Mathematical Problems in Low Frequency Electromagnetic Inversion and in Inverse Scattering in Random Media

William Marsh Rice University, Houston TX

Investigators

Abstract

We consider theoretical and numerical studies of two inverse problems: (a) Low frequency electromagnetic inversion, where we seek unknown coefficients (electrical conductivity/permittivity) in elliptic systems of equations, inside a bounded domain, given the Neumann to Dirichlet map at the boundary. We explore the use of variational principles in the solution of such problems. In particular, we wish to develop new, variational reconstruction algorithms and to study resolution limits (distinguishability). In numerical inversion, having a proper discretization is paramount. We propose a finite difference discretization of the problem, on optimal grids. We have demonstrated that optimal grids give stable and efficient inversion algorithms, in one dimensional (Sturm-Liouville) problems. We wish to study further the optimal grids and to extend them to higher dimensions. (b) Inverse scattering in random media, where we wish to image the reflectivity of targets buried in clutter, which we model as a random medium. We are interested in remote sensing regimes, with significant multipathing (multiscattering) of the waves, by the inhomogeneities in clutter. The proposed work starts with a close look at wave propagation in random media and it strives to develop statistically stable imaging algorithms, which give reliable images, independent of one's lack of knowledge of the details of the clutter. We consider theoretical and numerical studies of two inverse problems: (a) The first problem considers the recovery of properties such as the electrical conductivity and permittivity of a body, given measurements of electric currents and voltages, or the electric and magnetic fields, at the surface of the body. Because different materials display different electrical properties, these problems are important whenever we wish to infer the internal structure of a body, by gathering data at its periphery. Examples of applications are: (1) In medicine: for detection of pulmonary emboli, monitoring of heart function and blood flow, detection of breast tumors, etc. (2) In environmental sciences: for detection of leaks in underground tanks, monitoring of underground flows, etc. (3) Nondestructive testing of materials: detection of corrosion, cracks and voids in metals, etc. Our research focuses on using state of the art variational techniques and optimal grids (parametrizations), to obtain efficient and reliable recoveries of the unknown electrical properties inside the body. (b) The second problem considers the detection and imaging of targets in clutter, via active arrays of antennas (transducers) which send probing signals in the medium and record the scattered echoes. Imaging in clutter is not well understood, so far, but it has important applications in ultrasound imaging, land or shallow water mine detection, ground or foliage penetrating radar, etc. Our approach to imaging in clutter is based on knowledge of wave propagation in random media and it considers the development of statistically stable imaging algorithms, which give reliable results, independent of one's uncertainty of the clutter.