The high resolution sampling methods for acoustic sources from multi-frequency far field patterns at sparse observation directions

TL;DR

The paper addresses the inverse source problem in the plane, aiming to recover the support $\Omega$ of a source from multi-frequency far-field patterns measured at sparse directions $\Theta_L$. A key lemma links the discontinuities of the derivative of an integral transform $I_{\hat{x}}$ to lines that pass through corners or are tangent to boundary components, enabling uniqueness results. The authors prove that when $\Omega$ is composed of annuli and polygons, the entire support and, under piecewise-constant $f$, the values on the boundary can be uniquely determined from sparse multi-frequency data, with explicit lower bounds on $L$ depending on the structure. Building on this theory, two direct sampling indicators, $\mathcal{I}^-$ and $\mathcal{I}^+$, are introduced to reconstruct $\Omega$ with high resolution; the second indicator can also recover $f$ under sufficient directional data. Numerical experiments in 2D demonstrate robustness to noise, effectiveness for complex geometries, and performance under limited aperture, highlighting practical applicability and potential extensions to obstacle and medium scattering problems.

Abstract

This work is dedicated to novel uniqueness results and high resolution sampling methods for source support from multi-frequency sparse far field patterns. With a single pair of observation directions $\pm\hat{x}$, we prove that the lines $\{z\in\mathbb R^2|\, \hat{x}\cdot z = \hat{x}\cdot y, \,y\in A_{\hat{x}}\}$ can be determined by multi-frequency far field patterns at the directions $\pm\hat{x}$, where $A_{\hat{x}}$ denotes a set containing the corners of the boundary and points whose normal vector to the boundary is parallel to $\hat{x}$. Furthermore, if the source support is composed of polygons and annuluses, then we prove that the support can be determined by multi-frequency far field patterns at sparse directions. Precisely, the lowest number of the observation directions is given in terms of the number of the corners and the annuluses. Inspired by the uniqueness arguments, we introduce two novel indicators to determine the source support. Numerical examples in two dimensions are presented to show the validity and robustness of the two indicators for reconstructing the boundaries of the source support with a high resolution. The second indicator also shows its powerful ability to determine the unknown source function.

Figures (12)

• Figure 1: Illustration for $\alpha_j$ and $\beta_j$.
• Figure 7: Reconstructions of the source support with $L=1,\Lambda=30$. Left: The true source functions. Middle: reconstructions by plotting $\mathcal{I}^-$, here $\partial\Omega$ is plotted by red curves. Right: reconstructions by plotting $\widetilde{\mathcal{I}}^{-}$, here $\partial\Omega$ is plotted by red lines.
• Figure 8: Reconstructions of the source support with $L=25,\Lambda=30$. Left: The true source functions. Middle: reconstructions by plotting $\mathcal{I}^-$. Right: reconstructions by plotting $\widetilde{\mathcal{I}}^{-}$.
• Figure 9: Reconstructions of a kite structured source with $\Lambda=30$. Top row: results by plotting $\mathcal{I}^-$. Bottom row: results by plotting $\mathcal{I}^+$.
• Figure 10: Left: True source support $\Omega$ in this subsection. Right: Reconstruction by plotting $I_{ALHS}$ with $L=15,\Lambda=20$.

Theorems & Definitions (14)

• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof