Earth impedance model for through-the-earth communication applications with electrodes



[1] Through-the-earth (TTE) communications are relevant in applications such as caving, tunnel and cave rescue, mining, and subsurface radiolocation. The majority of the TTE communication systems use ground electrodes as load antenna. Wires, electrode contact, and earth impedances are the major contributors to the impedance observed by the transmitter. In this paper, state-of-art models found in the literature are reviewed, and an improved method to measure the earth impedance is presented. The paper also proposes an optimal circuit model for earth impedance between electrodes as a function of frequency, as a consequence of the particular conditions of the application. The model is validated with measurements for different soil conditions, showing a good agreement between empirical data and the simulation results.

1. Introduction

[2] Through-the-earth (TTE) communications are usually employed in environments such as tunnels, mines, caves, that force the electronics system to work in hostile operating conditions under the presence of mud, water and hazardous transportation. The transmission channel is the ground, propagating the electromagnetic energy through a dissipative medium such as the earth, rock or concrete. The main problem of TTE communications is the signal attenuation due to the earth skin effect. The effect increases with the frequency and the earth conductivity, therefore, subsurface communications operate in low frequency (ELF, VLF and LF). In this frequency range, the wavelength values are large and the dimensions of the antennas for far field operation would be excessive for applications in confined areas. However, for communication distances between 100 and 1000 m, the system works in near field conditions. Due to the limited range of available frequencies, the bandwidth is also reduced, restricting the amount of information transmitted. Two possible solutions for TTE communications can be adopted: inductive coupling or current injection by means of ground electrodes. In order to be efficient, inductive loops usually operate in resonance, offering a high-quality factor. This reduces the transmission bandwidth, making these load antennas unsuitable for low-frequency or high-rate data communication systems. Therefore, ground electrodes represent the best medium access solution for these applications.

[3] In TTE communication systems using electrodes, the transmitter injects a current into the earth by means of the galvanic coupling established between the power stage and the earth between the electrodes. The signal is captured by a second pair of electrodes in the receptor (see Figure 1).

Figure 1.

Scheme of through-the-earth communication with electrodes.

[4] The earth may be considered as a conductor, with lines of current linking both electrodes. In DC, equipotential surfaces are created around the electrodes. The receptor, with high input impedance, detects a voltage difference between its electrodes. The skin effect affects the current propagation, reducing the penetration depth when the frequency or the conductivity of the earth increases. Communication through the earth with electrodes has been studied by Wait [1982] and Durkin [1991], among others. These studies offer expressions for the coupling between emitter and receptor electrodes as a function of the frequency, the position of the electrodes and the earth characteristics.

[5] It is of paramount importance to quantify the load impedance seen by the power stage in order to optimally design the matching between them. The ground electrode impedance depends on many factors that can modify its value, even in two orders of magnitude. A very high value of ground contact impedance results in a small current injected into the earth and a shorter communication range. This paper explores the earth impedance, among the ground electrode impedance components. The state of the art of available models has been studied and an improved novel model for these applications is proposed in this paper and validated, obtaining good matching between theory and measurements.

2. Electrodes Impedance Model

[6] Figure 2 shows the total impedance of load seen by the power stage of a TTE communication system that includes three components: the wire impedance (Zw), the impedance of the contact between the electrodes and the earth (Zc) and the impedance of the earth between the electrodes (Ze).

Figure 2.

Ground electrodes impedance model with its three components: wires impedance (Zw), electrode contact impedance (Zc), and earth impedance (Ze).

2.1. Wire Impedance

[7] The wires connecting the electrodes to the system are usually copper multifilar wires between 20 and 50 m long. The inner diameter of the conductor varies between 0.8 (AWG 20) and 1 mm (AWG 18). The extended impedance of such a wire, Zw, presents an inductive-resistive series model [Bataller et al., 2008]. If the wire is coiled in a cable spool, a winding-dependent additional inductive component is added to the total impedance of the wire. Also a capacitive coupling between windings appears, resulting in a more complicated electrical model. In most cases a fraction of the wire is coiled, while the rest remains uncoiled, so a combination of the previous electrical models must be considered. In the situation under study, a resistive-inductive series model is considered. From authors experiences, the capacitive effect can be neglected for the frequency range of study.

