A constructive solution to the inverse scattering problem of the wave equation at one

frequency and plane wave irradiation

F. Natterer

August 1989

Summary: This is a tutorial paper reporting essentially on a method suggested by [2], [3] and [1]. The conclusion is that the method is useless for numerical purposes.

1 Introduction

Let Ω be a bounded domain in R^{n}, n ≥ 3, and let supp(q) ⊆ Ω. Let k ≥ 0 be a fixed wave number. For each Θ ∈ S^{n−1} let u_{Θ} be the solution to the reduced wave equation

(Δ + k^{2} + q)u = 0  (1)

such that

u_{Θ} = e^{ikx·Θ} + w_{Θ}

where w_{Θ} satisfies Sommerfelds radiation condition. The problem is to recover q from the knowledge of u_{Θ} outside Ω.

2 The NSU method

An essential ingredient of the Nachmann - Sylvester - Uhlmann method is a certain - non-physical - solution ψ_{ζ} of (1) depending on the parameter ζ ∈ C^{n}.

Proposition 1: Let ζ^{2}(= ζ · ζ) = k^{2}, Imζ ≠ 0 and δ > 1/2. Then there is a unique solution ψ_{ζ} of (1) such that

ψ_{ζ} = e^{ix·ζ}(1 + v_{ζ})

where v_{ζ} ∈ L_{2,−δ} (= L_{2}(R^{n}, (1 + x^{2})^{−δ})) and

∥v_{ζ}∥_{L_{2,−δ}} ≤ \frac{c}{|ζ|} .

Proof: With ψ_{ζ} = e^{ixζ}(1 + v_{ζ}), (1) assumes the form

Δv_{ζ} + 2iζ · ▽v_{ζ} = −q(1 + v_{ζ}) .

In [2] it is shown that for a sufficiently large and δ > \frac{1}{2} there is a constant c(δ, a) such that for Imζ ≠ 0 and |ζ| ≥ a

Δv + 2iζ · ▽v = f

has a unique solution for each f ∈ L_{2,δ}, and

∥v∥_{L_{2,−δ}} ≤ \frac{c(δ, a)}{|ζ|}∥f∥_{L_{2,δ}} .  (2)

It follows that v_{ζ} satisfies

\begin{aligned}∥v_{ζ}∥_{L_{2,−δ}} & ≤ & \frac{c(δ,a)}{|ζ|}∥q(1 + v_{ζ})∥_{L_{2,δ}} \\ & ≤ & \frac{c(δ,a)}{|ζ|} \{∥q∥_{L_{2,δ}} + ∥q(1 + x^{2})^{δ}∥_{L_{∞}(Ω)}∥v_{ζ}∥_{L_{2,−δ}}\} . \end{aligned}

For |ζ| > c(δ, a)∥q(1 + x^{2})^{δ}∥_{L_{∞}(Ω)} the estimate on v_{ζ} follows.

□

The NSU method starts out from Green‘s formula

∫_{Ω} {(Δ + k^{2} + q)ψ · φ − ψ · (Δ + k^{2} + q)φ} dx

= ∫_{∂Ω} (\frac{∂ψ}{∂ν} · φ − ψ \frac{∂φ}{∂ν}) dσ(x) .

With ψ = ψ_{ζ}, φ = e^{−ix·(ζ+ξ)}, ξ ∈ R^{n}, (ξ + ζ)^{2} = k^{2}, this reads

− ∫_{Ω} ψ_{ζ}qe^{−ix(ζ+ξ)}dx = t(ξ, ζ)

where

t(ξ, ζ) = ∫_{∂Ω} (\frac{∂ψ_{ζ}}{∂ν} + i(ξ + ζ) · νψ_{ζ}) e^{−ix·(ζ+ξ)}dσ(x) .

Making use of Proposition 1 we get

− ∫_{Ω} (1 + v_{ζ})qe^{−ix·ξ}dx = t(ξ, ζ) .

As an immediate consequence we obtain

Theorem 2.1 The Fourier-transform q̂ of q can be obtained from t by

q̂(ξ) = −(2π)^{−n/2} \begin{matrix}lim t(ξ, ζ) \\ ζ^{2} = k^{2} \\ ξ^{2} + 2ζ · ξ = 0 \\ |ζ| → ∞ . \end{matrix}

We are left with the problem of determining t.

