A new finite-difference scheme for frequency-domain seismic modeling on non-uniform grids Quanli Li*, Xiaofeng Jia and Wei Zhang, Laboratory of Seismology and Physics of Earth’s Interior, School of
Earth and Space Sciences, University of Science and Technology of China
Summary
In this paper we introduce a new finite-difference scheme
for 2D frequency-domain acoustic wave modeling on
irregular grids. It is developed from the average-derivative
optimal nine-point scheme (ADM). The ADM scheme
breaks the limitation on directional sampling intervals but
can’t be implemented on non-uniform grids. Our new
scheme overcomes this restriction and is more generalized
than the ADM scheme. It can be proved that the ADM
scheme is a particular case of our new scheme. A simple
two-layer model is tested for verifying the feasibility and
efficiency of our new scheme.
Introduction
Seismic wave forward modeling is an important part for full
waveform inversion (FWI) and migration and also an
important tool to understand wave propagation phenome-
non in realistic underground media. Besides time-domain
forward modeling, frequency-domain forward modeling is
widely used, especially for multi-scale FWI, because it is
convenient to manipulate single frequency wavefield.
Frequency-domain operators have stronger stability than
time-domain operators as it is not limited by time-marching
step and no accumulative error occurs. Therefore, it’s a
reasonable choice to employ frequency-domain operators to
perform numerical modeling when dealing with frequency-
dependent physical problems with limited frequency
bandwidth.
Nevertheless, a crucial disadvantage of frequency-domain
operators is that it would cost memory and time
prohibitively when we handle large model areas. Different
from the time-domain operators, every single frequency is
computed by solving equations independently in frequency-
domain operators. A huge sparse impedance matrix needs to
be conducted when we solve the system of linear equations.
Although influenced by the solver we used, the
computational cost increases with the grid number
exponentially no matter which solver is chosen (Li et al.,
2015). Hence, to reduce the memory and time cost, less grid
nodes are used in the numerical computation.
In order to reduce the number of grid nodes on the premise
of ensuring the accuracy of modeling, many kinds of finite-
difference scheme are proposed in recent years. Pratt and
Worthington (1990) propose conventional second-order
central finite-difference scheme. For this scheme, thirteen
grid nodes per wave-length is needed in order to suppress
the numerical dispersion and achieve the satisfying result. A
rotated optimal nine-point scheme is presented by Jo et al.
(1996) which is more efficient than conventional second-
order finite-difference scheme. Four grid nodes per
wavelength is enough to obtain the accurate modeling result.
An average-derivative optimal nine-point scheme (ADM)
proposed by Chen (2012) improves the rotated nine-point
scheme. Same to the former, four grid nodes per wavelength
is necessary for this scheme to suppress the numerical
dispersion greatly. This scheme can overcome the
restriction on directional sampling intervals to which the
scheme of Jo et al. (1996) is limited.
In addition to these schemes which are aimed to improve the
finite-difference accuracy, other approaches are employing
non-uniform grids spatially to decrease the overall grid
number. In case of complex underground media, using non-
uniform grids is a reasonable strategy. Quite a few papers
about non-uniform grids have been published in recent years
(Oprsal and Zahradnik, 1999; Tessmer, 2000; Adriano and
Oliveira, 2003; Zhang and Chen, 2006; Liu et al., 2011; Liu
and Sen, 2011; Chu and Stoffa, 2012; Zhang et al., 2013).
However, almost all studies focus on the time-domain
operators and frequency-domain operators are hardly
investigated.
In this paper, we propose a new finite-difference scheme for
frequency-domain acoustic wave propagator on non-
uniform grids. The mathematical principle is deduced and
the numerical dispersion is analyzed. A simple numerical
example is presented to demonstrate the validity of this new
scheme.
