2m þ 3
3
3
3
5
5
5
7
7
7
2m þ 1
2m þ 1
2m þ 1
Li et al.
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n
n−1
2
n
Unþ1
i;j;k ¼ 2U i;j;k − U i;j;k þ ðcΔtÞ ðΔUÞi;j;k
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þ
ðcΔtÞ4 2 n
ðΔ UÞi;j;k :
12
fourth-order accuracy in space. This can be seen as doing Taylor
expansions in space. Then, equation 5 becomes
(5)
Then, following the methodology that has been previously described for the 2D case of NACD (Yang et al., 2012), we apply
the central difference discretization to the second-order spatial
derivatives and keep the high-order terms required to achieve
n
n−1
U nþ1
i;j;k − 2U i;j;k þ U i;j;k
¼c
Δt2
n
U
− 2U ni;j;k þ U ni−1;j;k
2 iþ1;j;k
U ni;jþ1;k − 2Uni;j;k þ Uni;j−1;k
Δy2
4 n
c Δx c Δt2
∂ U
−
þc
−
12
12
∂x4 i;j;k
Δz2
2 2
c Δy c4 Δt2
∂4 U n
c2 Δz2 c4 Δt2
∂4 U n
−
−
−
−
12
12
12
12
∂y4 i;j;k
∂z4 i;j;k
c4 Δt2
∂4 U n
∂4 U n
∂4 U n
: (6)
þ
þ
þ
6
∂x2 ∂y2 i;j;k
∂x2 ∂z2 i;j;k
∂y2 ∂z2 i;j;k
Un
2 i;j;kþ1
Δx2
− 2U ni;j;k þ U ni;j;k−1
þ c2
2
2
4
The key now is how to solve the high-order (≥ 4) spatial derivatives;
this is the main difference from the conventional FD method. Note
that although equation 6 appears to have maximum fourth-order
derivatives, fifth-order derivatives are required to get ∂u∕∂x;
∂u∕∂y; ∂u∕∂z as shown in Appendix A. The discretization of fourthand fifth-order derivatives with respect to x, y, and z are the same as
those in the previous work (Yang et al., 2007). Then, following the
directional derivatives approach, which was introduced by Yang
et al. (2012), we get the stereo-modeling type expressions for all
the mixed spatial partial derivatives needed, which are different
from the previous work (Yang et al., 2007, 2012). For convenience,
we list the expressions of all the derivatives needed in computation
in Appendix A. This scheme achieves accuracy in time and space of
the fourth-order. It has a symmetric structure, which guarantees that
it is more stable than other stereo-modeling methods that are not
symmetric (e.g., Yang et al., 2007). Also, it has all the beneficial
characteristics of stereo-modeling methods that enable effective
wave propagation modeling on a large scale (Yang et al., 2003,
2004, 2007).
NUMERICAL RESULTS
We will first show forward wavefield propagation simulations to
demonstrate the advantage of our method with regard to numerical
dispersion. We will then show several 3D RTM examples using our
modeling method on synthetic and field data.
To demonstrate the numerical dispersion of the method in the 3D
case, we consider the following 3D scalar wave equation:
Figure 2. Snapshots obtained with (a) NACD (Δx ¼ Δy ¼
Δz ¼ 40 m), (b) fourth-order LWC (Δx ¼ Δy ¼ Δz ¼ 40 m),
(c) eighth-order LWC (Δx ¼ Δy ¼ Δz ¼ 40 m), (d) fourth-order
LWC (Δx ¼ Δy ¼ Δz ¼ 20 m), and (e) eighth-order LWC (Δx ¼
Δy ¼ Δz ¼ 32 m). Background velocity is 4 km∕s,and source is a
15-Hz Ricker.
2
∂2 u
∂ u ∂2 u ∂2 u
2
− c0
þ
þ
¼ fðtÞδðxs ; ys ; zs Þ:
∂t2
∂x2 ∂y2 ∂z2
(7)
We first test a simple homogeneous case, choosing the acoustic
velocity to be 4 km∕s and the Courant number (α ¼ cΔt∕h;
Table 2. Comparison of the computational cost for NACD, fourth-order LWC, and eighth-order LWC.
