## 1. Introduction

Ground-penetrating radar (GPR) technology finds applications in many areas such as geophysical prospecting, archaeology, civil engineering, environmental engineering, and defence applications as a non-invasive sensing tool [3], [6], [18]. One key component in any GPR system is the receiver/transmitter antenna. Desirable features for GPR antennas include efficient radiation of ultra-wideband pulses into the ground, good impedance matching over the operational frequency band, and small size. As the attenuation of radio waves in geophysical media increases with frequency [9], [13], ground-penetrating radars typically operate at frequencies below

In this book chapter, the full-wave analysis of electromagnetic coupling mechanisms between resistively loaded wideband dipole antennas operating in realistic GPR scenarios is carried out. To this end, a locally conformal finite-difference time-domain (FDTD) technique, useful to model electromagnetic structures having complex geometry, is adopted [1], [2]. Such a scheme, necessary to improve the numerical accuracy of the conventional FDTD algorithm [19], [21], by avoiding staircase approximation, is based on the definition of effective material parameters [14], suitable to describe the geometrical and electrical characteristics of the structure under analysis. By doing so, the losses in the soil, as well as the presence of ground-embedded inhomogeneities with arbitrary shape and electrical properties, are properly taken into account. Emphasis is devoted to the investigation of the antenna pair performance for different Tx–Rx separations and elevations over the ground, as well as on scattering from dielectric and metallic pipes buried at different depths and having different geometrical and electrical characteristics. Novelty of the analysis lies in the fact that at the lowest operational frequency both the receive antenna and a pipe are situated in the near-field, whilst at the highest operational frequency only the far field is playing the role. The obtained numerical results provide a physical insight into the underlying mechanisms of subsurface diffraction and antenna mutual coupling processes. This information in turn can be usefully employed to optimize the performance of detection algorithms in terms of clutter rejection.

Finally, a frequency-independent equivalent circuit model of antenna pairs is provided in order to facilitate the design of the RF front-end of ground-penetrating radars by means of suitable software CAD tools. The procedure employed to extract the equivalent circuit is based on a heuristic modification of the Cauer’s network synthesis technique [10] useful to model ohmic and radiation losses. In this way, one can obtain a meaningful description of the natural resonant modes describing the electromagnetic behaviour of antenna pairs for GPR systems.

## 2. Locally conformal finite-difference time-domain technique

The analysis and design of complex radiating structures requires accurate electromagnetic field prediction models. One such widely used technique is the FDTD algorithm. However, in the conventional formulation proposed by Yee [19], [21], each cell of the computational grid is implicitly supposed to be filled by a homogeneous material. For this reason, the adoption of Cartesian meshes could result in reduced numerical accuracy when structures having curved boundaries have to be modelled. In this case, locally conformal FDTD schemes [1], [2] provide clear advantages over the use of the stair-casing approach or unstructured and stretched space lattices, potentially suffering from significant numerical dispersion and/or instability [19]. Such schemes, necessary to improve the numerical accuracy of the conventional algorithm, are based on the definition of effective material parameters suitable to describe the geometrical and electrical characteristics of the structure under analysis.

In this section, a computationally enhanced formulation of the locally conformal FDTD scheme proposed in [1] is described. To this end, let us consider a three-dimensional domain

As a consequence, the edge lengths between adjacent vertices in

The secondary or dual mesh

As usual, the electric field components are defined along each edge of a primary lattice cell, whereas the magnetic field components are assumed to be located along the edges of the secondary lattice cells. In this formulation, the relationship between

where:

as

Hence, combining the equations above yields:

where we have introduced the normalized field quantities:

and the averaged effective permittivity

The time derivative in (7) is then evaluated using a central-difference approximation that is second order-accurate if

where:

denotes the finite-difference expression of the normalized

with:

The update equations of the remaining components of the electric and magnetic field can be easily derived by permuting the spatial indices

As it can be readily noticed, the computation of position-dependent coefficients (14)-(16) can be carried out before the FDTD-method time marching starts. As a consequence, unlike in conformal techniques based on stretched space lattices, no additional correction is required in the core of the numerical algorithm. Furthermore, the resulting FDTD update equations (12)-(13) have a very convenient structure, leading to a

## 3. The full-wave antenna modelling

It is our intention to focus the attention on the full-wave analysis of a resistively loaded dipole antenna pair located above a ground modelled as a lossy homogeneous half-space having relative permittivity

