# Calibration of near-surface multi-frequency electromagnetic induction data

Cecile Finco, Fayçal Rejiba, Cyril Schamper, Luis Henrique Cavalcante Fraga, Ao Wang

► To cite this version:

Cecile Finco, Fayçal Rejiba, Cyril Schamper, Luis Henrique Cavalcante Fraga, Ao Wang. Calibration of near-surface multi-frequency electromagnetic induction data. Geophysical Prospecting, 2023, 71 (4), pp.765-779. 10.1111/1365-2478.13344. hal-04290098

HAL Id: hal-04290098

https://hal.science/hal-04290098v1

Submitted on 7 Aug 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
ORIGINAL ARTICLE

Geophysical Prospecting

<figure><img src="image_2.png" /></figure>

# Calibration of near-surface multi-frequency electromagnetic induction data

Cécile Finco<fnref n="1" /> | Fayçal Rejiba<fnref n="1" /> | Cyril Schamper<fnref n="2" /> | Luis Henrique Cavalcante Fraga<fnref n="1" /> | Ao Wang<fnref n="1" />

<figure><img src="image_3.png" /></figure>

**Correspondence**

C. Finco, Univ Rouen Normandie, UNICAEN, CNRS, M2C UMR 6143, F-76000 Rouen, France.

Email: cecile.finco@univ-rouen.fr

**Funding information**

European Office of Aerospace Research & Development (EOARD), Grant/Award Number: FA9550-19-1-7041; University of Dayton Research Institute contract RSC18040; US Air Force Prime Contract Number FA8650-18-C-2808

**Abstract**

The frequency-domain electromagnetic induction method is a very efficient and convenient method to investigate a site of interest and for a large range of applications including hydrogeology and archaeology. Besides an easy infield acquisition, the data can be used to estimate the electrical, magnetic and, in certain cases, dielectric properties of the ground. It does, however, require calibration to get reliable quantitative values of these properties. We propose here an original and reliable self-calibration method, specific to multi-frequency measurements in the range of (5–100 kHz)-based only on the consistency between the data at different acquisition frequencies. Contrary to other existing cross-calibration procedures, no additional measurements with other methods are necessary. This self-calibration (quadrature and in-phase components of the electromagnetic induction signal) procedure is tested at a test site located at the centre for studies on risks, the environment, mobility and urban planning Normandie-Centre facilities (Rouen, France). The developed quadrature self-calibration has been successfully compared with the usual electrical resistivity tomography – electromagnetic induction cross-calibration procedure. The main results reveal the potential of systematic assessment of all electromagnetic parameters from multi-frequency and multi-configuration electromagnetic induction data: The electrical resistivity and the magnetic susceptibility are estimated from the low-frequency range, whereas the highest part of the frequency range is used for the dielectric permittivity.

**KEYWORDS**

calibration, electromagnetics, multi-parameter Interpretation, multi-frequency measurements

## INTRODUCTION

The electromagnetic induction (EMI) method, also called frequency-domain electromagnetic, is used with Slingram geometry devices for many applications where mapping the electrical resistivity of the subsurface is of interest, like soil studies, hydrogeology or archaeology (Cameron et al., 1981; Christiansen et al., 2016; Corwin & Lesch, 2005; De Smedt et al., 2014; Delefortrie et al., 2019; Donati et al., 2017; Falco et al., 2021; Rejiba et al., 2018).

The EMI method does not require galvanic contact with the ground like direct current resistivity methods and it is therefore much more efficient to cover large areas in the same amount of time. It is then possible to use the acquired EMI maps to locate appropriately further prospections – geophysical or not – for example at zones with high contrasts in the EMI measurements. Another advantage of the EMI method, particularly used in archaeology studies, is its sensitivity to additional subsurface properties that complete the electrical resistivity mapping: the complex magnetic susceptibility

<figure><img src="image_4.png" /></figure>

<footnote n="1">¹Univ Rouen Normandie, UNICAEN, CNRS, M2C UMR 6143, Rouen, France</footnote>

<footnote n="2">²Sorbonne Université, CNRS, EPHE, UMR 7619 METIS, Paris, France</footnote>
$\kappa = \kappa_{ph} - i\kappa_q$ comprising the in-phase magnetic susceptibility ($\kappa_{ph}$) and the quadrature magnetic viscosity ($\kappa_q$); and the complex dielectric permittivity $\varepsilon = \varepsilon' - i\varepsilon''$ with $i = \sqrt{-1}$ (Benech et al., 2016; McFadden & Scott, 2013; Tabbagh et al., 2021). In the measured EM fields, the contributions of the dielectric losses $\varepsilon''$ cannot be separated from the conductivity response; hereafter, we will only consider the real part of the effective relative dielectric permittivity $\varepsilon_r$ where $\varepsilon' = \varepsilon_0 \varepsilon_r$ (with $\varepsilon_0$ the permittivity in vacuum: $\varepsilon_0 = 8.854 \times 10^{-12} \text{ F m}^{-1}$).

The devices used for EMI measurements are operating at frequencies above a few kHz and below 100 kHz. Two coils (at least) are necessary for the measurement: A transmission coil ($T_x$) to generate a primary magnetic field that diffuses in the ground and a reception coil ($R_x$) where the total magnetic field, including the secondary field, can be measured. The secondary magnetic field is the ground's response, in reaction to the applied primary field. Detailed descriptions of the EMI method can be found in Everett (2013) and Telford et al. (1990).

