Introduction
For decades, radar sounding has been a method of choice in the large-scale characterization of ice sheets, ice caps and glaciers (de Robin, Reference Robin1975; Bingham and Siegert, Reference Bingham and Siegert2007). Not only can radar sounding measurements be qualitatively interpreted to yield estimates of the ice mass thicknesses and underlying bed topographies but the quantitative interpretation of radar bed echo strengths is used to characterize subglacial hydrology and identify bodies of subglacial liquid water (i.e. lakes) (Shabtaie and Bentley, Reference Shabtaie and Bentley1988; Wright and Siegert, Reference Wright and Siegert2012; Young and others, Reference Young, Schroeder, Blankenship, Kempf and Quartini2016; Rutishauser and others, Reference Rutishauser2018). These interpretations provide critical boundary conditions for glacial dynamic and subglacial hydrology models as well as potential targets for extreme habitat microbiological studies.
In all studies relying on radar sounding data, it is important to define the reflectors chosen to represent particular horizons of interest (i.e. the surface, bed or englacial layers) consistently, which is complicated by the wide beam patterns associated with conventional dipole antennas (Peters and others, Reference Peters2007). Coherent (magnitude and phase) acquisition systems and subsequent synthetic aperture radar (SAR) focusing can be used to improve along-track reflector resolution (Leuschen and others, Reference Leuschen, Gogineni and Tammana2000; Legarsky and others, Reference Legarsky, Gogineni and Akins2001; Peters and others, Reference Peters, Blankenship and Morse2005, Reference Peters2007; Hélière and others, Reference Hélière, Lin, Corr and Vaughn2007), but the cross-track beam pattern remains broad. The end effect is a radar image (the radargram) that may contain anomalous signals reflected from cross-track off-nadir surface and subsurface features (i.e. clutter) that can be misinterpreted as a reflector of interest.
Several techniques have been proposed to discriminate surface cross-track clutter in airborne radar sounding data including radargram reprojection and cluttergram simulation (Holt and others, Reference Holt, Peters, Kempf, Morse and Blankenship2006) as well as interferometry (Castelletti and others, Reference Castelletti2017; Haynes and others, Reference Haynes, Chapin, Moussessian and Madsen2018). Radargram reprojection involves the cross-track mapping of radargram energy onto a digital elevation model (DEM) as if it were all derived from surface scatterers. Surface clutter is then discriminated by comparing where the signal in the radargram would have been reflected from had it originated on the surface. Energy mapping to distinct topographic features on the surface can then be used to infer if that signal represents surface clutter. Discrimination through cluttergram simulation represents the opposite approach; whereby a DEM is used to simulate a radargram assuming no subsurface reflectors are present. Similar echo features in both the simulated cluttergram and the measured radargram are then likely to originate from surface reflectors. Finally, interferometry discriminates clutter by measuring the phase difference between an echo reflecting from the same surface feature but arriving at two antennas separated in the cross-track direction. Ignoring any systematic phase offset between the antennas due to the acquisition system, the differential phase between the two antennas is solely a function of the cross-track look angle to the surface reflector (Castelletti and others, Reference Castelletti2017; Haynes and others, Reference Haynes, Chapin, Moussessian and Madsen2018). Including any of these clutter discrimination methodologies in radar data processing can enhance the consistency of the interpreted surface reflector.
In contrast to the surface, there is often no a priori insight into bed topography prior to the acquisition and interpretation of radar sounding data. As such, the radargram reprojection and cluttergram simulation approaches cannot be used to discriminate cross-track bed clutter. In this paper, we demonstrate how radar interferometry can be used to produce a consistent interpretation of a glacier bed in areas containing additional ambiguous echoes from cross-track basal reflectors. We are not concerned with demonstrating whether the interpretation of the bed can be shown to be at nadir (ideally with a zero interferometric phase difference between the received echoes, Castelletti and others, Reference Castelletti2017; Haynes and others, Reference Haynes, Chapin, Moussessian and Madsen2018) as, even for a nadir target, internal acquisition system phase offsets, as well as cross-track surface and subsurface slopes, can introduce a differential phase between the received echoes. For our method, precise correction for these effects is not necessary as we are interested in producing an interpretation of the bed that is consistent within the larger context. This approach can be considered a sparse antenna array version of the array processing methods of Li and others (Reference Li2012).
