Abstract
Although the direct sampling method (DSM) has demonstrated its feasibility and robustness for imaging of small inhomogeneities, mathematical analyses of DSM have been conducted only on the full-aperture inverse scattering problem. Numerous studies have shown that DSM can also be applied in the limited-aperture inverse scattering problem, but most of its applications are still heuristic. This study considers an application, mathematical analysis, and improvement of DSM with a single-incident field only in the limited-aperture inverse scattering problem. First, we introduce a traditional indicator function of DSM at a single frequency, establish its mathematical structure, and examine its inherent limitation. To demonstrate the theoretical result, various results of numerical simulations with synthetic and experimental data are presented. Next, we consider the multi-frequency indicator function of DSM with a single-incident direction to improve imaging performance. For this, we design a multi-frequency indicator function of MDSM, analyze its mathematical structure, and theoretically explain the improvement of the imaging of single inhomogeneity and the limitation on the identification of multiple inhomogeneities. Various numerical simulations with synthetic and experimental data are presented to validate our results.
Highlights
The main goal of the inverse scattering problem is to determine the unknown targets, such as location, shape, and/or other physical properties, from the collected scattered fields or far-field patterns
To explain the theory behind this, we investigate a mathematical theory of a multi-frequency indicator function by establishing a relationship with Bessel functions, Struve functions, Legendre polynomials, generalized hypergeometric functions, the range of incident and observation directions, and physical property of targets
SIMULATION RESULTS: SINGLE-FREQUENCY CASE we validate the theoretical results by conducting various numerical simulations using both synthetic and experimental data
Summary
The main goal of the inverse scattering problem is to determine the unknown targets, such as location, shape, and/or other physical properties, from the collected scattered fields or far-field patterns. Seismic imaging [19]–[21], the range of incident and observation directions must be restricted This is known as a limited-aperture inverse scattering problem. Some authors investigated the mathematical structure of the DSM indicator function, proposed an improved version, and verified its relationship with the Kirchhoff migration in full-aperture inverse scattering problem [52]. It has been applied heuristically in the limited-aperture inverse scattering problem; there is a need to perform a rigorous analysis to establish the mathematical structure of DSM. We consider a DSM with a single-incident direction to identify two-dimensional small or large dielectric inhomogeneities in the limited-aperture inverse scattering problem.
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