Radar inference of the bulk properties of glacier beds, most notably identifying basal
melting, is, in general, derived from the basal reflection coefficient. On
the scale of an ice sheet, unambiguous determination of basal reflection is
primarily limited by uncertainty in the englacial attenuation of the radio
wave, which is an Arrhenius function of temperature. Existing bed-returned
power algorithms for deriving attenuation assume that the attenuation rate is
regionally constant, which is not feasible at an ice-sheet-wide scale. Here we
introduce a new semi-empirical framework for deriving englacial attenuation,
and, to demonstrate its efficacy, we apply it to the Greenland Ice Sheet. A
central feature is the use of a prior Arrhenius temperature model to estimate
the spatial variation in englacial attenuation as a first guess input for the
radar algorithm. We demonstrate regions of solution convergence for two input
temperature fields and for independently analysed field campaigns. The
coverage achieved is a trade-off with uncertainty and we propose that the
algorithm can be “tuned” for discrimination of basal melt (attenuation loss
uncertainty

Ice-penetrating radar (IPR) data provide valuable insights into
several physical properties of glaciers and their beds including ice
thickness (e.g.

Bulk material properties of glacier beds can, in principle, be identified
from their basal (radar) reflection coefficient

In this study we introduce a new ice-sheet-wide framework for the
radar inference of attenuation and apply it to IPR data from the Greenland
Ice Sheet (GrIS). A central feature of our approach is to firstly estimate
the spatial variation in the attenuation rate using an Arrhenius model, which
enables us to modify the empirical bed-returned power method. Specifically,
the estimate is used to (i) constrain a moving window for the algorithm
sample region, enabling a formally regional method to be applied on a ice-sheet-wide scale and (ii) to standardise the power for local variation in
attenuation within each sample region when deriving attenuation using
bed-returned power. We demonstrate regions of algorithm solution convergence
for two different input temperature fields and for independently analysed IPR
data. The coverage provided by the algorithm is a trade-off with solution
accuracy, and we suggest that the algorithm can be “tuned” for basal melt
discrimination in restricted regions, primarily in the southern and eastern
GrIS. This is supported by the decibel range for the basal reflection
coefficients (

The airborne IPR data used in this study were collected by the Center for
Remote Sensing of Ice Sheets (CReSIS) within the Operation IceBridge project.
Four field seasons from 2011 to 2014 (months March–May) have been analysed
in this proof of concept study. These field seasons are the most spatially
comprehensive to date, with coverage throughout all the major drainage basins
of the GrIS and relatively dense across-track spacing toward the ice margins
(Fig.

Flow diagram for the components of the radar algorithm.

A flow diagram for the separate components of the radar algorithm is shown in
Fig.

The processing of the IPR data, based upon the method developed by

Using the waveform processing method of

Waveform processing using the power depth integral method,
Eq. (

The incoherently averaged basal waveforms range from sharp pulse-like returns
associated with specular reflection, to broader peaks associated with diffuse
reflection (refer to

It is well established that the dielectric conductivity and radar attenuation
rate in glacier ice is described by an Arrhenius relationship where there is
exponential dependence upon inverse temperature and a linear dependence upon
the concentration of soluble ionic impurities
(

The Arrhenius relationship is empirical and the dielectric properties of
impure glacier ice (pure ice conductivity, molar conductivities of soluble
ionic impurities, and activation energies) need to be measured with respect
to a reference temperature and frequency. Two Arrhenius models for the
dielectric conductivity and the attenuation rate were applied to the GrIS by

Temperature dependence of estimated attenuation rate,

Estimated spatial dependence of depth-averaged attenuation rate for
the GrIS using the Arrhenius model.

Constraining the target window boundaries.

Maps for target window radii vector length using the GISM temperature
field.

The temperature fields for GISM and SICOPOLIS were used to estimate the
spatial variation in the depth-averaged attenuation rate for the GrIS and
were interpolated at 1 km grid resolution. Both the GISM and SICOPOLIS
models provide temperature profiles as a function of relative depth, and
these were vertically scaled using the 1 km Greenland Bedmap 2013 ice
thickness data product

Radar inference of the depth-averaged attenuation rate, using the
relationship between bed-returned power and ice thickness, requires sampling
IPR data from a local region of the ice sheet

Figure

A primary consideration for the moving target window is that the dimensions,

The method of using the relationship between ice thickness and bed-returned
power to infer the radar attenuation rate and basal reflection coefficient
has been employed many times to local regions of ice sheets

As discussed here and in Sect.

An example of a

Bed-returned power vs. ice thickness pre- and post-local
attenuation correction, Eq. (

When applying the linear regression approach described in this section, IPR
data from each field season were considered separately. To ensure that there
were sufficiently dense data within each sample region, a minimum threshold of
20 measurements was enforced, where each measurement corresponds to a
separate along-track averaged waveform as described in
Sect.

