Journal of Geophysics and Engineering

PAPER

A frequency-domain element-free method forseismic modeling and reverse-time migrationTo cite this article: Xiaofeng Jia and Xiaodong Qiang 2018 J. Geophys. Eng. 15 1719

 

View the article online for updates and enhancements.

Related contentElement-free method and its efficiencyimprovement in seismic modelling andreverse time migrationZhibin Fan and Xiaofeng Jia

-

The small-scale forward modeling methodfor large models based on the waveequationYitao Pu, Xingyao Yin, Danping Cao et al.

-

Adaptive 9-point frequency-domain finitedifference scheme for wavefield modelingof 2D acoustic wave equationWenhao Xu and Jinghuai Gao

-

This content was downloaded from IP address 222.195.74.144 on 22/05/2018 at 05:04

A frequency-domain element-free methodfor seismic modeling and reverse-timemigration

Xiaofeng Jia1,2 and Xiaodong Qiang1,2

1 Laboratory of Seismology and Physics of Earths Interior, School of Earth and Space Sciences, Universityof Science and Technology of China, Hefei 230026, Peopleʼs Republic of China2National Geophysical Observatory at Mengcheng, University of Science and Technology of China,Anhui, Peopleʼs Republic of China

E-mail: xjia@ustc.edu.cn

Received 22 October 2017, revised 25 March 2018Accepted for publication 3 April 2018Published 16 May 2018

AbstractIn this paper, a frequency-domain element-free method (EFM) for seismic modeling and reverse-timemigration is presented. The application of time-domain EFM has been demonstrated successfully inseismic data processing and its advantages are shown. The absence of elements makes the EFMmoreflexible than the finite element method and it can be applied to more complex problems such as forirregular surfaces. Because of the utilization of the moving-least-squares fitting method, thedependent variable and its derivative are both continuous and precise in EFM. However, the largecomputation time of time-domain EFM limits its usefulness. We have developed an algorithm forfrequency-domain EFM that reduces its computation time. Unlike time-domain propagators, thedeveloped algorithm solves array equations to calculate a single-frequency wavefield. Without timeiteration, there is very little accumulated error and, because of the spectrum of the source, we onlyneed to calculate parts of the frequencies. The independence of each frequency makes it convenientto manipulate single-frequency wavefields and easy to accelerate computation with parallelization.For the frequency-domain algorithm feature, we can solve multiple shots at the same time. Theimplementation of frequency-domain EFM is shown for the full scalar wave equation. Based on thistheory, numerical examples of seismic process and time acceleration are presented.

Keywords: element-free method, migration, wave propagation, modeling, frequency domain

(Some figures may appear in colour only in the online journal)

Introduction

The wave-equation-based method is widely used in seismicmodeling and imaging. Various numerical algorithms suchas the finite difference method (FDM) and the finite elementmethod (FEM) have been developed (Claerbout 1971,Marfurt 1984). However, the low stability and precision ofFDM and huge amounts of computation required by theFEM are major shortcomings. In recent years, meshlessmethods have become popular for solving partial differ-ential equations because they are relatively economical andconvenient to use (Melenk and Babuška 1996). The ele-ment-free method (EFM) (Belytschko et al 1994) is a

meshless method that could be a possible solution forseismic modeling and migration. The element-free preciseintegration method (Jia and Hu 2006) gives a useful way toapply EFM to seismic modeling and imaging. The numer-ical result depends on factors such as the basis functions,node distribution, node numbering, stabilization condition,and boundary condition. To reduce the computation mem-ory and time required, several efficient improvements havebeen implemented (Fan and Jia 2013, Zhou et al 2018). Forexample, large sparse linear system equations are solvedinstead of performing matrix inversion. Further, a decom-posed element-free Galerkin method (Katou et al 2009) wasshown to exhibit good performance in terms of computation

Journal of Geophysics and Engineering

J. Geophys. Eng. 15 (2018) 1719–1728 (10pp) https://doi.org/10.1088/1742-2140/aabb2b

1742-2132/18/0401719+10$33.00 © 2018 Sinopec Geophysical Research Institute Printed in the UK1719

time and numerical accuracy compared with the FDM forelastic wave propagation.

