In this paper, we apply radar tomography methods to very-high-frequency, airborne synthetic-aperture radar data to measure the ice thickness field and to construct three-dimensional basal image maps of a 5 × 20 km study area located along the southern flank of the Jakobshavn Glacier, Greenland. Unlike ice radar measurements typically made at nadir, our approach uses radar-echo phase and amplitude measured across an antenna array to determine the propagation angle and signal strength of pixel elements distributed on each side of the aircraft flight path. That information, combined with knowledge of aircraft position and the assumed dielectric properties of the glacier, can be used to measure ice thickness and radar reflectivity across a 3-km wide swath. Combining ice thickness and surface topography data, we estimate basal topography and basal drag. We conclude that the glacier is sliding over the bed. We use the three-dimensional image maps of the bed to inspect the modern subglacial geomorphology and find for the first time beneath the Greenland Ice Sheet assemblages of long ridge-groove landforms that are oriented in the direction of the ice flow. Spatial dimensions (10 to 30 m depths, 150 to 500 m spacing and lengths of 10 km or more) and correlation with the current ice flow direction suggest that these are glacial erosional features similar to mega-grooves observed on deglaciated terrain.
 Images of the surface of the polar ice sheets provide graphic evidence of changes in the polar environment and contribute to a quantitative understanding of the forces presently shaping the glacial cover. The shifting margins of the polar ice sheets and the catastrophic collapse of ice shelves have been documented in imagery from spaceborne optical and microwave synthetic aperture observations [Jezek et al., 2003]. Changing patterns of surface melt are detected in time-series of satellite passive microwave data [Bhattacharya et al., 2009]. Although the surface properties of the ice sheets are well documented, the nature of the glacier bed remains obscured by its icy cover. Revealing basal properties, such as the topography and the presence or absence of subglacial water, is important for understanding the processes driving observed, rapid changes, accurately estimating ice flux from the interior of the ice sheet to the sea, and forecasting anticipated changes in the size of the ice sheets.
 Today, radars operating between 1-500 MHz are the primary tools used for measuring ice sheet thickness and basal topography, as well as for inferring basal properties over large areas [Allen et al., 1997; Gogineni et al., 1998]. These radars typically acquire profile data along nadir tracks that are often separated by 5 km or more. Consequently, we were motivated to develop new approaches [Jezek et al., 2006; Gogineni et al., 2007; Paden et al., 2010] that, analogous to spaceborne imaging of the ice sheet surface, yield quantitative, three-dimensional imagery of the ice sheet base. In this paper, we present images of the base of the Greenland Ice Sheet constructed by applying radar tomography to airborne synthetic-aperture radar data. Our 5 km × 20 km study area is located along the southern flank of the Jakobshavn Glacier. We briefly discuss our ice thickness and echo-amplitude measurement techniques and then compare basal topography and reflectivity with independent measurements of ice sheet motion. We use the comparisons to estimate local mass balance, to estimate the important stresses acting on the glacier, and to infer information about the subglacial geomorphology.
2. Radar Tomography
 Four technical obstacles confront the airborne acquisition of wide-swath, three-dimensional images of the ice sheets. The first is that scattered signals from the upper surface of the ice sheet can mask the weaker, off-nadir signals from the ice sheet base. The second obstacle involves separating signals from the left and right sides of the aircraft. Side looking radars, for example, overcome this obstacle by steering the antenna array. However, left/right signal isolation is difficult with arrays of only 6–8 elements. The third obstacle is obtaining sufficient signal strength to reconstruct the basal topography across the swath and map the spatial reflectivity. Finally, sub-wavelength knowledge of antenna position is required all along the flight track. Based on a series of experiments conducted from 2006–2008 [Jezek et al., 2008], we selected tomographic processing of multi-aperture radar data as the most suitable processing approach for dealing with these technical obstacles and for robustly imaging the base of the polar ice sheets.