Theory
Consider the 2D acoustic wave equation in frequency
domain
((𝜕2
𝜕𝑥2+
𝜕2
𝜕𝑧2) +
𝜔2
𝑣2) 𝑝(𝑥, 𝑧, 𝜔) = 𝑆(𝑥0, 𝑧0,𝜔), (1)
where p denotes frequency-domain wavefield, and S
denotes the source and (𝑥0, 𝑧0) is the source position. We
introduce a new finite-difference scheme for equation 1
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∆𝑥1(�̅�𝑚+1,𝑛 − �̅�𝑚,𝑛) − ∆𝑥2(�̅�𝑚,𝑛 − �̅�𝑚−1,𝑛)
∆𝑥1∆𝑥2(∆𝑥1 + ∆𝑥2)/2
+∆𝑧1(𝑝𝑚,𝑛+1 − 𝑝𝑚,𝑛) − ∆𝑧2(�̃�𝑚,𝑛 − 𝑝𝑚,𝑛−1)
∆𝑧1∆𝑧2(∆𝑧1 + ∆𝑧2)/2
+𝜔2
𝑣𝑚,𝑛2 ( ∑ 𝑐𝑖,𝑗𝑝𝑚+𝑖,𝑛+𝑗
1
𝑖,𝑗=−1
) = 𝑆,
(2)
where
�̅�𝑚+1,𝑛 =1−𝛼2
2𝑝𝑚+1,𝑛+1 +
𝛼1+𝛼2
2𝑝𝑚+1,𝑛 +
1−𝛼1
2𝑝𝑚+1,𝑛−1,
�̅�𝑚,𝑛 =1 − 𝛼2
2𝑝𝑚,𝑛+1 +
𝛼1 + 𝛼2
2𝑝𝑚,𝑛 +
1 − 𝛼1
2𝑝𝑚,𝑛−1,
�̅�𝑚−1,𝑛 =1−𝛼2
2𝑝𝑚−1,𝑛+1 +
𝛼1+𝛼2
2𝑝𝑚−1,𝑛 +
1−𝛼1
2𝑝𝑚−1,𝑛−1,
𝑝𝑚,𝑛+1 =1−𝛽2
2𝑝𝑚+1,𝑛+1 +
𝛽1+𝛽2
2𝑝𝑚,𝑛+1 +
1−𝛽1
2𝑝𝑚−1,𝑛+1,
𝑝𝑚,𝑛 =1 − 𝛽2
2𝑝𝑚+1,𝑛 +
𝛽1 + 𝛽2
2𝑝𝑚,𝑛 +
1 − 𝛽1
2𝑝𝑚−1,𝑛,
𝑝𝑚,𝑛−1 =1−𝛽2
2𝑝𝑚+1,𝑛−1 +
𝛽1+𝛽2
2𝑝𝑚,𝑛−1 +
1−𝛽1
2𝑝𝑚−1,𝑛−1,
(3)
where 𝛼1, 𝛼2, 𝛽1, 𝛽2 and 𝑐𝑖,𝑗 are weighted coefficients and
∑ 𝑐𝑖,𝑗1𝑖,𝑗=−1 = 1. Figure 1 is shown for more details.
It can be testified that the ADM scheme is a special case of
this scheme when the following constraints are fulfilled,
∆𝑥1 = ∆𝑥2, ∆𝑧1 = ∆𝑧2, 𝛼1 = 𝛼2, 𝛽1 = 𝛽2,
𝑐−1,−1 = 𝑐−1,1 = 𝑐1,−1 = 𝑐1,1,
𝑐−1,0 = 𝑐1,0 = 𝑐0,−1 = 𝑐0,1. (4)
Therefore, we call our new scheme as the generalized ADM
(GADM) for convenience.
Figure 1: Schematic of the GADM. Weights are shown by rows and
columns.
