Lossy Half-space Research Articles

Helmholtz–Galerkin Technique in Dipole Field Scattering from Buried Zero-Thickness Perfectly Electrically Conducting Disk

Non-invasive concealed object detection, identification, and discrimination have been of interest to the research community for decades due to the needs to preserve infrastructures and artifacts, guarantee safe conditions for the detection and location of landmines, etc. A modern approach is based on the use of an unmanned aerial vehicle equipped with ground-penetrating radar, which has the advantage of not requiring direct contact with the ground. Moreover, high-resolution underground images are obtained by coherently combining measurements by using a synthetic aperture radar algorithm. Due to the complexity of the real scenario, numerical analyses have always been welcomed to provide almost real-time information to make the best use of the potential of such kinds of techniques. This paper proposes an analysis of the scattering from a zero-thickness perfectly electrically conducting disk buried in a lossy half-space surrounded by air and illuminated by a field generated by a Hertzian dipole located in the air. It is carried out by means of a generalized form of the analytically regularizing Helmholtz–Galerkin technique, introduced and successfully applied by the authors to analyze the plane-wave scattering from a disk and a holed plane in a homogeneous medium. As clearly shown in the numerical results, the proposed method is very effective and drastically outperforms the commercial software CST Microwave Studio 2023.

The electromagnetic field of a current conductor in the presence of lossy half-space

In this paper, the electromagnetic field problem of a line current conductor in the presence of homogenous and ISO-tropic lossy half-space is analyzed. The main difficulty in this analysis is to take correctly into account the influence of ground finite conduct since the use of integral transformations in Maxwell's equations leads to Sommerfeld integral, hard to evaluate even numerically. Charge Simulation Method, based on the equivalence theorem of different electromagnetic systems, is proposed here for the numerical approach. Two independent, equivalent systems created of appropriately chosen, shaped, and arranged fictitious sources are suggested for the determination of EM field components , in the air and in the lossy half-space. Point-matching method based on the boundary conditions between media is applied for the determination of the fictitious source intensities. The proposed procedure has the advantage that it takes into account the influence of the finite ground conductivity, without calculation of Sommerfeld integral and it ensures excellent accuracy with a small number of fictitious sources. Also, the features of the proposed method are simple impel-mentation on a standard PC, fast and accurate simulation procedure, and easy generalization for some other cases of arbitrarily oriented wires of finite lengths above a lossy half space.

Accurate Estimation of the Lateral Wave’s Contribution to the Radiated Fields of a Vertical Electric Dipole Embedded in a Lossy Half-Space

Accurate Estimation of the Lateral Wave’s Contribution to the Radiated Fields of a Vertical Electric Dipole Embedded in a Lossy Half-Space

Human Exposure to Field Radiated by Vertical Dipole Antenna over a Lossy Half-Space using Analytical Approach

Human Exposure to Field Radiated by Vertical Dipole Antenna over a Lossy Half-Space using Analytical Approach

On the Deficiency of the First-Order Ott–Clemmow Saddle Point Method as Applied to the Sommerfeld Half-Space Problem

The accuracy of the first-order multiplicative (Ott–Clemmow) and additive (van der Waerden) modified saddle point integration methods for the Sommerfeld problem of a vertical Hertzian dipole over a lossy half-space is investigated using two theoretically equivalent formulations with either a positive or a negative image term extracted. It is demonstrated that whereas the additive method leads to the same asymptotic field representation irrespective of the sign of the image, the multiplicative variant yields distinct results in these two cases, both of which differ from the unique result of the additive method. It is further found that the positive-image Ott–Clemmow method yields the well-known Norton formula, but the negative-image variant is inaccurate and predicts a zero field in the on-surface transmitter–receiver configuration. It is also demonstrated that the first-order multiplicative method yields an incomplete representation of the first order in the inverse distance.

Calculation of Complex Power Generated by a Transmitting Thin Wire Antenna Radiating above a Lossy Half-space

The paper deals with an efficient approach to determine complex power generated by a thin wire antenna above a lossy ground. Once the current distribution along a thin wire is obtained by numerically solving the Pocklington integro-differential equation the complex power of the antenna can be obtained by solving the integral over the inner product of tangential component of the electric field and current distribution along the wire stemming from Poynting theorem. Numerical calculation procedure uses the vectors and matrices already constructed within the calculation procedure for the current distribution. Some illustrative numerical results for the active power, reactive power and apparent power of the dipole antenna radiating over a lossy half-space are presented in the paper for different values of input parameters.

