Tackling capacitive coupling in broad-band spectral electrical impedance tomography (sEIT) measurements by selecting electrode configurations

SUMMARY

Electromagnetic (EM) coupling effects including both inductive and capacitive coupling have long been an essential problem in broad-band spectral electrical impedance tomography (sEIT) measurements at the field scale. Efforts have been made to remove EM coupling numerically or to suppress the effects by modified data acquisition strategies. For near-surface applications with relatively small survey layouts, inductive coupling can be well removed in the mHz to kHz frequency range. With the use of shielded coaxial cables and so-called active electrodes where the amplifiers are mounted at the electrodes, capacitive coupling in sEIT measurements can also be reduced. However, it remains challenging to cope with capacitive coupling between the cable shield and the ground, especially in resistive field conditions. The aim of this study is to deal with this type of capacitive coupling effect by identifying and filtering out sEIT measurements that are strongly affected by capacitive coupling. Based on a correction method for capacitive coupling proposed in a previous study, an approach to estimate measurement errors due to capacitive coupling is presented first. In the second step, a workflow was proposed to calculate the capacitive coupling strength (CCS) for each electrode configuration, which is defined as the ratio of the imaginary part of the impedance induced by capacitive coupling and the imaginary part of the impedance due to the subsurface electrical conductivity. In the final step, measurements with low CCS were selected for inversion and the results were compared with inversion results obtained using the previously developed correction approach. It was found that the filtering method based on CCS is more capable in tackling capacitive coupling compared to using model-based corrections. Spectrally consistent sEIT results up to kHz were obtained using the newly developed filtering method, which were not achieved in previous work using model-based correction.

1 INTRODUCTION

Spectral electrical impedance tomography (sEIT) is a non-invasive geophysical imaging technique that provides the spectral complex electrical conductivity distribution in the subsurface by measuring complex transfer impedances between different electrodes in a broad frequency range (i.e. mHz to kHz). Laboratory studies have revealed relations between spectral induced polarization (SIP) measurements and material properties of interest, such as electrical–hydraulic relationships (Slater & Lesmes 2002; Binley et al. 2005; Zisser et al. 2010) and links with biochemical and biogeochemical properties (Atekwana & Slater 2009; Mellage et al. 2018; Strobel et al. 2023). This has resulted in an increased interest in determining the complex electrical conductivity in the field using sEIT to support environmental, hydrological and biogeochemical investigations (Williams et al. 2009; Flores Orozco et al. 2011, 2012; Attwa & Günther 2013).

Many applications at the field scale were limited to a lower frequency range up to 100 Hz (Hördt et al. 2009; Flores Orozco et al. 2012; Attwa & Günther 2013; Martin et al. 2020) due to electromagnetic (EM) coupling effects, including both inductive coupling and capacitive coupling. Inductive coupling refers to the unwanted voltage in the wires for voltage measurements induced by the magnetic field generated by the injected alternating current. The amount of inductive coupling mainly depends on the measurement frequency and the geometry of the cable layout. Early work on the removal of inductive coupling was limited to low frequencies and single measurements (Pelton et al. 1978; Coggon 1984; Wait & Gruszka 1986; Routh & Oldenburg 2001; Schmutz et al. 2014). For near-surface applications where the survey scale is relatively small, inductive coupling in sEIT measurements can be corrected by numerical modelling using the known geometry of the cable layout (Zimmermann et al. 2019; Wang et al. 2021) or by calibration measurements when the position of the cables is unknown or difficult to control during field acquisition (Zhao et al. 2013, 2015; Kelter et al. 2018; Weigand et al. 2022). Wang et al. (2021) proposed a novel index called inductive coupling strength (ICS), which is defined as the ratio of the imaginary part of the impedance due to inductive coupling (i.e. the frequency-dependent mutual inductance) and the imaginary part of the impedance due to the polarization of the subsurface. Through proper selection of electrode configurations based on the ICS, consistent phase inversion results up to kHz without correction of inductive coupling were achieved.

Capacitive coupling in sEIT measurement usually refers to the leakage of electric current through possible capacitances wherever a potential difference exists. Typically, three types of capacitive coupling are considered. The first type of capacitive coupling occurs between cables (Dahlin & Leroux 2012; Radic 2004; Zhao et al. 2013). This type of capacitive coupling is usually the most important when non-shielded multicore cables are used. Dahlin & Leroux (2012) proposed a parallel cable layout separating the current injection and potential measurement in different cable bundles to reduce this type of capacitive coupling in time-domain induced polarization (TDIP) measurements. Alternatively, coaxially shielded cables can be used to reduce this type of capacitive coupling (Zhao 2017; Flores Orozco et al. 2021). The second type of capacitive coupling occurs between the ground and electrodes, which can be largely reduced using the model and methods presented in Zimmermann et al. (2008), Kelter et al. (2015) and Huisman et al. (2016). The third type of capacitive coupling occurs between the cable shield and the ground (or the materials of interest).

