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On the detection of medium inhomogeneity by contrast agent: wave scattering models and numerical implementations

Zhe Wang, Ahcene Ghandriche, Jijun Liu

Abstract

We consider the wave scattering and inverse scattering in an inhomogeneous medium embedded a homogeneous droplet with a small size, which is modeled by a constant mass density and a small bulk modulus. Based on the Lippmann-Schwinger integral equation for scattering wave in inhomogeneous medium, we firstly develop an efficient approximate scheme for computing the scattered wave as well as its far-field pattern for any droplet located in the inhomogeneous background medium. By establishing the approximate relation between the far-field patterns of the scattered wave before and after the injection of a droplet, the scattered wave of the inhomogeneous medium after injecting the droplet is represented by a measurable far-field patterns, and consequently the inhomogeneity of the medium can be reconstructed from the Helmholtz equation. Finally, the reconstruction process in terms of the dual reciprocity method is proposed to realize the numerical algorithm for recovering the bulk modulus function inside a bounded domain in three dimensional space, by moving the droplet inside the bounded domain. Numerical implementations are given using the simulation data of the far-field pattern to show the validity of the reconstruction scheme, based on the mollification scheme for dealing with the ill-posedness of this inverse problem.

On the detection of medium inhomogeneity by contrast agent: wave scattering models and numerical implementations

Abstract

We consider the wave scattering and inverse scattering in an inhomogeneous medium embedded a homogeneous droplet with a small size, which is modeled by a constant mass density and a small bulk modulus. Based on the Lippmann-Schwinger integral equation for scattering wave in inhomogeneous medium, we firstly develop an efficient approximate scheme for computing the scattered wave as well as its far-field pattern for any droplet located in the inhomogeneous background medium. By establishing the approximate relation between the far-field patterns of the scattered wave before and after the injection of a droplet, the scattered wave of the inhomogeneous medium after injecting the droplet is represented by a measurable far-field patterns, and consequently the inhomogeneity of the medium can be reconstructed from the Helmholtz equation. Finally, the reconstruction process in terms of the dual reciprocity method is proposed to realize the numerical algorithm for recovering the bulk modulus function inside a bounded domain in three dimensional space, by moving the droplet inside the bounded domain. Numerical implementations are given using the simulation data of the far-field pattern to show the validity of the reconstruction scheme, based on the mollification scheme for dealing with the ill-posedness of this inverse problem.

Paper Structure

This paper contains 9 sections, 2 theorems, 135 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Reduced Acoustic Model
  3. The direct scattering for inhomogeneous medium
  4. Generated fields for the unperturbed medium
  5. Generated fields for the perturbed medium
  6. Numerical implementations for direct scattering
  7. The unperturbed medium case
  8. The perturbed medium case
  9. Recovering the bulk modulus function $k_{0}(\cdot)$ inside $\Omega \, \equiv \, B({\bf 0},1)$

Key Result

Lemma 2.1

Denote by $\{ (\lambda_{n}, e_{n}),\;n \in \mathbb{N}\}$ the eigensystem related to the Newtonian operator $N_{B}$, defined from $\mathbb{L}^{2}(B)$ to $\mathbb{L}^{2}(B)$ by i.e., $N_{B}(e_{n}) = \lambda_{n} e_{n}$ in $B$. Then, for $z\in\Omega$ such that $D_z\subset\subset\Omega$, the far-filed patterns admit uniformly for all $\theta,\hat{x}\in \mathbb{S}^{2}$, under the condition $\omega^{

Figures (12)

  • Figure 1: Comparison of $\phi-\tilde{\theta}$ distributions at polar radius $r=0.3$.
  • Figure 2: Comparison of $\phi-\tilde{\theta}$ distributions at polar radius $r=0.6$.
  • Figure 3: Comparison of far-field for $\hat{x}\in \mathbb{S}^2$.
  • Figure 4: Comparison of $\phi-\tilde{\theta}$ distributions at polar radius $r=0.2$.
  • Figure 5: Comparison of $\phi-\tilde{\theta}$ distributions at polar radius $r=0.4$.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.5
  • Remark 2.6
  • Remark 3.1
  • Remark 3.2
  • Remark 4.1