Contrast source inversion methods in elastodynamics - Acoustical ...

Contrast source inversion methods in elastodynamics George Pelekanos Department of Mathematics and Statistics, Southern Illinois University, Edwardsville, Illinois 62026

Aria Abubakar Schlumberger-Doll Research, 36 Old Quarry Road, Ridgefield, Connecticut 06877

Peter M. van den Berg Center for Technical Geoscience, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands

共Received 5 May 2001; accepted for publication 25 August 2003兲 In this paper a nonlinear inversion method is presented for determining the mass density of an elastic inclusion from the knowledge of how the inclusion scatters known incident elastic waves. The algorithm employed is an extension of the multiplicative regularized contrast source inversion method 共MR-CSI兲 to elasticity. This method involves alternate determination of the mass density contrast and the contrast sources 共the product of the contrast and the fields兲 in each iterative step. The simple updating schemes of the method allow the introduction of an extra regularization term to the cost functional as a multiplicative constraint. This so-called MR-CSI method 共MR-CSI兲 has been proven to be very effective for the acoustic and electromagnetic inverse scattering problems. Numerical examples demonstrate that the MR-CSI method shows excellent edge preserving properties by robustly handling noisy data very well, even for more complicated elastodynamic problems. © 2003 Acoustical Society of America. 关DOI: 10.1121/1.1618751兴 PACS numbers: 43.60.Pt, 43.10.Nq, 43.10.Sv 关VWS兴

I. INTRODUCTION

As described in Pao,1 the essential properties of a medium are defined by the stiffness coefficients and the mass density. Direct or forward problems are concerned with the determination of the field when the source and medium properties are given. Inverse problems deal with the determination of the material properties when the sources are known and some data measuring the responses are given. Not much is known about solutions of inverse elasticity problems. Interested readers may find an introduction to the subject in the monograph by Claerbout.2 Ayme´-Bellegarda and Habashy3 introduced a generalized noniterative inverse solution for 2D and 3D scatterers in a multilayered solid elastic medium; they used an exact dyadic contrast-source observation equation simplified in the Born framework, variations in the two Lame´ constants and the density being arbitrary, while numerical results are given for a 2D variation in density. We need not emphasize the importance of inverse problems since many of the interesting applications of the theory of elastic waves are inverse problems. They include the wellknown problem of locating an earthquake source from seismograms, determining the nature of an earthquake source, finding the oil-trapping porous layer in the earth, and determining the ocean bottom from reflection data. Motivated by the importance of these problems, Pelekanos et al.4 used an iterative technique to reconstruct the spatially dependent density of a two-dimensional object from measurements of the elastic field scattered when the object is illuminated by known sources. The inverse problem is formulated as an optimization problem, in which the cost functional is the sum of two terms: one is the defect in matching measured field data with the field scattered by a body of particular real denJ. Acoust. Soc. Am. 114 (5), November 2003

Pages: 2825–2834

sity 共error in data equation兲 and the second is the error in satisfying the equations of state, a system of integral equations for the field due to each excitation 共error in the object equation兲. The density and the fields are updated by a linear iterative method in which the updating directions are weighted by parameters which are determined by minimizing the cost functional. The method we implement here is an extension of the ideas presented in van den Berg and Kleinman,5 and then extended by van den Berg et al.6 and van den Berg and Abubakar.7 This so-called contrast source inversion 共CSI兲 method reconstructs the mass density contrast and the contrast sources alternatingly in each iterative step by minimizing the cost functional consisting of errors in the data and object equations. The method is very attractive, since it does not need to solve any full forward problem in each iteration. Later on, the CSI method has been modified to include a multiplicative regularization factor by van den Berg et al.6 and extended by Abubakar et al.8 to the full-vectorial electromagnetic 3D problem. This multiplicative technique allows the method to use a regularization factor without the necessity of determining an artificial weighting parameter. This regularization parameter is determined by the iterative process itself, which makes the method very suitable to invert experimental data as shown by Bloemenkamp et al.9 In the present paper the improved version of the method, the so-called multiplicative regularized CSI 共MR-CSI兲 method, described in van den Berg and Abubakar,7 is applied to the even more challenging elasticity inverse scattering problem. Numerical examples demonstrate that the MR-CSI method shows excellent edge-preserving properties by robustly handling noisy as well as limited data very well.

0001-4966/2003/114(5)/2825/10/$19.00

© 2003 Acoustical Society of America

2825

usct,( j) 共 x兲 ⫽ 共 k S 兲 2 ␮

冕

D

⌫共 x,x⬘ 兲 ␹ 共 x⬘ 兲 u( j) 共 x⬘ 兲 dv共 x⬘ 兲 , 共4兲

x苸S.

