We utilise airborne gravity data from OIB and the ITGC campaign, together with marine gravity data from cruise NBP19-02 (Fig. 1b and c). The OIB free-air gravity data were collected from a DC-8 aircraft travelling at ∼120 m s-1 at an altitude of ∼450 m above the ice surface, using the Sander Geophysics AirGrav system (Studinger et al., 2008). These data have an error of ∼1.67 mGal in this region and resolve anomalies with a ∼10 km full wavelength (Cochran and Bell, 2010, updated 2018; Tinto and Bell, 2011). The ITGC campaign utilised a Twin Otter aircraft, flying at ∼60 m s-1, on average 340 m above the ice, and a different “strapdown” gravity approach based around an iMAR Inertial Navigation System (INS) (Becker et al., 2015; Wei and Schwarz, 1998). The resulting data have an internal error from crossover analysis of 1.56 mGal and resolve wavelengths down to ∼5 km (Jordan et al., 2020c) (see Supplement Sect. S2 for details).

Airborne gravity data were restricted to lines flown at <1500 m above the surface. Of this subset over 95 % of the data were collected at 450m±200m above the surface. Upward and downward continuation of the gravity data to a common altitude was neglected as continuation by ∼200 m will have little impact on the amplitude of the gravity anomalies (∼1 mGal) given the ∼1000 m range to the key bathymetric sources. Downward continuation can also introduce unnecessary artefacts, and neglecting upward continuation preserves short wavelength gravity information. The data collected >650 m from the ice surface may give rise to an artificially smooth bathymetry but are spatially restricted (Fig. S1 in the Supplement) and do not appear to give rise to any anomalous signals in the integrated free-air gravity dataset (Fig. 1c).

Marine gravity data from cruise NBP19-02 matched the pattern of the airborne anomalies but were offset by ∼7.14 mGal above the level of the airborne data. The majority of this offset is due to the difference between geoid (marine) and ellipsoid (airborne) references used for the different systems. In the area of overlap the geoid–ellipsoid difference results in a ∼9 mGal discrepancy, based on the GOCO3s satellite gravity model (Pail et al., 2010). The residual ∼2 mGal difference may reflect drift in the marine system, and potential discrepancies in base station ties between the different surveys. Alternatively, unconsidered shorter wavelength variability in the gravity field not resolved by the GOCO3s model (<∼160 km) or temporal changes in the geoid associated with glacio-isostatic adjustment and mass loss may account for the residual shift. Such features do not impact the locally recovered bathymetry and are beyond the scope of this paper. The average measured shift of 7.14 mGal was therefore subtracted from the marine line data as a single offset value. All line data were then merged into a single database, interpolated onto a 1 km mesh raster, and filtered with a 5 km low-pass filter removing residual line-to-line noise. This filter wavelength is justified, as anomalies with wavelengths <5 km are not resolved by the airborne gravity systems used. The final, integrated free-air gravity map (Fig. 1c) shows a clear pattern of high and low anomalies, which first-order match the main 5–10 km wavelength features visible in the available onshore subglacial topography and offshore bathymetry (Fig. 1d).

The topographic observations onshore were taken from OIB line radar data (Paden et al., 2010, updated 2018), augmented with new depth sounding radar collected along with the gravity data during the ITGC campaign (Fig. 1d). These new bed elevation data were collected using a 600–900 MHz accumulation radar provided by the Center for Remote Sensing of Ice Sheets (CReSIS). Bed elevations were picked from synthetic aperture radar (SAR)-processed radargrams in a semi-automated fashion. Although the primary target of this radar system was shallow ice sheet structures, bed elevation was resolved through ice up to ∼1900 m thick. Visual inspection revealed a few incorrect onshore bed picks in the OIB dataset on Bear Island, which gave bed elevations above the highly accurate Reference Elevation Model of Antarctica (REMA) surface digital elevation model (DEM) (Howat et al., 2019). These points were deleted from the integrated line bed elevation dataset. The line bed elevation data were corrected to the GL04c Geoid (Forste et al., 2008), and the data were then interpolated onto a 1 km mesh raster. This gridded dataset was carefully masked to remove regions that are now covered by the floating ice shelf based on the most up-to-date grounding lines (Rignot et al., 2014; Milillo et al., 2019). Bed elevation values over local sub-shelf pinning points were also excluded. This masking mitigates the risk of the base of a floating ice shelf being misidentified as a bed elevation point and biasing the inversion. Beyond the ice shelves we took the values constrained by a new compilation of shipborne multibeam swath bathymetric data (Hogan et al., 2020), which was downsampled to a 1 km mesh raster for this study (Fig. 1d).