The dipoles are denoted as dipole

has been applied to the flairs of the considered radiators. In (17),

The FDTD characterization of the structure has been carried out by using a non-uniform computational grid with maximum cell size

where

In (18),

where:

with:

denotes the time-discretised nominal current delivered by the generator. Similarly, the electric field distribution within the feed point of dipole

where:

and with

where

where

As it appears from Fig. 4a, the return-loss level is slightly affected by the Tx–Rx antenna separation that, on the other hand, is primarily responsible for the parasitic coupling level between the radiating elements. The impact of the antenna elevation above the ground has been also analyzed (see Fig. 4b). It is worth noting that, as

In the performed numerical computations, a ten-cell uniaxial perfectly matched layer (UPML) absorbing boundary condition for lossy media [19] has been used at the outer FDTD mesh boundary to simulate the extension of the space lattice to infinity. As outlined in [19], the UPML is indeed perfectly matched to the inhomogeneous medium formed by the upper air region and the lossy material modelling the soil. So, no spurious numerical reflections take place at the air-ground interface. In particular, a quartic polynomial grading of the UPML conductivity profile has been selected in order to have a nominal reflection error

## 4. The radar detection of buried pipes

In this section, emphasis is devoted to the analysis of the dipole antenna pair located above a lossy homogeneous/inhomogeneous material half space where an infinitely long pipe is buried (see Fig. 5). In such configuration, the transmit element of the radar unit emits an electromagnetic pulse that propagates into the ground, where it interacts with the target, modelled as a

The parasitic coupling level between transmit and receive antennas is a critical parameter in the design of ground-penetrating radars and satisfactory levels are usually achieved by empirical design methods. Anyway, the prediction of coupling levels already at the design stage enhances structure reliability, while also improving design cycle. To this end, the locally conformal FDTD model presented in Section 2 has been usefully adopted. In this way, as it can be noticed in Fig. 6, it has been found that the antenna return-loss parameter

### 4.1. Analysis of sub-surface scattering processes

The considered antenna pair has been used to analyze the sub-surface scattering phenomena arising from the field interaction with a PVC pipe buried at a depth

### 4.2. Impact of ground-embedded inhomogeneities

An invariable feature of real-life soils is heterogeneity. Without taking into account the inhomogeneities altering the idealized nature of the considered ground model, it becomes a futile effort to design a complex GPR system that will perform well over a real-life soil. To overcome this limitation, a realistic ground model has been developed by simulating small ellipsoidal scatterers embedded in the soil (see Fig. 5). The size, location and electrical properties of these inhomogeneities are determined randomly within preset limits. In particular, the maximum dimensions of the scatterers are

with mean

As outlined in [11], the rigorous analysis of the aforementioned subsurface scattering phenomena is very important in order to asses the suitability of detection and imaging algorithms for GPR applications in realistic scenarios.

## 5. Frequency-independent equivalent circuit model

Typically, electromagnetic field solvers and measurement systems, such as network analyzers, generate

The proposed procedure for the equivalent circuit extraction is based on a heuristic modification of the Cauer’s network synthesis technique [10]. Resistive elements are introduced to model metal, dielectric, and radiation losses. The scattering matrix

where

## 6. Conclusion

The full-wave analysis of electromagnetic sensing of buried pipes with GPR in realistic scenarios has been carried out. An enhanced locally conformal FDTD technique, useful to accurately model complex electromagnetic structures as well as ground-embedded inhomogeneities with arbitrary shape and material parameters, has been adopted. By using this scheme, an extensive parametric analysis of the antenna scattering parameters and radiated near-field spatial distribution has been performed for different Tx–Rx antenna separations and elevations over the ground, taking into account the presence of buried metallic and dielectric targets, as well as soil-embedded ellipsoidal inhomogeneities with arbitrary size, location and electrical properties. The obtained numerical results provide a physical insight into the underlying mechanisms of subsurface scattering and antenna mutual coupling processes. Finally, a frequency-independent equivalent circuit, useful to be employed in CAD tools, has been derived from the antenna scattering parameters, showing that including the effect of just a few resonant modes yields high numerical accuracy.

## 7. Appendix

In order to validate the accuracy of the proposed locally conformal FDTD scheme a number of test cases have been considered. Here the results obtained for the computation of the fundamental resonant frequency of a dielectric resonator enclosed in a metallic cavity are presented. The structure under consideration (see Fig. 13a) has been already analyzed in [5]. It consists of a perfectly conducting metallic cavity of dimensions

height