For convenience, in the solution of the inverse problem the P 共dilatation兲 and SV 共rotation兲 components of the twodimensional scattered field are computed 共Eringen and Suhubi12兲. The scattered P component is given by j) j) u P,( j) 共 x兲 ⫽ ⳵ 1 u sct,( 共 x兲 ⫹ ⳵ 2 u sct,( 共 x兲 , 1 2

x苸S,

共5兲

j⫽1,...,J.

Consequently, the scattered SV component is given by j) j) u SV,( j) 共 x兲 ⫽ ⳵ 1 u sct,( 共 x兲 ⫺ ⳵ 2 u sct,( 共 x兲 , 2 1

x苸S, FIG. 1. The geometry of the scattering experiments.

II. NOTATION AND PROBLEM STATEMENT

Consider a bounded, simply connected domain D located in an unbounded homogeneous background medium with constant Lame´ coefficients ␭ and ␮. Let a scattering object 共or objects兲 B, whose location and contrast are unknown, be embedded in D and assume that B is made by an inhomogeneous elastic material, also characterized by the constant Lame´ coefficients ␭ and ␮. The vector x denotes the vectorial position in R2 . We irradiate the object by a number of known incident fields of one single frequency, uinc,( j) (x) ⫽uinc(x,x⬘j ), j⫽1,...,J, and source points x⬘j . The sources are located in a domain 共or on a curve兲 S outside of and surrounding D, where the scattered field is measured as well. This scattering configuration is shown in Fig. 1. It has been shown 共see Kupradze10兲 that the total field in D satisfies the following domain integral equation: u( j) 共 x兲 ⫽uinc,( j) 共 x兲 ⫹ 共 k S 兲 2 ␮ ⫻u( j) 共 x⬘ 兲 dv共 x⬘ 兲 ,

冕

D

⌫共 x,x⬘ 兲 ␹ 共 x⬘ 兲

x苸D,

␳ 共 x⬘ 兲 ⫺ ␳ e , ␳e

共1兲

i 1 i 1 P IH (1) 共 k S R 兲 ⫺ ““ 关 H (1) 0 共k R兲 4 ␮ 0 4 ␻ 2␳ e S ⫺H (1) 0 共 k R 兲兴 ,

共3兲

where R⫽ 兩 x⫺x⬘ 兩 , while k P and k S are the wave numbers for the P- and the S-wave, respectively and I is a unit matrix. Note that if x⬘ is not in B, then the contrast function ␹ vanishes outside B. If u( j) solves the equation above, then the scattered field is obtained from the integral representation 2826

J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003

w( j) ⫽ ␹ u( j) ,

共7兲

which is called the contrast source. Multiplying both sides of 共1兲 by ␹, and using 共7兲, we obtain the object equations in terms of the contrast sources as follows: w( j) ⫺ ␹ ZD w( j) ⫽ ␹ uinc,( j) ,

x苸D,

j⫽1,...,J,

共8兲

where ZD w( j) 共 x兲 ⫽ 共 k S 兲 2 ␮

冕

D

⌫共 x,x⬘ 兲 w( j) 共 x⬘ 兲 dv共 x⬘ 兲 ,

x苸D. 共9兲

Using 共7兲, the data equations become ZSP •w( j) ⫽u P,( j) ,

on S,

共10兲

( j) SV,( j) ZSV , S •w ⫽u

on S,

共11兲

( j) are set equal to the right-hand where ZSP •w( j) and ZSV S •w side of Eqs. 共5兲 and 共6兲, respectively.

共2兲

is the normalized contrast between the mass density ␳ ⫽ ␳ (x) of the scattering object and the mass density ␳ e of the homogeneous embedding. The Green’s displacement tensor ⌫(x,x⬘ ) is given by Morse and Feshbach11 as ⌫共 x,x⬘ 兲 ⫽

where ⳵ 1 and ⳵ 2 denote the spatial differentiations. It is easily observed that in the equations above the contrast and fields occur as a product, hence, as in van den Berg and Kleinman,5 we introduce the quantity

and

where

␹ 共 x⬘ 兲 ⫽

共6兲

j⫽1,...,J,

III. CONTRAST SOURCE INVERSION METHOD

The application of the contrast source inversion in the inverse elasticity problem is based on the construction of sequences of sources w(nj) and contrast ␹ n in order for a cost functional to be minimized. In our specific problem there are three error measurements involved; the first is the defect in matching the measured P data in L 2 (S), the second is the defect in matching the measured SV data also in L 2 (S), and the third is the error in the object equation in L 2 (D). We combine these error measures into a normalized cost functional F n 共 w( j) , ␹ 兲 ⫽F S 共 w( j) 兲 ⫹F D,n 共 w( j) , ␹ 兲 ,

