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A frequency-domain method to inverse moving source problem with unknown radiating moment

Guanqiu Ma, Hongxia Guo, Guanghui Hu

Abstract

This paper introduces a multi-frequency factorization method for imaging a time-dependent source, specifically to recover its spatial support and the associated excitation instants. Using far-field data from two opposite directions, we establish a computational criterion that characterizes both the unknown pulse moments and the narrowest strip (perpendicular to the direction) enclosing the source support. Central to our inversion scheme is the construction of indicator functions, defined pointwise over the spatial and temporal sampling variables. The proposed inversion scheme permits the recovery of the $Θ$-convex support domain from far-field data at sparse observation directions. Uniqueness in determining the convex hull of the support and the excitation instants-using all observation directions-is also established as a direct consequence of the factorization method. The effectiveness and feasibility of the approach are examined through comprehensive numerical simulations in two and three dimensions.

A frequency-domain method to inverse moving source problem with unknown radiating moment

Abstract

This paper introduces a multi-frequency factorization method for imaging a time-dependent source, specifically to recover its spatial support and the associated excitation instants. Using far-field data from two opposite directions, we establish a computational criterion that characterizes both the unknown pulse moments and the narrowest strip (perpendicular to the direction) enclosing the source support. Central to our inversion scheme is the construction of indicator functions, defined pointwise over the spatial and temporal sampling variables. The proposed inversion scheme permits the recovery of the -convex support domain from far-field data at sparse observation directions. Uniqueness in determining the convex hull of the support and the excitation instants-using all observation directions-is also established as a direct consequence of the factorization method. The effectiveness and feasibility of the approach are examined through comprehensive numerical simulations in two and three dimensions.
Paper Structure (13 sections, 12 theorems, 71 equations, 15 figures)

This paper contains 13 sections, 12 theorems, 71 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Problem formulation in the time domain
  3. Problem reduction in the frequency domain
  4. Literature review
  5. Range of far-field operator
  6. Test functions
  7. Indicator functions and uniqueness
  8. Numerical examples
  9. Reconstruction of the pulse moment $t_0$
  10. Reconstruction of the source location and shape at time $t_0$
  11. Reconstruction of the trajectory of a moving source
  12. Noise test
  13. Reconstruction of a moving extended source in ${\mathbb R}^3$

Key Result

Theorem 2.1

We have $F=L\mathcal{T}L^*$, where $L=L_{D}^{(\hat{x})}: L^2({D})\rightarrow L^2(0, K )$ is defined by for all $u\in L^2(D)$, and $\mathcal{T}: L^2(D)\rightarrow L^2(D)$ is a multiplication operator defined by Further, the operator ${L} : L^2({\color{mgq}D}) \to L^2(0, K)$ is compact with dense range.

Figures (15)

  • Figure 1: Radiating signals $U(x_0,t)$ versus time $t$ at a fixed observation position $x_0$.
  • Figure 2: Illustration of the strips $K_{D}^{(\hat{x})}$ (green area), ${K}_{D,\eta}^{(-\hat{x})}$ and ${K}_{D,\eta}^{(\hat{x})}$ (blue area) with $\eta = 2.75$ and $\hat{x} = (1,0,0)$ in the $Ox_1x_2$-plane. The strip $K_{D, \eta}^{(\hat{x})}$ lies on the right hand side of $K_{D}^{(\hat{x})}$ if $\eta>t_0$ and on the left hand side if $\eta<t_0$.
  • Figure 3: Reconstruction for $K_{D,\eta}^{(-\hat{x})}$ using multi-frequency far-field data from a single observation direction $\hat{x}=(1,0)$ with the auxiliary indicator function $1/I_\eta^{(\hat{x})}(y)$. Various $\eta$ are tested and the pulse moment is set at $t_0=4$.
  • Figure 4: Reconstruction for $K_{D,\eta}^{(\hat{x})} \cap K_{D,\eta}^{(-\hat{x})}$ using multi-frequency far-field data from a pair of observation directions $\hat{x}=(\pm1,0)$ with the indicator function $W_\eta^{(\hat{x})}(y)$. Various $\eta$ are tested and the pulse moment is $t_0=4$.
  • Figure 5: Determination of the pulse moment $t_0$ using multi-frequency far-field data from a pair of observation directions $\hat{x}=(\pm1,0)$ with the function $h_{t_0}^{(\hat{x})}(\eta)$. The pulse moment $t_0$ is set to be $2,4,6$, respectively.
  • ...and 10 more figures

Theorems & Definitions (19)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2: Determination of $t_0$
  • ...and 9 more