Toward a Complete Forward Model for Underground Imaging Using RF Tomography Lorenzo Lo Monte, Michael C. Wicks Sensors Directorate Air Force Research Laboratory Wright-Patterson AFB, OH - USA Lorenzo.lomonte.ctr@wpafb.af.mil Michael.wicks@us.af.mil Francesco Soldovieri Istituto per il Telerilevamento Elettromagnetico dell’Ambiente National Research Council Naples, Italy Soldovieri.f@irea.cnr.it Danilo Erricolo Department of Electrical and Computer Engineering University of Illinois at Chicago Chicago, IL - USA Erricolo@ece.uic.edu Abstract—The forward model for underground imaging using RF tomography is extended to account for elementary loops as transmitters and/or receivers. The theory of RF tomography remains applicable, provided that opportune dyadic Green’s functions are derived and expressed explicitly. We will present new formulas and derivations that consent the use of RF Tomography for any arbitrary choice of the Tx/Rx sensor. I. INTRODUCTION Radio frequency (RF) tomography has been proposed for imaging underground dielectric/conducting anomalies [1]. A system using RF tomography employs a set of low-cost, narrowband electromagnetic transmitters and receivers placed on top, above, or shallowly buried in the ground at arbitrary positions. During the first stage, sensors accurately identify their position, orientation and time reference. During the second stage, a transmitter radiates a known waveform using a suitable polarization. The probing wave impinges upon a target (represented in terms of dielectric / conducting anomaly), thus producing scattered fields. Spatially distributed receivers collect samples of the total electric/magnetic field, remove noise, clutter and the direct path, and store the information concerning only the scattered field. In the next iteration, a different transmitter is activated, or different waveforms / polarizations are used. During the third stage, the collected data is relayed to the control post for processing and imaging, (see Figure 1 for a pictorial representation). The system operates using ultra-narrowband, adaptive waveforms, thus ensuring low noise, low dispersion and affordable cost of instrumentation. To reach deep targets, the system shall operate at the range of HF. In this range, efficient small antennas are constructed using ferrite loaded coils, which can be mathematically approximated as electrically small loops. Therefore, in RF tomography it is important to know the dyadic half-space Green’s functions of the investigated scenario, whether the source/receiver is either an electric/magnetic dipole. Formulas for computing half-space Green’s functions have been known since the time of Sommerfeld, and more recently generalized by Tai [2], King [3], and Michalski [4-6]. However, these formulations are generally difficult to be computed, difficult to be integrated, expressed in the unpractical polar form, and numerically unstable (see remarks in [4]). Much effort has been spent recently to derive formulations that: - Have explicit solutions. - Are numerically stable (less singular). - Are naturally expressed in Cartesian coordinates. - Can be easily computed using FFT. - Can be easily integrated (particularly over a cubic region). - Do not involve approximations or specific applications, such as most of the closed-form solutions [7-10]. - Are directly applicable for arbitrarily oriented dipoles. We believe that Green’s functions expressed in Cartesian and spectral form (such as [11-13]) meet these requirements; however, most of these expressions are: - Incomplete (usually reporting only the electric dyadic Green’s function of electric type, computed between the air-earth interface only). - Not optimized for the reduction of singular points. We are aware that much research is ongoing, but we believe that the strategy pioneered by Chew and Cui [14-24] is certainly adequate for the problem we are addressing. In this work, we took inspiration from Chew-Cui derivations so that we were able to extend the derivation of the dyadic half-space Green’s functions (providing explicit expressions for all possible values of r and ' r ) to the cases of: 1) electric field due to an electrically small dipole 2) magnetic field due to an electrically small dipole, 3) electric field due to an electrically small loop, and 4) magnetic field due to an electrically small loop. With the knowledge of these new formulas, the step between theory and real-world application for RF tomography will be further reduced. II. MATHEMATICAL FORMULATION The 3D geometry depicted in Fig. 2 is considered. The scene is modeled as two layered media separated by a planar interface whose shape is assumed known a priori. Mathematically, the scene is divided into two planar layered (ideal) half-spaces with the interface at 0 z = . The lower half-space is modeled as a homogeneous medium with relative 978-1-4577-0333-7/11/