The see reference list for a link to web page is a series of airborne campaigns carried out using the institute's two Basler BT-67 research aircraft Polar 5 and 6 to measure Arctic sea ice and its change since 2009. The campaigns operate from airfields extending from Longyearbyen in Svalbard to Utqiaġvik (Barrow) in Alaska and coincide closely with the Arctic sea-ice summer minimum (August) and winter maximum (April). The primary scientific instrumentation on the aircraft includes an electromagnetic (EM) induction sounding instrument (EM-Bird) to measure total (i.e. ice + snow) thickness, an airborne laser scanner (ALS) for surface topography and freeboard measurements, a microwave radar to measure snow depth, and an infrared radiation pyrometer to record surface temperature (Fig. ). We describe each instrument in the following sections. The low altitude of 200 ft (≈60 m) and slow speed of 110 kn (≈60 ms-1) during the nominal surveys are beneficial for high-resolution data acquisition.

In this study, we used data collected during the IceBird winter campaigns in early April of 2017 and 2019 (Table , Fig. ), utilising the unique data set of simultaneous total thickness, snow freeboard, and snow depth measurements. From 2017, we used measurements from four survey flights that took place over the Beaufort and Chukchi seas as part of the Polar Airborne Measurements and Arctic Regional Climate Model Simulation Project PAMARCMiP;. From 2019, we included five survey flights that covered regions in the Lincoln Sea and the Arctic Ocean in addition to an overlap with the measurements in the Beaufort Sea in 2017.

We measured total (ice + snow) thickness of sea ice (htot) using the towed EM induction sounding instrument, the EM-Bird, suspended below the aircraft 10–20 m above the sea-ice surface as illustrated in Fig. . The EM-Bird utilises the contrast between resistive snow and ice layers and conductive sea water by transmitting an EM field that induces eddy currents only in the latter . The EM-Bird measures the phase and amplitude of a secondary EM field induced by those eddy currents in relation to the primary field. The phase and amplitude of the secondary field depend on the distance between the instrument and the ice–water interface and decrease negative exponentially with increasing distance. Subtracting the instrument height above the surface, measured by an integrated laser altimeter, from the distance to the sea water gives the total thickness . Sea-ice thickness (hi) was derived by subtracting snow depth from total thickness (see Sect. ). The EM-Bird sampling rate was 10 Hz, which translated to 5–6 m point spacing at the nominal survey speed. Approximately every 15–20 min brief ascents to more than 100 m were carried out to monitor the sensor drift during post-processing . Comparison to drill-hole measurements over level ice have indicated an accuracy of 0.1 m, whereas ridge peak thicknesses are generally underestimated by up to 50 % as a result of mass-conserving averaging effects within the approximately 40 m diameter footprint of the instrument . For our analysis, we disregarded measurements of total thickness (i) less than the instrument accuracy of 0.1 m and (ii) less than the mean snow freeboard or snow depth and (iii) where the surface temperature was above -5 ∘C within the footprint of the instrument to avoid open or newly frozen leads with total thickness below the accuracy of the EM-Bird (see Sect.  and ).