To deal with capacitive coupling between the cable shield and the ground, a straightforward strategy is to put the cables in the air, for example by using Styrofoam to elevate the cables from the ground (Weigand et al. 2022). For monitoring purposes and relatively small survey layouts, the required extra work to elevate cables is likely worth the effort to reduce capacitive coupling. However, for one-time surveys with larger survey layouts and borehole sEIT surveys, field measurements with elevated cables are not feasible anymore. Zhao et al. (2013) proposed to consider capacitive coupling in forward modelling and inversion by integrating capacitances at the nodes corresponding with the cable positions, which improved the inversion results in the kHz range. When the subsurface is relatively conductive (Zhao et al. 2013, 2015), the electric currents preferably flow through the ground rather than the capacitances between the cable shield and the ground. Therefore, the leakage current is relatively small and can be neglected. In more resistive environments where the leakage currents are typically higher and can no longer be neglected, Zimmermann et al. (2019) considered both integrated capacitances and leakage currents in the forward modelling for surface sEIT measurement and improved the phase accuracy at 71 Hz substantially. However, the corrections were only partly successful for higher frequencies. The results presented in Zimmermann et al. (2019) were limited to a small set of electrode configurations from the so-called complete configuration (Xu & Noel 1993), and the strength of capacitive coupling was not systematically investigated.

Following our previous work on dealing with inductive coupling (Wang et al. 2021), the aim of this study is to deal with the capacitive coupling between the cable shield and the ground by identifying and filtering out measurements that are strongly affected by capacitive coupling. In the first step, a new strategy to model the impedances induced by capacitive coupling will be presented. A novel index to quantify the capacitive coupling strength (CCS) will then be proposed. The CCS of all possible electrode configurations will then be examined by assuming that only capacitive coupling needs to be considered and that other sources of measurement error are of secondary importance. In the final step, electrode configurations with low CCS will be selected for inversion of sEIT data and the results will be compared to inversion results obtained using the complete configuration considered in Zimmermann et al. (2019) to show the feasibility of tackling capacitive coupling by selecting electrode configurations instead of model-based corrections.

2 MATERIALS AND METHODS

2.1 Definition of leakage current and sEIT measurement equipment

The investigation of capacitive coupling in this study is focused on the capacitance between the cable shield and the ground. The fundamental cause of this type of capacitive coupling is leakage currents flowing through stray capacitances. To account for the leakage current for a two-point excitation, the injected current I* is decomposed into two parts, including the symmetric current I_s and the leakage current I_L (Zimmermann et al. 2019):

Fig. 1 shows a schematic illustration of the different current paths. The symmetric current only flows through the current electrodes C+ and C−, while the leakage current can travel between the ground and the cable shield and eventually return to the system through the cable shield. A custom-made sEIT measurement system (Zimmermann et al. 2008) was used in this study to acquire the field sEIT measurements. For each excitation, this system measures the excitation current |${I}_1$| through the positive current pole (C+ in Fig. 1) and |${I}_2$| through the negative current pole (C− in Fig. 1). The symmetric current |${I}_s$| and the leakage current |${I}_{\mathrm{L}}$| can then be calculated by (Zimmermann et al. 2019)

The sEIT system used here relies on electrode modules with a relay and an amplifier integrated at each electrode for current excitation and voltage measurements, respectively (Fig. 1). The system measures the voltage |${U}_\mathrm{ p}$| relative to the system ground (cable shield) at all electrodes except the two current electrodes. The potential difference for a four-pole electrode configuration can then be calculated in a post-processing step. The measured single-point voltages can be used to derive the total capacitance between the ground and the cable shield, which will be elaborated in the following section.

2.2 The total capacitance and normalized leakage current

To obtain the total capacitance between the cable shield and the ground, additional measurements with single-pole excitation were conducted in Zimmermann et al. (2019). A current |${I}_i$| was injected at a single electrode so that it can only flow back to the system as leakage currents through the capacitances between ground and cable shield (no symmetric current). The potential measurements |${U}_\mathrm{ p}$| were carried out at all other electrodes relative to system ground (i.e. cable shield). The total capacitance |${C}_\mathrm{ T}$| can then be calculated using

where f is the measurement frequency. To achieve high accuracy of the estimated total capacitance, the same procedure to estimate the total capacitance was repeated at each electrode and the mean value of all estimated total capacitances was then used. The idea behind estimating the total capacitance with eq. (4) is to obtain the applied current and voltage that act on the capacitances. In this study, an alternative approach to directly derive the total capacitance from field sEIT measurements was used. In the case of actual sEIT measurements, the leakage current is the current applied flowing through the capacitances and can thus be used to replace |${I}_i$| in eq. (4). For each current excitation at two current electrodes, the voltages |${U}_\mathrm{ p}$| measured at all other electrodes relative to system ground can be used to determine the total capacitance using eq. (4) without the need for additional measurements. This is possible since the leakage currents through the capacitances and the potentials relative to the system ground are available for regular sEIT measurements with the system used here.

The normalized leakage current is defined as the ratio of the leakage current and the symmetric current expressed as a percentage (⁠|$100{\rm{\ per\ cent}} \times {I}_\mathrm{ L}/{I}_\mathrm{ s}$|⁠). The real part of the normalized leakage current reflects the loss in the real part of the injected currents |${I}_1$| and |${I}_2$|⁠, whereas the imaginary part of the normalized leakage current indicates how large the phase shift between the excitation currents is. Usually, the real part of the normalized leakage current is much smaller than the imaginary part because capacitive coupling mainly induces a phase shift in the applied currents. The absolute value of the normalized leakage current as used in Zimmermann et al. (2019) represents the total loss in the injected currents.

