Detection of a buried wire with two resistively loaded wire antennas

Authors


Abstract

[1] The use of two identical straight thin-wire antennas for the detection of a buried wire is analyzed with the aid of numerical calculations. The buried wire is located below an interface between two homogeneous half-spaces. The detection setup, which is formed by a transmitting and a receiving wire, is located above the interface. The transmitter is excited by a pulsed voltage in a small gap. A resistance profile according to Wu and King [1965] has been used to suppress the reflections at the end of the wires. This enhances the visibility of properties of the lower half-space directly from the time response of the current along the receiving antenna. The problem is solved in several steps. First, the electric field integral equation for the total current along a single thin-wire antenna in a homogeneous space is formulated. The result is then used to construct a set of coupled integral equations to describe the currents along all three wires without the resistive load. The integral equations contain the transmitted and reflected fields due to the interface. Next, the set of integral equations is adapted for the resistance profile along the wires of the detection setup. The reflected and transmitted fields in both half-spaces are treated as secondary incident fields in the integral equation for the currents along the wires. In these equations, the response from a pulsed dipole source in the same configuration occurs as a Green's function. The inverse spatial Fourier transformation that occurs in the transmitted and reflected fields is carried out with the aid of a frequency independent, composite Gaussian quadrature rule. The set of coupled integral equations is solved by using the continuous-time discretized-space approach, where the space discretization is kept fixed for all frequencies. This results in a linear system of equations with a fixed dimension which is solved by the conjugate gradient-fast Fourier transform (CG-FFT) method. With the aid of a marching-on-in-frequency scheme, the system is solved for a number of frequencies. Time domain results are obtained by applying an inverse Fourier transformation. Representative numerical results are presented and discussed.

1. Introduction

[2] Rubio Bretones and Tijhuis [1997] and Rubio Bretones et al. [2002] investigated the properties of a dielectric half-space with the aid of two wire antennas. In fact, the signal on the passive wire has a strong relation with the properties of the dielectric half-space. A logical next step is to see whether or not a buried object can be detected with two wire antennas. To keep the mathematical formulation relatively simple, a third wire antenna is chosen as the buried object. The half-space surrounding the buried wire is kept homogeneous.

[3] A transmitting and a receiving wire form the detection setup and are located in a homogeneous half-space denoted as medium 1. The buried wire is to be detected and is located in the homogeneous half-space denoted as medium 2. All wire antennas are directed parallel to each other and to the interface between the two homogeneous half-spaces. The transmitting antenna is excited by a pulsed voltage, while the induced current along the receiving antenna is used for detection. Each wire in the detection setup gives two secondary contributions to the current along each of the other wires; one direct contribution and one via a reflection at the interface. The reflection only exists when the medium parameters of both half-spaces are different. The buried wire gives an additional interaction to both wire antennas in the detection setup and vice versa. This particular contribution is in the form of a transmitted field and makes detection in principle possible.

[4] The idea of using two wire antennas for the detection of a buried wire came after studying previous work by Rubio Bretones and Tijhuis [1997] where the transient excitation of two wire antennas over a inhomogeneous dielectric half-space is considered. In their wire antenna setup, one antenna acts as a transmitter and one as a receiver. It was shown that the current flowing along the transmitting wire antenna is significantly affected by the presence of the interface especially for small distances between the wire and the interface. For a certain choice of the distance between both wire antennas, the lower half-space has a strong influence on the time response of the receiving antenna.

[5] For perfectly conducting wires this response is quite complicated due to repeated reflections at the ends of both wires. Therefore, in later work [Rubio Bretones et al., 2002] the wire antennas were loaded with a resistance distribution as proposed by Wu and King [1965] (Wu-King profile). By loading the wire antennas with the Wu-King profile, unwanted electric current reflections at the ends of the antennas were avoided. With this modification to the wire antennas, properties of the entire configuration under study could be recognized, directly from the current along the receiving antenna. The success of this modification inspired us to also consider two loaded wire antennas above an interface to detect a buried wire.

