element-free precise integration method and its ...staff.ustc.edu.cn/~xjia/publications/gji.pdf ·...

Geophys. J. Int. (2006) 166, 349–372 doi: 10.1111/j.1365-246X.2006.03024.x

GJI

Sei

smol

ogy

Element-free precise integration method and its applicationsin seismic modelling and imaging

Xiaofeng Jia∗ and Tianyue HuSchool of Earth and Space Sciences, Peking University, Beijing 100871, China

Accepted 2006 March 24. Received 2006 March 23; in original form 2005 July 23

S U M M A R YIn this paper the theory of the element-free precise integration method (EFPIM) is presentedas well as its applications in seismic modelling and imaging. The key point of this methodis the absence of elements, which makes nodes free from the elemental restraint. Due tothe moving least-squares (MLS) fitting instead of interpolation, the EFPIM results in highaccuracy for both the dependent variable and its gradient. The EFPIM improves the implicitelement-free method (EFM) by cutting the computational cost significantly. At the same time,the accuracy of this method keeps as good as that of the implicit EFM. The scheme of EFPIMis shown for the full scalar wave equation. Numerical stability is examined for the schemesubsequently. Based on the theory, a simple example of vibrant film is discussed in details toindicate the effectivity of the EFPIM. Main factors affecting the accuracy of the method areillustrated. Furthermore, we show some synthetic examples to demonstrate good performanceof the EFPIM in seismic modelling and imaging problems. Both post-stack and pre-stack casesare considered. Combined with appropriate absorbing boundary conditions, the EFPIM cangenerate sections with accurate traveltimes and amplitudes. Complex structures can be imagedclearly such as high-angle dip and embedded high-velocity anomalies.

Key words: absorbing boundary, accuracy, element-free method, imaging, modelling, preciseintegration.

1 I N T RO D U C T I O N

The wave-equation-based method is a well known seismic modelling and imaging method. Quite a few numerical algorithms are developed

such as the finite difference method (FDM) and the finite element method (FEM) (Claerbout 1971; Marfurt 1984). In recent years, meshless

approximations have become promising in solving partial differential equations due to their economy and convenience in applications (Lucy

1977; Nayroles et al. 1992; Liu et al. 1995; Melenk & Babuska 1996). The element-free method (EFM) (Belytschko et al. 1994; Lu et al.1995), one of these meshless approaches, could be a possible solution for seismic wave modelling and imaging. As a matter of fact, the EFM

has been demonstrated successful in elasticity, heat conduction and fatigue crack growth problems (Belytschko et al. 1994). In this paper we

will apply the improved EFM to solve the seismic wave equation.

There are many common features in the theories of the EFM and the FEM; however, the two methods are also very different. First, only

nodal data in the EFM are required in contrast with more knowledge of element connectivity in the FEM. The element mesh is unnecessary.

Consequently, its pre-processing saves much more time and computer resources than that of the FEM. The absence of elements gives this

method kind of flexibility as well. Second, in the FEM it is a hard work to determine the shape function due to the complex restraints on it.

But in the EFM, these restraints are mostly released, which makes the shape function easily generated with the moving least-squares (MLS)

criterion originally cited by Lancaster & Salkauskas (1981). Third, it is usually difficult to improve the accuracy locally in the FEM allowing

for the global influence of local accuracy change. However, in the EFM the weight function is introduced so that one can treat local cases

conveniently. The weight function has some important features such as non-negativity, monotone decreasing and differentiability up to a

certain order. Furthermore, both the dependent variable and its gradient obtained by the EFM are continuous and therefore post-processing to

obtain a smooth gradient field is unnecessary.

In spite of these exclusive merits, the EFM has rarely been discussed in seismic exploration yet. Compared with the FDM, which has

become very common in applications, the EFM has some advantages such as high accuracy and good stability. However, considering its cost

∗Now at: Department of Earth Sciences, University of California, Santa Cruz, CA 95064, USA. E-mail: [email protected]

C© 2006 The Authors 349Journal compilation C© 2006 RAS

350 X. Jia and T. Hu

burden which is still much heavier than that of the FDM, this method seems difficult to be developed in seismic modelling and imaging. The

implicit time integration mostly used in the EFM is one of the main reasons. In this paper, we will employ an explicit precise integration

instead of implicit one to yield the time recursion relations. This alternative method is very time saving with its accuracy still high enough.

Some synthetic results are presented for seismic modelling and imaging by this improved EFM.

2 G E N E R AT I O N O F T H E D I S C R E T E S Y S T E M I N T H E E L E M E N T - F R E E M E T H O D

We consider the following wave propagation problem in the domain � bounded by �:

∂2u

∂t2= D

∂2u

∂x2+ D

∂2u

∂y2, (1)

where u is the displacement field; t, x and y denote the temporal and spatial coordinates, respectively; D is the square of wave velocity in the

media.

