Introduction
Modeling of a seismic wave field is accomplished by simulating the pattern
of the seismic waves as they propagate through various geologic media and
computing the simulated measurements at observation points on the Earth's
surface or underground, given that the underground medium's structure and
the relevant physical parameters are known. Numerical modeling of a seismic
wave field is an important tool for seismic data processing and
interpretation and for geodynamic studies of the Earth's interior. In recent
years, many full waveform inversion methods have been widely proposed and
applied to seismic exploration. In the waveform inversion process, wave
field modeling is one of the key algorithms because it must be performed
first to obtain the predicted wave field that is used to compute the
residual errors between the predicted and the actual wave field records. In
addition, the information provided by the residual errors, which is required
for refinement of the initial model, is actually calculated by a modeling
algorithm that uses the residual errors as virtual sources. After many
iterations of the above processes, an optimized approximate model of the
underground medium can be acquired. Numerical modeling of a wave field will
be executed thousands of times throughout the waveform inversion process, so
a wave field modeling algorithm is crucial in many ways when performing a
waveform inversion algorithm, such as for computational precision, speed, and
storage requirements.
The main numerical techniques for seismic wave field modeling include the
finite-element method (Marfurt, 1984; Yang et al., 2008), the
pseudo-spectral method (Kreiss and Oliger, 1972; Dan and Baysal, 1982), and
the finite-difference method (Kelly et al., 2012; Virieux, 1984; Yang et
al., 2002; Moczo et al., 2007; Zhang et al., 2013). Due to its easy
implementation and the satisfactory compromise between accuracy and
efficiency, the finite-difference method is the preferred method. For a
comprehensive overview of applications of the finite-difference methods, see
Moczo et al. (2014). Over the last several decades, many studies have
focused on determining the coefficients of the finite-difference method and
designing computational templates (Li et al., 2017).
Schematic of our method: (a) the entire region and (b) partial
enlargement of the yellow rectangle in panel (a).
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(a) Velocity profiles in depth and (b) distribution of source
and receivers.
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Comparison of the analytical solution (red solid line) with the
proposed (second-order conventional grid (CG) scheme) and classic staggered-grid (SG) perfectly matched layer (PML) methods (second-order
SG scheme) (blue dotted line) at different receivers, d=12 m. (a) Proposed method at receivers 1, 2, and 3 for the first 2 s;
(b) proposed method at receivers 1, 2, and 3 after 2 s; (c) classic SG
PML method at receivers 1, 2, and 3 for the first 2 s;
(d) classic SG PML method at receivers 1, 2, and 3 after 2 s.
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According to the formulation of the wave equations, the finite-difference
methods can be implemented based on the first-order velocity–stress
equations or the second-order displacement equations, which lead to
different computational templates. A staggered grid (SG) is usually set up
for the first-order wave equations and has been widely used with the
acoustic and elastic wave equations (Virieux, 1984, 1986; Moczo et al., 2014;
Madariaga, 1976; Gold et al., 1997; Saenger et al., 2000;
O'Brien, 2010). Many methods of optimizing the differential coefficients,
based on a SG, have been proposed to increase the accuracy of the numerical
solution, such as the time–space domain dispersion-relation-based method
(Liu and Sen, 2011), the simulated annealing algorithm (Zhang and Yao,
2013), and the least-squares method (Yang et al., 2015). However, a
conventional grid (CG) is often directly obtained from the second-order wave
equation. These methods include the central scheme (Alford et al., 1974;
Igel et al., 1995), the high-order compact finite-difference method
(Fornberg, 1990), the Lax–Wendroff correction (LWC) scheme (Lax and
Wendroff, 1964; Dablain, 1986; Blanch and Robertsson, 2010), the nearly
analytical discrete method (Yang et al., 2003), and the nearly analytical
central difference method (Yang et al., 2012).
d=10 m; the rest is the same as in Fig. 3.
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Comparison of the relative errors between the analytical solutions
and the proposed method (second-order CG scheme) (red solid line) or the
classic SG PML method (second-order SG scheme) (blue dotted line) at
different receivers and different grid spacings. (a) d=12 m, receivers
1, 2, and 3; (b) d=10 m, receivers 1, 2, and 3.
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The algorithm design of the CG scheme is easier to use than that of the SG
scheme because the variable definition is uniform throughout the grid.
However, it is hard to determine which of the two schemes is more accurate
and efficient. Although the SG scheme has sometimes been regarded as more
precise than the CG scheme (Huang and Dong, 2009), there is also some
theoretical and experimental proof in the literature that does not support
this proposition. Moczo et al. (2011) compared the accuracy of the different
finite-difference schemes with respect to the P-wave to S-wave speed ratio
using theoretical analysis and numerical experiments. Their investigation
showed that the relative local errors of the CG scheme are almost equal to
those of the SG scheme when modeling planar S waves propagating in an
unbounded homogeneous elastic isotropic medium with a low P-wave to S-wave
velocity ratio (Vp/Vs=1.42). They showed that only at higher P-wave to
S-wave velocity ratios (Vp/Vs=5,10) will the relative local error of the
CG scheme increase faster than that of the SG scheme, but the difference in
the relative local errors of the two schemes will decrease when using a
higher-order spatial scheme, i.e., from second-order to fourth-order in
space. Moczo et al. (2011) also showed that the insufficient accuracy of the
CG scheme at higher P-wave to S-wave speed ratios can be compensated for by
using a higher spatial sampling ratio, i.e., a smaller grid size. This means
that a CG scheme with a sufficiently small grid size will be as precise as
the SG scheme or better, even if the P-wave to S-wave speed ratio is high.
