research on wave field numerical simulation of high order
TRANSCRIPT
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
American Journal of Earth Sciences 2018; 5(1): 10-18 13
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)
14 Zhang Xiaodan et al.: Research on Wave Field Numerical Simulation of High Order Finite Difference in
Multi-Scale Grid Wave Equations
)]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):
American Journal of Earth Sciences 2018; 5(1): 10-18 15
∑
∑
=−−++
−+
−
=
+−∆
+
+−∆
=∂∂
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.
16 Zhang Xiaodan et al.: Research on Wave Field Numerical Simulation of High Order Finite Difference in
Multi-Scale Grid Wave Equations
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.
American Journal of Earth Sciences 2018; 5(1): 10-18 17
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).
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