This paper deals with the issue of monitoring the spatial
distribution of bulk electrical conductivity,

Soil water content and salinity vary in space both vertically and
horizontally. Their distribution depends on management practices and on the
complex nonlinear processes of soil water flow and solute transport,
resulting in variable storages of solutes and water (Coppola et al.,
2015). Monitoring the actual distribution of water and salts in the soil
profile explored by roots is crucial for managing irrigation with saline
water, while still maintaining an acceptable crop yield. For water and salt
monitoring over large areas, there are now non-invasive techniques based on
electromagnetic sensors which allow the bulk electrical conductivity of
soils,

Electromagnetic induction (EMI) sensors provide measurements of the
depth-weighted apparent electrical conductivity (EC

With regard to EC

To obtain reliable vertical distributions of electrical conductivity, the
EC

Unfortunately, like every other physical measurement, frequency-domain
electromagnetic measurements are sensitive to noise that is very hard to
model effectively. Moreover, as discussed, for example, in Lavoué et
al. (2010), Mester et al. (2011) and Von Hebel et al. (2014), an
instrumental shift in conductivity values could be observed due to system
miscalibration and the influence of surrounding conditions such as
temperature, solar radiation, power supply conditions, the presence of the
operator, zero-leveling procedures, cables close to the system and/or the
field setup (see, amongst others, Sudduth et al., 2001; Robinson et al.,
2004; Abdu et al., 2007; Gebbers et al., 2009; Nüsch et al., 2010).
Hence, the EC

When relatively shallow depths have to be explored (1–2 m), direct soil
sampling and ERT can be effectively replaced by TDR observations. TDR
devices are designed to measure the dielectric properties of soils. More
precisely, they measure the apparent electrical permittivity, from which
not only the dielectric constant but also the effective electrical
conductivity can be deduced (e.g., Dalton et al., 1984; Topp et al., 1988;
Weerts et al., 2001; Noborio, 2001; Robinson et al.,
2003; Lin et al., 2007, 2008; Thomsen et al., 2007; Huisman et al., 2008;
Koestel et al., 2008; Bechtold et al., 2010). In general, TDR
measurements might be difficult to use to recover the electrical
conductivity with the desired accuracy. However, in the literature, many
examples are reported in which, within the range 0.002–0.2 S m

Schematic view of the experimental field.

In the present research, we focus on the use of TDR data to absolutely
calibrate the conductivities obtained by inverting the EMI measurements. To
do this, a dataset collected during an experiment carried out along four
transects under different salinity and water content conditions (and
monitored with both EMI and TDR sensors) is utilized. We first tackle the
problem of inferring the soil electrical conductivity distribution from
multi-height EC

The experiment was carried out at the Mediterranean Agronomic Institute of
Bari (MAIB) in south-eastern Italy. The soil was pedologically classified as
Colluvic Regosol, consisting of a silty-loam layer of an average depth of
0.6 m on fractured calcarenite bedrock. The experimental setup (Fig. 1)
consisted of four transects of 30 m length and 2.8 m width, equipped with a
drip irrigation system with five dripper lines placed 0.35 m apart and
characterized by an inter-dripper distance of 0.2 m. The dripper discharge
was 2 L h

The four transects were irrigated with water at two different salinity
levels and with two different water volumes. Transect 1: 100 % of the
irrigation water at 1 dS m

EMI readings – in vertical magnetic dipole configurations – were collected
by using a Geonics EM38 device (Geonics Limited, Ontario, Canada). The EM38
operates at a frequency of 14.6 kHz with a coil spacing of 1 m, and with a
nominal measurement depth of

At the beginning of the measurement campaign, the EMI sensor was “nulled” according to the manufacturer's manual. Readings were taken just after each irrigation application at 1 m step, along the central line of each transect, for an overall total of 26 soundings per transect. Multi-height EM38 readings were acquired at heights of 0.0, 0.2, 0.4 and 0.6 m from the ground. Taking measurements just after irrigation allowed relatively time-stable water contents to be assumed at each site throughout the monitoring phases.

Just after the EM38 measurements, a TDR probe was inserted vertically at the
soil surface in 26 locations, each corresponding to the central point of an
EM38 reading. A Tektronix 1502C cable tester (Tektronix Inc., Baverton, OR, USA)
was used in this study. It enables simultaneous measurement of water content

Nonlinear 1-D forward modeling, which predicts multi-height EMI readings from a loop-loop device, can be obtained by suitable simplification of Maxwell's equations that takes the symmetry of the problem into account. This approach is described in detail in Hendrickx et al. (2002), and is based on a classical approach extensively described in the literature (Wait, 1982; Ward and Hohmann, 1988). The predicted data are functions of the electrical conductivity and the magnetic permeability in a horizontally layered medium.

When the coils of the recording device are vertically oriented with respect
to the ground surface, the reading at height

A least squares data fitting approach leads to the minimization of the function:

Examples of sharp and smooth inversions applied to the dataset 100-6dS. The results are shown together with their corresponding data misfit.

We solve the nonlinear minimization problem by the inversion procedure described in Deidda et al. (2014). The algorithm is based on a damped regularized Gauss–Newton method. The problem is linearized at each iteration by means of a first order Taylor expansion. The use of the exact Jacobian (Deidda et al., 2014) makes the computation faster and more accurate than using a finite difference approximation. The damping parameter is determined in order to ensure both the convergence of the method and the positivity of the solution. The regularized solution to each linear subproblem is computed by the truncated generalized singular value decomposition (TGSVD – Díaz de Alba and Rodriguez, 2016) employing different regularization operators. Besides the classical regularization matrices based on the discretization of the first and second derivatives, in all the cases characterized by sharp interfaces, we tested a nonlinear regularization stabilizer promoting the reconstruction of blocky features and thus to improve the spatial resolution of EMI inversion results. (Zhdanov et al., 2006; Ley-Cooper et al., 2015; Vignoli et al., 2015, 2017). The advantage of this relatively new regularization is that, when appropriate prior knowledge about the medium to reconstruct is available, it can mitigate the smearing and over-smoothing effects of the more standard inversion strategies. This, in turn, can make the calibration of the EMI data against the TDR data more effective. For this reason, in the following, the EMI results used for our assessments are those inferred by means of this sharp inversion. The differences between the “standard” smooth (based on the first derivative) reconstruction and the sharp one are clearly shown in Figs. 2 and 4. In all cases, the inversions are performed with a 100-layer homogeneous discretization, down to 8 m, with fixed interfaces. We opted for such a parameterization to be able to (i) control the inversion results by acting merely on the regularization parameters and (ii) remove the regularization effects possibly originated by the discretization choice (e.g., the number of layers, interfaces locations). In this way, it was possible to use an automatic strategy for the selection of the regularization parameters. In Figs. 2 and 4, the sharp results (upper panels) associated with the cases 100-6dS and 50-6dS are compared against the corresponding smooth inversions (middle panels). Even if the data misfit levels largely match (lower panels in Figs. 2 and 4, but also Figs. 3 and 5), the two inversion strategies produce reconstructions that differ significantly. This is due to the inherent ill-posedness of the EMI inversion. By solely considering the geophysical observations, it is impossible to decide which model is the best. In this research, based on the fact that, just after the irrigation, the effect of the water is supposed to remain localized in the shallowest portion of the soil section, the sharp inversion was found to provide more reliable results. Moreover, to some extent, the general better agreement of the data calculated from the sharp model supports the idea that the electrical property distributions are better inferred via the sharp regularization. In any case, since in this research we calibrate the EMI-derived models (and not the data), the final calibrated result will reflect the assumptions made in the first place, when the EMI data are inverted (specifically, the regularization assumptions).

Comparison of the data fitting associated with the sharp and smooth inversions applied to the dataset 100-6dS (Fig. 2). The calculated data corresponding to the sharp and smooth results are shown together with the observations for each of the four measured channels (heights).

Examples of sharp and smooth inversions applied to the dataset 50-6dS. The results are shown together with their corresponding data misfit.

Comparison of the data fitting associated with the sharp and smooth inversions applied to the dataset 50-6dS (Fig. 4). The calculated data corresponding to the sharp and smooth results are shown together with the observations for each of the four measured channels (heights).

A possible alternative way to still effectively use the TDR data to
calibrate the EMI measurements (and not the associated conductivity model)
could consist of performing the calibration in the data space (and not in
the model space). In the data-space calibration, the measured TDR
conductivity could be used as input model to calculate the EC

It is worth noting that the constant magnetic permeability assumption is not always valid. Inverting for the magnetic permeability is sometimes not only necessary but it can also provide an additional tool for soil characterization (e.g., Beard and Nyquist, 1998; Farquharson et al., 2003; Sasaki et al., 2010; Guillemoteau et al., 2016; Noh et al., 2017; Deidda et al., 2017).

For the sake of clarity, hereafter, the

The Tektronix 1502C can measure the total resistance

The

The agreement between

Scatter plots of the

Because of their relatively small observation volume (

Hence, in order to make the two datasets comparable, the original spatial TDR data series need to be filtered to remove the variation from small-scale heterogeneities (recorded only by the TDR probe). In this way, only the information on a spatial scale equal to or larger than the observation volume of both sensors is preserved.

Thus, a simple filter based on the Fourier transform (FT) is applied to the
TDR series. So, a low-pass frequency filtering is performed on the TDR data
to remove all components related to the small-scale heterogeneities and make
it comparable with the EMI measurements. More specifically, for each
transect, we consider the

Standard deviation of the EMI series (horizontal black line) for the
50-6dS transect at 0.2–0.4 m depth. The squares show the corresponding
standard deviations for the TDR series for different levels of filtering. The
intersection of the EMI line with the TDR curve allows for identifying the optimal
cut-off frequency range (

Hereafter, the original and filtered data will be labeled ORG
and FLT, respectively.
The graphs in the top panels in Fig. 7 compare

The general conclusion is that, in all four transects and for all three
considered depth layers, the

Comparison between

Table 1 shows the MRA coefficients (

Von Hebel et al. (2014) found a similar behavior when comparing their
EMI and ERT datasets. In that case, the EC

Concordance parameters for the four transects for the TDR_ORG and
EMI_ORG data. The table reports the concordance,

Here, we follow a different approach to calibrate the

In nearly all of the graphs in the top panels in Fig. 7, the discrepancies
between

The distance along the

The difference in the slope of the MRA and of the 1 : 1 lines, which stems
from the different variability in

The scatter of the data around the MRA line, which may come from different sensors' noise and the influence of surrounding conditions (e.g., temperature).

The distance of the MRA from the 1 : 1 line is mostly due to the difference
in the observed means. The plot in Fig. 8a compares the means for the two
original series (squares and solid line for TDR, circles and dashed line for EMI).
Figure 8b reports the same comparison on a 1 : 1 plot (triangles and solid
regression line). The mean values confirm the general underestimation of TDR
by the EMI data. However, the trends are evidently similar, which is
reflected in the high correlation between the means of the two series, with
a significantly high

The different slope of the two lines has to be ascribed to the different
variability in the two series. Figure 9a compares the standard deviations
for the two original series (squares and solid line for TDR, circles and dashed line
for EMI). Figure 9b reports the same comparison on a 1 : 1 plot
(triangles and solid regression line). Conceptually, the different variability
in the two series can be related to the different sensor observation volumes
(originated from the different spatial sensitivity of the sensors – Coppola
et al., 2016). For TDR probes, most of the measurement
sensitivity is close to the rods (Ferré et al., 1998b). Conversely,
the spatial resolution of inverted EMI EC

The scatter is consistently reduced by the spatial filtering (as similarly discussed in Von Hebel et al., 2014).

Concordance parameters for the four transects for the TDR_FLT and
EMI_ORG data. The table reports the concordance,

Maps of bulk electrical conductivity for the

The bottom panels in Fig. 7 show the results of the application of the
linear mapping. In particular, they compare the calibrated EMI data (EMI rg)
with the filtered TDR (TDR FLT) measurements. The MRA parameters and the
concordance coefficients in the case of filtered TDR data are reported in
Table 2. Clearly, considering the (calibrated) EMI and (filtered) TDR
standard deviations turns the MRA line to be practically matching the 1 : 1
line, with the coefficient

Figure 10 shows, on the left, the original

As already discussed, the high correlation of the means and the standard
deviations of the two series are central for this procedure to be of
practical interest. In short, the procedure can be summarized as follows:
(i) an area is monitored via EMI survey and a few TDR calibration
measurements are collected concurrently. (ii) The availability of the two
different datasets allows for performing of the regression for the mean and the
standard deviation of the original EMI inversion results and the filtered
TDR data, like those shown in Figs. 8b and 9b. (iv) These statistical
parameters can be promptly used for the calculation of the coefficients

The proposed workflow enables us to translate the original non-calibrated

The objective of the paper is to infer the bulk electrical conductivity
distribution in the root zone from multi-height (potentially non-calibrated)
EMI readings. TDR direct measurements are used as ground-truth

The proposed analysis allows for discussing the physical reasons for the
differences between EMI- and TDR-based electrical conductivity and
developing an approach to calibrate the original

Our approach is based on the MRA coefficients and, hence, on the statistical
parameters (mean and standard deviation) of the two series. Specifically,
the approach looks for a systematic correction of the bias based on
well-defined statistical sources of the discrepancies. A low-pass filtering
has been carried out on the TDR data to obtain a significantly high
correlation between the standard deviations of the two data series. After
that, a simple linear transformation can be applied to the originally
inverted EMI section

On the one hand, the proposed strategy relies on the assumption that TDR direct measurements supply absolutely calibrated observations of the electrical conductivity of the soil and can be effectively used to calibrate the conductivity distributions inferred from EMI data. The availability of EMI calibrated data paves the way to reliable reconstructions of the electrical conductivity distribution over large areas (typical for EMI surveys, but not for TDR campaigns) unaffected by the usual EMI miscalibrations. This, in turn, can result in the possibility of effective time-lapse surveys and/or consistent merging of subsequent surveys.

On the other hand, the proposed statistical workflow for making the TDR measurement comparable with the associated EMI results provides a more sophisticated approach than simple smoothing to upscale the TDR data. Thus, from the opposite perspective, the approach in question can be used to tackle the problems connected with handling the TDR data characterized by excessively high spatial resolution.

The dataset used in this paper and the code for data filtering are available on request to antonio.coppola@unibas.it. The code for smooth inversion is available on request at giuseppe.rodriguez@unica.it. The code for sharp inversion is available on request at gpdeidda@unica.it.

The authors declare that they have no conflict of interest.

This work was supported by the Mediterranean Agronomic Institute of Bari (IAMB – Italy) and by the Doctoral funds of the School of Agricultural, Forestry and Environmental Sciences (University of Basilicata – Italy). The suggestions and the constructive comments from Giorgio Cassiani and the anonymous reviewer have significantly improved the early version of the paper. Edited by: Erwin Zehe Reviewed by: Giorgio Cassiani and one anonymous referee