Comparison of the Born Iterative Method and Tarantola’s Method for an Electromagnetic Time-Domain Inverse Problem

M. Moghaddam and W. C. Chew

Electromagnetics Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61 801

M. Oristaglio Schlumberger-Doll Research, Old Quarry Road, Ridgefield, Connecticut 06877

ABSTRACT Two methods of solving the nonlinear two-dimensional electromag- netic inverse scattering problem in the time domain are considered. These are the Born iterative method and the method originally proposed by Tarantola for the seismic reflection inverse problems. The former is based on Born-type iterations on an integral equation, whereby at each iteration the problem is linearized, and its solution is found via a regularized optimization. The latter also uses an iterative method to solve the nonlinear system of equations. Although it linearizes the problem at each stage as well, no optimization is carried out at each iteration; rather the problem as a whole is posed as a (regularized) optimization. Each method is described briefly and its computational complexity is analyzed. Tarantola‘s method is shown to have a lower numerical complexity compared to the Born iterative method for each iteration, but in the examples considered, required more iterations to converge. Both methods perform well when inverting a smooth profile; however, the Born iterative method gave better results in resolving localized point scatterers.

I. INTRODUCTION Inverse scattering has been the subject of a host of recent studies. In one spatial dimension, several methods of recon- structing unknown profiles have been developed, among which are the iterative method developed by Tijhuis [l], the distorted Born approximation proposed by Chew and Chuang [2] and Habashy et al. [3], and the method proposed by Bolomey et al. [4], based on a space-time discretization. A few other methods, including the method of characteristics, are discussed by Chew [5]. In higher dimensions, many solu- tions have been developed for linear inverse problems, includ- ing diffraction tomography [6-81, reverse-time migration [9 ] , and methods involving the Born approximation [5, 10-121. Recently, a number of methods have been proposed for the solution of nonlinear inverse scattering problems. For exam- ple, Johnson et al. [13-151 developed a method in which the unknown profile and the total field are expanded using sinc basis functions. The distorted Born approximation has also been used in an iterative procedure for the reconstruction of two-dimensional objects 1161. Another iterative approach

~~

Received May 2, 1991; revised manuscript received September 12, 1991

using a successive relaxation technique on an integral equa- tion was reported by Kleinman and van den Berg [17]. There have also been solutions which are not of an iterative nature. Perhaps the most prominent is the one proposed by Newton [18, 191; however, these methods have not yet resulted in satisfactory numerical results, possibly due to instabilities. Bolomey and Pichot have discussed several electromagnetic inversion methods and describe some actual imaging systems [20]. Bolomey [21] and Bolomey and Hawley [22] have also discussed recent active microwave imaging systems and their applications.

In this article we compare two successful methods of solving the nonlinear two-dimensional inverse scattering prob- lem in the time domain. These are the Born iterative method and the functional analysis method proposed by Tarantola. The reader is referred to Refs. 23-28 for detailed treatments of each method. To solve the nonlinear inverse scattering problem, both methods implement iterative techniques. In the Born iterative method, the problem is linearized at each iteration; a regularized optimization is carried out to update the solution (the object profile). This optimization is carried out using a conjugate gradient algorithm, which is itself a finite iterative method. Thus this method consists of two nested iterative processes: the inner loop finds an optimum solution to a linearized problem to minimize a predefined norm; the main (or outer) loop uses the output from the inner loop to update the object profile and its interior field. Taran- tola’s method attempts to solve the nonlinear problem by a direct iterative process without involving an optimization pro- cedure within each linearization. The steepest descent method or any Newton-type algorithm would be a possible candidate for this purpose. Both methods require the solution of the forward problem at each iteration once the unknown has been updated. In the Born iterative algorithm, matrix algebra concepts are utilized in the solution procedure, whereas Tarantola’s method uses the ideas of functional analysis to replace the matrix-vector multiplications by a forward prob- lem solver, e.g., a finite difference code.

We begin by giving a brief summary of each method in Sec. 11, followed by a comparison of their computational complexi- ty in Sec. 111. Finally, in Sec. IV, we present some results for

International Journal of Imaging Systems and Technology, Val. 3, 318-333 (1991) @ 1991 John Wiley & Sons, Inc. CCC 0899-9457/91/040318- 16%04.00

identical problems using each of two solution methods and discuss the extent of their validity.

