Analysis of Kirchhoff Migration and Direct Sampling Method within Far-Field Approximation: From the Multi-Static to the Mono-Static Configuration

A recently developed direct sampling method (DSM) is studied in the framework of multi- and mono-static configurations within the far-field hypothesis. A full analysis of the behavior of the DSM has not been provided yet in both cases even if, due to a lack of information compared to multi-static data, the mono-static configurations need an adapted analysis. Thanks to the theoretical analysis based on the asymptotic formula for far-field patterns and small obstacles hypothesis, the behavior of DSM is explained and the reason for which it failed in the mono-static configuration is identified. A modified DSM is then proposed and compared to the classical Kirchhoff migration technique (KM) in order to improve the imaging efficiency of the DSM approach. The theoretical results are supported by numerical experiments with synthetic data.

1 Introduction

The two-dimensional inverse scattering problem is an important topic highly related to many issues in modern human life [1]. Yet, it is hard to solve because of its inherent non-linearity and ill-posedness. Various non-iterative reconstruction methods have been investigated to overcome those issues, for example, MUltiple SIgnal Classification (MUSIC), linear sampling method (LSM), and Kirchhoff migration (KM), direct sampling method (DSM), etc. Related works can be found in [2]–[5] and references. Even though good results can be obtained with multi-static data, full analysis of these methods is needed for a proper comparison of their behaviors. However, since the mono-static configuration is of great interest in various applications a deep understanding and development of effective algorithms is still needed. Existing methods have been adapted and applied as MUSIC [6], LSM [7], KM [8], DSM [9], etc.