research on wave field numerical simulation of high order

American Journal of Earth Sciences 2018; 5(1): 10-18

http://www.openscienceonline.com/journal/ajes

ISSN: 2381-4624 (Print); ISSN: 2381-4632 (Online)

Research on Wave Field Numerical Simulation of High Order Finite Difference in Multi-Scale Grid Wave Equations

Zhang Xiaodan1, 2

, She Yichong1, Liu Guizhong

2, Zhang Zhiyu

3, Zhu Lei

1

1College of Electronic and Information, Xi`an Polytechnic University, Xi’an, China 2College of Electronics and Information Engineering (College of Microelectronics), Xi`an Jiaotong University, Xi’an, China 3College of Automation and Information engineering, Xi'an University of Technology, Xi’an, China

Email address

To cite this article Zhang Xiaodan, She Yichong, Liu Guizhong, Zhang Zhiyu, Zhu Lei. Research on Wave Field Numerical Simulation of High Order Finite

Difference in Multi-Scale Grid Wave Equations. American Journal of Earth Sciences. Vol. 5, No. 1, 2018, pp. 10-18.

Received: March 27, 2018; Accepted: April 10, 2018; Published: May 29, 2018

Abstract

In the numerical simulation of seismic wave field, the problem of how to ensure both high efficiency and precision has always

been one of the hot spots of seismic exploration scholars. The traditional method used the constant small step length in finite

difference, which greatly reduces the calculation efficiency. A method of adopt different scale grids according to the

characteristics of the geological model and optimize the transition zone has been proposed. firstly, analysis the speed model of

the research object to determine the scale of grid; secondly, determine the scope of the transition zone; finally, calculate the

coefficient and the differential points of both inside and outside the transition zone, gain the wave field value of every grid

point of the model. According to the experimental results in the paper, the calculation efficiency of multi-scale grid method can

be improved obviously, and the case’s results of this article can as high as 25.16% in average.

Keywords

Multi-scale Grid, Step Length, Transition Zone, Wave Field Simulation, Computational Efficiency

1. Introduction

There are the problems of to explore the more complex

geologic and deeper regions on the surface which exist in

modern oil and gas exploration, it makes high requirements

for the collection, processing and interpretation of the seismic

data under complex conditions. The numerical simulation

method is widely used because of its economy and accuracy, it

contains finite difference method, pseudo-spectral method and

finite element method, Finite difference method are most

popular one in this three methods with its advantages of easy

implement, high calculation speed and high simulation results.

[1-3] Traditional finite difference method use a constant scale

grid to discretization the whole model area, [4-6] it act well in

the simple geologic body, but it is not suitable for the region of

the complex geologic body and the lithology structure,

because the medium distribution is highly uneven in this kind

of areas, and the medium usually contains low-speed

interlayer, it calls for improve the simulation accuracy, restrain

the dispersion and ensure the stability with Smaller sampling

intervals. [7-9] However, there will have more grid points to

calculation if the whole model area are smaller subdivided,

and the computation will be multiplied, which greatly reduces

the simulation efficiency, what’s more, it, ll oversampling the

high-speed layer.

Therefore, in order to solve the above problem, scholars

have done a lot of researches, some of the scholars adopt the

method of improve the difference order to avoid dispersion:,

Li Bin using high precision difference in both time and space

domains, [10] it has improved the precision of forward

modeling, Yue Xiaopeng use the space 2N order, time 4 order

difference accuracy for forward modeling. [11-12] Some other

scholars from the perspective of the staggered grid, Wei

Zhong Wang proposed the multi-scale rotating staggered grids,

successfully reduced the computing time, [13] Hongyong puts

American Journal of Earth Sciences 2018; 5(1): 10-18 11

forward a optimal grid finite difference by make a

combination of minimax approximation and Taylor series

expansion, [14] Wang Jianmake improvement of staggered

grid algorithm by cosine function modified binomial window,

effectively controlled of the numerical frequency dispersion;