In the lowest part of the frequency range (below 30 kHz) and for a metric-sized instrument, the quadrature (out-of-phase) component of the measurement is mainly sensitive to the electrical resistivity, whereas the in-phase component (once the primary field is removed) is mainly influenced by the magnetic susceptibility. When increasing either the frequency or the inter-coil separation, the dielectric permittivity contribution can no longer be neglected in the in-phase measurements. It is, namely the case on grounds with unusually large values of dielectric permittivity (Benech et al., 2016) or with measurements acquired at frequencies above 30 kHz. When acquiring the data at a frequency close to 100 kHz, the dielectric permittivity is predominant in the in-phase component and often cannot be neglected in the quadrature component either. Therefore, the use of a multi-frequency EMI device allows an estimation of apparent resistivity and magnetic properties at the lowest frequencies (below 30 kHz) and of apparent resistivity and apparent dielectric permittivity at the highest frequencies (~90–100 kHz) as proposed by Simon et al. (2015, 2019). For even higher frequencies of the middle frequency range (300 kHz–3 MHz), only electrical resistivity and dielectric permittivity can be considered (Kessouri et al., 2016).

The quantitative interpretation of EMI data, however, requires the estimation of accurate values for each electromagnetic parameter, and to this end, a calibration of the data is mandatory. Historically, such calibration is performed either using electrical resistivity soundings – ES (1D) or tomography – ERT (2D) or direct measurements on samples when available (Lavoué et al., 2010; Tabbagh, 1982). Using auger soundings or a conductive metallic sphere as a reference is also an alternative for calibration (Thiesson et al., 2014). The determination of these calibration offsets to correct the posi-

tion of the zeros of the measurement can be complex as the quantity measured in the receiver coil(s), that is the value of the ratio of the secondary magnetic field over the primary magnetic field ($H_s/H_p$), depends on the electromagnetic properties of the ground below the device, but also on its geometry as well as on electronic parameters whose values can drift with time (Thiesson et al., 2014). However, the main problem when cross-calibrating EMI with other geophysical methods is that the spatial sensitivity of each method is different, which adds a bias that is difficult to bypass. In addition, even if the comparison between results acquired with multiple methods is recognized as a good practice, it is not always possible.

Considering a multi-frequency EMI device with a unique $T_x-R_x$ spacing (distance), the different measurements acquired at frequencies below 30 kHz have a similar depth of investigation (DOI). In fact, in this range of frequencies, the DOI is mainly dependent on the distance between the transmitting and receiving coils (McNeill, 1996). Consequently, the apparent resistivity deduced from these measurements should be very similar. This consistency between the measurements will be the basis of the self-calibration of the quadrature component process presented here. This calibration method has been, as a first step, assessed on a test site build in the context of the Odyssey project aiming to assess the ground compaction level using geophysical standoff measurements. This test site, of known material and geometry and built at scale 1:1, is ideal to develop and test a calibration protocol.

In the next section, the GEM-2 characteristics will be described along with a detailed presentation of the self-calibration process. The geometry and characteristics of the test site will be described and the calibration results and multi-parameter interpretation of the measurements will be exposed and discussed.

# MATERIALS AND METHODOLOGY

## Acquisition device

The geophysical device used to acquire the data presented in this study is the GEM-2 from Geophex Ltd. (Won et al., 1996). The GEM-2 has one receiver coil 1.66 m apart from the transmitter coil (Figure 1) and can be used with the acquisition coils in the horizontal coplanar (HCP) or the vertical coplanar configurations by simply rotating the device board by 90°. This device has two main specifications compared to other electromagnetic induction (EMI) instruments:

1. A bucking coil ($B_x$) has been added to compensate exactly for the influence of the primary field in the receiver coil.
<figure><img src="image_5.png" /><figcaption>FIGURE 1 Schematic geometry of the GEM-2 device (Geophex Ltd.).</figcaption></figure>

Both receiving coils ($R_x$ and $B_x$) have the same area but the number of turns (woven in the opposite directions) has been divided by 4 in the bucking coil which is located 1.035 m apart from the transmission coil (see Figure 1). The measured quantity is therefore (Won et al., 1996, 2003):

$$ \frac{H_{sRx} - H_{sBx}}{H_p}, \qquad (1) $$

where $H_{sRx}$ is the secondary field in the receiver coil, $H_{sBx}$ is the secondary field in the bucking coil and $H_p$ is the primary field at the $R_x$ location.

The addition of the bucking coil allows a greater depth of investigation (DOI). As an example, above a homogenous half-space with an electrical resistivity of 100 Ωm in the HCP configuration and at an altitude of 0.2 m, the estimated DOI at 20 kHz is around 2.05 m without the bucking coil and 2.45 m with it. The algorithm used in these DOI estimations was proposed by Christiansen and Auken (2012). It consists in over-discretizing a model, originally the inverted model, but can be used with any numerical ground model in general, and applying the forward model with the geometry of interest on it to serve as a reference. The electrical resistivity of each layer is then sequentially modified by 1%. By comparing the forward modelling results to the previously calculated reference, the sensitivity and cumulative sensitivity with depth are calculated. A threshold is then used on the cumulative sensitivity to define the DOI, the authors recommend using threshold values between 0.6 and 1.2 and use themselves 0.8 in all examples. The DOI values previously given are calculated using the 0.8 DOI threshold and the forward model used here is the analytical solution proposed by Schamper et al. (2012).

2. It is possible to acquire data, simultaneously or sequentially, at several frequencies in the range of (5–100 kHz). As mentioned above, in this frequency range, for an instrument whose size is of the order of the meter and

for grounds with common properties (in particular, not extremely conductive like sea water), the DOI is governed by the distance between the transmission and reception loops (McNeill, 1996; Simon et al., 2015).

## Calibration process: offset determination at each frequency

The calibration consists in shifting the data to align the measured instrument zeros with the theoretical zeros that would be obtained during an acquisition in free space. The objective is therefore to determine calibration offsets that will be used to shift the data, two per frequency, one for the quadrature component and one for the in-phase component of the response.

For the quadrature component, the calibration process presented here is based on two main assumptions:

1. The DOI for the low induction number (LIN) apparent electrical resistivity (deduced from the quadrature component of $H_s/H_p$) is stable at all the frequencies used, and the estimated values of electrical resistivity should therefore be similar across the frequency range. Using the same algorithm from Christiansen and Auken (2012), the DOI variations, calculated over a homogeneous half-space at 100 Ωm in the HCP configuration between 5 and 100 kHz, are less than 10 cm with estimated values ranging from 2.44 to 2.37 m.