The accuracy of the radar-inferred attenuation rate solution from
Eq. (

In general,

Ice-sheet-wide properties of the radar algorithm using the GISM
temperature field.

The attenuation rate solution from the radar algorithm,

Attenuation solution convergence for the SE GrIS.

With a view toward identifying regions of the GrIS where the radar
attenuation algorithm can be applied, we firstly consider ice-sheet-wide
properties for the linear regression correlation parameters
(Sect.

Ice-sheet-wide maps for the linear regression correlation parameters are
shown in Fig.

Examples of algorithm coverage for three different sets of (

The ice-sheet-wide properties of the algorithm are preserved using the
SICOPOLIS temperature field as an input (refer to Supplement for a repeat
plot of Fig.

To demonstrate the convergence of the attenuation solution for different
input temperature fields (convergence is defined here as a normally
distributed difference distribution about zero), we compare the solution
differences for the (input) Arrhenius models,

The inset region we consider is shown in (Fig.

Corresponding difference distributions for the attenuation loss are shown in
Fig.

Attenuation solution and basal reflection.

If a similar analysis for the attenuation solution differences is applied to drainage basins 3, 5, and 6 (southern and eastern Greenland) we observe algorithm solution convergence (in the sense of a normally distributed difference centred on zero) and an associated reduction in the solution bias from the Arrhenius model input. In drainage basins 1, 2, 7, and 8 (northern and western Greenland), we do not observe analogous solution convergence for the radar-inferred values. We do, however, typically see a reduction in the mean systematic bias for the attenuation rate/loss solution relative to the Arrhenius model input. In the Supplement we provide additional plots and discuss the potential reasons for the algorithm non-convergence, which are thought to relate primarily to the more pronounced temperature sensitivity of the algorithm target windows in the northern GrIS.

For regions of the GrIS where the attenuation rate solution converges and
there is algorithm coverage overlap for the different temperature field
inputs, it is possible to define the mean radar-inferred attenuation rate
solution

Inset maps for the depth-averaged attenuation rate and basal reflection
coefficient are compared with balance velocity

The probability distribution for the relative basal reflection coefficient,

Relationship between algorithm coverage and uncertainty as measured
by attenuation solution difference distributions.

Evaluation of temperature bias for ice sheet models using
attenuation rate differences.

There are two metrics, both as a function of the quality threshold parameters

Attenuation rate and loss solution difference distributions for three

A similar relationship between the choice of

Since for both uncertainty metrics, the solution differences are a function
of

The recent study by

The evaluation of the temperature bias of a thermomechanical ice sheet model
using attenuation rates inferred from IPR data was recently considered for
the first time by

Arrhenius model-radar algorithm attenuation solution differences are shown
for the M07 model (GISM Fig.

A final caveat to our approach here is that it does not include layer
stratigraphy in the Arrhenius model. The analysis in

In this study, we considered the first application of a
bed-returned power radar algorithm for englacial attenuation over the
extent of an ice sheet. In developing our automated ice-sheet-wide approach,
we made various refinements to previous regional versions of the algorithm

The converged radar algorithm attenuation solution provides a means of
assessing the bias of forward Arrhenius temperature models. Where temperature
fields are poorly constrained, and where the algorithm has good coverage, we
suggest that it is preferable to using a prior Arrhenius model calculation.
With this in mind, the potential problems with using a forward Arrhenius
model for attenuation were illustrated (Sect.

We have yet to consider an explicit classification of the subglacial
materials and quantification of regions of basal melting. In future work, we
aim to combine IPR data from preceding CReSIS field campaigns to produce a
gridded data product for basal reflection values and basal melt. It is
anticipated that, as outlined by Oswald and Gogineni (

Finally, we envisage that the framework introduced in this paper could be
used for radar inference of radar attenuation, basal reflection and basal
melt for the Antarctic Ice Sheet. Given that for high solution accuracy the
radar algorithm requires high topographic roughness and relatively warm ice,
we suggest that IPR data in rougher regions toward the margins should be
analysed first (refer to

List of principal symbols.

In ice, a low loss dielectric, the radar attenuation rate,

Following the assumptions in Sect.

Both the W97 model and the M07 model assume that the dielectric
conductivity/attenuation rate is frequency independent between the medium
frequency (MF), 0.3–3

For the MCoRDS system that is considered in this study and by

The W97C model with

Estimated basal reflection coefficient,

A prior estimate for the basal reflection coefficient,

In this appendix we describe the rms integral measure that we use to define
the sample region boundaries, as described conceptually in
Sect.

Tuning the rms tolerance, Eq. (

This study was supported by UK NERC grant NE/M000869/1. We would like to thank our reviewers J. Macgregor and M. Wolovick for their valuable comments which greatly improved this manuscript, and K. Matsuoka for handling our manuscript submission. We would also like to thank J. Macgregor for supplying ice core temperature profiles. Edited by: K. Matsuoka Reviewed by: J. MacGregor and M. Wolovick