In general, the theory of the EFM is based on the MLS criterion and the variational principle. Define the MLS approximant by

uh(x) =m∑j

p j (x)a j (x) ≡ pT(x)a(x), (2)

where m is the dimension of the basis vector p(x) and in the 2-D case p(x) can be defined by

pT(x) = [1, x, y] or [1, x, y, x2, xy, y2]. (3)

In eq. (2) a(x) is an unknown coefficient vector which is to be determined by minimizing the norm as

J =Ninf∑

I

w(x − x I )[

pT(x I )a(x) − uI

]2, (4)

where Ninf is the number of nodes in the neighbourhood of x, called the influence domain of x, in which the weight function w(x − x I ) > 0,

and uI is the nodal value at x I . The weight function is defined in such a way that the further x I is away from x, the closer this weight would

become to zero.

From ∂ J/∂a = 0, we have

a(x) = A−1(x)B(x)U , (5)

where

A(x) =Ninf∑

I

w(x − x I )p(x I )pT(x I ), (6)

B(x) = [w(x − x1)p(x1), w(x − x2)p(x2), . . . , w

(x − xNinf

)p(xNinf

)], (7)

U = [u1, u2, . . . , uNinf

]T. (8)

Substituting a(x) obtained above into eq. (2) will yield

uh(x) = p(x)A−1(x)B(x)U ≡ φ(x)U , (9)

where φ (x) is the shape function. Due to the fitting criterion, the approximant uh(x I ) is not exactly equal to the nodal displacement uI . This

feature of the shape function in the EFM is quite different from that in the FEM. Eq. (9) can be directly called by eq. (1), which means solving

the wave equation just by the MLS fitting. We can also combine eqs (1) and (9) using the variational principle. If the penalty Galerkin method

(Zhu & Atluri 1998) is used, the discrete system under essential boundary conditions will be obtained as

KU + MU + F = 0, (10)

in which K is the stiffness matrix, M is the mass matrix and F is the equivalent load vector. These large and sparse matrices are defined by

K =∫

�

[(∂φ

∂x

)T

D

(∂φ

∂x

)+

(∂φ

∂y

)T

D

(∂φ

∂y

)]d� + β

∫�u

φTφ d�, (11)

M =∫

�

φTφ d�, (12)

F = −β

∫�u

φu� d�, (13)

where u|�u = u� is the prescribed boundary value and β is the penalty factor. Since the penalty factor affects both the diagonal and off-diagonal

entries of the stiffness matrix, a very large penalty will probably make the matrix ill-conditioned. Based on our experience, the penalty factor

should be controlled within (103 ∼ 106) ·E/L , where E is the Young’s modulus of the media and L is the dimension of the region under

consideration.

C© 2006 The Authors, GJI, 166, 349–372

Journal compilation C© 2006 RAS

EFPIM in seismic modelling and imaging 351

In the EFM, nodes are jointed one another by influence domains. When computing the quadratures in eqs (11)–(13) by Gauss method, the

independent variable in the shape function, i.e. x, represents the Gauss point. Due to the local weight function and consequently the local shape

function, only the nodes x I (I = 1, . . . , Ninf) in the influence domain of the Gauss point x would be selected to compute the corresponding

term in the Gauss quadrature. The determination of Ninf depends on the complexity of special problems. For the 2-D case where the quadratic

basis is used, Ninf can be chosen as 5∼9.

Besides the penalty method, the essential boundary conditions can also be enforced in other ways such as direct collocation method,

Lagrange multipliers method, and using the weak form of essential boundary conditions (Lu et al. 1995). However, due to the MLS fitting

instead of interpolation, few of these methods could treat the boundary problem in the EFM very well. Allowing for both accuracy and cost,

the penalty method is a convenient choice in most cases.

3 T I M E I M P L E M E N TAT I O N B Y P R E C I S E I N T E G R AT I O N

The discrete system (10) is semi-discrete actually because there is the acceleration U still in it. The time recursion relations could be obtained

by integrating U using the average acceleration algorithm:{un+1 = un + un�t + (�t)2(un + un+1)/4

un+1 = un + �t(un + un+1)/2, (14)

where n denotes the time step. The average acceleration method is used for implicit time integration and it is the special case of Newmark

integrator. Inserting eq. (9) into eq. (14) and noting that the shape function is independent of time, we have⎧⎨⎩ U n+1 = U n + �tUn + (�t)2(U

n + Un+1

)/4

Un+1 = U

n + �t(Un + U

n+1)/2

. (15)

From the discrete eq. (10), we get

Un + U

n+1 = −M−1 K (U n + U n+1) − M−1(Fn + Fn+1). (16)

Substituting eq. (16) into eq. (15) yields⎧⎪⎨⎪⎩U n+1 =

[M + (�t)2

4K

]−1 [M − (�t)2

4K

]U n + �t

[M + (�t)2

4K

]−1

MUn − (�t)2

4

[M + (�t)2

4K

]−1

(Fn + Fn+1)

Un+1 = U

n − �t2

M−1 K (U n + U n+1) − �t2

M−1(Fn + Fn+1)

, (17)

which gives a very complex time recursion. In eq. (17), inversions and multiplications of large-scale matrices need to be implemented.

