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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