[15-16] And the others have taken a different approach, Ma

Jihaoand Li Yusheng make a combination of different seismic

data for the forward modeling, [17-18] Liang Quanwen

determine the finite difference coefficient of the new template

by improved linear method, and improve the calculation

accuracy. [19-20] All the algorithms above are based on the

grid of constant size, so some scholars chose to solve the

problem by change the step length of grid: Moczo firstly use

the grid continuous variation method to reduce the amount of

calculation and improve the efficiency of the numerical

simulation, [21-22] in 1994, Jastram and Behle propose a

two-dimensional wave equation for variable step length grid

of a certain depth algorithm, [23] Wang and Schuster apply the

variable grid method into elastic wave equation, and using the

double step length on the grid boundary, [24] zhang

Jianfengpropose a irregular grid difference method for

crisscross calculate the stress and velocity base on the

stress-elastic wave equations, [25] Huang Chao and Dong

liangguo proposes a high order finite difference simulation

method which spatial grid size and time step can be change

freely, [26] it is based on technology of combination of the

alterable space grid and time step. The above algorithms have

greatly improve the computational efficiency and storage

space usage, but there is a transition zone which exist in the

neighbourhood of the grid step mutations area while

calculation, some of the algorithms utilize interpolation in the

transition area to solve this problem, it will increase the

instability of the algorithm; the others smooth the transition

zone area by smooth function, but it makes the algorithms

more difficult to implement. Therefore, a multi-scale grid

algorithm without interpolation in transition zone has been

presented in this paper, which can effectively solve the above

problems.

2. Wave Equation Difference Scheme

Using the 2-dimentions wave equation for example, it’s

equation of the propagation underground medium can be

expressed as follows:

2 2 2

2 2 2 2

1( , , )

( , )

u u us x z t

x z V x z t

∂ ∂ ∂+ = +∂ ∂ ∂

(1)

Where u is the wave filed velocity function, ),( yxV is

the velocity of the propagation of longitudinal wave in

medium. ),,( zyxs is the seismic focus function, It is usually

a Ricker wavelet with a frequency of 20-40Hz. The x , z and

t items of formula (1) are respectively carried out in N order

Taylor series expansion, and the 2N order difference format as

formula (2) can be obtained after sorting:

22 2

2

122 2

2

1

( , , 1) 2 ( , , ) ( , , 1)

[ ( , , ) 2 ( , , ) ( , , )]

t[ ( , , ) 2 ( , , ) ( , , )]

z

N

n

nN

n

n

u i j k u i j k u i j k

t VC u i n j k u i j k u i n j k

x

VC u i j n k u i j k u i j n k

=

=

+ = − −∆+ + − + −∆

∆+ + − + −∆

∑

∑

(2)

Where, nC is the difference coefficient, it’s value can be

obtained by Taylor series expansion

3. Perfectly Marched Layer (PML) and

Numerical Dispersion

Because the existed of the artificial boundary, It is

necessary to solve the problem of the reflection of the

seismic wave, absorbing boundary condition is one of the

widest used solutions, Bérenger proposed an efficient

absorbing boundary condition named Perfectly matching

layer (PML), [27] It is a special medium layer by truncating

the boundary in the finite different time domain (FDTD)

region, The wave impedance of the medium is perfectly

matched with the wave impedance of the adjacent medium,

set the appropriate parameters, the incident wave from any

conditions will pass through the interface without reflection

and enter PML. PML has been widely applied in seismic

modeling since it was proposed, and scholars has also make

it a great development in practical application: Haiqiang

Lan utilize PML in simulating seismic wave propagation in

elastic media with an irregular free surface; [28-30] Yingjie

Gao compare the 3 kinds of the boundary conditions’

absorbing performance via theoretical analyses and

numerical experiments; [31] Weijuan Meng and Li-Yun Fu

make a Seismic wavefield simulation by a modified finite

element method with a perfectly matched layer absorbing

boundary; [32] Michael Brun proposed a hybrid

Asynchronous Perfectly Matched Layer for seismic wave

propagation in unbounded domains. [33-34]