The near-infrared (1064 nm), line-scanning Riegl VQ-580 airborne laser scanner (ALS) measured ellipsoidal elevations of ice surfaces with a 60∘ field of view, resulting in a swath width approximately equal to the aircraft's altitude above ground (nominally ≈60 m). We obtained freeboard from the ALS data by subtracting the local sea-surface height from the ice surface elevations. The height of the sea surface along the flight track is sporadically observed by the ALS at open fractures (leads) of the sea-ice cover or newly frozen leads with negligible freeboard, and we manually selected the corresponding elevations. The spacing of leads was diverse and depended on the ice regime varying between less than 10 km (FYI) and potentially more than 30 km (densely packed MYI). We subtracted the mean sea surface DTU15 MSS; from the surface elevations to remove large-scale variations and reduce interpolation errors before applying a spline interpolation between the sea-surface height tie points along the flight track. Data before the first lead detection and after the last lead detection during a survey flight were discarded to avoid extrapolation errors. Subtracting the interpolated sea-surface height from the ice elevations results in snow freeboard (hfs), as the elevation measurement of the ALS always includes the snow layer (Fig. ). Sea-ice freeboard (hfi), i.e. the location of the snow–ice interface in relation to the local sea level, was derived by subtracting snow depth from snow freeboard (see Sect. ). We then interpolated the obtained point cloud data of snow freeboard onto a regular grid with a 0.25 m resolution. Freeboard uncertainties are dominated by the accuracy of the interpolation of the instantaneous sea surface anomaly that depends on the abundance of leads. Therefore, especially regions with packed MYI and low lead density are associated with high uncertainties , and we manually masked out such areas. Supervising the along-track interpolation, we estimated an overall uncertainty of 0.1 m. For each total thickness measurement, we calculated the corresponding mean snow freeboard within the EM-Bird footprint (Fig. ).

We used an ultra-wideband (2–18 GHz) frequency-modulated continuous-wave (FMCW) quad-polarised microwave radar, hereafter Snow Radar, to measure snow depth on sea ice (hs). The radar was developed by the Center of Remote Sensing of Ice Sheets (CReSIS) at the University of Kansas, and similar radars have been operated as part of NASA's Operation IceBridge (OIB) since 2009 . Similar to the ALS, the Snow Radar transmitter and receiver antennae were mounted under the aft floor of the aircraft looking nadir (Fig. ). Due to the broad bandwidth of the radar, its range resolution in snow was 1.14 cm when assuming a snow density of 300 kgm-3. The low altitude and slow speed of the IceBird surveys resulted in an approximately 4–5 m sample spacing and a theoretical smooth surface footprint diameter of only 2.6 m across-track and 1.0 m along-track (Fig. ). A detailed description of the radar is given in and .

To calibrate the raw Snow Radar data, we used a workflow described in including coherent noise removal and system impulse deconvolution. Using an open-source Python package pySnowRadar , we detected air–snow and snow–ice interfaces with the algorithm by that is based on a pulse peakiness approach by used in satellite altimetry. Taking into account the decreased wave propagation speed in snow by assuming a snow density, the distance between the identified interfaces determined the snow depth estimate. In post-processing, we filtered out values which were acquired during the EM-Bird calibration manoeuvres with a simple altitude threshold of 100 m and when the absolute roll or pitch of the aircraft exceeded 5∘. Additionally, for each snow depth estimate we calculated a surface topography estimate (htopo) as the difference between the 95th and 5th percentiles of the ALS surface elevation data within the radar footprint to disregard potentially erroneous snow depth estimates over heavily deformed sea ice using a threshold value of 0.5 m .

A validation exercise over level, landfast FYI yielded a negligible mean bias of 0.86 cm, which was below the radar resolution and within the accuracy of the ground truth data, and a root-mean-square error (RMSE) of 6.9 cm for the radar-derived snow depth estimates . For each total thickness measurement, we calculated the corresponding mean snow depth requiring at least five valid snow depth estimates within the EM-Bird footprint (corresponds to approximately 50 % of the values; Fig. ). The averaging reduced the error by a factor of N, where N is the number of averaged estimates. We disregarded averaged snow depth values where the surface temperature within the EM-Bird footprint was above -5 ∘C and where the mean snow freeboard was less than the mean snow depth (negative sea-ice freeboard) to avoid potentially erroneous snow depth retrievals due to changes in the dielectric properties of snow induced by liquid water (; ; ; ). However, it has to be noted that the studies of , , and use different radar versions and retrieval algorithms compared to this study, which may limit the direct applicability of their results.