[6] The paper is organized as follows. In section 2, we first consider two wire antennas above an interface and a buried wire below that interface. A frequency domain version of Hallén's integral equation for the current along one of the wire antennas is formulated. This equation is then used to construct a set of three coupled integral equations. The presence of the interface is accounted for by introducing reflected and transmitted field terms according to Rubio Bretones and Tijhuis [1997]. The reflected and transmitted field expressions in this configuration are computed by using a special spectral technique. The response from a pulsed dipole source in the same configuration occurs as Green's functions in the expressions for the reflected and transmitted fields. The involved spatial Fourier inversions are evaluated by a fixed, composite quadrature rule [Rubio Bretones and Tijhuis, 1995]. The set of integral equations is solved by using the continuous-time discretized-space (CTDS) approach according to Rubio Bretones and Tijhuis [1995]. The successive introduction of a fixed space discretization and a temporal Fourier transformation results in a linear system of equations of a fixed dimension for the sampled currents, in which the frequency occurs as a parameter. The system is solved repeatedly for increasing frequencies, with the aid of a conjugate gradient-fast Fourier transform (CG-FFT) method [Sarkar et al., 1985; van den Berg, 1985] and a dedicated extrapolation procedure [Tijhuis and Peng, 1991]. The time domain result is then obtained by performing an inverse temporal Fourier inversion. Representative numerical results are presented and discussed.

[7] In section 3, we investigate which choice of parameters results in the best possible detection of a buried wire. As observed by Rubio Bretones and Tijhuis [1997], the current along a wire may change significantly when the wire is located near an interface. Whether or not a buried wire can be detected with a similar setup is also studied.

[8] In section 4, loaded wires in the detection setup are considered as was previously studied by Rubio Bretones et al. [2002]. Again representative numerical results are presented and discussed for some practical configurations.

2. Formulation of the Problem

[9] We consider the problem of the transient excitation of a detection setup consisting of two straight thin-wire antennas, denoted as wire Rx and wire Tx. The detection setup is located in the upper homogeneous half-space z < 0 with permittivity ε1, permeability μ1 and conductivity σ1, see Figure 1. The length of each wire is LTR, the cross section is circular with radius a and the distance in the positive y-direction between both wires is d. The central axes of the wires are located at positions rTx = xux + z1uz and rRx = xux + duy + z3uz, with 0 < x < LTR, z1 < 0 and z3 < 0, for wire Tx and wire Rx, respectively. Here, ux, uy and uz are the unit vectors in a Cartesian coordinate system. Wire Tx is the transmitting antenna and is excited by an impressed voltage equation image(t) across the gap xg − Δx < x < xg + Δx with vanishing Δx. The impressed voltage is negligible prior to the initial instant t = 0. Wire Rx is the receiving antenna. The object to detect is a straight thin wire, denoted as wire B, with length LB and with a circular cross-section with radius a. The index B stands for “buried.” Wire B is embedded in the lower homogeneous half-space z > 0 with permittivity ε2, permeability μ2 and conductivity σ2. The central axis of wire B is located at position rB = xux + d2uy + z2uz, with xoff < x < xoff + LB, z2 > 0 and where the offset is given by xoff = (LTRLB)/2. The dimensions of the wires are chosen such that aLTR,B and the wires are assumed to be perfectly conducting. The integral equations that describe the total currents along the wires will be formulated in the frequency domain. By considering the various interactions between the wires and the interface, a set of coupled integral equations is constructed.

Figure 1.

The configuration.

[10] The set of integral equations is solved numerically by using the CTDS approach [Rubio Bretones and Tijhuis, 1997] and by applying a marching-on-in-frequency scheme [Tijhuis and Peng, 1991]. After applying an inverse temporal Fourier transformation, the time domain currents are obtained.

[11] To transform the time domain equations to the frequency domain and vice versa, the following temporal Fourier transformation and its inverse are introduced:

equation image

where we have used the property that in the time domain, only causal and real-valued, functions equation image(t) are considered. Once more, the effect of the voltage pulse prior to t = 0 may be neglected in applying the temporal Fourier transformation and its inverse.

2.1. Hallén's Equation for a Single-Wire Antenna

[12] Before we formulate the set of coupled integral equations for the currents along the three wires, we devote attention to the integral equation for a single wire.

[13] We consider one of the wires with a generalized length L in a homogeneous embedding with permittivity ε1, permeability μ1 and conductivity σ1. In a Cartesian coordinate system, the central axis of the wire is located at 0 < x < L, y = 0 and z = 0. A solution for the current along a perfectly conducting wire antenna in the frequency domain is obtained by solving the so-called reduced form of Pocklington's thin-wire equation [Pocklington, 1897]. For a wire antenna with the dimensions described in the previous section, the thin-wire equation is given by:

equation image

where

equation image

The latter equation is valid for 0 < x < L. The impressed voltage and the total current are denoted as V(ω) and I(x, ω), respectively. The external incident field component Exi(xux, ω) in (2) is the electric field in absence of the wire. The only approximation in this equation amounts to neglecting the radial currents on the end faces at x = 0, L [Tijhuis et al., 1992].