To recover a gravity-enhanced bathymetry we follow an algorithmic approach, rather than a pure inversion (Hodgson et al., 2019). We refer to this as the topographic shift method, as an initial topographic estimate derived from the gravity data is shifted to match observed topographic tie points. Summarising the method, the initial 3D topographic estimate (Fig. S2a) was calculated from the free-air anomaly (Fig. 1c) using an iterative forward modelling method (von Frese et al., 1981). Differences between the initial topographic estimate (Fig. S2b) and the observed bathymetry and onshore topography were calculated (Fig. 1d) and interpolated using a tensioned spline (Smith and Wessel, 1990). This difference grid was then subtracted from the initial topographic estimate to provide the final bathymetric estimate (Fig. 2a). For the full details of the method, see Sect. S1 in the Supplement. The topographic shift method is conceptually similar to the gravity shift method developed and applied along the Greenland coast, where the initial free-air gravity data were shifted to match the variable gravity field from models of known topography prior to inversion for bathymetry (An et al., 2019). This gravity shift method was subsequently employed to fill the sub-ice shelf bathymetry in the Thwaites Glacier region of the recent BedMachine Antarctica compilation (Fig. S5c in the Supplement) (Morlighem et al., 2020). The advantage of both the topographic and gravity shift techniques is that features in the gravity field due to variations in crustal thickness, sedimentary basins, or intrusions are implicitly taken into account, as long as they overlap with the topographic control points. This assumption is most robust for long wavelength features, such as variations in crustal thickness or regional sedimentary basins, where the associated errors will impact multiple topographic control points, allowing good control of the resulting error field. The impact of more localised geological features that only partially overlap constraining topographic data will be less well defined, and we make the assumption that such errors fall off smoothly away from the affected control points. Geological features that have no overlap with constraining topographic observations can still introduce artefacts, distorting the recovered bathymetry in proportion to their size and density contrast.

The depth of the ice shelf base and the thickness of the sub-ice-shelf water-filled cavity (Fig. 2b) were calculated assuming the ice shelf is in hydrostatic balance (Griggs and Bamber, 2011). Hydrostatically defined draft is typically a good approximation to radar-measured ice thickness (Griggs and Bamber, 2011) and provides seamless coverage of our study area. The input surface elevation data were taken from the REMA digital elevation model (DEM) (Howat et al., 2019), corrected to the GL04c Geoid (Forste et al., 2008), and re-sampled onto a 500 m grid cell size raster. In the study area the REMA DEM is based on satellite observations between 2014 and 2016 and therefore reflects the surface elevation after widespread ungrounding between 1993 and 2014 (Rignot et al., 2014). Ice and water densities were assumed to be 917 and 1027 kg m-3, respectively, and a 16 m firn correction was applied (Griggs and Bamber, 2011). Uncertainties in these assumed values may have an impact on the precise values of ice shelf draft but are unlikely to significantly distort the calculated pattern of water cavity thickness. Comparison between ice shelf base calculated from the higher resolution DEM and the longer wavelength bathymetry resolved by the gravity (>5 km) will introduce high-frequency features into the estimated cavity depth, and unresolved bathymetric features will generate errors in cavity thickness. However, the regional trends in cavity thickness will not be affected and can be discussed. Errors in ice shelf draft of up to 80 m in the 10–25 km most proximal to the grounding line may occur due to the rigidity of the ice shelf (Rignot et al., 2011), but in Thwaites glacier this “bending zone” appears to be narrow (<5 km) (Milillo et al., 2019), which we attribute to the highly fractured nature of the ice shelf in this region.