Method
Grid size (m)
CPU time (s)
Memory usage (array)
NACD
Fourth-order LWC
Eighth-order LWC
40
43.88
120 × 120 × 120 × 12
20
568.76
240 × 240 × 240 × 3
32
80.06
150 × 150 × 150 × 3
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3D weak-dispersion RTM with NACD
h ¼ Δx ¼ Δy ¼ Δz) to be 0.2. The computational domain is
0 ≤ x ≤ 4.8 km, 0 ≤ y ≤ 4.8 km, and 0 ≤ z ≤ 4.8 km. A 15-Hz
peak frequency Ricker wavelet explosive source is located at the
center of the computational domain. Figure 2a, 2b, and 2c shows
wavefield snapshots at t ¼ 0.54 s on the vertical plane containing
the source when using a coarse computational grid (Δx ¼ Δy ¼
Δz ¼ 40 m; equivalent to approximately six grid points per dominant wavelength) generated by the (a) NACD, (b) fourth-order
LWC (fourth-order in time and space), and (c) eighth-order LWC
(fourth order in time and eighth order in space), respectively. To
compare the ability of the three methods for producing a comparable wavefield quality, we compute the same wavefield using LWC
with finer spatial increments. Figure 2d and 2e shows the wavefield
snapshot at t ¼ 0.54 s on finer grids, generated by the fourth-order
LWC (Δx ¼ Δy ¼ Δz ¼ 20 m) and eighth-order LWC (Δx ¼ Δy
¼ Δz ¼ 32 m) using the same Courant number. We can see that the
wavefronts of the seismic waves shown in Figure 2a, 2d, and 2e,
simulated by the NACD, the fourth order LWC and the eighth order
LWC, respectively, are similar. Comparing Figure 2a, 2b, and 2c,
we can see that the NACD has no available numerical dispersion
S23
even though the space increment is 40 m. However, when the LWC
and the NACD have similar accuracy, their computational costs are
quite different. It took the NACD about 43 s to generate Figure 2a
on a workstation, whereas the fourth-order and eighth-order LWC
method took about 568 and 80 s, respectively, to generate Figure 2d
and 2e under the same condition. For this simple simulation, the
computational speed of the NACD is roughly 14 times and 2 times
faster than that of the fourth-order and eighth-order LWC, respectively. Furthermore, their memory requirements are different. The
Figure 5. The RTM results obtained using (a) NACD and
(b) fourth-order LWC for the SEG/EAGE phase A classic data
set 1. White ellipses outline regions in which the NACD image
is better than the LWC image.
Figure 3. The phase A classic 1 acquisition over the SEG/EAGE
salt model. (a) The white line in the white box indicates the zone
containing the shot positions, and the white box indicates the position of the migrated volume. Panel (b) shows the layout of receivers
for one shot.
Figure 4. A vertical plane of the SEG/EAGE velocity model taken
directly below the source line for phase A classic data set 1.
Figure 6. The layout of shot and receivers for our selected subset of
SEAM model. The black star represents the source, and the receivers are distributed throughout the black box. The velocity in the
horizontal plane at a depth of 0.98 km is shown, and the shot and
receivers are at a depth of 0.62 km. Velocities in this plane range
from 1.5 to 4.48 km∕s.
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S24
Li et al.
NACD needs 12 arrays to store the wave displacement and gradient
at each spatial grid point, and the number of grid points is 121 ×
121 × 121 on the coarse grid (40 m). The fourth-order LWC needs
only three arrays to store the wave displacement at each grid point,
but the number of grid points on the fine grid (20 m) goes up to
241 × 241 × 241. Thus, the NACD requires only about half of
the storage space for the fourth-order LWC for comparable quality.
A comparison of the computational requirements for each method is
given in Table 2.
3D SEG/EAGE PHASE A CLASSIC DATA SET 1
We now present RTM results using the 3D SEG/EAGE phase A
classic data set 1 (Biondi, 2006) using the NACD method and compare with results obtained when using the fourth-order LWC
method. In phase A, two 138-shot 3D shot lines that are oriented
perpendicular to each other with their intersection at the crest of the
salt were acquired. The phase A classic data set 1 was extracted
from the acquisition along line 1. For this data set, each shot
has six streamers with a maximum of 65 groups per streamer.
Figure 7. Two different 3D views of the subset of SEAM velocity
model with (a) crossline section at x ¼ 4.8 km, inline section at
y ¼ 4.8 km, and horizontal plane at z ¼ 2.4 km and (b) crossline
section at x ¼ 3 km, inline section at y ¼ 3 km, and horizontal
plane at z ¼ 2.36 km.