Although there are usually some special methods to treat these sparse and band matrices, the time formula (17) seems awkward and the

recursion may consume much computational resource. The central difference method, one of the explicit methods, is too simple to be used

in the time recursion with many steps. Therefore, we will consider a new explicit time integration method rather than the central difference

method.

3.1 Time-marching scheme of the element-free precise integration method (EFPIM)

The discrete system (10) is composed by a group of separate equations such as

Kkkuk + Mkkuk + Fk +N∑

l,l �=k

(Kklul + Mkl ul ) = 0, (18)

where N is the scale of the matrices K or M and also the number of nodes in the whole spatial domain; k is the ordinal number of the

equation and l enumerates the terms in the equation. From eqs (11)–(13), due to the locality of the shape function, most of the terms in eq. (18)

are equal to zero except those corresponding to the nodes near or in the influence domain of the node numbered as k. In other words, each

individual equation of eq. (10) is corresponded to a special node which could be named ‘central node’. The central node of eq. (18) is the kth

of the N nodes.

Eq. (18) can be rewritten as

uk + Kkk

Mkkuk = − 1

Mkk

N∑l,l �=k

(Kklul + Mkl ul ) − Fk

Mkk. (19)

In the Taylor expansions of ul and ul around t = tn, if the zeroth-order term is kept only, we have

ul = unl and ul = un

l . (20)

Combining eq. (19) and (20), we obtain

uk + Kkk

Mkkuk = − 1

Mkk

N∑l,l �=k

(Kklu

nl + Mkl u

nl

) − Fk

Mkk. (21)

C© 2006 The Authors, GJI, 166, 349–372



Eq. (21) is an ordinary differential equation (ODE) in which the dependent variable is uk . Its solution is given as

uk = c1eat + c2e−at + b0/a2, (22)

where c1 and c2 are two undetermined coefficients, and

a2 = −Kkk/Mkk, (23)

b0 = 1

Mkk

N∑l,l �=k

(Kklu

nl + Mkl u

nl

) + Fk

Mkk. (24)

Note that a could be a pure imaginary. To determine c1 and c2, the special expressions of the solution (22) at tn−1, tn and tn+1 are written as

un−1k = c1eatn−1 + c2e−atn−1 + b0/a2, (25)

unk = c1eatn + c2e−atn + b0/a2, (26)

and

un+1k = c1eatn+1 + c2e−atn+1 + b0/a2. (27)

From eqs (25) and (26), we obtain

c1 = unj,i − un−1

j,i e−a�t − b0(1 − e−a�t )/a2

eatn − eatn−2and c2 = un

j,i − un−1j,i ea�t − b0(1 − ea�t )/a2

e−atn − e−atn−2. (28)

Inserting eq. (28) into eq. (27), we get the time-marching formula as follows

un+1k = (

unk − b0/a2

)(ea�t + e−a�t ) − un−1

k + 2b0/a2. (29)

Eq. (29) is much simpler than eq. (17). This method is a hybrid of the EFM and the precise integrator (see Jia et al. 2004), respectively,

in the spatial and temporal domain. Therefore, it can be named as the EFPIM. Under the same numerical conditions, the accuracy of this

method is almost as high as that of the EFM using the average acceleration integration. The computational cost of the EFPIM, however, has

been saved significantly. In addition, this method has good stability and high rate of convergence.

For eq. (29), unl in the expression of b0 is still unavailable. If the central difference is used for b0, eq. (29) will become divergent. Here

we borrow eq. (10) to determine unl , in the form

Un = −M−1(KU n + F). (30)

Higher-order methods could be considered for the EFPIM. Instead of eq. (20), we have the first- and second-order expansions of ul given

as

ul = unl + un

l (t − tn), (31)

and

ul = unl + un

l (t − tn) + unl (t − tn)2/2. (32)

From eq. (31), we can obtain the same scheme as eq. (29). On the other hand, eq. (32) will lead to more complicated result. In this case we

have

un+1k = (

unk − b0/a2 − 2b2/a4

)(ea�t + e−a�t ) − un−1

k + 2b0/a2 + 4b2/a4 + 2b2(�t)2/a2, (33)

where

b2 = 1

2Mkk

N∑l,l �=k

Kkl unl . (34)

If the time step is small enough, eq. (33) will be degraded into eq. (29). Therefore, in the case of small time step, high-order EFPIM could not

improve the accuracy of the method too much.

A comment should be made concerning the zeroth-order time scheme, i.e. eq. (29). Substituting eq. (24) into the time scheme, we get

un+1k = un

k (ea�t + e−a�t − 2)/a2 − un−1k + 2un

k , (35)

If we then use the second-order approximation of ea�t and e−a�t , in the form

ea�t = 1 + a�t + (a�t)2 and e−a�t = 1 − a�t + (a�t)2, (36)

eq. (35) will become

unk = (

un+1k − 2un

k + un−1k

)/(�t)2, (37)

which means the central difference is a O[(�t)2] approximation of the time scheme (29).

