Radar/GPS profiling

A 42 km long profile was made from the ice-rise promontory to the adjacent ice shelf (Figs 1 and 2). We used a 5MHz ground-based radar-profiling system positioned with a geodetic L1 GPS receiver (Reference Pattyn, Matsuoka and BertePattyn and others, 2010). Radio-wave propagation speed in ice is assumed to be 168.4mμs^–1, which corresponds to permittivity of 3.17. Surface topography was determined by a kinematic GPS survey using a temporary base station located near the crest of the ice promontory. The position of the base station in the International Terrestrial Reference Frame was determined by the Precise Point Positioning method. Earth Gravitational Model 1996 (EGM96) shows that the geoid height is ~24m above the World Geodetic System 1984 ellipsoidal elevation (WGS84) in the vicinity of the grounding zone and the floating ice that we surveyed. Therefore, we added 24 m to the GPS-measured elevations in order to derive elevations relative to regional sea level.

Fig. 2.

Observed and modeled properties along the radar profile. The location marker is referred to in Figure 4. (a) Radargram is adjusted to account for surface elevation variations. (b) Detailed surface topography of the grounding zone and floating ice. (c) δz = z _s – z _float, where z _s is the measured ice-surface elevation and z _float is the thickness of freely floating ice (Equation (4)). Vertical gray dashed line shows where ice becomes freely floating. (d) Depth-averaged attenuation rates ⟨N⟩ estimated using ice-flow models for geothermal fluxes GF = 42 and 54 mWm^–2 and for ice-flow enhancement factors mof 0.8 and 1.0. Results for different m are nearly identical. Color legend is as shown in (e). (e) Estimated relative reflectivity [R _basal]_dB using attenuation models. Also shown are anomalous basal returned power (Δ[P^c _basal]_dB) values (Equation (5)) estimated using a widely used radar method for four data ensembles (Fig. 4): the black solid curve is for the entire dataset, the two dashed curves are for grounded and floating ice, respectively, excluding the vicinity (5 km) of the grounding line, and the gray curve is for the vicinity (±5 km) of the grounding line. [R _basal]_dB and Δ[P^c _basal]_dB shown here are filtered with a 1km running average.

The measured radar power P _basal returned from the base of the ice column is affected by instrumental factors S, geometrical factors G _basal, two-way englacial attenuation L _ice and basal reflectivity R _basal. The relationship between these factors in the decibel scale ([x]_dB = 10 log₁₀ x) can be written as (e.g. Reference Matsuoka, Morse and RaymondMatsuoka and others, 2010)

(1)

Figure 2a shows the absence of englacial scatters such as buried crevasses, so losses from L _ice and G _basal are expected to dominate. The ratio of the ice thickness (up to 700 m) to the radio-wave wavelength (~30 m) is small, so extinction of power due to birefringence caused by alignments of ice crystals is probably insignificant (Reference Matsuoka, Gades, Conway, Catania and RaymondMatsuoka and others, 2009a). Therefore, Equation (1) is a reasonable approximation for our analysis.

The geometric factor G _basal is a function of ice thickness H and depth-averaged permittivity ε (assumed to be 3.17): . G _basal varies by ~10 dB for ice-thickness variations between 200 and 700 m, but it is negligibly affected by possible variations of the depth-averaged permittivity (i.e. density). Therefore, assuming that instrumental factors S remain stable over time, variations in the geometrically corrected returned power P^c _basal are caused by variations in englacial attenuation and basal reflectivity:

(2)

Values for P^c _basal come from measurements of returned power and ice thickness (calculated from the two-way travel time). Here we derive an accurate estimate L _ice in order to calculate [P^c _basal]_dB + [L _ice]_dB along the profile, thereby providing a proxy of spatial variations of [R _basal]_dB.

Attenuation estimates

We estimated depth-averaged englacial attenuation rates in two ways. The first method uses measurements of the geometrically corrected returned power as a function of ice thickness (e.g. Reference Winebrenner, Smith, Catania, Conway and RaymondWinebrenner and others, 2003; Reference Jacobel, Welch, Osterhouse, Pettersson and MacGregorJacobel and others, 2009). The thickness dependence of the returned power can be used as a proxy for englacial attenuation if the basal reflectivity is independent of ice thickness, that is when , (Equation (2)). This method is appropriate when englacial losses (e.g. scattering from buried crevasses) other than G _basal and L _ice are significant (e.g. Reference Raymond, Catania, Nereson and Van der VeenRaymond and others, 2006).