Figure 8. Three-dimensional views of the RTM slices obtained by
(a) NACD, (b) fourth-order LWC, and (c) eighth-order LWC for the
SEAM model with crossline section at x ¼ 4.8 km, inline section at
y ¼ 4.8 km, and horizontal plane at z ¼ 2.4 km that is displayed in
Figure 7a.
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3D weak-dispersion RTM with NACD
S25
The group interval is 40 m, the near offset is 160 m, and the far
offset is 2720 m. The sample interval is 8 ms, and the recording
time is 5 s. The shot and receiver layouts are shown in Figure 3.
We choose a Ricker wavelet with a peak frequency of 18 Hz as
the source wavelet to provide a close match to the spectra measured
from example data set waveforms. For migration, we choose grid
sizes of 40 and 20 m in the horizontal and vertical directions, respectively. The vertical plane of the velocity model, directly below
the source line, is shown in Figure 4. Figure 5 shows the results of
applying RTM to this data set computed using NACD and fourthorder LWC, respectively. The two images in Figure 5 demonstrate
that the NACD method produces a better image than the fourth-order LWC method when the same grid spacing is used, especially in
the regions above the left and right salt flanks. The white ellipses
superimposed on the images highlight regions in which the images
have significant differences. In Figure 5a, the whole image is very
clear and each interface above the salt is well imaged, but in the
corresponding regions in Figure 5b, the interfaces near the salt
can hardly be distinguished. Moreover, the salt is correctly positioned in Figure 5a, but in Figure 5b, the salt is a little shifted. Comparison between Figure 5a and 5b shows that most structure of the
model can be well imaged by the NACD method even using a
coarse grid size, consistent with the results found for the 2D case
(Li et al., 2013).
SEAM MODEL
Next, we apply the 3D RTM using the NACD and LWC methods
to the SEG Advanced Modeling project (SEAM) model (Fehler,
2008). We use NACD to generate a synthetic data set on a subset
of the SEAM model using a grid spacing of 40 m in the x- and ydirections and 20 m in depth. For convenience, we relabel the axes
of our selected study region located within the SEAM model
to make a new model that has dimensions of 0 ≤ x ≤ 6 km,
0 ≤ y ≤ 6 km, and 0 ≤ z ≤ 4.36 km. The southwest corner of
our model is at (12 km, 23.96 km) in the original SEAM model.
We use only one shot that is located at the center of an array of
receivers. The shot and receiver layout is shown in Figure 6, in
which the black star represents the source and the receivers are distributed on a 40 × 40 m grid within the box outlined by a solid black
line. The sample interval is 8 ms, and the recording time is 3 s. The
migrated volume is the entire subset of the SEAM model. In this
experiment, we choose a Ricker wavelet with a peak frequency of
20 Hz as the source wavelet. Two different 3D views of the velocity
model are shown in Figure 7. Figures 8a and 9a show 3D views of
image slices obtained by NACD for the slices that are shown in
Figure 7a and 7b, respectively. Corresponding slices obtained from
using fourth-order and eighth-order LWC are shown in Figures 8b
and 9b and Figures 8c and 9c. The white ellipses in Figures 8 and 9
indicate regions in which the salt and sediments are better imaged
by NACD than by LWC, and where we see a lot of dispersion and
noise in Figures 8b, 8c, 9b, and 9c, especially in regions in which
the velocity is complicated. To look into the details in the images,
we extract representative 2D crossline and inline sections from the
image volume and show them in Figures 10 and 11, respectively.
Comparing Figures 10 and 11, the differences among images migrated by NACD and LWC are very obvious, particularly in areas
within the white ellipses. We can see significant numerical dispersion in the images generated by LWC in the complex velocity
area, for example, the sharp salt body and the rough salt boundary.
Figure 9. Three-dimensional views of the RTM slices obtained by
(a) NACD, (b) fourth-order LWC, and (c) eighth-order LWC for the
SEAM model with crossline section at x ¼ 3 km, inline section at
y ¼ 3 km, and horizontal plane at z ¼ 2.36 km, which is displayed
in Figure 7b.
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S26
Li et al.
Figure 10. Vertical 3D migrated crossline sections at (a-c) x ¼ 3 km and (d-f) x ¼ 5 km obtained by (a and d) NACD, (b and e) fourth-order
LWC, and (c and f) eighth-order LWC for the subset of the SEAM model.
Figure 11. Vertical 3D migrated inline sections at (a-c) y ¼ 3 km and (d-f) y ¼ 5 km obtained by (a and d) NACD, (b and e) fourth-order
LWC, and (c and f) eighth-order LWC for the subset of the SEAM model.