2.2. Electrode Contact Impedance

[8] Modeling the contact of the electrode with the earth is highly complex. The value of the contact impedance depends on many factors as form, size and material of the electrode, electrical properties of the surrounding earth, penetration depth into the earth and frequency. Three components can be distinguished in the total contact impedance as reported by Okyere and Eduful [2006]. The first is the resistance of the conductor that forms the electrode. The second component is generated by the electro-chemical interface between the metal and the soil “solution”. The last is the contact surface of the electrode with the earth, referred as ground impedance. Only the third component is considered in most of the models found in the literature. In general, only the earth surrounding the electrode within a short distance is taken into account.

[9] The contact electrode impedance has been studied for different fields of interest, grounding installations, lightning, geophysical prospecting and biomedical applications [Okyere and Eduful, 2006; Rudemberg, 1968; Sunde, 1949; De la Vega, 2002; Gasulla, 1999; Marshall and Madden, 1959; Wang, 2005]. Each approach offers models for the contact but many of them are not suitable for the application reported in this paper either due to the size of the electrodes or to the frequency range considered. The electrodes used in this study, steel rod electrodes, fit the Gasulla model [Gasulla, 1999], as demonstrated by Bataller et al. [2008].

2.3. Earth Impedance

[10] The impedance of the earth placed between the electrodes is the third component of the impedance seen by the power stage of a TTE communication system. The value of this impedance depends on the earth conductivity, the working frequency and the electrodes gap. The conductivity is complex and frequency dependent.

[11] In the current conduction through the earth two types of path can be distinguished: the polarizable and the nonpolarizable. The nonpolarizable paths are due to an electrolytic conduction associated to a free path for ions dissolved in the water contained in the pores, and an electronic conduction where the electrons flow through the structure of certain metallic components of the terrain [Kiberu, 2002]. Furthermore, the polarization path presents two contributions: the membrane and the electrode polarization [Keevil and Ward, 1962]. The polarization makes the conductivity varying with frequency. It depends on some parameters analyzed by Marshall and Madden [1959]. Many models have been developed to simulate the electrical conductivity of the earth. The most important models can be found in Table 1 with their circuit schemes shown in Figure 3.

Figure 3.

Circuit schemes of the earth models included in Table 1.

Table 1. Earth Impedance Models With Their Reference, the Circuit Model Associated With the Impedance, and the Expressions of the Model Components With a Special Dependency on Frequency That Cannot be Modeled With Usual Circuit Elements
ModelReferenceCircuit ModelaObservations
  • a

    This code represents the correspondent circuit model shown in Figure 3.

WaitDias [2000]M1 
Madden and CantwellDias [2000]M1Z′ = a/()1/4
Cole-ColeDias [2000]M1Z′ = a/()c0 ≤ c ≤ 1
Davidson-ColeDias [2000]M1Z′ = a/(ωL + )c(R + R1)/(a/ωLc) ≫ 1 0 ≤ c ≤ 1
Generalized Cole-ColeDias [2000]M1Z′ = a/(ωLc + ()c)k(R + R1)/(a/ωLc) ≫ 1 0 ≤ (c; k) ≤ 1
WongDias [2000]M1 
Ward and FraserDias [2000]M2Zw = a/()1/2
WarburgDias [2000]M3Zw = a/()1/2
DiasDias [2000]M4Zw = a/()1/2
DebyeDias [2000]M5 
Multi Cole-ColeDias [2000]M6 
Marshall and MaddenMarshall and Madden [1959]M7 
Keevil and WardKeevil and Ward [1962]M8Electrode impedance included
MontañaMontaña [2006]M9Experimental model
ZongeDias [2000]M1equation image

[12] All models consider a complex resistivity that contains the earth permittivity in its expression. So we obtain an earth impedance value with frequency-dependent real and imaginary components.