2.3 Field sEIT measurements

The field data were acquired in Senna Lodigiana, Italy in the Lodi plain along a terrace of the Po river and were already presented in Zimmermann et al. (2019). For more information about the test site, the reader is referred to Inzoli (2016). Fig. 2 shows the geometry of the survey layout. The survey line has 30 electrodes with 1 m spacing. The 30 electrodes were connected to the measurement system using 30 individual shielded coaxial cables, which were arranged in a fan shape. The system is located at 5 m distance from the survey line. Since the system measures the potential at all remaining electrodes, only the dipoles used for current injection need to be specified a priori. In this study, a circulating scheme was adopted for current injection (Xu & Noel 1993) where 16 electrodes were skipped between the two electrodes for each injection (i.e. current injections at 01–18, 18–05, 05–22, …, 14–01), which was repeated until electrode 01 was reached again. In the end, there were 30 current injections and 11 340 four-pole electrode configurations that can potentially be obtained in post-processing. The measurements were conducted at 16 frequencies logarithmically distributed over a broad frequency range from 0.1 Hz to 10 kHz (0.10, 0.23, 0.52, 1.18, 2.68, 6.10, 13.9, 31.3, 71.4, 164, 366, 850, 1950, 4400 and 10 000 Hz). The analysis in this study is focused on several selected frequencies up to 1950 Hz, because the real part of the measured impedance was more affected by the leakage current for frequencies above 1950 Hz. Some general data filters were applied before further analysis. First, modelling accuracy was evaluated using a homogeneous half-space model. The modelled responses for a homogeneous model are compared with the analytic solution and measurement configurations with modelling error larger than 1 per cent were removed (more details about the forward modelling approach can be found in Section 2.4). Then, configurations with large geometric factors were also excluded. In particular, a threshold geometric factor of 1000 was applied for the so-called alpha and beta type configurations and a more strict threshold value of 100 was adopted for gamma type configurations. Measurements with a negative real part of the apparent electrical conductivity were also removed.

2.4 Forward modelling

The forward problem for the EIT method involves the solution of the Poisson equation (Weller et al. 1996):

where |${{\boldsymbol{\sigma }}}^*$| is the complex conductivity, |${\boldsymbol{j}}$| is the current density and |${{\boldsymbol{u}}}^*$| is the complex potential. Eq. (5) can be solved numerically with proper boundary conditions, which results in a discretized form of eq. (5) that can be written as

where |${{\boldsymbol{Y}}}^*$| is the complex admittance matrix, |${{\boldsymbol{U}}}^*$| is the complex potential vector to be solved and |${{\boldsymbol{I}}}_{\mathrm{s}}$| is the applied symmetric unit current. Considering the real and imaginary components of |${{\boldsymbol{Y}}}^*$| and |${{\boldsymbol{U}}}^*$|⁠, eq. (6) can also be written as

where i is the imaginary unit, |${\boldsymbol{Y}}_{\mathrm{s}}^{\boldsymbol{^{\prime}}}$| and |${\boldsymbol{Y}}_{\mathrm{s}}^{{\boldsymbol{^{\prime\prime}}}}$| represent the real and imaginary parts of the electrical conductivity distribution of the subsurface, |${\boldsymbol{U}}_{\boldsymbol{\ }}^{\boldsymbol{^{\prime}}}$| is the real part of the electrical potential and |${\boldsymbol{U}}_{\mathrm{s}}^{{\boldsymbol{^{\prime\prime}}}}$| is the imaginary part of the electrical potential. Since the applied symmetric unit current |${{\boldsymbol{I}}}_{\mathrm{s}}$| is real-valued, the imaginary terms in the left-hand side of eq. (7) should cancel each other, and the following relations can be obtained:

and

The second term |${\boldsymbol{Y}}_{\mathrm{s}}^{{\boldsymbol{^{\prime\prime}}}}{\boldsymbol{U}}_{\mathrm{s}}^{{\boldsymbol{^{\prime\prime}}}}$| in the left-hand side of eq. (8) is negligible for near-surface applications with small phase angles below 200 mrad (Johnson & Thomle 2018), which means that the real part of the electrical potential field mainly depends on the real part of the complex conductivity distribution and is largely independent of the imaginary part of the subsurface electrical conductivity. Eq. (9) illustrates that the imaginary part of the potential field (⁠|${\boldsymbol{U}}_{\mathrm{s}}^{{\boldsymbol{^{\prime\prime}}}}$|⁠) is determined by the real part of the potential field and the complex electrical conductivity distribution.