3 The Dirichlet - to - Neumann map

This is an operator, denoted by Λ_{q}, which acts on suitable function spaces on ∂Ω. For a function f on ∂Ω, Λ_{q}f is defined as follows: Solve (1) in Ω with Dirichlet boundary values f on ∂Ω and put Λ_{q}f = \frac{∂u}{∂ν} on ∂Ω.

Knowing Λ_{q} we can reconstruct q. In fact, since q = 0 in Ω′ = R^{n} \ Ω, ψ_{ζ} satisfies

(Δ + k^{2})ψ_{ζ} = 0  in Ω′
\frac{∂ψ_{ζ}}{∂ν} = Λ_{q}ψ_{ζ} on ∂Ω′  (3)
ψ_{ζ} = e^{ix·ζ}(1 + v_{ζ}), v_{ζ} ∈ L_{2,−δ} .

This is an exterior Helmholtz problem with a non-standard condition at ∞ and with an operator boundary condition. It can be solved, preferably by boundary equation methods, once Λ_{q} is given. Then, t can be found, whence q by Theorem 2.1.

In order to find Λ_{q} from the plane-wave scattering data we represent a function f on ∂Ω as

f(x) = ∫_{S^{n−1}} u_{Θ}(x)w(Θ)dΘ , x ∈ ∂Ω .  (4)

Then,

(Λ_{q}f)(x) = ∫_{S^{n−1}} \frac{∂}{∂ν} u_{Θ}(x)w(Θ)dΘ .

Since u_{Θ}(x) is known for x ∉ Ω and Θ ∈ S^{n−1} this determines Λ_{q}.

4 Numerical considerations

The method requires 3 steps:

i) Determine Λ_{q} from u_{Θ}(x), x ∉ Ω, Θ ∈ S^{n−1}.

ii) For each ξ ∈ R^{n} find a sequence (ζ) of points ζ ∈ C^{n} with ζ^{2} = k^{2}, ξ^{2} +2ζ ·ξ = 0, |ζ| → ∞ and determine ψ_{ζ} as the solution of the exterior Helmholtz problem (3).

iii) Compute t(ξ, ζ) from ψ_{ζ} and q̂ from t. Do an inverse Fourier transform to obtain q.

Step (i) calls for the solution of the integral equation (4). In the - trivial - case q = 0 we have u_{Θ}(x) = e^{ikΘ·x}, hence (4) reads

f(x) = ∫_{S^{n−1}} e^{ikΘ·x}w(Θ)dΘ, x ∈ ∂Ω .

For Ω the unit ball, ∂Ω = S^{n−1}, this equation can be solved by spherical harmonics: Let

f = ∑_{ℓ,m} f_{ℓm}Y_{ℓm} , w = ∑_{ℓ,m} w_{ℓm}Y_{ℓm} .

Since

\begin{aligned}∫_{S^{n−1}} e^{ikΘ·x}Y_{ℓm}(Θ)dΘ & = & (2π)^{n/2}i^{ℓ}k^{(2−n)/2}J_{ℓ+(n−2)/2}(k)Y_{ℓm}(\frac{x}{|x|}) \\ & = & c_{ℓ}(k)Y_{ℓm}(\frac{x}{|x|}) \end{aligned}

the solution is

w_{ℓm} = \frac{f_{ℓm}}{c_{ℓ}(k)} .

Unfortunately, c_{ℓ}(k) ≪ 1 if ℓ > k. This means that w_{ℓm} is poorly determined for ℓ > k. Thus, Λ_{q} can be determined stably only on the subspace ⟨Y_{ℓm} : ℓ ≤ k⟩ of L_{2}(∂Ω). This does not come as a surprise since we expect the resolution in an inverse scattering experiment with wave number k to be of the order k.

On the other hand, in step (ii), we have to apply Λ_{q} to the high frequency function ψ_{ζ} (|ζ| is large!). Thus the information on Λ_{q} we get from step (i) is not sufficient for step (ii). All we can expect is an O(\frac{1}{k}) - approximation to q. But this can be achieved much easier by other methods. Besides, the computational cost of step (ii) is probably prohibitive, unless one finds expedient asymptotic methods for solving (3) for |ζ| large.

References

[1] Calderon, A.P. (1980). On an inverse boundary value problem. Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Mathematica, Rio de Janerio.

[2] Nachmann, A.I. (1988). Reconstructions from boundary measurements. Annals of Mathematics 128, 531-576.

[3] Sylvester, J. - Uhlmann, G. (1987). A global uniqueness theorem for an inverse boundary value problem. Annals of Mathematics 125, 153-169.

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