II. SUMMARY OF EACH METHOD In this section, we briefly describe the Born iterative method and the method of Tarantola. For the sake of simplicity and consistency, we consider the scalar wave equation with trans- verse magnetic (TM) incidence for both cases and assume two-dimensional media, that is

where J,( p, t ) is an electric line source with a time-dependent excitation. Then, 4 ( p , t ) would be the z component of the electric field. In this case, the scattered field satisfies the time-dependent volume integral equation [5 , 231

where the composite subscript (i, j , k) refers to the data point measured at the ith receiver and jth time sample due to the kth transmitter. The superscripts r and t refer to receiver and transmitter locations, respectively. S, is the area over which +,, is nonzero.

To avoid the expensive evaluation of the time convolution integral in Eq. (3b), we Fourier transform both sides of Eq. (3a) using a standard fast Fourier transform (FFT) routine to convert the convolution in the time domain to a multiplication in the frequency domain. If the number of time (or frequency) samples is denoted by N,, then for each receiver and transmit- ter location and for each unknown, the time convolution would require O(N;) operations, whereas the FFT followed by multiplication requires O(Nf log, N,) operations. Thus, when N, is large, the savings in computation time resulting from this procedure far exceeds the additional time required to carry out the FFT operation. Equations (3a) and (3b) can now be rewritten as

a' a t t 2

x ~ 4 ( p ' , t ' ) O ( p ' ) dp' dt'

in the presence of a scatterer defined by O ( p ' ) . Here, p ' denotes the points in the domain of the support of the object, 4""( p, t ) is the scattered field, g( p, p' , t - t ' ) is the time- domain Green's function, and (6( p', t ' ) is the total field inside the scatterer. The object function, O ( p ' ) , is the unknown to be sought.

A. Born Iterative Method. In the Born iterative method, we define the object function to be

where H r ) ( k , I p: - p ' l ) is the Hankel function of the first kind of order zero with k, = w I a . The tildes denote transformed quantities.

The-object profile is unknown, and so is the interior object field, 4 ( p ' , p i , 0,). To estimate the unknown coefficients 0, from Eq. (4a), we start by setting e ( p ' ) = E,, representing a eomogeneous profile. Using this initial profile, the object field 4 is found via a forward problem solver (in this case, a finite-difference time-domain code) followed by an FIT, and used inside the integral in Eq. (4b). With the knowledge of the scattered field measured at the receivers, Eq. (4a) is solved for 0, resulting in an estimate for the object profile. This 2tep is carried pu t through minimizing the norm llM *

0 - @:::, 1 1 , where contains the Fourier transform of the scattered field from an experimental measurement. The

where e, is the known permittivity of the background. We choose e, to be the permittivity of a homogeneous medium, so that g(p, p ' , t - t ' ) = g,(p, p ' , t - t ' ) . Hence, Eq. (lb) be- comes

newly found object profile is then used again to calculate the object field in Eq. (4b). The process continues until the scattered field calculated from the latest object function is acceptably close to the measured scattered field. Notice that the Green's function of Eq. (4b) and the background permit- tivity eb remain fixed for all iterations; only the object profile ~ ( p ' ) and the interior object field are updated.

To avoid problems with the null space of M and the nonuniqueness of the solution at each iteration, a regulariza- tion procedure is employed [29] and 0 is found by solving

X [ ~ ( p ' ) - e,] dp' dt' . (2b)

Upon discretizing the Object function as O ( p ) = c z = ~ O,@,(f"), and assuming the expansion functions, 'kP> to be pulse functions, we arrive at the matrix equation:

-

M - 0 . (3a)

The vector WCat contains the time record of scattered field at all the receiver locations due to each of the incident waves.

stored in the vector 0. The elements of matrix M are given by

a-1 =

IIM. o - ~:;:,II~ + y llo\l = minimum , ( 5 )

The unknown coefficients O , describing the scatterer are where is the regularization parameter. Equivalently, the object function, which is a solution to ( 5 ) , is found from

I - - Mt.M.o+ yo=Mt.G:,"-,. (6)

a2 To reduce computational cost, the conjugate gradient method is used to solve for the unknown 0 in Eq. (6). Note that x - atf2 4(p', p i , t ' ) dp' dt' , (3b)

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although at each iteration the problem is linearized, the final outcome is the solution to the nonlinear inverse problem.

Defining E and pb to be the vectors containing the discret- ized forms of E ( p ' ) and e b , respectively, we can summarize this algorithm symbolically as follows:

where C, is the model covariance operator describing our degree of confidence in the initial model, and is part of the a priori information supplied to the algorithm. Similarly, C, is the data covariance operator, containing statistical informa- tion about the data and measurement errors.

To arrive at the solution using the steepest descent meth- od, the gradient of the functional S with respect to E is first computed. At the nth iteration, given E , , we can write

= E , + S E , , and

6, = i ( p ' , p', w ; object field

apt = f(p', p: t ; En) - ainc A = IIQ,Tt - ascat

s ( e , + % , ) = S ( ~ , , ) + (u,,~E,) + O ( ( I ~ E , I I ~ ) , ( W measll

Mr, 'M, . On+l + yo,,, = Mr, . Q,:,":,

where u, is the gradient of S, and ( u , , S E , ) = u,'. C i ' . &,. With the definition of the weighted norms of Eqs. (9) and (lo), and letting F, = (aQ,scat/ae)co denote the Frechet deriva- tive operator, the gradient of S can be derived as

If A < tolerance, then stop (7) - I

conjugate gradient solution + 0, + , - -7 --,

U, = C, . F, * C, . (@Yt - Q,~ ,"~ , ) + ( E , - eb) . ( l lb )

Here, Q,'"' is the vector containing the time samples of the incident fiei_d values at the receivers due to each transmitter, and f and f are the solutions of the time-domain forward problem and its Fourier-transformed version, respectively. The forward solver is invoked only once in each iteration. The iterations are stopped when the error A becomes less than a

Here, F,T is the transpose of the Frechtt derivative operator, and *;at is as defined in Eq. (7). Next, the step length to reach a minimum in the gradient direction is found. Thus, the Hessian of s must be ~i~~~ the above gradient, the Hessian is

prescribed tolerance value. ( 124 ' n

B. Tarantola's Method. This method [24] was originally de- veloped for acoustic and/or elastic waves to simultaneously invert the three independent medium parameters (the two Lame constants and density, or any other independent set of three parameters). To compare this method to the Born iterative method, we will restrict it to the case of a scalar electromagnetic problem with one unknown, E ( p ' ) .

Tarantola's method differs from the Born iterative method in two major aspects. First, it does not solve a regularized minimization of the linearized problem at each iteration; instead, the solution is found by directly posing the entire problem as a (regularized) minimization. Here, the system of equations would be given by Eq. (1). The unknown is found by using any iterative method 9f solving nonlinear systems, such as the steepest descent method or the preconditioned conjugate gradient method.

Consider the solution using the steepest descent technique. The algorithm is initialized by specifying an a priori object profile, eb(p' ) . The goal is to find the true profile, ~ ( p ' ) , such that

In the above, W"' = f(p', p', t ; E ) - @"'. The vectors E and

Assuming Wcat to be linear in S E , , we have ( a F / d E ) e u = 0. Denoting the identity operator by I, Eq. (12a) then becomes

Equivalently, this analysis follows from the distorted Born approximation, where [5]

aWCa' (x) C" . ( E - E , ) = F , . S e n . (13)

In this case, AQ,Tat = @""' - Q,rat=Q,-Q,,, where Q,= f (p' , pf, t ; E ) and Q," is the total field in the presence of E , .

The higher-order terms in SE, are neglected on the right-hand side of Eq. (13). Again (aQ,scat/aE)en =F, is the Frechet derivative, which is independent of E under the distorted Born approximation. This renders the functional S in Eq. (8) quadratic in E , and results in the Hessian given in Eq. (12b). For our problem, Eq. (13) can be written explicitly as [16]

AQ,rat( p', p', t ; S E , ) = F, . SE,

a' at"

X __ +,(p', pr, t ' ) S E , ( p ' ) dp' dt' . (14) eb contain the discretized values of ~ ( p ' ) and ~ ~ ( p ' ) , respec- tively. The norms are defined as The term g,( p', p' , t - t', E,,) is the Green's function calcu-

lated using the most recent object profile. Notice that Eqs. - - (14) and ( ib) from the Born iterative method are quite similar in form, except that in Eq. (2b), we use a fixed background permittivity instead of ~ ( p ' ) , and hence the Green's function does not change.