2.5 Decoupling of capacitive coupling

Zimmermann et al. (2019) proposed to consider both the capacitances and leakage currents in the forward model used for inversion to remove the effect of capacitive coupling. In particular, the total estimated capacitance between the cable shield and the ground was distributed along the known positions of the cables in the 3-D forward model. In addition, the leakage current can be integrated in the modelling by using the two measured excitation currents |${I}_1$| and |${I}_2$| for the current term instead of the unit symmetric current. Ignoring the influence of capacitive coupling on the measured real part of the complex impedance, the real part of the current term can be simply considered as the symmetric part of the current and the imaginary part represents the leakage current in the modelling. Taking the capacitances and leakage currents into account, eq. (7) can then be extended as

where |${\boldsymbol{Y}}_{\mathrm{c}}^{{\boldsymbol{^{\prime\prime}}}}$| is a diagonal matrix containing information on the distributed capacitances and |${\boldsymbol{U}}_{\mathrm{c}}^{{\boldsymbol{^{\prime\prime}}}}$| is the imaginary part of the potential distribution induced by capacitive coupling. Since |${{\boldsymbol{I}}}_{\mathrm{s}}$| is real-valued, the imaginary terms in the left-hand side of eq. (10) should be equal to the leakage current:

If it is assumed that the resulting real and imaginary parts of complex voltages associated with subsurfaces (⁠|${\boldsymbol{U}}_{\boldsymbol{\ }}^{\boldsymbol{^{\prime}}}$| and |${\boldsymbol{U}}_{\mathrm{s}}^{{\boldsymbol{^{\prime\prime}}}}$|⁠) are not affected by capacitive coupling, these two terms in eq. (9) will be identical to those in eq. (11) and eq. (9) can be used to simplify eq. (11) to

Eq. (12) clearly shows that the imaginary part of the electrical potential due to capacitive coupling (⁠|${\boldsymbol{U}}_{\mathrm{c}}^{{\boldsymbol{^{\prime\prime}}}}$|⁠) depends on the real part of the electrical conductivity, the resulting real part of the electrical potential distribution, the capacitances and the leakage currents. Therefore, |${\boldsymbol{U}}_{\mathrm{c}}^{{\boldsymbol{^{\prime\prime}}}}$| can be obtained by solving

without considering the imaginary part of the electrical conductivity distribution. Validation of the proposed method was presented in the  Appendix. It was shown that the results obtained by the approximated method using eq. (13) corresponds well with the true values obtained using eq. (10).

To ensure that all leakage currents flow through the distributed capacitances only, a Neumann (no-flow) boundary condition must be applied to all the surrounding boundaries. To avoid boundary effects, the no-flow boundaries need to be placed sufficiently far away from the survey area of interest. Nine additional nodes were added between two neighbouring electrodes to refine the modelling grid around the electrodes and the element size gradually increased with increasing distance from the electrodes.

2.6 Capacitive coupling strength

Assuming that all other sources of measurement error are negligible and that only capacitive coupling needs to be considered, the measured complex impedance |${Z}^{\boldsymbol{*}}$| can be written as

where |$Z^{\prime}$| is the real part of the impedance, |$Z^{\prime\prime}$| is the imaginary part of the impedance, |$Z_0^{^{\prime\prime}}$| is the imaginary part of the impedance due to polarization of the subsurface, and |$Z_{\mathrm{c}}^{^{\prime\prime}}$| is the imaginary part of the impedance induced by capacitive coupling which can be obtained using the method described in Section 2.5. Corrections for capacitive coupling in sEIT measurements can be simply done by extracting the term |$iZ_{\mathrm{c}}^{^{\prime\prime}}$| from the measurements (i.e. |${Z}^{\boldsymbol{*}} - iZ_{\mathrm{c}}^{^{\prime\prime}}$|⁠). The CCS (in percentage) for a measurement with a given electrode configuration can now be defined as

A small CCS value indicates that the contribution of capacitive coupling to the measured impedance is low, which is desirable for sEIT measurements. However, it is difficult to infer which type of configuration is more sensitive to capacitive coupling a priori because the influence of capacitive coupling depends on the distribution of the real part of the electrical conductivity and the resulting potential distribution, both of which are heterogeneous and unknown a priori. Therefore, CCS values for different electrode configurations are not general and vary for each survey. They should be analysed on a case-by-case basis using the actual on-site sEIT measurements.

Fig. 3 illustrates the workflow to calculate the CCS. Based on the decoupling approach described above, the distribution of the real part of the electrical conductivity must be obtained first in order to calculate |$Z_{\mathrm{c}}^{^{\prime\prime}}$| from the finite element method (FEM) modelling. For this, the real part of the impedances is therefore inverted to obtain the 2-D electrical conductivity distribution. In the second step, the 2-D distribution is mapped to a 3-D model to obtain a rough estimate of the electrical conductivity distribution below the fan-shaped layout, which is required for the forward modelling with distributed capacitances. It should be noted that the total capacitance was distributed homogeneously along the cable layout in the 3-D model and the corresponding nodes of the capacitances were defined in the 3-D model with 0.3 m spacing for generating the grid. In a third step, 3-D FEM modelling was performed by solving eq. (12) to obtain |$Z_{\mathrm{c}}^{^{\prime\prime}}$|⁠. In the final step, the CCS values of all electrode configurations were calculated using eq. (14). It should be noted that the measured data can also be corrected for capacitive coupling by removing |$Z_{\mathrm{c}}^{^{\prime\prime}}$| from the measurements.