共12兲

where Pelekanos et al.: Contrast source inversion

inner product on L 2 (D). The expression for the gradient is found to be

J

F S 共 w 兲 ⫽w S ( j)

兺 共储u

P,( j)

j⫽1

⫺ZSP •w( j) 储 2S

쐓

( j) 2 ⫹ 储 u SV,( j) ⫺ZSV S •w 储 S 兲 ,

( j) 쐓 ( j) ⫹w D,n⫺1 共 rn⫺1 ⫺ZD 兲兲 , 共 ¯␹ n⫺1 rn⫺1

and

P쐓

J

F D,n 共 w , ␹ 兲 ⫽w D,n⫺1 ( j)

兺

储␹u

inc,( j)

j⫽1

⫺w

( j)

2 ⫹ ␹ ZD w( j) 储 D .

共14兲 The normalization factors in 共13兲 and 共14兲 are chosen as

冋兺 J

w S⫽

j⫽1

and

共 储 u P,( j) 储 2S ⫹ 储 u SV,( j) 储 2S 兲

冋兺 J

w D,n⫺1 ⫽

j⫽1

2 储 ␹ n⫺1 uinc,( j) 储 D

册

册

⫺1

쐓

P,( j) SV,( j) ⫹ZSV g(n,j)v ⫽w S 共 ZSP q n⫺1 S q n⫺1 兲

共13兲

共23兲

SV 쐓

쐓 with ZD and ZS , ZS being the adjoint operators mapping from L 2 (D) and L 2 (S), respectively, into L 2 (D), and the overbar denotes complex conjugate. Once the update directions are chosen, the constant parameter ␣ n is determined to minimize the cost functional

F n 共 w( j) , ␹ n⫺1 兲 J

⫽w S

共15兲

( j) 2 兺 共 储 u P,( j) ⫺ZSP •w(nj) 储 2S ⫹ 储 u SV,( j) ⫺ZSV S •wn 储 S 兲

j⫽1

⫺1

J

⫹w D,n⫺1

.

The subscripts S and D are included in the L 2 norm to indicate the domain of integration. This is a quadratic functional in w( j) , but highly nonlinear in ␹. Note that the term ␹ ZD w( j) is responsible for the nonlinearity of the inverse problem. We propose an iterative minimization of this cost functional using an alternating method which first updates w( j) and then updates ␹.

兺

j⫽1

2 储 ␹ n⫺1 u(nj) ⫺w(nj) 储 D

J

⫽w S

P,( j) ⫺ ␣ n ZSP • v (nj) 储 2S 兺 共 储 q n⫺1

j⫽1

SV,( j) ( j) 2 ⫹ 储 q n⫺1 ⫺ ␣ n ZSV S •vn 储S兲 J

⫹w D,n⫺1

兺

j⫽1

( j) 2 储 rn⫺1 ⫺ ␣ n 共 v (nj) ⫺ ␹ n⫺1 ZD v (nj) 兲储 D ,

共24兲

and is found to be A. Updating the contrast sources

The method begins with the updating of the contrast sources w( j) in the following manner: Define the data errors to be q nP,( j) ⫽u P,( j) ⫺ZSP •w(nj) ,

n⫽1,2,...,

␣ n⫽

n⫽1,2,... .

The object error is r(nj) ⫽ ␹ n u(nj) ⫺w(nj) ,

共18兲

n⫽1,2,...,

共25兲

,

P,( j) SV,( j) ( j) f n, j ⫽w S 共 具 q n⫺1 ,ZSP • v (nj) 典 S ⫹ 具 q n⫺1 ,ZSV S •vn 典S兲

共16兲 共17兲

兺 j h n, j

where

( j) ⫹w D,n⫺1 具 rn⫺1 , v (nj) ⫺ ␹ n⫺1 ZD v (nj) 典 D ,

and j) ( j) q SV,( ⫽u SV,( j) ⫺ZSV n S •wn ,

兺 j f n, j

共26兲

and ( j) 2 ( j) h n, j ⫽w S 共 储 ZSP • v (nj) 储 2S ⫹ 储 ZSV S • v n 储 S 兲 ⫹w D,n⫺1 储v n 2 ⫺ ␹ n⫺1 ZD v (nj) 储 D