Quantification of the errors associated with gravity inversions is challenging, as a combination of intrinsic but quantifiable uncertainties in the gravity data, the inversion assumptions, and the poorly understood variability of sub-surface geology all contribute to the error budget. Errors in the gravity field of ∼1.56 mGal defined from crossover analysis directly contribute to ∼23 m uncertainty in the recovered bathymetry. The modelled rock density of 2670 kg m-3 assumes no sediments are present at the sea floor. This is reasonable given the generally rugged morphology observed across many parts of the Amundsen Sea inner shelf (Nitsche et al., 2013; Graham et al., 2009; Larter et al., 2009). However, assuming all bathymetry was carved into lithified sediment, the total amplitude of the sub-ice-shelf topography could be underestimated by ∼11 % (∼130 m), assuming a typical sediment density of 2500 kg m-3 (Telford et al., 1990). Lower-density unlithified sediment could lead to even larger underestimates of topographic amplitude, but such material would not be expected to form all of the >1000 m high ridges recovered by our inversion and imaged by recent swath data (Hogan et al., 2020). Other geological factors, such as dense gabbroic intrusions, or local sedimentary basins could further distort the recovered bathymetry if they are away from the direct bathymetric observations, which would mitigate the impact of such features on the final bathymetric model. Underlying geological factors can, in some cases, be revealed by coincident aeromagnetic data, as in the case of the Brunt Ice Shelf (Jordan and Becker, 2018; Hodgson et al., 2019) and Ross Ice Shelf (Tinto et al., 2019). In our study, tight correlation between high-amplitude magnetic (Jordan et al., 2020b) and gravity anomalies is only seen beneath the grounded part of Thwaites Glacier (Fig. 3b). Such tight correlation is indicative of a significant geological feature distorting the gravity signature (Jordan and Becker, 2018) but is not seen on profiles across the offshore regions (Fig. 3). This favours a model where underlying geological factors do not dominate the inversion results.

In addition to quantifying the errors, it is important to note that the resolution of the bathymetry recovered from gravity data is limited by the wavelengths resolved by the gravity systems and the survey line spacing. For this study, the gravity systems resolved minimum wavelengths of 5 to 10 km, and a minimum line spacing of ∼5 km is achieved outboard of Thwaites Glacier, while a minimum line spacing of ∼7.5 km was achieved over the Dotson and Crosson ice shelves. This study therefore only recovers bathymetric features with a wavelength of ∼5 km and upwards.

To best quantify the uncertainty in the sub-ice-shelf bathymetric estimate in our study region we utilised the new shipborne multibeam bathymetric data collected predominantly by a recent ITGC cruise, NBP19-02 (Fig. 1a) (Hogan et al., 2020). The topographic shift method was rerun with these multibeam data excluded from the constraining bathymetric dataset (Fig. 4a). The difference between the results with and without this test dataset (Fig. 4a and b) provides a snapshot of the errors associated with our recovered bathymetry (Fig. 4c). In this region the mean error is -40 m, with a standard deviation of 100 m. We take this standard deviation to be representative of the expected error in our modelled bathymetry. This error is within the typical range for that quoted for gravity-derived bathymetry, for example error estimates of ∼60 m have been suggested in Greenland (An et al., 2019), while errors of up to ∼160 m are suggested for the Larsen Ice Shelf where, unlike our study, no account had been made for the underlying geology (Brisbourne et al., 2014; Cochran and Bell, 2012). The mean error we find indicates that the bathymetry constrained by the swath data is deeper than predicted by the gravity inversion alone, and hence that there are geological features in this region distorting the recovered bathymetry. It is apparent that the largest errors are associated with higher-frequency topography revealed by the new multibeam data. Such errors resulting from comparison of datasets with fundamentally different resolutions is to be expected, highlighting the need for multibeam bathymetry in regions where sub-kilometre-scale resolution of bathymetry is required. This is particularly relevant in areas where the seabed topography includes high-amplitude variations at short wavelengths. In addition, this pattern of errors means that single seismic observations of cavity depth may not be ideal tie points for gravity inversions in rugged regions such as near Thwaites Glacier. A single such seismic measurement typically relies on a receiver array ∼250 m long (Brisbourne et al., 2014) and hence could image a local high or low, biasing the wider gravity inversion.

