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Selective focusing of multiple particles in a layered medium

Jun Lai, Jinrui Zhang

TL;DR

The paper tackles inverse scattering in a two-layer medium and introduces a hybrid approach that leverages the time reversal operator T = F^* F to achieve global and selective focusing on buried scatterers from limited aperture data. By conducting a detailed asymptotic and spectral analysis of the layered Green's function, it shows that, in the small, well-separated regime, each sound-soft particle yields one significant eigenvalue of T while each sound-hard particle yields three, with corresponding eigenfunctions generating incident waves that selectively target individual scatterers. These insights underpin a two-stage imaging strategy: use time reversal to obtain an initial indicator of scatterer locations, followed by Bayesian inversion to recover the shapes of extended obstacles with higher resolution. Numerical experiments in the paper demonstrate the method's efficiency and robustness, including selective focusing on multiple particles and improved reconstructions via Bayesian techniques, highlighting its potential for geophysical, medical, and remote-sensing applications in layered media.

Abstract

Inverse scattering in layered media has a wide range of applications, examples including geophysical exploration, medical imaging, and remote sensing. In this paper, we develop a selective focusing method for identifying multiple unknown buried scatterers in a layered medium. The method is derived through the asymptotic analysis of the time reversal operator using the layered Green's function and limited aperture measurements. We begin by showing the global focusing property of the time reversal operator. Then we demonstrate that each small sound-soft particle gives rise to one significant eigenvalue of the time reversal operator, while each sound-hard particle gives three. The associated eigenfunction generates an incident wave focusing selectively on the corresponding unknown particle. Finally, we employ the time reversal method as an initial indicator and propose an effective Bayesian inversion scheme to reconstruct multiple buried extended scatterers for enhanced resolution. Numerical experiments are provided to demonstrate the efficiency.

Selective focusing of multiple particles in a layered medium

TL;DR

The paper tackles inverse scattering in a two-layer medium and introduces a hybrid approach that leverages the time reversal operator T = F^* F to achieve global and selective focusing on buried scatterers from limited aperture data. By conducting a detailed asymptotic and spectral analysis of the layered Green's function, it shows that, in the small, well-separated regime, each sound-soft particle yields one significant eigenvalue of T while each sound-hard particle yields three, with corresponding eigenfunctions generating incident waves that selectively target individual scatterers. These insights underpin a two-stage imaging strategy: use time reversal to obtain an initial indicator of scatterer locations, followed by Bayesian inversion to recover the shapes of extended obstacles with higher resolution. Numerical experiments in the paper demonstrate the method's efficiency and robustness, including selective focusing on multiple particles and improved reconstructions via Bayesian techniques, highlighting its potential for geophysical, medical, and remote-sensing applications in layered media.

Abstract

Inverse scattering in layered media has a wide range of applications, examples including geophysical exploration, medical imaging, and remote sensing. In this paper, we develop a selective focusing method for identifying multiple unknown buried scatterers in a layered medium. The method is derived through the asymptotic analysis of the time reversal operator using the layered Green's function and limited aperture measurements. We begin by showing the global focusing property of the time reversal operator. Then we demonstrate that each small sound-soft particle gives rise to one significant eigenvalue of the time reversal operator, while each sound-hard particle gives three. The associated eigenfunction generates an incident wave focusing selectively on the corresponding unknown particle. Finally, we employ the time reversal method as an initial indicator and propose an effective Bayesian inversion scheme to reconstruct multiple buried extended scatterers for enhanced resolution. Numerical experiments are provided to demonstrate the efficiency.

Paper Structure

This paper contains 17 sections, 12 theorems, 111 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Global focusing in a layered medium
  4. Time reversal operator
  5. Global focusing
  6. Selective focusing of multiple buried particles
  7. Asymptotic behavior of far field pattern
  8. Selective focusing of sound-soft particles
  9. Selective focusing of sound-hard particles
  10. Bayesian inversion for the shape of extended buried obstacles
  11. Numerical experiments
  12. Selective imaging results of multiple particles
  13. Imaging of extended obstacles based on time reversal operator
  14. Reconstruction of extended obstacles improved by Bayesian technique
  15. Conclusion
  16. ...and 2 more sections

Key Result

Proposition 2.1

For layered medium scattering, the far field in the upper half space is where $Q(\hat{x})$ is given in equation transcoef. Furthermore, $u^{\infty}(\hat{x},d)$ is an analytic function on $-\widehat{\mathbb{S}}\times\widehat{\mathbb{S}}$ and satisfies the reciprocity relation

Figures (6)

  • Figure 1: Inverse scattering of multiple buried particles using the far field measurement.
  • Figure 2: Eigenvalues of the time reversal operator. (a) Sound-soft boundary condition. (b) Sound-hard boundary condition. Imaging of two sound-soft particles. (c)(d) The Herglotz wave with the eigenfunction of the first and second eigenvalues, respectively.
  • Figure 3: Imaging of two sound-hard particles. The Herglotz wave with the eigenfunction of the first six eigenvalues.
  • Figure 4: Imaging of 9 particles. (a) Exact locations. (b) Imaging of sound-soft particles. (c) Imaging of sound-hard particles.
  • Figure 5: Imaging of two extended obstacles. (a) The black dotted line represents the exact shape. (b) The Herglotz wave with eigenfunctions corresponding to the first eighty eigenvalues of sound-soft obstacles. (c) The Herglotz wave with eigenfunctions corresponding to the first eighty eigenvalues of sound-hard obstacles.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Proposition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • Lemma 4.1
  • Theorem 4.2
  • proof
  • Remark 4.3
  • ...and 13 more