The EFM has several advantages over the FEM. Forexample, EFM only requires nodal data; however, moreknowledge regarding the elements and mesh connectionsare needed in the FEM. Further, less pre-processingtime and computer resources are required for the EFM. Theabsence of mesh also makes the EFM more flexible andapplicable to more complex problems such as for irregularsurfaces. It is also easy to generate shape functions inthe EFM, with the only restraint being that of movingleast-squares (MLS) (Lancaster and Salkauskas 1981,Levin 1998), whereas the shape function in FEM has tosatisfy many requirements. Furthermore, the shape functionand its gradient in the EFM both have better continuity andsmoothness (Jia et al 2005).

The advantage of the EFM over the FDM is not in termsof efficiency, but rather in terms of accuracy and stability (Fanand Jia 2013). First, nodes are arranged more flexibly in theEFM than in the FDM. Second, the EFM can more easily dealwith heterogeneous media and irregular boundary problems.Further, the dependent variable and its derivative are con-tinuous and have higher precision owing to the MLS fitting.

Although the EFM has several advantages, the highcomputation time required limits its usefulness. The govern-ing equation of the EFM contains matrices. For time-domainwavefield modeling, several array equations have to be solvedat every time step. The cross-correlation imaging condition(Whitmore and Lines 1986) used in Reverse-time migration(RTM) (Baysal et al 1983) requires the forward-propagatingsource wavefield and the backward-propagating receiverwavefield to be available at the same time. This means thatthe array equations have to be solved at each time step againif RTM is performed. Multiple-shots data would result in thecomputation time being compounded.

Frequency-domain EFM is different from time-domainEFM and is also more efficient. For time-domain propagators,every time step wavefield is necessary; for the frequency-domain method, only single-frequency wavefields are needed.A single frequency is equivalent to a sinusoidal component inthe time domain (Sirgue and Pratt 2004). When a range offrequencies is used, the frequency-domain method isequivalent to the time-domain method when using the samerange of frequencies. Because each frequency is independent,this method conveniently manipulates the single-frequencywavefield and conducts frequency analysis and also flexiblychooses the frequency range. Because there is no iteration,there is very little accumulated error. Therefore, the stabilityof the frequency-domain algorithm is better than that of thetime-domain algorithm (Marfurt 1984). In addition, it is easyto implement attenuation (Jo et al 1996) and deal with mul-tiple shots. Another advantage of the frequency-domain RTMis that no wavefield-storage issue occurs as in the time-domain method (Symes 2007) because we calculate the for-ward-propagating wavefield and the backward-propagatingwavefield at the same time.

The remainder of this paper is organized as follows. Wefirst describe the theory of the frequency-domain EFM within

the acoustic wave equation framework. Then, based on thetheory, numerical examples are presented. The computationtime of the time-domain EFM and frequency-domain EFMare then compared in detail. Subsequently, the processemployed to deal with multiple shots is discussed. Furthercomputation acceleration with parallelization is also applied.

Theory

Consider solving the following two-dimensional (2D) fullscalar constant-density wave equation,

c

u

t

u

x

u

zf

10, 1

2

2

2

2

2

2

2

¶¶

-¶¶

+¶¶

- =⎛⎝⎜

⎞⎠⎟ ( )

where u is the displacement field; t is the temporal coordinate;x and z denote the spatial coordinates; c is the media velocity;and f is the source term. To match the real displacement u, wedefine the MLS approximant as follows:

u x p x a x P x a x , 2h

j

m

j jTå= =( ) ( ) ( ) ( ) ( ) ( )

where P(x) is the m-dimensional basis vector. For the 2Dcase, P(x) usually has the form

P x z x xz z1, , , , , . 3T 2 2= [ ] ( )

a(x) in equation (2) is an unknown coefficient that can bedetermined by MLS criterion. This means that the followingfunction is minimized:

J w x x P x a x u , 4I

N

IT

I I2

inf

å= - -( )[ ( ) ( ) ] ( )

where Ninf is the number of nodes in the influence domain ofx; w(x−xI) is the weight function. uI is the displacement atxI. Figure 1 shows the influence domain and weight function.For node xI, which is inside the influence domain, w(x−xI)decreases with the increase in the distance from xI to x. Fornodes that are outside the influence domain, w(x−xI)=0.