[13] As shown in Figure 3, most of the models have a resistive component that accounts for the electronic path and another branch in parallel with a frequency-dependent component that models the polarization. These models are described in detail in the corresponding references, with expressions for the resistivity in terms of the constants a, c and k, that are function of the electrochemical parameters such as the relaxation time (τ), the chargeability (m) and the DC resistivity (ρo) [Dias, 2000]. For the sake of clarity, only the circuit model and some observations about it are presented here.

[14] Most of these models are valid from DC to 1 MHz and for homogeneous terrain, except Keevil and Ward model [Keevil and Ward, 1962] which is valid from DC to 1 kHz and Montaña model [Montaña, 2006], valid from 100 Hz to 20 MHz.

[15] Each component of the total electrode impedance must be measured separately in order to find the model that better describes the dependency of the earth impedance with the frequency. The focus of this paper is to analyze only the earth impedance, so contact and wire impedance measurement methods are not presented here.

3. Earth Impedance Measurement Methods

[16] Measurement methods of the earth impedance have been studied for geophysical applications. The value of the earth conductivity/resistivity is the main object of these studies. The earth resistivity and its variation with the depth or the distance in surface can be determined by means of electrical, magnetic or electromagnetic methods. The impedance between two points at the working frequency can be calculated with the resistivity data with geometrical factors being applied. The state of the art measurement techniques are described in detail by Bataller [2009].

[17] After a careful analysis of these methods, the most suitable has been selected for the current study. The method is based on the complex resistivity method [Nelson et al., 1982], where a sinusoidal continuous wave current of known frequency is injected by means of two electrodes and a voltage difference is measured between another pair of electrodes. The earth transimpedance value is obtained computing the ratio between the voltage and the current. The resistivity can be calculated from this value applying a geometric factor K that depends on the configuration with the following equation:

equation image

[18] If the electrodes are deployed in the Wenner array configuration [Wenner, 1916] with the same separation between them of value a, the geometric factor takes a value of 2πa. At DC and low frequencies, the current penetrates into the earth easier than at high frequencies, when the current circulates mostly near the surface.

[19] The equipment required for the complex resistivity method includes a signal generator, amplifiers for voltage and current measurements, a current sense resistor or any alternative current sensing method, a data acquisition module and a personal computer (PC).

[20] The measurement system, shown in Figures 4a and 4b, utilizes a sinusoidal voltage generator of 30 Vpp, specifically designed for this application. An AC passive current probe and an amplifier are used for the current measurements. For voltage measurement a low noise amplifier is used. A high-speed data acquisition module captures the analogue signals and converts them into digital, communicating the data to the PC. A Matlab script specifically programmed for the application controls the generator and the acquisition module. It also processes the received digital data, providing as an output the impedance values as a function of the frequency. Several known methods for impedance calculation from voltage and current measurements have been studied [Angrisani et al., 1996; Mahmud, 1989; Liu et al., 1997, 1999]. The Discrete Fourier Transform (DFT) method described by Liu et al. [1997] was selected for this application. Otherwise, the current probe has not a constant response with the frequency; therefore the system calibration has been necessary. The calibration is performed with resistances of known value according to the method presented by Liu et al. [1999]. After calibration, impedance data are compared with the output of the different earth impedance models with an optimization process that allows the extraction of the model parameters that better fit the data.

Figure 4.

(a) Scheme of the measurement system configuration for the field trials. (b) Picture of the measurement system in Esjamundo cave field experiment.