[14] In Pocklington's equation (2), the differentiations with respect to x can cause problems in the numerical evaluation, especially near x = x′, where Ga(xx′, ω) becomes almost singular. Therefore a different approach, which was first proposed by Hallén [1930, 1938], will be used. Pocklington's equation (2) is used as a starting point for the derivation of Hallén's equation.

[15] The combination of the space differentiation and k1 is recognized as the differential operator and k1 governing the propagation of plane waves in a homogeneous, lossless medium. By using Green's function for this operator as given by Tijhuis et al. [1992], (2) reduces to

equation image

for 0 < x < L. In (3), equation image is the complex admittance of the homogeneous embedding. The terms containing F0(ω) and FL(ω) represent two independent homogeneous solutions of (2). They can be found as a combination of source terms by invoking the boundary condition I(0, ω) = I(L, ω) = 0 on Hallén's equation (3). In the present derivation these factors are solved numerically as extra unknowns. The result (3) is known as Hallén's integral equation for the straight thin wire [see also Bouwkamp, 1942].

2.2. Interactions Between the Wires

[16] In the previous section, Hallén's equation was found for a single thin-wire antenna in a homogeneous embedding. This equation is now used as a basis for the coupling problem of three perfectly conducting wires in two half-spaces. In Figure 2, we show the interactions between the various elements of the total configuration. The dots represent the wires and a pair of arrows represents an interaction. It is easily seen that an increase in the number of wires increases the number of interactions.

Figure 2.

Exchange of radiation between the 3 wires of the configuration.

[17] The secondary field contributions along the different wires are as follows: (1) wire Tx and interface, (2) wire Rx and interface, (3) wire Tx and wire Rx, (4) wire Tx, wire Rx and interface, (5) wire B and interface, (6) wire Tx, wire B and interface, and (7) wire Rx, wire B and interface. These interactions are used to construct the set of coupled integral equations to describe the currents along the wires. Therefore, we assume all currents apart from the one that is to be solved in the integral equation to be known.

2.3. Integral Equation for the Current Along Wire Tx

[18] As an example, the integral equation to describe the current along the transmitting wire Tx in the detection setup will be derived because this is the only wire that is voltage driven. Deriving the integral equations for the other two wires in an analogous way leads to a set of three coupled integral equations.

[19] The secondary field along wire Tx is composed of the reflected field at the interface due to its own emitted field and the scattered fields from the other two wires (see the interactions containing Tx in the list in section 2.2). The secondary field along wire Tx can thus be written as:

equation image

where the superscripts r, t and e denote a reflected, transmitted and emitted field originating from the wire that is indicated with the additional superscripts Tx, Rx and B, respectively.

[20] The emitted field from wire Rx is accounted for by using a special form of Green's function [Rubio Bretones and Tijhuis, 1997]. The other terms in (4) represent transmitted and reflected fields.

[21] To interrelate the current along a wire with the appropriate reflected or transmitted field in (4), the current is written as

equation image

where L is again the generalized length of a wire. Next, the expressions to describe the reflected and transmitted fields at an interface due to a current point source are adopted from Rubio Bretones and Tijhuis [1995]. Using superposition with these expressions and the latter equation gives the desired relationships. Rubio Bretones and Tijhuis [1995] use a current point source in the upper half-space z < 0 to derive expressions for the transmitted and reflected field in the lower and upper half-space, respectively. The current point source is defined as

equation image

with zs < 0 and the subscript s represents “source.” The reflected and transmitted field due to the dipole current given in (6) can be represented as

equation image
equation image

with equation image, r = ρ cos ϕux + ρ sin ϕuy + zuz and

equation image
equation image

are the electric and magnetic reflection and transmission coefficients, respectively. The parameter ν = kTc0/ω is the dimensionless, normalized version of the transverse wavenumber equation image. The expressions for ξ21r(r, zs, ω), ξ21t(r, zs, ω) due to a current point source in the lower half-space z > 0, i.e. zs > 0, are found analogously. With these definitions, the remaining parts of (4) are found as

equation image
equation image
equation image

for 0 < x′ < LTR.