Once the gradient is known, en+, (the profile in the next iteration) is related to E , via E , + ~ = E , - a,u,. In the steepest descent method, the step length a, is found from

Jl@saat - Q,~SC;,JJZ ~ (Q,SC,I - a s c a t ) T . C;l . (,scat - scat meas Q,',,,,) ,

(9)

and

IJC - E b ( I 2 = (. - € b ) T * C : l . ( E - E b ) , (10)

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where (u, , u, ) = uz . C,' . u,, and similarly for the inner product in the denominator. With the knowledge of the direction of the gradient and the step length, the unknown profile is updated, and a new estimate to the scattered field is found. The process is repeated until the functional in Eq. (8) is sufficiently small.

These steps can be written explicitly as

(i) 'Bn = f(p', p', t ; E , ) .$object field

'Brat = f(p', p : t ; €,) - ainc (ii) A = Il'B",'"' - 'Bz2is[l

(iii) If A < tolerance, then stop

(iv) u, = C, . F,T. C j l * (@rat - ' B ~ E , ) + (6, - (v) b, = F, . U,

- (16)

(vii) en+] = E, - a,u, , go to (i)

The vector f(p', p', t ; a ) is the solution of the forward problem given the latest object profile. Step (i) invokes the forward solver once. As shall be shown, the actions of F, and Fi on a vector are also effected by using the forward solver. Therefore, the forward problem is solved three times in each iteration.

The second point of departure of this method from the Born iterative method is the way in which the actions of operators M and F and their transposes are interpreted. We saw that except for the definitions of the Green's functions, these operators are identical. In the Born iterative method, M and MT were treated as matrices, and the algorithm was implemented using matrix algebra. Consequently, a number of matrix-vector multiplications were performed in the conju- gate gradient algorithm, in addition to one forward modeling run to update the object field. The present solution, however, avoids such matrix manipulations and (explicit) discretization of operators. The action of F, on any vector v, in accordance with Eq. (14), yields a vector h, given explicitly by

h,(p', p', t ; P,) = F, . v = -po -

which is interpreted as the solution of the forward problem with the quantity inside the brackets as the source, and using the most recent Green's function. However, F, . v, is com- puted more efficiently with a finite-difference solver (with the quantity inside the brackets as the source) than with the explicit form (17).

The transpose operator, F,, operating on the vector ,,,,), can be derived to have the explicit form

- T

6;' . (Qyt - @scat

[FZ . C,' . (a?' -@,=a'

n,= 1

Again, the convolution integral is interpreted as the solution of the forward problem using the Green's function for the most recent E, with "virtual" sources given by the residual fields at the receiver positions (quantity inside the square brackets). This step can again be more efficiently performed by a finite-difference forward solver. The propagation is backward in time, resulting in the fields, which, if propagated forward in time, would give rise to the residual field C,'. ('By' - 'Bzzis) at the receivers. At each point in the medium, this backpropagated field is time-correlated with the field of the physical transmitter calculated from the current object function. Finally, this procedure is repeated for and summed over all the transmitters. If the above operation were per- formed using matrix algebra, the dot product would have implied the summation over time and over all the transmitter locations.

As we will see in the next section, if efficient methods of solving the forward problem are available, great savings in computer time and storage could result from the use of this algorithm over the Born iterative techniques discussed previ- ously.

111. COMPUTATIONAL COMPLEXITY An important issue in evaluating any valid numerical method is its computational complexity or operation count. Even with the aid of present-day supercomputers, many realistic prob- lems cannot be solved within a reasonable length of time because of their large size. Limited memory is another signifi- cant problem. Hence, it is desirable to devise numerical schemes with the least computational complexity without sacrificing the quality of results.

A. Born Iterative Method. Consider the Born iterative al- gorithm (7). Each iteration consists of two main procedures: the solution to the forward problem, and the linearized inversion process. To obtain the solution to the forward problem, we use a finite-difference time-domain (FDTD) scheme with appropriate absorbing boundary conditions to eliminate reflections from the artificial boundaries [5,30,31]. Assuming the number of unknowns to be N u , it can be shown [5] that in a FDTD solution, the number of multiplications increases as N i l 2 . Furthermore, the forward problem needs to be solved for each transmitter location (or each incident wave). Hence, for N, transmitters, the total operation count of the forward problem increases as N,N;I2.

