Full-waveform Velocity Inversion Based on the Acoustic...

American Journal of Computational Mathematics, 2013, 3, 13-20 doi:10.4236/ajcm.2013.33B003 Published Online September 2013 (http://www.scirp.org/journal/ajcm)

Full-waveform Velocity Inversion Based on the Acoustic Wave Equation

Wensheng Zhang, Jia Luo Institute of Computational Mathematics and Scientific/Engineering Computing, LSEC

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, P. R. China Email: [email protected], [email protected]

Received June, 2013

ABSTRACT

Full-waveform velocity inversion based on the acoustic wave equation in the time domain is investigated in this paper. The inversion is the iterative minimization of the misfit between observed data and synthetic data obtained by a nu-merical solution of the wave equation. Two inversion algorithms in combination with the CG method and the BFGS method are described respectively. Numerical computations for two models including the benchmark Marmousi model with complex structure are implemented. The inversion results show that the BFGS-based algorithm behaves better in inversion than the CG-based algorithm does. Moreover, the good inversion result for Marmousi model with the BFGS-based algorithm suggests the quasi-Newton methods can provide an important tool for large-scale velocity inver-sion. More computations demonstrate the correctness and effectives of our inversion algorithms and code. Keywords: Finite Difference; Acoustic Wave Equation; Full-waveform Inversion; CG Method; BFGS Method;

Marmousi Model

1. Introduction

The velocity of underground media is an important pa-rameter for resources exploration. It may indicate the structure and position of reservoir. Full-waveform inver-sion uses travel time, amplitude and phase information simultaneously to inverse media velocity. It is a powerful way in reconstructing complex velocity structures. The inversion can be performed in the time-space domain [1- 4] or in the frequency-space domain [5-9]. The fre-quency-domain inversion approach is equivalent to the time-domain inversion approach if all of the frequency data components are used in the inversion process [9].

Full-waveform inversion can be formulated as a nonlinear optimization problem with an objective func-tion consisting of the difference between observed and synthetic wave field in a suitable norm, usually L2 norm. This choice results in least-squares formulation. However, solving full-waveform inversion problems using nonlin-ear optimization methods remains a challenge. This is mostly because of mathematical and numerical difficul-ties, which include multiple minima and ill-posedness. The objective function is relatively insensitive to wave number components of the model that are shorter than wavelength. The eigenvalues of the linearized forward operator corresponding to those components are nearly zero. Recently, the technique of multifrequency inversion of full-waveform has been investigated. For frequency

domain inversion, the important issue is how to choose the various frequencies for inversion [10]. Moreover, the wave modeling in the frequency domain requires solving a large-scale system at each iteration which is time-consuming for large-scale problem. Comparing with the frequency-domain inversion methods, the time-domain methods have the advantage of high effi-ciency in forward modeling.

The algorithms used to solve the optimization problem include linearized approximation, the steepest descent algorithm, the conjugate gradient (CG) [11] and the quasi-Newton methods [12-15]. However, these methods behave differently as their own characters and the essen-tial ill-posedness of the problem. In this paper, we im-plement full-waveform inversion with the CG method which is a typical gradient-based method, and with the BFGS method which is a representative of the quasi- Newton methods. As to the forward method, the finite- difference method is employed as its high efficiency. The discretization schemes including the absorbing boundary conditions and corner conditions are given.

2. Theoretical Methods

2.1. Finite Difference Simulation

The wave equations can be solved in either the time or the frequency domain. The usual numerical methods

Copyright © 2013 SciRes. AJCM

W. S. ZHANG, J. LUO 14

such as the finite difference method, the finite volume method and the finite element method can all be em-ployed. Here we use the finite difference method for its high efficiency. The 2-D acoustic wave equation in the time domain can be written as

2 2 2

2 2 2 2

1 ( )

( , )

( ) ( ) ( ),s s

u u u

v x z t x z

f t x x z z

(1)

where is the pressure, is the velocity, ( , , )u x z t ( , )v x z( )f t is the source time function, ( , )s sx z is the source

location. Usually the initial conditions are zero, i.e.