[22] Now that all secondary incident fields are described, we arrive at the following integral equation for the current along wire Tx:

equation image

with equation image and where the additional superscript in F0Tx and FLTx indicates the homogeneous solutions of (2) for wire Tx in particular.

2.4. Method of Solution

[23] For the other two wires, similar integral equations are found. With this set of three coupled integral equations, the currents along the wires can be found. To evaluate the set of coupled integral equations numerically, each equation is discretized. Again the integral equation for wire Tx will be used to demonstrate the discretization. To this end, we follow the Continuous-Time Discretized-Space (CTDS) approach as described by Rubio Bretones and Tijhuis [1997].

[24] First, a frequency-independent space discretization is carried out which reduces (12) to a linear system of equations for a fixed number of sampled currents. The interval 0 < x < LTR is subdivided into M subintervals with a fixed mesh size ΔxTR = LTX/M. The grid points are taken at the boundaries of the subintervals xm = mΔxTR, with m = 0, …, M, and the observation points are restricted to the grid points x = xm, with m = 0, …, M. The interval 0 < x < LB is subdivided in a similar way into MB subintervals with fixed mesh size ΔxB = LB/MB. Although the mesh sizes ΔxTR and ΔxB may be different, large differences should be avoided.

[25] The integrals over the unknown currents of (12) are approximated by weighted sums over the sampled values at the interior grid points x′ = xm with m′ = 1, …, M − 1. To this end, the numerator containing ITx in the left-hand side of (12) is approximated piecewise-linearly by

equation image

with equation image and where ϕm(x) is the triangular expansion function which is defined as

equation image

This approximation implicitly accounts for the boundary conditions ITx(0, ω) = ITx(LTR, ω) = 0. The integral of 1/Ra is obtained in closed form. The remaining integrals in (12) are evaluated directly with the aid of a repeated trapezoidal rule over the abcissae xm. The discretization described above results in a linear system of M + 1 equations for the M + 1 unknowns F0Tx(ω), FLTx(ω) and {ITx(xm, ω) | m = 1, …, M − 1}.

[26] The discretized version of (12) can thus be written as

equation image

with equation image for m = 0, …, M and where xB,m = m″ΔxB. The weighting coefficients in the trapezoidal rules are defined as

equation image
equation image

The reflected and transmitted field expressions ξ12r and ξ21t, respectively, are evaluated by using a composite quadrature rule as described by Rubio Bretones and Tijhuis [1995].

[27] The discretized equation (15) is supplemented with similar discretized equations for the currents IRx and IB yielding a system of 2M + MB + 3 unknowns. This system is then solved for increasing frequencies with the aid of a CG-FFT method. The initial estimates for which the conjugate gradient is started at a certain frequency are obtained by applying a dedicated extrapolation procedure to previously computed final results. For a detailed explanation, the reader is referred to work by Tijhuis and Peng [1991]. The extrapolation procedure reduces the computational effort to a few iterations for each frequency step. Finally, the time domain currents equation imageTx,Rx,B(x, t) are obtained by carrying out the inverse temporal Fourier transformation (1).

2.5. Numerical Results

[28] In this section, we demonstrate the potential of the approach presented in the previous section by analyzing some representative results. The relative permittivities in the examples correspond to various types of soils.

[29] All examples consider the wires to have length LTR = LB = 1 m, radius a = 2 mm and M = MB = 30 subdivisions. The distance between the wires in the detection setup is d = 1 m and the distance between wire B and wire Tx in the positive y-direction is d2 = 0.5 m. Wire B is located at a depth of z2 = 0.1 m. From now on, the wires in the detection setup are chosen at equal height, i.e. z1 = z3. The height z = z1 of the detection setup is varied. The medium parameters, except ε2r, are equal to those of vacuum. Wire Tx is excited at the center by an impressed Gaussian voltage of the form

equation image

with τ = 0.5 ns and tmax = 2 ns. In all examples, the total current is observed at the center of the wires, and results for the case without wire B, as calculated by Rubio Bretones and Tijhuis [1997], are used as a reference.