2. The quadrature offsets must be proportional to the frequency. This hypothesis comes from the data themselves: In LIN approximation, the calibrated $H_s/H_p$ quadrature data should be proportional to the frequency. As the medians of the measured values were proportional to the frequency in all our past experiments, the only possibility to preserve this proportionality property is that the offsets must be proportional to the frequency.

The proposed calibration process is exposed in Figure 2a. The forward model algorithm used here is based on the one developed by Schamper et al. (2012). It allows the consideration of a wide range of acquisition geometries and frequencies, as well as both the magnetic viscosity and the dielectric permittivity of a layered ground, in addition to the electrical resistivity and magnetic susceptibility.

As this calibration is based on the similarity in the apparent electrical resistivity estimated across the frequency range, the data used to determine the calibration offset must not be influenced by any other ground parameter. Therefore, only the low-frequency data (<30 kHz) will be used in the determination of the quadrature calibration offsets as the dielectric permittivity has a noticeable influence on the quadrature at higher frequencies. The amplitude of this effect, illustrated
<figure><img src="image_6.png" /><figcaption>**FIGURE 2** Self-calibration process (a) for the quadrature part of the multi-frequency electromagnetic induction (EMI) measurements and (b) for the in-phase part. LF, low frequency.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**FIGURE 3** Forward modelling illustrating the influence of the relative dielectric permittivity in the quadrature part of the measurement with the GEM-2 geometry in the horizontal coplanar (HCP) configuration over a homogeneous half-space within an electrical resistivity of (a) 500 Ωm and (b) 100 Ωm. The acquisition frequencies range from 5 to 100 kHz. ppt, part per thousand.</figcaption></figure>

with forward modelling in Figure 3, is more important if the electrical resistivity is higher as the induction response from the ground diminishes.

However, previous studies have shown that the magnetic viscosity has an influence on the quadrature part of the measurements (Dabas et al., 1992; Mullins & Tite, 1973) and could in a similar fashion distort the calibration results. Consequently, the part of the response originating from the magnetic viscosity should be removed before the calibration process. This will be possible thanks to the laboratory experiments made by Dabas and Skinner (1993) and Mullins and Tite (1973) on multiple soil samples showing that the magnetic viscosity value is frequency independent. Using the difference between the data measured at two frequencies will therefore eliminate the influence of the magnetic viscosity leaving only that of the electrical resistivity (Simon et al., 2015).

The data used to determine the quadrature calibration offsets will therefore be the differences for every pair of frequencies below 30 kHz. Another step of this calibration process is to create an abacus by using the forward model with the geometry of interest, in the present case, the geometry of the GEM-2. We consider the acquisition frequencies used in the field acquisition (in the range (5–100 kHz)) and a range of plausible values of electrical resistivity, here (10–1000 Ωm). This abacus is then used for all conversions from $H_s/H_p$ raw data to apparent electrical resistivity using the full forward response (Schamper et al., 2012) rather than the LIN approximation (McNeill, 1980). The quadrature calibration is an iterative optimisation process. Because the offsets values will be proportional to the frequency, only the offset value for one frequency (here the lowest but it can be any of the frequencies used below 30 kHz) is determined by optimization and used to deduce the offsets of the other frequencies. So, a
first calibration offset is arbitrarily chosen as a starting value, for example 0, and used to calculate the offsets of other frequencies. These first offsets are used to shift the data before using the abacus to convert it to an apparent electrical resistivity. The offset value is then iteratively modified to minimize the difference in apparent electrical resistivity calculated for each pair of frequencies. This process is repeated until the difference in apparent resistivities for each pair of frequencies (<30 kHz) reaches a minimum. Once the calibration offsets are determined, the data at all frequencies (and not the difference for each pair of frequencies) is shifted and will later be used to generate apparent resistivity maps. It is then possible to estimate a value of apparent magnetic viscosity from the difference for each pair of frequencies of the quadrature component (Simon et al., 2015).

The calibration of the in-phase part of the measurement comes in a second time with measurements at a fixed point (base) location carried out at least at two different heights above the ground (Figure 2b), using three heights would provide additional information with different sensitivity for each parameter and should be considered when possible. For each frequency, the difference between the data at both heights is calculated, thus removing the influence of the calibration offset as it is the same at both heights. The apparent electrical resistivity estimated from the previously calibrated quadrature part of the measurement is used to establish a ground model where the only unknown parameter is, depending on the frequency, either the magnetic susceptibility (for frequencies below 30 kHz) or the dielectric permittivity (for frequencies above 80 kHz). At this point, the intermediate frequencies (30–80 kHz) are not used as the magnetic susceptibility and dielectric permittivity are not considered simultaneously. An in-phase abacus with the GEM-2 geometry is then used to determine the unknown parameters so that the ground model at the base location is complete. This model is used to run a simulation at one of the heights (usually the lowest) and the calibration offset is determined from the difference between this simulated data and the field measurement at the same height.

## Determination of electromagnetic apparent properties

Once the data is ‘offset corrected’, the apparent electrical resistivity is estimated by confronting the difference of low-frequency quadrature data (without the potential contribution of magnetic viscosity) with the corresponding abacus. After that, the response generated by electrical resistivity modelling is calculated at all positions for all the frequencies and subtracted from the data both for the quadrature and the in-phase components.