[21] Inductive and capacitive coupling between the injection and voltage measurement wires [Gasulla, 1999] is a factor for the frequency range under consideration. The coupling depends on the type of wires, their length and the measurement setup. In order to minimize the coupling, coaxial wires were used. The wires connect the current generator to the external electrodes and the potential ones to the low noise amplifier [Ott, 1988]. Their length was minimized in order to reduce their self impedance. The wire shields are connected together to a central point between the electrodes, reducing the capacitive and inductive couplings [Ingeman-Nielsen and Baumgartner, 2006].

[22] Another source of error is the contact electrode impedance that depending on its value, it may or may not be neglected. Even, for very high contact impedance values, this prevents valid measurements from being obtained. This impedance must be minimized in order to achieve the minimal possible influence in the earth impedance measurements, allowing more current to be injected into the earth and the detector electrodes to have low impedance values. Therefore, the optimal electrodes for the specific terrain are chosen according to previous experiments [Bataller et al., 2008]. They are also wetted with salty water reducing in that way its contact impedance. In Figure 5 it can be seen a comparative of measurements of steel rod electrode impedance for different contact conditions. The best behaviors, i.e., those with the lowest impedance, correspond to the situation where bentonite or salty water is added to the contact surface. Other trials have demonstrated a contact impedance reduction when several electrodes are connected in parallel [Bataller, 2009]. In spite of applying these techniques to minimize the contact electrode impedance, in certain earth surfaces as granitic rocks and some type of clays, the impedance presented by the electrodes was so high that the earth impedance measurements were not possible to carry out.

Figure 5.

Earth measurement system circuit model with electromagnetic coupling.

[23] An additional problem is the common mode voltage in the voltage measurement. It has been solved by means of a low-noise preamplifier that offers a differential output from two input signals.

[24] Electrodes are settled in a Wenner array configuration with 4 m separation between them. This separation has been chosen in order to simplify the wires deployment. This configuration offers the earth impedance measurement up to 2 m in depth. If we would want to study the impedance in deeper terrain layers, more separation between the electrodes should be used.

[25] When it comes to describe the measurement setup electrically, the wire self impedances and the electrode contact impedances have been taken into consideration trying to minimize the error sources. Thus, a more realistic model of the measurement setup has been developed. Figure 6 depicts the general circuit network description of the complete system. In this model, the electromagnetic coupling between wires and the capacitive coupling to the earth are also represented.

Figure 6.

Earth measurement system circuit model without electromagnetic coupling.

[26] As it has been explained above, several solutions have been applied to minimize electromagnetic coupling including coaxial wires with shield to earth and minimum wires length. It has been proved experimentally that the earth impedance measured without the coaxial cables and the shields connection depends on the wires distribution, presenting incoherent measurement values. To further prove that the coupling is not affecting our system, a synthetic earth impedance value has been generated, introducing it in the circuit model also including different levels of coupling. Then the earth impedance with CR1mod software [Ingeman-Nielsen and Baumgartner, 2006] has been correctly deembedded. It has been proved that the response of the circuit model that takes into account the coupling fits to the data calculated with the software. However, the data measured in the earth do not follow the same trend, concluding that the coupling has been eliminated from the measurement with the employed shielding and grounding techniques. Assuming this, the model presented in Figure 6 is not adequate for our measurements, and the simplified model reported in Figure 7 is proposed.

Figure 7.

Magnitude of contact electrode impedance for dry, wet with salty water, and with bentonite conditions.

[27] The contact electrode model proposed by Gasulla has been found optimal for the type of electrode employed [Bataller et al., 2008]. The wire and contact impedance parameter values are estimated from measurements of the electrode contact impedance and total impedance between current electrodes, captured in the same experiment. These values have been considered as constant parameters in the optimization process for each data set. The optimization of earth impedance model parameters has been made in two phases: firstly by tuning in the AWR Design Environment Software and a secondly with a finer optimization with the Matlab Optimization toolbox. The Nelder-Mead Simplex method [Lagarias et al., 1998], an unconstrained nonlinear optimization method, is used in the optimization function. The different models have been fitted to the measured data calculating the mean quadratic error between model and data in real and imaginary part.