The remainder of this paper is structured as follows, after first presenting the fundamentals of radar sounding interferometry we then introduce the dataset on which we have tested our approach. Following this we present the results of our analysis, discuss the larger implications of a consistent bed definition, and finally summarize our main conclusions.
Fundamentals of radar sounding interferometry
Radar sounding interferometry is functionally similar to conventional interferometric SAR only that instead of leveraging the differential phase between two radar measurements acquired at different times to characterize intervening line-of-sight deformation in the surface or to estimate surface elevation, the two radar sounding measurements are contemporaneous (i.e. single-pass interferometry) (Rosen and others, Reference Rosen2000; Reis and others, Reference Reis, Hensley, Willams and Woods2009). A differential phase then relates to a difference in the path length travelled by a reflected echo between a cross-track reflector and two antennas separated by some cross-track distance (the interferometric baseline, L) (Castelletti and others, Reference Castelletti2017; Haynes and others, Reference Haynes, Chapin, Moussessian and Madsen2018). The two cross-track antennas will henceforth be referred to as ‘A’ and ‘B’.
The problem of cross-track bed clutter is illustrated in Figure 1. Figure 1a presents one position along a transect where both the surface and bed interfaces are flat and no cross-track clutter is generated. Figure 1b then presents a second position along the same transect; however, there is now significant cross-track topography in the ice–bedrock interface adjacent to the point directly below the aircraft. Under such conditions, two echoes could be generated; one from the portion of the interface below the platform (represented by the blue line in Fig. 1b), and one from the preferentially oriented areas on the cross-track topographic slope (the red line in Fig. 1b). These two reflections may be recorded at different times but with similar amplitudes and, without knowledge of the shape of the bed interface, it is not immediately evident which represents a continuation of the reflector observed at a different position along the transect (i.e. Fig. 1a).
Two measurement positions along a transect (flight direction is perpendicular to the plane of the page); one (a) with a flat bed interface that does not generate cross-track bed clutter and only a single echo from below the aircraft is recorded (blue line) and a second (b) with significant cross-track bed topography that generates an additional reflection in the form of cross-track bed clutter (red line).
Under the conditions presented in Figure 1a, and assuming no internal phase offsets in the electronics of the acquisition system, the phase difference between the bed reflections in the radargrams associated with the A and B antennas is expected to be close to zero. It is important to note though that cross-track slopes in both the surface and bed can lead to non-zero phase differences between the A and B radargrams even in the absence of cross-track clutter. In Figure 1b, unless occurring at a phase ambiguity, the reflection from the cross-track slope would present with a different differential phase compared to the reflection from directly below the aircraft. Ambiguity occurs when the differential phase of the cross-track slope reflection is an integer multiple of 2π compared to that from below the aircraft. At this point, even while the reflector is off-nadir, its interferometric phase response is indistinguishable from that of the nadir reflection. Here, assuming each individual reflection in the radargram is the response of a single, continuous target, we argue that by comparing the differential phase of the two reflections (Fig. 1b) with a measure of differential phase in an area of the radargram that is considered free from ambiguous bed clutter (Fig. 1a), the reflection that yields the more consistent interpretation of the bed interface can be identified.
Differential phases between A and B echoes for a particular reflection of interest can be investigated using a roll-corrected interferogram (Ψc), as explained below. The uncorrected radar sounding interferogram (Ψu) is constructed from the two SAR-focused, single-look complex-valued (SLC), and co-registered A and B radargrams through (Castelletti and others, Reference Castelletti2017)
Individual pixels in Ψu can be considered to be an average of the differential phase values from the A and B radargrams within a moving window. For our analysis, we used a rectangular window that encompassed two sequential fast-time samples (n) across 15 adjacent range lines (m). Range lines were aligned (but not yet co-registered) prior to focusing using known variations in aircraft elevation along the transect.