The coefficients in equation 2 are still to be determined. We
obtain the optimal weighted coefficients by minimizing the
numerical dispersion. We perform a plane wave analysis to
illustrate the numerical dispersion properties. A plane wave
that propagates in the mesh as Figure 1 shows can be written
as
𝑝𝑚,𝑛 = 𝑝0𝑒𝑥𝑝(−𝑖(𝑘𝑥𝑥 + 𝑘𝑧𝑧)),
𝑝𝑚,𝑛+1 = 𝑝0𝑒𝑥𝑝 (−𝑖(𝑘𝑥𝑥 + 𝑘𝑧(𝑧 + ∆𝑧2))),
𝑝𝑚,𝑛−1 = 𝑝0𝑒𝑥𝑝 (−𝑖(𝑘𝑥𝑥 + 𝑘𝑧(𝑧 − ∆𝑧1))),
𝑝𝑚+1,𝑛 = 𝑝0𝑒𝑥𝑝(−𝑖(𝑘𝑥(𝑥 + ∆𝑥2) + 𝑘𝑧𝑧)),
𝑝𝑚−1,𝑛 = 𝑝0𝑒𝑥𝑝(−𝑖(𝑘𝑥(𝑥 − ∆𝑥1) + 𝑘𝑧𝑧)),
𝑝𝑚−1,𝑛−1 = 𝑝0𝑒𝑥𝑝 (−𝑖(𝑘𝑥(𝑥 − ∆𝑥1) + 𝑘𝑧(𝑧 − ∆𝑧1))),
𝑝𝑚+1,𝑛−1 = 𝑝0𝑒𝑥𝑝 (−𝑖(𝑘𝑥(𝑥 + ∆𝑥2) + 𝑘𝑧(𝑧 − ∆𝑧1))),
𝑝𝑚−1,𝑛+1 = 𝑝0𝑒𝑥𝑝 (−𝑖(𝑘𝑥(𝑥 − ∆𝑥1) + 𝑘𝑧(𝑧 + ∆𝑧2))),
𝑝𝑚+1,𝑛+1 = 𝑝0𝑒𝑥𝑝 (−𝑖(𝑘𝑥(𝑥 + ∆𝑥2) + 𝑘𝑧(𝑧 + ∆𝑧2))),
(5)
where 𝑘𝑥 is the horizontal wavenumber, 𝑘𝑧 is the vertical
wavenumber and 𝑝0 is the wave amplitude.
Let
θ = arcsin(𝑘𝑥 𝑘⁄ ),
𝑟1 =𝐷𝑚𝑎𝑥
Δ𝑥1, 𝑟2 =
𝐷𝑚𝑎𝑥
Δ𝑥2, 𝑟3 =
𝐷𝑚𝑎𝑥
Δ𝑧1, 𝑟4 =
𝐷𝑚𝑎𝑥
Δ𝑧2,
G =2𝜋
𝑘𝐷𝑚𝑎𝑥, (6)
where θ is the angle between wavenumber direction and
vertical axis, 𝐷𝑚𝑎𝑥 is the maximal grid space in non-
uniform mesh and G is the number of grid points (respect to
maximal space) per wavelength.