3D weak-dispersion RTM with NACD
S27
Once again, our results illustrate that the NACD can give us very
clean images with great quality, even using a coarse grid size and
high frequency.
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3D FIELD DATA
In this section, we use NACD and fourth-order LWC to perform
3D RTM on a 3D marine field data set. This data set has been sorted
so that the receivers for one shot are positioned only along the line
belonging to the shot. We choose two neighboring shot lines, each
consisting of 234 shots, for the 3D migration study. The shot spacing and receiver interval are both 25 m. Correspondingly, the migration grid spacing is 25 and 20 m in horizontal and vertical
directions, respectively. We choose a 40-Hz Ricker source to attempt to match the high frequency content of the data. The velocity
model that we were provided is relatively smooth. We show the
vertical plane directly below one shot line in Figure 12. The corresponding images directly below the same shot line are shown
in Figure 13. Images shown are the results obtained after stacking
the 3D migrations using the two shot lines giving a total of 468
shots used to construct the image. Comparison between the two images in Figure 13 indicates many improvements to the structure;
better continuity and better event positioning are seen in Figure 13a.
As highlighted by the black arrows, faults are more clearly imaged
and cross-dipping events are placed more properly in the image obtained by NACD, than those in Figure 13b. There is a slight phase
shift in the image of Figure 13b compared with Figure 13a, which
means there is an error in the numerical velocity of fourth-order
LWC. The phase shift is particularly clear in the lower left side
of the images in the region inside the black boxes. These images
also tell us that NACD works well even using field data with such
a high frequency.
Figure 12. A vertical plane of the velocity model for the field data
set taken directly below the source line for the field data set. Velocities range from 1.5 to 5.2 km∕s.
NUMERICAL DISPERSION
The results above indicate that NACD has superior performance
compared with LWC for 3D RTM on synthetic and field data
sets. To better understand the reason why the NACD method performs better than the LWC method when using a coarse grid size
and why it provides a better image under the same condition as the
LWC method, we provide a simple numerical dispersion analysis.
Numerical dispersion causes the phase velocity to vary with the spatial and temporal frequencies. The computational merit of most
numerical schemes always hinges on their ability to minimize this
effect. Following the analysis approach presented by Moczo et al.
(2000), we investigate the numerical dispersion of the 3D NACD
and fourth-order LWC methods. The dispersion relation as a function of the sampling rate (grid spacing per wavelength) is shown in
Figure 14. The dispersion curves corresponding to different propagation directions tell us that numerical velocity for fourth-order
LWC has bigger error and more numerical anisotropy than that
of NACD. This observation is in agreement with our previous discussion about the advantages of stereo-modeling methods. A simple
spectral analysis shows that the spectrum of the field data set mainly
ranges from about 20 to 80 Hz. To obtain an estimate of the
dispersion for RTM of the field data set, we choose a Courant number of α ¼ 0.2, the spatial grid size to be 20 m, and the velocity to be
3 km∕s. The dispersion relation shown in Figure 14 is then transformed from being a function of the grid sampling rate into a function of frequency. Figure 15 shows representative dispersion
Figure 13. The 3D migrated inline sections obtained by (a) NACD
and (b) fourth-order LWC for the field data.
S28
Li et al.
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relation curves, as a function of frequency, corresponding to different propagation directions. The curves would not change if we
choose the grid size to be 40 m, but the frequency scale would be
divided by 2, so the range would be from 0 to 40 Hz, which fits the
case of the SEG/EAGE data set. These curves show that the maxi-
mum phase-velocity error of NACD is less than 8%, whereas the
maximum error of the fourth-order LWC is as high as 28% over
the frequency range of the data sets, which explains the dispersion
and the phase shift in those images generated by fourth-order LWC.
Comparing the curves for various propagation angles shows that there
is significantly more variation in numerical dispersion with propagation angle for fourth-order LWC than for NACD. The numerical
velocities for both methods are slower than the actual velocities when
the frequency is high, but the fourth-order LWC is worse. That error
will cause the events in the image to shift to shallower depths, which
explains the results obtained by the fourth-order LWC method in
Figures 5b and 13b compared with those for NACD. The differences
in depth of events are clearly shown in the regions highlighted by the
two boxes in Figure 13b compared with those in Figure 13a. This
means that, in practical cases, conventional RTM may yield an image
that appears to be in focus, but interfaces will be at incorrect locations.