The second method uses an attenuation model that depends on depth profiles of ice temperature and chemistry. Englacial attenuation is a result of contributions from pure ice, acidity and sea salt in the ice column, which have different temperature dependences. However, the lack of information on depth profiles of chemistry precludes calculating the contributions from acidity and sea salt, and here we consider only the pure-ice contribution. Acidity and sea salt contribute 26% and 21% of the total depth-averaged attenuation rate at Siple Dome and Vostok ice-core sites, respectively (Reference Matsuoka, MacGregor, Pattyn and CroccoMatsuoka and others, 2011). The local one-way attenuation rate N (dBkm^–1) is

(3)

Here σ _pure = 9.2 μsm^–1, E _pure = 0.51 eV, T _r = 251 K and k is the Boltzmann constant (Reference MacGregor, Winebrenner, Conway, Matsuoka, Mayewski and ClowMacGregor and others, 2007; Reference MatsuokaMatsuoka, 2011). These values have significant uncertainties because there are few published measurements of the temperature dependence of dielectric properties. Consequently, uncertainty in attenuation sensitivity to ice temperature and in estimated attenuation is also large. We ignore this uncertainty in the analysis below.

The ice temperature T for the attenuation model (Equation (3)) is calculated using a two-dimensional (2-D) thermomechanical high-order model (Reference PattynPattyn, 2002, Reference Pattyn2003). The model calculates the horizontal velocity field in the ice sheet using higher-order physics, taking into account vertical shearing as well as longitudinal stress gradient. The velocity fields for both the ice sheet and ice shelf are derived simultaneously. Vertical velocities are obtained from mass conservation and by considering the present-day ice topography in steady state (Reference HindmarshHindmarsh, 1999). The velocity field is fully coupled to the temperature field, based on vertical diffusion, horizontal and vertical advection and friction. Flow enhancement factor m is assumed to be uniform at 0.8 or 1.0. Boundary conditions to the ice-flow field are obtained from balance velocity calculations at either side of the profile presented in Figure 2, and a basal friction parameter at the bottom, relating basal velocities to basal shear stress (Reference Pattyn, Huyghe, de Brabander and De SmedtPattyn and others, 2006). Basal friction in the ice shelf is set to be zero.

Surface temperature T _s over the entire profile and geothermal flux GF over the grounded ice are assumed to be uniform: T _s = –12˚C and GF = 42 or 54 mWm^–2. Since the model is not coupled with the ocean, a temperature of –2˚C, an approximate melting point of sea water corrected for pressure at depth, is prescribed at the ice–water interface of floating ice. In this area, neither measurements nor model estimates of ocean temperature have been made. Surface accumulation rates are determined from the shallowest radar-reflector pattern and a regional mean of the recent accumulation rate of 0.27ma^–1 from a continental mass-balance model (Reference PattynPattyn and others, in press). For each geothermal flux and flow-enhancement factor, the model was run more than 20 000 years until the ice-flow and temperature fields reached a steady state.

The radar transect across the ice promontory was aligned perpendicular to surface contours, which is an approximation of the ice-flow direction. However, the radar transect on the ice shelf may deviate from the flowline. There are no depth variations of ice-flow speed over the ice shelf (plug flow), so the vertical velocity profile is linear so that the steady-state depth profile of temperature is a function of vertical advection rates and constrained by the upper and lower surface temperatures, regardless of the upstream ice-flow field (e.g. Reference HindmarshHindmarsh, 1999). Therefore, the effects of transverse stress coupling and the deviation of the radar profile from the true flowline on the estimated temperature field are expected to be small, as horizontal (advection) effects are negligible.

Radar and GPS measurements

Figure 2a shows a radargram with surface elevation adjustments using GPS-measured surface elevations z _s. Figure 2b shows details of the surface topography in the vicinity of the grounding zone, and Figure 2c shows δz = z _s – z _float, where z _float is the thickness of freely floating ice that can be derived from the ice thickness H and densities of ice ρ _i and of ocean water ρ _w:

(4)

When ice is in hydraulic equilibrium (and all ρ _i, ρ _w and H are accurate), δz is zero. In earlier work we showed that δz is minimal over the downstream side (x _a > 25.6 km) of this profile (Reference PattynPattyn and others, in press. At landward sites (x _a < 25.6 km), δz increases rapidly, so the ice is no longer in hydrostatic equilibrium. Therefore, we determined that the most landward site of freely floating ice is at x _a = 25.6 km.