[30] In the first example we have a relative permittivity of ε2r = 3 and a height of z1 = −0.2 m. It is seen from Figure 3 that the current along wire Tx is hardly influenced by the presence of wire B. The contribution of the voltage pulse to the current along wire Tx is much higher than the other contributions. Therefore the influence of the interface and the other wires on the current along wire Tx is minor. The observation that the current along wire Tx is hardly affected by the presence of the half-space and the buried wire also holds for the other examples. The current along wire Rx, however, shows a small deviation in its amplitude. Therefore, only results for wire Rx are plotted in the remaining part of this section.

Figure 3.

The induced currents at the centers of wire Tx and Rx versus time. Configuration parameters: LTR = LB = 1 m, a = 2 mm, d = 1 m, d2 = 0.5 m, z1 = −0.2 m, z2 = 0.1 m and ε2r = 3.

[31] In Figure 4, the current along wire Rx is plotted for a relative permittivity ε2r = 3 and heights z1 = −0.1 m and z1 = −1 m, respectively. Closer to the interface, the difference between the currents with and without wire B is very clear while further away it is hardly visible. In both cases, the current shows an attenuating oscillatory behavior.

Figure 4.

The induced currents at the center of wire Rx versus time for z1 = −0.1 m and z1 = −1 m, respectively. Configuration parameters: LTR = LB = 1 m, a = 2 mm, d = 1 m, d2 = 0.5 m, z2 = 0.1 m and ε2r = 3.

[32] For εr2 = 9 and height z1 = −0.1 m and z1 = −1 m, respectively, the same behavior is observed in Figure 5. Again, wire B affects the current along wire Rx significantly only when the detection setup is close to the interface.

Figure 5.

The induced currents at the center of wire Rx versus time for z1 = −0.1 m and z1 = −1 m, respectively. Configuration parameters: LTR = LB = 1 m, a = 2 mm, d = 1 m, d2 = 0.5 m, z2 = 0.1 m and ε2r = 9.

[33] At z1 = −1 m, the current along wire Rx is hardly influenced by wire B and is comparable in magnitude for both permittivities ε2r. This means that the interface does not significantly affect the current at this distance. In general, the current along wire Rx is affected less by the interface for ε2r = 3 than for ε2r = 9. When the permittivity ε2r increases, the contribution from the field reflected at the interface to the current along wire Rx increases as well. This was also concluded by Rubio Bretones and Tijhuis [1997] and Rubio Bretones et al. [2002].

[34] The results for z1 = −0.1 m for both permittivities show that the difference in magnitude between the current along wire Rx with and without wire B is larger for ε2r = 9 then for ε2r = 3. While for increasing permittivity ε2r the reflected field Exr,Rx,Tx increases, the transmitted fields Ext,Tx,Rx decrease. Consequently, the induced current along wire B is smaller but the transmitted field Ext,B increases. The combination of all effects causes wire B to have a bigger influence on the current along wire Rx.

[35] Inspired by these results, we want to see for what height z1 the detection of the buried wire is optimal. Therefore, we will define a detection criterion in the next section.

3. Detection Criterion

[36] The presence of an interface has an effect on the detection of a buried wire. To examine these effects and to establish optimum detection of the buried wire, the configuration from the previous section is used. In other words, again, LTR = LB = 1 m, a = 2 mm, d2 = 0.5 m and z1 = −0.1 m.

[37] The current along the wires is obtained for −2 < z1 ≤ 0 m in steps of 0.05 m for distances d = 1 m and d = 2 m between the wires in the detection setup. The relative permittivity of the lower half-space was chosen ε2r = 3 and ε2r = 9, respectively. The remaining medium parameters are chosen equal to those of vacuum. After obtaining the currents for the configurations as described above, the current along wire Rx is subjected to a detection criterion which is defined as

equation image

The values of the criterion for the various configurations are visualized in Figure 6.

Figure 6.

The detection criterion as a function of the height −z1 of the detection setup above the interface for several configurations.

[38] The results show that, in general, the best detection is obtained when the detection setup is close to the interface. Surprisingly, we find some local minima and maxima in the graphs. Apparently, standing waves exist between the wires of the detection setup and the interface for some specific choices of the parameters of the configuration. In case of a standing wave, less of the radiated field penetrates the interface and consequently the detection of the buried wire is more difficult.

[39] Our goal is to recognize the presence of a buried wire directly from the shape of the current along wire Rx. In the present model, this is hardly possible without signal processing because of the presence of currents reflected at the end faces of the wires. These reflections are responsible for the slowly attenuating, oscillatory behavior of the current along a wire.