( , , 0) 0, ( , , 0) 0.tu x z t u x z t (2)

Let , be the pressure at the time and at the spatial position , and ,n m be the velocity at

, then the difference scheme of (1) is

ln mu

m z

l t( ,n x m z )

)v

( ,n x1 1

, , ,

1, , 1,2 2, 2

1, , 1,

2

2

2 (

2 )

( ) ( ) ( ),

l l ln m n m n m

l lln m n m n m

n m

l lln m n m n m

s s

u u u

u u uv t

x ru u u

x rf l t n n m m

(3)

with the initial conditions 1 0, ,0, 0n m n mu u (4)

The numerical stability condition for (3) and (4) is

min( , ) 2 max( ).x z t v (5)

In the wave simulation, the unsuitable choice of time and spatial steps may cause serve dispersion and wave distortion. In order to suppress numerical dispersion, it is usually required that there are 10 sampling points in a wavelength for the second-order difference scheme, i.e.,

max

min( )min( , ) .

10

vx z

f (6)

Obviously for the same frequency the numerical dis-persion increases as the spatial steps rise. However, the dispersion can be depressed if the high-order accurate computational schemes are used.

In numerical computations, the computational domain is usually truncated to a finite region. So absorbing bound-ary conditions (ABCs) are required to absorb boundary reflections of outgoing waves. A number of ABCs have been introduced in wave simulation, for example, the paraxial approximation method [16] and the perfectly matched layer (PML) method [17]. The PML technique was proposed for electromagnetic fields equations, but can be applied to other wave equations and can fit ir-regular boundary.

Suppose the computational domain is a rectangular area

{( , ),0 ,0 }.x z x X z Z

We set the boundary condition on surface ( z 0 ) be the Dirichlet boundary condition as the data are gathered near the surface in our problem. For the boundary condi-tions of other three sides and four corners, we select the following Clayton and Engquist [16] conditions.

For the top boundary:

, 0, 0u x z t . (7)

For the bottom boundary:

2 2 2

2 2

1, ,

2

u u v ux z Z t

z t v t x

0.

(8)

For the left boundary:

2 2 2

2 2

10, , 0.

2

u u v ux z t

x t v t z

(9)

For the right boundary:

2 2 2

2 2

1, , 0.

2

u u v ux X z t

x t v t z

(10)

The absorbing conditions for four corners can be de-rived through the 45°rotation of the ABCs above. The results are the following:

For the left-upper corner:

20.

u u u

x z v t

(12)

For the right-upper corner:

20.

u u u

x z v t

(13)

For the left-down corner:

20.

u u u

x z v t

(14)

For the right-down corner:

20.

u u u

x z v t

(15)

The difference schemes for the absorbing boundary condition (7)-(10) can also be obtained by the second- order central finite difference method, i.e.,

1, 0,l

n mu (16)

0 ,, ,2

,

, 1 1, 1 ,2

1( )

2 2

( ) 0,4

tz ln m lt t l

n m n mi j

i j lx x ln m n m

uu u

t z t v

vu u

x

1

(17)



0 ,, 1,2

,

, 1 11, ,2

1( )

2 2

( ) 0,4

tx ln m lt t l

n m n mi j

i j lz z ln m n m

uu u

t x t v

vu u

z

(18)

0 ,, 1,2

,

, 1 11, ,2

1( )

2 2

( ) 0,4

tx ln m lt t l

n m n mi j

i j lz z ln m n m

uu u

t x t v

vu u

z

(19)

where and are the forward and backward differ-ence ope ators with first-order accuracy respectively, r 0 is th ral rence operator with second-curacy,

e cent diffe order ac-x and z are the spatial steps in x and

directions respectively. Similarly, the difference schemes rs

z

for four corne conditions (12)-(15) can be obtained in the following:

For the left-upper corner: ( , ) (0, 1)n m , 1 1

1 1, 2 , 1 3 ,1,

1 2 3

.l l ln m n m n ml

n m

r u r u r uu

r r r

upper corner:

(20)

For the right- ( , ) ( 1, 1)xn m N , 1

1 1, ,1

ln m ml

r u

12 , 1 3

,1 2 3

.l ln m n

n m

r u r uu

r r r

(21)

For the left-down corner:

z( , ) (0, 1), ( 2),n m N 1 1

1 1, 2 , 1 3 ,1,

1 2 3

1, 1), (0,

.

z z

l l ln m n m n ml

n m

N N

r u r u r uu

r r r

(22)

For the right-down corner:

1 11 1, 2 , 1 3 ,1

,1 2

n m r r 3

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

( 1, 2),

.

x z x z

x z

l l ln m n m n ml

n m N N N N

N N

r u r u r uu

r

where

(23)

1 2 3,

2, , .

n m

t tr r r

x z v

nversion Algorithms

Full-waveform inversion is minimization of the follow-ing objective function

(24)

2.2. Two Full-waveform I

2 21 1

( ) || || || || ,2 2obs calf v u u u (25)

n form

the

where obsu is the observed wave field and calu is the synthetic data. The functional (25) can be written the following discriminatio

