Inverse Obstacle Scattering

One of the basic inverse problems in scattering theory for time-harmonic acoustic waves is to determine the shape of an obstacle D with boundary B from a knowledge of the incident wave u_i and far field pattern u_inf of the scattered wave u_s

The scattering of time-harmonic acoustic waves by an infinite cylindrical obstacle with bounded cross section D a subset of R² leads to an exterior boundary value problem for the Helmholtz equation u +k² u = 0 with positive wave number k and Dirichlet boundary condition in R²/D. The total wave u is decomposed into the given incident wave u_i and the unknown scattered wave u_s which is required to satisfy the Sommerfeld radiation condition v_r - i k v = o(||x||) as ||r-x|| -> uniformly in all directions. Usually the incident disturbance is given by a plane wave, u_i(x) = exp(i k d.x), where d is a unit vector giving the direction of propagation.

The radiation condition (Sommerfeld) ensures the uniqueness for the exterior Dirichlet problem and leads to an asymptotic behavior for the scattered wave of the form

u_s= exp(i k||x||)/

||x|| { u_infx + O(1/||x||)} as ||x|| ->

. The function u_inf defined on the unit circle in R²/D is the (complex-valued) far field pattern of the scattered wave. The most comprehensive and readable reference here is to the book [CK]

In addition to the above, there is a boundary condition on the scatter itself. The case of an impenetrable sound-soft obstacle requires u=0 on B and u_n=0 is the sound hard case. The boundary need not be homogeneous and the condition u_n+iµ(t)u=0 governed by the impedance parameter µ(t) is realistic. There is also the case where the wave passes into the interior and the boundary condition is of transmission type, u_n|₊ - µ u_n|_- = 0

The inverse problem is, given the far field pattern u_inf of the scattered wave for a set of incoming plane waves, to determine the shape of the scatterer D.

Denoting by F the operator which, for a fixed u_i maps B onto the far field pattern u_inf the inverse problem consists in solving the nonlinear and ill-posed equation F(B)= u_inf for the unknown scattering object. Over the last few years we (this is joint work with Rainer Kress) have been interested in resolving the minimum amount of data required to reconstruct an obstacle from acoustic scattering data. The results of [KR1] show that even one incident wave is overdetermining, in the sense that the formulation requires the recovery of a real curve B from complex-valued far-field data. This points to directions of further inquiry. Under most circumstances it is much easier to measure amplitude information than it is to measure phase information, so this issue has considerable practical import. This lead to work on the recovery of an obstacle from amplitude-only data [KR2]. A further issue of considerable practical interest is to determine the type of boundary of the unknown obstacle for this cannot always be assumed as known. There is some analytical and numerical evidence indicating that the full far field pattern contains information on both the shape and the impedance condition µ(t) The conjecture is that a far field pattern from a single incident wave uniquely determines the pair . We are currently working on proving a local uniqueness result in the neighborhood of a known scatterer and known impedance value.

In a slightly different direction, we have completed a paper [KR3] that showed unique recovery of a scattering obstacle with boundary B from measurement of the far field at isolated points. First, when a complete sequence of incident waves from directions dj is used, and the far field pattern u_inf is evaluated at a single point xj equal to the incident direction dj plus a fixed offset angle gamma. Uniqueness and constructibility was shown under certain conditions. If the obstacle D is characterised by a curve of the form

q(t)=SUM(a_kcos(2 Pi k t)+b_ksin(2 Pi k t)) then we have

Theorem. Let 0 < gamma < 2 Pi. Then if the wavenumber k is sufficiently small the derivative map F' is injective. In the finite dimensional problem with M incident waves from distinct directions and a finite trigonometric basis with frequencies up to order N the resulting Jacobian matrix has trivial nullspace provided M >= 2N+1.

This configuration of measuring the scattered wave at a single point from a sequence of sources from different directions is common in medical imaging.

If the incident wave comes from a single direction d but at a range of frequencies k forming the incident fields and the scattered wave is measured at a single point d+gamma, then the nullspace of the map F(B) to u_inf(d+gamma) can be determined,

Theorem. For any gamma with 0 < gamma < 2 Pi, the nullspace of F' consists of the odd numbered cosine and the even numbered sine coefficients (when expanded with the origin at t=½gamma).

This allows us to prove,

Theorem. From the values of the far field pattern measured at two angles gamma₁ and gamma₂, we can recover all Fourier coefficients of the curve q characterising B provided sin(½m(gamma₁-gamma₂)) is non zero for m = 1,.....,N.

The original motivation for this problem was to recover sunken mines by means of reflected sonar waves from the deck of a ship.

References

David L. Colton and Rainer Kress Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edition, Springer-Verlag, Berlin, Heidelberg, 1998.
Kress, Rundell 1 "A Quasi-Newton Method in Inverse Obstacle Scattering", Inverse Problems, 10, (1994), 1145-1157.
Kress, Rundell 2 "Inverse obstacle scattering with modulus of the far field pattern as data."
Kress, Rundell 3 "Inverse Obstacle Scattering Using Reduced Data." SIAM J. Appld. Math, to appear.