We minimize the above norm J to obtain a(x) using

J

a x0. 5

¶¶

=( )

( )

Then, we obtain

a x A x B x U, 61= -( ) ( ) ( ) ( )

where

A x w x x P x P x

B x w x x P x w x x P xw x x P x

U u u u

,

, , ,,

, , , . 7

I

N

I IT

I

N N

NT

1 1 2 2

1 2

inf

inf inf

inf

å= -

= - - ¼-

= ¼

( ) ( ) ( ) ( )

( ) [ ( ) ( ) ( ) ( )( ) ( )]

[ ] ( )

Substituting equations (6) into (2), we get

u P x A x B x U x U, 8h 1 j= =-( ) ( ) ( ) ( ) ( )

where j(x) is called the shape function.

1720

J. Geophys. Eng. 15 (2018) 1719 X Jia and X Qiang

Using the variational principle for equation (1) isequivalent to finding the minimum of the following function(Zienkiewicz and Taylor 1989):

Jc

uu fuu

x

u

z

1¨

1

2

1

2d . 90 2

2 2

= - +¶¶

+¶¶

W⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥∬ ( )

Substituting equations (8) into (9), we get

10

Jc

U U f Ux

Uz

U1 ¨ 1

2

1

2d .0 2

2 2

j j jj j

= - +¶¶

+¶¶

W⎜ ⎟ ⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥∬

( )

We minimize function J0 using

J

U cU f

x xU

z zU

1 ¨

d 0. 11

T TT

T

02j j j

j j

j j

¶¶

= - +¶¶

¶¶

+¶¶

¶¶

W =

⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

∬

( )

This gives us,

KU MU G¨ , 12+ = ( )

where K is the stiffness matrix, M is the mass matrix, and G isthe equivalent load source term. The matrices are defined by

Kx x z z

Mc

G f

d ,

1d ,

d . 13

T T

T

T

2

j j j j

j j

j

=¶¶

¶¶

+¶¶

¶¶

W

= W

= W

⎜ ⎟⎡⎣⎢⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥∬

∬

∬ ( )

Equation (12) is the governing equation of EFM.The acoustic wave euqation is also written as

u

tc

u

x

u

zf 0. 14

2

22

2

2

2

2

¶¶

-¶¶

+¶¶

- =⎛⎝⎜

⎞⎠⎟ ( )

J0 can also be defined in the following form:

J uu fuc u

x

c u

zcu

u

x

c

x

cuu

z

c

z

¨2 2

2

2 d .

15

0

2 2 2 2

= - +¶¶

+¶¶

+¶¶

¶¶

+¶¶

¶¶

W

⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

∬

( )

Following the same derivation, we have

K cx x

cz z

cx

c

xc

z

c

z

M

G f

2 2 d ,

d ,

d . 16

T T

T T

T

T

2 2j j j j

jj

jj

j j

j

=¶¶

¶¶

+¶¶

¶¶

+¶¶

¶¶

+¶¶

¶¶

W

= W

= W

⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

∬

∬∬ ( )

Equation (16) (which we hereafter call Method II) is differentfrom equation (13) (which we hereafter call Method I), butthey are both correct. The gradient of the velocity has a sig-nificant influence on the modeling result and makes Method IIcomplex.

For the frequency-domain algorithm, we solve the gov-erning equations to calculate the single-frequency wavefield.Fourier transform is applied to equation (12) to give the fol-lowing new frequency-domain equation:

K M U G , 172w w w- =( ) ( ) ( ) ( )

where ω is angular frequency; U(ω) is the vector of thewavefield in the frequency domain; and G(ω) is the sourcevector in the frequency domain. We assemble matrices Kand M into a single matrix S(ω) for each ω, called the

Figure 1. (a) Influence domain of point x. The little red circles arenodes; the blue circle is the influence domain of point x; rinf is theradius of this influence domain. (b) Weight function.

1721

J. Geophys. Eng. 15 (2018) 1719 X Jia and X Qiang

complex-valued impedance matrix, such that

S U G . 18w w w=( ) ( ) ( ) ( )

In the frequency-domain EFM, there are several sparsematrices such as the mass matrix, the stiffness matrix and thecomplex-valued impedance matrix. The cost burden is mainlycaused by improper storage of these sparse matrices anddirect operations on them. In order to save computer resourcesand make the frequency-domain EFM more easily used, wecompress sparse matrices by the compressed sparse columnformat (Duff et al 1989), and solve the linear equationsequation (18) with a linear equations solver. A variety oflinear sparse solvers have been tested and it is found thatPARDISO is usually the most efficient solver (Gouldet al 2007), which we employ here to solve the linearequations in our method. PARDISO is a memory efficient andeasy-to-use software for solving large sparse linear systems ofequations on shared-memory and distributed-memorymultiprocessors.

