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 . These space-domain methods extract the Hurst exponent using the variogram (roughness verses profile length) and deviogram (roughness versus horizontal lag). Our motivation for use of these methods, rather than the spectral (frequency-domain) methods previously applied in studies of subglacial roughness , is that they better reveal self-affine scaling behaviour . Since the theory of self-affine roughness and related space-domain methods are not widely discussed in the glaciological literature – the only example being – we now provide a review of the key concepts.

Topographic roughness can be measured by means of statistical parameters that are, in general, a function of horizontal length scale . Two different interfacial roughness parameters – the root mean square (rms) height and rms deviation – are typically employed in self-affine roughness statistics . The rms height is given by ξ(L)=1N-1∑i=1N(z(xi)-z¯)212, where N is the number of sample points within the profile window of length L, z(xi) is the bed elevation at point xi, and z¯ is the mean bed elevation of the profile. ξ represents the standard deviation in bed elevation about a mean surface and models the topographic roughness as a Gaussian-distributed random variable . The rms deviation is given by ν(Δx)=1N∑i=1N[(z(xi)-z(xi+Δx)]212, where Δx is the horizontal step size (lag). ν has a particular significance in the parameterisation of radar scattering models with self-affine statistics , and we focus upon this roughness parameter when integrating topographic-scale roughness with radar scattering data. The rms slope, which is proportional to the rms deviation, is also widely used in self-affine statistics, but we do not do so here.

Self-affine scaling is a subclass of fractal scaling behaviour and can be parameterised using the Hurst exponent, H . H quantifies the rate at which roughness in the vertical direction increases relative to the horizontal length scale (and is defined for 0 ≤ H ≤ 1). For a self-affine interface the following power-law relationships hold: ξ(L)=ξ(L0)LL0H, and ν(Δx)=ν(Δx0)ΔxΔx0H, where L0 is a reference profile length and Δx0 is a reference horizontal lag . Three limiting cases of self-affine scaling are typically discussed . Terrain with H=1 (where the roughness in the vertical direction increases at the same rate as the horizontal length scale) is referred to as “self-similar”. Terrain with H=0.5 (where the roughness in the vertical direction increases with the square root of horizontal length scale) is referred to as “Brownian”. Terrain with H=0 (where the roughness in the vertical direction is independent of horizontal length scale) is referred to as “stationary”. For a stationary (H=0) interface it follows from Eqs. () and () that ξ and ν are independent of L and Δx respectively (i.e. the roughness parameters are independent of horizontal length scale).

We will later demonstrate that subglacial terrain exhibits near-ubiquitous self-affine scaling behaviour with pronounced spatial structure and variation for H. Examples of OIB ice-penetrating radargrams (Z scopes) and associated bed elevation profiles for terrain with different H are shown in Fig. . Clear differences are apparent between the different terrain examples. The black (H ≈ 0.9) terrain (Fig. a) and red (H ≈ 0.7) terrain (Fig. b) are between Brownian and self-similar scaling behaviour. This terrain exhibits “persistent trends”, where neighbouring measurements tend to follow a general trend of increasing or decreasing elevation (refer to for a full discussion). A feature of terrain with higher H is that it tends to appear relatively rough at larger length scales (low frequency) and smooth at smaller length scales (high frequency). By contrast, the green (H ≈ 0.3) terrain (Fig. d) is in the sub-Brownian scaling regime and exhibits “anti-persistent trends”, where neighbouring measurements tend to alternate between increasing and decreasing elevation. A feature of lower H terrain such as this example is that it tends to have similar roughness across length scales. The blue (H ≈ 0.5) terrain (Fig. c) is close to an ideal Brownian surface and exhibits no overall persistence (with some sections of the profile following an increasing/decreasing elevation trend and other sections alternating). The 10 km profile windows in Fig.  represent the length of flight-track data over which H is calculated (see Sect. ).