SAR-focused radargrams are used to build the interferogram as focusing minimizes the along-track beam pattern; isolating clutter to predominantly the cross-track direction. SLC radargrams are required in order to preserve the phase information that is destroyed during incoherent multi-looking (Hélière and others, Reference Hélière, Lin, Corr and Vaughn2007). Finally, following Castelletti and others (Reference Castelletti2017), a 1/10-pixel precision co-registration is used to align the two radargrams as finely as possible prior to interferogram creation. Co-registration is accomplished using a constant shift for each pair of range lines.
The correction that must be applied to the differential phases in the uncorrected interferogram (Ψu) before individual reflectors can be investigated is due to the roll experienced by the aircraft along the transect (Castelletti and others, Reference Castelletti2017). Roll (ϕ) in the acquisition platform is synonymous with a change in look angle (the angle between nadir and the cross-track target) and introduces an additional differential phase (ψ) shift between when a reflection is received by the A and B antennas
L in (2) represents the interferometric baseline (cross-track distance between the A and B antennas) while λ is the wavelength associated with the center frequency of the radar signal. Implicit in (2) is the assumption that expected reflections in the radargrams are the result of nadir surface and subsurface targets. The interferometric roll correction is constant for each range line (change in roll during the collection of an individual range line is negligible) and the roll-corrected interferogram (Ψc) is generated by adding the mean roll correction from (2), defined over the same 15-sample window used in (1), to each fast-time sample of the associated range line in the uncorrected interferogram, Ψu.
Datasets
The data used in this study were collected in 2018 over Devon Ice Cap (DIC) in the Canadian Arctic (Fig. 2a) as part of a collaborative study between the University of Alberta and the University of Texas Institute for Geophysics (UTIG). Data were collected using UTIG's MARFA acquisition system (Young and others, Reference Young, Schroeder, Blankenship, Kempf and Quartini2016). MARFA is a dual-phase center version of the HiCARS system (Peters and others, Reference Peters2007) operating with a 60 MHz center frequency, a 15 MHz signal bandwidth, a 50 MHz pulse sampling frequency and a 19 m interferometric baseline between the antennas. These data are part of a larger investigation into unique subglacial water bodies originally described in Rutishauser and others (Reference Rutishauser2018). Of interest in this study are portions of six transects in the region of the T2 water body located in the low-lying (darkly shaded) trough in the center of Figure 2b (Rutishauser and others, Reference Rutishauser2018). Figure 2b corresponds to the area within the dashed box overlain on the ArcticDEM (Porter and others, Reference Porter2019) composite basemap shown in Figure 2a. An updated bed topography model considering the results of this analysis is shown in Figure 2b. The five colored, preferentially north-south, lines (DEV/JKB2t/X75a, DEV/JKB2t/X76a, DEV/JKB2t/X77a, DEV/JKB2t/X78a and DEV/JKB2t/X79a) all cross the trough at an approximate right angle. In contrast, DEV2/JKB2t/Y82a crosses the trough more obliquely.
Overview map of (a) Devon Ice Cap as well as an outline of the area of interest and (b) specific radar transects in the area of interest overlain on a bed topography map corrected for the results of this analysis. All coordinates are presented in UTM zone 17N, bed elevations are presented in meters above sea level.
With the advantage of the consistent bed topography of Figure 2b, it is possible to recognize where bed clutter may manifest along the radargrams (i.e. when flight lines approach steep basal topography at an oblique angle). The most obvious location is along DEV2/JKB2t/Y82a as the aircraft approaches the steep northern slope of the trough. In the center of the trough (between where DEV2/JKB2t/Y82a crosses DEV/JKB2t/X78a and DEV/JKB2t/Y77a), the geometric situation is similar to what is presented in Figure 1a with a limited cross-track topographic variation. However, as cross-track bed topography increases as the aircraft begins to approach the north slope of the trough (a situation similar to what is presented in Fig. 1b), so does the potential to generate cross-track bed clutter.
The portion of the DEV2/JKB2t/Y82a radargram corresponding to the transect in Figure 2b is presented in Figure 3a. The radargram in Figure 3 is the coherent combination of the individual A and B radargrams. The locations where the five DEV/JKB2t/Xxxa intersecting lines cross DEV/JKB2t/Y82a are marked along with the semi-automatically picked surface and bed reflections prior to performing any refinement for potential cross-track bed clutter. The general criteria followed by the human interpreter during the initial horizon picking were to select the first, strongest and most laterally continuous reflection.