In the process of finite difference numerical simulation, the

most common problem is the numerical dispersion. It is

mainly due to the numerical error which occurs in process of

the difference operator approximates the differential operator

in the numerical calculation, It would strongly affect the

accuracy of Numerical simulation. The main factors which

play roles on the numerical dispersion include Precision of

finite difference operator, Sampling points in a single

wavelength, time sampling interval, the source frequency and

The incident Angle of the seismic wave.

It is an effective method to suppress dispersion that improve

the approximate degree of the wave equation by using the

higher order finite difference algorithm and reduce the time

12 Zhang Xiaodan et al.: Research on Wave Field Numerical Simulation of High Order Finite Difference in

Multi-Scale Grid Wave Equations

and space sampling step length. Generally speaking, the

difference operator with more than eight order precision can

reach the requirement of reduce numerical dispersion, and the

space sampling step length should no more than the min value

required in formula (3):

min

max

0.8

10≤ V

hf

(3)

where h is the max step length value of both x and z

directions, vmin is the littlest velocity in different mediums

which contain in the model f max is the max frequency of the

seismic focus. For keep the calculation efficiency, we usually

choice the space step length as large as we can under the

premise of the accuracy of numerical simulation.

4. Multi-scale Grid Finite Difference

Algorithm

It is usually using small step grid to sampling the

low-velocity layer and using a large step grid for high speed

sampling in the transverse isotropic medium, in this way, it

can effectively reduce the time needed for calculation and

improve the computational efficiency under the premise of

guarantee the calculation precision, the transition zone

appears in a symmetric N-1 grids with the center of the

velocity interface while perform the N-order difference.

As shown in figure 1, figure zero as a transitional zone in the

diagram, where black dot z1 and z2 is demarcation point between

transition zone and conventional grid, the 9 points represent the

transition zone, the low-velocity layer adopt small step sampling,

with dense dots in figure, and the sparse part represents the high

speed layer, sampling by double step length.

Figure 1. Differential schematic diagram of transition zone between high - speed and low - velocity layer.

Suppose the velocities are moderately changed or the same

in anywhere in x direction of the model, Sampling with

constant step size x∆ , conventional finite difference method

can reach the requirement of the calculation on the derivative

of x direction, but the step length of grid in the z direction is

discontinuous, therefore, conventional method is no longer

applicable there, a multi-scale difference algorithm of the

transition zone was presented as flowing:

The 1th column to 11th column in figure 1 shows the

transformation rule of difference points in the transition zone

from high velocity layer to low velocity layer, the red and

black dots represent the derivative points, dots in light color

represent the point which need to be calculated when

calculating the transition zone, choice time second order and


space 10th order difference accuracy, the 1st column shows

the situation of the difference point in the region of the high

speed layer but not entered the transition zone, the difference

points and the wave field values as is shown in formula (4):

)]2([

}][][

][][][{2

,

5

1

,,2

22

,01,1,12,2,2

3,3,34,4,45,5,52

221

,,

1

,

k

ji

k

k

jmi

k

jmik

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

uuugx

tV

uauuauua

uuauuauuaz

tVuuu

−+∆

∆+

+++++

+++++∆

∆+−=

∑=

−+

−+−+

−+−+−+−+

(4)

Where )5,1,0( ⋯=iai is the difference coefficient, the

numbers of foregoing and subsequent items of the difference

scheme is symmetric with the derivation point.