2.7 Inversion of sEIT data

Although inversion algorithms to consider the spectral nature of sEIT measurements are available (Loke et al. 2006; Günther & Martin 2016), the inversion of sEIT data is usually done separately for each individual frequency because a simultaneous inversion of all frequencies requires a large computer memory. The inversion of single frequency complex EIT measurements can be carried out using different strategies (Wang et al. 2023). In the case of small phase angles, the inversion can be conducted efficiently in the real number domain by ignoring the cross-sensitivity, which is sometimes referred to as a two-step real-valued inversion strategy (Martin & Günther 2013; Johnson & Thomle 2018; Wang et al. 2023) based on a Gauss–Newton approach. In the implementation of this inversion strategy used here, the first step is the well-known ERT inversion problem which considers the logarithm of the real part of the electrical conductivity as the model parameter and the logarithm of the measured real part of the transfer impedances as data. Given the Cauchy–Riemann condition (Kwok 2010) and ignoring the cross-sensitivity, the second step uses the final Jacobian from the first step and linearly inverts the imaginary part of the impedances to obtain the imaginary part of the electrical conductivity distribution. A custom-made Matlab program (Wang et al. 2023) extended from Zimmermann (2011) was used to invert the complex EIT measurements in this study. A smoothness regularization was applied in the inversion and the regularization strength was determined using the approximated L-curve method (Li & Oldenburg 1999; Günther 2004) and the damping factor was fixed through the inversion.

3. RESULTS AND DISCUSSION

3.1 Total capacitance and leakage current

Fig. 4(a) shows the relationship between the mean value of the recorded voltages |${U}_{\mathrm{p}}$| and the leakage currents |${I}_{\mathrm{L}}$| for all excitations at 1950 Hz, and a nearly perfect linear relation was obtained. The slope of this relationship reflects the left term in eq. (4), and is thus determined by the angular frequency multiplied with the total capacitance. This results in an estimated total capacitance of 10.67 nF using the actual EIT measurements, which is in good agreement with the estimated value of 10.50 nF obtained from calibration measurements in Zimmermann et al. (2019). This new approach was also tested with measurements at other frequencies, and the estimated total capacitances are presented in Fig. 4(b). The estimated total capacitance decreased with increasing frequency, which is attributed to a decrease in the dielectric permittivity of the cable insulation. A similar decrease was also observed in Zhao et al. (2013) for the measured capacitance of multicore cable in the frequency range from 10 Hz to 10 kHz. It should be noted that the estimated total capacitance using measurements at lower frequencies is more uncertain because the leakage currents at low frequencies are much smaller. The estimates at higher frequencies where capacitive coupling is expected to affect the measurement accuracy have comparably small uncertainty and are thus more reliable. These results show that it is feasible to estimate the total capacitance from the EIT measurements directly. In the following, the estimated value of 10.67 nF at 1950 Hz was used for the total capacitance in the FEM modelling.

An assumption in the proposed approach (eq. 13) is that the real part of the excitation current can represent the symmetric current. To evaluate this assumption, the real parts of the measured excitation currents I₁ and –I₂ are shown in Fig. 5(a). It can be seen that the measured excitation currents match very well. Fig. 5(b) shows the relationship between the real and imaginary parts of the normalized leakage current. Again, a strong relationship is observed, which shows that the loss in the real part of the excitation currents correlates well with the imaginary part of the normalized leakage current. This indicates that the filtering of large normalized leakage currents will remove measurements with a large loss in both the real part and phase shift of the excitation currents I₁ and –I₂. Moreover, we expect that the real part of the normalized leakage current is smaller than the imaginary part of the normalized leakage current, which means that the capacitance dominates the leakage of currents and there is no additional grounding connection in the measurement circuit. In the case that the real part of normalized leakage current is much larger than the imaginary part, outliners will be presented in both plots in Fig. 5. Although it is not the case in this study, it should be noted that measurements associated with such outliners should be discarded from further analysis.

3.2 Evaluation of CCS

Measurements at 1950 Hz were used for the analysis of the CCS. For the data acquisition strategy used here, there are 11 340 measurements that can be reconstructed in post-processing. A threshold value of 10 per cent as used in Zimmermann et al. (2019) for the absolute normalized leakage current was first adopted, which results in 8316 measurements (i.e. 22 current excitations) to be considered for the CCS evaluation. Additional data filtering based on modelling error, geometric factor, and negative apparent electrical conductivity values further reduced to number of measurements to 7745. Following the workflow shown in Fig. 3, the distribution of the real part of the electrical conductivity was obtained for these 7745 measurements. The resulting 2-D inversion result is presented in Fig. 6(a), and the relative RMS error is low with 2.67 per cent. The inverted electrical conductivity ranges from 0.1 to 10 mS m⁻¹ and the distribution shows a very resistive region at the right side of the profile with an electrical conductivity below 1 mS m⁻¹. Next, the 2-D inversion result was mapped to a 3-D distribution (Fig. 6b) and the estimated total capacitance of 10.67 nF was distributed homogeneously along the fan-shaped cable layout. By solving eq. (13) for this extended 3-D grid while only considering the real part of the subsurface electrical conductivity, the imaginary part of the impedance |$Z_{\mathrm{c}}^{^{\prime\prime}}$| induced by the capacitances was obtained and used to calculate the CCS values of all 7745 measurements using eq. (15).

