# Kirchhoff Migration for Identifying Unknown Targets Surrounded by Random Scatterers
## 1 Introduction  
Inverse scattering problems often involve identifying the specific characteristics (for example, location, geometry, or material properties) of an unknown target from scattered or far-field data. Such problems are of great practical importance in fields such as mathematics, physics, engineering, and biomedical science. However, due to their inherent nonlinearity and ill-posedness, they remain challenging. Related work can be found in [1; 2; 3; 4; 5; 6; 7; 8; 9; 10] and references therein. Attempts to address these types of problems have led to a search for fast and effective identification techniques, and various approaches have been developed. Those include the MUltiple Signal Classification (MUSIC) algorithm [11; 12; 13; 14; 15], the linear [16; 17; 18; 19; 20; 21] and direct [22; 23; 24; 25; 26] sampling methods, and topological derivatives [27; 28; 29; 30; 31]. We also refer to various non-iterative imaging techniques [32; 33; 34; 35; 36; 37].  
Kirchhoff and subspace migration are also well-known non-iterative techniques for finding location/shape of unknown inhomogeneities, and they have been applied to a variety of problems (see [38; 39; 40; 41; 42; 43], for instance). Several studies have confirmed that they are fast, stable, and effective methods for finding various kinds of defects without _a priori_ information of unknown targets. However, most of these researches was performed for homogeneous background media, and further research is still needed on the imaging performance of Kirchhoff and subspace migrations when the unknown targets are surrounded by random scatterers or embedded in an inhomogeneous medium. Notice that for subspace migration, some studies have considered inverse scattering problems in random media [44; 45; 46; 47], random scatterers [48; 49], and mathematical theory for detecting point-like scatterers withrandom noise [50]. However, Kirchhoff migration-based imaging in random media and related mathematical theory has not been considered yet.  
In this paper, we consider Kirchhoff migration for localizing small electromagnetic anomalies surrounded by small random scatterers. We carefully analyze its imaging functions by discovering a relationship with the Bessel function of the first kind of order zero and one. This enables us to discuss its various properties and compare the imaging performance with subspace migration. The analysis is based on the structure of the singular vectors associated with non-zero singular values of so-called multi-static response (MSR) matrix and asymptotic expansion formula in the presence of small electromagnetic defects [51].  
This paper is organized as follows: In Section 2, we introduce the two-dimensional direct scattering problem and asymptotic expansion formula that holds in the presence of small inhomogeneities surrounded by random scatterers. In Section 3, we introduce the imaging function of Kirchhoff migration and its mathematical structure by establishing a relationship with Bessel function of order zero and one. On the basis of analyzed structure, we discuss its intrinsic properties for several cases of anomalies and random scatterers, and compare the imaging performance with subspace migration. In Section 4, we present some numerical simulation results to support the analytical discussion. We apply various non-iterative techniques for imaging of several extended targets completely hidden in an inhomogeneous medium and discuss their imaging performances in Section 5. Finally, we present our conclusion in Section 6.