As is shown in the 2nd column in figure 1, the derivation

point get into the transition zone. The numbers of difference

points on both sides of the derivation point are no longer

symmetrical, and the difference coefficient were changed as

well, set it’s difference coefficient as )5,1,0( ⋯=ibi , the

wave field values as is shown in formula (5). The

transformation law of difference points and wave field values

of the 3rd to 5th column as is shown in formula (6) to formula

(8), and the difference coefficients are respectively ii dc , and

ie , the derivation point enters the low-speed layer after the 6th

column, and the sampling step changes to half time as the

former one. The wave field value of the 6th column is shown

in formula (9), Then the difference coefficients of the 7th to

10th column are contrary with the former’s, respectively is

iii cde ,, and ib , and the transformation rule of difference

points and wave field values shown in formula (10) to (13),

after the 11th column, the derivation point leave the transition

zone, and it’s wave field can calculated by formula (4) with the

sampling step length changed.

)]2([

}][][

][][][{2

,

5

1

,,2

22

,01,1,12,2,2

3,3,34,4,45,6,52

221

,,

1

,

k

ji

k

k

jmi

k

jmik

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

uuugx

tV

ubuubuub

uubuubuubz

tVuuu

−+∆

∆+

+++++

+++++∆

∆+−=

∑=

−+

−+−+

−+−+−+−+

(5)

)]2([

}][][

][][][{2

,

5

1

,,2

22

,01,1,12,2,2

3,3,34,5,45,7,52

221

,,

1

,

k

ji

k

k

jmi

k

jmik

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

uuugx

tV

ucuucuuc

uucuucuucz

tVuuu

−+∆

∆+

+++++

+++++∆

∆+−=

∑=

−+

−+−+

−+−+−+−+

(6)

)]2([

}][][

][][][{2

,

5

1

,,2

22

,01,1,12,2,2

3,4,34,6,45,8,52

221

,,

1

,

k

ji

k

k

jmi

k

jmik

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

uuugx

tV

uduuduud

uuduuduudz

tVuuu

−+∆

∆+

+++++

+++++∆

∆+−=

∑=

−+

−+−+

−+−+−+−+

(7)

)]2([

}][][

][][][{2

,

5

1

,,2

22

,01,1,12,3,2

3,5,34,7,45,9,52

221

,,

1

,

k

ji

k

k

jmi

k

jmik

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

uuugx

tV

ueuueuue

uueuueuuez

tVuuu

−+∆

∆+

+++++

+++++∆

∆+−=

∑=

−+

−+−+

−+−+−+−+

(8)



)]2([

}][][

][][][{2

,

5

1

,,2

22

,01,2,12,4,2

3,6,34,8,45,10,52

221

,,

1

,

k

ji

k

k

jmi

k

jmik

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

uuugx

tV

uauuauua

uuauuauuaz

tVuuu

−+∆

∆+

+++++

+++++∆

∆+−=

∑=

−+

−+−+

−+−+−+−+

(9)

)]2([

}][][

][][][{)*5.0(

2

,

5

1

,,2

22

,01,1,12,3,2

3,5,34,7,45,9,52

221

,,

1

,

k

ji

k

k

jmi

k

jmik

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

uuugx

tV

ueuueuue

uueuueuuez

tVuuu

−+∆

∆+

+++++

+++++∆

∆+−=

∑=

−+

−+−+

−+−+−+−+

(10)

)]2([

}][][

][][][{)*5.0(

2

,

5

1

,,2

22

,01,1,12,2,2

3,4,34,6,45,8,52

221

,,

1

,

k

ji

k

k

jmi

k

jmik

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

uuugx

tV

uduuduud

uuduuduudz

tVuuu

−+∆

∆+

+++++

+++++∆

∆+−=

∑=

−+

−+−+

−+−+−+−+

(11)

)]2([

}][][

][][][{)*5.0(

2

,

5

1

,,2

22

,01,1,12,2,2

3,3,34,5,45,7,52

221

,,

1

,

k

ji

k

k

jmi

k

jmik

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

uuugx

tV

ucuucuuc

uucuucuucz

tVuuu

−+∆

∆+

+++++

+++++∆

∆+−=

∑=

−+

−+−+

−+−+−+−+

(12)

