The linear sampling method for data generated by small random scatterers

TL;DR

This work addresses 2D sound-soft inverse scattering when data are generated by randomly distributed small scatterers. It advances the linear sampling method (LSM) by embedding a single controlled source within a random medium, enabling a cross-correlation–based approach to image the obstacle $D$. The authors develop a rigorous asymptotic model and a modified Helmholtz–Kirchhoff identity, and implement a boundary-element framework with SVD, Tikhonov regularization, and Morozov's discrepancy principle to compute sampling indicators from near-field data. Numerical experiments demonstrate robust obstacle reconstruction under full and limited aperture configurations and even with multiple small scatterers, with code available on GitHub. This provides a fast, data-driven initializer for inverse scattering and passive imaging, with clear pathways to 3D extensions and elastic-wave generalizations.

Abstract

We present an extension of the linear sampling method for solving the sound-soft inverse scattering problem in two dimensions with data generated by randomly distributed small scatterers. The theoretical justification of our novel sampling method is based on a rigorous asymptotic model, a modified Helmholtz--Kirchhoff identity, and our previous work on the linear sampling method for random sources. Our numerical implementation incorporates boundary elements, Singular Value Decomposition, Tikhonov regularization, and Morozov's discrepancy principle. We showcase the robustness and accuracy of our algorithms with a series of numerical experiments.

Figures (10)

• Figure 1: In subsurface imaging, the objective is to capture an image of a specific subsurface area to identify a defect, denoted as $D$---this could involve, for instance, locating shallow or deep geothermal reservoirs. In active imaging (on the left), both the sources and sensors are controlled. Vibroseismic trucks on the surface can generate signals, and sensors can be strategically placed and moved as desired. In passive imaging (on the right), controlled sensors are still utilized, but the signal originates from uncontrolled, random sources within the subsurface, such as microseisms and ocean swells.
• Figure 1: The incident field $\phi(\cdot,\boldsymbol{z}_\epsilon)$ is generated by a point source located at $\boldsymbol{z}_\epsilon=\lambda\epsilon^{-q}e^{i\theta_z}$. It is scattered by a small disk $D_\epsilon$ of radius $\mathcal{O}(\epsilon)$ centered at $\boldsymbol{y}_\epsilon=\lambda\epsilon^{-p}e^{i\theta_y}$ and an obstacle $D$ of radius $\mathcal{O}(1)$ centered at the origin. Measurements are taken near the obstacle $D$ in a volume $B$, whose size is also $\mathcal{O}(1)$. The resulting scattered field, $w^s_\epsilon$, can be approximated with an error $\mathcal{O}(\vert\log\epsilon\vert^{-1}\epsilon^p\epsilon^{q/2})$ in the $H^1(B)$-norm by the three-term sum $u^s+v^i_\epsilon+v^s_\epsilon$. As $\epsilon\to0$, the radius of $D_\epsilon$ goes to $0$, and $\boldsymbol{y}_\epsilon$ and $\boldsymbol{z}_\epsilon$ shoot to infinity. The scattering sequences, along with the amplitudes of the scattered fields, are also shown in \ref{['tab:asymptotics']}.
• Figure 1: In passive imaging with a random source (left), the defect $D$ is illuminated by an uncontrolled, random source located at points $\boldsymbol{z}$, which we assume to be distributed on a surface $\Sigma$ that encloses the defect. The medium's response is recorded at points $\boldsymbol{x}$ and $\boldsymbol{x}'$, contained in some measurement volume $B$. In passive imaging with a small random scatterer (right), the defect is illuminated by a single controlled point source located at $\boldsymbol{z}_\epsilon$. The incident field is scattered by a small random scatterer $D_\epsilon(\boldsymbol{y}_\epsilon)$ whose center $\boldsymbol{y}_\epsilon$ is located on $\Sigma_\epsilon$, creating a secondary, random source. The medium's response is recorded in some volume $B$.
• Figure 1: The left picture displays the imaginary part of the modified cross-correlation matrix \ref{['eq:mod-cross-cor-mat']} and the right picture displays the indicator function (values outside of the disk of radius $5\lambda$ were zeroed out). The measurement points are represented by crosses, while the true boundary is depicted in a solid black line. Our sampling method, based on cross-correlations and a small random scatterer, successfully identified $D$.
• Figure 2: The scattered field $w^s_\epsilon$ (top left) can be approximated with an error $\mathcal{O}(\vert\log\epsilon\vert^{-1}\epsilon^p\epsilon^{q/2})$ by the sum of three terms: $u^s$ (top right), $v^i_\epsilon$ (bottom left), and $v^s_\epsilon$ (bottom right). The relevant signal for the LSM is stored in the total field $v_\epsilon=v^i_\epsilon+v^s_\epsilon$, which satisfies a modified Helmholtz--Kirchhoff identity (\ref{['sec:mod-HK-identity']}).

Theorems & Definitions (40)

• Lemma 2.1: Estimates on $u^s$
• Proof 1
• Lemma 2.2: Estimates on $v^i_\epsilon$
• Proof 2
• Lemma 2.3: Estimates on $v^s_\epsilon$
• Proof 3
• Theorem 2.4: Estimates on $e_\epsilon$
• Proof 4
• Theorem 2.5: Estimates on $\tilde{e}_\epsilon$
• Proof 5