Fig. 7(a) shows the percentage of measurements with a CCS value above 50 per cent as a function of the absolute normalized leakage current for each current excitation. For excitations with a normalized leakage current larger than 3 per cent, about 80 per cent or more of the measurements have a CCS higher than 50 per cent. This clearly illustrates that high CCS values are more likely to occur when the normalized leakage current is large. It is also clear from Fig. 7(a) that measurements with small normalized leakage current can nevertheless also have relatively large CCS values above 50 per cent, which is due to the real part of the potential distribution that acts on the capacitances and induces a secondary current source term, that is |${\boldsymbol{Y}}_{\mathrm{c}}^{{\boldsymbol{^{\prime\prime}}}}{\boldsymbol{U}}_{\boldsymbol{\ }}^{\boldsymbol{^{\prime}}}$| in eq. (12). Although the normalized leakage current can be very small, this secondary source term applied to the electrical conductivity distribution may also result in a sufficiently high imaginary part of the potential such that the accuracy of the measured imaginary part of the impedance is noticeably affected. Based on these findings, it was decided to use a more strict threshold value of 3 per cent for the normalized leakage current, which resulted in a reduction from 7745 to 5288 measurements.

Fig. 7(b) shows the number of electrode configurations below a given CCS threshold value for the remaining 5288 measurements. It can be seen that there are about 4000 measurements with a CCS value below 50 per cent, 1433 measurements with a CCS value below 15 per cent (CCS15 data set), and 463 measurements with a CCS value below 5 per cent (CCS5 data set). It is reasonable to assume that the imaginary part of the impedance induced by capacitive coupling can be neglected for the CCS5 data set, but the number of remaining electrode configurations is low. Although the measured imaginary part of the impedance of the CCS15 data set may need to be corrected for capacitive coupling, this data set is expected to provide better sensitivity coverage than the small CCS5 data set. Based on these considerations, the following data sets were considered for further analysis. First, the uncorrected measurements of the CCS5 and CCS15 data sets were analysed (CCS5 and CCS15). Next, the CCS5 and CCS15 data sets were analysed after applying an additional correction as described in Section 2.6. As a reference, the results are compared to results obtained in Zimmermann et al. (2019) using a complete set of electrode configurations (Xu & Noel 1993). This data set was filtered with a threshold value of 10 per cent for the normalized leakage current as in Zimmermann et al. (2019) and the general filters described above were also applied, which resulted in a final data set with 544 measurements. Data sets without and with corrections (EZ and EZ-C) were considered.

Fig. 8 shows the imaginary part of the impedances of the different data sets over a broad frequency range. The measurements of the EZ data set (Fig. 8a) are noisy and have many positive imaginary part impedances at higher frequencies. After correction for capacitive coupling, the measurements are significantly improved (Fig. 8d) as already reported in Zimmermann et al. (2019). However, measurements at the highest frequency of 1950 Hz still seem to be noisy, and a small amount of measurements with positive imaginary part impedance remained. The CCS values of the EZ data set ranged from 0.15 per cent to about 4000 per cent. About 20 per cent of the measurements have a CCS value larger than 100 per cent and about 40 per cent of the measurements have a CCS value larger than 50 per cent. Again, the high CCS values above 500 per cent were associated with normalized leakage currents larger than 3 per cent. This suggests that many measurements of the EZ data set need considerable correction. For the CCS5 and CCS15 data sets, no apparent improvements could be identified after correction. However, the spectra of the CCS15 data set seem to show more spectral consistency after correction.

3.3 Inversion results of sEIT measurements

Fig. 9 presents the sensitivity coverage of the different configurations based on a homogeneous model. The CCS15 data set has a comparable coverage as the EZ data set, suggesting that the CCS15 data set has a similar information content and capability to resolve subsurface features. The CCS5 data set has less sensitivity at depth due to the limited number of electrode configurations.

The inversion results for the real part of the electrical conductivity distribution using different data sets were all nearly identical to the results presented in Fig. 6(a). No differences in the inversion results were observed for the raw and corrected data. Therefore, only the results for the imaginary part of the electrical conductivity distribution are presented here (Fig. 10) for four selected frequencies. The inversion results at 6.1 and 71.4 Hz showed no apparent differences between uncorrected and corrected data sets for all considered data sets, although the relative RMS error for the EZ data set at 71.4 Hz was slightly higher compared to other results at this frequency. At 367 Hz, the inversion result for the uncorrected EZ data set showed different features compared to the results at lower frequencies and the relative root mean square error (rRMSE) was high. After correction, the relative RMS error was reduced considerably and the inversion result was more consistent with the results at lower frequencies. This indicates that the correction at this frequency was successful for the EZ data set. At the highest considered frequency of 1950 Hz, the inversion result showed anomalies with unrealistically high and low values and the relative RMS error was high. Both the inversion result and the relative RMS error were improved after correction for capacitive coupling. However, the final inversion did not agree well with results at lower frequencies and the relative RMS was still high. This suggests that the correction for capacitive coupling improved the data quality but that it was not possible to reasonably correct all measurements. This is attributed to several high CCS values in the EZ data set.