2, )} ,n (26) 0 0 0

( ) { ( , , ) ( ; , ,s t r

s t

N N N

obs t s cal r r t sn n nr

f v u r n n u v n m n

where number of spatial points is x zM N N＝ , which is also the dimension of unknown velocity v. As-sume that the receiver number is rN and the source number is sN

tNthen the dimension of ( , , )obs t su r n n is

r sN N , here is the time

t fnegative

tN sampling points. r conjugateThe optimization methods such as nonlinea

gradient and BFGS can be implemented. The gradient- based methods are the commonly used methods in multi- dimensional unconstrained optimization. The CG method is one and i uses the conjugate directions instead o the

ient direction as search direction, whichgrad can be written as

1,

, 0,

1.k

kk k k

g kp

g p k

(27)

Different choices of k lead to different conjugate gradient methods. For example, the FR method [11], the HS method [18] and the PRP method [19, 20]:

21

2

|| ||, 1

|| ||

Tk kHSFR

k k Tk k k

g,kg y

g p y (28)

1

2,

|| ||

Tk kPRP

kk

g y

g (29)

where k1k ky g g partial derivative

. Here we adopt the PRP mThe of with resp

known velocity v is

ethod. ect to un- ( )f v

( ),T

f vJ u

v

(30)

where J et derivative matrix with dimension is the Fréch( )t r

M N N :

0 1 1

, .Mv

, ,cal cal cal calu u u uJ

v v v

(31)

The gradient g is a vector with dimension M , which can as be written

.Tg J u (32)

For the line search method we may use the Aigorithm [21] or the strong Wolfe algorithm [22, 23]. The A

,

mijo al-

imijo algorithm requires

1) ( ) ( )TJJ v p J v b p v （ (33)

where 1 (0,1)b and we select 1 0.001b in compu-tations. And the strong Wolfe algorithm requires the fol-lowing condition besides condition (33) above

T

2| ( )| (J Jp v p b p v ),T (34)

where 2 1( ,1)b b ow we g

and we select in e

1: Give the initial velocity model and the iterative

2 0.9b full-wav

computa-tions. N ive the following form inver-sion algorithm combining with the CG method.

Algorithm 1:

0v



|| || stopg . Set itera-stopping condition of the gradient: tion num ;

2: Compute the objective function 0f and the gradient g. If the gradient satisfies the stopping condition || ||

0k ber

g

stop then stop the iteration. Otherwise set p0 0g ; 3: Set 1k k . If maxk k then continue the itera-

tion; 3.1: Use the strong Wolfe line sea thm to get

the

rch algori step k ; 3.2 : Update the velocity model: 1k k k kv v p ; 3.3: Compute the objective function (f v an 1 1k k ) d

the gradient 1kg ; e gradi3.4: If t ent sa he stopping criterion then

goh tisfies t

k

out of the loop; 3.5: If the updating quantity of velocity satisfies the

stopping condition then goes out of the loop; 3.6: Calculate, 1 1, k k k k ks v v y g 3.7: Calculate

g ; PRPk and update

1kp : 1 1PRP

k k k kp g p .

Set 1k k , return step 3. The algorithm above doesn’t use the information of

second-order derivate of objective function. The BFGS m asi-Newton oposed de roy farb and Shanno [1 key of the qu method is ho con g t so-called qu

ethod is a qu methodpendently by den, Fletcher

which prand Gold

asi-Newtonhe following

B2-15]. Tstruc

he w to t 1kB satisfyinasi-Newton condition:

1 ,k k kB s y (35)

where 1k k ks v v and 1 .k k ky g g The BFGS method modifies matrix 1kB according to the follow-ing expression

1

T T

k k

B ss B yyB B ,k k

Tk

Ts B s (36)

in

s y

and its n by verse is give

1

1[ 1] [ ],

TTT T

k k

y Hss kk kT T T

yH H sy H H ys

y s y s y s

where

(37) 1

k kH B . If kH is positive definite then 1kH following

based is positive definite as long as . The algo form gorithmon the BFG thod.