Let Nd represent the total number of measured data points. If there are N, receivers, and N, tipe-samples are recorded, then Nd = N,N,N,. Consequently, M, 'Bzlls, and 0 have dimensions of Nd X Nu, Nd X 1, and Nu X 1, respectively. To solve for On+l from (iv) in Eq. (7), the conjugate gradient method is used, which involves several matrix-vector multipli- cations. Each of these has a numerical complexity of NdNu, or N,NrNfNu. As the number of unknowns, Nu, increases, the number of time samples required grows proportional to the linear dimension of the unknown, that is, N, a , : I 2 . Also, the effective aperture, created by the combination of transmitters

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and receivers, must increase as the perimeter of the scatterer, or N,N, N ; “ . (The number of data points needs to increase at least as N,, to avoid underdeterminancy, which is consistent with our analysis.) Thus the computational complexity of the inversion process is N t .

If the numbers of receivers and transmitters grow at the same rate, then N, cc N : ’ 4 . Thus the forward problem will have a complexity of N t . 7 5 . This is subdominant to the number of operations required by the inversion process, resulting in a final computational complexity of N : for this algorithm per iteration in Eq. (7).

B. Tarantola’s Method. As was discussed in Sec. ILB, this method mainly consists of three forward modeling runs at each iteration: referring to the first three equations in (14), the scattered field is calculated by solving the forward prob-

I I I I I I I I I I I I

R ? T 9

T I I 1

I I I I I I I

R R R l I I I

T Figure 1. Data acquisition geometry for full-angle inversion. Rs and Ts denote receivers and transmitters, respectively. The receivers collect the time record of the scattered field due to each receiver. The two-dimensional profile E ( p ’ ) is the unknown to be reconstructed.

lem using the current object profile; the actions of operators F and FT are also interpreted as forward modeling solutions. Since we have used the FDTD method to solve the forward problem, the complexity of the algorithm grows as N; .7s . We assume the model covariance operator to be diagonal. How- ever, if it includes cross-covariance terms, then the term v: . C i ’ . u, in the computation of a, requires O ( N 2 ) oper- ations, dominating the algorithm.

Hence, if no cross-covariance is assumed between the elements of the unknown object, each iteration of this method has lower computational complexity ( N t . 7 5 ) over the Born iterative algorithm ( N ; ) . If the number of iterations necessary for convergence is comparable for both algorithms, this al- gorithm would be especially useful as the size of the unknown increases.

IV. COMPARISON OF RESULTS In all the examples that follow, the object consists of 169 unknowns, and is illuminated by four incident pulses. The duration of the pulses is chosen such that the wavelength corresponding to the highest important frequency in the spec- trum is about 10 times the linear size of each unknown. Hence, the diagonal size of the object is about 1.5 wave- lengths, unless otherwise noted. Two data acquisition geomet- ries are considered: Figure 1 shows the configuration for full-angle inversion, where transmitters and receivers sur- round the scatterer in a symmetric fashion. There are four transmitters and eight receivers. The transmitter-receiver configuration for a limited-angle (here, subsurface) inversion scheme is shown in Fig. 2. In this case, the transmitters and receivers are confined to one side of the object.

A. Born ltetative Method. The results of this method have been given extensively [23]. A few examples are repeated here for comparison. For full-angle inversion, we start by inverting the smooth profile shown in Fig. 3(a). Figures 3(b)-3(e) depict the four iterations leading to the convergent reconstruction of the original profile. The peak contrast in this case is 1.5:l. In each successive iteration, the regularization parameter of Eq. (6) is decreased in order to allow finer features of the profile to be resolved. The effect of varying this parameter is discussed in great detail in Ref. 23. In each

Figure 2. Data acquisition geometry for limited-angle inversion. Rs and Ts denote receivers and transmitters, respectively.

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(4 (e)

Figure 3. Inversion of a general but smooth permittivity profile using the Born iterative method. (a) True profile. (b)-(d) Results of first through third iterations. (e) Fourth and final iteration. Peak contrast is 1.5:l and object diameter is one wavelength. At each iteration, a more refined profile is obtained. The final reconstruction shown results in scattered field that are within less than 1 % of the measurements.