The computational cost of the SG scheme is significantly higher than that of
an equal-sized CG scheme, as two variables (velocity and stress) have to be
calculated in the SG scheme and only one variable (displacement) have to be
computed in the CG scheme.
Comparison of the analytical solution (red solid line) with the
proposed (fourth-order CG scheme) and classic SG PML methods (fourth-order
SG scheme) (blue dotted line) at different receivers and d=12 m.
(a) Proposed method at receivers 1, 2, and 3 during the first 2 s;
(b) proposed method at receivers 1, 2, and 3 after 2 s; (c) classic SG
PML method at receivers 1, 2, and 3 during the first 2 s;
(d) classic SG PML method at receivers 1, 2, and 3 after 2 s.
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d=10 m; the rest is the same as in Fig. 6.
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Reflection from the artificial boundaries introduced by the limited
computational area is another numerical source of error. Over the past
30 years, many techniques have been developed for boundary processing:
paraxial conditions (Clayton and Engquist, 1977; Reynolds, 1978; Higdon,
2012), the sponge boundary (Cerjan et al., 1985; Sochacki et al., 1987), the
perfectly matched layer (PML) (Berenger, 1994), and the hybrid absorbing
boundary conditions (hybrid ABC) (Ren and Liu, 2012). Among these, the PML
is one of the most efficient and most commonly used methods. The PML was
first introduced for boundary processing of electromagnetic wave equation
modeling, after which, it was applied to the elastic–dynamic problem (Chew
and Liu, 1996) and acoustic simulations (Liu and Tao, 1998). Many modified
versions of the PML, such as the convolutional PML (Komatitsch and Martin,
2007), were subsequently proposed. Gao et al. (2017) compared most of the
typical artificial absorbing boundary processing approaches for use with
acoustic wave equations and came to the conclusion that a 20-layer PML is
ideal for most practical applications using general size models, even in the
presence of strong nearly grazing waves, which demonstrates the high
performance and efficiency of the PML approach.
Comparison of the relative errors between the analytical solutions
and the proposed method (fourth-order CG scheme) (red solid line) or the
classic SG PML method (fourth-order SG scheme) (blue dotted line) at
different receivers and different grid spacings. (a) d=12 m, receivers
1, 2, and 3; (b) d=10 m, receivers 1, 2, and 3.
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Comparison of the analytical solutions of the proposed (10th-order
CG scheme) (red solid line) and the classic SG PML methods (10th-order SG
scheme) (blue dotted line) at different receivers, d=10 m. (a) Proposed
method at receivers 1, 2, and 3 during the first 2 s; (b) proposed
method at receivers 1, 2, and 3 after 2 s; (c) classic SG PML method
at receivers 1, 2, and 3 during the first 2 s; (d) classic SG
PML method at receivers 1, 2, and 3 after 2 s.
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Comparison of the relative errors between the analytical
solutions and the proposed method (10th-order CG scheme) (red solid line)
or the classic SG PML method (10th-order SG scheme) (blue dotted line) at
different receivers and different grid spacings (d=10 m, receivers 1, 2,
and 3).
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In the field of real wave field simulation, most researchers are devoted to
unifying the format of the boundary processing algorithm and the wave
equation within the computational region. The classic PML is naturally
formulated based on the first-order wave equations for velocity and stress
(Collino and Tsogka, 1998), which has proven to be very efficient. It is
easy to integrate PML boundary processing into a SG finite-difference
algorithm. So, some scholars use the SG scheme in the computational region to
match the PML equations, while for many CG-based schemes, they need to adopt
other boundary processing methods, such as the hybrid ABC method. However,
in recent years, some scholars have also made efforts to formulate a PML for
a second-order system to match the second-order wave equation. Komatitsch
and Tromp (2003) reformulated the classic PML conditions in order to use it
with numerical schemes that are based on the elastic wave equation written
as a second-order system with displacement. Grote and Sim (2010) proposed a
PML formulation for the acoustic wave equation in its standard second-order
form, while Pasalic and McGarry (2010) extended the convolutional PML to
accommodate the second-order acoustic wave equation. Nevertheless, all of
these second-order PML formulations require the derivation of complicated
formulas, the introduction of auxiliary variables, and the modification of
existing second-order numerical codes in order to handle the first-order
equations describing the auxiliary variables, which increases the
computational cost and complexity.
(a) Velocity profiles in depth; (b) distribution of source and
receivers.
