Applying the Laplace transform, the exact distributed image current function is obtained for the classical Sommerfeld half-space problem with vertical magnetic current source. The resulting field integral is well behaved when the image current is situated in complex space. Unlike previous approximate images, the present theory is valid for any distance, height of the source, frequency, and half-space parameters. It is demonstrated that the present image theory reduces to the well-known dipole image at complex depth for large dielectric parameters of the half-space. Also, the reflection-coefficient method is obtained as a farfield approximation. Calculation of fields through exact image integration is seen to be simple and accurate and require modest computer capacity and time. In an appendix, some properties of the multivalued Green's function arising from a dipole source in complex space are also studied.

Exact image theory for the Sommerfeld half-space problem, part II: Vertical electrical dipole

The general Sommerfeld problem with both \epsilon and \mu discontinuous and a source consisting of arbitrarily oriented electric and/or magnetic dipoles at the same location is considered in terms of image theory. The problem is formulated with electric and magnetic fields instead of potential quantities resulting in a vector transmission-line interpretation of the Fourier transformed problem. The image sources are seen to be located in complex space expressable in terms of a certain basic image current function, which was encountered in part II of this paper on the vertical electric dipole problem. The horizontal electric/magnetic dipole image is solved and found to consist of both vertical and horizontal current components. The image concept is generalized to the most general three-dimensional sources. As a check, the well-known reflection coefficient method is obtained as the far-field approximation of the present theory.

Exact image theory for the Sommerfeld half-space problem, part III: General formulation

Ultra-wideband (UWB), short-pulse (SP) radar is investigated theoretically and experimentally for the detection and identification of targets buried in and placed atop soil. The calculations are performed using a rigorous, three-dimensional (3D) method of moments algorithm for perfectly conducting bodies of revolution. Particular targets investigated theoretically include anti-personnel mines, anti-tank mines, and a 55-gallon drum, for which the authors model the time-domain scattered fields and the buried-target late-time resonant frequencies. With regard to the latter, the computed resonant frequencies are utilized to assess the feasibility of resonance-based buried-target identification for this class of targets. The measurements are performed using a novel UWB, SP synthetic aperture radar (SAR) implemented on a mobile boom. Experimental and theoretical results are compared.

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Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement

A quasi-historical sketch of complex image theory for low-frequency electromagnetic fields is presented from a personal perspective. Studies using complex image theory, on launching TM (transverse magnetic) modes into the Earth-ionosphere waveguide from a horizontal insulated cable, grounded at the end points, are described. Subsequent implementation and further development of the concept by other authors is summarized. >

Complex image theory-revisited

During the past two decades, complex image theory techniques have successfully been utilized to determine simple form expressions for the electric and magnetic fields produced by antennas located near the earth's surface, for both single-layered and multilayered earths. It is the purpose of this paper to provide a tutorial overview of complex image theory. Particular attention will be paid to its range of validity. Specific applications will also be discussed.

Applications of complex image theory

The image theory for dipoles above an imperfectly conducting earth which has been developed by Wait, among others, is used to compute the change in the input impedance of a vertical magnetic dipole (VMD) due to the presence of the earth. Comparison with a previously developed rigorous solution is made and a good agreement is obtained, particularly when the conduction currents in the earth dominate over the displacement currents. Extension of the-single image theory is made in two ways. First, multi-images which are spread in space are used to represent the effect of the earth. Second, multipole sources having the same location are used. Both representations have shown a noticeable improvement over the single image in relation to computational results on the input impedance of a VMD.

New image representation for dipoles near a dissipative earth 1. Discrete images

A modified image theory for vertical magnetic and electric dipoles above a dissipative earth is presented Namely, the Fresnel reflection coefficient for horizontally polarized waves is approximated by an exponential term plus a new term which extends the validity of the approximation to high values of the horizontal wave number While the exponential term represents a single image inside the earth, the new added term represents a set of continuous distribution of images A similar approximation is applied to the Fresnel reflection coefficient of the vertically polarized waves These image representations are applied for the determination of input impedances of vertical magnetic and electric dipoles and horizontal large loops at a very low height above the earth Excellent agreement with previous exact results is reported in all cases

New image representation for dipoles near a dissipative earth 2. Discrete plus continuous images

Exact solutions are derived for the input impedances of vertical magnetic and electric short dipoles over a lossy half space. The exact solution is expressed as a simple formula and a remainder, which takes the form of a rapidly converging series. Reduction to available approximate solutions in the literature under conditions of large dipole heights, large refractive index of the half space, or both is demonstrated. The derivation of new simple formulae that tend to the exact solutions when the refractive index of the half space approaches both limits of infinity or unity is of particular interest. Numerical examples are presented and compared with published results.

Ahmed D. Metwally

Papers

New image representation for dipoles near a dissipative earth 1. Discrete images

New image representation for dipoles near a dissipative earth 2. Discrete plus continuous images

Input impedance of vertical dipoles near a lossy half space