For each frequency, we can solve equation (18) to get thesingle-frequency wavefield. For a given source, not all fre-quencies are needed. A frequency range that containsthe most energy, for example, the dominant frequency range, issufficient. The computation time of the frequency-domainmethod is directly proportional to the number of frequencies.Meanwhile, because of the independence of each frequency,parallelization can be easily implemented.

In the frequency-domain RTM, with governingequation (18), it is necessary to define the right-hand sourceterm G(ω). For the forward-propagating problem, G(ω) isthe true source signal at each shot location. For the back-ward-propagating problem, G(ω) is the virtual sourceassociated with every receiver. For both the forward-pro-pagating problem and the backward-propagating problem atfrequency ω, the complex-valued matrix S(ω) is the same.For different shots, the complex-valued matrix S(ω) is alsothe same. This means that we can calculate the forward-propagating wavefield and the backward-propagatingwavefield of different shots at the same time if we use adirect sparse solver via LU decomposition.

In the time domain, the cross-correlation imaging con-dition has the following form:

I u t u T t td , 19T

s r0ò= -( ) ( ) ( )

where I is the image amplitude; T is total length of time; andus(t) and ur(T−t) are the forward-propagating sourcewavefield and the backward-propagating receiver wavefield,respectively. Both wavefields are extrapolated in the sametime-coordinate system. Transformation of the time-domainimaging condition to the frequency domain (Xu et al 2010)gives

I u u1

2e d , 20s r

Tiòpw w w= w

-¥

+¥ˆ ( ) ˆ ( ) ( )

where us wˆ ( ) and ur wˆ ( ) are the corresponding forward-pro-pagating source wavefield and backward-propagating receiver

wavefield, respectively, at frequency ω. The imaging resultcan be obtained using equation (20) if every single-frequencywavefield needed is obtained.

Boundary condition

Every numerical method for wave modeling has the samedrawback—the existence of an artificial boundary. To avoidthis issue, in time-domain wavefield modeling, variousmethods can be used to enlarge the computation zone in orderto delay the backward reflections or prevent the forwardwavefront from reaching the boundary even at the maximumtime. Although the computation time is increased, thesemethods are very useful. This approach does not apply tofrequency-domain methods because each frequency implicitlycontains information for all times. Another importantreason is that finite-length inverse Fourier transform hastime aliasing. Consequently, boundary reflections cannot beremoved by simple time windowing.

A simple way to eliminate time aliasing is to utilizecomplex-valued frequency (Mallick and Frazer 1987, Sirgueand Pratt 2004). The complex-valued frequency comes fromthe time-domain damping wavefield. For a time-domainwavefield with a damping factor σ, u t e ts-( ) has the followingFourier transform:

u t t u t t

U

e e d e d

i , 21

t t t

0

i

0

i iò òw s

=

= -

s w w s+¥

- -+¥

- -( ) ( )

( ) ( )

( )

where ω−iσ is the complex-valued frequency. However, asignificant disadvantage of the complex-valued frequency isthat the wavefield of the entire model damps over time.

To eliminate the artificial boundary reflection and pre-serve the amplitude of the wavefield, we employ the perfectlymatched layer (PML) method (Berenger 1994). With damp-ing functions α and β incorporated into equation (1), the 2Dscalar wave equation becomes

c

u

t

u

x

u

zf

1 1 10. 22

2

2

2

2

2

2

2a b¶¶

-¶¶

+¶¶

- =⎛⎝⎜

⎞⎠⎟ ( )

The damping function may assume different forms; we utilizethe cosine type (Moreira et al 2014):

id x

Lx L

id z

Lz L

1 cos2

, 0

1 cos2

, 0 . 23

pml pmlpml

pml pmlpml

aw

p

bw

p

= +

= +

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠ ( )

In equation (23), L denotes the width of the PML layer; xpml isa local coordinate denoting the distance to the horizontalboundary; and zpml is a local coordinate denoting the distanceto the vertical boundary. The scalar coefficient dpml dependson the width of the PML layer and i 1= - . In the initialmodel zone, α=1 and β=1, such that equation (22) isequivalent to equation (1).