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4

‘p i

Figure 4. Full-angle inversion of a general but smooth permittivity profile by the Born iterative method. (a)-(e) Results of first, third, sixth, ninth, and twelfth iterations. Peak contrast is 3:1, and the object diameter is 1.5 wavelength.

324 Vd. 3, 318-333 (1991)

iteration, a linearized problem is solved using the conjugate gradient method. If the peak contrast is increased, the prob- lem becomes more and more nonlinear, and thus requires a larger number of iterations. This is shown in Fig. 4, where for a peak contrast of 3:l the convergent solution is obtained after 12 iterations. Iterations 1, 3, 6, 9, and 12 are shown. If the contrast is raised beyond about 4:1, the solution will not converge. We consider a solution convergent if the error ll@:at - @ ~ ~ ~ s ] J / J l @ ~ & l ] is less than a prescribed value, which was chosen to be 1% here.

Next, consider the point spread function, that is, the inversion of a point scatterer. This is shown in Fig. 5, where the eighth iteration results in a convergent reconstruction of the point object. The contrast of scatterer with respect to background is 2:1, although equally good results can be obtained for contrasts as high as 5:l.

Figure 6 demonstrates the resolution capability of this algorithm, where it is seen that two point objects separated by 0.1 wavelength are well resolved. Here the contrast is also 2:l . The superresolution feature of this method can be attrib- uted to multiple scattering and mode conversions taking place between the two objects and also within each of them.

In the case of limited-angle inversion, the ill-conditioning of the problem is increased due to large differences between the magnitudes of incident fields from near and far transmit- ters and also between the magnitudes of measured scattered fields at near and far receivers. To eliminate this problem, the transmitted and received field magnitudes are scaled by ap- propriate factors. It is shown in Refs. 23 and 32 that if the transmitters (receivers) are in the far field of the object, the scaling factor is m, where p is the distance between each transmitter (receiver) and the center of the scatterer, and it is llln(lp1) if they are in the near field of the scatterer.

The results of the limited-angle inversion problem are shown in Figs. 7 and 8. The transmitters and receivers are on the lower right-hand side of the object. Figures 7(a)-7(e) are the results of iterations 1, 2, 5, 9, and 14 for the reconstruc-

Figure 5. Point spread function for Born iterative method using full-angle data. The eighth iteration converges to the final result. In order to recover the sharp discontinuity without smoothing the object function, the tuning parameter y is made vary small. This will in turn produce a small amount of background noise.

Figure 6. Full-angle resolution of two point objects placed 0.1 wavelength apart. Born iterative algorithm is used. The contrast is 2:l. The ninth and final iteration is shown.

tion of the smooth profile of Fig. 3(a) with a peak contrast of 3: 1. Figures 8(a)-8(d) demonstrate the resolution capability of the limited-angle inversion algorithm, where we see iterations 2, 4, 6, and 9. The level of background noise remains less than 10% of the peak profile contrast.

We saw in Sec. 111 that the computational complexity of Tarantola’s method is lower than that of the Born iterative method, which is a consequence of evaluating integrals of the form (15) as solutions of forward problems as opposed to matrix-vector operations. This idea can, in principle, be used in the Born iterative method as well to achieve lower compu- tational complexity. We implemented this idea, that is, re- placed all matrix-vector operations appearing in (7) with their equivalent as given by Eqs. (15) and (16), after modifying the Green’s functions. However, we observed that the results are contaminated by an unacceptable amount of noise. The noise is generated as a result of a large number of forward modeling runs required at each iteration in the conjugate gradient algorithm.

B. Tarantola’s Method. This algorithm was implemented and tested for the same cases discussed in Sec. 1V.A. In the full-angle case, due to the uniform data collection scheme and assuming uncorrelated errors (if any) in the measured data, the data covariance operator C, was set to the identity operator. This method is successful in inverting smooth pro- files, such as the one shown in Fig. 3(a). The profile converges after 15 iterations in this case, as opposed to four iterations in the Born iterative method. The results of iterations 1,4, 8, 12, and 15 are shown in Figs. 9(a)-9(e) for a peak contrast of 1.5:l. The CPU time for each iteration of the algorithm was 2 seconds on the CRAY-2; each (main) iteration of the Born iterative method, which includes a complete conjugate gradi- ent solution, took about 4 CPU seconds on the same machine. The range of contrasts for which convergent results are ob- tained are larger than that of the Born iterative method. We were able to invert profiles with peak contrasts as high as l O : l , as shown in Fig. 10. The figure shows iteration 35. The initial profile was chosen to be the same as free space. The

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Figure 7. Limited-angle inversion of a general but smooth permittivity profile by the Born iterative method. (a)-(e) Results of iterations 1, 2, 5, 9, and 14. Peak contrast is 3:1, and the object diameter is 1.5 wavelength.