This may lead to incorrect decisions about well positioning that could
lead to large economic losses.
CONCLUSIONS
Figure 14. The ratio R of the numerical wave velocity to the phase
velocity versus the sampling rate (grid spacing per wavelength) for
(a) NACD and (b) fourth-order LWC with a Courant number of
α ¼ 0.2, in which ϕ is the wave propagation angle relative to
the z-axis, and θ is the propagation angle of the wave projection
on the xy-plane relative to the x-axis.
We have proposed and applied a fourth-order 3D stereo-modeling
method to RTM and obtained a weak-dispersion prestack depth migration method that allows large extrapolation grid size to be used.
Numerical results illustrate that the stereo-modeling method, which
uses the wave displacement and its gradient, can greatly increase the
computational efficiency and save computer memory through the
use of the large spatial increments and the resulting large time steps.
Tests on synthetic and field data sets have demonstrated that the
stereo-modeling method is effective in imaging even using coarse
grids, compared with conventional methods such as the fourth-order
and the eighth-order LWC. The stereo-modeling method has significantly less numerical anisotropy than the conventional methods.
These results imply that stereo-modeling methods have a promising
future in 3D imaging.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science
Foundation of China (grant no. 41230210) and the ERL Founding
Members Consortium of MIT. J. Li was supported by the China
Scholarship Council. We thank Maersk for permission for the
SEAM model and ENI for the field data set. We thank the comments
and suggestions of the editor E. Slob and the associate editor F. Liu.
We thank the anonymous reviewers whose comments helped to improve the manuscript. We thank E. Liu who helped to revise the
paper. This study was also supported by the Statoil Company (contract no. 4502502663).
APPENDIX A
APPROXIMATION TO HIGH-ORDER
DERIVATIVES
Figure 15. The ratio R of the numerical wave velocity to the phase
velocity versus the frequency for (a) NACD and (b) fourth-order
LWC with velocity to be 3 km∕s and grid size to be 20 m.
To aid in implementing the 3D NACD method, we provide the
approximations of the high-order derivatives related to the wave
displacement and its gradient. For convenience, here we present
2
the expressions used in the computation. First, we define E−1
x , δx ,
and the other operators as follows:
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3D weak-dispersion RTM with NACD
n
n
E−1
x f i;j;k ¼ f i−1j;k ;
E1x f ni;j;k ¼ f niþ1j;k ;
n
n
E−1
y f i;j;k ¼ f i;j−1;k ;
E1y f ni;j;k ¼ f ni;jþ1;k ;
n
n
E−1
z f i;j;k ¼ f i;j;k−1 ;