C© 2006 The Authors, GJI, 166, 349–372



3.2 Stability analysis for the EFPIM

Eq. (35) can be rewritten as the acceleration form

unk = hk

(un+1

k − 2unk + un−1

k

), (38)

where

hk = a2/(ea�t + e−a�t − 2). (39)

Considering all the nodes in the spatial domain, eq. (38) can be extended as

Un = HU n+1 − 2HU n + HU n−1, (40)

where

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

h1

h2

. . .

hN

⎤⎥⎥⎥⎥⎥⎥⎥⎦. (41)

We then apply the acceleration integrator, i.e. eq. (40), into the discrete equation (10), which gives

MHU n+1 = (2MH − K )U n − MHU n−1 − F. (42)

Then

U n+1 = [2 − (MH)−1 K ]U n − U n−1 − (MH)−1 F. (43)

Eq. (43) is equivalent to the original time scheme (29). We may examine the stability of this scheme using Von Neumann method (Carey

& Oden 1984). Let ω l (l = 1, 2, . . . , N ) denote the eigenvectors of the matrix (MH)−1 K . Assuming these vectors to be distinct and linearly

independent, the error vector en at the nth time step can be expressed as

en =N∑

l=1

ηnl ωl , (44)

where ηnl are constants. Substituting eq. (44) into the homogeneous form of eq. (43) and simplifying, we are left with

N∑l=1

[ηn+1

l − (2 − λl )ηnl + ηn−1

l

]ωl = 0, (45)

where λl are the eigenvalues of (MH)−1 K . Eq. (45) means for each ω l ,

ηn+1l − (2 − λl )η

nl + ηn−1

l = 0, (46)

and hence we obtain[ηn+1

l

ηnl

]=

[2 − λl −1

1 0

][ηn

l

ηn−1l

]. (47)

From eq. (47), the principal equation can be determined as

γ 2 − (2 − λl )γ + 1 = 0. (48)

Since the product of the two roots γ1 and γ2 for eq. (48) satisfies γ1γ2 = 1, the general solution which is the linear combination of γ 1 and γ 2

will contain a growth term unless γ1 and γ2 are complex. This implies that the discriminant of eq. (48) cannot be positive if the scheme is to

be stable. In this sense we require

(2 − λl )2 − 4 ≤ 0. (49)

Thus the stability condition becomes

λl ≤ 4. (50)

In the case of linear diffusion problems, the precise integration scheme like eq. (29) is unconditionally stable. When dealing with wave

equations, some development on the scheme is required to release the stability constraint.

4 A N E X P E R I M E N T O F V I B R A N T F I L M

Before using the EFPIM to solve seismic modelling and imaging problems, a non-dimensional example of vibrant film will be shown to

demonstrate the good performance of the method. The vibrant film is fixed in the region of 0 ≤ x ≤ 2 and 0 ≤ y ≤ 2. The parameter D in

eq. (1) is 3. Assuming initial displacement as u = x(2 − x)y(2 − y) and initial velocity as zero, we can obtain the displacement anytime at

C© 2006 The Authors, GJI, 166, 349–372



Figure 1. Displacements for three typical nodes on the vibrant film, obtained, respectively, by exact solution, the FDM, the implicit EFM and the EFPIM. The

FDM is second order in time and fourth order in space. For the implicit EFM, the average acceleration method is used to handle time integration. The three

insets show details at the special time t = 1.

any point by solving this 2-D scalar wave equation problem. Since the wave equation for this example is almost the same as that for seismic

modelling and imaging, it will give us a preview on seismic applications of the EFPIM.

4.1 Test of the EFPIM on the accuracy and cost

Both the FDM and the EFM are employed to handle this problem. The results of some typical nodes are shown in Fig. 1 for displacements and

Fig. 2 for displacement gradients. The grid used in both the fourth-order FDM and the EFM is 33 × 33. The time step, which is also the same

for these methods, is given as 0.001. In the EFM, the Gauss cells used for Gauss quadrature are 20 × 20 and in each cell 3 × 3 quadrature is

employed. Besides, the power function is adopted as the weight function shown in eq. (4), which is

w(x − x I ) =⎧⎨⎩

r2inf

r2I +ε2r2

inf

(1 − r2

Ir2inf

)4

(rI ≤ rinf)

0 (rI > rinf)

, (51)

where rI is the distance between x and x I , rinf is the radius of the influence domain of x, and ε is a constant. In fact, other functions such as

exponential functions and spline functions (Belytschko et al. 1994; Dolbow & Belytschko 1999) can also be chosen as the weights, provided

that they are characteristic of non-negativity and monotone decreasing. Figs 1 and 2 show that in addition to good accuracy of displacements,

the EFM has especially high accuracy of displacement gradients. This merit has made the EFM more popular in mechanics than in other fields

since the gradient of displacement is usually related to the stress and strain.

Figs 1 and 2 also show the comparison between the implicit EFM and the EFPIM results. They are almost the same in terms of accuracy.

