Transient excitation of a resistive loaded electric dipole on the upper interface of a horizontally layered lossy half space

Abstract

Extensive research effort has been directed toward the numerical modeling of electromagnetic wave fields for GPR applications. Several versions of codes for modeling piecewise homogeneous horizontally stratified media exist. For more general earth models, numerical codes based on fmite differences and finite elements also exist. In the implementation, the electric current in the source antenna is always assumed to be known, while the receivers are usually taken as point devices. We have implemented the theory for the electromagnetic radiation and scattering problem for GPR applications where the electric current in the antenna is not known, but is computed using the model of a voltage-gap driven, resistive loaded, electric dipole as the source antenna. The receiver antenna is identical to the source except, of course, for the voltage gap. This is a more realistic approach. To model the earth we take a lossy layered half space to study the effect of the permittivity/conductivity ratio on the current that is excited in the source antenna, given a fixed voltage-gap excitation in the source. This is done for practical sources with different center frequencies. Then the electric field is computed at all positions in the receiving antenna and is used to compute the induced current in the receiving antenna. INTRODUCTION The increasing commercial use of GPR systems is based on their portability and the resolution of the images that can be obtained. The data usually is obtained in a mono-static or bi-static mode, which makes it difficult to use processing sequences to come to a tial image of the subsurface. For proper migration of the data, one needs a velocity model which cannot be obtained from the data unless CMP measurements have been carried out also. The direct interpretation of the data usually is complicated, more so since the interaction of the transmitting and/or receiving antennas with the subsurface is unknown. It is our aim to understand how the permittivity and conductivity of the earth close to the surface affects the signal that is sent into the earth. To this end, we have developed a model that consists of a resistive loaded electric dipole as transmitting antenna that is excited by a voltage gap in the middle of the wire. The current that is generated in the wire can then be computed with the aid of a one-dimensional integral equation for the current along the center of the wire using a thin-wire approximation. The first formulation is due to Pocklington (1897) where the antenna is placed in a homogeneous, lossless full space. Many other formulations followed after Pocklington, of which Hallen’s (1938) procedure is one of the most important developments. Recently, Tijhuis et al. (1992) have given a new almost exact formulation which turned out to yield an integral equation that is identical to the reduced form of Pocklington’ s equation. Here we use that integral equation formulation but instead of a full space, we take a lossy half space where the antenna is placed on the air/earth interface. Bretones et al. (1996) formulated the problem of two loaded wire antennas above a horizontal, piecewise continuously inhomogeneous half space. They also gave a method on how to numerically solve the obtained integral equation. Our approach is slightly different, the difference in the formulation itself is not fundamental but our approach for the numerical handling is more suited for an antenna directly on top of the interface and we use a new and optimized numerical form of Hallen’s method that is used by them. This makes our scheme less complicated and computationally more efficient. INTEGRAL EQUATION FORMULATION We assume in all our equations an exp(iwt) time dependence. The transmitter is a straight wire, directed along the &axis, of length L with a circular cross section of radius a. The embedding is a homogeneous lossy half space where the antenna is placed on top of the interface in the air. The source antenna is excited by an impressed voltage across the gap -6 < < the origin of the Cartesian reference frame being in the middle of the antenna. The antenna satisfies the inequalities, a The impressed voltage across the gap is taken constant over the cross-sectional area of the wire. The electric field is non-zero inside the antenna because it is not a perfect conductor, in stead, the electric field equals the product of the resistance and the electric current in the wire. Further, since the antenna radius is very small compared to its length, the assumption that only the z-component of the electric field contributes to the current generated in the wire is justified. Hence the electric scalar function of x only. current inthewireisan x-directed,