Identification, by the intersecting canonical domain ... - Semantic Scholar

Inverse Problems 16 (2000) 1709–1726. Printed in the UK

PII: S0266-5611(00)10131-5

Identification, by the intersecting canonical domain method, of the size, shape and depth of a soft body of revolution located within an acoustic waveguide James L Buchanan†, Robert P Gilbert‡, Armand Wirgin§ and Yongzhi Xu
† Department of Mathematics, US Naval Academy, Annapolis, MD 21402, USA ‡ Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA § Laboratoire de Mecanique et d’Acoustique, UPR 7051 du CNRS, 13402 Marseille C´edex 20, France
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA Received 6 December 1999, in final form 3 April 2000 Abstract. The Rayleigh hypothesis is employed to solve the forward problem of scattering of a known, arbitrary acoustic wave from a body of revolution immersed in a shallow body of water. The acoustic wavefields obtained in this way are then employed as simulated data for the inverse problem of the determination of the (unknown) size, shape and depth of the immersed body, which is known to be acoustically soft, axisymmetric in shape and located on the vertical axis of a Cartesian reference system within the shallow-water waveguide. This fully 3D inverse problem is solved by the intersecting canonical domain, using multifrequency scattered field data, for two types of body: a non-convex (indented) spindle and a conical seamount. (Some figures in this article are in colour only in the electronic version; see www.iop.org)

1. Introduction This work is concerned with the identification (i.e. determination of size, shape and depth) of 3D objects in a shallow-water environment by inversion of the scattered acoustic wavefield. The area of applications is the identification of mines, submarines or submerged navigational obstacles (wrecks, reefs, seamounts, etc) in harbours and other shallow bodies of water. All these objects are essentially acoustically passive, in contrast to noisy ships, which are often located (but rarely characterized) by matched-field processing techniques employing a point source model of the active body [9]. When the object is passive and its shape is the sought-for information, the radiating point source description is no longer appropriate. At the very least, a whole set of induced point sources must be employed for this purpose, and the identification problem reduces to the much more complicated task of the determination of the positions and intensities of these sources. This method has been implemented for 2D models of the scattering object in a homogeneous waveguide with impenetrable boundaries [20,21] and for a 3D object in free space [1], but not for identifying objects in layered media or waveguides. Another approach for classifying a submerged object from its shape is based on the socalled resonance scattering technique [16], but is not easy to apply unless the target has a simple shape and is in an unrealistically simple environment. It is conceivable to identify the object by more complicated iterative techniques appealing to boundary or domain integral 0266-5611/00/061709+18$30.00

© 2000 IOP Publishing Ltd

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representations of the scattered wavefield [17], but these techniques are so computer intensive that it is dubious that they would be of practical use in the above-mentioned applications. Nevertheless, some authors [11, 13, 15] have investigated, by integral equation methods, propagation and shape reconstruction of an object in acoustic waveguides with pressure release sea surface and rigid or elastic sea floors. Subsequent research [7,12,14] in this area employed the intersecting canonical domain (ICD) approximation procedure to invert the field in a numerically more efficient manner, the synthetic data being constructed by the same method or by a perturbation technique to construct the field. In this paper, the sea floor is rigid, the sea surface is pressure release, the immersed axisymmetric body is acoustically soft and the interrogating wave is arbitrary so that the inverse problem is fully 3D. The synthetic data are generated by a Rayleigh-hypothesis-inspired scheme and the image of the body is obtained by the ICD technique. 2. Problem description In the absence of the body, the waveguide occupies the open domain
included between the two horizontal flat-plane boundaries z = 0 (called ‘top’ or ST ) and z = h (called ‘bottom’ or SB ), the Oxyz Cartesian reference system being such that the top occupies the xy-plane and the z-axis points towards the bottom. This unbounded (in the horizontal direction) water-filled waveguide, whose depth is h, is in contact with an acoustically-soft medium (such as air) at the top and with an acoustically-hard medium (such as rock) at the bottom. The boundary conditions at the top and bottom are therefore of the Dirichlet and Neumann types respectively. A monochromatic acoustic wave radiated by known sources in
is assumed to propagate in the waveguide. This wave strikes and is scattered off the acoustically-soft body of revolution, whose axis is the z-axis (this is a priori information on the range of the object). The body occupies the domain
1 and what remains of
is denoted by
0 (see figure 1). The boundary S = ∂
1 is described by a circle in any horizontal cross-cut (parallel to the xy) plane and by the curve  in any vertical cross-cut (ϑ = const, with r, ϑ, z the cylindrical coordinates) plane such that r = f (z);

x ∈ ;