4. Wu-King Resistance Loading

[40] To avoid these unwanted reflections, the wires in the detection setup are continuously loaded with a resistance distribution RL(x). The current attenuates at the end faces of the wire by choosing the specific value of RL(x) according to Wu and King [1965]. Consequently, the transmitting as well as receiving characteristics are improved. It was found by Rubio Bretones et al. [2002] that the Wu-King resistance loading greatly enhances the visibility of the properties of the lower dielectric half-space directly from the time response of the current along wire Rx.

4.1. Specification of the Profile

[41] The wire antennas Tx and Rx are loaded with a resistance profile to avoid the unwanted reflections at the end faces of the wires. According to Wu and King [1965], a pure outward traveling current wave exists when for a cylindrical wire antenna of finite length, and a radius much smaller than the wavelength λ, the antenna has an impedance of

equation image

where Ψ is a constant, for given LTR, a and λ. This constant is given by

equation image

with

equation image

where equation image. Note that the wire is embedded in a homogeneous medium. As argued before, the current along the wire antenna is basically a traveling wave and a loaded wire antenna therefore exhibits broadband characteristics. At frequencies such that LTR < λ, the imaginary part of ZL(x) can be neglected. Therefore, we consider a resistive loading of the wire antennas only.

[42] The wire antennas of the detection setup are thus loaded with a resistance profile RL(x) according to

equation image

This is a good approximation of the Wu-King profile for wire antennas whose length is smaller than the minimum wavelength in the spectrum of the excitation signal.

[43] The resistance is a function of the position along the wire and increases towards the end points in such a way that no current is reflected. The resistance distribution is interpreted as an extra induced electric field. Therefore, we first multiply the resistance distribution with the current along wire Tx

equation image

where the superscript WK indicates the Wu-King profile. The extra induced electric field is an addition to the incident field along wire Tx in (4). For wire Rx, the same procedure is applied.

4.2. Numerical Results

[44] In the examples that will be given, we will comment on the advantages of loading the wires of the detection setup with the Wu-King resistance profile for the detection of a buried wire. We will demonstrate that the detection of a buried wire is possible with two loaded wire antennas in the detection setup by analyzing some examples. At the end of this section, we will discuss whether the detection criterion as described in section 3 is still a good choice.

[45] All examples consider the two wires of the detection setup to have length LTR = 1 m, radius a = 2 mm, M = 30 subdivisions. The length of the buried wire is LB = LTR = 1 m with MB = M subdivisions unless stated otherwise. The radius of the buried wire is a = 2 mm. The distance between the wires of the detection setup is d = 1 m and the distance between wire B and wire Tx in the positive y-direction is d2 = 0.5 m. The medium parameters, except ε2r, are equal to those of vacuum. Wire Tx is excited at its center by the impressed Gaussian voltage given by (17) and the total current is observed at the center of the wires. Again, all wires are assumed to be perfectly conducting.

[46] In the first example, see Figure 7, the depth of the buried wire z2 is varied and the height of the detection setup is kept fixed at z1 = −0.25 m. Thus, we can compare the current along wire Tx and wire Rx for various depths z2 to the reference result without wire B. The relative permittivity of the lower half-space is ε2r = 9. Again, the current along wire Tx is hardly influenced by the presence of the buried wire. This also holds for the other examples. Therefore only results for wire Rx will be shown in the remainder of this section. The results for wire Rx clearly show that the presence of the buried wire results in a small waveform in the current IRx. The second additional waveform occurs about 10 ns later than the first one, which can be explained from the difference in path length due to the increased depth z2 of wire B. In fact, the geometrical difference in path length corresponds to a time difference of 9.5 ns.

Figure 7.

The induced currents at the centers of wire Tx and Rx with Wu-King profile versus time, respectively. Configuration parameters: LTR = LB = 1 m, a = 2 mm, d = 1 m, d2 = 0.5 m, z1 = −0.25 m and ε2r = 9.

[47] In Figure 8, the total current at the center of wire Rx has been plotted for ε2r = 9 and for z1 = −0.1 m and z1 = −1 m, respectively. The depth is kept fixed at z2 = 0.1 m. When the detection setup is close to the interface, the presence of the buried wire causes the current along wire Rx to deviate strongly. Further away, a difference is hardly noticeable which was also found in section 2.5. However, in this case there is a slight deviation in the amplitude of the current around 15 ns.

Figure 8.