 Tomography relies on a distributed signal-transmitting and receiving array to localize the position of an echo source in three dimensions. The approach is widely used in medicine for imaging the body and has been extensively used in ocean acoustics and seismologic studies of the Earth. It is also used to a lesser extent in radar studies [Reigber and Moreira, 2000; Valle et al., 1999; Mast and Johansson, 1994]. Unlike the related method of radar-interferometry, radar-tomography relies on the absolute phase measured by each receiving antenna. The amplitude and phase of signals measured by each receiving antenna element are related to the range and arrival angle of echoes from individual targets at a constant range from the antenna elements. In this method, each range bin containing possible targets is represented by matrices that relate the total signal measured at each antenna element to the phase and amplitude of signals arriving from the left and right sides of the aircraft and from the surface and the base. (X. Wu, manuscript in preparation, 2010).
 The calculation is initialized by the nadir returns. Measured data are then inverted using the maximum likelihood method to solve for signal amplitude and arrival angle ranging over both the left and right sides of the flight path. Given these parameters and an estimate of the index of refraction for ice, the ice thickness can be determined. Subtracting the ice thickness from surface elevation data yields the basal topography. For measurements made at elevations a few hundred meters above the ice sheet surface and supplemented with independent surface elevation data, the number of equations can be reduced by assuming that surface clutter is weak compared to the basal return, which is justified by the relatively small basal incidence angle (maximum of about 20 degrees) and the large surface incidence angle (about 80 degrees) for coincident surface and basal returns. We also assume that scattering from internal layers is weak which is justified both on the weak dielectric contrast reported for the layers and the specular scattering behavior of the layers [Paren and Robin, 1975].
 We developed a radar for operation at 150 MHz with a bandwidth of 20 MHz and with multiple receivers for sounding and imaging polar ice sheets [Rodriguez-Morales et al., 2008]. In July 2008, we flew the 150-MHz radar mounted in a Twin-Otter Aircraft. The antennas consisted of two, six-element dipole-arrays mounted under each wing of the aircraft. Eight elements were used as receivers and four elements were used as transmitters. Data were collected along parallel flight tracks located just south of the main channel of Jakobshavn Glacier when surface melting was occurring. Adjacent tracks were flown during the inbound and outbound legs of the experiment. Swath widths for each track were about 3 km wide, and the two swaths were balanced to remove residual biases and mosaicked. The average off-nadir ground-range resolution is about 50 m in both azimuth and range. Approaching nadir, the ground range resolution degrades to about 75 m on either side of the flight path resulting in ambiguous signals directly beneath the aircraft.
3. Basal Imaging
 We applied synthetic aperture radar processing techniques to range and azimuth compress 150- MHz data from each receiver channel. We used radar tomography algorithms to construct three-dimensional images of the bed. The mosaicked data collected along two adjacent flight lines over our study area are shown in Figure 1. Figure 1a shows a RADARSAT-1 spaceborne synthetic aperture radar image acquired during the winter of 2000. Locations of summer surface lakes appear as irregular bean-shapes. Lake locations are more or less stationary with time, but the same lake does not always fill/drain to the same level each year, resulting in some dark shapes and some bright shapes. Narrow streams connecting the lakes and crevasses are also located in the scene. The red box outlines the study area. Surface elevation data in meters above the ellipsoid were provided by B. Csatho (personal communication, 2007) who fused ICESat laser altimetry data with a photoclinometry digital elevation model [Scambos and Haran, 2002] and validated the model against Airborne Topographic Mapper data. The elevation contours, within the red box, document the gently sloped topography of this part of the ice sheet.
Figure 1b shows the measured ice thickness, which is about 1400 m on average. The ice thins by about 400 m across the study area. As validation, we compared independently-derived, nadir ice thicknesses measured along flight lines that were oriented obliquely to our flight lines. We found 16 m root mean square differences and no detectable bias between our cross-swath ice thickness measurements and previously-acquired nadir data. Also shown in Figure 1b are interferometrically derived [Goldstein et al., 1993] surface velocity vectors (I. Joughin, personal communication, 2008, and OSU processed data at www.bprc.osu.edu/rsl). Average speed on the upstream end of the area is about 350 m/yr. The speed increases to about 425 m/yr at the downstream end of the study area.