326 Vol. 3, 318-333 (1991)

Figure 8. Limited-angle resolution of two point objects placed 0.1 wavelength apart. Born iterative algorithm is used. The contrast is 2:i (a)-(d) The results of iterations 2, 4, 6, and 9.

model covariance was chosen to be so large as to make / /en - ebllZ much smaller than first term in the second equa- tion in (8). This would imply that the profile can vary arbitrarily from the eb, and minimizes the effect of this a priori model in determining the final results.

The situation becomes different for the inversion of point scatterers. Choosing the same u priori model and covariance operator as above to find the point spread function, we found that convergence, if at all possible, is extremely slow. Figure ll(a) shows the outcome of the 50th iteration, where the peak contrast has barely reached 20% of its true value. The rate of convergence has become quite small at this point. such that in the last few iterations, the percentage of error has nearly remained fixed at a large value. A remedy can be devised by altering the model covariance operator. To simplify matters,

let us assume that C, is diagonal (no cross-covariance between the elements of E ) , and furthermore, that the diagonal ele- ments are all equal to the same value, say a. If this value is decreased such that the term (en - eb) becomes significant, the implication is that the true profile very closely resembles the u priori model at every point. Of course, this is not true at the point where the scatterer is located. This contradiction causes the results to diverge. We observed that although the location of the object was found correctly, the contrast con- tinued to grow at a large rate beyond the desired value. Varying the value of a did not improve the results: very small values led to immediate divergence; increasing a gradually caused a slow movement towards the true profile, however, after a few iteration and before converging, the results di- verged. For much larger values of a, extremely slow conver-

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Figure 9. Full-angle inversion of the smooth profile shown in Fig. 4(a) by Tarantola's method. (a)-(e) Results of first, fourth, eighth, twelfth, and fifteenth iterations.

328 VIA. 3, 318-333 (1991)

Figure 10. Tarantola's method is capable of inverting smooth pro- files of higher contrast than the Born iterative method. Here, the final convergent iteration (35th) is shown. The peak contrast is l O : l , and the object diameter is 1.5 wavelengths.

gence rate was observed, as discussed previously. Next, more a priori information was supplied to the algorithm by changing the diagonal elements of 6, such that those elements corre- sponding to the ones "near" the location of the scatterer were made larger, so that at those points the profile could vary arbitrarily from the background. The remaining elements were kept small, resulting in significant dependence on the background. Consequently, the convergent profile shown in Fig. l l(b) was obtained after 18 iterations (twice as many as the Born iterative method).

Similarly, for the problem of resolution of two closely spaced point objects, using a model covariance operator with large diagonal elements resulted in the nonconverging results of Fig. 12(a). When C, was modified in the vicinity of the scatterers, the algorithm converged in 25 iterations, as shown in Fig. 12(b).

For limited-angle inversion, the data collection scheme is nonuniform, as described in Sec. 1V.A. Hence, the data covariance matrix, C d , must be modified in a manner to normalize the scattered field measurements relative to-each other. This is done by scaling the diagonal elements of C, by the corresponding (mm)' or

('-- )* In(lp'l) WlP'I) '

where the superscripts r and t refer to receivers and transmit- ters, respectively [23,32]. The scaling factor is squared, since, as can be seen from the definition of the norm in (9), the data covariance matrix is multiplied by the scattered field vector once from the right and once from the left.