For the frequencies below 30 kHz, the remaining part of the response for the quadrature part is generated by the magnetic viscosity as the influence of the dielectric permittivity is limited in this frequency range, as illustrated through forward modelling in Figure 3. These residuals (in ppm in the measured data) are then confronted with a previously calculated abacus to estimate the apparent magnetic viscosity at each location. Simon et al. (2015) note, however, through numerical calculations, that there is a slight dependence on the electrical resistivity values in the magnetic viscosity response. This dependence is more important as the frequency increases and for low values of magnetic viscosity. The values of magnetic viscosity should therefore be estimated using the lowest frequency available in the measurements even if this estimation would be unreliable for low viscosity values, below $2.10^{-5}$ SI. Intermediary or high values of apparent magnetic viscosity can, however, be determined with the procedure proposed.

The contribution of the estimated apparent resistivity to the in-phase component is then estimated through forward modelling and removed. The residual is then used similarly to determine the magnetic susceptibility and the dielectric permittivity from the low and high-frequency data, respectively. The relative influence of the magnetic susceptibility and relative dielectric permittivity is illustrated through numerical modelling in Figure 4: Below 30 kHz, the influence of the dielectric permittivity is negligible compared to that of the magnetic susceptibility. The abacus of the in-phase response is once again used to estimate the apparent magnetic susceptibility for the frequencies below 30 kHz. Once the apparent electrical resistivity, magnetic susceptibility and magnetic viscosity are estimated, the dielectric permittivity can be estimated from the highest frequency measurements. From the low-frequency magnetic susceptibility estimation and its known variability with frequency (Dabas et al., 1992), the response generated by the magnetic susceptibility is calculated for all the frequencies and subtracted from the field data as well. The apparent relative dielectric permittivity is estimated from an abacus for the highest frequencies.

## Test site description

This calibration process will then be validated on a test site located in the CEREMA (Centre for Studies on Risks, the Environment, Mobility and Urban Planning) facilities near Rouen, Normandie, France at position 358020E, 5472975N (the coordinate system is WGS84 UTM31N in meter). It was built in December 2019, specifically for the Odyssey project whose objective is to establish a relation between geotechnical and geophysical measurements in order to estimate the
<figure><img src="image_8.png" /><figcaption>**FIGURE 4** Forward electromagnetic induction (EMI) modelling illustrating the influence of the relative dielectric permittivity and magnetic susceptibility of a homogeneous half-space at acquisition frequencies varying from 5 to 100 kHz. The electrical resistivity is fixed at 100 Ωm, and the horizontal coplanar (HCP) configuration was used with the GEM-2 geometry. ppt, part per thousand.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**FIGURE 5** Geometry of the test site located near Rouen (Normandy, France). (a) Top view; (b) lateral view. The compaction level is different in each area, decreasing from A1 to A3. The grey areas are dead zones (no compaction).</figcaption></figure>

soil bearing capacity for a potential use in the assessment of unimproved landing zones. The material chosen to build this test site is a clayey-sandy silt of type A1 (French standard NF P11-300). The site is 29 m long and 5 m wide with a thickness of 75 cm (Figure 5) for the compacted silt layer.

The geometry of the test site was chosen so that it is possible to carry out measurements with limited influence from the edges and a sufficient thickness to integrate the response of the chosen material when making measurements with the existing commercial geophysical apparatus we wanted to test.
The site is divided into three 7 m long zones (A1, A2 and A3) with 4 m dead zones in between. Each zone was built to have a differing compaction level, decreasing from A1 to A3. The silt used to build the test site has an electrical resistivity of the order of 60 Ωm. X-ray diffraction analysis indicates a high proportion of quartz and feldspar. A few per cent of clays are also present, mostly kaolinite and illite.

Series of geotechnical tests, both in situ and on samples, as well as geophysical measurements have been carried out on the test site (Rejiba et al., 2020). Electrical, EMI and seismic methods as well as dynamic cone penetration and deflection tests were all applied in situ. Samples were also taken from each of the 3 zones at different depths and used for particle size analysis, water content measurements, bender elements tests to estimate shear wave velocity and methylene blue tests for clay content estimations. Time-domain reflectometry measurements were also made on coring samples to estimate the dielectric permittivity.

The geometry and properties of this test site are therefore very well-known making it ideal to develop and test a calibration protocol for multi-frequency EMI data.

# RESULTS ON THE ROUEN TEST SITE

## Data processing

This calibration process has been tested on data acquired on the site with the GEM-2. The frequencies used in this case were 10,425, 15,625, 31,225, 44,825 and 93,025 Hz. The histograms of the raw data in quadrature at all the frequencies in the horizontal coplanar (HCP) (Figure 6a) and vertical coplanar (VCP) configuration (Figure 6b) show a similar range of variations for each frequency but vastly different values.

Fixing the first decile to 0 is the first necessary step to reduce the offset differences between the different frequencies. After this first shift of the raw data and to verify our hypothesis of proportionality of data to frequency, the data median (Figure 7) was fitted with a linear regression against the acquisition frequencies and the coefficient of determination ($R^2$) obtained was 0.999 for the HCP configuration and 0.995 for the VCP. In both cases, the measured data is proportional to the frequency which agrees with the hypothesis previously made. This refocus of the data is done on both the quadrature and in-phase parts of the data. There is, however, a particularity for the in-phase part of the HCP configuration acquired close to the ground (10 cm height), the values are indeed very different for each frequency but follow a trend with the negative data amplitudes getting higher when the frequency increases until the last frequency of 93,025 Hz where the sign changes, whereas the absolute values keep increasing (Figure 8). This sign change results from the increased influence of the dielectric permittivity over the magnetic sus-

ceptibility at the highest frequencies. Forward modelling with a homogenous ground with an electrical resistivity of 100 Ωm and a 30,000 Hz frequency shows opposite variations: a negative slope permittivity dependence for a magnetic susceptibility fixed at of $50 \cdot 10^{-5}$ SI, Figure 9a, and, a positive slope magnetic susceptibility dependence for a dielectric permittivity fixed at 1000, Figure 9b. This opposite behaviour, specific to the HCP configuration (Benech et al., 2016), explains the sign change (positive values below 90 kHz and negative for the highest frequency) observed in the data: The magnetic susceptibility is the main parameter causing the low-frequency in-phase response, whereas the dielectric permittivity dominates the high-frequency response. To keep this opposite variations and sign change in the data processing, the last decile is set to 0 for the highest frequency (so most of the data remains negative), whereas the first is set to 0 for the others (and most of the data remain positive).