Multi-Frequency GPR Microwave Imaging of Sparse Targets through a Multi-Task Bayesian Compressive Sensing Approach.

An innovative inverse scattering (IS) method is proposed for the quantitative imaging of pixel-sparse scatterers buried within a lossy half-space. On the one hand, such an approach leverages on the wide-band nature of ground penetrating radar (GPR) data by jointly processing the multi-frequency (MF) spectral components of the collected radargrams. On the other hand, it enforces sparsity priors on the problem unknowns to yield regularized solutions of the fully non-linear scattering equations. Towards this end, a multi-task Bayesian compressive sensing (MT-BCS) methodology is adopted and suitably customized to take full advantage of the available frequency diversity and of the a-priori information on the class of imaged targets. Representative results are reported to assess the proposed MF-MT-BCS strategy also in comparison with competitive state-of-the-art alternatives.

Scattering by a chiral sphere above a half-space.

Scattering from a chiral sphere above a lossy half-space, which could be of interest in remote sensing and optics, is analytically examined. The proposed method combines the vector Mie solution and the field transformations between vector spherical functions (VSFs) and plane waves (PWs). Using the reflection coefficients of the half-space and vector Mie solution for the chiral sphere, the first-order Mie field together with a relation between the Mie fields of successive orders are derived. The total Mie field is obtained as a series solution which is next converted to a non-recursive formulation. The scattered field is written as the sum of the total Mie field and its reflection from the half-space. The derived expressions are numerically validated. Some explanations based on the series solution are given and numerical results for different cases are presented and briefly discussed.

An Analytical Solution of the Electric Field Excited by a Vertical Electric Dipole Above a Lossy Half-Space: From Radio to Microwave Frequencies

In this article, an analytical solution of the electromagnetic radiation from a vertical electric dipole (VED) above a lossy half-space is proposed. The expression for the traditional solution is associated with the Sommerfeld-type integral, which is difficult to evaluate due to its highly oscillating and slowly decaying integrand. The solution for the intermediate Hertz potential is decomposed into two integrals and a rigorous closed-form solution in the near and far-field regions are presented for each term. Then, the scattered electric-field components are calculated from the intermediate Hertz potential. A numerical evaluation of the solution for different lossy half-spaces, such as seawater, wet earth, dry earth, and lake water, validates the accuracy of the proposed solution at various frequencies, from radio to microwave frequencies, as well as different distances from the antenna in the near and far-field regions.

Direct Solution of Finite Element-Boundary Integral System for Electromagnetic Analysis in Half-Space

An efficient direct solution scheme is presented to solve the finite element-boundary integral (FEBI) system for the analysis of open-region electromagnetic problems in half-space. Data-sparse representation based on hierarichal (H-) matrix is employed to construct the direct solution framework. The FEBI system matrix is first represented to be an H-matrix form, in which the FE submatrices are accurately represented by H-matrices, while the H-matrix approximations of the BI submatrices are constructed by the adaptive cross approximation algorithm combined with singular value decomposition recompression (ACA-SVD). Based on the data-sparse formatted algorithm, the lower and upper (LU) triangular factors of the FEBI system matrix can be computed and stores as H-matrix forms with low computational costs. The resulting method, refered to as the direct FEBI method (D-FEBI), is free of convergence problem and well-suited for multiple right-hand side (RHS) computations. The performance of the D-FEBI is examined by two examples of monostatic electromagnetic scattering above a lossy half-space.

Vertical Dipole Above the Lossy Dielectric Half-Space—A Characteristic Mode Analysis

This work introduces a theoretical extension of the characteristic mode formulation for analyzing the vertical electric dipole lying above a lossy dielectric half-space. As the conventional characteristic formulation fails to maintain the orthogonality of the characteristic field modes over the infinite sphere, an alternate modal formulation is proposed here to maintain the orthogonality for both the current and field modes. The modal results are found to match closely with its method of moment counterparts. Later, the modes of an isolated dipole with no ground plane have been used to predict the role of the lossy ground plane through a theory of the linear combination of the eigenvectors. The proposed formulations have been studied with different heights from the ground plane and are compared with the direct modal solutions to validate its accuracy. It helps to provide a thorough understanding of how the isolated modes interact among each other to constitute the perturbed modes in the presence of the lossy half-space. It can find application to include the lossy earth effect in the study of the lightning fields and the path loss modeling of the antennas over the lossy ground.