E1z f ni;j;k ¼ f ni;j;kþ1 ;
δ2x f ni;j;k ¼ f niþ1;j;k − 2f ni;j;k þ f ni−1;j;k
δ2y f ni;j;k ¼ f ni;jþ1;k − 2f ni;j;k þ f ni;j−1;k
e; g ¼ x; y; z;
If ni;j;k ¼ f ni;j;k ;
f ¼ u; ux ; uy ; uz :
n
¼
i;j;k
n
6
12 2 n
1 − E−1 Þ ∂u
ðE
−
δ u ;
g
g
∂g i;j;k Δg4 g i;j;k
Δg3
g ¼ x; y; z;
n
∂4 u
2
∂g ∂e
2
3
n
δ2g ðE1e þ E−1
e − 2IÞui;j;k
Δg Δe2
n
1
2 ðE1 − E−1 Þ ∂u
−
δ
e
g
g
∂g i;j;k
2ΔgΔe2
n
1
∂u
−
; e;g ¼ x;y; z;
δ2g ðE1e − E−1
e Þ
∂e i;j;k
2Δg2 Δe
¼
i;j;k
2
(A-2)
∂5 u
∂g5
n
¼−
i;j;k
þ
90 1
n
ðEg − E−1
g Þui;j;k
Δg5
n
30 1
∂u
−1
ðE
þ
E
þ
4IÞ
;
g
g
4
∂g i;j;k
Δg
∂5 u
∂g4 ∂e
n
6
n
¼ − 4 δ2g ðE1e − E−1
e Þui;j;k
Δg Δe
i;j;k
3
−1
−1 1
1 −1
ðE1g E1e þ E−1
g Ee − Eg Ee − Eg Ee Þ
Δg3 Δe
n
∂u
;
e; g ¼ x; y; z;
(A-4)
×
∂g i;j;k
þ
∂5 u
∂g∂e4
n
¼−
i;j;k
6
n
δ2e ðE1g − E−1
g Þui;j;k
ΔgΔe4
3
−1
−1 1
1 −1
ðE1g E1e þ E−1
g Ee − Eg Ee − Eg Ee Þ
ΔgΔe3
n
∂u
;
e; g ¼ x; y; z;
(A-5)
×
∂e i;j;k
þ
n
i;j;k
−1
δ2e ð3ðE−1
g Ed
1
1 1
1 −1
− E−1
g Ed − Eg Ed þ Eg Ed Þ
−1
1 1
−1 1
1 −1
þ α2 · δ2e ðE−1
g Ed þ Eg Ed − Eg Ed − Eg Ed Þ
∂u
∂g
∂u
−1
1 1
−1 1
1 −1
þ α3 · δ2g ðE−1
e Ed þ Ee Ed − Ee Ed − Ee Ed Þ
∂e
−1
δ2e ðE−1
g Ed
1
− 2ðE−1
d þ Ed ÞÞ
þ
∂u
∂d
E1g E1d
n
þ
1
E−1
g Ed
;
þ
n
i;j;k
n
i;j;k
E1g E−1
d
(A-7)
i;j;k
1
1
when g ¼ x, e ¼ y, d ¼ z, α1 ¼ 4Δx2 Δy
α2 ¼ 4ΔxΔy
2 Δz,
2 Δz,
1
1
α3 ¼ 4Δx2 ΔyΔz, and α4 ¼ 4Δx2 Δy2 ; when g ¼ x, e ¼ z, d ¼ y,
1
1
1
1
α1 ¼ 4Δx2 Δy
2 Δz, α2 ¼ 4ΔxΔy2 Δz, α3 ¼ 4Δx2 Δy2 , and α4 ¼ 4Δx2 ΔyΔz;
1
1
when g ¼ y, e ¼ z, d ¼ x, α1 ¼ 4Δx2 Δy
,
α
¼
,
2 Δz
2
4Δx2 ΔyΔz
1
α3 ¼ 4Δx12 Δy2 , and α4 ¼ 4ΔxΔy
,
where
e;
g
¼
x;
y;
z
corresponds
2 Δz
to six cases: g ¼ x, e ¼ y; g ¼ x, e ¼ z; g ¼ y, e ¼ x; g ¼ y,
e ¼ z; g ¼ z, e ¼ x; and g ¼ z, e ¼ y, and where we have used
ð∂l u∕∂gp ∂eq ∂dl−p−q Þni;j;k ¼ ð∂l u∕∂gp ∂dl−p−q ∂eq Þni;j;k
¼ ð∂l u∕∂eq ∂gp ∂dl−p−q Þni;j;k
l;p;q ∈ N:
l
q
l−p−q p n
¼ ð∂ u∕∂e ∂d
∂g Þi;j;k ¼ ð∂l u∕∂dl−p−q ∂gp ∂eq Þni;j;k
l
l−p−q q p n
¼ ð∂ u∕∂d
∂e ∂g Þi;j;k ;
g ¼ x; y; z;
(A-3)
(A-6)
1
n
− 6ðE−1
d − Ed ÞÞui;j;k
þ α4 ·
(A-1)
¼−
i;j;k
∂5 u
∂g2 ∂e2 ∂d
¼ α1 ·
With the notation above and following the directional approach, we
can get expressions of the high-order mixed spatial derivatives for
3D NACD. The expressions of the other derivatives that are the
same as the previous work are also given (Yang et al., 2007).
∂4 u
∂g4
n
3
n
δ2e ðE1g − E−1
g Þui;j;k
2Δg3 Δe2
n
3
∂u
2
1
−1
þ
δ ðEg þ Eg Þ
;
2
2 e
∂g i;j;k
2Δg Δe
δ2z f ni;j;k ¼ f ni;j;kþ1 − 2f ni;j;k þ f ni;j;k−1
∂5 u
∂g3 ∂e2
S29
REFERENCES
Alford, R. M., K. R. Kelly, and D. M. Boore, 1974, Accuracy of finite difference modeling of the acoustic wave equation: Geophysics, 39, 834–842,
doi: 10.1190/1.1440470.
Baysal, E., D. D. Kosloff, and J. W. C. Sherwood, 1983, Reverse time migration: Geophysics, 48, 1514–1524, doi: 10.1190/1.1441434.