1722

J. Geophys. Eng. 15 (2018) 1719 X Jia and X Qiang

Figure 2. Boundary test on a homogeneous model. (a) Real part of single-frequency wavefield with fixed boundary. (b) Time-domainsnapshot with fixed boundary. (c) Real part of single-frequency wavefield with complex-valued frequency. (d) Time-domain snapshot withcomplex-valued frequency. (e) Real part of single-frequency wavefield with PML boundary. (f) Time-domain snapshot with PML boundary.

1723

J. Geophys. Eng. 15 (2018) 1719 X Jia and X Qiang

Figure 3. (a) Two-layer velocity model. (b) Real part of single-frequency wavefield with frequency 10 Hz. (c) Time-domain snapshot at time0.4 s. (d) Seismic record of receiver A. (e) Seismic record of receiver B.

1724

J. Geophys. Eng. 15 (2018) 1719 X Jia and X Qiang

Figure 2 shows the modeling results obtained for thehomegeneous model with different boundary conditions.The first row shows the results for a fixed boundary.Figure 2(a) shows the real part of the single-frequencywavefield with frequency 10 Hz. Figure 2(b) is the time-domain snapshot at 0.8 s. The second row shows the resultsfor complex-valued frequency. The time-domain snapshothas a normal waveform but low amplitude. The thirdrow shows the results for PML boundary with normalwaveform, normal amplitude, and marginal boundaryreflection.

To apply the PML boundary condition to the frequency-domain EFM, the matrices in equation (13) must be

recalculated with new forms:

Kx x z z

Mc

G f

1 1d ,

1d ,

d . 24

T T

T

T

2

aj j

bj j

j j

j

=¶¶

¶¶

+¶¶

¶¶

W

= W

= W

⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥∬

∬

∬ ( )

Matrices K, M, and G are calculated by the respectiveGaussian quadratures (Golub and Welsch 1969). The inte-gration points within the entire model, including the PMLlayer, are first partitioned. Then, we calculate the localmatrices on the integration points. A summation of all local

Figure 4. Left column: time-domain snapshots. Middle column: seismic record of receiver A. Right column: seismic record of receiver B.First row: modeling result ignoring gradient of velocity. Second row: modeling result of second-order difference gradient of velocity. Thirdrow: modeling result of eight-order difference gradient of velocity.

1725

J. Geophys. Eng. 15 (2018) 1719 X Jia and X Qiang

matrices is made to obtain global matrices K, M, and G.Because the matrices in equation (24) are obtained using theGaussian quadratures, an efficient solver is used to obtain thesingle-frequency wavefield. When all single-frequencywavefields needed have been calculated, the time-domainsnapshot and imaging result are easily obtained.

Numerical examples

In this section, several examples for seismic modeling andRTM are shown to verify the accuracy of the frequency-domain EFM. Comparison of the computation time with thetime-domain EFM and other characteristics of frequency-domain algorithm are presented.

Modeling and imaging examples

Figure 3(a) shows a two-layer velocity model with a300×240 grid. The grid interval is 10 m. The first layer hasa velocity of 2000 m s−1 and the second has velocity2500 m s−1. The dominant frequency of the source located atpoint S is 20 Hz and the frequency range used is between 0and 60 Hz. Two receivers are located at point A and point B.Figure 3(b) shows the real part of a single-frequency wave-field with frequency 10 Hz. Figure 3(c) shows a time-domainsnapshot at time 0.4 s.

To illustrate the accuracy of frequency-domain EFM, wecompared the modeling result with that of the FDM. Figures 3(d)and (e) show that both the arrival time and the waveform offrequency-domain EFM are consistent with that of FDM.

Method II was also tested with different gradients ofvelocity. In the first test, the gradient of velocity was ignored.In the second and third tests, the gradient of velocity wascalculated using the second-order difference and the eight-order difference, respectively. Figure 4 shows the resultsobtained. For Method II, all of the transmitted waves arecorrect; however, only the reflected wave with gradient ofvelocity calculated by the eight-order difference is consistentwith that of FDM.

The frequency-domain EFM not only requires less timethan the time-domain EFM, it returns high-quality results forRTM. We tested the 2D SEG/EAGE salt model depicted infigure 5(a). This model has a 676×210 grid with gridinterval 10 m. The imaging results of time-domain EFM,frequency-domain EFM and FDM are presented infigures 5(b)–(d) with 28 shots stacked, respectively. Thesethree images are all show the main structures of the 2D SEG/EAGE salt model clearly, except for the slight artifacts nearthe surface in the FDM result. The result for frequency-domain EFM, showing clear spike reflector and salt boundarywith good resolution, is at the same level as that of time-domain EFM, with some sections even better than in the timedomain. The subsalt section has less noise.