The low-frequency data used for the calibration of the quadrature part of the measurement and the estimation of the electrical resistivity are 10,425, 15,625 and 31,225 Hz. The self-calibration process (Figure 2a) of the HCP quadrature data gives a calibration offset of 259 ppm for the lowest frequency, 10,425 Hz. Then, by considering the linearity with respect to the frequency, it is used to deduce the offsets for the other frequencies, 388 ppm for 15,625 Hz, 775 ppm for 31,225 Hz, 1113 ppm for 44,825 Hz and 2311 ppm for 93,825 Hz. Once the data is calibrated, the apparent electrical resistivity is deduced by confronting the low-frequency data to the corresponding abacus and the values obtained (Figure 10), in the range of (100–300 Ωm). These values and the extension range are of the order of magnitude which was expected on a site with a 75 cm silt layer over a more resistive substratum, and are much closer than the resistivity values that can be obtained from the uncalibrated data (Figure 10a): between 300 Ωm and almost 1500 Ωm. These uncorrected values are very high, and their range is too wide for this kind of ground, especially when compared to the other resistivity measurements, in particular those directly made on the silt samples (values close to 60 Ωm).

The in-phase part of the data is then calibrated using measurements made at a base location ($x = 2.5$ m, $y = 0$ m) at two heights $h_1 = 0$ m and $h_2 = 2$ m. The calibration offsets obtained for the different frequencies are very close, ranging from 37 to 47 ppm across the frequency range with of course a different sign for 93,025 Hz. The estimations of the apparent magnetic susceptibility at the low frequency of 10,425 Hz and of the apparent dielectric permittivity at 93,025 Hz (Figure 11) show a media with a rather low magnetic susceptibility, and no values (including the resistivity) specific to each of the three zones with different compaction levels.

The same calibration process has been applied to measurements acquired on the Rouen test site with the GEM-2 at the
<figure><img src="image_10.png" /><figcaption>**FIGURE 6** Histograms of raw quadrature data acquired on the Rouen test site in the (a) horizontal coplanar (HCP) and (b) vertical coplanar (VCP) configurations for each frequency with the corresponding fitted normal distribution.</figcaption></figure>

same frequencies but in the VCP configuration. The apparent electrical resistivity, magnetic viscosity and relative dielectric permittivity are presented in Figure 12. The apparent electrical resistivity map is particularly interesting as the more resistive area starting from around $y = 22$ m corresponds to the zone A3 which is the less compacted. All other geophysical and geotechnical campaigns on the test site (dynamic cone penetration, electrical resistivity tomography, electrical soundings, seismic and electromagnetic induction) also show a different response from A3 compared to A1 and A2. This is not observed in the HCP maps probably because the depth of investigation (DOI) exceeds 2 m, whereas the test site itself

is only 75 cm deep. The DOI estimated for the VCP configuration with the Christiansen and Auken (2012) method for a two-layer ground with a 75 cm thick conductive layer at 60 Ωm over a resistive layer at 500 Ωm was shallower, close to 1.50 m. The apparent susceptibility and permittivity maps show no significant difference from the HCP ones.

## DISCUSSION

This self-calibration method has the distinct advantages of neither relying on other measurements, like the
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>**FIGURE 7** Median of the quadrature component of the electromagnetic induction (EMI) data from the Rouen test site after shifting the first decile to zero against a frequency. The dashed line represents the result of a linear regression.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**FIGURE 8** Histograms of raw in-phase data acquired on the Rouen test site in the horizontal coplanar (HCP) configuration for each frequency with the corresponding fitted normal distribution.</figcaption></figure>

cross-calibration using electrical resistivity tomography (ERT) profiles or electrical soundings (ES), nor requiring a lot of sample tests in the field. One base point acquisition is necessary to identify and correct any possible time drift. The

use of this test site near Rouen is particularly well-suited to test a calibration methodology as multiple acquisitions with series of both geophysical and geotechnical methods have been carried out and facilitate cross-checking. In particular,
<figure><img src="image_14.png" /></figure>

FIGURE 9 Variation of the in-phase component with the (a) relative dielectric permittivity and (b) the magnetic susceptibility over a homogenous half-space considering the GEM-2 geometry and an electrical resistivity of 100 Ωm at 30,000 Hz.

<figure><img src="image_15.png" /></figure>

FIGURE 10 (a) Apparent resistivity maps for the 3 lowest frequencies in the horizontal coplanar (HCP) configuration before the self-calibration process (note that the colour scale at 10,425 Hz is different) and (b) after the self-calibration process. The hatched areas correspond to the test site dead zones.

a high resolution ERT profile in the centre of the site along the entire length of the profile $x = 2.5$ m (Schlumberger array, 96 electrodes with a spacing of 30 cm). The inversion was carried out using the framework pyGIMLI (Python Geophysical Inversion and Modelling Library; Rücker et al., 2017) with a final RMS error of 1.1% after 4 iterations. This ERT profile has been used to cross-calibrate the data acquired with the GEM-2. This approach considers the ERT inverted profile as ground truth and uses it to run forward models with the acquisition geometry of the electromagnetic induction (EMI) data set, it has been commonly applied in field studies (Cavalcante Fraga et al., 2019; Lavoué et al., 2010; Rejiba et al., 2018). The difference in sensitivity between the electrical and electromagnetic methods is, however, a difficulty when using this cross-calibration approach, as the diffusion of the EM field differs from the distribution of the injected current of ERT. Another issue is that a 'ground truth' never exists in fact; the interpretation of inverted ERT results is in particular limited by the equivalences inherent to the electrical method. However, in the present case, as the
<figure><img src="image_16.png" /><figcaption>**FIGURE 11** Apparent resistivity at 15,625 Hz, magnetic susceptibility at 15,625 Hz and relative permittivity at 93,025 Hz (from left to right) estimated from measurements on the test site with the GEM-2 (Geophex Ltd.) in the horizontal coplanar (HCP) configuration. The hatched areas correspond to the test site dead zones.</figcaption></figure>

test site was specifically built for this project, the thickness of the first layer is known to be 75 cm, limiting the equivalence issue. Using the cross-calibration against ERT to access the results of the self-calibration of the quadrature GEM-2 data is therefore relevant. The comparison of the two calibrations (Figure 13) for the horizontal coplanar and vertical coplanar configurations shows a better agreement at the low frequencies (lowest offset values) than at higher frequencies. This difference can be explained by a lower reliability of the cross-calibration at frequencies above 40 kHz where the dielectric permittivity can have a substantial effect on the quadrature data (Figure 3). In addition, the cross-calibration against the ERT profile can only consider the electrical resistivity and not any other electromagnetic property. This is not an issue for the self-calibration process, as the quadrature offset is estimated using the low frequencies data, whereas the higher frequencies calibration offsets are deduced from those at low frequencies. The agreement of the low frequency offset obtained from both methods (self-calibration and cross-calibration with ERT) is a good indication that the method is reliable. The apparent electrical resistivity maps obtained after self-calibration are in good agreement with data acquired with electrical methods, both ERT and ES

acquired in the centre of each area of the test site (Figure 14). The electrical resistivity of the test site (first layer) of A3 at 71 Ωm is slightly higher than A1 and A2 at 59 and 56 Ωm, respectively.

This self-calibration method, however, relies exclusively on the GEM-2 measurements and the data quality is therefore crucial in getting reliable results. In particular, estimating a measurement error and integrating it into the process would be very suitable. However, the GEM-2 does not provide an estimate of the measurement error within the data.

Magnetic susceptibility measurements on 10 ground samples from the test site have been realized with the Bartington MS2 at two frequencies 476 Hz and 4760 Hz. The measured values are centred around $13.10^{-5}$ SI and range from $12.3 \times 10^{-5}$ to $17 \times 10^{-5}$ SI with a standard deviation of $1.1 \times 10^{-5}$ SI with no observable variation with the frequency. This confirms the results of the EMI measurements indicating low magnetic susceptibility and no viscosity (Figures 11 and 12). The mean magnetic susceptibility estimated from the GEM-2 measurements is indeed very low at $38 \times 10^{-5}$ SI with a standard deviation of $12 \times 10^{-5}$ SI.

Dielectric permittivity measurements were also made on samples from the test site material at different volumet-
<figure><img src="image_17.png" /></figure>

**FIGURE 12** Apparent resistivity at 15,625 Hz, magnetic susceptibility at 15,625 Hz and relative permittivity at 93,025 Hz (from left to right) estimated from measurements on the test site with the GEM-2 (Geophex Ltd.) in the vertical coplanar (VCP) configuration. The hatched areas correspond to the test site dead zones.

<figure><img src="image_18.png" /></figure>

**FIGURE 13** Comparison of the offset obtained from the self-calibration and from the cross-calibration with an electrical resistivity tomography (ERT) for both horizontal coplanar (HCP) and vertical coplanar (VCP) configurations of the GEM-2 device.

ric water contents and over a wide range of frequencies from 100 Hz to 10 MHz using a capacitive cell (Kessouri, 2012). Moreover, the volumetric water content in the test

site is measured over time via a time-domain reflectometry probe SoilVue (Campbell Scientific Inc.) and was at that time around 0.36. Using these two measurements, the relative dielectric permittivity of the test site material measured with the capacitive cell at 100 kHz with a volumetric water content of 0.36 should be around 200 in the range of values mapped in Figure 10. Of course, the material used for the laboratory measurements has been disturbed compared to the in situ EMI acquisition and its microstructure has an important influence on the dielectric permittivity of soils (Mitchell & Soga, 2005).

A crucial part of this self-calibration process is the hypothesis that the quadrature calibration offsets are proportional to the acquisition frequency. This hypothesis came from looking at all the data sets acquired with the GEM-2. The difference between the theoretical zero (measurement with no influence from the ground) and the actual zero measured with the instrument is interpreted as the effect of the EM induction by disturbing objects close to the coils and probably corresponding to different pieces (mechanical, electronic...) of the instrument itself. As data acquired over the same ground should be proportional to the acquisition frequencies, it is imperative to make sure that the raw data themselves appear proportional to the frequency before attempting the presented
<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /><figcaption>FIGURE 14 Electrical resistivity soundings acquired on the Rouen test site along line x = 2.5 m in the centre of each area at y = 3.5 m on A1, y = 14.5 m on A2 and y = 22.5 m on A3.</figcaption></figure>

calibration process. It is easy to check this by visualizing the data distribution after the initial refocus of the data by setting the first (or last) decile to 0. In a different manner, the in-phase calibration offsets appeared to be constant across the frequency range (5–100 kHz). Although this behaviour should be verified with more data sets, it could be explained by a mechanical deformation of the instrument, to which the in-phase part is always very sensitive. This difficulty exists for all EMI instruments and would affect the in-phase data at all the considered frequencies in the same way.

Before the determination of the apparent EM properties, it appeared when looking at the raw data from this data set and other acquired on the same test site that the lowest frequency is always noisier than the others (Figure 10). The measurements at the different frequencies are carried out simultaneously and the lowest frequency is logically the most disturbed, because not only the amplitude of the induced signal is lower, but the number of periods sampled for each measurement is smaller than for higher frequencies. It is also possible to acquire the data at the different frequencies sequentially, but this would require to significantly reduce the acquisition speed.