Billette, F., and G. Lambaré, 1998, Velocity macromodel estimation from
seismic reflection data by stereotomography: Geophysical Journal
International, 135, 671–690, doi: 10.1046/j.1365-246X.1998.00632.x.
Biondi, L. B., 2006, 3D seismic imaging: SEG, Investigations in Geophysics
no. 14.
Blanch, J. O., and J. O. A. Robertsson, 1997, A modified Lax-Wendroff
correction for wave propagation in media described by Zener elements:
Geophysical Journal International, 131, 381–386, doi: 10.1111/j.1365246X.1997.tb01229.x.
Chen, S., D. H. Yang, and X. Y. Deng, 2010, A weighted Runge-Kutta
method with weak numerical dispersion for solving wave equations:
Communications in Computational Physics, 7, 1027–1048, doi: 10
.4208/cicp.2009.09.088.
Dablain, M. A., 1986, The application of high-order differencing to the
scalar wave equation: Geophysics, 51, 54–66, doi: 10.1190/1.1442040.
Fehler, M., 2008, SEG advanced modeling (SEAM): Phase I first year update: The Leading Edge, 27, 1006–1007, doi: 10.1190/1.2967551.
Fei, T., and K. Larner, 1995, Elimination of numerical dispersion in finite
difference modeling and migration by flux-corrected transport: Geophysics, 60, 1830–1842, doi: 10.1190/1.1443915.
Downloaded 03/26/15 to 18.51.1.3. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
S30
Li et al.
Fleury, C., and I. Vasconcelos, 2010, Investigating an imaging condition for
nonlinear imaging principles and application to reverse-time-migration
artifacts removal: 80th Annual International Meeting, SEG, Expanded
Abstracts, 3338–3343.
Fornberg, B., 1987, The pseudo-spectral method: Comparisons with finite
difference for the elastic wave equation: Geophysics, 52, 483–501, doi: 10
.1190/1.1442319.
Fornberg, B., 1988, Generation of finite difference formulas on arbitrarily
spaced grids: Mathematics of Computation, 51, 699–706, doi: 10.1090/
S0025-5718-1988-0935077-0.
Fornberg, B., 1990, High-order finite differences and pseudo-spectral method
on staggered grids: SIAM Journal on Scientific Computing, 27, 904–918.
Igel, H., P. Mora, and B. Riollet, 1995, Anisotropic wave propagation
through finite-difference grids: Geophysics, 60, 1203–1216, doi: 10
.1190/1.1443849.
Kelly, K., R. Ward, S. Treitel, and R. Alford, 1976, Synthetic seismograms: A
finite-difference approach: Geophysics, 41, 2–27, doi: 10.1190/1.1440605.
Konddoh, Y., 1991, On thoughts analysis of numerical scheme for simulation using a kernel optimum nearly-analytical discretization (KOND)
method: Journal of the Physical Society of Japan, 60, 2851–2861, doi:
10.1143/JPSJ.60.2851.
Konddoh, Y., Y. Hosaka, and K. Ishii, 1994, Kernel optimum nearly analytical discretization algorithm applied to parabolic and hyperbolic equations: Computers & Mathematics with Applications, 27, 59–90., doi: 10
.1016/0898-1221(94)90047-7.
Kosloff, D., and E. Baysal, 1982, Forward modeling by a Fourier method:
Geophysics, 47, 1402–1412, doi: 10.1190/1.1441288.
Lax, P. D., and B. Wendroff, 1964, Difference schemes for hyperbolic equations with high order of accuracy: Communications on Pure and Applied
Mathematics, 17, 381–398, doi: 10.1002/cpa.3160170311.
Lele, S. K., 1992, Compact finite difference schemes with spectral-like resolution: Journal of Computational Physics, 103, 16–42, doi: 10.1016/
0021-9991(92)90324-R.
Li, J., D. H. Yang, and F. Q. Liu, 2013, An efficient reverse-time migration
method using local nearly analytic discrete operator: Geophysics, 78,
no. 1, S15–S23, doi: 10.1190/geo2012-0247.1.
Ma, X., D. H. Yang, and F. Q. Liu, 2011, A nearly-analytic symplectic partitioned Runge-Kutta method for 2D elastic wave equations: Geophysical
Journal International, 187, 480–496, doi: 10.1111/j.1365-246X.2011
.05158.x.
Mizutani, H., R. J. Geller, and N. Takeuchi, 2000, Comparison of accuracy
and efficiency of time-domain schemes for calculating synthetic seismograms: Physics of the Earth and Planetary Interiors, 119, 75–97, doi: 10
.1016/S0031-9201(99)00154-5.