In order to calculate H, and identify the scale regime over which glacial terrain exhibits self-affine behaviour, ξ and ν are plotted as functions of L and Δx, respectively, on double-logarithmic-scale plots, referred to as the variogram and deviogram . Variogram and deviogram plots for ξ(L) and ν(Δx) for the four terrain examples in Fig.  are shown in Fig. a and b respectively. It follows from Eqs. () and () that, upon this double-logarithmic scale, a straight line relationship is predicted for glacial terrain that is self-affine with the gradient equal to H. In practice, a single self-affine relationship only holds over a limited scale regime and a “break-point” transition is often observed . We describe how we assess the break points for glacial terrain in Sect. , along with further details regarding the application of the variogram and deviogram to along-track RES data. Figure  clearly demonstrates the significance of the Hurst exponent and horizontal length scale when assessing the relative roughness of different terrain. For example, the black (H ≈ 0.9) terrain is rougher than the red (H ≈ 0.7) terrain at larger length scales but is smoother at smaller length scales.

The space-domain variogram and deviogram have an approximate correspondence to the frequency-domain power spectrum . In frequency space, self-affine scaling occurs when the power spectrum, S, has a relationship of the form S(k)∝k-β, where k is the spatial frequency and -β is the spectral slope. The relationship between β and H is dimensionally dependent and for along-track data is given by H=12(β-1) . Despite this correspondence, the space-domain methods are recommended to calculate H as they are less noisy and less likely to bias slope estimates than the power spectrum method . The study by observed self-affine scaling in the roughness power spectrum over length scales from ∼ 10-3 to ∼ 10 m for different sites across recently deglaciated terrain in the immediate foreground of Tsanfleuron glacier, Switzerland. Their range for measured values of β corresponds to 2.27<β<2.48, which implies H ≈ 0.7.

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 ∼ 0.87 m in ice. After accounting for pulse shaping and windowing, this results in a depth-range resolution in ice of ∼ 4.3 m . For the flight lines considered, the along-track resolution after synthetic aperture radar (SAR) processing and multi-looking is ∼ 30 m with an along-track-sample spacing of ∼ 15 m . The 2011 and 2014 field seasons were used since they have a higher along-track resolution than other recent field seasons and hence enable a clearer connection to be made between radar scattering and topographic-scale roughness.

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. ). The first reason for selection of this region is that the data coverage for the 2011 and 2014 field seasons is of high density relative to most other regions of the ice sheet. The second reason is that confidence regarding the basal thermal state is high near to the ice cores and thus enables the validity of the basal water RES analysis by to be tested.

Measurements from MCoRDS are supplied as data products with different levels of additional processing . Level 2 data correspond to ice thickness, ice surface, and bed elevation data and are used to calculate topographic-scale roughness and the Hurst exponent (Sect. ). Details regarding the semi-manual picking procedure are described by , and only the highest-quality picks were used. Level 1B data correspond to radar-echo strength profiles and are used to extract the waveform abruptness parameter from the bed echo (Sect. ). Basal reflection values can also be extracted from Level 1B data, but we do not do this here because we do not wish to bias our interpretation due to uncertainty in radar attenuation. The pre-processing of the combined channel Level 1B data is also described by . Sequentially this involves channel compensation between each of the antenna phase centres, pulse compression (using a 20 % Tukey window in the time domain), coherent-averaging of the channels, SAR processing with along-track frequency window, channel combination, and waveform combination.

The along-track spacing (∼ 15 m) of the Level 2 data is half the horizontal resolution (∼ 30 m), which represents the spacing at which bed elevation measurements are considered as independent. Therefore, to remove local correlation bias, the Level 2 data were down-sampled, considering every second data point (corresponding to a ∼ 30 m along-track spacing). Each flight track was then divided into 10 km along-track profile windows, as shown in the examples in Fig. a. The windows overlap with a sample spacing of 1 km, with the centre of each window defined to be the point to which H and the roughness parameters are geolocated. This “moving window” approach was employed as it enables greater continuity in the estimates for H. Prior to estimating H, ξ(L) and ν(Δx) were computed following Eqs. () and () respectively. These calculations used the “interleaving” sampling method described in , which enables all of the data points to be sampled effectively. The windowing method is similar to that described in for the self-affine characterisation of Martian topography, where a non-overlapping 30 km window was assumed. The choice of 10 km for the profile window and 1 km for the effective resolution represents a good tradeoff between resolution and the smoothness of the derived data fields.