共27兲

and where 具 .,. 典 S denotes the inner product on L (S). After the contrast sources w(nj) are obtained, the total fields u(nj) can be found via 共19兲 2

where u(nj) ⫽uinc,( j) ⫹ZD w(nj) . We now assume that update w( j) by

共19兲

( j) wn⫺1

and ␹ n⫺1 are known and we

( j) ⫹ ␣ n ZD v (nj) . u(nj) ⫽un⫺1

共20兲

B. Updating the contrast

( j) w(nj) ⫽wn⫺1 ⫹ ␣ n v (nj) ,

where ␣ n is a constant and the update directions v (nj) are chosen as the Polak–Ribie`re conjugate gradient directions, namely v (0j) ⫽0,

( j) v (nj) ⫽g(nj) ⫹ ␥ n v n⫺1

n⭓1,

共21兲

If the contrast sources w(nj) and the fields u(nj) are known, the contrast ␹ n is now obtained by minimizing the second term in 共12兲, i.e., J

F D,n 共 w(nj) , ␹ n 兲 ⫽w D,n⫺1

with

␥ n⫽

( j) 兺 j 具 g(nj) ,g(nj) ⫺gn⫺1 典D 2 j) 兺 j 储 g(n⫺1, v储 D

,

共22兲

where g(n,j)v is the gradient of the cost functional with respect ( j) , ␹ n⫺1 while 具 .,. 典 D denotes the to w( j) evaluated at wn⫺1 J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003

共28兲

兺

j⫽1

2 储 ␹ n u(nj) ⫺w(nj) 储 D .

共29兲

The normalized error F D,n (w(nj) , ␹ n ) is minimized by choosing5

␹ n⫽

兺 j Re共 w(nj) •u(nj) 兲 兺 j 兩 u(nj) 兩 2

.

Pelekanos et al.: Contrast source inversion

共30兲 2827

Note that this result is identical to the result obtained by updating the contrast as

␹ n ⫽ ␹ n⫺1 ⫹ ␣ ␹n d n ,

共31兲

⫺1 where ␣ ␹n ⫽w D,n⫺1 and d n is the preconditioned gradient of ( j) F D,n (wn , ␹ n ), i.e.,

d n⫽

⫺w D,n⫺1 兺 j Re关共 ␹ n⫺1 u(nj) ⫺w(nj) 兲 •u(nj) 兴 兺 j 兩 u(nj) 兩 2

.

␣ 0⫽

쐓

쐓

,

共35兲

u(0j) ⫽uinc,( j) ⫹ZD w(0j) ,

共36兲

2 兺 j 兩 u(0j) 兩 D

is given in 共33兲. This completes the description of the nonregularized version of the contrast source inversion 共CSI兲 algorithm, where in each iteration an update of the contrast sources is followed by an update of the contrast. Note that this alternating scheme allows us to introduce a multiplicative constraint in a simple fashion. w(0j)

D. Inclusion of a multiplicative constraint

Recent work with image enhancement has shown that minimization of the total variation of the image can significantly improve the quality of the reconstruction, see, e.g., Acar and Vogel,13 Blomgren et al.,14 Dobson and Santosa.15,16 Van den Berg and Kleinman17 incorporated the total variation 共TV兲 in an inverse scattering problem by enhancing the modified gradient algorithm. In the latter approach a total variation term was added to the cost functional, resulting in a substantial improvement of the performance of the reconstruction method, both for ‘‘blocky’’ and smooth contrast configurations. The addition of the total variation to the cost functional has a very positive effect on the quality of the reconstructions for both blocky and smooth profiles, but a drawback is the presence of an artificial weighting parameter in the cost functional, which can only be determined through considerable numerical experimentation; see Hansen,18 and a priori information of the desired reconstruction. van den Berg et al.6 have suggested including the total variation as a multiplicative constraint, and hence the origi2828

where

P쐓 P,( j) SV,( j) 2 SV,( j) 2 储 S ⫹ 储 ZSV 兺 j 储 ZSP • 共 ZSP u P,( j) ⫹ZSV ⫹ZSV쐓 兲储 S S u S • 共 ZS u S u

兺 j Re共 w(0j) •u ¯ (0j) 兲

共33兲

쐓

where

and

SV,( j) w(0j) ⫽ ␣ 0 共 ZSP쐓 u P,( j) ⫹ZSV쐓 兲, S u

SV,( j) 2 兺 j 储共 ZSP u P,( j) ⫹ZSV 兲储 S S u

As far as the starting value of ␹ 0 is concerned, we obtain

␹ 0⫽

We now need to indicate starting values for w(0j) . Clearly zero is not a good choice, since then the cost functional 共12兲 will be undefined for n⫽1. To this end we choose as starting values the contrast sources that minimize the data error, that is the contrast sources obtained by backpropagation

共32兲

In case we have a priori information that the mass density is a positive quantity, we remark that this positivity constraint is easily implemented by enforcing a negative value to zero after each update of the contrast.

