Why the high-frequency inverse scattering by topological sensitivity may work.

Area of Science:

• Electromagnetics
• Mathematical Physics
• Inverse Problems

Background:

• Reconstructing anomalies from scattered fields is crucial in various scientific domains.
• High-frequency scattering problems present unique challenges for anomaly detection.
• Topological sensitivity offers a potential pathway for characterizing complex obstacles.

Purpose of the Study:

• To decipher topological sensitivity (TS) for reconstructing and characterizing impenetrable anomalies in the high-frequency regime.
• To express the TS formula as nested surface integrals for detailed analysis.
• To investigate the role of asymptotic approximations in TS-based reconstruction.

Main Methods:

• Formulation of TS as nested surface integrals over obstacle and measurement surfaces.
• Utilization of multipole expansion to simplify integral forms involving Green's function.
• Application of Kirchhoff approximation and asymptotic expansion for scattered field evaluation.
• Employing catastrophe theory and stationary phase approximation for TS analysis.

Main Results:

• The TS formula is expressed as nested integrals, simplified by multipole expansion and asymptotic analysis.
• Three key asymptotic approximations (near-boundary, uniform expansions, stationary phase) are identified as crucial for TS.
• Topological sensitivity is dominated by the near-boundary term, explaining reconstruction capabilities.
• The study demonstrates how to expose anomaly character (Dirichlet or Neumann) using TS analysis.

Conclusions:

• Topological sensitivity is a potent tool for high-frequency anomaly reconstruction and characterization.
• Asymptotic analysis provides critical insights into the behavior and application of TS.
• The method offers a robust framework for identifying obstacle properties even when unknown.