In this study we are interested in calculating H at the length scale of the Fresnel zone (∼ 100 m), since this enables the most accurate parameterisation of the radar scattering model described in Sect. . Additionally, due to the break point transitions that occur at larger length scales, the focus on smaller length scales is a robust approach to calculate H . For the data we consider, the lower bounds of the horizontal length scales are ∼ 90 m for ξ(L) (since three elevation measurements are the minimum required to calculate ξ(L) using Eq. ) and ∼ 30 m for ν(Δx). ν(Δx) therefore better enables the estimation of H at smaller length scales and we primarily focused on the deviogram method (Fig. b). Additionally, as suggested in Fig. , the relationships for ν(Δx) are, in general, significantly smoother than ξ(L). The upper length scales in the deviogram and variogram were set to be Δx=1 km and L=1 km respectively, which follows from the recommendation by that at least 10 independent sections of track are used in the calculations. As shown in Fig. , the gradients (H) were calculated using the first five data points (which, for the deviogram, are over the range Δx∼ 30–150 m). Self-affine scaling behaviour often extends beyond these smaller length scales and we estimated the break points for ξ(L) and ν(Δx) using a segmented linear regression procedure. Briefly, this involved firstly calculating the gradient (H) for the first five data points. Additional data points at increasing length scales were then added into each linear regression model, and the gradient was recalculated. Finally, break points in the linear relationship were identified by testing if the new gradient exceeded a specified tolerance from the original estimate.

The post-processing of the Level 1B data (analysis of the basal waveform) uses the procedure described in , which, in turn, is largely based upon . Firstly, this involved performing an along-track average of the basal waveform, where adjacent basal waveforms are stacked about their peak power values and arithmetically averaged. This averaging approach is phase-incoherent and acts to smooth power fluctuations due to electromagnetic interference . The size of the averaging window varies as a function of Fresnel zone radius, and subsequently each along-track averaged waveform corresponds to approximately a separately illuminated region of the glacier bed (see , for details). The degree of radar scattering is quantified using the waveform abruptness A=PpeakPagg, where Ppeak is the peak power of the bed echo and Pagg is the aggregated power, which is calculated by a discrete summation of the bed-echo power measurements in each depth range bin. Pagg was introduced by since, based upon energy conservation arguments, it is argued to be more directly related to the predicted (specular) reflection coefficients than equivalent peak power values. In radar altimetry, the waveform abruptness is commonly called “pulse peakiness” (e.g. ).

Observed values of A range from ∼ 0.03 to 0.60, and in Sect.  we theoretically constrain the maximum value to be ∼ 0.65. Three examples of basal waveforms, along with their corresponding A values, are shown in Fig. . Higher A values are associated with specular reflections from smoother regions of the glacier bed (e.g. the blue waveform), whilst lower A values are associated with diffuse reflections from rougher regions (e.g. the green waveform) . The positions of the peak power were established by firstly using Level 2 data picks, then applying a local re-tracker to centre over the peak power. When calculating the summation for Pagg (both fore and aft of the peak power so as to best capture the energy contained in the echo envelope), a signal-noise-ratio threshold was implemented by testing for decay of the peak power to specified percentage above the noise floor. Thresholds of 1, 2, and 5 % were considered and 2 % was found to give the best coverage, whilst excluding obvious anomalies. Due to this quality-filtering step there are therefore sometimes small gaps in the along-track A data.

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 A depends upon the exact character and roughness of the surface. As a first example, if the specular/nadir component of the echo is assumed to be coherent, whilst the diffuse/off-nadir component is assumed to be incoherent (e.g. ), then coherent processing would cause the specular component of the signal to increase with coherent gain but not the diffuse (incoherent) signal. Therefore the measured A value would decrease with gain. As a second example, if both the specular/nadir and diffuse/off-nadir components of the echo are assumed to be coherent (e.g. ), then for small SAR processing angles (coherent pre-summing) the waveform abruptness should be largely unaffected. However, for larger angles (exceeding the angle spanned by the specular component of the echo in the scattering function) the A value will decrease with coherent pre-processing.

The basal waveform (and hence the calculated values of A) results from a superposition of along-track and cross-track energy . Subsequently, the anisotropy of radar scattering (and inferences regarding the anisotropy of subglacial roughness) is not explicitly revealed by A. Hence, the studies of treat A as an isotropic parameter, and we follow this approach here.