쐓

C. Choice of the initial estimate

J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003

共34兲

.

nal cost functional is the weighting parameter, i.e., determined by the inversion problem itself. This eliminates the choice of the artificial regularization parameters completely. In each iteration, a multiplicative cost functional is introduced as Fn 共 w( j) , ␹ 兲 ⫽ 关 F S 共 w( j) 兲 ⫹F D,n 共 w( j) , ␹ 兲兴 F TV n 共 ␹ 兲,

共37兲

where the first factor is the original cost functional 共12兲 of the CSI method, and where the second factor is the so-called TV factor. Recently, van den Berg and Abubakar7 have considered the TV factor as a weighted norm on L 2 (D), in which the weighting favors flat parts and nonflat parts of the contrast profile almost equally. This weighted TV factor has been shown to be more effective than the one given in van den Berg et al.6 Thus, in this paper we choose to use a TV factor similar to the one used in van den Berg and Abubakar.7 This weighted total-variation factor is given by F TV n 共 ␹ 兲⫽

1 V

冕

2 兩 “ ␹ 共 x⬘ 兲 兩 2 ⫹ ␦ n⫺1

D 兩 “ ␹ n⫺1 共 x⬘ 兲 兩

2

2 ⫹ ␦ n⫺1

dv共 x⬘ 兲 ,

共38兲

where V⫽ 兰 D dv (x⬘ ) denotes the area 共two-dimensional volume兲 of the test domain D. The quantity ␦ 2 in 共38兲 is introduced for restoring differentiability to the total variation and is chosen as 2 ˜ 2, ␦ n⫺1 ⫽F D,n⫺1 ⌬

共39兲

˜ denotes the reciprocal mesh size of the discretized where ⌬ domain D, and F D,n⫺1 is the normalized norm of the object error of the previous iteration, cf. 共29兲. We have chosen ␦ 2 to be large in the beginning of the optimization and small towards the end; hence, optimization will reconstruct the contrast in the first iterations in a normal way before it will apply the minimization of variation to shape the image further. Pelekanos et al.: Contrast source inversion

The choice of the multiplicative cost functional 共37兲 is based on two things: the objective of minimizing the error in the data and object equations and the observation that the weighted TV factor, when minimized, converges to 1. The structure of this cost functional is such that it will minimize the total variation with a large weighting parameter in the beginning of the optimization process, because the value of cost functional F n (w( j) , ␹ ) is still large, and that it will gradually minimize more and more the error in the data and object equations when the weighted TV factor has reached a nearly constant value close to 1. If noise is present in the data, the data error term will remain at a large value during the optimization; therefore, the weight of the weighted TV factor will be more significant. Hence, the noise will at all times be suppressed in the reconstruction process and we automatically fulfill the need of a larger regularization factor

g ␹n ⫽

when the data contains noise as suggested by Chan and Wong19 and Rudin et al.20 By introducing this cost functional Fn , the TV factor does not change the updating of the contrast sources w(nj) and the fields u(nj) , because F TV n ( ␹ n⫺1 )⫽1 at the beginning of each iteration. The updating scheme for ␹ n is given by

␹ n ⫽ ␹ n⫺1 ⫹ ␣ ␹n d n ,

where the update directions d n are taken as Polak–Ribie`re conjugate gradient directions of the cost functional 共37兲, viz., d 0 ⫽0,

d n ⫽g ␹n ⫹

␹ Re具 g n␹ ,g n␹ ⫺g n⫺1 典D ␹ ␹ ,g n⫺1 具 g n⫺1 典D

d n⫺1 ,

n⭓1, 共41兲

while the preconditioned gradient is determined as, cf. 共32兲

⫺w D,n⫺1 兺 j Re关共 ␹ n⫺1 u(nj) ⫺w(nj) 兲 •u(nj) 兴 ⫹F n 共 w( j) , ␹ n⫺1 兲 g TV n 兺 j 兩 u(nj) 兩 2