Comparison between our topographic shift method and previous gravity inversions in this region show the broad sub-ice-shelf features are resolved by all methods, but differences in the detailed results are clear (Fig. S5). The OIB Level 3 data product (Tinto et al., 2011; Tinto and Bell, 2011) shows the largest discrepancies (Figs. S5a, d and 3a, b), with our new inversion showing bathymetry 200 to 300 m shallower at the grounding line. This in part reflects the fact that the OIB bathymetric estimate was limited to using gravity data from 2011 and earlier. In addition this bathymetric model relied on integrating the results of a series of 2D forward gravity models, incorporating observed bathymetry and radar-derived topography beneath the grounded ice. These gravity models did not factor in any regional trends in the gravity field but rather corrected for a single DC shift at the outboard end of each profile and modified the upper crustal density at the inland end of the profile to achieve a good fit. Unmodelled regional trends could therefore be a factor distorting the recovered bathymetry. The Millan et al. (2017) inversion of bathymetry from gravity data shows the same general pattern of sub-ice-shelf bathymetry as our topographic shift method (Fig. S5b). However, differences are observed, most clearly beneath the inboard parts of the Dotson Ice Shelf (Fig. S3e). In addition significant undulation in the recovered bathymetry that is not associated with any gravity signal is seen, for example, from 100 to 120 km in Fig. 3d. Such variability is indicative of artefacts due to the inversion approach. As our topographic shift method is different, and we incorporate additional new gravity, bathymetric, and radar data, it is not immediately clear what the source of these discrepancies are. To independently assess the results of Millan et al. (2017) we compare their results with BedMachine Antarctica (Morlighem et al., 2020) (Fig. S4c), which used the same input data as Millan et al. (2017), and the gravity shift method previously applied to the Greenland margin (An et al., 2019). The key difference between the An et al. (2019) and Millan et al. (2017) methods is that the newer approach applies a variable rather than single DC shift to the gravity prior to inverting for the bathymetry. The residuals between the Millan et al. (2017) result and BedMachine Antarctica (the gravity shift method) (Fig. S4f) show a similar pattern to the residuals between the Millan et al. (2017) result and our topographic shift method (Fig. S4e). This indicates that the use of a single DC shift was a significant issue in the older inversion (Millan et al., 2017), which may have led to an overestimate of the depth of some near-shore features.

Comparing BedMachine Antarctica and our topographic shift results reveals that differences of over 250 m are still present (Fig. S5g). We suggest that these remaining differences reflect the additional multibeam bathymetric, ITGC radar, and gravity data used in our topographic shift result. In addition, the different bed topography onshore (OIB and ITGC line radar data here vs. mass conservation in BedMachine Antarctica; Morlighem et al., 2020) and exclusion of sub-ice-shelf pinning points from our topographic shift result also likely contributed to the differences. For example, topography with no associated gravity signal is seen in the BedMachine profile in Fig. 3d, indicating the method and tie points used introduced some artefacts. This highlights the need for caution when using gravity-derived bathymetry and the value of high-resolution gravity data with tight line spacing, such as the integrated OIB/ITGC dataset, together with additional well-constrained and well-distributed observational tie points beneath the ice shelves and around their margins.