Subglacial water modulates glacier-bed friction and therefore is of
fundamental importance when characterising the dynamics of ice masses. The
state of subglacial pore water, whether liquid or frozen, is associated with
differences in electrical resistivity that span several orders of magnitude;
hence, liquid water can be inferred from electrical resistivity depth
profiles. Such profiles can be obtained from inversions of transient
(time-domain) electromagnetic (TEM) soundings, but these are often
non-unique. Here, we adapt an existing Bayesian transdimensional algorithm
(Multimodal Layered Transdimensional Inversion – MuLTI) to the inversion of TEM data using independent depth constraints
to provide statistical properties and uncertainty analysis of the
resistivity profile with depth. The method was applied to ground-based TEM
data acquired on the terminus of the Norwegian glacier, Midtdalsbreen, with
depth constraints provided by co-located ground-penetrating radar data. Our
inversion shows that the glacier bed is directly underlain by material of
resistivity 10

Subglacial structure and material properties are one of several important controls on ice flow, both through composition and ice–material interactions. The potential for subglacial sediments to store and pressurise water is a key element in predicting the evolution of ice masses of all sizes, from small mountain glaciers to large polar ice sheets (Christoffersen et al., 2014; Siegert et al., 2018). Currently, the ability to develop accurate ice-flow models is limited by poor understanding of processes acting at the ice–bed interface and the composition of subglacial material. With increased knowledge of subglacial structure and sediment liquid water content, our ability to predict glacier retreat patterns would be greatly increased.

Non-invasive geophysical imaging methods are widely and successfully applied to characterise the internal properties of glacier ice and its immediate basal environment. Such methods (including reflection seismology and ground-penetrating radar) can underperform when characterising material properties beyond the first few metres of the glacier bed (Booth et al., 2012), yet subglacial aquifers, sediment accumulations and permafrost can extend to much greater depths (e.g. Mikucki et al., 2015; Hauck et al., 2001). Further still, inversions of isolated geophysical datasets are unconstrained and non-unique, with many models of the subsurface matching the observed dataset. Joint inversions using multiple independent datasets can constrain the model space, combining depth and resolution sensitivities from multiple datasets. Glaciological surveys often involve the acquisition of multiple geophysical datasets: given the typical absence of ground-truth data, imaging the target with several methods provides a more robust interpretation (e.g. Merz et al., 2016). However, these datasets are seldom combined numerically. In this paper, we provide a mechanism for the constrained inversion of transient electromagnetics (TEM), with depth constraints derived from ground-penetrating radar (GPR), to provide geophysical insight into the structure and water characteristics of the subglacial environment. This method is adapted from a transdimensional Bayesian framework, termed “MuLTI” (Multimodal Layered Transdimensional Inversion) and described in Killingbeck et al. (2018), originally applied to characterise subglacial sediment distribution from seismic surface wave data (Killingbeck et al., 2019). Here, we explore a similar concept for the TEM method.

Time-domain electromagnetic methods use electromagnetic fields to sound the subsurface structure. Here, we use the transient time-domain method (TEM) which indirectly probes the subsurface resistivity structure by measuring transient eddy currents induced by current transmitted through either a grounded wire, coincided loop or offset coil. The method has a depth sensitivity ranging from a few metres to kilometres, depending on the survey parameters used. Of particular relevance here is that electrical resistivity increases by several orders of magnitude when water in pores freezes (Hoekstra and McNeill, 1973), allowing resistivity methods to indicate the liquid water content of subsurface materials. TEM methods have been extensively applied for hydrogeophysical exploration to map groundwater resources (Auken et al., 2003), mapping permafrost on mountainous regions under debris-covered glaciers (Hauck et al., 2001), mapping arctic permafrost in Alaska (Minsley et al., 2012) and more recently mapping deep saline groundwater zones in Antarctica's Taylor Valley (Mikucki et al., 2015). These studies illustrate that characterising the resistivity of the subsurface offers a promising means of distinguishing material type and water content within the subglacial environment.

In common with most geophysical inversions, such resistivity profiles are non-unique: many profiles fit the data within error tolerance, and smoothing is usually employed to recover a single solution. Early inversion techniques for TEM data included non-linear least squares (Barnett, 1984) and an Occam-type regularisation method to obtain a smooth solution (Constable et al., 1987), but these were prone to being trapped in local minima with any large resistivity variations becoming smoothed. More recent inversion methods include laterally and spatially constrained algorithms to regularise the inversion and obtain solutions that agree with the expected geological variations (e.g. Christensen and Tølbøll, 2009; Vignoli et al., 2015; Auken et al., 2015). Yet, these methods do not provide detailed uncertainty analysis of the estimated model parameters and require a fixed number of layers in the model. The maximum depth of investigation (DOI) is generally estimated using methods such as half-space skin depth (Spies, 1989) or the Jacobian sensitivity matrix (Christiansen and Auken, 2012), though these do not consider the non-linear sensitivity of the DOI to conductivity structure. These limitations in uncertainty quantification, fixed model space and DOI estimation can be mitigated by transdimensional Bayesian sampling-based inverse methods. These produce an ensemble of models from which statistical properties of the model parameters, including model dimensions, can be inferred (Mosegaard and Tarantola, 1995; Blatter et al., 2018). The computed posterior probability density function (PDF) provides a robust measure of DOI, highlighting model uncertainty at each depth (Blatter et al., 2018). To further reduce the parameter space and improve vertical resolution, the inversion can be constrained with complementary depth information (e.g. from borehole records or other geophysical sources), which is most useful in the event that any internal layer represents a discontinuity in properties to which all techniques are sensitive.

In this paper, we derive the implementation of “MuLTI-TEM” (Multimodal
Layered Transdimensional Inversion of Transient ElectroMagnetics) and its
use for characterising the subglacial environment. After testing the method
on a synthetic dataset, we analyse a TEM dataset acquired on the Norwegian
glacier, Midtdalsbreen, an outlet of the Hardangerjøkulen ice cap, using
complementary GPR data for constraining the ice thickness. Since the glacier
bed represents a transition in both dielectric constant and electrical
resistivity, the GPR depth constraint can be used directly in the TEM
inversion. Recent results from Killingbeck et al. (2019) interpret the
Midtdalsbreen subsurface from seismic shear-wave velocity (

In TEM surveying, an electromagnetic field is generated by sending a periodic, modified square-wave current through a transmitter coil. When the current is on, a static electromagnetic field is established in the ground. The electromagnetic field is varied by terminating the current abruptly at the first quarter period, being reduced to zero for the second quarter period; the current is then reversed for the third quarter period, before being reduced to zero again for the final quarter period. This switch off induces eddy currents in the subsurface, initiating within the immediate vicinity of the transmitter, then spreading downwards. The eddy currents produce a secondary electromagnetic field which propagates back up through the subsurface, inducing a current in a receiver coil located at some distance from the transmitter. The receiver typically measures the induced secondary electromagnetic field in the transmitter off periods. The response of the subsurface is measured in terms of the decaying amplitude of the secondary electromagnetic field. This is recorded as a function of time, with later responses originating from greater depths. With regards to conductivity of the subsurface, the more conductive the subsurface, the larger the eddy currents and the larger the measured secondary electromagnetic field will be, implying a slower transient decay. By taking repeated measurements, a sounding curve, similar to DC resistivity soundings, is obtained (Geonics, 1994). The measured voltages versus time from the receiver coil are then used to constrain the resistivity profile with depth.

The maximum depth of investigation (DOI) (

MuLTI-TEM is a Bayesian inversion MATLAB code that determines the posterior
distribution of resistivity as a function of depth. It is adapted from the
MuLTI algorithm, developed
for seismic surface wave inversions (see detailed information in Killingbeck et al.,
2018). The data input,

We describe the 1-D variation of resistivity with depth as a piecewise
constant function using Voronoi nuclei (see Killingbeck et al., 2018). Any
available depth constraints separate the resistivity into different depth
layers. In our case, in which constraints are drawn from high-resolution GPR
data, we consider depth constraints to be exact since the accuracy of GPR
depth estimation is at decimetre scale (Killingbeck et al., 2019) compared to
the metre-scale resolution of the TEM. Within each layer, we define a single
confined nucleus; aside from being confined to the given layer, this nucleus
is otherwise unconstrained in depth. The number of confined layers,

The model vector, that describes the resistivity profile, is then

For the choice of prior distribution in transdimensional calculations, it is
worth noting that usually the geophysical properties of the cells (here the
resistivity) and the cell depths are assumed independent, allowing a simple
separated analytic form for the prior distribution (e.g. Bodin and
Sambridge, 2009). This is followed in our simplest geometry with no GPR
constraints, for which the prior distribution on the resistivity is
depth independent and uniform with wide bounds on

Lastly, the likelihood is defined by assuming that the measurements are
normally distributed about values calculated from a forward model of TEM
response (assuming a given resistivity profile) and the estimated standard
deviation

MuLTI-TEM numerically approximates the posterior distribution by creating an ensemble of models, traversing the model space and sampling the models with greater likelihood more often than models with a poor fit to the observed data. Provided the ensemble is sufficiently sampled, the numerically obtained posterior distribution will converge to the true posterior. This is achieved by constructing a Markov chain, each model in the chain being based on the previous model but randomly perturbed, the size of the perturbation being controlled by the user. We thin the Markov chain by using every 100th model when computing the distribution statistics, which suppresses any localised correlations of neighbouring models and speeds up convergence. MuLTI-TEM produces a variety of statistics of the resistivity ensemble including, but not limited to, the mean and mode (the most likely) solution and 95 % credible intervals as an estimate of its uncertainty, thus giving a profile with a quantified uncertainty.

Data acquisition was performed on Midtdalsbreen, a NE-flowing outlet glacier
of the Hardangerjøkulen ice cap in central-southern Norway
(60.59

Survey configuration testing at the intersection of lines B and D.

GPR, seismic and TEM surveys were performed around and over the glacier front (Fig. 1). All methods were acquired at each line highlighted, A–D, in the same field season. Lines B and C are located entirely on the glacier, whereas line A shows no glacier ice. Line D traverses through each of lines A, B and C, and extends beyond the glacier terminus. At the time of acquisition, the subsurface comprised snow (2–4 m thick) overlying a varying thickness (0–25 m) of glacier ice and a substrate of unknown subglacial material. This layered interpretation is based on the interpretation of the GPR dataset, which also suggests that the snow and ice layers show little variation in any of lines A, B and C.

Killingbeck et al. (2019) used MuLTI to jointly interpret the seismic and
GPR data, defining regions of partially frozen sediment and hard bedrock
based on subglacial shear-wave velocities (

TEM data were acquired with a Geonics PROTEM47 system, consisting of a
three-channel digital time-domain receiver, a TEM47 battery-powered transmitter
and a 3-D multi-turn receiver coil. All survey parameter are listed in Table 1.
For cross-glacier lines (A, B and C), the system was moved along the lines
in 4 m intervals; for the longer down-glacier line (D), this was increased to
8 m. Multiple survey configurations were initially tested at the
intersection of lines B and D to determine the optimal survey configuration
for imaging the subglacial environment at Midtdalsbreen. These tests
comprised the following (the maximum DOI of each test is estimated using Eq. 1,
with voltage noise level 1 nV m

37 m

10 m

5 m

it had a fast turn-off time (for imaging the shallow surface);

the raw signal (received voltage) recorded was sufficiently greater than the background noise (Fig. 2d);

it was easily deployed on the glacier and could be moved rapidly between the points of the survey lines (Fig. 3); and

the estimated DOI (ranging from 64 to 400 m depending on the underlying
resistivity) is sufficient for imaging our target depth, subglacial
sediments below

TEM survey parameters.

Synthetic TEM responses from a variety of models representing different possible glacial and subglacial structures of the Midtdalsbreen glacier were input into MuLTI-TEM for validation (models a–e in Fig. 4). Each model included layers of snow, ice and bedrock, with models (b)–(e) also including saturated subglacial sediment. Each layer was populated with representative resistivities from previous TEM studies (Mikucki et al., 2015). Certain models were designed to test particular aspects of the inversion: model (b) tested the maximum DOI using our specified survey design and synthetic, and model (c) tested whether the inversion can resolve a 5 m thin layer.

1-D synthetic block models created to simulate different subsurface scenarios expected at Midtdalsbreen.

Synthetic TEM responses were calculated from the 1-D block models using the
Leroi forward modelling algorithm (Raiche, 2008), then normally distributed
random noise, with a magnitude of 5 % of the signal at each time gate, was
applied to all time gates, a similar noise model to Blatter et al. (2018);
see Fig. 5. The simulated TEM survey configuration assumed a 10 m

Forward modelled responses for 1-D synthetic block models (a)–(e) with 5 % random noise applied. The lines within the circles represent the 5 % error bars.

The inversions were run using resistivity ranges shown in Table 2 using a maximum depth of 80 m, consistent with the estimated maximum DOI for this configuration. To highlight the benefit of additional depth constraints, MuLTI-TEM is run separately for an unconstrained case and a case in which the depths of snow-ice and ice-bed horizons are fixed. In total, 1 million iterations were sufficient for the posterior distribution to converge (a test of 2 million iterations produced the same posterior; some example test results are shown in Fig. A2). Additional figures in Fig. A3 show that the modal model from the depth-constrained inversion fits the data better than that from the non-constrained inversion, with, in general, a lower data misfit. Multiple chains were also tested with 1 million iterations using different initial conditions. For the constrained case, these produced similar posterior distributions with identical interpretation, indicating that only one chain was needed. For the unconstrained case, the posterior distributions differ slightly but are nevertheless qualitatively the same, suggesting that the unconstrained case is not yet converged (Fig. A4). More detailed inversion parameters used are documented in Table A2.

Resistivity parameter boundaries used in MuLTI-TEM for the glacier feasibility study.

Posterior PDFs of the synthetic
resistivity profiles produced from MuLTI-TEM are shown in Fig. 6. These are
shown, within their 95 % credible interval, as coloured contours, where
red indicates the most likely values. Consistent with many previous studies,
both the unconstrained and constrained inversions indicate that the TEM
method can resolve conductive structures much more accurately than resistive
ones, highlighted by the much tighter PDF over the conductive sediment layer
compared to the resistive ice layer. The unconstrained inversions (Fig. 6a)
capture a similar structure to the true model, but they struggle to resolve
true layer depths. The simple synthetic model (a) and thick resistive
layered model (b) are relatively well resolved; however, the more complicated
synthetic models with thin layers and large resistivity contrasts (c, d and
e) are not. The depth-averaged resistivity errors within the subglacial
layer (calculated from the difference of the modal and true solutions) of
each synthetic non-constrained solution are (a) 275

Posterior distributions of resistivity determined from MuLTI-TEM
inversion:

The TEM method is generally more sensitive to conductance (the product of
conductivity and thickness) rather than the layer conductivity or thickness
alone (Geonics, 1994). Therefore, modelling is challenged by a non-unique
problem, for example, with thinner, more conductive layers producing a similar
TEM signal to thicker, less conductive layers. The addition of depth
constraints greatly reduces this non-uniqueness, enabling more accurate
solutions to be obtained at all depths. However, an example where this TEM
inversion struggles is when a thin conductive layer exists above a resistive
basement (Fig. A6). In this example (Fig. A6a), a thin layer 1 m
thick and resistivity 1

This feasibility study highlights the significant added value of depth constraints when a complex resistivity structure is expected. It demonstrates that MuLTI-TEM is promising for the potential distributions of resistivity beneath Midtdalsbreen.

Using the data collected at Midtdalsbreen, we produced 1-D resistivity
profiles for the soundings acquired at the midpoint of lines A, B and C
(Fig. 7). Anomalous data points either induced by residual current still in
the transmitter (at early time gates) or below the background noise level (at
later time gates) were removed, shown as “X” symbols in Fig. 7b. Inversions were
run with depth constraints taken from the GPR dataset for the snow-ice and
glacier-bed interfaces (red and blue horizons, respectively, in Fig. 7). The
same inversion parameters were used as the synthetic study (Table A2) but
with the maximum depth extended to 160 m to test the limit of DOI. The data
variance (

Results of the 1-D soundings acquired at the midpoint of lines A, B
and C inverted using MuLTI-TEM.

Posterior resistivity distributions are shown in Fig. 7a; comparisons of
data fit with 200 randomly chosen forward models from the model ensemble are
shown in Fig. 7b, and posterior distributions of the number of nuclei are
shown in Fig. 7c. The estimated uncertainty for the most likely solution
is calculated as one-half of the interquartile range at each depth, used due
to non-normal PDFs. As is clear from Fig. 7a, conductive layers identified
within the subglacial material are well resolved, with a tight PDF and low
uncertainty. However, resistive layers (e.g. ice) have a wide PDF with
large uncertainty estimates. For example, the uncertainty of the resistive
ice layer is typically estimated as

The 1-D inversions show a

MuLTI-TEM is used to invert multiple independent 1-D soundings acquired along lines A, B, C (4 m intervals) and D (8 m intervals). Again, anomalous data points either induced by residual current in the transmitter (at early time gates) or below the background noise level (at later time gates) were removed, shown as the grey regions in Fig. 8 (left column). The raw signal acquired is generally above the background noise level for all time gates, except some anomalous points in the centre of line A, corresponding with anomalous points at the NE end of line D, highlighted in the left column of Fig. 8.

2-D inversion outputs for lines A–D from multiple 1-D MuLTI-TEM inversions. Left column: received voltages input to MuLTI-TEM; central column: most likely 2-D resistivity profiles; right column: estimated uncertainty (half the interquartile range of the posterior distribution). Snow and ice horizons are plotted in blue and red, respectively.

When using a central-loop TEM survey configuration, the 1-D response is simply located at the centre of the transmitter, where the receiver is located. However, with an offset transmitter–receiver survey configuration used in this study, the location of the 1-D sounding is a subject of debate. Some place the location below the receiver and others midway between the transmitter and receiver (Hoekstra and Blohm, 1990). The entire section between the transmitter and receiver is expected to influence the measurements, especially at late times, as the current is diffuse. In what follows, we assume the 1-D location of each sounding to be at the centre point between the transmitter and receiver, although we note subsurface conditions near the transmitter may have a slightly larger influence on the received voltage measure at early times, when the current loop radius is approximately the same as the transmitter loop radius and not overlapping the receiver offset position.

Inversions were run with depth constraints supplied from snow and ice
horizons picked from the GPR data and using the same parameters as the
synthetic study. We verified convergence of the solutions by running another
Markov chain and increasing the chain length to 1.5 million iterations: all
tests reproduced the same posterior distribution. Consistent with initial
observations in the 1-D analysis, the 2-D resistivity profiles (Fig. 8,
central column) highlight a wide range of subglacial resistivity values,
from 10

a

a high-resistivity layer, 10

a lowermost layer of highly conductive material,

The variability of resistivity (10

Joint interpretation of

The initial interpretation of the seismic data was presented in Killingbeck et al. (2019). This study excluded certain phase velocities on the grounds that they were too high (Fig. A7), but this merits re-evaluation when compared with the co-located observation of high resistivity. Therefore, in this integrated interpretation, the high-phase velocities are included, thus providing broader bandwidth dispersion curves.

Observing trends of

zones of low

zones of high

zones of high

The resistivity and

The joint interpretation shows zones of mainly sediment and thick permafrost
(

MuLTI-TEM combines a probabilistic approach with external depth constraints to mitigate ambiguous, non-unique solutions found in conventional TEM inversions. It provides a robust quantitative uncertainty analysis of any chosen model at all depth levels, also providing an accurate estimate of DOI using the posterior distribution. The addition of depth constraints improves the characterisation of material, particularly beneath conductive layers and enables a faster convergence of the solution, as demonstrated in the synthetic study. We note other methods could be used to enhance the efficiency of the transdimensional inversion, potentially providing better convergence rates, such as proposing the birth parameters from the prior (instead of a Gaussian distribution), e.g. Dosso et al. (2014). Further still, having access to the full posterior distribution enables subsets of the posterior model probabilities to be selected, testing various hypothesis about the model structure (Ray and Key, 2012).

Nevertheless, the success of MuLTI-TEM depends fundamentally on the input data quality and its suitability for the specific target imaged. With TEM methods, it is often not possible to separately determine the conductivity and thickness; only the conductance (product of thickness and conductivity) can be determined. Therefore, thorough synthetic modelling should be undertaken before acquiring data in the field to determine if the survey design and time range of measurements are sufficient and suitable to detect specific targets.

Although this paper focuses on the specific TEM survey design used in this
study, a ground-based

This paper has presented how a joint analysis of three geophysical datasets
can increase our understanding of the material in the subsurface and provide
a more detailed interpretation. Using GPR information as a depth constraint,
we have combined insight from TEM and seismic shear-wave methods to provide
a detailed characterisation of the material beneath the margins of
Midtdalsbreen. Critically, TEM data reveal hydrological properties to which
the seismic analysis was insensitive, whereas the seismic data indicate the
varying stiffness of the subglacial material. Future extensions of this
interpretative strategy could include petrophysical relationships to obtain
and/or guide interpretations of the volumetric proportions of water, ice and
air in the subsurface (e.g. Hauck et al., 2008). A further promising
extension would be a modification to calculate the joint distribution of
resistivity and

The material properties of the subglacial environment, in particular their water content and saturation, can be characterised by inferences of their resistivity, obtained from TEM measurements. However, conventional TEM inversions provide solutions that are non-unique with no quantification of uncertainty estimates in depth and resistivity. This paper has presented an inversion algorithm, MuLTI-TEM, used to overcome such problems. Our method uses a transdimensional Bayesian inversion approach adapted from the MuLTI algorithm (Killingbeck et al., 2018), which incorporates independent depth constraints to limit the solution space reducing ambiguity. Synthetic testing of multiple different scenarios representing a small glacier underlain by sediment showed the addition of depth constraints greatly improves numerical convergence. This results in constrained solutions having a large improvement in the depth-average uncertainty of the output model, an average factor of 15 improvement on their unconstrained equivalents, with little computational power needed to obtain these results.

A joint interpretation, using

MuLTI-TEM is highly versatile, being compatible with most TEM survey designs, ground-based or airborne, as the Leroi forward modelling code can model most transmitter–receiver combinations, along with the depth constraints being provided from any external source. This study presents novel methodologies, through MuLTI-TEM and MuLTI, by which other glacier and ice-sheet subglacial material can be explored, highlighting the importance of acquiring multiple geophysical datasets for accurately characterising the subglacial environment.

MuLTI-TEM can be found at

SFK, ADB, PWL and LJW designed the project. SFK and ADB acquired the data. SFK and PWL developed MuLTI TEM. CRB provided advice and support while using the TEM equipment. SFK prepared the manuscript with contributions from all co-authors.

The authors declare that they have no conflict of interest.

The time-domain electromagnetic equipment was supplied by NERC Geophysical Equipment Facility, loan 1090. Fieldwork was greatly assisted by Emma Pearce, James Killingbeck and Kjell Magne Tangen. Alan Hobbs and all NERC GEF staff are thanked for their support and advice throughout the project.

This research has been supported by the UK NERC SPHERES DTP (grant no. NE/L002574/1). Fieldwork was funded by the research project “Snow Accumulation Patterns on Hardangerjøkulen Ice Cap (SNAP)”, itself funded by the European Union's Horizon 2020 project INTERACT, under grant agreement no. 730938.

This paper was edited by Ulrike Werban and reviewed by Anandaroop Ray and one anonymous referee.