共40兲

共42兲

,

冋

where

F⫽ F S 共 w(nj) 兲 ⫹F D,n 共 w(nj) , ␹ n⫺1 兲

冋

册

1 ⵜ ␹ n⫺1 . g TV 2 n ⫽ “• V 兩 “ ␹ n⫺1 兩 2 ⫹ ␦ n⫺1

⫹2 ␣ w D,n Re

共43兲

⫹ ␣ 2 w D,n Note that the gradient tends to the direction d n of 共32兲 of the original CSI method as the gradient, g TV n , tends to zero. The weighting of the gradients clearly depends on the errors in the cost functional F n and the weighted TV factor F TV n . Similar to an additive regularization, the present multiplicative regularization decreases the chance that the gradient has a zero direction, which reduces the possibility to arrive at in a local minimum. Comparing this gradient with the one of the nonweighted TV factor as in van den Berg et al.,6 we immediately observe that apart from a constant factor, these gradients are identical. Hence, this weighted TV factor combines the features of minimization of the TV in the L 2 norm and in the L 1 norm through its gradient. With the Polak–Ribie`re update directions completely specified, the real-valued constant ␣ ␹n in 共40兲 is found as

␣ n␹ ⫽arg min兵 关 FS 共 w(nj) 兲 ⫹F D,n 共 w(nj) , ␹ n⫺1 ⫹ ␣ real ␣ ⫻F TV n 共 ␹ n⫺1 ⫹ ␣ d n 兲 其 .

d n 兲兴 共44兲

The minimization of the multiplicative cost functional 共44兲 can be performed analytically. The cost functional is a fourth-degree polynomial in ␣ ␹ , viz., J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003

兺j 具 d n u(nj) , ␹ n⫺1 u(nj) ⫺w(nj) 典 D

兺j 储 d n u(nj) 储 D2

册

⫻ 关 1⫹2 ␣ Re具 b n⫺1 ⵜ ␹ n⫺1 ,b n⫺1 ⵜd n 典 D 2 ⫹ ␣ 2 储 b n⫺1 ⵜd n 储 D 兴,

共45兲

where 2 b n⫺1 ⫽V ⫺ 1/2共 兩 ⵜ ␹ n⫺1 兩 2 ⫹ ␦ n⫺1 兲 ⫺ 1/2.

共46兲

Differentiation with respect to ␣ yields a cubic equation with one real root and two complex conjugate roots. The real root is the desired minimizer ␣ ␹n . In our numerical examples we use this weighted TV factor as the multiplicative regularization of the CSI method, and we name this method the MRCSI method.

IV. NUMERICAL RESULTS

In this section some numerical examples of inversion using the MR-CSI method will be presented. The discretized versions of the various operators used in the algorithm can be found in Pelekanos et al.21 Further, in order to be able to compare the reconstruction results using the data excited either by P- or SV-waves, we introduce the quantity error in contrast as follows: ERRn ⫽

2 储 ␳ 共 x兲 ⫺ ␳ exact共 x兲储 D 2 储 ␳ exact共 x兲储 D

.

Pelekanos et al.: Contrast source inversion

共47兲 2829

FIG. 2. The approximate original profile of the circular cylinder.

In all of our numerical examples, after generation of synthetic data by either the Bessel series solution 共see White22兲 or the conjugate gradient FFT method 共see Pelekanos et al.21兲, 10% additive random white noise is added. Moreover, the simple inclusion of a priori positivity constraint is used. We excite our object by either P- or SV-waves. The object will scatter both P- and SV waves. Subsequently, we reconstruct the object by measuring the contributions from both kinds of scattered waves. A. Reconstruction of a circular cylinder

In our first numerical example the scatterer was taken to be a circular cylinder of radius a⫽0.35 m and density ␳ ⫽2.0, while the outer medium’s density was ␳ e ⫽1. Hence, the contrast is ␹ ⫽1.0. The circular cylinder is centered at (⫺0.35,⫺0.35) m. The scatterer is located in the test domain D. This test domain was divided into 61⫻61 subsquares of 0.0344⫻0.0344 m2 . Note that this fine discretization grid is needed in order to adequately model the boundary of the circular cylinder. The wave numbers of the P- and SV waves are k P ⫽3 and k S ⫽6, respectively. This means that the side length of the test domain was equal to about one wavelength for the P waves and two wavelengths for the SV waves in the exterior medium. The measurement surface S was chosen to be a circle with a radius of 2 m. Twenty-nine stations (J⫽29) were located uniformly on this circle, with each station serving successively as a line source and all stations acting as receivers. Throughout the description of the solution for both direct and inverse problems, it became apparent that the method used is essentially the same. There is a suspicion, however, that the use of ‘‘measured’’ data in the inverse problem, which are produced by the same numerical method 共forward problem兲 can result in an ‘‘inverse crime.’’ Therefore in this first example we use the exact data obtained from the Bessel series solution; see White.22 Furthermore, after generation of synthetic data we add 10% random additive white noise to the scattered fields for both P- and SV waves. The approximate model of the circular cylinder is given in Fig. 2. For P-wave incidence excitation, the reconstruction after 512 iterations using the CSI method is shown in Fig. 3共a兲. Note that although the number of iterations is large, we do not solve any full-forward problem in each iteration. One iteration of the CSI method for this example takes only 10 s on a personal computer with a 600-MHz Pentium II proces2830