However, the implicit EFM requires more computer memory and computational time. Table 1 shows the computational time consumed by the

two methods in the same numerical and computer circumstances. Our computer is the SUN sparc workstation and its RAM is 2048M. From

Table 1 we find that with increasing the size of grid used in this problem, the EFPIM will save more and more computational time compared

with the implicit EFM.

Numerical features of the EFM, the EFPIM, the FEM and the FDM will be discussed more here. The principles of the EFM and the

FEM are very similar except the way constructing the trial function. Consequently, the computational costs of these two methods are almost

of the same order. However, due to the absence of elements, the EFM can still save much computing time and memory compared with the

FEM, especially for the large-scale problems. Without the restriction of elements, the EFM can easily change the distribution of the nodes,

which makes it more flexible than the FEM. The FDM has good performance in the aspect of efficiency. Although the EFPIM improves the

C© 2006 The Authors, GJI, 166, 349–372



Figure 2. The gradients of displacement in the x direction for two typical nodes on the vibrant film, obtained, respectively, by exact solution, the FDM, the

implicit EFM and the EFPIM. The FDM is second order in time and fourth order in space. For the implicit EFM, the average acceleration method is used to

handle time integration. The three insets show details at the special time t = 1.

efficiency of the EFM significantly, it is still quite time consuming. Fortunately, the EFM has another exclusive merit, which can compensate

for the disadvantage of cost more or less. Since the MLS fitting is employed for the derivation of the shape function, the solution obtained

by the EFM has good differentiability up to high orders. Therefore, it can deal with the problems related to stress or strain such as fracture

modelling and volumetric locking much more easily and precisely than the FDM and the FEM. In addition, the rate of convergence for the

EFM can exceed that of the FDM or the FEM greatly.

4.2 Main factors affecting the accuracy of the EFPIM

The accuracy of the EFPIM is influenced by several factors although the method is quite stable. The first factor is the weight function. The

choice of weights depends on the estimation of the solution form. In Fig. 3 we see that both power weight and exponential weight yield good

results for this special problem of vibrant film. Basis function is the second factor which is shown in Fig. 4. The basis can be chosen as

the linear, quadratic, cubic or orthogonal basis. In principle the quadratic basis is preferable considering the balance between accuracy and

cost. The orthogonal basis has been developed to eliminate the burden of inverting the matrix at quadrature points. However, the process of

constructing the orthogonal basis is rather complicated and requires even more computational time.

In 2-D cases, the radius of the influence domain can be defined as

rinf =√

αm

πd, (52)

where m is the dimension of the basis vector, d is the local density of nodes, and α is an adjustable constant.

Table 1. Computational time for the implicit EFM and the EFPIM to model the vibrant film in the same circum-

stances.

Grid size Time for the implicit EFM (s) Time for the EFPIM (s)

21 × 21 16 14

41 × 41 217 148

61 × 61 1129 610

81 × 81 12 803 5025

C© 2006 The Authors, GJI, 166, 349–372



Figure 3. Displacements for three typical nodes on the vibrant film, respectively, with the spline function, the piecewise spline function, the exponential

function and the power function as the weight. The spline weight is fourth order and the piecewise spline weight is third order. The three insets show details at

the special time t = 1.

Figure 4. Displacements for three typical nodes on the vibrant film, respectively, with the linear basis, the quadratic basis and the cubic basis. The three insets

show details at the special time t = 1.

C© 2006 The Authors, GJI, 166, 349–372



Figure 5. (a) Error function of displacement versus size of influence domain. The figure shows for three typical nodes on the vibrant film at the time t = 1. The

argument α in the figure is directly proportional to the square of the size of influence domain, which is shown in eq. (52). (b) Error function of displacement

versus density of Gauss cells. The cases for three nodes on the film at the time t = 1 are shown. N G in the figure denotes the number of Gauss cells.

Figure 6. (a) Error function of displacement versus density of nodes. In the figure, Nx is the number of nodes in x direction which is inversely proportional to

�x. Large Nx indicates high density of nodes. (b) Error function of displacement versus time step. The horizontal axis is logarithmically scaled. Both (a) and

(b) show for just the special time t = 1.

C© 2006 The Authors, GJI, 166, 349–372



Figure 7. Pre-stack modelling by the EFPIM for a model including two horizons under different absorbing boundary conditions. (a) Velocity model. The

source is marked in the model. (b) Modelling result obtained without any absorbing boundary condition. Note that the artificial boundary reflections on both

sides of the model are very strong. (c) Modelling result for which the first-order paraxial wave equation was used to absorb the boundary reflections. (d) Same

as (c) except that the multitransmitting formula was employed as the absorbing boundary condition. (e) Same as (c) except that the damping method, where the

damping coefficient in the absorber was directly defined, was used to eliminate the boundary reflections. (f) Same as (e) except that the discrete equations in

the absorber were defined for the damping method. (g) Same as (c) except that the multitransmitting formula and the damping method used in (f) were jointly

exploited to weaken the boundary reflections.