Figures 13(b)-13(e) show the reconstruction of the smooth profile of Fig. 13(a), where the peak contrast is 5:l. Iterations 1, 5, 15, and 22 are shown. The transmitters and receivers are located on the upper left-hand side of the object. The model

A

Figure 11% (a) Full-angle point spread function for Tarantola's meth- od before the modification of the model covariance operator. The true contrast is 2:l. Here, only 4 of this height is reached after 50 iterations. (b) The point scatterer is perfectly reconstructed after the modei covariance operator is modified to include a priuri information about the location of the object.

covariance matrix elements are increased once again, to pro- vide arbitrary independence of the true profile from the initial background profile. Figure 14(a) is the 50th iteration of the algorithm to resolve two point scatterers, with C, the same as that for Fig. 13. The resulting profile does not converge to the true profile. By modifying C, such that the diagonal elements corresponding to the area around the point scatterers are very large and the other elements are small, we obtain the results

Vol. 3, 318-333 (1991) 329

Figure 12. (a) Similar to the point spread function, the full-angle resolution of two closely spaced objects is not possible without modifying the model covariance operator in Tarantola’s method. Here, iteration 40 is shown, where no sign of convergence to the true object profile is detected. (b) After specifying the approximate loca- tion of the object in the covariance operator, perfect resolution of the two point scatterers is obtained using the functional analysis method with full-angle data.

of Fig. 14(b) in 20 iterations, where the two point objects are almost exactly recovered. Hence, in general, the success of this method greatly depends on the quality of the a priori model and our knowledge of its covariance operator. Since in many cases this information is not available before solving the problem, this algorithm does not always yield the reliable solution.

As stated in Ref. 33, the least-squares criterion used in the inversion algorithm of Eq. (16) is not a good choice if the model probability distribution is not Gaussian. The reason could be that in a least-squares inverse problem, it is assumed that the error is mostly due to the error in the data, and the goal is to minimize the L, norm of the difference between the scattering data and the calculated scattered fields; however, the error resulting from the forward modeling of the problem may be many times larger than that in the measured data [34]. A more robust criterion could be to minimize the L, norm of the quantity ,Tat - Another method for circumvent- ing this problem has been proposed recently by Berryman [351.

V. CONCLUSION It has been shown that Tarantola’s method has a lower operation count than the Born iterative method. We have shown that its computational complexity increases as N : 75 for each iteration as opposed to the more costly N : for the Born iterative method. Therefore, if both methods produce results of same quality and the number of iterations are comparable, Tarantola’s method would be more appropriate for large problems. Given the size of the problems considered here, although each iteration of this method takes half as much CPU time as the Born iterative method, it takes more than twice the number of iterations to converge to the final result.

For the case of a scalar wave problem with TM incidence, it was observed that, starting from the same initial profile, both methods produced excellent results for smooth, non- localized scatterers. The range of profile contrasts for which valid results were obtained was almost twice as large for Tarantola’s method. However, this method was not successful in recovering nonsmooth and localized profiles corresponding to point scatterers. The Born iterative method produced very good results in this case as well. It was further observed that in order for Tarantola’s method to yield satisfactory results, it must be given more a priori information in the form of a better initial object profile or a descriptive covariance operator. If one has access to this type of a priori knowledge, then given the lower computational complexity of each itera- tion, this might be the method of choice. But since this information is not always available, the Born iterative al- gorithm may be considered to be the more flexible method in terms of applicable geometries and resolution.

We believe the failure of Tarantola’s method to reconstruct point objects satisfactorily using a least-squares criterion is related to the number of forward problem runs per iteration in this method. The forward runs produce numerical noise which exacerbates the reconstruction of high-resolution ob- jects, which is difficult to begin with.

ACKNOWLEDGMENTS This work was supported by the Army Research Office under Contract No. DAAL03-87-KO006 to the University of Illinois Advanced Construction Technology Center, by the Office of Naval Research under Grant No. N00014-89-5-1286, and by the National Science Foundation under Grant No. NSF ECS- 85291. The computation time was provided through a grant from the National Center for Supercomputing Applications (NCSA) at the University of Illinois, Urbana-Champaign.

330 Vol. 3, 318-333 (1991)

(4 (el

Figure 13. Limited-angle inversion of the smooth profile shown in (a) by Tarantola’s method. (b)-(e) Results of iterations 1, 5, 15, and 22.

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I

Figure 14. (a) Limited-angle resolution of two closely spaced ob- jects is not possible without modifying the model covariance operator in Tarantola’s method. Here, iteration 50 is shown, where no sign of convergence to the true object profile is detected. (b) After specifying the approximate location of the object in the covariance operator, perfect resolution of the two point scatterers is obtained using the functional analysis method with limited-angle data.

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