0
z
h,

(1)

wherein f (z) (hereafter termed the ‘profile function’, which comprises shape, size and depth information) is a single-valued, continuous function of z (which can vanish in subintervals of [0, h]) and x the vector joining the origin O to a generic point in
. Let ω designate the angular frequency of the incident wave whose time dependence is assumed to be exp(−iωt). The latter is communicated to the scattered field and is henceforth suppressed. Let the incident, scattered and total pressures in
be designated by uI , us , and u = uI + us respectively. The governing equations (in which s(x) is the source density, ∂ν the normal derivative and k = ω/c the wavenumber in water, with c the celerity in water) are ( + k 2 )ui (x) = −s(x);

x ∈
0 ,

(2)

( + k )u (x) = 0; x ∈
0 , u(x) = 0; x ∈ ST , ∂ν u(x) = 0; x ∈ SB ,

(3) (4)

u(x) = 0;

(6) (7)

2

s

x ∈ S,

lim u (x) = outgoing waves; s

r→∞

(5) 0
z
h;

0
ϑ
2π.

Identification, by the intersecting canonical domain method

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0

−2

−4

−6

−8

−10 2 2

1 1

0 0 −1

−1 −2

−2

Figure 1. An indented parabolic spindle.

The forward problem is: given ui (x) or s(x), S and h find u(x); x ∈
0 . The inverse problem is: given ui (x) or s(x), u(x); x ∈
D ⊂ (
0 ∪ ST ∪ SB ∪ S) and h, find S. Note should be taken of the fact that (2)–(7) and the geometry of the problem imply that the field is 2π periodic in terms of ϑ. 3. Field representations in cylindrical coordinates Let R ≡ max f (z), z∈(0,h)

and
+0 ≡ {r  R; z ∈ [0, h], ϑ ∈ [0, 2π )},
− 0 ≡ {f (z)
r < R; z ∈ [0, h], ϑ ∈ [0, 2π )}. By solving the Helmholtz equation (3) by separation of variables in cylindrical coordinates one obtains the following representation of the field, satisfying (4), (5), (7) and the periodicity relation: us (x) =

∞ ∞

n=1 m=−∞

anm Hm (kαn r)φn (z)eimϑ ;

∀x ∈
+0 ,

(8)

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wherein Hm(1) (·) is the Hankel function of the first kind (the superscript is henceforth dropped) of order m and   2 πz φn (z) = sin (2n − 1) , h 2h   π 2 , Re αn + Im αn  0. αn = 1 − (2n − 1) 2kh For the sake of convenience, it will be assumed that the sources are in
+0 , so that, by the same means, one can show that ui (x) =

∞ N

n=1 m=−∞

bnm Jm (kαn r)φn (z)eimϑ ;

∀x ∈
+0 ,

with bnm = im φn (z0 )e−imϑ0 wherein Jm (·) is the Bessel function of order m. 4. The Rayleigh hypothesis When the body is a circular cylinder (whose axis is the z-axis) occupying the full height of the waveguide, (8) holds at all points outside and on S. If the body is not such a circular cylinder than the scattered field does not admit a representation as simple as (8) in the domain
+0 . The Rayleigh hypothesis (RH) consists in assuming that a finite subset of the modes in (8) + can nevertheless be employed to represent the scattered field in
− 0 as well as in
0 : us (x) =

M L



anm Hm (kαn r)φn (z)eimϑ ;

n=1 m=−M

∀x ∈
+0 ∪
− 0.