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 . Coherent scattering models are applicable where the reflecting region is orientated nearly perpendicular to the incident pulse (the nadir regime) and the reflecting region is fairly smooth at the scale of the illuminating wavelength , which is normally assumed to be a good approximation for the RES of glacier beds (e.g. ). Volume (Mie) scattering is typically neglected from basal RES scattering analysis and would hypothetically require scatterer dimensions of the order of the radar wavelength (∼ 0.5 to 5 m dependent on the bed dielectric and radar system). This neglection of volume scattering is justified given the ∼ 10-6 to 10-3 m scale of water pore radii in typical bed materials . Moreover, even in the extreme case of planetary ice regoliths (which are colder than terrestrial ice and will therefore sustain larger heterogeneities), scatterer dimensions are ∼ 10-3 to 10-2 m and volume scattering losses are small .

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 model is parameterised using the Hurst exponent values derived from the subglacial topography (Sect. ) and thus enables a connection to be made between the topographic roughness and radar scattering. Coherent scattering models can be used to model a decrease in specularly reflected power as a function of rms roughness , and this is the central aspect of the model which we focus upon here. Specifically, we show that, under assumptions of energy conservation, this power decrease can be used to theoretically predict the relationship between the Hurst exponent and waveform abruptness.

The physical assumptions behind the self-affine scattering model are summarised in . The central assumption that differentiates the model from coherent stationary (H=0) models is that the rms height increases as a function of radius, r, about any given point, following the self-affine relationship ξ(r)=12νλrλH, where νλ=ν(Δx=λ) is the wavelength-scale rms deviation. Equation () assumes radial isotropy for H and ξ. Since we are focusing upon constraining the (near-) isotropic abruptness parameter, this is a justifiable approximation. The statistical distribution for ξ(r) is assumed to be Gaussian, which is similar to most H=0 models (but with an additional radial dependence). Via νλ, the self-affine model is explicitly formulated with respect to the horizontal scale of rms roughness. The radio wavelength of MCoRDS in ice is ∼ 0.87 m, and hence wavelength-scale rms deviation is approximately equivalent to metre-scale rms deviation. An unavoidable caveat to the parameterisation of the radar scattering model using Eq. () is that the H values derived from the topography (length scale ∼ 30–150 m) are extrapolated downwards to the wavelength scale.

An expression for the radar backscatter coefficient (radar cross section per unit area) is then derived by considering a phase variation, 4πξ(r)λ, integrated across the Fresnel zone . For nadir reflection the radar backscatter coefficient is given by σ0=16π2Re2r^max⁡2∫0r^max⁡exp⁡-4π2λ2νλ2r^2Hr^dr^2, where r^=rλ is the wavelength-scaled radius, r^max⁡ is the wavelength-scaled radius of the illuminated area (the Fresnel zone), and Re is the reflection coefficient for the electric field . The coherent power, P, can then be obtained by dividing Eq. () by 4π2r^max⁡2 (a geometric factor which follows from the backscatter coefficient of a flat conducting plate; ) to obtain P=4Re2r^max⁡4∫0r^max⁡exp⁡-4π2λ2νλ2r^2Hr^dr^2. For the case where H=0, ξ(r) in Eq. () is independent of radius. It follows that ξ2=12νλ2 and the exponent in Eq. () is also independent of radius, which gives P=Re2exp⁡-16π2λ2ξ2. Equation () is the same power decay formula as coherent H=0 models , where it is sometimes multiplied by a first-order Bessel function (which enables some of the incoherent energy contribution to be captured; ). Thus the stationary limit of the self-affine model is consistent with previous glacial basal scattering models. It is clear that the coherent power for the self-affine model, Eq. (), has two roughness degrees of freedom: H and νλ, which can be conceptually related to the gradient and the intercept of the deviogram (Fig. ). This contrasts with the stationary model, Eq. (), which has one degree of freedom: ξ.