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FIG. 3. The reconstruction results of the circular cylinder from excitation with P waves 共a兲 and SV waves 共b兲 using the CSI method. The data sets are generated using series of Bessel function solutions with 10% additive random white noise.

sor. For SV-wave incidence, the reconstruction after 512 iterations using the CSI method is shown in Fig. 3共b兲. The top values of the reconstructed circular cylinder for P- and SV-wave incidence are approximately equal to ␹ ⫽1.15 and ␹ ⫽1.05, respectively. Further, we observe that the boundary of the circular cylinder is better reconstructed from data generated by SV incident waves. Next, we use the CSI method using the multiplicative regularization factor 共the MR-CSI method兲. The reconstructions after 512 iterations are shown in Fig. 4共a兲 for P-wave incidence, while for SV-wave incidence in Fig. 4共b兲. The improvements are remarkable and the reconstruction time per iteration in the inversion algorithm is only increased by approximately 20%. Now, the top values of the reconstructed circular cylinder for both P- and SV-wave excitation are approximately equal to ␹ ⫽1.05. Nevertheless, the SV-wave incidence still gives a more accurate result. In order to have some idea about the convergence behavior of the reconstruction results from both excitations, we present in Fig. 5 the error in contrast ERRn , see 共47兲, as a function of the number of iterations. The errors in contrast for P-wave incidence are given by the dotted line for the CSI method and the dashed line for the MR-CSI method. The errors in contrast for SV incident waves are given by the dashed-dotted line for the CSI method and the solid line for the MR-CSI method. Note that, even at the very beginning of the iterative process 共after backpropagation兲, the errors in contrast of the reconstructed profile from P-wave excitations for both methods are always larger than the ones from SV-wave excitations. This is an indication that the bad rePelekanos et al.: Contrast source inversion

FIG. 6. The original profile of the three distinct square cylinders.

FIG. 4. The reconstruction results of the circular cylinder from excitation with P waves 共a兲 and SV waves 共b兲 using the MR-CSI method. The data sets are generated using series of Bessel function solutions with 10% additive random white noise.

constructed boundary of the circular cylinder from P-wave incidence is due to the physical behavior of the incident waves; it is not due to the behavior of the inversion algorithm. B. Reconstruction of three distinct square cylinders

In our second numerical example we consider the inversion of multiple scatterers. The scatterers were taken to be three square cylinders of diameter 3/4 m. The mass densities

for the scatterers were ␳ ⫽2.0, ␳ ⫽2.5, and ␳ ⫽3.0, while the outer medium’s density was ␳ e ⫽1.0. Hence, the contrasts are ␹ ⫽1.0, ␹ ⫽1.5, and ␹ ⫽2.0. The scatterers are assumed to be located in a test domain D. This test domain was divided into 29⫻29 subsquares of 0.1034⫻0.1034 m2 . The wave numbers of the P- and SV-waves are k P ⫽3 and k S ⫽6, respectively. This means that the side length of the test domain was equal to about one and one half wavelengths for the P waves and three wavelengths for the SV waves in the background medium. The measurement surface S was chosen to be a circle of radius 3 m. Twenty-nine stations (J⫽29) were located uniformly on this circle, with each station serving successively as a line source and all stations acting as receivers. The data are generated numerically using the CG-FFT method described in Pelekanos et al.21 The actual profile which has been used to generate synthetic data is given in Fig. 6. After generation of synthetic data we added 10% random additive white noise to the scattered fields for both P- and SV-waves. The reconstructed results after 512 iterations from P-wave incidence using the CSI method are presented in Fig. 7共a兲. One iteration of the CSI method for this example takes only 2.5 s on a personal computer with a 600-MHz Pentium II processor. However, poor reconstruction results are observed. The contrast value levels for the square cylinders are equal to ␹ ⫽0.85, ␹ ⫽1.2, and ␹ ⫽1.45. Inversion of the SV-wave data yields a minor improvement. The results are shown in Fig. 7共b兲. The contrast value levels for the square cylinders are approximately the same as the ones obtained from inverting the P-wave data. Again, we observe a remarkable improvement by using the MR-CSI method. The results for both excitations are given in Fig. 8. Now, the levels for the contrast values for the square cylinders for both excitation types are, ␹ ⫽1.0, ␹ ⫽1.45, and ␹ ⫽1.95. Note that also here reconstructions using incident SV waves are better compared to the ones obtained from P-wave excitations. C. Reconstruction of concentric square cylinders