The size of the influence domain plays a significant role on the accuracy of the EFPIM, which is shown in Fig. 5(a). Too large influence

domain will lead to cost-push; on the other hand, quite small influence domain harms the continuity of the solution and may cause the method

invalid. In some sense, the influence domain can be called equivalent element, which is not so rigid as the element in the FEM and can be

overlapped one another. Fig. 5 also shows how the accuracy varies with the number of Gauss quadrature cells. We can find that the accuracy

tends to change little when the Gauss cells get too dense. Besides, like many other numerical methods, the accuracy of the EFPIM is also

affected by the space interval and the time step. Fig. 6 indicates that the EFPIM has good convergence and stability.

5 A B S O R B I N G B O U N DA RY C O N D I T I O N S I N T H E E F P I M

Due to the MLS fitting instead of interpolation, the EFM, as well as the EFPIM, could not treat essential boundary conditions as easily as the

FEM does. However, this is not a severe problem in seismic exploration since the essential boundary conditions are seldom used. Actually

C© 2006 The Authors, GJI, 166, 349–372



Figure 7. (Continued.)

the absorbing boundaries play an important role in attenuating the artificial boundary reflections in seismic modelling and imaging. In this

section, the combination of the EFPIM and several absorbing boundary methods will be presented in details.

Paraxial wave equation (Clayton & Engquist 1977) is one of the most common methods used for absorbing boundary condition. When the

upgoing wave equation is enforced to the down-going incident wave at the bottom boundary, no reflections would be generated in theory. The

principle holds for other boundaries, i.e. the right-going wave equation for the left boundary, the left-going wave equation for the right boundary

C© 2006 The Authors, GJI, 166, 349–372




and so on. The paraxial approximation may be zeroth order, first order or higher order. In principle, the high-order paraxial approximations

have good accuracy at the cost of sacrificing the flexibility in computation.

When the incident wave reaches a point on the boundary, the wave motion at this point can be simulated by the previous motions at its

adjacent points so that the incident wave seems to transmit through the boundary without any reflection. Assuming x = x L and x = x R to be

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the left and right boundaries, the multitransmitting formulae (Liao 1996, 2001) can be expressed as

u(xL, y, t + �t) =Nmul∑i=1

(−1)i+1C Nmuli u(iva�t, y, t − (i − 1)�t), (53)

u(xR, y, t + �t) =Nmul∑i=1

(−1)i+1C Nmuli u(xR − iva�t, y, t − (i − 1)�t), (54)

where �t is the time step, va is the artificial transmitting velocity, Cni is the binomial coefficient and N mul is the order. Usually va can be

assigned a value identical to the lowest velocity of the media. High-order formulae can deal with the boundary reflections at large incidence

angles, but high orders will deteriorate the stability of the method.

In the damping boundary condition, a shock absorber surrounding the model is attached to decay the incident wave. The damping

coefficient of the absorber should be determined reasonably and expressed as a function of material properties of the media. Since the effect of

absorbing boundary conditions is sensitive to the damping coefficient, one needs to define it carefully. The distribution of damping coefficient

in the absorber can be directly defined (Sochacki et al. 1987). In other words, seismic wave in the absorber follows the same wave equation as

that outside the absorber except for the addition of a damping term. An alternative method (see Sarma et al. 1998) is to present the discrete

system in the absorber as

KU + CU + MU + F = 0, (55)

where C is the damping matrix just for the absorber. The perfectly matched layer (PML) method is also an effective method to construct a

sponge layer where the outgoing waves at all angles and frequencies would decay (Berenger 1994; Komatitsch & Tromp 2003; Vay 2002).

For the damping layer, the wave equations need to be solved by staggering the dependent variables in space as well as in time.

To test the effect of these absorbing boundary methods in the EFPIM, we design a simple model (Fig. 7a) consisting of two horizons.

The media velocities are v1 = 1500 m s−1, v2 = 2500 m s−1 and v3 = 4000 m s−1. The grid interval is 10 m, the time step is 0.0012 s and the

dominant frequency of the source is 30Hz when not mentioned elsewhere.

Without using any absorbing boundary, the artificial boundary reflections are so strong that the primary reflection from the lower horizon

could be hardly recognized (Fig. 7b). In contrast, the modelling result using the first-order paraxial absorbing boundary conditions (Fig. 7c)

shows the primary reflection from the deep horizon clearly. The multitransmitting formulae we use to attenuate boundary reflections is second

order. It can be seen from Fig. 7(d) that the absorbing effect of multitransmitting formula is a little better than that of the first-order paraxial

equations. Figs 7(e)–(f) show the modelling results using two damping methods to decay the outgoing wave. The grid interval is 12.5 m. The

thickness of the absorber is 75 m on either side and 150 m at the bottom. The implementation of defining the damping discrete system in the

absorber provides higher absorption rate since the EFPIM employs a similar system shown as eq. (10).