We did not find a topographic minimum or a break in slope associated with the non-hydrostatic ice flexure response typically found in grounding zones (Reference Brunt, Fricker, Padman, Scambos and O’NeelBrunt and others, 2010). Although our repeat GPS profiles between the ice promontory and the ice shelf do show some changes in elevation at sites landward of freely floating ice, they do not allow us to determine the most landward site affected by tidal forcing.

Attenuation estimates using thermomechanical model

Figure 3 shows englacial attenuation rates N estimated using the thermomechanical model for (GF,m) = (54mWm^–2, 1.0). In the model, we prescribe the grounding line at x _a = 25.6 km, although in reality it is slightly landward of this position. Ice temperature increases with depth monotonically, as does the attenuation rate. The model predicts that the basal temperature of grounded ice is at the pressure-melting point, except for the vicinity of the ice-rise crest when the geothermal flux is 54 mWm^–2 regardless of the flow-enhancement factor. For a geothermal flux of 42 mWm^–2, the basal temperature of the grounded ice is below the pressure-melting point by several degrees.

Fig. 3.

Variations in local attenuation rate N estimated for geothermal flux of 54 mWm^–2 and flow-enhancement factor of 1.0. Red and blue markers along the abscissa show locations where the basal ice is at the (pressure-)melting point over grounded and floating ice, respectively.

Figure 2d shows the spatial pattern of depth-averaged attenuation rate ⟨N⟩, which is derived by integrating local attenuation rates N (Fig. 3) vertically at sites along the profile. Total attenuation [L _ice]_dB is proportional to the ice thickness H and the depth-averaged attenuation rate ⟨N⟩: [L _ice]_dB = 2⟨N⟩ H[ (the factor of 2 is to account for the two-way travel). Although ⟨N⟩ increases with geothermal flux GF and decreases with flow-enhancement factor m, the amplitudes of ⟨N⟩ variations along the profile are 1.3– 1.7 dBkm^–1 and the spatial pattern remains similar regardless of GF and m (Fig. 2d). Differences in the flow-enhancement factors have negligible effects on ⟨N⟩.

It is still an open question whether the temperature in Equation (3) is relative to the pressure-melting point, because all dielectric properties used to determine σ _pure and E _pure were measured with meteoric or laboratory-grown ice at the standard atmospheric pressure, so the (pressure-) melting point of the ice specimen for those measurements is close to 0˚C. If an attenuation approximation (Equation (3)) is more appropriate for temperature relative to the pressure-melting point, then the estimated variations in ⟨N⟩ over the shelf ice would be larger than those shown in Figure 2 by ~1dBkm^–1. This difference is nearly uniform along the profile, so this uncertainty in the attenuation parameterization (Equation (3)) does not significantly affect our interpretations of the basal environment.

Basal returned power dependence on ice thickness

Figure 4 shows basal returned power dependence on ice thickness for four linear fits: (1) for the entire dataset; (2) for floating ice 5 km or more seaward of the prescribed grounding line; (3) for grounded ice 5 km or more landward of the grounding line; and (4) in the vicinity (±5 km) of the grounding line. The attenuation rate 〈N〉 varies by a factor of 2, depending on the choice of data ensembles used in the fit (Fig. 4): 58 dB km^–1 (all data), 56 dB km^–1 (floating ice), 38 dB km^–1 (grounding zone) and 28 dB km^–1 (grounded ice). In contrast, results from the temperature-dependent attenuation model (Equation (3)) indicate relatively uniform depth-averaged values, 30–33 dB km^–1 (Fig. 2d). As discussed earlier, the temperature-dependent attenuation model used here does not account for contributions from ice chemistry, which could increase these calculated 〈N〉 by ~10 dB km^–1 if sea salt of 17 μmol L^–1 is included in the ice.

Fig. 4.

Ice-thickness dependence of geometrically corrected radar power returned from the basal interface. Colors represent the location along the profile (Fig. 2). Solid black line shows linear fit for the entire dataset. Black and gray dashed lines show fits for grounded and floating ice, respectively, excluding the grounding-line vicinity (5 km). Solid gray line shows fit in the vicinity (±5 km) of the grounding line. Inset shows the slope of these lines (unit: dBkm^–1 for round trip); half of these values are frequently used as one-way attenuation rate ⟨N⟩.

The thickness dependence of the basal returned power does not give a regional mean attenuation rate when the attenuation rates vary significantly in the data ensemble or the bed reflectivity depends on ice thickness (Reference MatsuokaMatsuoka, 2011). Along our profile, the attenuation rate estimated from the attenuation temperature model is nearly uniform within ±1dBkm^–1 (Fig. 2d). However, the grounded ice is generally thicker than the floating ice, and a significant thickness variation is found in the floating ice (Fig. 4; color of markers show the locations along the profile (see Fig. 2)). These features yield apparent thickness dependence of the basal reflectivity, which is a likely reason why attenuation rates estimated with the thickness dependences of the basal returned power do not match well with the model estimates.

