Inverse obstacle scattering with a single moving emitter

TL;DR

This work addresses time-domain inverse obstacle scattering using a single moving point emitter with $|v(t)|<c$, deriving approximate forward-field solutions centered at the scatterer and proposing two direct sampling indicators, $I_1$ and $I_2$ (with a practical $\tilde{I}_2$), to reconstruct both point-like and extended scatterers from data on $\Gamma_m$. The approach leverages a boundary-integral formulation to obtain center-based representations and employs a time-convolution indicator for robust inversion, validated by extensive 2D and 3D numerical experiments. The results show accurate reconstructions from a single moving emitter, with strong robustness to noise and limited aperture data, illustrating a non-iterative, data-driven alternative to full time-domain solvers. The methods have potential implications for geoacoustic inversion and Doppler-tomography-type applications where moving sources are natural probes.

Abstract

This paper is concerned with time domain forward scattering and inverse scattering problems with a single moving point source as the emitter. Approximate solutions are provided for the forward scattering problem with a moving emitter. Regarding the inverse problem, in addition to a basic indicator function based on the approximate solutions, a novel indicator function is developed to construct the direct sampling method to recover both point-like and extended scatterers. Numerical experiments demonstrate that the proposed algorithms are effective in reconstructing both two-dimensional and three-dimensional scatterers with a single moving emitter.

Figures (13)

• Figure 1: Two-dimensional geometrical setting of the problem.
• Figure 2: The signal function $\lambda_N(t)$.
• Figure 3: The inversion performance of the indicator functions $I_1(z)$ and $\tilde{I}_2(z)$ for multiple point-like scatterers with the signal function $\lambda_N (t)$ under different numbers of periods. The noise level is $\sigma=5\%$.
• Figure 4: The inversion performance of the indicator functions $I_1(z)$ and $\tilde{I}_2(z)$ for an extended scatterer with the signal function $\lambda_N (t)$ under different numbers of periods. The noise level is $\sigma=5\%$.
• Figure 5: The inversion performance of the indicator functions $I_1(z)$ and $\tilde{I}_2(z)$ with different moving speeds of the emitter. The noise level is $\sigma=5\%$.

Theorems & Definitions (6)

• Lemma 2.1: cf. Bo Chen & Yao Sun CB point-like scatterer
• Proposition 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2