The induced currents at the center of wire Rx with Wu-King profile versus time for z1 = −0.1 m and z1 = −1 m, respectively. Configuration parameters: LTR = LB = 1 m, a = 2 mm, d = 1 m, d2 = 0.5 m, z2 = 0.1 m and ε2r = 9.

[48] Figure 9 shows the total current at the center of wire Rx for ε2r = 3 and z2 = 0.1 m for z1 = −0.1 m and z1 = −1 m, respectively. Again, close to the interface the current along wire Rx is strongly affected by the presence of wire B. Further away, there is hardly any difference.

Figure 9.

The induced currents at the center of wire Rx with Wu-King profile versus time for z1 = −0.1 m and z1 = −1 m, respectively. Configuration parameters: LTR = LB = 1 m, a = 2 mm, d = 1 m, d2 = 0.5 m, z2 = 0.1 m and ε2r = 3.

[49] A comparison between Figures 8 and 9 shows that for z1 = −0.1 m, the influence of wire B is stronger for lower permittivity. For ε2r = 3 in particular, the current along wire Rx shows again the attenuating oscillatory behavior that was seen in section 2. Since wire B is a target, the wire is not loaded with the Wu-King profile, reflections of the current at the end faces of this wire are still present. These repeated reflections can be clearly seen in the current along wire Rx which indicates the strong coupling between wire Rx and wire B. Further away from the interface, there is still a difference between the currents. However, it reduces to an amplitude effect again. To investigate the coupling between wire B and wire Rx, we take a closer look at the currents along these wires for ε2r = 3. In Figure 10, the absolute values of the currents along wire Rx and wire B are plotted on a logarithmic scale versus time. The behavior of the current along wire B can immediately be recognized from the current along wire Rx.

Figure 10.

The magnitude of the currents at the centers of wire Rx with Wu-King profile and wire B versus time. Solid lines: |IRx| without wire B. Dashed lines: |IRx| with wire B. Dotted lines: |IB|. Configuration parameters: L = 1 m, a = 2 mm, d = 1 m, d2 = 0.5 m, z1 = −0.1 m, z2 = 0.1 m and ε2r = 3.

[50] Next, we consider the case of the buried wire in a lossy medium. In Figure 11, the total current at the center of wire Rx has been plotted for σ = 0, σ = 0.01 S/m and σ = 0.03 S/m. The remaining variables are ε2r = 9, z1 = −0.25 m and z2 = 0.25 m. The height z1 and the depth z2 are chosen in such a way, that the additional waveform as observed is clearly visible in the current along wire Rx. The conductivity of the lower half-space does not have an influence on the arrival time and the general shape of the additional waveform due to the presence of wire B. It does have an attenuating effect on the effects of the buried wire. This is in agreement with the fact that the time domain current along a wire in a homogeneous embedding is attenuated by the conductivity and that the conductivity does not introduce extra delays.

Figure 11.

The induced currents at the center of wire Rx with Wu-King profile versus time for σ = 0 S/m, σ = 0.01 S/m and σ = 0.03 S/m, respectively. Configuration parameters: LTR = LB = 1 m, a = 2 mm, d = 1 m, d2 = 0.5 m, z1 = −0.25 m, z2 = 1 m and ε2r = 9.

[51] In the next two examples, the length of the buried wire will be varied. The medium parameters of the lower half-space are given by ε2r = 3, μ2r = 1 and σ2 = 0. The distance between the wires of the detection setup is d = 1 m and the distance between wire B and wire Tx in the positive y-direction is d2 = 0.5 m. The height of the detection setup is z1 = −0.1 m and the depth of the buried wire is z2 = 0.1 m. In Figure 12, the total current at the center of wire Rx has been plotted for LB = 0.36 m and LB = 0.5 m. The number of subdivisions along wire B is MB = 12 and MB = 15, respectively. The results for wire Rx clearly show that the presence of the buried wire results in an additional waveform in the current IRx. Comparing both plots shows that the oscillating behavior of the current is a direct effect of the length of wire B. The period in the case that LB = 0.5 m is longer than in the case that LB = 0.36 m. This is easily explained from the traveling wave nature of the current along the wire. When the wire is longer, the reflections at the end faces of the wire occur later.

Figure 12.

The induced currents at the center of wire Rx with Wu-King profile versus time for LB = 0.36 m and LB = 0.5 m, respectively. Configuration parameters: LTR = 1 m, a = 2 mm, d = 1 m, d2 = 0.5 m, z1 = −0.1 m, z2 = 0.1 m and ε2r = 3.

