Subglacial roughness can be determined at a variety of length scales from radio-echo sounding (RES) data either via statistical analysis of topography or inferred from basal radar scattering. Past studies have demonstrated that subglacial terrain exhibits self-affine (power law) roughness scaling behaviour, but existing radar scattering models do not take this into account. Here, using RES data from northern Greenland, we introduce a self-affine statistical framework that enables a consistent integration of topographic-scale roughness with the electromagnetic theory of radar scattering. We demonstrate that the degree of radar scattering, quantified using the waveform abruptness (pulse peakiness), is topographically controlled by the Hurst (roughness power law) exponent. Notably, specular bed reflections are associated with a lower Hurst exponent, with diffuse scattering associated with a higher Hurst exponent. Abrupt waveforms (specular reflections) have previously been used as a RES diagnostic for basal water, and to test this assumption we compare our radar scattering map with a recent prediction for the basal thermal state. We demonstrate that the majority of thawed regions (above pressure melting point) exhibit a diffuse scattering signature, which is in contradiction to the prior approach. Self-affine statistics provide a generalised model for subglacial terrain and can improve our understanding of the relationship between basal properties and ice-sheet dynamics.

With the development of the newest generation of thermomechanical ice-sheet
models, there has been a growing awareness that better constraining the
physical properties of the glacier bed is essential for improving their
predictive capability (e.g.

RES data analysis methods for determining subglacial physical properties can
be categorised in two ways: those which determine bulk properties (including
the discrimination of basal water) and those which determine interfacial
properties (subglacial roughness). Bulk material properties of the glacier
bed can, in principle, be determined using the basal reflectivity

Degrees of radar scattering can be mapped either using the waveform
properties of the bed echo – e.g. the waveform abruptness (pulse-peakiness)

In this study, we explore the connection between self-affine subglacial
roughness and radar scattering using recent airborne Operation IceBridge
(OIB) RES data from the north-western Greenland Ice Sheet (GrIS). Firstly we
review the theory of self-affine roughness statistics, using examples from
ice-penetrating radargrams and bed elevation profiles to demonstrate its
applicability to subglacial terrain (Sect.

Statistical methods to calculate the Hurst exponent, and thus to quantify
self-affine scaling behaviour, are well established in the earth and
planetary science literature

Example radargrams (top panel) and 10 km bed elevation profiles
(bottom panel) for subglacial terrain with different Hurst exponent,

Topographic roughness can be measured by means of statistical parameters that
are, in general, a function of horizontal length scale

Self-affine scaling is a subclass of fractal scaling behaviour and can be
parameterised using the Hurst exponent,

We will later demonstrate that subglacial terrain exhibits near-ubiquitous
self-affine scaling behaviour with pronounced spatial structure and variation
for

In order to calculate

The space-domain variogram and deviogram have an approximate correspondence
to the frequency-domain power spectrum

The airborne RES data used in this study were collected by the Center for
Remote Sensing of Ice Sheets (CReSIS) within the OIB
project, over the months March–May in years 2011 and 2014. For all
measurements the radar instrument, the Multichannel Coherent Radar Depth
Sounder (MCoRDS), was installed upon a NASA P-3B Orion aircraft. The sounder
has a frequency range from 180 to 210 MHz, corresponding to a centre
wavelength

The study focused on flight-track data from north-western Greenland and
encompassed measurements close to three deep ice cores: Camp Century, NEEM,
and NorthGRIP (Fig.

Data coverage map for OIB flight tracks and region of interest. The
locations of the Camp Century, NEEM, and NorthGRIP ice cores are indicated,
along with the terrain profile sections in Fig.

Measurements from MCoRDS are supplied as data products with different levels
of additional processing

The along-track spacing (

In this study we are interested in calculating

The post-processing of the Level 1B data (analysis of the basal waveform)
uses the procedure described in

Observed values of

As RES over ice employs a nadir-facing sounder, the scattering contribution
toward the waveform abruptness is mainly from coherent reflection (as opposed
to side-looking SAR instruments which would be mainly diffuse scattering).
Whether coherent pre-processing (either coherent pre-summing of Doppler
focusing) of the raw data acts to increase or decrease the value of

The basal waveform (and hence the calculated values of

Examples of bed-echo waveforms and their abruptness (pulse
peakiness). Observed values for

The waveform abruptness has previously been discussed without reference to
roughness statistics, and here we do this using a self-affine radar
scattering model. Radar scattering models from natural terrain fall into two
different categories: “coherent”, which incorporates deterministic phase
interference, and “incoherent”, which incorporates random phase interference

Below we describe and adapt a coherent scattering model, first developed for
the nadir regime of planetary radar sounding measurements, which incorporates
self-affine roughness statistics

The physical assumptions behind the self-affine scattering model are
summarised in

An expression for the radar backscatter coefficient (radar cross section per
unit area) is then derived by considering a phase variation,

The utility of the waveform abruptness in quantifying different degrees of
scattering rests upon the assumption that the majority of the overall energy
is contained within the echo envelope

Parametric dependence of the self-affine radar scattering model.