The inversion results for the CCS5 data set before and after correction showed very similar distributions of the imaginary part of the electrical conductivity. The relative RMS errors after correction showed slight improvement and were comparably low for the results before and after correction. The results for the different frequencies showed good spectral consistency and no anomalies were observed. The inversion results for the CCS15 data set are very similar to the results produced by the CCS5 configuration, even without correction for capacitive coupling. This indicates that the inversion can tolerate a higher CCS threshold, and also confirms that the CCS5 configuration in this study is able to capture the subsurface complex electrical conductivity distribution despite the low amount of electrode configurations. However, especially at higher frequencies, apparent decreases in the relative RMS error after correction for capacitive coupling were observed for the CCS15 data set. Overall, the inversion results for the CCS-filtered data sets are very promising. Without correction for capacitive coupling, reliable and spectrally consistent inversion results were achieved.

4 IMPLICATIONS FOR sEIT MEASUREMENTS WITH OTHER DEVICES

Using filtering based on the newly proposed CCS value, reliable inversion results for the imaginary part of the electrical conductivity up to kHz frequencies were achieved, even in the case of a resistive subsurface. To achieve this, a customized sEIT measurement system was used that allows to measure leakage currents and potentials relative to a common ground point, which are both essential to address capacitive coupling. It is not straightforward to apply the method developed in this work to commercial devices, but some useful insights on tackling capacitive coupling with such devices can nevertheless be obtained. First of all, it is important to identify the possible paths for leakage current as shown by Wang & Slater (2019) for laboratory SIP measurements using a commercially available instrument. The possible paths for leakage current or the types of capacitive coupling can be quite different depending on the design of the measurement instruments, survey set-up and types of cable. When measuring with four electrodes and a single recording channel in a laboratory environment, it is possible to determine the exact current and voltage applied to the sample by additional measurements, for example using the equivalent electric circuit model in Wang & Slater (2019). In the case of multichannel measurement strategies in the field with multiple electrodes, it becomes very challenging to determine the exact amount of current between two arbitrary receivers. In this case, the modelling method and the approach for data filtering proposed in this work are advantageous but require information about the total leakage current and the capacitances. Obtaining this necessary information is not yet possible with commercial devices. Therefore, future development of commercial devices for accurate sEIT measurements should focus on changing the system design such that both leakage currents and the total capacitance can be determined. One strategy to determine the leakage current would be to measure the current with shunt resistors at both ends of the voltage or current sources. It is important to note that shielded cables with a common ground point for all shields are also recommended for accurate sEIT measurements. Without shielded cables with a common ground, it is highly challenging to determine the total capacitance and to model the capacitive effects. From a practical perspective, it would also be beneficial to know the leakage current for on-site determination of the current excitations. As is well known and also shown in this study, measurements with low normalized leakage current are preferred. If measured leakage currents would be available, the following strategy to obtain measurements with small leakage current could be applied to obtain accurate field sEIT measurements with multiple electrodes. In the first step, a measurement at a high frequency (e.g. 1 kHz) with all possible combinations of the excitation pairs could be conducted. After this, the normalized leakage current for all excitations could be obtained and electrode combinations with low normalized leakage current could be considered for actual spectral measurements.

This study dealt with capacitive coupling between the ground and cable shield. The capacitance of the cable itself was eliminated by using the so-called active electrodes where the amplifiers were mounted at the electrodes (Zimmermann et al. 2008). For commercial systems, a central multiplexer is typically used, which is connected to the electrodes using shielded cables. Therefore, the cable capacitance should additionally be considered when a centralized multiplexer is used. With the approach developed in this study, it is also possible to address this type of capacitive coupling by integrating the cable capacitances at the electrodes in the modelling. Considering the same 30-electrode layout with 1 m electrode spacing, the total capacitance will be much higher after taking the cable capacitance into account. In this case, the idea of using an initial test measurement at a high frequency to identify current excitations with low normalized leakage currents would be particularly beneficial. Additional research is warranted to test this approach with the use of a centralized multiplexer and commercial measurement systems.

5 CONCLUSIONS AND OUTLOOK

In this study, a method to evaluate the CCS of sEIT measurements was proposed. The CCS is defined as the ratio of the imaginary part of the impedance induced by capacitive coupling and the imaginary part of the impedance due to polarization of the subsurface. Based on an analysis of the FEM forward modelling considering leakage currents and capacitances, it was shown that the imaginary part of the impedance due to capacitive coupling can be separately obtained without knowing the imaginary part of the electrical conductivity distribution. With the use of a customized sEIT measurement system that can measure the excitation current at both the positive and negative current electrodes, a method was proposed to derive the total capacitance directly from actual sEIT measurement instead of from previously used calibration measurements based on a single-point excitation. It was shown that the total capacitance obtained from the actual sEIT measurement was similar to the value obtained in previous work using single pole measurements.