[52] In Figure 13, the total current at the center of wire Rx has been plotted for LB = 1.8 m and LB = 4 m. The number of subdivisions along wire B is MB = 54 and MB = 120, respectively. In both cases the amplitude of the current along wire Rx is raised at t = 7.5 ns. The current along wire Rx is oscillating slightly in both cases. The period of the oscillation of the additional waveform in the current along wire Rx again increases linearly with the length of the buried wire. Comparing the plots in Figures 12 and 13 shows that the amplitude of the additional waveforms due to the buried wire are smaller when the buried wire is longer than the wires of the detection setup.

Figure 13.

The induced currents at the center of wire Rx with Wu-King profile versus time for LB = 1.8 m and LB = 4 m, respectively. Configuration parameters: LTR = 1 m, a = 2 mm, d = 1 m, d2 = 0.5 m, z1 = −0.1 m, z2 = 0.1 m and ε2r = 3.

[53] Finally, the detection criterion of section 3 is studied for wire Rx with the Wu-King profile. The computation has been carried out for the same parameters as given in section 3. From Figure 14, it again follows that wire B is better detected close to the interface. Compared to the results in Figure 6, the loaded wires of the detection setup have a higher value for the criterion only for z1 = −0.1 m. In general, for z1 < −0.5 m, the results for ε2r = 3 and ε2r = 9 are almost parallel to each other for both distances d = 1 m and d = 2 m, respectively. Another interesting observation is that the local minima and maxima are less pronounced for the loaded wires in the detection setup. This confirms our assumption that standing waves exist between the unloaded wires of the detection setup and the interface. The two wires from the detection setup and the interface behave as a multi conductor transmission line. The Wu-King profiles along wire Tx and wire Rx reduces the influence of standing waves between the detection setup and the interface.

Figure 14.

Detection criterion for the loaded receiving wire.

5. Conclusions

[54] In this paper, we have generalized the approach for modeling two wires above a half-space described by Rubio Bretones and Tijhuis [1997] and Rubio Bretones et al. [2002] to the case where a buried wire is present in the lower half-space. To obtain the currents along the wires, first, Pocklington's integral equation is reduced to an equivalent form of the Hallén type. The interaction with the other wires and the homogeneous half-space are taken into account by treating the different contributions as secondary incident fields. In particular, the transmitted and reflected fields are discretized with the same frequency independent composite quadrature rule that was derived by Rubio Bretones and Tijhuis [Rubio Bretones and Tijhuis, 1995]. This results in three coupled integral equations which are solved by using the CTDS approach [Rubio Bretones and Tijhuis, 1995]. Eventually a linear system of equations with a fixed dimension is obtained for each frequency content of the transient voltage excitation, which is solved by marching on in frequency.

[55] We showed some numerical results in which all wires are of length L = 1 m and radius a = 0.02 m. The distance between the wires of the detection setup is d = 1 m and the distance in the y-direction from the buried wire with respect to the transmitting wire is d2 = 0.5 m. The material parameters, except ε2r, are equal to those of vacuum. The height z1 and depth z2 are also varied. In the examples it is shown that it is possible to detect a buried wire with a detection setup consisting of two conducting wires. This is difficult by merely looking at the current along the receiving wire because the buried wire mainly causes an amplitude effect. Therefore we tried to find a detection criterion to see if this effect originates from the parameters of the total configuration or from our approach in general. It was found that the choice of parameters affects the detection. In general the detection is better when the detection setup is close to the interface. Because only the amplitude of the current along the wires in the detection setup is affected, we tried a different detection setup.

[56] The wires of the detection setup were considered to be loaded with the Wu-King profile. Then, the presence of a buried wire can be observed immediately from the current along the receiving wire Rx. It is still necessary to keep the detection setup close to the interface to obtain this result. With this particular detection setup, the effects on the current along wire Rx due to different lengths of the buried wire was studied. The buried wire results in an oscillatory behavior of the current along wire Rx. The period of this oscillation varies linearly with the length of the buried wire. When the buried wire is longer than the wires from the detection setup, the amplitude of the additional waveform in the current along wire Rx decreases.

[57] After studying the detection criterion for both cases, the loaded wires are preferred over the unloaded wires if the detection setup is very close to the interface only. It is clear that the criterion presented here is fairly simple and should be improved.

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