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Receiver records of the four methods for a different number of
absorbing layers: (a) 0 absorbing layers, (b) 10 absorbing layers, and
(c) 20 absorbing layers.
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Values of the absorption coefficient R at each receiver for our
method (red line), the classic SG PML method (black line), the second-order
PML method (blue line), and the hybrid ABC method (magenta line): (a) 10
absorbing layers and (b) 20 absorbing layers.
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In order to preserve the original efficiency of the PML boundary processing
method as well as the accuracy and efficiency of the CG scheme, it is worth
trying to integrate the classic first-order PML algorithm into the CG
finite-difference scheme in a second-order system and make it easy to
implement. In this paper, we propose a new boundary-matched algorithm that
uses a CG finite-difference scheme within a limited computational area and
an SG finite-difference scheme in a PML area. Our approach enables the inner
area and the PML condition to be independent during computation, while
preserving the individual advantages of the two methods. The algorithm
matches the computational area and the absorbing boundary layers simply by
point updating along the boundaries of the computational area and avoids
complex formula conversion. Thus, none of the original formulas of the CG
scheme or the PML equations are modified and no unnecessary variables are
added. The assessment of the proposed algorithm is composed of two parts.
First, we compared the accuracy and efficiency of the proposed algorithm
with those of the classic SG PML method (SG scheme both in computational
area and PML area), which demonstrated the rationality of our decision to
use the CG scheme in the computational area. To simulate the actual
underground medium, a medium with a linearly increasing velocity gradient
was selected for the experiment. The experimental results indicated that the
accuracy of the two methods for equal grid sizes is almost equal, but the
efficiency of our method is approximately 30 %–50 % higher than that of
the classic SG PML method. Next, the proposed algorithm was evaluated by
comparing its absorption efficiency and computational cost with those of the
classic SG PML method, the second-order PML method (CG scheme both in
computational area and PML area) introduced by Pasalic and McGarry (2010),
and the hybrid ABC method (CG scheme in computational area and hybrid ABC
scheme in boundary area) introduced by Ren and Liu (2012). The numerical
experimental results indicated that our algorithm provides an excellent
absorption effect and was easier to implement.
(a) Velocity profiles in depth; (b) distribution of source and
receivers.
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Receiver records for the four methods for a different number of
absorbing layers: (a) 0 absorbing layers, (b) 10 absorbing layers, and
(c) 20 absorbing layers.
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Methodology
Although the elastic wave equation can describe the propagation of seismic
waves more comprehensively, modeling an elastic wave field is complex and
computationally expensive. In practice, the acoustic wave equation is also
popularly used to approximate seismic wave propagation. For the convenient
error analysis of these methods, we consider a scalar wave field p
propagating through an unbounded three-dimensional medium where the wave
field satisfies Eq. (1) (Engquist and Runborg, 2003):
∂2p∂x2+∂2p∂y2+∂2p∂z2=1c2⋅∂2p∂t2,
where the wave field p is a function of the space variables x, y, z, and the time
variable t, and c is the velocity of the medium. Numeric modeling of Eq. (1) is
expressed as follows.
Conventional-grid finite-difference scheme
The discretization of the acoustic wave (Eq. 1) with a 2M-order
finite-difference scheme is (Chu and Stoffa, 2012)
pi,j,kn+1=2pi,j,kn-pi,j,kn-1+c2Δt2Δx2c0pi,j,kn+∑m=1Mcmpi-m,j,kn+pi+m,j,kn+c2Δt2Δy2c0pi,j,kn+∑m=1Mcmpi,j-m,kn+pi,j+m,kn+c2Δt2Δz2c0pi,j,kn+∑m=1Mcmpi,j,k-mn+pi,j,k+mn,
where cm for all m are finite-difference coefficients. i,j, and k denote the
discrete spatial variables, and n denotes the discrete time variable. The
increments Δx, Δy, and Δz
are grid spacings, and Δt is the time step. In many
applications, a regular rectangular grid with a grid spacing Δx=Δy=Δz=d is a natural and reasonable choice (Moczo et al.,
2007).
Values of the absorption coefficient R at each receiver for our
method (red line), the classic SG PML method (black line), the second-order
PML method (blue line), and the hybrid ABC method (magenta line): (a) 10
absorbing layers and (b) 20 absorbing layers.
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Marmousi velocity model.
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Numerical analyses show that grid dispersion increases with increasing grid
size, but decreasing the grid size increases the computational cost.
High-order finite-difference schemes are able to control this numerical
dispersion using a larger grid spacing compared with low-order schemes (Tan
and Huang, 2014).
Because the subscripts i,j, and k used in Eq. (2) have integer values, it is
convenient to define and calculate the medium's parameters and wave field p for
the same grid points, which leads to the CG scheme. The pressure source s is
an additive item (Hustedt et al., 2004); i.e., it can be directly added in
the corresponding equations.