Acceleration of computation

The huge computation time is a major shortcoming of the time-domain EFM for seismic modeling and imaging. Time stepiteration accounts for the majority of the time. The discreteequation, equation (12), is actually semi-discrete because it stillcontains acceleration U . The time recursion relations can beobtained by integrating U using the average acceleration algo-rithm (Jia and Hu 2006). After this process, there are two array

Figure 5. (a) 2D SEG/EAGE velocity model. (b) Imaging result oftime-domain EFM. (c) Imaging result of frequency-domain EFM.(d) Imaging result of FDM.

1726

J. Geophys. Eng. 15 (2018) 1719 X Jia and X Qiang

equations that should be solved and other matrix operationssuch as multiplication and addition in each time loop (Fan andJia 2013). In these matrix operations, solving the arrayequations requires the most amount of time. Consideringbackward-propagating wavefield, the number of time loopswill double. The frequency-domain EFM has its specificadvantage to save computation time. Matrix S(ω) is the samefor forward-propagating wavefield as backward-propagatingwavefield. We can solve equation (18) directly to obtain bothforward-propagating single-frequency wavefield and back-ward-propagating single-frequency wavefield using a directsolver. The only problem is that matrix K contains ω. Thismeans that we have to recalculate matrix K for every frequencyloop. Table 1 compares the computation time of the time-domain EFM, the frequency-domain EFM and the FDM forRTM. The velocity model for this test is shown in figure 3(a).The selected frequency range is 0–60 Hz for the frequency-domain EFM. The total number of time steps is N=8001 andtime interval is Δt=0.0005 s.

We know that the computation efficiency in solving largesparse array equation systems depends on the matrix band-width and the number of non-zero coefficients (Hustedtet al 2004, Liu et al 2013). The PML boundary conditionincreases the matrix bandwidth and the number of non-zerocoefficients, and also makes the impedance matrix S(ω)complex-valued. Therefore, the time required to solve anarray equation once in the frequency-domain EFM is morethan that required for the time-domain EFM. Recalculatingmatrix K in every loop also requires a lot of time. However,the reduction in the number of loop results in the frequency-domain EFM to be more efficient than the time-domain EFM.In this test, the total time taken by the time-domain EFM forthe entire RTM is approximately 27 times that required by thefrequency-domain EFM. The computation speed of FDM (an8th-order classical finite difference scheme adopted) is muchhigher than both the time-domain EFM and the frequency-domain EFM. However, it is possible to make the frequency-domain EFM more efficient by some computation strategies.

There are two other ways to reduce the computation timefor the frequency-domain EFM. Matrix S(ω) in equation (18) isthe same for all shots. For a particular frequency, once a directsolver based on LU decomposition is used, the single-frequencywavefield for all shots can be calculated at the same time.Figure 6(a) shows the relative computation time of multipleshots. It is clear that even when 32 shots are solved in onefrequency loop, the computation time expended is virtually 1.45times that of one shot.

The second method to reduce the computation time isparallelization. The wavefields of different frequencies areindependent. We can calculate the wavefield of each frequencyusing a different thread on a multicore machine. Figure 6(b)shows the relative computation time with parallelization. It isobvious that parallelization reduces the computation timesignificantly.

Table 1. Computation time comparison of the time-domain EFM, the frequency-domain EFM and the FDM for RTM.

Time taken to solve onearray equation (s)

Number of arrayequations in one loop

Number ofloops

Time for allloops (s)

Time for the entireprocess (s)

Time-domain EFM 3.46 2 16000 116163 116202Frequency-domain EFM

9.95 1 240 4023 4261

FDM — — — — 31

Figure 6. (a) Relative computation time of multiple shots. (b) Relativecomputation time of parallelization with multiple threads.

1727

J. Geophys. Eng. 15 (2018) 1719 X Jia and X Qiang

Conclusions

In this paper, we described the frequency-domain EFM usinga 2D full scalar wave equation and applied it to seismicmodeling and imaging. On the basis of this theory, we pre-sented numerical examples and obtained accurate wavefieldmodeling results and high-quality imaging results. The fre-quency-domain EFM significantly reduces the computationtime because it involves fewer array equations than the time-domain EFM. The characteristics of the frequency-domainalgorithm enable us to deal with multiple shots at once, withonly a small time penalty. Furthermore, the independence ofthe frequencies in the frequency-domain EFM means thatparallelization can be implemented for different frequencies,which effectively reduces the computation time. The GPU-accelerated technical platform may provide the best method toimplement parallelization in the future.