Algorithm1: Give t tial velocity model and the iter

l co the

0Tky s

inversion alrithm is the full-S me

2: he ininditio

wave

0v ative termina n of gradient: || || stopg . Set -tion number 0k ;

2: Compute the objective function 0f and the gradient g. If the gradient satisfies the stopping rule || ||

itera

stopg the ation. n stop the iter Otherwise set

tio

0 0 ; p g 3: Set 1k k . If maxk k then continue the itera-n;

3.1: Use the strong Wolfe line search algorithm to ob-tain the step k ;

3.2 : Update the velocity model: 1k k k kv v p ; 3.3: Compute the objective function 1kf

the gradient 1k

1k ( )v andg ;

3.4: If ient the stopping condition, then go o

th satisfiesut of the loop;

iti , then go out of the loop;

e grad

3.5: If the updating quantity of velocity satisfies the stopping cond on

3.6: Calculate 1k k ks v v , ky 1k

3.7: Calculate variable scaling factor witkg g ;

h

and let ;k kk k k kT

k k k

B Bs B s

Ty s

3.8: Update 1kB with (36). 3.9: Use the Cholesky method to solve the equation

1 1 1 1 to get .k k k kB p g p

1k k Set , continue with step 3. In step 3.7 above, the scaling factor k is introduced

which can improve convergence [24]. CG method is on nve he BFGS method is su-peeach iteration.

3. merica

ary conditions are used. Figure 2 is the 1 but the ABCs are ndary reflections in

The ly linearly co rgent while trlinear convergent but requires solving a system at

Nu l Computations

First of all, we test the validity of the finite difference method for wave modeling. Figure 1 is the wave filed received near the surface for a fixed source. The bound-ary reflections are very serious in Figure 1 since only Dirichlet boundwave field corresponding to Figureused. We can see the serious bouFigure 1 are eliminated obviously in Figure 2.

Figure 1. Wavefield received near the surface by the finite difference method with Dirichlet boundary conditions.



Next we test two inversion algorithms for a simple model shown in Figure 3. The model has a homogene-ous background media with velocity 2000 m/s and two square areas with greater abnormal velocity. The left square media has velocity 2600 m/s and the right square media has the velocity 2100 m/s. The configuration of sources and receivers are shown in Figure 4. There are 16 sources totally and 31 receivers for each source. The source step is 10 m and the receiver step is 5 m. The source line and receiver line are 10 m and 5 m below the surface respectively. The spatial discretization points are

and . The spatial steps are 17xN 33zN x z 5m . The parameter stop is The maxi

a is 200. The initial velocit

710 . mum num-y 0v is ber of iter

2000m/s. tions maxk

Figure 2. Wavefield received near the surface by the finite difference method with absorbing boundary conditions.

Figure 3. True model for testing two inversion algorithms.

re-lt after 20 iterations and Figure 6 is that after 200 it-

erations. For the BFGS-based algorithm the inversion results are shown in Figures 7-9, which are the results after 20 iterations, 100 iterations and 200 iterations, re-spectively. Comparing Figures 7-9 with Figures 5 and 6, we can see that the BFGS-based algorithm has higher inverse accuracy than the CG-based algorithm. In order to test the ability of the BFGS-based algorithm further, we inverse the Marmousi model next.

The Marmousi model is a benchmark model for testing the ability of migration or inversion methods [25]. The velocity model is shown in Figure 10. We can see it has many complex structures. The synthetic data is also gen-erated by the finite difference code. The following com-putations parameters are adopted:

The inversion results with the CG-based algorithm are shown in Figures 5 and 6. Figure 5 is the inversion su

493xN 249zN ,

Figure 4. The configuration of sources and receivers for full-waveform inversion.

Figure 5. Inversion result after 20 iterations with the CG- ased Algorithm 1. b



Figure 6. Inversion result after 200 iterations with the CG-based Algorithm 1.

Figure 7. Inversion result after 20 iterations with the BFGS- based Algorithm 2.

Figure 8. Inversion result after 100 iterations with thBFGS-based Algorithm 2.

e

Figure 9. Inversion result after 200 iterations with the BFGS-based Algorithm 2.

Figure 10. Marmousi model with complex structures.

Figure 11. Initial velocity model for inversion.

6x z m , 0.6t ms , 3501N . t is initial model and Figure 12 is the inversion resu

Figure 11lt

fter 200 iterations. This result clearly shows that BFGS- based algorithm has obtained very good inversion accu-racy.

tahe



FigureFGS-

tic data. The algorithm is independent of the mensionality of the problem. Two full-waveform inver-sion algorithms based on the CG method and BFGSmethod are proposed. Numerical computations for twomodels including complicated Marmousi model shthat the BFGS-based method has higher inversion accu-racy than the CG-based algorithm because the later usesthe second-order derivative information in computations.The good inversion results for complicated Marmmodel show that the BFGS-based algorithm can be ex- pected to inverse real data in the future.

5. Acknowledgements

This research is supported by the State Key project with grant number 2010CB731505 and the National Center

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