Boundary conditions
Due to limitations in the capacity and speed of computer facilities, the
numerical simulation of a wave field can only be implemented for a limited
area. The computational area is surrounded by artificial boundaries, except
for the free surface. As described above, the PML boundary condition can
effectively absorb the wave field reflections from the artificial boundaries
in order to simulate wave field propagation in an open space. In a PML
medium, the wave field p is assumed to be decomposed into subcomponents. The
PML formulation based on the acoustic equations is as follows (Liu and Tao,
1998):
∂vx∂t+αxvx=-1ρ∂p∂x,∂vy∂t+αyvy=-1ρ∂p∂y,∂vz∂t+αzvz=-1ρ∂p∂z,∂px∂t+αxpx=-c2ρ∂vx∂x,∂py∂t+αypy=-c2ρ∂vy∂y,∂pz∂t+αzpz=-c2ρ∂vz∂z,p=px+py+pz.
Here, αx, αy, and αz are the attenuation
coefficients in the PML medium. In this paper, the attenuation coefficient
was set using the following function (Wang, 2003):
αij=B1-sinjπ2Pml,i=xyz;j=0,1,…,Pml.
B is the amplitude of attenuation coefficient, i.e., the maximum value of
the coefficient, which we set as 400 in the numerical experiment; Pml
is the thickness of the PML layer.
Using the SG finite-difference scheme to discretize Eq. (3), the results are as
follows:
vxn+12i+12,j,k=vxn-12i+12,j,k-αxΔtvxn-12i+12,j,k-ΔtρΔxpxni+1,j,k+pyni+1,j,k+pzni+1,j,k-pxni,j,k-pyni,j,k-pzni,j,k,vyn+12i,j+12,k=vyn-12i,j+12,k-αyΔtvyn-12i,j+12,k-ΔtρΔypxni,j+1,k+pyni,j+1,k+pzni,j+1,k-pxni,j,k-pyni,j,k-pzni,j,k,vzn+12i,j,k+12=vzn-12i,j,k+12-αzΔtvzn-12i,j,k+12-ΔtρΔzpxni,j,k+1+pyni,j,k+1+pzni,j,k+1-pxni,j,k-pyni,j,k-pzni,j,k,pxn+1i,j,k=pxni,j,k-αxΔtpxni,j,k-c2ρΔtΔxvxn+12i+12,j,k-vxn+12i-12,j,k,pyn+1i,j,k=pyni,j,k-αyΔtpyni,j,k-c2ρΔtΔyvyn+12i,j+12,k-vyn+12i,j-12,k,pzn+1i,j,k=pzni,j,k-αzΔtpzni,j,k-c2ρΔtΔzvzn+12i,j,k+12-vzn+12i,j,k-12.
Implementation of our new boundary-matched algorithm
A finite-difference scheme based on a CG requires no computation of
intermediate variables, and thus the computational cost is lower than that
of an SG scheme. We will show in the next section that the accuracy of the
CG scheme can reach the same level as that of the SG scheme but with lower
computational costs. However, it is difficult to incorporate a naturally
formulated PML boundary processing algorithm based on an SG scheme into a
CG finite-difference scheme. In this paper, we propose a new
boundary-matched algorithm that can bridge the gap between an SG-based PML algorithm
and a CG-based numerical simulation of a seismic wave field with neither
introduction of intermediary variables nor reformulation of the PML
equations. The core idea of the scheme is to interface the wave field
reasonably along the boundaries between the CG area and the SG absorbing
layers. A detailed description of the method is given below.
As shown in Fig. 1, the entire domain consists of two parts: the
computational area and the boundary absorbing area. The computational area
is located in the center and is surrounded by the absorbing layers. The
algorithm uses a CG finite-difference scheme within the computational area
and an SG finite-difference scheme within the boundary absorbing area. If we
can reasonably interface the computed values of the wave field between the
computational area and the boundary absorbing area, then the scheme can
perform satisfactorily. For a clearer explanation, we start with a
two-dimensional model.
Receiver records for the four methods for a different number of
absorbing layers: (a) 0 absorbing layers, (b) 10 absorbing layers, and
(c) 20 absorbing layers.
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We let the computational area and the PML area overlap each other for one
layer. As shown in Fig. 1a, the bold red boundary line is both the
outermost boundary of the computational area and the innermost boundary of
the PML area. On this overlapped layer, both the particle velocity
v and the wave field p in the PML area are calculated using
the value of wave field p in the computational area. Using this method, the
two areas can be connected. This avoids the introduction of intermediate
variables and saves storage space. In order to distinguish, we use
pi,jn+1 to represent the wave field value in the computational area,
and pxn(ij) and pzn(ij) to represent the wave field value in
the PML area. In the PML area, the values of attenuation coefficients
αx and αz can be calculated using Eq. (4). When the
grid points are located on the four corners of the PML area, the values of
αx and αz are not zero. When they are on the left- and
right-hand sides of the PML area, αx≠0 and αz=0;
when they are on the upper and bottom sides of the PML area, αx=0
and αz≠0. The specific steps of our method are as follows.
