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The monotonicity method for the inverse elastic scattering on unbounded domains

Bastian Harrach, Jianli Xiang

TL;DR

The paper addresses inverse elastic scattering on unbounded domains to locate inhomogeneities in the Lamé parameters and density from far-field data. It extends the monotonicity method to the time-harmonic Navier equation by establishing a monotonicity relation for the far-field operator $F_c$ and coupling it with localized potentials to obtain constructive shape tests. The main contributions are monotonicity inequalities for the Navier system in unbounded settings, a localized-potential theory on unbounded domains, and a monotonicity-based reconstruction criterion that detects the support osupp$(D)$. This yields a non-iterative, data-driven approach for robustly identifying elastic inclusions from scattering measurements with potential applications in nondestructive testing and geophysical imaging.

Abstract

We discuss a time-harmonic inverse scattering problem for the Navier equation with compactly supported penetrable and possibly inhomogeneous scattering objects in an unbounded homogeneous background medium, and we develop a monotonicity relation for the far field operator that maps superpositions of incident plane waves to the far field patterns of the corresponding scattered waves. Combining the monotonicity relation with the method of localized potentials, we extend the so called monotonicity method to characterize the support of inhomogeneities in the Lamé parameters and the density in terms of the far field operator.

The monotonicity method for the inverse elastic scattering on unbounded domains

TL;DR

The paper addresses inverse elastic scattering on unbounded domains to locate inhomogeneities in the Lamé parameters and density from far-field data. It extends the monotonicity method to the time-harmonic Navier equation by establishing a monotonicity relation for the far-field operator and coupling it with localized potentials to obtain constructive shape tests. The main contributions are monotonicity inequalities for the Navier system in unbounded settings, a localized-potential theory on unbounded domains, and a monotonicity-based reconstruction criterion that detects the support osupp. This yields a non-iterative, data-driven approach for robustly identifying elastic inclusions from scattering measurements with potential applications in nondestructive testing and geophysical imaging.

Abstract

We discuss a time-harmonic inverse scattering problem for the Navier equation with compactly supported penetrable and possibly inhomogeneous scattering objects in an unbounded homogeneous background medium, and we develop a monotonicity relation for the far field operator that maps superpositions of incident plane waves to the far field patterns of the corresponding scattered waves. Combining the monotonicity relation with the method of localized potentials, we extend the so called monotonicity method to characterize the support of inhomogeneities in the Lamé parameters and the density in terms of the far field operator.
Paper Structure (5 sections, 11 theorems, 103 equations)

This paper contains 5 sections, 11 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. A monotonicity relation for the far field operator
  4. Localized potentials for the Navier equation
  5. Monotonicity based shape reconstruction

Key Result

Lemma 3.1

Let $\lambda,\mu,\rho\in L^{\infty}_{+}(\mathbb{R}^2)$, and let $B_R(O)$ be a ball containing $\Omega$. Then If $\lambda_{j}$, $\mu_{j}$, $\rho_{j}\in L_{+}^{\infty}(\mathbb{R}^2)$, then for any $j,l\in\{1,2\}$ we have

Theorems & Definitions (21)

  • Lemma 3.1
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • Lemma 3.3: Theorem 2 in S2005
  • Remark 3.4
  • Lemma 3.5
  • Proof 3
  • Theorem 3.6
  • Remark 3.7
  • ...and 11 more