DEV2/JKB2t/Y82a radargram corresponding to (a) the groundtrack shown in Figure 2b and (b) the immediate vicinity of the northern slope of the main bedrock trough (dashed box in (a)). The positions where the DEV2/JKB2t/Y82a radargram is intersected by the five DEV/JKB2t/Xxxa radargrams are marked in white. Intersecting bed picks from DEV/JKB2t/X77a and DEV/JKB2t/Y76a are highlighted with yellow starts in (b). Black lines correspond to the initial picks for the surface and bed reflections. The unambiguous lake reflection and four features possibly related to cross-track basal clutter are labeled in (b). A different colorbar is used in (b) than in (a) to highlight basal reflections.
The section of the DEV2/JKB2t/Y82a in the vicinity of the northern slope of the main trough highlighted by the dashed box in Figure 3a is reproduced as Figure 3b. Locations where DEV/JKB2t/X77a and DEV/JKB2t/X76a cross DEV2/JKB2t/Y82a are also presented. Four reflections are clearly visible near the northern slope and labeled as features 1 through 4. These features have all been interpreted to be basal in origin due to the absence of rugged surface topography near the summit of DIC (Fig. 2a). However, it is not clear which feature is consistent with the bed reflection in the center of the trough where the T2 lake has been inferred. Initial picking of the bed reflection (black line) followed from the lake reflection, along feature 1 and continued toward the north. The stars represent the position at which the picked DEV/JKB2t/X77a and DEV/JKB2t/76a beds intersect with the DEV2/JKB2t/Y82a radargram. While the Y82a/X76a (northern-most) intersection exhibits little offset between the two interpretations, the Y82a/X77a bed interpretations are significantly different suggesting that these picks may have identified cross-track basal clutter instead of the actual bed.
Results
In order to assess which feature identified in Figure 3b most likely represents the consistent continuation of the lake reflection, we derived the roll-corrected interferogram following the procedure outlined above. Figure 4 presents the interferogram for the same portion of the DEV2/JKB2t/Y82a radargram shown in Figure 3b. The reflections of interest identified in Figure 3b are also highlighted in Figure 4. From the interferogram, the mean differential phase for each feature was estimated and the results are summarized in Table 1.
DEV2/JKB2t/Y82a interferogram in the immediate vicinity of the northern slope of the main bedrock trough. Initial bed picks (black line) and features identified for interferometric testing (cyan, red, blue, green and magenta lines) are highlighted.
Summary of mean interferometric phase and standard deviation associated with each feature of interest identified in the DEV2/JKB2t/Y82a radargram
The large change in mean differential phase relative to the lake reflection implies that feature 1, the reflection that was initially selected as the representative bed reflection (Fig. 3), is most likely cross-track bed clutter from a situation similar to what is shown in Figure 1b. This interpretation further justified considering the large offset observed between the initial DEV2/JKB2t/Y82a bed interpretation and that of DEV/JKB2t/X77a (Fig. 3b). However, it should be noted that if based solely on the expectation of Gaussian interferometric phase distributions (Castelletti and others, Reference Castelletti2017; Haynes and others, Reference Haynes, Chapin, Moussessian and Madsen2018), the feature 1 probability density function (as defined by the mean and standard deviation in Table 1) would overlap significantly with that of the lake reflection. Without the near-by intersection with DEV/JKB2t/X77a, statistical methods such as those presented in Castelletti and others (Reference Castelletti2017) and Haynes and others (Reference Haynes, Chapin, Moussessian and Madsen2018) would be required to discriminate feature 1 as an off-nadir basal reflection that is inconsistent with that of the lake.
Feature 4 also exhibits a significant difference in mean differential phase relative to the lake reflection and, even though it exhibits a large standard deviation, as it is spatially limited, it is also likely related to cross-track bed clutter. While the bed pick from DEV/JKB2t/X77a more closely matches features 2 and 3, the interferometry results suggest that feature 2 is likely cross-track clutter. This is a result of the large discrepancy in mean differential phase and the fact that the associated Gaussian distributions do not overlap at one standard deviation. Feature 3 exhibits the smallest difference in mean differential phase relative to the lake reflection as well as most overlap with the associated probability density function. It is therefore the feature 3 reflection that is interpreted as the reflection most likely to represent a consistent extension of the bed as it has been defined in the center of the trough.