Wave field snapshots with different PML at different time; the
three coordinate axes represent the number of grids in three directions:
(a) PML is 0 at 250, 350, and 450 ms; (b) PML is 20 at 250, 350, and
450 ms.
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At the beginning of iteration, take n=1, and let the initial wave field
values pi,jn and pi,jn-1 in the
computational area, the particle velocity vxn-12i+12,j and vzn-12i,j+12, the wave field pxn(ij) and pzn(ij) in the PML area
all be zero.
Calculate the wave field pi,jn+1 in the computational area. In
this step, we do not calculate the value of wave field pi,jn+1
located on the red boundary line; instead, we only calculate them on the
blue boundary line and in its inner region in Fig. 1 using the
two-dimensional form of Eq. (2). For the high-order difference scheme in the
computational area, we use the second-order difference scheme for the grid
points on the blue rectangular line, the fourth-order difference scheme for
the grid points on the inner green rectangular line, the sixth-order
difference scheme on the inner layer, and so on, until we reach the required
order of difference. In this way, the computational area and the PML area
can be independent, and the number of overlapping layers does not increase
with the increase of order of difference. The numerical experiments in
Sect. 4.1 will prove that our treatment to the boundary does not bring
much additional dispersion and error.
Calculate the particle velocity and wave field in the PML area layer by
layer. Calculate the values of all particle velocity
vxn+12i+12,j and
vzn+12i,j+12 in the PML area
(including those on the red line) using the two-dimensional forms of the
first and third formulas in Eq. (5). Calculate all of the wave field
values pxn+1(ij) and pzn+1(ij) in the PML area (including
those on the red line) using the two-dimensional forms of the fourth and
sixth formulas in Eq. (5).
It is important to note that the particle velocities in the area between the
blue and red lines are calculated from the wave field pxn(irjr)
and pzn(irjr) on the red line in the PML area and the wave
field pib,jbn on the blue line in the computational area. As
shown in Fig. 1b, we can obtain vxn+12ir+12,jr for the line between the left-hand line of
the red rectangle and blue rectangle using Eq. (6); the subscript r stands
for the grid points on the red lines, and b stands for the blue lines.
Similarly, we can obtain vxn+12ir-12,jr between the right-hand line of two
rectangles using Eq. (7); for the line between the upper lines, we
obtain vzn+12ir,jr+12 using
Eq. (8); for the line between the lower lines, we obtain
vzn+12ir,jr-12 using Eq. (9):
vxn+12ir+12,jr=vxn-12ir+12,jr-αxΔtvxn-12ir+12,jr-ΔtρΔxpib,jbn-pxnir,jr-pznir,jr,vxn+12ir-12,jr=vxn-12ir-12,jr-αxΔtvxn-12ir-12,jr-ΔtρΔxpxnir,jr+pznir,jr-pib,jbn,vzn+12ir,jr+12=vzn-12ir,jr+12-αzΔtvzn-12ir,jr+12-ΔtρΔzpib,jbn-pxnir,jr-pznir,jr,vzn+12ir,jr-12=vzn-12ir,jr-12-αzΔtvzn-12ir,jr-12-ΔtρΔzpxnir,jr+pznir,jr-pib,jbn.
After calculating the complete PML area, let the value of the wave field
pi,jn+1 on the red line in the computational area be equal to the
sum of the wave field pxn+1(irjr) and
pzn+1(irjr) on the red line in the PML area.
Update the value of pi,jn-1 with the value of pi,jn, and
update the value of pi,jn with the value of pi,jn+1; then,
let n=n+ 1.
Repeat steps 2–4 until n reaches the required time length.
The two-dimensional algorithm described above can easily be generalized to
three-dimensional. In the three-dimensional model, we need to add a particle
velocity component vy and a space position label k. The red and blue
boundary lines become the red and blue boundary surfaces, respectively. In
addition, the computational area becomes a cube surrounded by the PML area.
Performance analysis
As described in the introduction, the errors in the wave field numerical
model are mainly caused by differential dispersion and reflected waves that
are not fully absorbed by the boundary processing algorithm. In order to
verify the validity of our algorithm, we used a variety of models to compare
the computational accuracy, the efficiency of the absorption of the
reflected waves, and the computational efficiency of the proposed algorithm
to the other methods.
Computational accuracy
In order to obtain a more convincing result when comparing the computational
accuracy, we used a constant-gradient velocity model, the velocity of which
increases linearly with depth. This model is closer to the actual velocity
distribution of an underground medium than a homogeneous model. We
calculated the relative error between our method and the classic SG PML
method using the analytical solutions for different grid spacings and the
order of difference, and then we performed a comparative analysis of the
two methods. The relative error between the two methods and the analytical
solution is defined by the following time function:
errort=20logpt-panaltmaxpanalt.
In Eq. (10), pt represents the value of wave field
calculated by the numerical methods at a receiver, and panal(t) is the
value of wave field calculated by the analytic solution at the same
receiver.
