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Solution of the Three-Dimensional Inverse Acoustic Scattering Problem on the Basis of the Novikovðhenkin Algorithm N

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Solution of the Three-Dimensional Inverse Acoustic Scattering Problem on the Basis of the Novikovðhenkin Algorithm N Acoustical Physics, Vol. 51, No. 4, 2005, pp. 367–375. Translated from AkusticheskiÏ Zhurnal, Vol. 51, No. 4, 2005, pp. 437–446. Original Russian Text Copyright © 2005 by Alekseenko, Burov, Rumyantseva. Solution of the Three-Dimensional Inverse Acoustic Scattering Problem on the Basis of the Novikov–Henkin Algorithm N. V. Alekseenko, V. A. Burov, and O. D. Rumyantseva Moscow State University, Vorob’evy gory, Moscow, 119992 Russia e-mail: [email protected] Received May 5, 2004 Abstract—A theoretical study of the practical abilities of the Novikov–Henkin algorithm, which is one of the most promising algorithms for solving three-dimensional inverse monochromatic scattering problems by func- tional analysis methods, is carried out. Numerical simulations are performed for model scatterers of different strengths in an approximation simplifying the reconstruction process. The resulting estimates obtained with the approximate algorithm prove to be acceptable for middle-strength scatterers. For stronger scatterers, an ade- quate reconstruction is possible on the basis of a rigorous solution. © 2005 Pleiades Publishing, Inc. METHODS FOR SOLVING INVERSE number of computing operations than in the iteration SCATTERING PROBLEMS methods. The functional analytical methods for solving one- The inverse scattering problem consists in the deter- dimensional inverse scattering problems appeared as mination of the characteristics of a scatterer located in ᑬ early as in the 1950s (Gel’fand, Levitan, Marchenko, a spatial domain and described by a function v(r), and others). The first comprehensive studies of the mul- ∈ ᑬ where r , from the measured scattered field. In the tidimensional inverse scattering problem were carried case of an infinite domain ᑬ, the function v(r) is out by Berezanskii in the 1950s and by Faddeev and assumed to be rapidly decreasing at infinity. Below, Newton in the 1960–1970s. A systematic investigation scatterers concentrated in a finite domain ᑬ are consid- of this problem in application to quantum mechanics ered. The inverse problem is reduced to estimating v(r) can be found in [7]. In the 1980s, the ideas put forward in the Schrödinger equation or the Helmholtz equation by Faddeev were developed by Nachman and Ablowitz, ∆ 2 ∂ u + k0 u = v(r)u, which coincides with the Schrödinger who proposed a functional approach called the equation in the monochromatic case, for the total wave method (here, the symbol ∂ means differentiation with field u. Here, k0 is the wave number in the homoge- respect to the variable that is the complex conjugate of neous nonabsorbing background medium. For acoustic the main argument). Henkin and Novikov continued problems, in the case of a constant density of the developing this approach by applying it to multidimen- medium, the function v(r) characterizes the deviation sional inverse scattering problems with monochromatic of the phase velocity of sound c(r) in the scattering data [6]. In such a problem, a plane monochromatic inci- ᑬ ≡ ω2 –2 dent wave is characterized by the wave vector k ∈ ޒn, region from its background value c0: v(r) [c0 Ð − while the scattered wave in the far-field zone is charac- c 2(r)]. If the scatterer possesses absorbing properties, ∈ ޒn 2 2 2 c(r) can be formally considered as a complex quantity. terized by the wave vector l , where k = l = k0 Two classes of approaches to solving the inverse scat- and n is the space dimension. The experimental data are tering problem can be distinguished: one of them is represented by the scattering amplitude f(k, l) related to based on iteration methods [1–3], and the other, on the asymptotics of the scattered field in the far-field functional analysis methods [4, 5]; a review of the latter zone according to the formula can be found in [6]. The advantage of the iteration (), () methods is that they can proceed from fragmentary data u rk = exp ikr obtained with different geometries of the experiment () exp ik0 r , r 1 and at different frequencies. These methods impose no + An--------------------------- f k l = k0----- + o ----------------- , n – 1 r n – 1 strict requirements on the completeness of the experi- r r mental scattering data and can use any a priori and π π π2 a posteriori information. The main advantages of the where An = 2 = Ð(1 + i) / k0 and An = 3 = Ð2 . The functional approach are the possibility to obtain a rig- reconstruction of v(r) is based on the following gener- orous solution and (in some cases) the use of a smaller alizations, which were first introduced by Faddeev [8]. 