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Figure 8. Post-stack modelling and imaging by the EFPIM for a simple model including a high-angle dip reflector. (a) Velocity model. Equivalent sources are

shown in the model to illustrate the exploding reflector theory used for post-stack modelling. (b) Modelling result (seismic section) during the first 2 s for the

model shown in (a). The inset compares the results from the EFPIM (solid) and the FDM (dashed) for two special traces. (c) Image for the model in (a). The

modelling result from the EFPIM is taken as the input of reverse-time migration to generate this image. Note the absence of energy on the left-hand side of the

section in the deep. (d) Same as (c) except that the input data of reverse-time migration is obtained by the finite difference modelling.

Although the absorbing boundary conditions have been studied for many years, few of them can solve the problem easily. The paraxial

equations and multitransmitting formula can weaken outgoing waves just at very limited incident angles. The damping boundaries increase

the cost of computation and the methods are sensitive to the definition of absorber. Nevertheless, it can be proved that the hybrid of different

methods mentioned above may generate better results. Fig. 7(g) shows the modelling result obtained by jointly using the damping absorber

and multitransmitting formulae. The absorbing capability has been improved.

6 S Y N T H E T I C S E I S M I C M O D E L L I N G A N D I M A G I N G B Y T H E E F P I M

The FEM and the FDM are very common in modelling and imaging of seismic exploration. Here it will be proved that the EFPIM can also

deal with seismic modelling and imaging problems very well. Four velocity models (Figs 8a, 9a, 10a and 11a) are designed, two for post-stack

case and the other two for pre-stack case. For all of them the receivers are distributed evenly on the surface. The sources sketched in the

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figures are expressed by Ricker wavelets. In order to weaken peg-leg multiples, the non-reflecting wave equation (Baysal et al. 1984) instead

of eq. (1) has been solved by the EFPIM, which makes little change in the scheme of the method.

We employ 41 × 41 regular nodes for all the models. The time step is 0.002 s and the dominant frequency of the source is 10Hz. For

each model 30 × 30 Gauss cells and 3 × 3 Gauss quadrature are used. The power function shown in eq. (51) is chosen as the weight function.

For seismic imaging by the EFPIM, if the input data is the modelling result obtained also by the EFPIM, the image will make little sense

probably. Usually it is necessary that the two methods used in modelling and imaging are different. Therefore, we also image by the EFPIM

based on the FDM modelling results.

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Figure 9. Post-stack modelling and imaging by the EFPIM for a model including a low-velocity dip layer and an embedded complex reflector. (a) Velocity

model, in which equivalent sources are sketched for the theory of exploding reflector. (b) Modelling result (seismic section) during the first 2 s for the model

of (a). Note that the event corresponding to the embedded reflector is distorted due to its wedge-shaped overburden. (c) Image for the model of (a), based on

the modelling result shown in (b). (d) Same as (c) except that it is based on the modelling result obtained by the FDM. For the deep high-angle reflector, the

image is fuzzy because the signals from this high-angle surface can hardly be recorded.

6.1 Post-stack seismic modelling and imaging

The two models for post-stack synthetic tests are shown in Figs 8(a) and 9(a). The first model (Fig. 8a) is a single high-angle dip reflector; the

second one (Fig. 9a) includes a low-velocity dip layer near the surface and an embedded complex reflector. In Fig. 8(a), v1 = 1000 m s−1 and

v2 = 2000 m s−1; in Fig. 9(a), v1 = 1000 m s−1, v2 = 2000 m s−1 and v3 = 4000 m s−1.

The theory of exploding reflector has been used to generate post-stack seismograms. Afterwards, we use reverse-time migration

(McMechan 1983; Harris & McMechan 1992) to obtain the images. From the seismograms (Figs 8b and 9b) we can recognize kinds of

events such as primary signals (direct arrivals) and residual boundary reflections. The inset of Fig. 8(b) shows two special traces obtained by

using both the EFPIM and the FDM. For the FDM, the grid is 81 × 81 and the algorithm is fourth order in spatial domain. The results of the

two methods agree with each other very well. In Fig. 9(b), the direct arrival from the deep complex reflector seems in segments due to the

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dip layer near the surface. Moreover, the images obtained by the EFPIM are very clear for both the EFPIM and the FDM modelling inputs.

As a matter of fact, since the grid we use for the FDM modelling is 81 × 81, the resolution of the images (Figs 8d and 9d) based on the FDM

modelling results is better than that of the EFPIM modelling counterpart (Figs 8c and 9c). This means the effect of imaging by the EFPIM

just depends on the quality of input data no matter which method has been used for the input. For the first model, the image of the high-angle

reflector gets absent in the deep because the signals from deep regions can hardly be recorded on the surface.

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Figure 10. Pre-stack modelling and imaging by the EFPIM for a model including a sloping interface. (a) Velocity model. A single shot is exploded at the

location of (Distance, Depth) = (500 m, 100 m). (b) Modelling result (snapshot of the wavefield) at the time t = 0.5 s for the model shown in (a). Note the

distribution of energy among direct arrival, transmitted wave, reflected wave and boundary reflected wave. (c) Image for the model of (a), based on the single

shot modelling result which has been obtained by using the EFPIM. (d) Image for the model shown in (a). Seven single shots at different locations were exploded

separately to yield seven shot gathers. The EFPIM was used to simulate this process. Based on each shot gather, reverse-time migration was implemented by

the EFPIM to obtain the single shot images individually. The image shown in (d) is the stack of these seven images. (e) Same as (c) except that the modelling

result obtained by the FDM was taken as the input of imaging. (f) Same as (d) except that the seven shot gathers were generated by the FDM instead of the

EFPIM.