FIG. 5. The error in contrast ERRn as the function of the number of iteration n for the reconstruction of the circular cylinder. J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003

In our last example, an inhomogeneous scatterer is located in a test domain that is subdivided into 29⫻29 subsquares of 0.1034⫻0.1034 m2 . The wave numbers of the Pand SV waves are k P ⫽3, k S ⫽6, respectively. This means Pelekanos et al.: Contrast source inversion

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FIG. 7. The reconstruction results of the three distinct square cylinders from excitation with P waves 共a兲 and SV waves 共b兲 using the CSI method. The data sets are generated using CG-FFT method with 10% additive random white noise.

that the side length of the test square was equal to about one and one half wavelengths for the P waves and three wavelengths for the SV waves in the background medium. The actual profile of the scatterer consists of an inner square with mass density ␳ ⫽1.9, a middle square with ␳ ⫽1.6, and an outer square with ␳ ⫽1.3; see Fig. 9. The contrasts from the inner square to the outer are equal to ␹ ⫽0.9, ␹ ⫽0.6, and ␹ ⫽0.3. The side length of the inner square is equal to 0.75 m, the one of the middle square is equal to 1.5 m, and the one of the outer square is equal to 2 m. The measurement setup is the same as in the previous examples. The data are generated numerically using the CGFFT method, and after generation, 10% additive random white noise was added. The reconstruction results after 512 iterations for both P- and SV-wave incidence using the CSI method are shown in Figs. 10共a兲 and 共b兲, while the ones using the MR-CSI method are shown in Figs. 11共a兲 and 共b兲. Note that for the MR-CSI method the contrast values are very well reconstructed. Comparing now the results of both excitations, we observe again that inversion from SV-wave data yields better reconstructions compared to the ones obtained from P-wave excitations. It is worthwhile to note that even though we only show reconstructions of objects with sharp edges, the MR-CSI method has also been shown to work properly with smooth contrast objects.23 2832

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FIG. 8. The reconstruction results of the three distinct square cylinders from excitation with P waves 共a兲 and SV waves 共b兲 using the MR-CSI method. The data sets are generated using CG-FFT method with 10% additive random white noise.

V. CONCLUSIONS

We have developed a nonlinear inversion method for the two-dimensional problem of scattering of an elastic ( P or SV) wave. This method, which is denoted as CSI, is based on the source-type integral equation which relates measured data to a source distribution in the scattering object. A cost functional is defined consisting of errors in the source-type equations and the state equations. The updates in the sources are found as a conjugate gradient step, after which the con-

FIG. 9. The original profile of the concentric square cylinders. Pelekanos et al.: Contrast source inversion

trast is updated by minimizing the error in the state equations in a very simple manner. The most interesting feature of the algorithm is that it does not solve any full forward problem at any stage of the iteration. This feature, as also expressed in Abubakar et al.,8 gives hope that a three-dimensional inverse problem can be handled with a moderate computational power. In addition, we have discussed a new type of regularization that, together with the contrast source inversion method, is an extremely effective and robust with respect to noisy data algorithm. This new multiplicative regularized CSI method is denoted as the MR-CSI method. Future work will concern further refinements to make the algorithm more applicable in real-world problems by removing the assumption that the same traction operator occurs in both media. This results in a very complicated hypersingular integral equation which also contains gradients of the displacement fields. This will give more degrees of freedom to the problem, and the object will be reconstructed from its Lame´ parameters. Future work should also be directed toward extending the method to include measurements at more than one frequency in order for larger contrasts to be treated effectively. FIG. 10. The reconstruction results of the concentric square cylinders from excitation with P waves 共a兲 and SV waves 共b兲 using the CSI method. The data sets are generated using CG-FFT method with 10% additive random white noise.

FIG. 11. The reconstruction results of the concentric square cylinders from excitation with P waves 共a兲 and SV waves 共b兲 using the MR-CSI method. The data sets are generated using CG-FFT method with 10% additive random white noise. J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003

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