)]2([

}][][

][][][{)*5.0(

2

,

5

1

,,2

22

,01,1,12,2,2

3,3,34,4,45,6,52

221

,,

1

,

k

ji

k

k

jmi

k

jmik

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

uuugx

tV

ubuubuub

uubuubuubz

tVuuu

−+∆

∆+

+++++

+++++∆

∆+−=

∑=

−+

−+−+

−+−+−+−+

(13)

The rule that the difference coefficient and the

transformation of the difference points to calculate the

transition zone’s change with the continuous advance of the

derivation point are summarized as follows: let

jiUzzxxU ,),( =∆+∆+ , where x∆ and z∆ is the space

sampling step length, Then, the second order derivative of z

direction can be approximate to Nth order difference by using

symmetrical finite difference coefficient:

∑=

−+ +−∆

=∂∂ N

m

mjijimjim UUUazz

U

1

,,,22

2

)2(1

(14)

And the corresponding two-step grid difference can be

defined as:

∑=

−+ +−∆

=∂∂ N

m

mjijimjim UUUazz

U

1

2,,2,22

2

)2()2(

1 (15)

Model of high speed layer to low speed layer as is shown in

figure 1, use double step grid sampling the area from surface

to depth of zn∆ , the rest part sampled by single length step,

then, in a finite difference method with the length of 2N,

formula (15) is applicable to the area from surface to

zNn ∆− )( , formula (14) was utilized in the depth under

zn∆ , the derivative of the rest 1−N points in the central can

be calculated by united formula (14) and (15), the derivative

of the grid points in depth of zcNn ∆+− )( can computed by

following formula (16):


∑

∑

=−−++

−+

−

=

+−∆

+

+−∆

=∂∂

N

m

cmjijicmjim

mjijimji

cN

m

m

UUUaz

UUUazz

U

1

)(2,,)(2,2

,,,

122

2

)2(1

)2()2(

1

(16)

Summarize a general formula for transitional zone from the

above formulas, and a difference point change function )(mc

is introduced, The transition zone’s difference points

transformation rule in 2N order accuracy as follow::

∑=

−+ +−∆

=∂∂ N

m

mcjijimcjim UUUazz

U

1

)(,,)(,22

2

)2(1

(17)

Where ma represents the difference coefficient factors

)5,1,0(,,,, ⋯=iedcba iiiii , it can be obtained by Taylor series

expansion, and the ma ’s value in tenth order difference

accuracy have 7 valid digits as is shown in table 1. the

)(mc ’s value in tenth order finite difference precision was

shown in table 2.

Table 1. The ma ’s values in tenth order difference accuracy.

a b c d e

0 -2.927222 -2.902777 -2.843083 -2.711806 -2.367730

1 1.666666 1.645714 1.595052 1.486077 1.121124

2 -0.238095 -0.225000 -0.194444 -0.133333 -0.029907

3 0.039682 0.033862 0.021267 0.003333 0.02769

4 -0.004960 -0.003214 -0.000365 -0.000181 -0.000252

5 0.000317 0.000027 0.000009 0.000007 0.000013

Table 2. The ( )c m ’s values in tenth order difference accuracy.

a b c d e f

c (0) 0 0 0 0 0 0

c (1) 1 1 1 1 1 1

c (2) 2 2 2 2 2 3

c (3) 3 3 3 3 4 5

c (4) 4 4 5 6 7 8

c (5) 5 6 7 8 9 10

Meanwhile, the transformation rule of difference points and

wave field value of the transition zone in the situation of from

low velocity layer to high velocity layer are very similar to the

former condition’s, but the absolutely opposite. It can be

backward derived from figure 1.

5. Example and Result Analysis

The computer used to modeling has a 4G - Byte memory

and Intel Core i5-2430 for processor, the seismic focus is

Ricker wavelet which has a frequency of 30Hz, using second

order in time and tenth order in space difference accuracy to

do numerical simulation, the model respectively is layered

velocity model and step speed model.

