The temporal domain derivative in inverse acoustic obstacle scattering

TL;DR

The paper tackles time-domain inverse scattering for a sound-soft obstacle by rigorously defining and analyzing the temporal domain derivative, which captures leading-order changes in measurements due to boundary perturbations. It develops a Laplace-domain boundary-integral framework, proves the existence and explicit structure of the domain derivative, and transfers these results to the time domain via a temporal Hilbert-space formalism. A Runge–Kutta convolution-quadrature approach provides a stable, convergent time discretization, enabling a Gauss–Newton reconstruction scheme that uses the discrete domain derivative to recover the obstacle boundary from time-domain measurements. Numerical experiments in two dimensions validate the approach, showing effective shape reconstruction under various incident waves and noise levels, while highlighting the role of receiver placement.

Abstract

This work describes and analyzes the domain derivative for a time-dependent acoustic scattering problem. We study the nonlinear operator that maps a sound-soft scattering object to the solution of the time-dependent wave equation evaluated at a finite number of points away from the obstacle. The Fréchet derivative of this operator with respect to variations of the scatterer coincides with point evaluations of the temporal domain derivative. The latter is the solution to another time-dependent scattering problem, for which a well-posedness result is shown under sufficient temporal regularity of the incoming wave. Applying convolution quadrature to this scattering problem gives a stable and provably convergent semi-discretization in time, provided that the incoming wave is sufficient regular. Using the discrete domain derivative in a Gauss--Newton method, we describe an efficient algorithm to reconstruct the boundary of an unknown scattering object from time domain measurements in a few points away from the boundary. Numerical examples for the acoustic wave equation in two dimensions demonstrate the performance of the method.

Figures (6)

• Figure 1: Scattering from a kite-shaped obstacle $D$. In the left and middle plot the solid black strip represents the support of the incoming wave $u^i$. Diamonds represent receivers that detect the scattered wave $u$. Visualizations of $u$ at the receivers $1$ and $2$ are found in the right plot.
• Figure 2: Visualization of the scattering object with boundary parametrized by the curve from \ref{['eq:parex1']}, together with the total wave $u + u^i$ corresponding to Example \ref{['ex:1']} and Example \ref{['ex:2']}.
• Figure 3: Convergence history starting with a circle with radius $r=0.5$ and center point ${\boldsymbol z} = [-1, \; -1.5]^\top\in {\mathbb{R}}^2$ (left). The middle plot shows the iterate $\ell = 5$. The right plot shows the final result after 17 steps.
• Figure 4: Convergence history starting with a circle with radius $r=0.5$ and center point ${\boldsymbol z} = [-1, \; -1.5]^\top\in {\mathbb{R}}^2$ (left). The middle plot shows the iterate $\ell = 5$. The right plot shows the final result after 20 steps.
• Figure 5: Visualization of the dove-shaped scattering object together with the total wave $u + u^i$ of Example \ref{['ex:3']} at different times. The incoming wave is zero for $t\leq 0.5$.

Theorems & Definitions (30)

• Remark 2.1: Notational convention
• Proposition 2.2
• proof
• Proposition 2.3
• Remark 2.4
• proof
• Remark 2.5
• Proposition 2.6
• proof
• Corollary 2.7