Moczo, P., J. Kristek, and L. Halada, 2000, 3D fourth-order staggered-grid
finite-difference schemes: Stability and grid dispersion: Bulletin of the Seismological Society of America, 90, 587–603, doi: 10.1785/0119990119.
Robertsson, J. O. A., J. O. Blanch, and W. W. Symes, 1994, Viscoelastic
finite-difference modeling: Geophysics, 59, 1444–1456, doi: 10.1190/1
.1443701.
Sei, A., and W. Symes, 1995, Dispersion analysis of numerical wave propagation and it computational consequences: Journal of Scientific Computing, 10, 1–27, doi: 10.1007/BF02087959.
Takeuchi, N., and R. J. Geller, 2000, Optimally accurate second order timedomain finite difference scheme for computing synthetic seismogram in
2D and 3D media: Physics of the Earth and Planetary Interiors, 119, 99–
131, doi: 10.1016/S0031-9201(99)00155-7.
Tong, P., and D. H. Yang, 2013, A high-order stereo-modeling method for
solving wave equations: Bulletin of the Seismological Society of
America, 103, no. 2A, 811–833, doi: 10.1785/0120120144.
Virieux, J., 1986, P-SV wave propagation in heterogeneous media: Velocitystress finite difference method: Geophysics, 51, 889–901, doi: 10.1190/1
.1442147.
Wang, L., D. H. Yang, and X. Y. Deng, 2009, A WNAD method for seismic
stress-field modeling in heterogeneous media: Chinese Journal of Geophysics, 52, 1526–1535 (in Chinese), doi: 10.1002/cjg2.1447.
Whitmore, N. D., 1983, Iterative depth migration by backward time propagation: 53rd Annual International Meeting, SEG, Expanded Abstracts,
827–830.
Yang, D. H., E. Liu, Z. J. Zhang, and J. W. Teng, 2002, Finite-difference
modelling in two-dimensional anisotropic media using a flux corrected
transport technique: Geophysical Journal International, 148, 320–328.
Yang, D. H., M. Lu, R. S. Wu, and J. M. Peng, 2004, An optimal nearly
analytic discrete method for 2D acoustic and elastic wave equations:
Bulletin of the Seismological Society of America, 94, 1982–1992, doi:
10.1785/012003155.
Yang, D. H., J. M. Peng, M. Lu, and T. Terlaky, 2006, Optimal nearly analytic discrete approximation to the scalar wave equation: Bulletin of
the Seismological Society of America, 96, 1114–1130, doi: 10.1785/
0120050080.
Yang, D. H., G. Song, and M. Lu, 2007, Optimally accurate nearly analytic
discrete scheme for wave-field: Bulletin of the Seismological Society of
America, 97, 1557–1569, doi: 10.1785/0120060209.
Yang, D. H., J. W. Teng, Z. J. Zhang, and E. Liu, 2003, A nearly-analytic
discrete method for acoustic and elastic wave equations in anisotropic
media: Bulletin of the Seismological Society of America, 93, 882–
890, doi: 10.1785/0120020125.
Yang, D. H., P. Tong, and X. Y. Deng, 2012, A central difference method
with low numerical dispersion for solving the scalar wave equation:
Geophysical Prospecting, 60, 885–905, doi: 10.1111/j.1365-2478.2011
.01033.x.
Yang, D. H., and L. Wang, 2010, A split-step algorithm with effectively
suppressing the numerical dispersion for 3D seismic propagation modeling: Bulletin of the Seismological Society of America, 100, 1470–1484,
doi: 10.1785/0120090200.
Zhang, Z. J., G. J. Wang, and J. M. Harris, 1999, Multi-component wavefield
simulation in viscous extensively dilatancy anisotropic media: Physics of
the Earth and Planetary Interiors, 114, 25–38, doi: 10.1016/S0031-9201
(99)00043-6.
Zheng, H. S., and Z. J. Zhang, 2005, Synthetic seismograms of nonlinear
seismic waves in anisotropic (VTI) media: Chinese Journal of Geophysics, 48, 660–671 (in Chinese).
Zheng, H. S., Z. J. Zhang, and E. Liu, 2006, Nonlinear seismic wave
propagation in anisotropic media using the flux-corrected transport technique: Geophysical Journal International, 165, 943–956, doi: 10.1111/j
.1365-246X.2006.02966.x.