 The general form of the bed topography expressed as contours is presented in Figure 1c. The lowest topography is found on the right side where depths are about 200 m below the ellipsoid. The topography rises in the right center, where a low hill (150 m elevation) flanks the upper part of the data set. Color codes represent relative radar reflectivity, where we have subtracted the scene-wide-mean angular decrease in backscatter. The darkest areas are on the right and the brightest are on the left. Brightness variations are driven by changes in the ice sheet surface properties, attenuation through the ice, and basal reflectivity properties. Based on the RADARSAT image, surface reflectivity is likely uniform over much of the scene, save for the surface lakes. The mean ice thickness changes by about 20%, thinning from right to left, so some of the brightening may be due to reduced radar-wave absorption through thinner ice, but we do not attempt to make a thermal absorption correction, which would require knowledge of the depth-varying physical temperature. We attribute the remainder of the intensity variation to changes in basal properties, which could either be changes in basal roughness or wetness. Based on the reflectivity data alone, we cannot discriminate between roughness and wetness. However, the measurable off-nadir returns suggest that water, if present, is most likely in the form of a film, as opposed to ponded water. Ponded water would result in a strong specular echo and very dark off-nadir returns [Niamsuwan, 2009, pp. 107–114].
 We highlighted the details of the basal topography by first vertically exaggerating the topography by a factor of 2 and then using an artificial light source located at a 45° elevation and 40° azimuth to depict shadowing (Figure 1d). We then draped the hill-shaded topography with the radar reflectivity as a color overlay. The combination yields a unique image of the base of the ice sheet. Although our data set only covers a small patch under the ice sheet, we can deduce several properties from the different data fields. At this location, the characteristic bed form is an assemblage of long ridge-groove landforms that trend from lower right to upper left and which are superimposed on the modest rises and depressions more noticeable in the contour plot. The bed topography and its roughness and structures have the appearance of a bedrock surface with little-to-no drift sediments present. The most striking aspects are long, slightly sinuous, positive-relief features and more discontinuous and straighter linear grooves, with depths of 10 to 30 m and spacings of 150 to 500 m. As with many bedrock surfaces the final morphology observed is often a mix of landforms produced by agents of erosion superimposed on, and in part influenced or controlled by, the underlying geological structures within the rock. This surface (Figure 1d) has such an appearance with the slightly sinuous positive relief features resembling outcropping of more resistant geological structures (perhaps foliation), and we speculate that where weaker elements exist, these appear to have been exploited in places by erosional grooving. Such morphology is reminiscent of glacier paleo-megagrooves observed in exposed crystalline bedrock and which have been reported to be kilometers long with depths of 10 to 20 m and groove separations of 50 to 200 m [Smith, 1948; Bradwell, 2005; Bradwell et al., 2008]. Abrasion or scouring by debris-rich basal ice or from migrating subglacial sediments or subglacial water are hypothesized to be the possible erosional agents for creating the grooves [Witkind, 1978; Gjessing, 1966; Kor et al., 1991]. We posit the landforms carved into the modern glacier bed are also glacier erosional features, in part because they have forms and dimensions similar to those of glacial mega-grooves reported in the literature (Figure 2). Additional evidence for this interpretation follows from the agreement between modern glacier flow lines overlain on the image. Flow lines are derived from synthetic aperture radar interferometric velocity measurements, and there is very good agreement between the trends of the flow lines and the trends of the basal landforms. Interpreting the landforms to be erosional features cut into bedrock implies that the glacier has been flowing persistently in the same direction. We note that this direction is consistent with slightly converging flow into the trough of the Jakobshavn ice stream.