The interferometry results summarized in Table 1 aid in discriminating the consistent DEV2/JKB2t/Y82a bed reflection (feature 3) from cross-track bed clutter (features 1, 2 and 4). This is supported by a comparison with the bed picks interpreted from the crossing DEV/JKB2t/X77a radargram (Fig. 3b). However, it is worthwhile repeating the analysis process to ensure that the DEV/JKB2t/X77a bed is consistently defined and itself not affected by cross-track bed clutter.
Figure 5a presents the portion of the DEV/JKB2t/X77a radargram corresponding to the transect shown in Figure 2b, while Figure 5b focuses on the northern slope of the main trough. As with Figure 3, the black lines represent the initial interpretation of the surface and bed reflections and the stars represent the points of intersection between the initial (yellow) and interferometry-revised (red) DEV2/JKB2t/Y82a bed picks and the DEV/JKB2t/X77a radargram. A reference reflection in the area of the T2 subglacial lake is selected (labeled again as lake) but unlike Figure 3b, in Figure 5b there is only one reflection in the vicinity of the northern slope (labeled as slope) already suggesting that cross-track bed clutter may not be a significant issue. Figure 6 presents the interferogram associated with the portion of the radargram in Figure 5b. The mean differential phase and standard deviation (in parentheses) associated with the lake (blue) and slope (green) reflections extracted from the interferogram are 24.36° (22.64°) and 26.95° (44.24°), respectively. The very close agreement between the mean differential phases of these two reflectors implies that they represent a consistent bed definition and the initial interpretation of DEV/JKB2t/X77a is unaffected by cross-track basal clutter.
DEV/JKB2t/X77a radargram corresponding to (a) the groundtrack shown in Figure 2b and (b) the immediate vicinity of the northern slope of the main bedrock trough (dashed box in (a)). The position where the DEV/JKB2t/X77a radargram is intersected by the DEV2/JKB2t/Y82a radargram is marked. Original and interferometry-revised intersecting bed pick from DEV2/JKB2t/Y82a are highlighted with yellow and red stars, respectively, in (b). Black lines correspond to the initial manual picks for the surface and bed reflections. The unambiguous lake reflection and the northern slope reflection are labeled in (b). A different colorbar is used in (b) than in (a) to highlight basal reflections.
DEV/JKB2t/X77a interferogram in the immediate vicinity of the northern slope of the main bedrock trough. Initial bed picks (black line) and lake and slope features identified for interferometric testing are highlighted.
Discussion
The results of our analysis have shown radar sounding interferometry is capable of discriminating basal clutter, leading to a more consistent interpretation of the bed. While demonstrating this technique is the goal of this study, it is also useful to examine how subsequent analyses based on the bed interpretation can be affected by cross-track basal clutter.
Figure 7a presents the central portion of the DEV2/JKB2t/Y82a radargram across the main trough containing the T2 water body and the final interpretations of both the surface and the bed (solid black lines). The dashed black line represents the original interpretation of the bed prior to adjusting for basal clutter. At its maximum, there is approximately a 2 μs two-way travel time difference between the original and revised bed reflection as determined from interferometry. For a representative relative dielectric permittivity of ice (E r equal to 3.15), this corresponds to a change of 168 m in the elevation of the bed; approximately one-quarter of the total ice thickness in the T2 trough.
Central portion of the DEV2/JKB2t/Y82a radargram (a) and the associated absolute (black) and change in (red) hydraulic head profiles (b) derived from the interpreted surface and bed picks. Solid black lines refer to the revised interferometry-corrected interpretation of the bed reflection. The black dashed line shows the original interpretation of the bed.
