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Passive inverse obstacle scattering problems for the Helmholtz equation

Thorsten Hohage, Meng Liu

TL;DR

The paper addresses passive inverse obstacle scattering for the Helmholtz equation with a Gaussian random source, showing that near-field correlation data uniquely determine both the obstacle and (when applicable) the source strength. It develops a forward model based on the Dirichlet Green function, defines the near-field and covariance operators, and proves uniqueness results for recovering $D$ and $q$ under precise assumptions. A Fréchet-derivative framework and its adjoint are derived for shape and source dependence, enabling iterative reconstructions via IRGNM, BERGN, and Tikhonov-type methods; discretization and synthetic data experiments demonstrate effective 2D and 3D reconstructions from correlation data. The work advances passive imaging by providing both theoretical identifiability results and practical inversion strategies that perform comparably to active scattering in suitable settings.

Abstract

Passive imaging involves recording waves generated by uncontrolled, random sources and utilizing correlations of such waves to image the medium through which they propagate. In this paper, we focus on passive inverse obstacle scattering problems governed by the Helmholtz equation in $\mathbb{R}^d\;(d=2,3)$. The random source is modelled by a Gaussian random process. Uniqueness results are established for the inverse problems to determine the source strength or shape and location of an obstacle, or both of them simultaneously from near-field correlation measurements. Finally, we present efficient methods for numerical reconstructions.

Passive inverse obstacle scattering problems for the Helmholtz equation

TL;DR

The paper addresses passive inverse obstacle scattering for the Helmholtz equation with a Gaussian random source, showing that near-field correlation data uniquely determine both the obstacle and (when applicable) the source strength. It develops a forward model based on the Dirichlet Green function, defines the near-field and covariance operators, and proves uniqueness results for recovering and under precise assumptions. A Fréchet-derivative framework and its adjoint are derived for shape and source dependence, enabling iterative reconstructions via IRGNM, BERGN, and Tikhonov-type methods; discretization and synthetic data experiments demonstrate effective 2D and 3D reconstructions from correlation data. The work advances passive imaging by providing both theoretical identifiability results and practical inversion strategies that perform comparably to active scattering in suitable settings.

Abstract

Passive imaging involves recording waves generated by uncontrolled, random sources and utilizing correlations of such waves to image the medium through which they propagate. In this paper, we focus on passive inverse obstacle scattering problems governed by the Helmholtz equation in . The random source is modelled by a Gaussian random process. Uniqueness results are established for the inverse problems to determine the source strength or shape and location of an obstacle, or both of them simultaneously from near-field correlation measurements. Finally, we present efficient methods for numerical reconstructions.
Paper Structure (21 sections, 11 theorems, 79 equations, 5 figures)

This paper contains 21 sections, 11 theorems, 79 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The forward problem
  3. Well-posedness of the obstacle scattering problem for deterministic distribution-valued sources
  4. The random source process
  5. Uniqueness results
  6. Identification of the source strength q
  7. Simultaneous identification of a source strength q and an obstacle D
  8. Fréchet derivative of the forward operator and its adjoint
  9. Formulation of the simplified forward operator
  10. Fréchet derivative
  11. Adjoint of Fréchet derivative
  12. Inversion methods
  13. Discrete forward operator
  14. Synthetic data
  15. Data fidelity term
  16. ...and 6 more sections

Key Result

Theorem 1.2

\newlabelsourceshape0 Let $D_1, D_2\subset \mathbb{R}^d$ be open, bounded sets such that $\mathbb{R}^d\setminus D_i$ are connected for $i\in\{1,2\}$. Let $\Omega\subset\mathbb{R}^d$ be another open, bounded set such that $\overline{\Omega}\cap(\overline{D_1\cup D_2})=\emptyset$. Moreover, let $M$

Figures (5)

  • Figure 1: \newlabelfig:geometry0 Sketch of the geometrical setup.
  • Figure 1: Reconstructions of source strength with $\kappa=\pi$, $N_{\mathrm{sample}}=10000$, ${N_{\mathrm{src}}}=288$, ${N_{\mathrm{meas}}}=32$, and $R=5$ from $\mathcal{C}^{\mathrm{obs}}$.
  • Figure 2: Reconstructions of non-star-shaped obstacle with $\kappa=2.5\pi/2$, $N_{\mathrm{sample}}=10000$, ${N_{\mathrm{src}}}=288$, ${N_{\mathrm{meas}}}=32$, and $R=5$ from $\mathcal{C}^{\mathrm{obs}}$.
  • Figure 3: Reconstructions of non star-shaped obstacle and source strength with $\kappa=2.5\pi/2$, $N_{\mathrm{sample}}=10000$, ${N_{\mathrm{src}}}=128$, ${N_{\mathrm{meas}}}=32$, and $R=4$ from $\mathcal{C}^{\mathrm{obs}}$.
  • Figure 4: Reconstruction of a star-shaped obstacle for $\kappa=\pi$, $N_{\mathrm{sample}}=10000$, ${N_{\mathrm{src}}}=64$ and ${N_{\mathrm{meas}}}=42$, and $R=4$ from $\mathcal{C}^{\mathrm{obs}}$.

Theorems & Definitions (25)

  • Theorem 1.2
  • Proposition 2.1
  • Proof 1
  • Lemma 3.1
  • Proof 2
  • Theorem 3.2
  • Proof 3
  • Proof 4: Proof of Theorem \ref{['sourceshape']}.
  • Proposition 4.1
  • Proof 5
  • ...and 15 more