5.1. Layered Velocity Model

The layered velocity model as is shown in figure 2. The

model’s size is mm 200100 × , and it has two different velocity

layers, the lower speed layer’s velocity is sm /1200 , the

velocity of the second layer is sm /2400 , the velocity

interface locate in the line of z=100.

Small grid step size is adopted at longitudinal of low speed

area and large grid step size is adopted at longitudinal of high

speed area. The small grid sampling step length is

mdzmdx 5.2,5 == , The large grid sampling step length is

mdzmdx 5,5 == , and the shot point’s coordinate is (50m,

75m).

Time sampling interval is 1 ms and the wave field snapshot

of t=30 ms as is shown in figure 3, the wave shape is clear and

there is no numerical dispersion in transformation zone, the

simulation result shows that the method can effectively

simulate the wave propagation in layered medium.

Figure 2. Layered velocity model.



Figure 3. T=30ms Wave field snapshot.

The comparison of computing time and storage space of 2

algorithms as is shown in table 3, it is obvious that, in layer

velocity model, multi-scale finite difference method can

effectively improve the calculation efficiency and save storage

space under the condition of the simulation precision.

Table 3. The efficiency ratio of the 2 algorithms.

computing time Storage space

Traditional method 1’30”18’” 231kb

This paper's method 1’8”58’” 185kb

Efficiency promotion 32% 22%

5.2. Step Speed Model

Figure 4 shows the step velocity model, the model size ismm 200100 × , the stage is in the point of the horizontal

ordinate is x=50 m and the vertical ordinate is z=125 m, the

height of the stage is 25m, it divided the whole model in to two

part with different speed, the velocity of first layer is sm /1200 ,

the velocity of the second layer is sm /2400 .

The first layer adopts the small grid step size, the sampling

step length is mdzmdx 5.2,5 == , the second layer adopts the

large grid step size. the large grid step is mdzmdx 5,5 == ,

the shot point coordinates is (50m, 75m), and calculation step

is 1ms. The numerical simulation results of t=20 ms as is

shown in figure 5.

The result shows that the waveform is correct in the left step

of the model, and there is no numerical dispersion in the

velocity interface, it is tested that the multi-scale grid finite

difference algorithm can effectively reduce numerical

dispersion, the comparison of the required storage space and

calculation efficiency with the multi-scale method and

traditional difference method as is shown table 4.

Figure 4. Step speed model.


Figure 5. T=20ms Wave field snapshot.

Table 4. The efficiency ratio of the two algorithms.

computing time Storage space

Traditional method 1’30”18’’’ 234.21kb

This paper's method 1’18’’25’’’ 198.56kb

Efficiency promotion 19.4% 17.96%

6. Conclusion

By experimental verification, due to the utilization of the small

step length to sampling the low speed region, and a symmetrical

finite difference method was introduced to the transition zone,

compared with the traditional finite difference method, the

multi-scale grid step finite-difference reduces the calculation

time by 25.16% in average, and the average saving storage

space ratio is 21.89%. Therefore, multi-scale grid step finite

difference algorithm has the following features: make the model

with small grid step sampling need less memory storage space;

obviously improve the calculation efficiency in layered speed

model and step speed model, avoid using interpolate in the

transition zone, which effectively reduce the accumulated error

caused by interpolation, and it can effectively reduce on the

numerical simulation of the layered velocity layer and step speed

layer.

Acknowledgements

The research work was supported in part by a grant from the

Natural Science Foundation of China (No. 61401347), a grant

from Natural Science Foundation of Shaanxi province

department of education of China under Grant (No.

16JK1322); Xi’an Polytechnic University Doctor Scientific

Research foundation of China under Grant (No. BS1118).