Substituting equation 5 into equation 2 and supposing the
constant velocity 𝑣 , we achieve the normalized phase
velocity as
𝑣𝑝ℎ
𝑣=
𝐺
2𝜋√−
∑ 𝑏𝑖,𝑗𝑎𝑖,𝑗1𝑖,𝑗=−1
∑ 𝑐𝑖,𝑗𝑎𝑖,𝑗1𝑖,𝑗=−1
, (7)
where
[
𝑎−1,−1 𝑎0,−1 𝑎1,−1
𝑎−1,0 𝑎0,0 𝑎1,0
𝑎−1,1 𝑎0,1 𝑎1,1
]
=
[ 𝑐𝑜𝑠 (
2𝜋𝑠𝑖𝑛𝜃
𝐺𝑟1+
2𝜋𝑐𝑜𝑠𝜃
𝐺𝑟3) 𝑐𝑜𝑠 (
2𝜋𝑐𝑜𝑠𝜃
𝐺𝑟3) 𝑐𝑜𝑠 (
2𝜋𝑠𝑖𝑛𝜃
𝐺𝑟2−
2𝜋𝑐𝑜𝑠𝜃
𝐺𝑟3)
𝑐𝑜𝑠 (2𝜋𝑠𝑖𝑛𝜃
𝐺𝑟1) 1 𝑐𝑜𝑠 (
2𝜋𝑠𝑖𝑛𝜃
𝐺𝑟2)
𝑐𝑜𝑠 (2𝜋𝑠𝑖𝑛𝜃
𝐺𝑟1−
2𝜋𝑐𝑜𝑠𝜃
𝐺𝑟4) 𝑐𝑜𝑠 (
2𝜋𝑐𝑜𝑠𝜃
𝐺𝑟4) 𝑐𝑜𝑠 (
2𝜋𝑠𝑖𝑛𝜃
𝐺𝑟2+
2𝜋𝑐𝑜𝑠𝜃
𝐺𝑟4)]
,
(8)
[
𝑏−1,−1 𝑏0,−1 𝑏1,−1
𝑏−1,0 𝑏0,0 𝑏1,0
𝑏−1,1 𝑏0,1 𝑏1,1
]
=
[
1 − 𝛼1
1𝑟1
(1𝑟1
+1𝑟2
)+
1 − 𝛽1
1𝑟3
(1𝑟3
+1𝑟4
)
1 − 𝛼1
−1𝑟1
1𝑟2
+𝛽1 + 𝛽2
1𝑟3
(1𝑟3
+1𝑟4
)
1 − 𝛼1
1𝑟2
(1𝑟1
+1𝑟2
)+
1 − 𝛽2
1𝑟3
(1𝑟3
+1𝑟4
)
𝛼1 + 𝛼2
1𝑟1
(1𝑟1
+1𝑟2
)+
1 − 𝛽1
−1𝑟3
1𝑟4
𝛼1 + 𝛼2
−1𝑟1
1𝑟2
+𝛽1 + 𝛽2
−1𝑟3
1𝑟4
𝛼1 + 𝛼2
1𝑟2
(1𝑟1
+1𝑟2
)+
1 − 𝛽2
−1𝑟3
1𝑟4
1 − 𝛼2
1𝑟1
(1𝑟1
+1𝑟2
)+
1 − 𝛽1
1𝑟4
(1𝑟3
+1𝑟4
)
1 − 𝛼2
−1𝑟1
1𝑟2
+𝛽1 + 𝛽2
1𝑟4
(1𝑟3
+1𝑟4
)
1 − 𝛼2
1𝑟2
(1𝑟1
+1𝑟2
)+
1 − 𝛽2
1𝑟4
(1𝑟3
+1𝑟4
)]
.
(9)
Consequently, the weighted coefficients are determined by
minimizing the phase error
E(𝛼1, 𝛼2, 𝛽1, 𝛽2, 𝑐𝑖,𝑗 ) =
∬(1 −𝑣𝑝ℎ(𝜃,�̃�;𝛼1,𝛼2,𝛽1,𝛽2,𝑐𝑖,𝑗)
𝑣)2
𝑑�̃�𝑑𝜃, (10)
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where
�̃� =1
𝐺. (11)
�̃� varies from 0 to 0.25 and θ varies from 0 to 𝜋 2⁄ , respect-
ively.
We employ a constrained nonlinear optimization approach
to achieve the optimal coefficients. Different grid space
ratios (i.e., different 𝑟1, 𝑟2, 𝑟3 and 𝑟4 ) lead to different
optimal coefficients. In this study, 𝑟𝑖 (𝑖 = 1,2,3,4) varies
from 1 to 5 with an interval of 0.5. Therefore, a “dictionary”
composed by 9 × 9 × 9 × 9 groups of optimal coefficients
is obtained. For every discrete point, we can select the
corresponding coefficients from the “dictionary” according
to the ratio of maximal grid space and four girds spaces
around it.