Quasi-Static Solution for ELF/SLF near-Zone Field of a Vertical Magnetic Dipole near the Surface of a Lossy Half-Space

Dipole antennas over the boundary between two different media have been widely used in the fields of geophysics exploration, oceanography, and submerged communication. In this paper, an analytical method is proposed to analyse the near-zone field at the extremely low frequency (ELF)/super low frequency (SLF) range due to a vertical magnetic dipole (VMD). For the lack of feasible analytical techniques to derive the components exactly, two reasonable assumptions are introduced depending on the quasi-static definition and the equivalent infinitesimal theory. Final expressions of the electromagnetic field components are in terms of exponential functions. By comparisons with direct numerical solutions and exact results in a special case, the correctness and effectiveness of the proposed quasi-static approximation are demonstrated. Simulations show that the smallest validity limit always occurs for component H2z, and the value of k2ρ should be no greater than 0.6 in order to keep a good consistency.

Analysis of Time-Domain Shielding Effectiveness of Enclosures Above Lossy Half-Space Using an Improved Half-Space FDTD Method

Since shielding enclosures which are widely used in electronic systems and devices are usually located close to the ground, the effect of ground reflection must be taken into account when analyzing the electromagnetic (EM) characteristics of enclosures. Consequently, the assessment of time-domain shielding effectiveness (SE) is a composite problem of the lower half-space and the enclosure above it. However, the application of existing half-space algorithms is often limited by the high computational demand and the complex half-space Green's function, which is difficult to derive and calculate. This article proposes an improved half-space finite-difference time-domain method without employing the half-space Green's function, while effectively reducing the computational memory and time. Based on this new method, the time-domain SE of enclosures above lossy half-space is analyzed; additionally, the difference between the coupling field and time-domain SE, on the one hand, and the free space and half-space, on the other hand, is discussed in detail. The numerical results show that the time-domain SE is reduced significantly owing to the reflection effect of the lower half-space. To sum up, this method provides an effective time-domain approach for the analysis of EM characteristics of objects above the complex half-space.

Evaluating the Ground Impedance—A New Methodology Based on EM Reciprocity

The electromagnetic reciprocity theorem of the time-convolution type is applied to introduce a general methodology for constructing approximate expressions for the ground impedance parameter. The methodology is first illustrated on expressing the ground impedance concerning a thin conductive strip above a homogeneous, lossy half-space, which yields the classic ground-impedance approximate expressions due to Carson and Vance. Second, the proposed methodology is applied to the corresponding strip-ground configuration in which the effect of the ground is modeled via the surface impedance approximation. This leads to a new ground-impedance approximate expression given in terms of the exponential integral. The validity of the latter approximation is assessed numerically and its relation to the well-known approximations is briefly discussed. Finally, the corresponding two-layer ground problem configuration is analyzed. The result is a correction of the ground impedance accounting for the contrast in soil stratification.

Time domain generalized telegrapher’s equations for the electromagnetic field coupling to finite-length wires buried in a lossy half-space

The paper deals with a derivation of a time domain variant of the generalized telegrapher’s equations for transient electromagnetic field coupling to buried wires. The formulation is based on the thin wire antenna theory in the time domain. The influence of a lossy half-space is taken into account by means of the time dependent reflection coefficient approximation arising from the modified image theory (MIT), while the per-unit-length surface impedance accounts for the conductor losses. The obtained space-time integral expressions are handled analytically. The concept of the scattered voltage is naturally included in the full model formulation. Computational examples are given for the induced currents along the buried wire and compared to the results obtained by using the transmission line (TL) approximation.

Direct time domain evaluation of the transient field transmitted into a lossy ground due to GPR antenna radiation

The paper deals with the direct time domain calculation of a transient electric field generated by a ground penetrating radar (GPR) dipole antenna and transmitted into the lossy half-space. The space–time dependent current along the dipole is governed by the Hallen integral equation which is numerically solved by means of the Galerkin–Bubnov scheme of the Indirect Boundary Element Method (GB-IBEM). Provided that the current distribution along the GPR antenna is determined, one may compute the related electromagnetic field transmitted into the lossy ground by solving the field integral formulas. Some illustrative computational examples related to the transmitted field into the lower half-space are reported in this paper.