For the two-dimensional scalar (Eq. 1), the wave field analytic
solution for a constant-gradient medium can be obtained from the integral
form of the three-dimensional solution using the dimension reduction method
(Cerveny, 2001). The velocity distribution is cz=cz01+γz1+γz0=c(0)(1+γz), where
z0 is the depth of source, cz0 is the velocity of
the source layer, c(0) is the velocity of the layer z=0, and
γ=1h, h is the distance from depth z=0
to the level where the propagation velocity is null (c-h=0). When the line source s=δ(t-t0)δ(x-x0)δ(z-z0) is located at (x0z0), and the
density in the method of Sanchez-Sesma et al. (2001) is a constant, a
two-dimensional scalar Green function can be obtained by
G(xzt)≈Λ⋅H(t-t0-τ)2π(t-t0)2-τ2,
where Λ=1+γz01+γz⋅c(z0)τRw.
H(t) is the Heaviside step function (equal to 0 when t < 0 and equal
to 1 when t > 0). For fixed t, the radius Rw of the wavefront
circle is Rw=(z0+h)sinh(γc(0)τ) and the
travel time τ can be computed by means (Cerveny, 2001) of τ=|1γc0arccosh[1+(γc(0)r)22c(z)c(z0)]|, r=(x-x0)2+(z-z0)2. This is an approximate solution, but
usually the error is less than 1 % (Sanchez-Sesma et al., 2001). In the
next numerical experiment, we use a symmetric Ricker wavelet with a peak
frequency of 20 Hz as the source. In this paper, the expression is
usually st={1-2[20π(t-t0-1/20)]2}e-[20π(t-t0-1/20)]2, and t0 is equal
to 200 ms. The final result of the analytic solution is
panalx,z,t=Gx,z,t×δ(x-x0)δ(z-z0)S(t).
Absorption efficiency of the reflected waves
When comparing the absorption efficiency, we used three different geological
models to determine the reflected wave absorption effect of our algorithm:
the homogeneous, constant-gradient velocity and the Marmousi models. We
compared the absorption effects of our algorithm with the classic SG PML
method, the second-order PML method, and the hybrid ABC method using the
same conditions to prove whether our algorithm can effectively combine the
CG scheme with the SG scheme PML boundary condition and achieve the same or
better effect as other methods do. In the computational area, the reflection
coefficient R of a receiver is defined as
R=20logmax[pt-preft]max[preft],
where the wave field value pt is calculated by the
numerical methods at a receiver, and preft is the wave
field value that has no boundary reflection on the same receiver calculated
using the numerical methods, which can be obtained by expanding the model.
The value of R reflects the reflected wave absorption effect of the
algorithm. The smaller the R value, the better the absorption effect.
Computational efficiency index
In the comparison of the computational accuracy and efficiency of the
absorption of the reflected waves, we determined the computation time of the
three methods separately, which can reflect the advantages and disadvantages
of all of the methods in terms of the computational efficiency.
Numerical experiment
Based on the discussion of the performance analysis, in this section, we
present the results of the numerical experiments. All of the numerical
experiments were run on a desktop personal computer with a 3.40 GHz Intel
Core i5-3570 processor, 32 GB of DDR3 memory, on a 64 bit Windows 7 operating
system, using algorithmic software written in C ++.
Computational accuracy and computation time
As shown in Fig. 2, the constant-gradient velocity model has a size of
6000 m × 6000 m, a velocity distribution of c=500 m s-1 for z=0 m and c=5300 m s-1 for z=6000 m, and a velocity gradient of 0.8. In
order to determine the stability of the differential form throughout the
computational process, the time step was set as 0.001 s and the total
simulation time as 10 s. The sources are located in the middle of the model
(3000, 3000 m), and the three receivers are located at (600, 1800 m),
(1800, 1800 m), and (3000, 1800 m). We compared the errors of the
numerical solution and the analytical solution at the receivers of the method
we proposed with the classic SG PML methods. The number of PMLs is set as 10.
Figures 3 and 4 show the comparison of the analytical solutions at the three
receivers of the proposed method (second-order CG scheme in the computational
area) and the classic SG PML method (second-order SG scheme in the
computational area) when the grid spacing (d) is 12 and 10 m,
respectively. Figure 5 shows the relative errors between the analytical
solutions and the two methods for the conditions described above where the
relative error was calculated using Eq. (10).
From Figs. 3 and 4, we can see that both of the methods have obvious errors
during the first 2 s. In particular, when the grid spacing is 12 m,
the error is the largest, and there is significant numerical dispersion.
Reducing the grid spacing can reduce the error and the dispersion. When the
grid spacing is 10 m, the result improves. In addition, the results for a
longer simulation time also prove the numerical stability of our method.
Further comparison of the relative error curves shown in Fig. 5 indicates
that although neither method is particularly good; the amplitude of relative
errors with their analytical solutions are almost the same.
In theory, the error of the numerical solution can be reduced by using a
higher-order difference. We compared the experimental results of the
proposed method (fourth-order CG scheme in the computational area) and
classic SG PML methods (fourth-order SG scheme in the computational area)
with the analytic solution, as shown in Figs. 6–8.