It is important to note that the predictions of the self-affine radar
scattering model are consistent with the specular RES scattering signature
that we would expect from electrically deep subglacial lakes. Under the
self-affine roughness framework, a large geometrically flat feature such as a
lake would have a negligible value of

The physical explanation for the strong dependence of the coherent power upon

Firstly, we describe maps for the rms deviation and Hurst exponent
(topographic-scale roughness) and the waveform abruptness (radar scattering)
in the northern Greenland (Sect.

Data maps for the northern GrIS:

In Fig.

Pronounced spatial variation in the Hurst exponent,

The basal thermal state prediction by

There are some clear discontinuities in the flight-track maps for

Before we consider a quantitative comparison between the predictions of the
radar scattering model and the RES-derived data, we first summarise the
statistics for the Hurst exponent,

Relationship between Hurst exponent,

The self-affine coherent scattering model (Sect.

Distributions from basal RES analysis in thawed and frozen regions
of the northern GrIS (corresponding to flight-track data in
Fig.

In order to test this prediction, we considered the statistics of three
separate

Here we summarise the statistics of the RES-derived roughness and scattering
data in predicted thawed and frozen regions of the glacier bed, with an
overall purpose of testing the basal water discrimination algorithm by

The distributions for all RES-derived data exhibit pronounced statistical
differences between thawed and frozen regions (Fig.

In RES data analysis, cross-over distributions at flight-track intersections
can give an indication of uncertainty based upon internal consistency (e.g.

As part of the analysis we also considered estimation of the breakpoint
transitions for

Our results demonstrate that self-affine scaling behaviour is a
near-ubiquitous property of the subglacial topography of northern Greenland.
Moreover, there is both spatial structure and variability in the Hurst
exponent, which can range from being near-self similar (

The Hurst exponent provides information about the relationship that exists
between vertical roughness and the horizontal length scale. Whilst it is
related to the slope of the roughness power spectrum, past spectral analysis
of glaciological terrain tends to obscure this information (since an
integrated “total roughness” metric is typically used)

The Hurst exponent has previously been shown to play a dynamical role in the
flow resistance of alluvial channels

The statistical analysis of the waveform abruptness in predicted frozen and
thawed regions (Sect.

Subglacial hydrological systems are understood to produce more complex and
variable scattering signatures than the specular lake-like reflection assumed
by

The anisotropy of the Hurst exponent was not considered in the radar
scattering model, which was justifiable because we were interested in
understanding how the Hurst exponent relates to the (near-) isotropic waveform
abruptness. However, in certain regions of the ice sheets, basal radar
scattering is known to be highly anisotropic, as revealed by maps of the
specularity content for Thwaites glacier

Finally, geostatistically based interpolation methods which employ aspects of
self-affine statistics

In this study we used recent OIB RES data to demonstrate that subglacial roughness in northern Greenland exhibits self-affine scaling behaviour, with pronounced spatial variation in the Hurst (roughness power law) exponent. We modified a planetary radar scattering model to predict how the Hurst exponent exerts control upon the degree of scattering, which we parameterised using the waveform abruptness. We then demonstrated an agreement between the predictions of the radar scattering model and the statistically distributed inverse relationship that is observed between the Hurst exponent and waveform abruptness. This enables us to conclude that self-affine statistics provide a valuable framework in understanding the topographic control which influences ice-penetrating radar scattering from glacier beds. Self-affine statistics also provide a generalised model for subglacial terrain and in the future could be used to further explore the relationship between bed properties, ice-sheet dynamics, and landscape formation.

An additional glaciological motivation behind our study was to establish
whether the
waveform abruptness could be used to aid in the discrimination of basal water
(and to test the prior assumption that subglacial hydrological systems in
Greenland produce abrupt bed echoes;

All data used for the preparation of this paper are openly available. The Level 1B and Level 2 OIB RES data
are available from CReSIS at

The authors declare that they have no conflict of interest.

T. M. Jordan, J. L. Bamber, and C. N. Williams were supported by UK NERC grant
NE/M000869/1 as part of the Basal Properties of Greenland project.
M. A. Cooper was supported by the UK NERC grant NE/L002434/1 as part of the
NERC Great Western Four