COMPARISON OF TIME-DOMAIN FINITE-DIFFERENCE, FINITE-INTEGRATION, AND INTEGRAL-EQUATION METHODS FOR DIPOLE RADIATION IN HALF-SPACE ENVIRONMENTS

In this paper we compare current implementations of commonly used numerical techniques - the Finite-Difference Time-Domain (FDTD) method, the Finite-Integration Technique (FIT), and Time-Domain Integral Equations (TDIE) - to solve the canonical problem of a horizontal dipole antenna radiating over lossless and lossy half-spaces. These types of environment are important starting points for simulating many Ground Penetrating Radar (GPR applications which operate in the near-field of the antenna, where the interaction among the antenna, the ground, and targets is important. We analysed the simulated current at the centre of the dipole antenna, as well as the electric field at different distances from the centre of the antenna inside the half-space. We observed that the results from the simulations using the FDTD and FIT methods agreed well with each other in all of the environments. Comparisons of the electric field showed that the TDIE technique agreed with the FDTD and FIT methods when observation distances were towards the far-field of the antenna but degraded closer to the antenna. These results provide evidence necessary to develop a hybridisation of current implementations of the FDTD and TDIE methods to capitalise on the strengths of each technique.

Advanced multi-frequency GPR data processing for non-linear deterministic imaging

In this paper, the quantitative imaging of the dielectric characteristics of unknown targets buried in a lossy half-space is performed by suitably processing wide-band ground penetrating radar (GPR) measurements. An innovative multi-frequency (MF) fully non-linear inverse scattering (IS) technique exploiting the integration of a conjugate-gradient (CG) solver within the iterative multi-scaling approach (IMSA) is proposed. Representative results from numerical test cases are presented to provide the interested readers with some indications on the effectiveness, as well as the current limitations, of the proposed approach when directly compared to a state-of-the-art frequency-hopping (FH) based method formulated in the same framework. Such a validation points out that if, on the one hand, the proposed MF strategy is computationally more efficient than the FH one, on the other hand, it turns out to be less reliable and accurate in several situations.

“Unlocking” the Ground: Increasing the Detectability of Buried Objects by Depositing Passive Superstrates

One of the main problems when trying to detect the position and other characteristics of a small inclusion into lossy Earth via external measurements is the object's poor scattering response due to attenuation. Hence, increasing the scattered power generated by the inclusion when using not an active but a passive material is of great interest. To this direction, we examine a procedure of “unlocking” the ground by depositing a thin passive layer of conventional material atop of it. The first step is to significantly enhance the transmission into a lossy half-space, in the absence of the inclusion, by covering it with a passive slab. The redistribution of the fields into the slab and the infinite half-space, due to the interplay of waves between the interfaces, makes it possible to determine the thickness and permittivity of an optimal layer. The full boundary value problem (including the inclusion and the deposited superstrate) is solved semianalytically via integral equation techniques. Then, the scattered power of the buried inclusion is compared to the corresponding quantity when no additional layer is present. We report a substantial improvement in the detectability of the inclusion for several types of ground and burying depths by using conventional lossy materials. Implementation aspects in potential applications as well as possible future generalizations are also discussed. The developed technique may constitute an effective “configuration (structural) preprocessing” which may be used as a first step in the analysis of related problems before the application of an inverse scattering algorithm concerning the efficient processing of the scattering data.

Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method.

A rigorous theoretical treatment to analyze the electromagnetic scattering of an inhomogeneous elliptically polarized plane wave by a sphere buried in a lossy half-space is presented. To consider the losses in the media an inhomogeneous plane wave is considered. The incident and the scattered electric field components are expanded in series of vectorial spherical harmonics using the Legendre functions generalized via hypergeometrical and gamma functions, with unknown expansion coefficients. The spectral-domain method to represent the scattered electric field is used in order to compute the scattered-reflected and scattered-transmitted fields, considering the reflection and transmission of each elementary plane wave by the interface. Finally, the unknown coefficients of the scattered field are computed by imposing the boundary condition on the spherical surface. In order to validate the model, a homemade code has been implemented. Comparisons with the simulations performed with a commercial software and the results in the literature are presented.