From Figs. 6 and 7, we can see that when the fourth-order difference is used,
the relative errors between the analytical solution and both methods are
significantly reduced compared with when the second-order difference is
used. In addition, as with the second-order result above, the relative error
also decreases as the grid spacing decreases. Figure 8 illustrates the fact
that the relative error curves of our algorithm and the classic SG PML
method are also very similar for the fourth-order difference, In addition,
it is difficult to distinguish the advantages and disadvantages of the two
algorithms. Although the results of the two methods still exhibit a small
error at this time, we can continue to use the higher difference order or we
can reduce the grid spacing to reduce the error. The laws of the two methods
are the same.
In Figs. 9 and 10, we adopt a 10th-order difference scheme and d=10 m. At
this time, the numerical results are very close to the analytical solution
and the relative error is very low. Based on this, we can conclude that the
accuracies of the proposed method and the classic SG PML method for the same
conditions with a constant-gradient velocity are similar. Meanwhile, it also
demonstrates an additional advantage of our method. When the computational
area and the PML area are independent of each other, we can easily optimize
the numerical algorithm of the computational area to improve the accuracy of
the algorithm. When the grid spacing and the order of difference are
appropriate, our method yields the expected results. Because our method uses
the CG scheme in the computational area, the experimental results also show
that in the scalar wave field simulation, the accuracy of the SG scheme is
not higher than that of the CG scheme. This conclusion is in agreement with
the experimental results of the elastic wave field simulated by Moczo et al. (2011) at a low P-wave to S-wave speed ratio (Vp/Vs=1.42).
Table 1 presents the computation times of the two methods at different grid
spacings and difference orders. The efficiency percentage is the total
computation time of our method divided by the total computation time of the
SG method. Under the same circumstances, the total computation time of our
method is only 57 %–70 % that of the classic SG PML method. It is
noteworthy that the result of our method for the 10th-order difference and
a grid spacing of 10 m is much better than that of the classic SG PML method
for the fourth-order difference and a 10 m grid spacing, while the
computation time is nearly the same. Also, the result of our method for the
fourth-order difference and a grid spacing of 12 m is much better than that
of the classic SG PML method for the second-order difference and a 10 m grid
spacing, while the former computation time is only 53.3 % of the latter.
Therefore, for the same computation time as the classic SG PML method, our
method always achieves a higher accuracy for a smaller grid spacing and a
higher-order difference. We obtained these conclusions in a
constant-gradient velocity medium. Therefore, the algorithm we propose works
well when the CG scheme is used in the computational area. Next, we discuss
the absorption efficiency of the reflected waves of our method in a series
of simple and complex models.
Computation time for our and the classic SG PML
methods.
Condition
Our method
The classic
Efficiency
SG PML method
percentage
Second-order
10 m 19 s
15 m 17 s
67.5 %
d=12 m
Second-order
14 m 41 s
21 m 01 s
69.8 %
d=10 m
Fourth-order
11 m 13 s
17 m 56 s
62.5 %
d=12 m
Fourth-order
16 m 58 s
25 m 54 s
64.3 %
d=10 m
10th-order
25 m 48 s
44 m 54 s
57.4 %
d=10 m
Computation times for the four methods.
Condition
Our method
Classic SG PML method
Second-order PML method
Hybrid ABC method
Homogeneous model
20 s
28 s
24 s
20 s
PML =10
Homogeneous model
24 s
34 s
28 s
25 s
PML =20
Constant-gradient velocity model
21 s
30 s
26 s
21 s
PML =10
Constant-gradient velocity model
25 s
35 s
30 s
26 s
PML =20
Marmousi model
6 m 57 s
10 m 31 s
8 m 20 s
7 m 14 s
PML =10
Marmousi model
6 m 05 s
9 m 05 s
7 m 18 s
6 m 25 s
PML =20
Absorption efficiency and computation time
Homogeneous model
First, we used a two-dimensional homogeneous model to verify the reflected
wave absorption efficiency of our new boundary-matched algorithm. As shown
in Fig. 11, the model size is 2000 m × 2000 m at a velocity of c=2500 m s-1
and a grid spacing of 10 m. The source is the same as previously
described and is located at (1000, 1000 m) with a time step of 0.001 s and
a total simulation time of 1.5 s. A total of 201 receivers are evenly
distributed on a horizontal line with a depth of 500 m, and the distance
between each receiver is set as 10 m. In Fig. 12, we compared the receiver
records of our method, the classic SG PML method, the second-order PML
method, and the hybrid ABC method for a different number of absorbing
layers. In general, the amplitude of the reflected wave will be reduced to
less than 1 % of that of the normal wave field after the boundary
conditions are processed. Thus, in order to illustrate the reflected wave
more clearly, we set the range of the color bar of the wave field to be
-0.001 to 0.001. For further comparison, we also calculated the value of
the reflection coefficient R using Eq. (13) and plotted the corresponding
curve in Fig. 13.