Inferring conditions at the basal interface

Although the radar instrument factor is unknown, the transmitter used here is known to be stable (Reference Matsuoka, Wilen, Hurley and RaymondMatsuoka and others, 2009b). So, [P^c _basal]_dB + [L _ice]_dB provides a proxy of spatial variations of [R _basal]_dB (Equation (1)). The anomalous basal returned power Δ[P^c _basal]_dB from values predicted with the traditional radar method (Fig. 4) also provides a proxy of spatial variations of [R _basal]_dB:

(5)

Figure 2e shows Δ[P^c _basal]_dB anomalies from the mean of Δ[P^c _basal]_dB for each data ensemble along the profile as a proxy of relative basal reflectivity. Attenuation-rate estimates depend on the geothermal flux (Fig. 2d), but the spatial patterns of [R _basal]_dB estimated here are not sensitive to the magnitude of the geothermal flux because the ‘pattern’ of ⟨N⟩ is controlled primarily by the ‘pattern’ of ice thickness and accumulation rate.

Figure 2e shows four distinct features in [R _basal]_dB estimated with attenuation modeling and in Δ[P^c _basal]_dB. First, Δ[P^c _basal]_dB for the entire dataset show similar values for the grounded ice and floating ice in general, although basal conditions for the grounded and floating ice are presumably distinct. Near the profile end on the floating ice (x _a >~35 km), Δ[P^c _basal]_dB for the entire profile and for the floating ice are smaller than for the grounded ice. However, radar-measured ice thickness gives surface elevation for the freely floating ice consistent with the GPS-measured elevation over this area (Fig. 2b). The presence of marine ice is therefore unlikely, and basal conditions should be more or less constant over the floating ice. Also, attenuation estimates used for these Δ[P^c _basal]_dB over the entire profile and over the floating ice are extremely high (56–58 dB km^–1 for one way). Therefore, we conclude that these Δ[P^c _basal]_dB values for the entire dataset and for the floating ice do not represent variations in the basal conditions.

Second, Δ[P^c _basal]_dB for the grounding-line vicinity varies ~20 dB. This variation happens in the horizontal distance of ~5 km. A similar feature is found in [R _basal]_dB estimated with attenuation modeling. In the grounding-line vicinity, the modeling predicts an attenuation variation less than 1 dB km^–1 for one way; the ice thickness varies <150m in this area, giving minimal variations in the total attenuation [L _ice]_dB within the grounding zone. Thus, the predicted variations in [R _basal]_dB are essentially identical to variations in [P ^c _basal]_dB. In practice, basal reflectivity is insensitive to the attenuation estimates in such areas where the ice thickness and attenuation rate vary little.

Third, [R _basal]_dB estimated with attenuation modeling has a broad minimum near the crest (x _a =~12 km). It becomes larger towards both ends of the profile over the grounded ice. This pattern is also found in the modeled basal temperature, when geothermal flux is 54 mWm^–2 (Fig. 3). The range of [R _basal]_dB over the grounded ice is ~20 dB from the crest to the western end of the profile and 10 dB from the crest to the grounding line. However, basal reflectivity is related to the local amount of water, rather than local melting rate, so our 2-D modeling cannot be used to constrain the flow-model parameters such as geothermal flux, which alter the ice temperature, using the pattern of the observed bed reflectivity.

Finally, variations of Δ[P^c _basal]_dB depend on data ensembles used for constructing the fit between returned power and ice thickness (Fig. 4). For example, 〈d[P ^c _basal]_dB /dH〉H derived for the floating ice is –144 or –112 dB km^–1 when the data ensemble includes or excludes, respectively, the 5 km portion of the profile close to the grounding line. However, 〈d[P ^c _basal]_dB /dH〉H derived for the grounded ice varies only 3dBkm^–1 if the data ensemble includes or excludes the grounding-line vicinity. Therefore, the analysis of the spatial pattern of Δ[P^c _basal]_dB is not rigorous. [R _basal]_dB derived from four modeled temperature-dependent attenuation rates show similar spatial patterns (Fig. 2e). These results suggest that basal conditions revealed by the temperature-dependent attenuation modeling are robust despite uncertainties in the modeling parameters and boundary conditions.