Acknowledgments

We would like to thank Zhen Zhou and Quanli Li for theirfruitful discussions and useful advice. This study receivedsupport from the National Natural Science Foundation ofChina (41774121) and the Fundamental Research Funds forthe Central Universities (WK2080000078).

References

Baysal E, Kosloff D D and Sherwood J W 1983 Reverse timemigration Geophysics 48 1514–24

Belytschko T, Lu Y Y and Gu L 1994 Element-free Galerkinmethods Int. J. Numer. Methods Eng. 37 229–56

Berenger J-P 1994 A perfectly matched layer for the absorption ofelectromagnetic waves J. Comput. Phys. 114 185–200

Claerbout J F 1971 Toward a unified theory of reflector mappingGeophysics 36 467–81

Duff I S, Grimes R G and Lewis J G 1989 Sparse matrix testproblems ACM Trans. Math. Softw. 15 1–14

Fan Z and Jia X 2013 Element-free method and its efficiencyimprovement in seismic modelling and reverse time migrationJ. Geophys. Eng. 10 025002

Golub G H and Welsch J H 1969 Calculation of Gauss quadraturerules Math. Comput. 23 221–30

Gould N I M, Scott J A and Hu Y 2007 A numerical evaluation ofsparse direct solvers for the solution of large sparse symmetriclinear systems of equations ACM Trans. Math. Softw. 33 10

Hustedt B, Operto S and Virieux J 2004 Mixed-grid and staggered-grid finite-difference methods for frequency-domain acousticwave modelling Geophys. J. Int. 157 1269–96

Jia X and Hu T 2006 Element-free precise integration method and itsapplications in seismic modelling and imaging Geophys. J. Int.166 349–72

Jia X, Hu T and Wang R 2005 A meshless method for acoustic andelastic modeling Appl. Geophys. 2 1–6

Jo C-H, Shin C and Suh J H 1996 An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolatorGeophysics 61 529–37

Katou M, Matsuoka T, Mikada H, Sanada Y and Ashida Y 2009Decomposed element-free Galerkin method compared withfinite-difference method for elastic wave propagationGeophysics 74 H13–25

Lancaster P and Salkauskas K 1981 Surfaces generated by movingleast squares methods Math. Comput. 37 141–58

Levin D 1998 The approximation power of moving least-squaresMath. Comput. 67 1517–31

Liu L, Liu H and Liu H 2013 Optimal 15-point finite differenceforward modeling in frequency-space domain Chin. J.Geophys. 56 91–100

Mallick S and Frazer L N 1987 Practical aspects of reflectivitymodeling Geophysics 52 1355–64

Marfurt K J 1984 Accuracy of finite-difference and finite-elementmodeling of the scalar and elastic wave equations Geophysics49 533–49

Melenk J M and Babuška I 1996 The partition of unity finite elementmethod: basic theory and applications Comput. Methods Appl.Mech. Eng. 139 289–314

Moreira R M, Cetale Santos M A, Martins J L, Silva D L,Pessolani R B, Filho D M and Bulcão A 2014 Frequency-domain acoustic-wave modeling with hybrid absorbingboundary conditions Geophysics 79 A39–44

Sirgue L and Pratt R G 2004 Efficient waveform inversion andimaging: a strategy for selecting temporal frequenciesGeophysics 69 231–48

Symes W W 2007 Reverse time migration with optimalcheckpointing Geophysics 72 SM213–21

Whitmore N and Lines L R 1986 Vertical seismic profiling depthmigration of a salt dome flank Geophysics 51 1087–109

Xu K, Zhou B and McMechan G A 2010 Implementation of prestackreverse time migration using frequency-domain extrapolationGeophysics 75 S61–72

Zhou Z, Jia X and Qiang X 2018 GPU-accelerated element-freereverse-time migration with Gauss points partition J. Geophys.Eng. 15 718

Zienkiewicz O C and Taylor R L 1989 The Finite Element Methodvol 1 (London: McGraw-Hill)

1728

J. Geophys. Eng. 15 (2018) 1719 X Jia and X Qiang