1063-7710/05/5104-0367 $26.00 © 2005 Pleiades Publishing, Inc. 368 ALEKSEENKO et al. (‡) (b) l ∈ ޒn, are in one-to-one correspondence with f(k, l) k k+ + I = I kI = kI according to the formula [4] (), (), π (), hg k l = f k l + 2 ih∫ g km lR n (3) p m ∈ ޒ kR ×θ[]δ()mk– , g ()m2 – k2 f ()m, l dm; k, l ∈ ޒn; kR here, θ[(m Ð k, g)] is the Heaviside function, whose Ð Ð kI = kI kI = kI argument is the scalar product of the vectors m Ð k and g. From condition (1), which is equivalent to condi- tions (2), it follows that, in the two-dimensional case, Fig. 1. Orientation of mutually orthogonal real and imagi- ± nary components of wave vectors in the (a) two-dimen- only two orientations are possible for the vector kI = kI sional and (b) three-dimensional cases. being orthogonal to a fixed vector kR (Fig. 1a). The mutual orientation of the vectors lR and lI is similar. In To solve the problem under consideration, it is pro- addition, it is assumed that | | | | | | | | posed to formally extend the wave vectors to the com- lR = kR and, hence, lI = kI . (4) ∈ ރn plex domain k, l by assuming that k = kR + ikI The limiting values of h (k, l) that correspond to these ≡ ≡ ≡ g and l = lR + ilI, where kR Rek, kI Imk, lR Rel, two orientations are denoted as h+(k, l) and hÐ(k, l), ≡ and lI Iml. This passage from real values to complex respectively. In the three-dimensional case, the vector 2 ⊥ ones should be performed without leaving the k con- kI (kI kR) can have any direction in the plane perpen- 0 dicular to the fixed vector k . At the same time, when stant-energy surface, which is expressed as R solving a three-dimensional problem, an additional requirement should be introduced: 2 2 2 , ∈ ރn k ==l k0; k l . (1) kI = lI. (5) Requirement (1) means that the following conditions From conditions (2) and (5), it follows that should be simultaneously satisfied: ⊥ ⊥ kI kR, kI lR, lR = kR . (6) ⊥ 2 2 2 ⊥ 2 2 2 Therefore, kI is orthogonal to the plane containing the kI kR, kR – kI ==k0; lI lR, lR – lI k0 . (2) ≡ two vectors kR and lR and, hence, the vector p k Ð l = kR Ð lR. Then, the number of possible orientations of the | |2 2 2 vector kI with respect to this plane becomes equal to Here, the value of k = kR + kI can be as large as one two (Fig. 1b), as in the two-dimensional space. (The | | | | || likes; however, when kR increases, kI should also degenerate case of kR lR is considered as the limiting increase, so that, according to conditions (2), the differ- situation for the case of noncollinear kR and lR.) ence in their squares should always be equal to the 2 “energy” of the incident wave k0 . The above passage is NOVIKOV–HENKIN THREE-DIMENSIONAL accompanied by the generalization of the classical ALGORITHMS FOR SOLVING THE INVERSE wave fields and Green functions to the case of complex SCATTERING PROBLEM wave vectors [6, 9]. In particular, the classical scatter- A functional algorithm for solving the two-dimen- ing amplitude f(k, l) (k, l ∈ ޒn) passes into a general- sional monochromatic inverse scattering problem ized scattering amplitude (generalized scattering data) with relatively strong scatterers was proposed by h(k, l), where k, l ∈ ރn. Novikov in [10] and by Grinevich and Manakov in [11]. Later, this algorithm was studied in application to The imaginary part of the wave vector can be repre- acoustic tomography [9] and was also implemented in sented as k ≡ |k |g, where the vector g ∈ ޒn (|g| = 1) char- a computer program [12, 13]. In the framework of the I I two-dimensional problem, a scatterer v r that causes acterizes the direction of the vector k . When |k | 0, ( ) I I no backscattering can be rigorously and stably recon- the limiting values of the generalized functions depend on structed by a simplified scheme with allowance for mul- the direction g of the infinitesimal but oriented imaginary tiple scattering processes on the basis of the known clas- part kI. The limiting values of the function h(k, l) are sical scattering amplitude and the two limiting functions ≡ | | | | | | ± ∈ ޒ2 hg(kR, lR) lim h (kR + i kI g, lR), where lR = kR = k0 h (k, l) (where k, l ) obtained from it. From the → kI 0 absence of backscattering, it follows that the high-fre- ≡ in the limit. The quantities hg(kR, lR) hγ(k, l), where k, quency components of the spatial spectrum of the scat- ACOUSTICAL PHYSICS Vol. 51 No. 4 2005 SOLUTION OF THE THREE-DIMENSIONAL INVERSE ACOUSTIC SCATTERING PROBLEM 369 1 l terer v˜ (x) ≡ ------------- ∫v ()r exp(Ðixr)dr, where n = 2, are p R ()2π n negligibly small. For weak (Born) scatterers, these are | | ≥ the components with x 2k0. As the scatterer strength increases, the corresponding threshold value for |x| α k becomes smaller than 2k0.
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