Inverse acoustic scattering for random obstacles with multi-frequency data

TL;DR

The paper tackles inverse obstacle scattering with geometric uncertainty by modeling the obstacle boundary as a Gaussian process on the angular domain and parameterizing it via a truncated KL expansion. A two-stage method first reconstructs the mean shape using multi-frequency data and a Recursive Linearization approach, then estimates the KL spectrum and covariance hyperparameters by analyzing the reconstructed fluctuations across realizations. Theoretical results establish well-definedness, local identifiability, and convergence estimates for the eigenvalues and hyperparameters, complemented by numerical experiments on circle, pear, and flower geometries showing stable recovery of geometry and statistics. The approach provides a statistically informed framework for random obstacle inversion with practical multi-frequency measurements, highlighting both its effectiveness for simple shapes and the challenges posed by higher geometric complexity.

Abstract

We study an inverse random obstacle scattering problems in $\mathbb{R}^2$ where the scatterer is formulated by a Gaussian process defined on the angular parameter domain. Equipped with a modified covariance function which is mathematically well-defined and physically consistent, the Gaussian process admits a parameterization via Karhunen--Loève (KL) expansion. Based on observed multi-frequency data, we develop a two-stage inversion method: the first stage reconstructs the baseline shape of the random scatterer and the second stage estimates the statistical characteristics of the boundary fluctuations, including KL eigenvalues and covariance hyperparameters. We further provide theoretical justifications for the modeling and inversion pipeline, covering well-definedness of the Gaussian-process model, convergence for the two-stage procedure and a brief discussion on uniqueness. Numerical experiments demonstrate stable recovery of both geometric and statistical information for obstacles with simple and more complex shapes.

Figures (18)

• Figure 1: schematic of the inverse obstacle scattering problem with a single incident direction.
• Figure 2: \ref{['fig:covariance']}: 3 periodic covariance functions with $\sigma=1$, $\ell = 0.5$, $N_{\theta}=400$ and 50 Fourier expansion coefficients $\{a_j\}$: $C_{\mathrm{geod}}(t)$ (red solid line) defined in \ref{['eq:cov-1d']}; $C_f(t)$ (black dashed line) defined in \ref{['standard']}; ${C}_{\delta r}(t)$ (blue dashed line with "o" markers) defined in \ref{['correc']}. \ref{['fig:covariance10000']}: 3 periodic covariance functions with $\sigma=1$, $\ell = 0.5$, $N_{\theta}=40004$ and 10000 Fourier expansion coefficients $\{a_j\}$: $C_{\mathrm{geod}}(t)$ (red solid line); $C_f(t)$ (black dashed line); ${C}_{\delta r}(t)$ (blue dashed line with "o" markers). \ref{['hinvv']}: decay of first 30 $\lambda_j$ of the covariance operator $\mathcal{C}$ when $\sigma=0.05$, $\ell=0.5$ and $N_{\theta}=400$: $\lambda_j$ under $C_{\mathrm{geod}}(t)$ (black line); $\lambda_j$ under ${C}_{\delta r}(t)$ (red line with "o" markers).
• Figure 3: the geometry of the scatterer
• Figure 4: verification of reciprocity relation \ref{['reciprocity']} plotted in polar coordinates. red dashed line: $|u^{\infty}(-d,-\hat{x})|$; blue solid line: $|u^{\infty}(\hat{x},d)|$.
• Figure 5: \ref{['fig:randomcircle']}: random realizations with 20 samples. dashed line: random circular scatterer; solid line: standard circular scatterer. \ref{['fig:samplecircle']}: geometric visualization of the $6$-th random sample and its corresponding far-field pattern data. left: dashed line represents the circle; solid line represents the true scatterer shape. right: far-field pattern $u^\infty(\hat{x}_m,d,k=8)$ for sample $6$ plotted in polar coordinates.

Theorems & Definitions (18)

• Theorem 2.1
• proof
• Remark 1
• Remark 2
• Proposition 3.1
• proof
• Theorem 4.1
• proof
• Proposition 4.1
• proof