As can be seen in Figs. 12 and 13, all of the four methods can absorb the
reflected waves to a certain degree. For the same number of absorbing
layers, the absorption performance of our method and that of the classic SG
PML method are almost the same and both methods are superior to the other
two methods, while the hybrid ABC method is the worst. Specifically, when
the number of absorbing layers equals 10, the absorption coefficients of our
method and classic SG PML method are both -60 dB, which means that the
amplitude of the reflected wave after absorbing is only 0.1 % of that
before absorbing. Increasing the number of absorbing layers can improve the
absorption effect of the four methods. In addition, the 20-layer
second-order PML method performs similarly to the 10-layer proposed method
and the 10-layer classic SG PML method. This indicates that the second-order
PML method always requires more absorbing layers than the first-order PML
does. Although the core idea of the second-order PML method is the same as
that of the first-order PML method, there are very different ways to deal with
mathematical equations. In the second-order method, the original PML
equations are transformed and some additional nonphysical variables are
added, which greatly reduces the efficiency of boundary conditions.
Moreover, the second-order PML method still needs to handle the first-order
system, i.e., a Euler or a Runge–Kutta time scheme has to be used
(Komatitsch and Tromp, 2003). So, in this case, the second-order PML method
is naturally less effective than our first-order PML method and requires
more absorbing layers. For hybrid ABC, it has poor absorbing performance
compared to the PML method since it is based on a one-way equation. So, it would
have limited improvement due to the phase and the amplitude errors that are
associated with the expansion of one-way wave equation.
Constant-gradient velocity model
Taking into account the fact that the homogeneous model is relatively simple
and quite different from the actual distribution of an underground
medium, the second model that we use is the constant-gradient velocity
model, as shown in Fig. 14. The velocity is 1500 m s-1 at z=0 m and 3500 m s-1
at z=2000 m, and the velocity gradient is 1. The source and receiver
locations are exactly the same as those in the first homogeneous model.
Figures 15 and 16 compare the receiver records and the reflection
coefficients R of the four methods for different boundary conditions. It can
be seen that the absorption effect of this model is not as good as that of
the homogeneous model, but the results are the same. For the same number of
absorbing layers, our method has the same absorbing ability as that of the
classic SG PML and performs better than the other two methods. Also, we
still need to use more layers for the second-order PML method instead of
using the thin method we proposed. In addition, we see that the 10-layer
proposed method is much better than the 20-layer hybrid ABC method.
Marmousi model
We next compared the absorption efficiency of the four methods for a complex
Marmousi model. The Marmousi model has a size of 9200 m × 3000 m, a
grid spacing of 12.5 m, a time step of 0.001 s, and a total recording time
of 8 s. The velocity distribution is shown in Fig. 17. Taking the first shot
of the Marmousi model as an example, we can see that the shot is located on
the ground surface at a horizontal distance of 3000 m and that the 185
receivers are evenly distributed between 0 and 9200 m on the surface. The
results in Fig. 18 show that the absorption effect of our method is equal to
or better than the absorption effect of the other methods. When the number
of PMLs is 20, the reflected wave is relatively small. Therefore, the method
we propose is also suitable for simulating complex models.
Based on the above numerical experiments, although the hybrid ABC method is
often used as the boundary condition of the CG-based method because it is
easy to deduce its second-order form, its absorption performance is
obviously worse than that of the other three PML methods since it is based
on a one-dimensional wave equation. Among the three PML methods, the
10-layer classic SG PML method (first-order PML is used inside) for the
first-order wave equation is enough to suppress the edge reflections, while
the 20-layer second-order PML method is sufficient for the second-order wave
equation. However, our first-order PML method only requires a thickness of
10 grid spacings to absorb the outgoing wave entirely. It may have a
significant advantage over the second-order PML method. Table 2 shows the
computation times of the four methods for different numbers of absorbing
layers. Among them, the computation time of our method is the shortest and
that of the classic SG PML method is the longest. Given that our method uses
the CG scheme in the computational area, it requires much less computation
time than the classic SG PML method does. In addition, the second-order PML
method requires the transformation of the original first-order PML equation
into a second-order form. The required complex formulas and extra variables
without physical meaning increase the computation time. In addition, our
method naturally implements high-order temporal discretization if necessary,
while the second-order PML method does not. Therefore, our method is ideal
for seismic wave forward modeling.
Three-dimensional homogeneous model
In order to facilitate the experiments and comparative analyses, we used the
two-dimensional models described in the above numerical experiments. To
further illustrate the effectiveness of our method, Fig. 19 shows the
experimental results of this method for a three-dimensional homogeneous
velocity model. The model size is 1000 m × 1000 m × 1000 m,
the grid spacing is 10 m, and the velocity is 2000 m s-1. The source is
located at (500, 500, 500 m) with a time step of 0.001 s. Figure 19
shows snapshots of the wave field at different times. From this, we find that
when the number of PMLs is 20, the wave field record is very clear, and
almost no reflected waves are seen.