## CONCLUSION

The calibration of the electromagnetic induction (EMI) measurements is a crucial step in processing the data to get reliable measurements and therefore to be able to compare and complete the data acquired from other methods. Previous methods of calibration, in particular cross-calibration against electrical measurements, whether soundings or (electrical resistivity tomography [ERT]) profiles have two main downsides: (1) the required ‘reference’ measurements with another method which can be – in the case of electrical profile – quite time-consuming and not always possible; (2) the inversion results from these electrical measurements are considered ‘ground truth’ which they are not.

The self-calibration method proposed here for the quadradure part of the multi-frequency EMI measurements relies

only on the EMI data and the hypothesis that the calibration offsets are proportional to the acquisition frequency. The in-phase part was also calibrated with an existing procedure with measurements at different heights at the same position. Once the data is calibrated, it can be used to estimate the apparent EM properties on-site (electrical resistivity, magnetic susceptibility and viscosity and dielectric permittivity).

This procedure has been tested on data acquired with a commercial multi-frequency EMI acquisition device on a test site of known properties. The estimated apparent EM properties proved to agree with all the other measurements previously done in situ as well as on samples in the laboratory. This gives confidence that the data calibration was correct and ready to be evaluated in real site conditions.

## ACKNOWLEDGEMENTS

This work was supported by the European Office of Aerospace Research & Development (EOARD) with the Grant FA9550-19-1-7041, and by the University of Dayton Research Institute contract RSC18040 in support of US Air Force Prime Contract Number FA8650-18-C-2808. The authors would also like to thank Professor Alain Tabbagh for his advice and support during this work.

Open Access funding enabled and organized by Projekt DEAL.

## DATA AVAILABILITY STATEMENT

The data that support the findings of this study is available from the corresponding author upon reasonable request.

## ORCID

Cécile Finco https://orcid.org/0000-0002-2085-3567

## REFERENCES

Benech, C., Lombard, P., Rejiba, F. & Tabbagh, A. (2016) Demonstrating the contribution of dielectric permittivity to the in-phase EMI response of soils: example of an archaeological site in Bahrain. *Near Surface Geophysics*, 14, 337–344.
Cameron, D.R., Read, D.W.L., Jong, E.D. & Oosterveld, M. (1981) Mapping salinity using resistivity and electromagnetic inductive techniques. *Canadian Journal of Soil Science*, **61**, 67–78.

Cavalcante Fraga, L.H., Schamper, C., Noël, C., Guérin, R. & Rejiba, F. (2019) Geometrical characterization of urban fill by integrating the multi-receiver electromagnetic induction method and electrical resistivity tomography: a case study in Poitiers, France. *European Journal of Soil Science*, **70**(5), 1012–1024. https://doi.org/10.1111/ ejss.12806.

Christiansen, A.V. & Auken, E. (2012) A global measure for depth of investigation. *Geophysics*, **77**(4), 171–177. https://doi.org/10.1190/ geo2011-0393.1.

Christiansen, A.V., Pedersen, J.B., Auken, E., Søe, N.E., Holst, M.K. & Kristiansen, S.M. (2016) Improved geoarchaeological mapping with electromagnetic induction instruments from dedicated processing and inversion. *Remote Sensing*, **8**(12), 1022. https://doi.org/10. 3390/rs8121022.

Corwin, D.L. & Lesch, S.M. (2005) Characterizing soil spatial variability with apparent soil electrical conductivity. *Computers and Electronics in Agriculture*, **46**, 103–133. https://doi.org/10.1016/j. compag.2004.11.002.

Dabas, M., Jolivet, A. & Tabbagh, A. (1992) Magnetic susceptibility and viscosity of soils in a weak time varying field. *Geophysical Journal International*, **108**, 101–109. https://doi.org/10.1111/j.1365- 246X.1992.tb00841.x.

Dabas, M. & Skinner, J.R. (1993) Time-domain magnetization of soils (VRM), experimental relationship to quadrature susceptibility. *Geophysics*, **58**(3), 326–333.

De Smedt, P., Van Meirvenne, M., Saey, T., Baldwin, E., Gaffney, C. & Gaffney, V. (2014) Unveiling the prehistoric landscape at Stonehenge through multi-receiver EMI. *Journal of Archaeological Science*, **50**, 16–23. https://doi.org/10.1016/j.jas.2014.06.020.

Delefortrie, S., Hanssens, D., Saey, T., Van De Vijver, E., Smetryns, M., Bobe, C. & De Smedt, P. (2019) Validating land-based FDEM data and derived conductivity maps: assessment of signal calibration, signal attenuation and the impact of heterogeneity. *Journal of Applied Geophysics*, **164**, 179–190. https://doi.org/10.1016/j.jappgeo.2019. 03.001.

Donati, J.C., Sarris, A., Papadopoulos, N., Kalayci, T., Simon, F.X., Manataki, M., Moffat, I. & Cuenca-García, C. (2017) A regional approach to ancient urban studies in Greece through multi-settlement geophysical survey. *Journal of Field Archaeology*, **42**(5), 450–467. https://doi.org/10.1080/00934690.2017.1365565.

Everett, M.E. (2013) *Near-surface applied geophysics*. Cambridge University Press. https://doi.org/10.1017/CBO9781139088435.

Falco, N., Wainwright, H.M., Dafflon, B., Ulrich, C., Soom, F., Peterson, J.E., Brown, J.B., Schaettle, K.B., Williamson, M., Cothren, J.D., Ham, R.G., McEntire, J.A. & Hubbard, S.S. (2021) Influence of soil heterogeneity on soybean plant development and crop yield evaluated using time-series of UAV and ground-based geophysical imagery. *Scientific Reports*, **11**(1), 7046. https://doi.org/10.1038/s41598-021- 86480-z.

