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A direct sampling method based on the Green's function for time-dependent inverse scattering problems

Qingqing Yu, Bo Chen, Jiaru Wang, Yao Sun

TL;DR

This work addresses time-domain inverse acoustic scattering with point-like and normal-size scatterers by introducing direct sampling methods that rely on the Green's function and time convolution. The authors formulate indicator functions, notably $I_1(z)$ and its variants, to identify scatterer locations without solving forward or inverse problems, and they provide theoretical justification (e.g., peak behavior at the true center $y_0$). A sequence of algorithms (Algorithms 1–3) is proposed to implement these indicators, with numerical demonstrations in 2D and 3D showing robustness to noise and limited aperture data and applicability to multiple scatterers and complex shapes. The methods are simple to implement, require only integral computations, and offer a practical, transportable approach for locating scatterers in time-domain inverse scattering tasks.

Abstract

This paper concerns the numerical simulation of time domain inverse acoustic scattering problems with a point-like scatterer, multiple point-like scatterers or normal size scatterers. Based on the Green's function and the application of the time convolution, direct sampling methods are proposed to reconstruct the location of the scatterer. The proposed methods involve only integral calculus without solving any equations and are easy to implement. Numerical experiments are provided to show the effectiveness and robustness of the methods.

A direct sampling method based on the Green's function for time-dependent inverse scattering problems

TL;DR

This work addresses time-domain inverse acoustic scattering with point-like and normal-size scatterers by introducing direct sampling methods that rely on the Green's function and time convolution. The authors formulate indicator functions, notably and its variants, to identify scatterer locations without solving forward or inverse problems, and they provide theoretical justification (e.g., peak behavior at the true center ). A sequence of algorithms (Algorithms 1–3) is proposed to implement these indicators, with numerical demonstrations in 2D and 3D showing robustness to noise and limited aperture data and applicability to multiple scatterers and complex shapes. The methods are simple to implement, require only integral computations, and offer a practical, transportable approach for locating scatterers in time-domain inverse scattering tasks.

Abstract

This paper concerns the numerical simulation of time domain inverse acoustic scattering problems with a point-like scatterer, multiple point-like scatterers or normal size scatterers. Based on the Green's function and the application of the time convolution, direct sampling methods are proposed to reconstruct the location of the scatterer. The proposed methods involve only integral calculus without solving any equations and are easy to implement. Numerical experiments are provided to show the effectiveness and robustness of the methods.
Paper Structure (7 sections, 2 theorems, 51 equations, 13 figures, 1 table)

This paper contains 7 sections, 2 theorems, 51 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem setting
  3. The direct sampling methods
  4. A direct sampling method
  5. The modified indicator functions
  6. Numerical experiments
  7. Conclusion

Key Result

Lemma 3.1

$($IPSE-1$)$ Assume that $u(x,t;y)$ is the solution to the scattering problem (wave_eq)-(initiall) with a point-like scatterer $D$, the incident point $y$ and the nontrivial signal function $\lambda(t)$. The incident surface $\Gamma_i$ and the measurement surface $\Gamma_m$ satisfy $\Gamma_i=\Gamma_ where $C$ is a constant depending only on $x'$ and $D$, $\varepsilon'(x;y)=\max\{\varepsilon(x), \v

Figures (13)

  • Figure 1: Reconstructions of a single point-like scatterer centered at $(0,0)$ with different indicators and noise levels.
  • Figure 2: Reconstructions of multiple point-like scatterers using different indicator functions, $\varepsilon=5\%$. (a-c) Reconstructions of $2$ point-like scatterers centered at $(-1,-1)$ and $(1,1.5)$, respectively. (d-f) Reconstructions of $3$ point-like scatterers centered at $(-1,-1)$, $(1,1.5)$ and $(1.5,-1)$, respectively. (g-i) Reconstructions of $5$ point-like scatterers centered at $(-1,-1)$, $(1,1.5)$, $(1.5,-1)$, $(-1.5,1.5)$ and $(0,0)$, respectively.
  • Figure 3: Reconstructions of circular scatterers using different indicator functions, $\varepsilon=5\%$. The circles are centered at $(0,0)$ in the first row and centered at $(1,1)$ in the second row.
  • Figure 4: Reconstructions of kite-shaped scatterers using different indicator functions, $\varepsilon=5\%$. The kite is centered at $(0,0)$ in the first row and centered at $(1,1)$ in the second row.
  • Figure 5: Reconstructions of starfish-shaped scatterers using different indicator functions, $\varepsilon=5\%$. The starfish is centered at $(0,0)$ in the first row and centered at $(1,1)$ in the second row.
  • ...and 8 more figures

Theorems & Definitions (10)

  • Lemma 3.1
  • Theorem 3.1
  • Remark 3.1
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Example 4.4
  • Example 4.5
  • Example 4.6
  • Example 4.7