4. Results

[28] The measurement method has been tested in several nonlaboratory environments with the equipment shown in Figure 4. Three of them are presented in this paper. Two measurements were taken near the village of Villanúa, in the open air and inside a cavity (Esjamundo cave). The third set was measured in Walqa Technological Park. The localization of the measurement points can be seen in Figure 8.

Figure 8.

Map of Spain with a zoom of Huesca region. Walqa and Villanúa locations are indicated.

[29] Real and imaginary parts of measured data have been compared with the transimpedance calculated from the electrical model of Figure 7, substituting the different earth model boxes by the earth impedance models found in the literature (Figure 3). These models assume the earth is formed by “microscopic elements” and not a single element. The inductance due to the virtual loop formed by current circulation through the earth plays a very important role for TTE communications. In these applications, electrodes can be separated several tens of meters. The effect of the virtual loop inductance makes the impedance between the electrodes follow an inductive pattern instead of a capacitive or resistive one, the typical pattern for small electrode separation situations. In the current study, the electrode separation is not large enough for the virtual current impedance to dominate. The wires impedance of the circuit model takes into account this virtual current loop contribution.

4.1. Villanúa Measurements

[30] A first set of measurements was obtained in the open air on a field near the village of Villanúa (Huesca, Spain). The second set of data was measured in the interior of the Esjamundo cave also situated in Villanúa. The open air location and the topography of Esjamundo cave are shown in Figure 9. The physical characteristics of the measurements are presented in Table 2, with the external conditions at the time measurements were taken.

Figure 9.

(top) Picture of Villanúa measurement location. The point of measure is located in a field of calcareous soil. (bottom) Topography of Esjamundo cave formed in limestone.

Table 2. Characteristics Parameters of Villanúa Measurement Location, at Open Air and Inside Esjamundo Cave
ParameterVillanúa Location
Open AirEsjamundo Cave
UTM coordinate30T 702120 472907630T 702120 4729076
Date25 Jan 200925 Jan 2009
Weather conditionsWet, cold and cloudyWet, cold and cloudy
Soil characteristicsShallow calcareous soil lying on limestoneLimestone. Soil formed by sandy gravel sediments of glacial origin.
OriginEocene ageEocene age

[31] Figure 10 shows the real and imaginary components of the earth impedance measured and the optimization results for the first measurements set captured in the open air. Comparing all the earth models previously proposed, Ward and Fraser model [Dias, 2000] offers the best fit to the current data. The value of the optimal parameters of the circuit model is presented in Figure 11 with the contact electrode impedances, both injector and detector types, and the wire injector impedance with inductance and resistance components, that also models the “virtual loop impedance” formed by the wires and the current circulating through the earth. The detector wires impedance has not been considered because its value is negligible and does not influence in the earth impedance.

Figure 10.

Real and imaginary parts of the earth impedance measured in Villanúa with the optimized circuit model response.

Figure 11.

Circuit model proposed with the optimal parameter values for Villanúa measurement set.

[32] Figure 12 represents the measurement set captured in Esjamundo cave, with the circuit response of the earth optimal model. In this case the model with the best adjustment is also Ward and Fraser model. The parameter values of the optimal circuit model are shown in Figure 13, with the same considerations commented for the previous measurement.

Figure 12.

Real and imaginary parts of the earth impedance measured in Esjamundo with the optimized circuit model response.

Figure 13.

Circuit model proposed with the optimal parameter values for Esjamundo measurement set.

4.2. Walqa Technological Park Measurements

[33] The third set of measurements was taken at open air in an esplanade near the Walqa Technological Park (Huesca, Spain). A picture of the measurement location is shown in Figure 14. The location characteristics are included in Table 3.

Figure 14.

Picture of Walqa measurement location. The point of measure is a plot of gravels over sandstone.