References

[1] Tian Renfei, Cao Junxing, The efficiency improvement of the pseudo-spectral method and the simulation of the cross-well seismic wave modeling [J]. Geophysical and Geochemical Exploration: 2008, 4: 430-433.

[2] Li Lin, Liu Tao, Hu Tianyue, Spectral element method with triangular mesh and its application in seismic modeling [J]. Chinese Journal Of Geophysics: 2014, 4: 1224-1234.

[3] DU Huakun, FENG Deshan, GPR simulation by finite element method of unstructured grid based on Delaunay triangulation [J]. Journal of Central South University (Science and Technology), 2015, 46 (4): 1326-1334.

[4] Lian Ximeng, Shan Lianyu, Sui Zhiqiang, An overview of research on perfectly matched layers absorbing boundary condition of seismic forward numerical simulation [J]. Progress In Geophysics: 2015, 30 (4): 1725-1733 1004-2903.

[5] Zhang Geng, Tuo Xianguo, Zhang Yi, Forward modeling study on data acquisition parameters of shallow seismic reflection method [J]. Geotechnical Investigation&Surveying: 2015, (1): 93-98, 1000-1433.

[6] Huang Jianping, Yang Jidong, Li Zhenchun, Gaussian beam seismic forward modeling for 3 dimensional undulating surface under the Effective neighborhood wave field approximation framework [J]. Oil Geophysical Prospecting: 2015, 50 (5) P896-904, 804 1000-7210.



[7] Jian-Ping Huang; Ying-Ming Qu, Variable-coordinate forward modeling of irregular surface based on dual-variable grid [J]. Applied Geophysics: 2015, 12 (1): 101-110.

[8] Bai, Chao-ying; Hu, Guang-yi, Seismic wavefield propagation in 2D anisotropic media: Ray theory versus wave-equation simulation [J]. Journal of Applied Geophysics: 2014, 104: 167-171.

[9] Cai Xiao-Hui, Liu Yang, Ren Zhi-Ming, Three—dimensional acoustic wave equation modeling based on the optimal finite-difference scheme [J]. APPLIED GEOPHYSICS: 2015, 12 (3): 409—420.

[10] Li Bin, Wen Mingming, Mu Zelin, Staggered-grid Finite-difference Method with High-order Accuracy in Time-space Domains for Acoustic Forward Modeling [J]. Computerized Tomography Theory And Application: 2017, 3: 259-266.

[11] Yue Xiaopeng, Bai Chaoyin, Yue Chongwang, Accuracy analysis of elastic wave field simulation based on high-order staggered grid finite difference scheme [J]. Coal Geology & Exploration: 2017, 1, 125-130.

[12] X Yue; C Yue, Simulation of acoustic wave propagation in a borehole surrounded by cracked media using a finite difference method based on Hudson’s approach [J]. Journal of Geophysics and Engineering. 2017, 14: 633–640.

[13] Wei-Zhong Wang, Tian-Yue Hu, x, Varia-ble-order rotated staggered-grid method for elastic-wave forward modeling [J]. Applied Geophysics: 2015, 12 (3): 389-400.

[14] Hongyong Yan, Lei Yang, Seismic modeling with an optimal staggered-grid finite-difference scheme based on combining Taylor-series expansion and minimax approximation [J]. Studia Geophysica et Geodaetica: 2017, 61 (3): 560-574.

[15] Wang Jian, Meng Xiaohong, Liu Hong, Cosine-modulated window function-based staggered-grid finite-difference forward modeling [J]. Applied Geophysics: 2017, 1: 115-124.

[16] Wang Jian; Meng Xiaohong; Liu Hong; Optimization of finite difference forward modeling for elastic waves based on optimum combined window functions [J]. Journal of Applied Geophysics, 2017, 138: 62-71.

[17] Feng Jihao, Cao Danping, Qin Haixu, Analysis of reflection characteristics of multi-scale seismic data based on the wave equation forward modeling [J]. Progress In Geophysics: 2016 (3): 1058-1065 1004-2903.

