chapter 6, The Inverse Scattering Problem for Buried Objects

Excerpt

Up to now we have only discussed the inverse scattering problem for obstacles situated in a homogeneous background. However, in most applications the unknown target is embedded in an inhomogeneous background. The use of electromagnetic fields to detect buried objects has a long history and continues to be an active area of research [3], [4], [9], [32], [61], [62]. Of particular interest is the use of such methods to detect chemical waste deposits, examine urban infrastructure, and locate land mines. However, from a practical point of view, there are two main reasons why such imaging problems are challenging. The first is the difficulty of distinguishing the scattered field due to the target from the scattered fields due to the earth, the antenna, and, in particular, the air-earth interface. The second reason is that the material properties of the target are in general unknown. For example, a land mine can be made of metal or plastic, whereas a rusted barrel of chemical waste deposits is typically modeled by a complicated mixed boundary value problem involving a dielectric of unknown permittivity. Because of these reasons, traditional methods of imaging, such as the use of weak scattering approximations and nonlinear optimization techniques, remain problematic.

The linear sampling method (LSM) has a number of features which make it attractive for the imaging of buried objects. In particular, it is a linear method that does not ignore multiple scattering effects and determines the shape of a target without requiring any a priori knowledge of the target's physical properties. However, until recently, the implementation of the LSM for a nonhomogeneous background media required knowledge of the Green's function for the background media. This is obviously an unattractive feature if it is desired to use this method for the detection of buried objects, particularly if the scattering effects due to the antenna play a significant role. In order to overcome the problem of needing to compute the Green's function for the background media, a new version of the LSM, based on the reciprocity gap functional, was introduced by Colton and Haddar [46] for the scalar case and by Cakoni, Fares, and Haddar [32] for the vector case. However, in imaging nothing is free, and the price paid for avoiding the need to compute the Green's function is that one now needs to measure both the electric and magnetic fields corresponding to time-harmonic electric dipoles as incident fields.

We begin this chapter with a brief discussion of the LSM for objects buried in a known inhomogeneous background using near field measurements.