It has been shown, albeit in a somewhat different context [2, 19], that, if S is a sufficiently small departure from the canonical circular cylindrical surface r = R, then (8) is a valid representation of the field even with L = M = ∞. We shall assume this to be the case in the forward problem context. By employing the RH representation in the boundary condition (6), we obtain, on account of (1), ui (r, ϑ, z) + us (r, ϑ, z) ∞ ∞ L N



= anm Hm (kαn f (z)) + bnm Jm (kαn f (z))φn (z)eimϑ = 0, n=1 m=−∞

n=1 m=−∞

∀z ∈ [0, h],

∀ϑ ∈ [0, 2π ),

(9)

wherein f ≡ f (z). For a surface S characterized by r = ρ(ϑ, z) we have upon multiplying (9) by φj (z)e−i+ϑ ; + ∈ [−M, M] and integrating over S ∞   L

anm Hm (kαn ρ(ϑ, z))φn (z)φj (z)ei(m−+)ϑ dS n=1 m=−∞

S

+

∞   N

n=1 m=−∞

S

bnm Jm (kαn ρ(ϑ, z))φn (z)φj (z)ei(m−+)ϑ dS = 0.

(10)

Identification, by the intersecting canonical domain method The area element is dS =

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 ρ(θ, z)2 + ρθ (θ, z)2 + ρ(θ, z)2 ρz (θ, z)2 dθ dz

and thus for a solid of revolution r = f (z) (10) becomes  h L
 an+ H+ (kαn f (z))φn (z)φj (z)f (z) 1 + f  (z)2 dz 0

n=1

= −

N


 bn+



2π

 J+ (kαn f (z))φn (z)φj (z)f (z) 1 + f  (z)2 dz

0

n=1

since

h

ei(m−+)θ dθ = 2πδ+m .

0

Hence the coefficients in the expansion can be obtained by solving the linear systems L
n=1

Ej(+) n an+ = −

N
n=1

Dj(+) n bn+ ,

j = 1, . . . , L,

+ = 0, ±1, . . . , M.

(11)

Introducing matrix notation a (+) = (an+ )Ln=1 ,

b(+) = (bn+ )N n=1 ,

where Ej(+) n = Dj(+) n

=



h

0 h

L E (+) = (Ej(+) n )j,n=1 ,

N D (+) = (Dj(+) n )j,n=1

 H+ (kαn f (z))φn (z)φj (z)f (z) 1 + f  (z)2 dz  J+ (kαn f (z))φn (z)φj (z)f (z) 1 + f  (z)2 dz

(12)

0

(11) takes the form E (+) a (+) = −D (+) b(+) . The matrices (12) will be computed using numerical integration, which is computationally intensive. To reduce the time required, note that for + > 0 bn,−+ = i−+ φn (z0 )ei+ϑ0 = bn+

E (−+) = (−1)+ E (+) ,

D (−+) = (−1)+ D (+)

and thus the matrix equations for the unknown coefficient vectors a (+) and a (−+) are E (+) [a (+) , a (−+) ] = −D (+) [b(+) , b(+) ] for + = 0, 1, . . . , M. The scattered field is then computed as us (r, ϑ, z, ϑ0 , z0 ) M L

= anm Hm (kαn r)φn (z)eimϑ n=1 m=−M

=

L


an0 H0 (kαn r)φn (z)

n=1

+

L M



[anm eimϑ + an,−m (−1)m e−imϑ ]Hm (kαn r)φn (z).

(13)

m=1 n=1

The above formulation is not directly applicable to a scatterer with a profile r = f (z) for which f (z) = 0 outside some interval 0 < a < z < b < h since the functions H+ (kαn f (z))

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−5

−6

−7

−8

−9

−10 1 1

0.5 0.5

0 0 −0.5

−0.5 −1

−1

Figure 2. A conical seamount.

in (12) are singular at such points. In order to obtain an approximate field for such a scatterer we convert the object to a full-height scatterer by replacing f (z) by  z