Table 3. Characteristics Parameters of Walqa Measurement Location
ParameterWalqa Location
UTM coordinate30T 710036 4665412
Date29 Oct 2009
Weather conditionsSunny, dry
Soil characteristicsFour meters of gravels lying on sandstone and clay with an aquifer in the contact surface.
OriginMiocene age

[34] For the Walqa measurement set a new model for earth impedance in the frequency range 500 Hz to 100 kHz is proposed. In a first optimization, the circuit model did not fit the data. The specific geology of the location with an aquifer, forming a more conductive path for the current, was not able to be represented by the available earth models. Therefore, an inductive-resistive series component, in parallel to existing earth models was added, trying to simulate the presence of this inner more conductive layer. The different literature models with the added inductance have been compared with the measured data.

[35] The real and imaginary parts of the earth impedance measured with the optimal modified model response are shown in Figure 15. The model that properly fits the data measured is Cole-Cole [Dias, 2000] modified with the R-L component. However, only a small difference is noted when the data are fitted to other earth models. Figure 16 shows the circuit model with the optimal values of the parameters for this measurement set.

Figure 15.

Real and imaginary parts of the earth impedance measured in Walqa with the optimized circuit model response.

Figure 16.

Circuit model proposed with the optimal parameter values for Walqa measurement set. The earth model contains the added R-L component.

[36] The error with and without the inductive-resistive component added to model the more conductive layer is calculated for all the available earth models. Figure 17 depicted the mean quadratic error for optimizations realized with the different earth models showing the improvement obtained when the R-L branch is added to the earth model.

Figure 17.

Mean quadratic error between measured data and optimization results for the different earth impedance circuit models (see Table 1) with and without the added resistive-inductive (R-L) component.

4.3. Results Analysis

[37] In order to check the measurement repeatability for each location, two set of measurements were taken with an elapsed time of 1 h. The maximum variation between both data for all the frequency range is a 5%. This difference presents a similar value in the three experiments carried out. Figure 18 shows the magnitude of these two earth impedance measurements for the Esjamundo location, proving the repeatability of the measurements. The variation between both curves is minimal.

Figure 18.

Magnitude of the earth impedance measured in Esjamundo, in two different field trials with an elapsed time of 1 h.

[38] Comparing the magnitude of the earth impedance models for the three measurements (Figure 19), it can be seen that they present a very different behavior depending on the type of soil. However, the range of values for the three measurements is in the same order of magnitude and is low in comparison with the contact electrodes impedance (Figure 5). As a result, the earth impedance cannot be altered in order to minimize its value, but due to its low magnitude, it is not relevant in the total impedance seen by a TTE transmitter with electrodes.

Figure 19.

Comparative of the magnitudes of the earth impedance measurements for the three different data sets (Villanúa, cross; Esjamundo, circle; Walqa, square).

5. Conclusions

[39] This paper explores the earth impedance models and corresponding measurement methods. An improved measurement method has been reported for the particular application of TTE communications. Various sets of measurements have been taken with this method. Two of them have been compared with the developed circuit model, including the different earth impedance models and the optimization of its parameters for each measurement. For one set of measurements, a novel model has been proposed including an inductive-resistive series component in parallel with the existing earth model. This component represents a more conductive singular path for low frequency: aquifers or metallic pipes. Future studies could focus on the application of this technique for the search of metallic ores or even underground aquifers.

[40] Among the several available, the earth models that better fit the data are the Cole-Cole and Ward and Fraser models for the different measurements. This method allows characterization of the earth impedance response and obtains a new circuit model. The earth impedance presents a much lower value than the electrode contact impedance and therefore it could be considered negligible in most of the TTE communications applications with electrodes.


[41] This work has been funded by the projects TESSEO DPI 2009-08126 (MCYT, Spanish Government) and 2008/0486 (CyT, Aragón Government), and by the agreement between IAF, Aragón Government, and the I3A (University of Zaragoza) 2008/0574 regarding the WALQA research laboratory.