Carrying this analysis further, Figure 7b presents the subglacial hydraulic head derived from the ice surface and the two interpretations of the bed (solid and dashed black lines) following Shreve (Reference Shreve1972); Rutishauser and others (Reference Rutishauser2018). The change in the hydraulic head after accounting for basal clutter (original minus revised) is also shown. Not only does the hydraulic head drop by almost 30 m in the region containing previously ambiguous basal clutter, there is also a significant change in the hydraulic head gradient. The slope in the hydraulic head is important in water routing models as it determines where water flows (down hydraulic gradients) and where it can pool (in the areas of near-zero hydraulic gradient, potentially forming subglacial lakes). In the original interpretation of the bed, the hydraulic gradient suggested water could potentially pond at the top of the trough's northern slope wall (between 8360.5 and 8360.8 km in Fig. 7b). In the revised bed interpretation, this area of potential ponding has been dramatically reduced, suggesting liquid water flowing along the bed from the north will preferentially concentrate in the base of the trough.
Finally, the revised interpretation of the bed reflection has significant implications for the possible extent of the lake occupying the trough (Rutishauser and others, Reference Rutishauser2018). Careful inspection of Figure 3b shows that the original DEV2/JKB2t/Y82a bed interpretation began to follow feature 1 before the strong, lower-relief bed echo it was following along the base of the trough ended (approximately at the intersection with the DEV/JKB2t/X77a transect). Subsequent analysis may suggest that this strong reflection omitted in the original interpretation of DEV2/JKB2t/Y82a bed exhibits reflectivities high enough to indicate the presence of liquid water (Rutishauser and others, Reference Rutishauser2018). Lake boundaries based on the original interpretation of the bed would have then underrepresented the size of the water body.
While this study demonstrates how relative differences in interferometric phase can be leveraged in order to define the most consistent bed reflection, it is worthwhile to revisit the absolute interferometric phase angles; specifically, those related to the lake reflections (Figs 3b and 5b). The interferometric phase and standard deviation (in parentheses) of the lake reflection as extracted from the DEV2/JKB2t/Y82a and DEV/JKB2t/X77a interferograms (Figs 4 and 6) are 38.98° (29.52°) and 24.36° (22.64°), respectively. It is clear that neither lake reflection exhibits the typical interferometric response that would be associated with a flat, specular target at nadir (i.e. 0° mean differential phase). Internal, systematic phase offsets between the A and B antennas within the MARFA acquisition system are expected to be on the order of single digits of degrees and unable to completely account for the observed absolute phase differences.
The most likely source of the differential phases observed in the lake reflections is related to the presence of cross-track slopes. Cross-track slopes introduce slight changes in the length of the radar propagation paths between A and B antennas and their respective points of reflection on the lake surface compared to a situation where both the ice and lake surfaces are flat (similar to Fig. 1a). These differences in path length then result in the phase offsets between the basal/lake surface reflections recorded by both the A and B antennas. As cross-track slopes at both the ice and lake surfaces introduce variations in radar path lengths, their respective contributions cannot be separated without considering additional sources of information (such as a surface digital terrain model).
Conclusions
Cross-track clutter represents a source of ambiguity and impediment in the consistent interpretation of surface and bed horizons from SAR-focused radar sounding data. Surface clutter in a radargram can be discriminated through either interferometry or by relying on available surface DEMs. In contrast, as there is often no a priori knowledge of bed topography prior to radargram interpretation, similar DEM-reliant techniques cannot be used to discriminate clutter due to cross-track bed topography. In this study, we demonstrate that interferometry can successfully discriminate basal clutter and be used to produce a continuous interpretation of the bed that is consistent with regions unaffected by clutter. Furthermore, we have shown that effectively accounting for cross-track clutter can have significant implications on derived ice thicknesses, basal water routing models and subglacial water body extents.
Acknowledgement
We thank The W. Garfield Weston Foundation who funded the aero-geophysical survey over Devon Ice Cap and subsequent standard data processing, The G. Unger Vetlesen Foundation who funded the subsequent interferometric data processing, and the UTIG Postdoctoral Fellowship program. We also thank the Polar Continental Shelf Program, Kenn Borek Air Ltd. and the field team for their excellent support, and NRI and the peoples of Grise Fjord and Resolute Bay for permission to conduct airborne surveys over Devon Ice Cap. This is UTIG contribution number 3650.
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