6.2 Pre-stack seismic modelling and imaging

For the pre-stack case, we also design two models shown in Figs 10(a) and 11(a). One model has a single dip reflector and the other includes

a complex layer, a high-velocity anomaly and a buried horizon. The media velocities are given by: v1 = 1000 m s−1 and v2 = 2000 m s−1 in

Fig 10(a); v1 = 1000 m s−1, v2 = 2000 m s−1 and v3 = 4000 m s−1 in Fig 11(a). A single split spread is adopted to create pre-stack seismograms,

and we still use reverse-time migration to obtain images. In order to test the EFPIM imaging intensively, we have taken both the EFPIM and

the FDM modelling results as the input of imaging.

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A single shot can hardly cover the whole region and consequently, the image from a single shot gather is very confusing caused by a large

area of blind zone. Therefore, several shots at different locations should be considered for modelling and imaging separately. The quality of

the image obtained by stacking these single shot images will be improved greatly. Here we repeat seven single shots with their depth fixed to

100 m; their distances on the surface are 200, 250, 400, 500, 600, 750 and 800 m, respectively.

Fig. 10(b) shows the snapshot of the wavefield at t = 0.5 s. The wave fronts can be easily traced for the direct arrival carrying the strongest

energy, the transmitted wave through the reflector, the primary reflection, and the downward boundary reflection from the surface. Both wave

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fronts of the boundary reflection and the direct arrival are circular, and their centres mirror each other along the surface. The seismogram

shown in Fig. 11(b) describes the full and exact wavefield on the surface in the first two seconds. The primary signal from the deep horizon

has been distorted strongly due to its overlapped structures. Additionally, comparison between the results of the EFPIM and the FDM for two

special traces is illustrated in the inset of Fig. 11(b).

Although the stacked images (Figs 10d, f, 11d and f) are much better than those (Figs 10c, e, 11c and e) yielded from single shot gathers,

they are still fuzzy in the deep. The lack of resolution is mainly caused by the complex overburdens which may rebuild and deform the deep

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Figure 11. Pre-stack modelling and imaging by the EFPIM for a model including high-velocity anomaly and embedded horizon with complex overburdens.

(a) Velocity model. The location of the source is the same as that in Fig. 10(a). (b) Modelling result (seismic section) for the model of (a). The inset compares

the results from the EFPIM (solid) and the FDM (dashed) for two special traces. (c) Image for the model of (a). The modelling result shown in (b) was taken as

the input of imaging. (d) Image for the model of (a), which is the stack of seven single shot images obtained by the EFPIM individually. Each single shot image

was based on the modelling result similar to that shown in (b). (e) Same as (c) except that it is based on the modelling result yielded by the FDM. (f) Same as

(d) except that the input of imaging is obtained by the FDM.

signals. Besides, these images are obtained without any noise attenuation of input data except manually muting the direct arrivals. The muting

is awkward because it needs only a visual check. Finally, the rough grid is another reason for this deficiency which can be refined to improve

the images.

7 C O N C L U S I O N S

In the EFM, one of the significant advantages is the absence of element mesh which makes the method cheaper and more flexible than the

FEM. Due to the MLS fitting, a key feature of the EFM, both the dependent variable and its gradient are continuous and precise. At the same

time, the advantages of the EFM do not come without any costs. The computational cost is still higher than that of the FDM. In order to

eliminate the burden of large matrices implementation, we develop an explicit EFM in which the precise integration is used to handle the

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time recursion. This integrator can also be borrowed by the other numerical methods such as the FDM if the discrete system is integrable in

temporal domain. The EFPIM requires less computational resources than the implicit EFM, with their accuracy almost the same.

The results presented here show that the EFPIM is very promising in seismic modelling and imaging. Even the buried high-velocity

anomaly and the structures under complex overburden conditions can be imaged successfully. Note that the imaging results have been obtained

in this paper without any prior processing such as noise attenuation for the input data. In terms of the absorbing boundary conditions, the

damping method in which the discrete equations in the absorber are defined can be combined with the EFPIM easily due to the similarity

between their numerical structures. In addition, different absorbing boundary methods can be jointly used to improve the absorbing effect.

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A C K N O W L E D G M E N T S

This work benefited from fruitful discussions with Runqiu Wang and Zhicheng Jing. We would like to thank Dr Yanick Ricard, Professor Jean

Virieux and an anonymous referee gratefully for their constructive comments. This study received support from National Natural Science

Foundation of China (No. 40274041) and Ministry of Science of China ‘973’ Programme (No. 2001CCA02300).

C© 2006 The Authors, GJI, 166, 349–372



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