Here we choose four groups of optimal coefficients among
the huge “dictionary” randomly. The normalized phase
velocity curves are illustrated in Figure 2 and their
corresponding grid spaces ratios is shown in Table 1. Due
to the limited space, values of the four groups of optimal
coefficients are not shown. From Figure 2 and Table 1, we
can observe that the phase error is influenced slightly by
different grid spaces ratios. Nevertheless, for all grid spaces
ratios, the phase errors in different directions are within
±0.5% even if the number of grid points is small as four.
Table 1: Grid spaces ratios corresponding to four groups of optimal
coefficients chosen randomly.
𝒓𝟏 𝒓𝟐 𝒓𝟑 𝒓𝟒
a) 1.5 1.5 2.0 3.0
b) 5.0 1.5 5.0 4.5
c) 3.5 4.0 1.5 5.0
d) 3.0 5.0 2.0 3.5
Numerical examples
We verify the GADM scheme by a simple two-layer model
(Figure 3). The size of this model is 3000m in x direction
and 1600m in z direction. The shot position is at 1500m on
the surface. The source is Ricker wavelet of which the
dominant frequency is 20Hz and the frequency bandwidth
is [5Hz, 60Hz]. The hybrid absorbing boundary conditions
introduced by Moreira et al. (2014) is appended around the
model area.
Figure 3: The two-layer velocity model. The upper layer is 2000m/s and the below layer is 3000m/s.
a)
b)
c)
d)
Figure 2: Normalized phase velocity curves defined by four groups
of optimal coefficients chosen randomly. Numerical phase velocity
𝑣𝑝ℎ is normalized with respect to the true velocity 𝑣 and plotted
versus 1 𝐺⁄ . Different colors of curves denote different directions of wavenumber.
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The grid space in the GADM scheme is non-uniform as
Figure 4 shows. The grid space in z direction (dz) varies
with depth and the grid space in x direction (dx) varies with
distance, respectively. dz is 6.25m in the upper area and
7.5m in the lower area. dz increases gradually from 6.25m
to 7.5m with an increment of 0.1m in Area 1. dx is 6.25m in
global model area except in Area 2. In Area 2, dx increases
from 6.25m to 7.5m with an increment of 0.1m and then
decreases from 7.5m to 6.25m with an increment of -0.1m.
In Area 3, dx varies in the same way as in Area 2 and dz
varies in the same way as in Area 1. The snapshot at 0.5s is
illustrated in Figure 5a.
Figure 4: The schematic of non-uniform mesh used in the GADM
scheme.
In order to test validity of the GADM scheme, an
experiment with the ADM is performed simultaneously.
The grid space in whole model area in this experiment is
dx = dz = 6.25m. Except that meshes are different, all the
other parameters in two experiments are coincident. The
snapshot at 0.5s is presented in Figure 5b. Compared to
modeling result of the ADM, the result of the GADM
scheme is satisfactory.
We compare the memory and time cost in two experiments
and the results are shown in Table2. From Table 2, we can
realize that the time and memory cost in the GADM scheme
is less than in the ADM scheme. The GADM scheme is
more efficient than the ADM scheme.
Table 2: Cost comparison between two methods
The GADM scheme The ADM scheme
Memory cost/G 0.54 0.65
Time cost/s 227 281
Conclusions
We propose a new finite-difference scheme for frequency-
domain propagator on non-uniform grids. Its validity and
efficiency have been certified through a simple numerical
example. Although complex model tests have not been
implemented so far, it’s promising to be applied on more
non-uniform problems. The method can be easily extended
to higher order and higher dimension form.
Acknowledgments
This study received support from National Natural Science
Foundation of China (41374006, 41274117).
a)
b)
Figure 5: Snapshot at 0.5s in two experiments a) the GADM scheme
b) The ADM scheme
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