 Low velocity gradients (less than 0.01 yr−1) in both the along and cross flow directions coupled with weak thickness gradients (less than 0.1 m/m) implies that there is little longitudinal or lateral resistance (less than 3 kPa for a conservative rate factor of 400 kPa yr1/3) to support the glacier at this location. Then, force balance implies that the gravitational driving stress (the product of the ice thickness, the surface slope relative to the geoid, the ice density, and the acceleration due to gravity) is resisted primarily by basal drag [Van der Veen, 1999, p. 36]. So, we calculate the driving stress in the direction of maximum surface slope, and hence directly infer the basal drag. The surface slopes, where geoidal elevations are averaged over a 2-km grid, are generally aligned with the flow direction. Driving stress/basal drag contours are shown in Figure 1e. As suggested by Figure 1a, the surface elevation contours bunch in the left center part of the image where there is an increase in surface slope and the driving stress approaching 200 kPa. The peak in driving stress or basal drag is also associated with the upstream flank of a subglacial hill. Driving stress is smaller on the ends of the study where the magnitude drops to about 100 kPa and where the basal elevations are lower. On average, the values are similar to those estimated by Bamber et al.  (about 120 kPa for this flank of the ice sheet). The higher peak values likely arise from the finer grid used in this analysis. Our basal drag estimates can be contrasted to the values for ice streams (10–20 kPa), which move by sliding over a well-lubricated, sedimentary bed. Our much higher basal drag values indicate that the base is resisting flow, but it is likely that some basal sliding is occurring, thus indicating the presence of basal water. Specifically, surface velocities computed using a simple laminar flow model [Paterson, 1994, p. 251] indicate that low values of the depth-averaged, temperature-dependent rate factor (200–250 kPa yr1/3 corresponding to about −5° C) are required to approximate measured surface velocities in the central part of the study area. Even lower rate factors are required on the eastern and western ends, unless basal sliding is taken into account. The presence of basal water is also suggested by the observation that the off-nadir radar return is weaker where there is lower basal drag and lower bedrock elevations.
 Following Thomas and Bentley , we use the mass continuity equation to compute the mass balance and its uncertainty across the study area. We estimate flux divergence over a flow band that extends across the whole study area and partitioned the band into two halves to check for consistency. Taking 10 cm/yr of ice as an upper bound on the net surface balance [Box et al., 2006], we find an average thinning rate of 1.6 ± 0.5 m/yr. The errors in mass flux are large because the study area is small, so any systematic error in ice thickness results in an appreciable increase in the mass balance error. Our thinning rate can be contrasted to the 1.5 ± 0.5 m/yr thinning rate derived from 2002–2005 Airborne Topographic Mapper (LIDAR altimeter) measurements [Krabill et al., 2004]. The statistically insignificant difference between the two results suggests that the ice sheet is primarily thinning by creep.
 Radar measurements of the polar ice sheets are evolving from nadir measurements of ice thickness typical of the past 50 years of polar research to three-dimensional imaging of the glacier bed using airborne radar with a cross-track array. In this paper, we demonstrate the ways in which radar tomography captures detailed information about the subglacial terrain, information that can be used to investigate small (100s of meters) scale details of subglacial topography and infer information about the conditions at the bed. Indeed we were able to capture, for the first time, images of long, sinuous landforms with 10 to 30 m depths, 150 to 500 m spacing and lengths of 10 km or more that we interpret as erosional features formed at the base of a modern ice sheet. There are two major limitations to the technique to note. First, the signal strength must be several dB above noise to satisfactorily process the data. This can be hard to achieve in areas of thick warm ice where absorption is high. The technique is also limited by the size of the antenna aperture, which essentially limits the ability to resolve the angular difference between closely separated objects. As pointed out above, range ambiguities will occur near nadir because of the small arrival angle differences across the aperture even for relatively flat terrain. At large angles, radar layover and foreshortening will confuse interpretation over steeply sloping terrain unless the number of antenna elements is increased. We believe that by sharpening the transmit beam by using three, independent, centered, transmitting dipoles and improving the tomographic calculation by using 18 receive elements, we can achieve 10 km wide swaths for ice thicknesses less than 2 km and a 6 km platform elevation above the ice surface, where our estimate assumes a severe, 38 dB/km absorption rate (E. Rodriguez, unpublished report, 2009). Where signal levels are strong (several dB above the radar noise floor), such a configuration can provide detailed information about the base of the ice sheet at resolutions comparable to currently available surface ice sheet topography and velocity measurements. The combination of such high resolution data sets can significantly advance measurements of ice sheet mass balance, improve ice sheet modeling by incorporating better information on the glacier bed, and further investigate geomorphologic structures underlying the modern ice sheet.
 This research was supported by grants from the NASA Earth Science and Technology Office, the NASA Cryosphere Program, and by the Office of Polar Programs of the National Science Foundation.