Stable Determination of an Acoustic Medium Scatterer by a Single Far-Field Pattern

In this chapter, we are concerned with the stability analysis for inverse problems associated with time-harmonic acoustic scattering, where we follow the treatment in [3]. Let k ∈ ℝ + $$k\in \mathbb {R}_+$$ be a wavenumber of the acoustic wave, signifying the frequency of the wave propagation. Let V ∈ L ∞ ( ℝ n ) $$V\in L^\infty (\mathbb {R}^n)$$ , n = 2, 3, be a potential function, which signifies the material parameter of the medium at the point x and is related to the refractive index in our setting. We assume that supp(V ) ⊂ BR, where BR is a central ball of radius R ∈ ℝ + $$R\in \mathbb {R}_+$$ in ℝ n $$\mathbb {R}^n$$ . That is, the inhomogeneity of the medium is supported inside a given bounded domain of interest. The inhomogeneous medium is often referred to as a scatterer. An incident wave ui is sent to interrogate the medium V , where the scattered wave us is generated. We let u denote the total wave field. The former is an entire solution to the Helmholtz equation (Δ + k2)ui = 0. The aforementioned acoustic scattering problem can be formulate by ( Δ + k 2 ( 1 + V ) ) u = 0 in ℝ n . $$\displaystyle {} \big (\varDelta + k^2(1+V)\big ) u = 0\quad \mbox{in}\quad {\mathbb R}^n. $$ Moreover, the scattered wave us = u − ui satisfies the Sommerfeld radiation condition | x | n − 1 2 ( ∂ r − i k ) u s → 0 $$\displaystyle {} \vert \mathbf x\vert ^{\frac {n-1}{2}} \big (\partial _r - \mathrm ik\big ) u^s \rightarrow 0 $$ uniformly with respect to the angular variable θ := x∕|x| as r : = | x | → ∞ $$r:= \vert \mathbf x\vert \rightarrow \infty $$ . Here, ∂r is the derivative along the radial direction from the origin. The radiation condition implies the existence of a far-field pattern. More precisely, there is a real analytic function on the unit sphere at infinity, namely, A u i : 𝕊 n − 1 → ℂ $$A_{u^i}: \mathbb {S}^{n-1}\rightarrow {\mathbb C}$$ satisfies u ( r θ ) = u i ( r θ ) + e i k r r ( n − 1 ) ∕ 2 A u i ( θ ) + O ( 1 r n ∕ 2 ) $$\displaystyle {} u(r\theta ) = u^i(r\theta ) + \frac {e^{ikr}}{r^{(n-1)/2}} A_{u^i}(\theta ) + \mathcal {O} \Big ( \frac {1}{r^{n/2}} \Big ) $$ uniformly along the angular variable θ. This function is called the far-field pattern or scattering amplitude of u.