Kessouri, P. (2012) Mesure simultanée aux fréquences moyennes et cartographie de la permittivité diélectrique et de la conductivité électronique du sol [PhD thesis]. Université Pierre et Marie Curie – Paris VI. https://tel.archives-ouvertes.fr/tel-00787908/document

Kessouri, P., Flageul, S., Vitale, Q., Buvat, S., Rejiba, F. & Tabbagh, A. (2016) Medium-frequency electromagnetic device to measure electric

conductivity and dielectric permittivity of soils. *Geophysics*, 81(1), 481–491. https://doi.org/10.1190/geo2014-0468.1.

Lavoué, F., van der Kruk, J., Rings, J., André, F., Moghadas, D., Huisman, J.A., Lambot, S., Weiherrnüller, L., Vanderborght, J. & Vereecken, H. (2010) Electromagnetic induction calibration using apparent electrical conductivity modelling based on electrical resistivity tomography. *Near Surface Geophysics*, 8(6), 553–561. https:// doi.org/10.3997/1873-0604.2010037.

McFadden, M. & Scott, W.R. (2013) Broadband soil susceptibility measurements for EMI applications. *Journal of Applied Geophysics*, 90, 119–125. https://doi.org/10.1016/j.jappgeo.2013.01.009.

McNeill, J. (1980) *Electromagnetic terrain conductivity measurement at low induction numbers (Technical Note TN-6; p. 13)*. Mississauga, Ontario: Geonics Ltd.

McNeill, J. (1996) *Why doesn't Geonics limited built a multi-frequency EM-31 or EM-38? (Technical Note TN-30; p. 6)*. Mississauga, Ontario, CANADA: Geonics Ltd.

Mitchell, J.K. & Soga, K. (2005) *Fundamentals of soil behavior*, 3rd edition, New York, NY: John Wiley & Sons.

Mullins, C.E. & Tite, M.S. (1973) Magnetic viscosity, quadrature susceptibility, and frequency dependence of susceptibility in single-domain assemblies of magnetite and maghemite. *Journal of Geophysical Research*, **78**(5), 804–809. https://doi.org/10.1029/ JB078i005p00804.

Rejiba, F., Fauchard, C., Schamper, C., Jacqueline, D., Bodet, L., Fleitz, P., Delage, P., Finco, C. & Willburn, Z. (2020) Relation between multi-configurations electromagnetic induction and geotechnical measurements (CBR/DCP) on a 1:1 standardized test site. In: *33rd Annual Symposium on the Application of Geophysics to Engineering and Environmental Problems*.

Rejiba, F., Schamper, C., Chevalier, A., Deleplancque, B., Hovhannissian, G., Thiesson, J. & Weill, P. (2018) Multiconfiguration electromagnetic induction survey for paleochannel internal structure imaging: a case study in the alluvial plain of the River Seine, France. *Hydrology and Earth System Sciences*, **22**(1), 159–170. https://doi.org/10.5194/hess-22-159-2018.

Rücker, C., Günther, T. & Wagner, F.M. (2017) pyGIMLi: an open-source library for modelling and inversion in geophysics. *Computers & Geosciences*, **109**, 106–123. https://doi.org/10.1016/j.cageo.2017. 07.011.

Schamper, C., Rejiba, F. & Guérin, R. (2012) 1D single-site and laterally constrained inversion of multifrequency and multicomponent ground-based electromagnetic induction data—application to the investigation of a near-surface clayey overburden. *Geophysics*, **77**(4), 19–35. https://doi.org/10.1190/geo2011-0358.1.

Simon, F.X., Sarris, A., Thiesson, J. & Tabbagh, A. (2015) Mapping of quadrature magnetic susceptibility/magnetic viscosity of soils by using multi-frequency EMI. *Journal of Applied Geophysics*, **120**, 36–47. https://doi.org/10.1016/j.jappgeo.2015.06.007.

Simon, F.X., Tabbagh, A., Donati, J.C. & Sarris, A. (2019) Permittivity mapping in the VLF–LF range using a multi-frequency EMI device: first tests in archaeological prospection. *Near Surface Geophysics*, **17**(1), 27–41. https://doi.org/10.1002/nsg.12022.

Tabbagh, A. (1982) L'interprétation des données en prospection électronique avec les appareils SH3 et EM15. *Revue d'Archéométrie*, **6**(1), 1–11. https://doi.org/10.3406/arsci.1982.1188.

Tabbagh, A., Rejiba, F., Finco, C., Schamper, C., Souffaché, B., Camerlynck, C., Thiesson, J., Jougnot, D. & Maineult, A. (2021)
<figure><img src="image_21.png" /></figure>

The case for considering polarization in the interpretation of elec-
trical and electromagnetic measurements in the 3 KHz to 3 MHz
frequency range. *Surveys in Geophysics*, 42(2), 377–397. https://doi.
org/10.1007/s10712-020-09625-1.

Telford, W.M., Geldart, L.P. & Sheriff, R.E. (1990) *Applied geophysics*, 2nd edition, Cambridge: Cambridge University Press.

Thiesson, J., Kessouri, P., Schamper, C. & Tabbagh, A. (2014) Calibration of frequency-domain electromagnetic devices used in near-surface surveying. *Near Surface Geophysics*, 12(4), 481–491. https://doi.org/10.3997/1873-0604.2014012.

Won, I.J., Keiswetter, D.A., Fields, G.R.A. & Sutton, L.C. (1996) GEM-2: a new multifrequency electromagnetic sensor. *Journal of Environmental and Engineering Geophysics*, 1(2), 129–137. https://doi.org/10.4133/JEEG1.2.129.

Won, I.J., Oren, A. & Funak, F. (2003) GEM-2A: a programmable broad-band helicopter-towed electromagnetic sensor. *Geophysics*, 68(6), 1888–1895. https://doi.org/10.1190/1.1635041.

**How to cite this article:** Finco, C., Rejiba, F., Schamper, C., Cavalcante Fraga, L.H. & Wang, A. (2023) Calibration of near-surface multi-frequency electromagnetic induction data. *Geophysical Prospecting*, 71, 765–779.
https://doi.org/10.1111/1365-2478.13344