Inverse wave-number-dependent source problems for the Helmholtz equation

TL;DR

This work develops a frequency-domain framework for imaging the support of wave-number-dependent time-domain sources in the Helmholtz setting. By establishing a factorization $F= LTL^*$ for the multi-frequency far-field operator and proving a new range identity, it derives a practical indicator-based method to recover the $\Theta$-convex hull of the source support from sparse multi-frequency data, with a near-field extension in 3D. The approach hinges on carefully chosen test functions that probe the range of the data-to-pattern operator, yielding sharp geometry via strips $K_D^{(\hat{x})}$ and their intersections across observation directions. The paper also discusses imaging of two disconnected components and provides extensive 2D and 3D numerical experiments that confirm accuracy, robustness to noise, and the relative strengths of far-field versus near-field measurements. Overall, it offers a frequency-domain, range-based algorithm for recovering time-dependent source supports, with clear links to time-domain inverse source problems and practical imaging capabilities.

Abstract

This paper is concerned with the multi-frequency factorization method for imaging the support of a wave-number-dependent source function. It is supposed that the source function is given by the inverse Fourier transform of some time-dependent source with a priori given radiating period. Using the multi-frequency far-field data at a fixed observation direction, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the observation direction. The far-field data from sparse observation directions can be used to recover a $Θ$-convex polygon of the support. The inversion algorithm is proven valid even with multi-frequency near-field data in three dimensions. The connections to time-dependent inverse source problems are discussed in the near-field case. Numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. This paper provides numerical analysis for a frequency-domain approach to recover the support of an admissible class of time-dependent sources.

Figures (17)

• Figure 1: Illustration of the strip $K_D^{(\hat{x})}$ with $\hat{x}=(1,0)$.
• Figure 2: Reconstructions using a single observation direction and multi-frequency far-field data for a kite-shaped support. We choose $t_{\min}=0$ and $t_{\max}=T=0.1$.
• Figure 3: Reconstructions of a kite-shaped support with $S=3(x_1^2+x_2^2- {4)(t+1)}$ and $\theta=3\pi/4$ with different inverse Fourier transform windows $(0, T)$.
• Figure 4: Reconstructions of a kite-shaped support with $S=3(x_1^2+x_2^2- {4)(t+1)}$, $\theta=3\pi/4$ and with different inverse Fourier transform windows of the form $(T_0, T_0+T)$.
• Figure 5: Reconstructions of a kite-shaped source for $S=3(x_1^2+x_2^2) {(t+1)}$ with $M$ observation directions. The inverse Fourier transform window $(t_{\min}, t_{\max})$ is chosen as $t_{\min}=0$ and $t_{\max}=0.1$.

Theorems & Definitions (24)

• Theorem 2.1
• proof
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Remark 2.1
• Lemma 3.1
• proof
• Corollary 3.1