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Identifying an acoustic source in a two-layered medium from multi-frequency phased or phaseless far-field patterns

Yan Chang, Yukun Guo, Yue Zhao

TL;DR

The paper tackles reconstructing an acoustic source buried in a two-layer medium from multi-frequency far-field data measured on the upper hemisphere. It extends a Fourier-based inverse source approach to two-layer media and introduces a phase retrieval step using auxiliary reference points to handle phaseless data, enabling recovery of Fourier coefficients $\hat{s}_{\vec{l}}$ and subsequent reconstruction. The main contributions are a uniqueness-based recovery of Fourier coefficients and a practical two-stage pipeline (phase retrieval followed by Fourier reconstruction) validated in 2D and 3D, showing accurate phase recovery and meaningful source reconstruction even with a limited observation aperture. The method enables rapid, non-iterative reconstruction without relying on a forward solver, with potential impact in applications involving layered media and phaseless measurements.

Abstract

This paper presents a method for reconstructing an acoustic source located in a two-layered medium from multi-frequency phased or phaseless far-field patterns measured on the upper hemisphere. The interface between the two media is assumed to be flat and infinite, while the source is buried in the lower half-space. In the phased case, a Fourier method is proposed to identify the source based on far-field measurements. This method assumes that the source is compactly supported and can be represented by a sum of Fourier basis functions. By utilizing the far-field patterns at different frequencies, the Fourier coefficients of the source can be determined, allowing for its reconstruction. For the case where phase information is unavailable, a phase retrieval formula is developed to retrieve the phase information. This formula exploits the fact that the far-field patterns are related to the source through a linear operator that preserves phase information. By developing a suitable phase retrieval algorithm, the phase information can be recovered. Once the phase is retrieved, the Fourier method can be adopted to recover the source function. Numerical experiments in two and three dimensions are conducted to validate the performance of the proposed methods.

Identifying an acoustic source in a two-layered medium from multi-frequency phased or phaseless far-field patterns

TL;DR

The paper tackles reconstructing an acoustic source buried in a two-layer medium from multi-frequency far-field data measured on the upper hemisphere. It extends a Fourier-based inverse source approach to two-layer media and introduces a phase retrieval step using auxiliary reference points to handle phaseless data, enabling recovery of Fourier coefficients and subsequent reconstruction. The main contributions are a uniqueness-based recovery of Fourier coefficients and a practical two-stage pipeline (phase retrieval followed by Fourier reconstruction) validated in 2D and 3D, showing accurate phase recovery and meaningful source reconstruction even with a limited observation aperture. The method enables rapid, non-iterative reconstruction without relying on a forward solver, with potential impact in applications involving layered media and phaseless measurements.

Abstract

This paper presents a method for reconstructing an acoustic source located in a two-layered medium from multi-frequency phased or phaseless far-field patterns measured on the upper hemisphere. The interface between the two media is assumed to be flat and infinite, while the source is buried in the lower half-space. In the phased case, a Fourier method is proposed to identify the source based on far-field measurements. This method assumes that the source is compactly supported and can be represented by a sum of Fourier basis functions. By utilizing the far-field patterns at different frequencies, the Fourier coefficients of the source can be determined, allowing for its reconstruction. For the case where phase information is unavailable, a phase retrieval formula is developed to retrieve the phase information. This formula exploits the fact that the far-field patterns are related to the source through a linear operator that preserves phase information. By developing a suitable phase retrieval algorithm, the phase information can be recovered. Once the phase is retrieved, the Fourier method can be adopted to recover the source function. Numerical experiments in two and three dimensions are conducted to validate the performance of the proposed methods.

Paper Structure

This paper contains 8 sections, 2 theorems, 59 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Fourier method for ISP
  4. Phase retrieval for PLISP
  5. Numerical examples
  6. Phase retrieval
  7. Source reconstruction
  8. Conclusion

Key Result

Theorem 1

Let $S$ be a function in $H^\sigma(V_0)\,(\sigma>0),$ then the following estimate holds:

Figures (3)

  • Figure 1: The source function $S$ is located in the lower half-space.
  • Figure 2: The exact source function and its reconstruction. (a) Surface plot (b) Reconstruction (c) Reconstruction by removing the restriction that $\theta_{\vec{l}}\in\Theta_c^n$ from \ref{['eq: XN']}
  • Figure 3: The exact source function and its reconstruction in 3D. Column 1: exact sources; Column 2: reconstructions; Row 1: $x_1=0.01,$ Row 2: $x_2=0.12,$ Row 3: $x_3=-0.5.$

Theorems & Definitions (4)

  • Theorem 1
  • Definition 1: Admissible wave numbers in the lower half-space
  • Theorem 2
  • proof