In the next step, a workflow to calculate the CCS values for each possible measurement was proposed. For this, the measurements with large modelling error, large geometric factor and large normalized leakage current were first filtered out. The remaining measurements after filtering were used to obtain the distribution of the real part of the electrical conductivity through inversion of the measured real part of the impedance. The obtained 2-D distribution was mapped to a 3-D model and the total capacitance was distributed along the cables by integrating the corresponding nodes in the 3-D forward model. The FEM modelling with the real part of the electrical conductivity distribution, integrated capacitances and measured leakage currents was conducted to obtain the imaginary part of the impedance induced by capacitive coupling. Finally, the CCS values for all considered measurements were calculated by assuming that all other sources of errors are negligible. The resulting CCS values showed that high normalized leakage currents are likely to produce measurements with a high CCS, while a low normalized leakage current does not ensure a small CCS value.

After the calculation of the CCS values, measurements with CCS values below 5 per cent (CCS5) and 15 per cent (CCS15) were selected for data inversion to obtain the imaginary part of the electrical conductivity distribution for a range of frequencies. The inversion results were compared with a complete configuration (EZ) that was considered in previous work. An analysis of the measured impedance spectra showed that significant improvements can be observed for the EZ data sets before and after correction for capacitive coupling. However, for CCS-filtered data sets, no apparent changes were observed. The inversion results for the EZ data sets showed that the correction for capacitive coupling was only partly successful, which was attributed to the presence of measurements with high CCS values and large leakage currents for several electrode configurations used in the EZ data set. The inverted imaginary part of the electrical conductivity distributions obtained with the CCS-filtered data sets showed good spectral consistency before and after correction for capacitive coupling for a broad frequency range up to kHz. It was found that the filtering method based on CCS is more capable in tackling capacitive coupling compared to using model-based corrections. Spectrally consistent sEIT results up to kHz were obtained using the newly developed filtering method, which were not achieved in previous work using model-based correction. In case of limited measurements with low CCS, a combination of filtering and correction could be the option to obtain improved sEIT results.

The results presented in this study showed that it is of great importance to measure both leakage currents and the total capacitance if accurate sEIT measurements are required in the presence of capacitive coupling in resistive environments. This has obvious implications for the future direction of system design for commercial sEIT equipment. Determination of the leakage current would, on the one hand, be useful to model capacitive coupling and calculate the CCS, which can then be used for data filtering as shown in this study. On the other hand, it could also be used for the on-site determination of an optimized current excitation scheme using a test measurement at a high frequency.

ACKNOWLEDGEMENTS

HW would like to thank the China Scholarship Council for funding this research. We acknowledge the reviewers for insightful comments that helped to improve the manuscript significantly.

AUTHOR CONTRIBUTIONS

Haoran Wang: Conceptualization, Methodology, Formal analysis, Writing – original draft. Johan Alexander Huisman: Conceptualization, Methodology, Formal analysis, Writing – review & editing. Egon Zimmermann: Conceptualization, Methodology, Formal analysis, Writing – review & editing. Harry Vereecken: Supervision

DATA AVAILABILITY

The data supporting the findings of this study are available upon reasonable request.

References

APPENDIX

In Section 2.5, we presented an approximation to calculate the error in the imaginary part of the impedance due to capacitive coupling. The idea is to carry out the forward modelling considering capacitances and leakage current with the real part of the electrical conductivity distribution only. Here, we use the inverted complex electrical conductivity model obtained by the CCS15 configurations at 1950 Hz to evaluate the validity of this approximation. Three sets of simulations were considered to obtain the different impedances. The first set of simulations considers the complex electrical conductivity distribution and a symmetric current to obtain the noise-free complex impedances. The resulting imaginary part of the impedance is referred to as |$Z_0^{^{\prime\prime}}$|⁠. The second set of simulations considers the complex electrical conductivity distribution, the capacitances, and both leakage and symmetric currents. This scenario is considered as the case where the sEIT measurements are affected by capacitive coupling. After removing the noise-free |$Z_0^{^{\prime\prime}}$| from the imaginary part of impedance obtained by this second set of simulations, the remainder is considered as the true value for the imaginary part of the impedance due to capacitive coupling (⁠|${Z}_{\mathrm{c}1}$|⁠). The third set of simulations considers the real part of the complex conductivity distribution with capacitances and both leakage and symmetric currents. The resulting imaginary part of the impedance obtained in this third set of simulations is the approximated value for the imaginary part of the impedance due to capacitive coupling (⁠|${Z}_{\mathrm{c}2}$|⁠).

Fig. A1(a) shows a comparison of the true imaginary part of the impedance due to capacitive coupling and the approximated values. The simulation results perfectly fall on the 1:1 line, which validates the approximation method used in this study to calculate the imaginary part of impedances due to capacitive coupling. We also examined the relative error between the true and approximated values. More than 95 per cent of the impedances have a relative error below 5 per cent. The remaining high relative errors are all associated with small |${Z}_{\mathrm{c}1}$| as can be seen from Fig. A1(b). Finally, we recalculated the CCS values for the CCS15 configurations using |${Z}_{\mathrm{c}1}$| and the resulting CCS values are still all below 15 per cent, which means that the higher relative errors for those measurements with small |${Z}_{\mathrm{c}1}$| have no significant effect on the CCS values.