[18] Li Yuesheng, Wu Guochen, Seismic modeling in three-dimensional monoclinic fractured media [J]. Progress In Geophysics: 2016, 2: 525-536 1004-2903.

[19] Liang Wenquan, Liao Yongming, Jiang Lin, Acoustic Wave Equation Modeling with a New Time-Space Domain Finite Difference Stencil and an Improved Linear Algorithm [J]. China Earthquake Engineering Journal: 2016, 5: 815-821.

[20] Wang, Yanfei; Liang, Wenquan; Nashed, Zuhair; Determination of finite difference coefficients for the acoustic wave equation using regularized least-squares inversion [J]. J. Inverse Ill-Posed Probl, 2016, 24 (6): 743-760.

[21] Peter Moczo, Finite-difference technique for SH-waves in 2-D media using irregular grids-application to the seismic response problem [J]. Geophysical Journal International: 1989, 99: 321-329.

[22] Emmanuel Chaljub; Emeline Maufroy; Peter Moczo; 3-D numerical simulations of earthquake ground motion in sedimentary basins: testing accuracy through stringent models [J]. Geophysical Journal International, 2015, 201 (1): 90-111.

[23] Jastram C, Tessmer E. Elastic modeling on a grid with vertically varying spacing [J]. Geophysics Prospecting: 1994, 42 (4): 357-370.

[24] Wang Y, and Schuster G T. Finite difference variable grid scheme for acoustic and elastic wave equation modeling [C]. 1996 SEG annual meeting., Soc. Expl. Geophysics Expanded Abstracted: 1996, 674-677.

[25] Zhang Jianfeng, Non-regular grid difference method for elastic wave numerical simulation. [J]. Chinese Journal Of Geophysics 1998, 41 (supplementary issue): 357-366.

[26] Dong Liangguo, Li Peiming, Dispersive Problem in Seismic Wave Propagation Numerical Modeling [J]. Nature Gas Industry, 2004, 24 (6): 53-56.

[27] Bérenger JP. A perfectly matched layer for the absorption of electromagnetic waves [J]. Journal of computational physics, 1994, 114 (2): 185. 200.

[28] Haiqiang Lan, Application of a perfectly matched layer in seismic wave field simulation with an irregular free surface [J]. Geophysical Prospecting, 2016, 64 (1): 112-128.

[29] Lan, Haiqiang; Zhang, Zhongjie, Comparative study of the free-surface boundary condition in two-dimensional finite-difference elastic wave field simulation [J]. Journal of Geophysics and Engineering, 2011, 8 (2): 275-286.

[30] Haiqiang Lan and Ling Chen, An upwind fast sweeping scheme for calculating seismic wave first-arrival traveltimes for models with an irregular free surface [J]. Geophysical Prospecting, 2017, DOI: 10.1111/1365-2478.12513.

[31] Yingjie Gao; Hanjie Song; Jinhai Zhang, Comparison of artificial absorbing boundaries for acoustic wave equation modelling [J]. Exploration Geophysics, 2017, 48 (1) 76: 93.

[32] Weijuan Meng and Li-Yun Fu, Seismic wavefield simulation by a modified finite element method with a perfectly matched layer absorbing boundary [J]. Journal of Geophysics and Engineering, 2017, 14 (4): 852.

[33] Michael Brun; Eliass Zafati; Irini Djeran-Maigre; Florent Prunier, Hybrid Asynchronous Perfectly Matched Layer for seismic wave propagation in unbounded domains [J]. Finite Elements in Analysis and Design, 2016, 122: 1-15.

[34] Michael Brun Jensen; José Alberto Manresa; Ken Steffen Frahm; Analysis of muscle fiber conduction velocity enables reliable detection of surface EMG crosstalk during detection of nociceptive withdrawal reflexes [J]. BMC neuroscience, 2013, 14: 39, doi: 10.1186/1471-2202-14-39.

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