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 . In other words, it is assumed that, for reflection from the same bulk material, the aggregated/integrated power from a rough interface (νλ>0) is equivalent to the peak power from a given smooth interface; i.e. Pagg≈P(νλ=0). This energy equivalence was demonstrated to hold well for the waveform processing procedure and Greenland RES systems by . It follows from this energy equivalence that the abruptness, A, can be expressed in terms of the coherent power, Eq. (), as A=PpeakPagg=CP(νλ)P(νλ=0), where C is a proportionality constant that corresponds to the theoretical maximum abruptness value, which occurs when the radar pulse is specularly reflected and Pagg=Ppeak. For a perfectly specular reflection the pulse is the shape of compressed chirp (absolute value of a sinc function with the width determined by the signal bandwidth). If the depth-range sample spacing of the waveform (Fig. ) were the same as the depth-range resolution then C would be near unity. However, C can be estimated from the ratio of the sample spacing (∼ 2.8 m) to the range resolution (∼ 4.3 m) to give C ∼ 0.65. Finally, substituting Eq. () into Eq. () gives A=4Cr^max⁡4∫0r^max⁡exp⁡-4π2λ2νλ2r^2Hr^dr^2. As is the case for P in Eqs. () and () A has two roughness degrees of freedom: H and νλ. note that the primary dependence for P (and hence A) is upon H, with a weaker secondary dependence upon νλ. In order to illustrate this dependency, we consider first the relationship between A and H for fixed νλ (Fig. a) and secondly the relationship between A and νλ for fixed H (Fig. b). Figure a demonstrates that higher values of νλ (the black curve) result in negligible A for all but the lowest values of H. Intermediate values of νλ (the red and blue curves) exhibit a sharp transition from higher to lower values of A as H increases. Low νλ (the green curve) has high A for all H. Figure b demonstrates a monotonic decrease in A with νλ for each value of H, with the decay length decreasing rapidly with increasing H.

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 H and νλ. This scenario occurs for the low H limit of the green curve in Fig. a, where predicted values for A are ∼ 0.65 (corresponding to a perfectly specular reflection).

The physical explanation for the strong dependence of the coherent power upon H, and the relationships which we observe in Fig. , is discussed by and . It relates to the fact that significant coherent returns can only occur from annular regions where ξ(r)<λ8 (the Rayleigh criterion). It follows from Eq. () that high values of H lead to a rapid increase in roughness with radius that rapidly exceeds this threshold. Subsequently, for high H interfaces, the roughness at the wavelength scale, νλ, must be a couple orders of magnitude smaller than the Rayleigh criterion to enable significant coherent returns (i.e. non-negligible A). The curves in Fig.  assume r^max⁡=100 (corresponding to a Fresnel zone radius ∼ 115 m for the ice wavelength ∼ 0.87 m). In general, the relationships in Fig.  are insensitive to this choice of radius. This is because the radii of the coherent annular regions are typically significantly less than the Fresnel zone and thus act as the dominant length scale for the integration limit in Eq. ().

In Fig.  flight-track maps for the RES-derived roughness and scattering data are compared with the Greenland bed DEM and the predicted basal thermal state (Fig. 11 in ). The flight-track maps all demonstrate a high degree of spatial structure, with some notable correlations present (both between each other and the DEM). There is a clear inverse relationship between the rms deviation, ν (shown at two different length scales in Fig. a and b), and the waveform abruptness, A (Fig. c), with higher abruptness (specular reflections) present in smoother regions of the ice-sheet bed and lower abruptness (diffuse scattering) present in rougher regions. For example, smoother regions (lower ν, higher A) occur for flight tracks in the region inland from the settlement of Qaanaaq and around Camp Century (including the green profile, Fig. d), around the NorthGRIP ice core, and a region ∼ 150 km ENE of the NEEM ice core. Whilst these smoother regions are at a range of bed elevations (ranging from ∼ 800 m NE of Qaanaaq to around sea level in the interior), they are all spatially correlated with flatter bed topography (Fig. e). Correspondingly, many rougher regions (higher ν, lower A) are spatially correlated with more complex topography – e.g. the region of the ice sheet inland from the Melville Bugt coast (including the red profile, Fig. b). However, some rougher regions of the bed have a less obvious correlation with higher contour gradients – e.g. the flatter regions inland from the Humboldt glacier.

Pronounced spatial variation in the Hurst exponent, H, is evident in Fig. d. H also has a inverse relationship with A and spatially correlates with the bed topography in a similar manner to ν. In other words, lower H is associated with higher A and flatter regions of the bed – e.g. near Camp Century – whilst higher H is associated with lower A and generally more complex bed topography – e.g. inland from the Melville Bugt coast and inland from Ryder glacier (including the black profile, Fig. a). In Sect.  a quantitative assessment of this relationship is made using the radar scattering model. The simple notion that, at the topographic scale, rougher regions of the bed correspond to higher H can be related back to the power-law scaling relationship in the deviogram (Fig. b). The length scales for the rms deviation maps ν(Δx=30 m) in Fig. a and ν(Δx=150 m) in Fig. b are chosen as they are the lower and upper bounds in the deviogram calculation for H. It is notable that, despite the clear spatial variation in H in Fig. d, the overall spatial distributions for ν(Δx=30 m) and ν(Δx=150 m) are remarkably similar. Thus, from a purely visual inspection of ν at different length scales, the pronounced spatial variation in H is not immediately apparent.

The basal thermal state prediction by (Fig. f) represents an up-to-date best estimate for the GrIS at a 5 km resolution. It is based upon a trinary classification: likely thawed/above pressure melting point (red), likely frozen/below pressure melting point (blue), and uncertain (grey). The mask was determined using four independent methods: thermomechanical modelling of basal temperature, basal melting inferred from radiostratigraphy, surface velocity, and surface texture. The mask is therefore independent of our RES-derived data fields. There are some obvious correlations between the basal thermal state prediction and the RES-derived roughness and scattering data. For example, many predicted thawed regions toward the margins – e.g. the region of the ice sheet inland from the Melville Bugt coast – correspond to rougher terrain (higher H and ν) and diffuse scattering (lower A). However, there are regions of predicted thaw that demonstrate the opposite behaviour (lower H and ν and higher A) – for example, the two interior regions previously identified as smooth around the NorthGRIP ice core and the region ENE of NEEM. The scattering signature of predicted thawed regions is therefore non-distinct and can be either specular or diffuse. Predicted frozen regions tend to be smoother with specular reflections (higher A), although it is clear that spatial variation is present with some regions exhibiting more diffuse scattering (lower A). Section  provides a more detailed statistical analysis.

There are some clear discontinuities in the flight-track maps for ν and H in Fig. . These can be explained by either roughness anisotropy or the self-affine terrain model breaking down in certain regions (e.g. a sharp terrain discontinuity such as a subglacial cliff). By contrast the map for A is smoother, which is consistent with its interpretation as an isotropic scattering parameter.

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, H. The total frequency distribution for H, corresponding to the flight-track data in Fig. d, is shown in Fig. a. The distribution is divided into three categories: (i) H>0.75 (“high” H), (ii) 0.5<H≤0.75 (“medium H”), and (iii) H≤0.5 (“low” H), which we later use to compare with the radar scattering model predictions. These categories correspond to approximately 30, 50, and 20 % of the total data respectively. Approximately 0.1 % of the H estimates are > 1 and none of the H estimates are < 0, representing near-ubiquitous self-affine scaling behaviour (0<H<1). An overall negative skew for the distribution of H is observed with a mean value of 0.65, indicating that the majority of the subglacial terrain along the flight tracks lies between Brownian (H=0.5) and self-similar (H=1) scaling regimes. The spatial coverage of the radar flight tracks in Fig. d is, however, more comprehensive in regions of higher H. Thus the mean value and skew of H in Fig. a are likely overestimates and underestimates of true (equal area) averaged values for the region of the northern GrIS in Fig. .

The self-affine coherent scattering model (Sect. ) predicts that there are two roughness degrees of freedom that control A: H (the primary control) and νλ (the secondary control). At metre scale, νλ is significantly smaller than the along-track resolution (∼ 30 m) and therefore cannot be observed directly. Additionally, given the theoretically predicted primary dependence of A upon H, a natural starting point is to compare with the observed relationship between A and H (Fig. ). Based upon the assumption that νλ varies spatially, a statistically distributed inverse relationship between H and A is predicted which corresponds to the family of predicted curves in H–A space in Fig. . This approach assumes a downward extrapolation of H from the topographic scale to the wavelength scale in the radar scattering model.

In order to test this prediction, we considered the statistics of three separate A distributions for each H category, which are shown for high H in Fig. b, medium H in Fig. c, and low H in d. A nearest-neighbour interpolation was used to pair each A value (∼ 100–150 m along-track spacing) with each H value (1 km along-track spacing). The lowest mean value, smallest variance, and strongest positive skew are observed for the high-H category. This supports the general prediction in Fig.  that higher A values (specular reflections) are suppressed in regions of higher H, with lower A values (diffuse scattering) being more probable. The highest mean value, greatest variance, and weakest positive skew are observed for the low-H category. Again, this supports the prediction in Fig.  that A is less constrained in regions of lower H, with a tendency toward higher values (specular reflections). As would be expected, the A-distribution statistics for the medium H category lie between the high-H and low-H categories with intermediate mean values, variance, and skewness. Finally, the observed values of A in Fig.  range from ∼ 0.03 to 0.60, which is in agreement with the theoretically constrained maximum value of 0.65.

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 . Conceptually, their approach assumes that water in thawed regions has a similar RES signature to deep subglacial lakes which exhibit brighter and more specular reflections than surrounding regions (e.g. ). In their algorithm wet regions are discriminated if (i) the relative bed reflectivity is above a threshold (using an attenuation model where the attenuation rate has an inverse relationship with surface elevation) and (ii) the abruptness is also above a threshold (around 0.3). Thus, in their approach, high abruptness (specular reflections) is a necessary, but not sufficient, criterion for identifying basal water. A further feature of their approach is that spatial continuity for water is imposed, i.e. only larger-scale regions (∼ 100s of km2 and upwards) are considered.

The distributions for all RES-derived data exhibit pronounced statistical differences between thawed and frozen regions (Fig. ). The mean value for H in thawed regions is 0.74 with a strong negative skew (Fig. a), whereas the mean value for H in frozen regions is 0.54 with a weak negative skew (Fig. b). The mean value for ν(Δx=30 m) in thawed regions is 6.36 m, which is over double the mean value of 2.80 m in frozen regions. A qualitatively similar distinction between thawed and frozen regions is also present for ν(Δx=150 m), with a mean value of 21.7 m in thawed regions and 7.2 m in frozen regions (not shown). The thawed distribution for A is similar to the high-H category in Fig. b, with a mean A value of 0.165 and strong positive skew. The frozen distribution is similar to the low-H category in Fig. d with a mean A value of 0.264 and a weak positive skew. These statistics demonstrate a contradiction with the basal water discrimination algorithm of . Lower abruptness (diffuse scattering) is more common in thawed regions where basal water is likely to be present. Moreover, the necessary high-abruptness (specular reflections) condition for water is generally not satisfied (particularly at the larger spatial scales that were considered by when mapping basal water).

In RES data analysis, cross-over distributions at flight-track intersections can give an indication of uncertainty based upon internal consistency (e.g. ). However, due to the anisotropy in Fig. d, cross-over analysis for H(ν(Δx)) cannot be applied directly. Hence repeat estimates were made using the variogram to calculate H(ξ(L)) (i.e. calculating H using rms height). The map for H(ξ(L)) (not shown) has a similar spatial distribution as Fig. d but with greater high-frequency noise apparent. Differencing the estimates as H(ν(Δx))–H(ξ(L)) and performing cross-over analysis gives a mean bias of -0.026 and a standard deviation of 0.10 (10 % of the parameter range). The small mean bias is potentially explained by the variogram estimates being at a slightly larger length scale (L ∼ 90–210 m). Additional cross-over analysis using different profile window sizes (e.g. 15 km) confirms that 0.10 serves a reasonable estimate for the uncertainty of H. Since A is assumed to be isotropic, the uncertainty can be estimated via cross-over analysis of flight-track intersections. This gives a cross-over standard of ∼ 0.05 (again ∼ 10 % the parameter range).

As part of the analysis we also considered estimation of the breakpoint transitions for H(ξ(L)) and H(ν(Δx)) using the segmented linear regression procedure described Sect. . The exact values of the breakpoints depend upon how strict the stopping criterion is, so here we just discuss some general trends. Firstly, the self-affine scaling relationships often extend over a much greater length scale than the upper length scale used in the calculation of H (often over 500 m as occurs in Fig. ). Secondly, the breakpoints for H(ν(Δx)) generally occur at greater length scales than for H(ξ(L)). Thirdly, the break points for both H(ν(Δx)) and H(ξ(L)) tend to be greater toward the ice-sheet margins where H is higher.