Simultaneous measurements of sea-ice thickness, snow depth, and freeboard enable us to calculate sea-ice bulk density using the so-called “freeboard and ice thickness technique” . Archimedes' principle dictates 1ρihi+ρshs=ρwhi-hfi, where ρ is density for ice, snow, and sea water denoted with subscripts “i”, “s”, and “w”, respectively. The terms can be rearranged to solve for ρi: 2ρi=ρw1-hfihi-ρshshi. By substituting measured total thickness and snow depth for sea-ice thickness (hi=htot-hs) in addition to snow freeboard and snow depth for sea-ice freeboard (hfi=hfs-hs), we obtain 3ρi=ρw1-hfshtot-hs+ρw-ρshshtot-hs. To solve Eq. (), we need to assume values only for the densities of sea water and snow, but their impact on the uncertainty of sea-ice bulk density is small (see Eq.  below). Here we took sea-water density and its uncertainty according to as ρw=1024 kgm-3 and σρw=0.5 kgm-3, respectively. For snow density in April, when measurements were carried out, we chose ρs=300 kgm-3 following and for the respective uncertainty σρs=34 kgm-3 from . These values and the uncertainties of the measured variables are summarised in Table . Assuming that the individual uncertainties are uncorrelated, we can derive uncertainty for sea-ice bulk density (σρi) using Gaussian error propagation: 4σρi=∂ρi∂ρw2σρw2+∂ρi∂ρs2σρs2+∂ρi∂htot2σhtot2+∂ρi∂hs2σhs2+∂ρi∂hfs2σhfs212=1+hs-hfshtot-hs2σρw2+hshtot-hs2σρs2+ρwhfs-hs+ρshshtot-hs22σhtot2+ρwhtot-hfs-ρshshtot-hs22σhs2+ρwhtot-hs2σhfs212. Following the uncertainty source analysis of and using the values summarised in Table , the largest contributors to the uncertainty of sea-ice bulk density are, in descending order of magnitude, snow freeboard, snow depth, and total thickness. We disregarded density values with uncertainty exceeding 100 kgm-3 from further analysis.

While the assumption of isostatic equilibrium may not necessarily be valid locally, e.g. close to pressure ridges, it holds true when averaging over a sufficient length scale. We varied the averaging length in 10 m increments and found that the mean bulk densities and the standard deviations for different ice types of the surveys did not change significantly beyond a length scale of about 200 m, if measurements of each ice type were abundant. Here, we computed sea-ice bulk density estimates at two length scales representing the sensor resolution of the CryoSat-2 satellite as well as the typical resolution of gridded sea-ice thickness data. In the first case, we approximated the scale of full-resolution altimeter footprint by the diameter of a circle with the same area as the 300m×1650 m pulse-Doppler-limited footprint of CryoSat-2 in the synthetic aperture radar (SAR) acquisition mode. The diameter of that circle is equal to approximately 800 m. In the second case, we chose the typical satellite product grid cell size of 25 km. We assumed that the sea-ice and snow layers are in isostatic equilibrium at both scales. We calculated an along-track weighted average using the squares of individual uncertainty values as weights: 5ρi‾=∑k=1N1σρi,k2ρi,k∑k=1N1σρi,k2, where k is the index and N equals the number of values within the length scale to be averaged. The resulting uncertainty was determined with 6σρi‾=1∑k=1N1σρi,k2. We also calculated inverse-uncertainty weighted averages for snow depth since its uncertainty varied spatially. For all other variables we calculated an arithmetic mean.

Figure  shows an example section of a measured sea-ice profile, roughly 30 km in length along a survey track and including different ice types in the Beaufort Sea in 2019. The bulk densities of FYI, SYI, and MYI, derived using the along-track length scale of 800 m, are stated in Table  and shown in Fig. . On average, FYI bulk density was higher than the value derived by (A10), also for individual surveys. FYI bulk density in 2017 was slightly higher than in 2019, and combined they resulted in an average density of 928.5±16.4 kgm-3. SYI and MYI bulk densities differed only a little from each other but were 23–30 kgm-3 lower than FYI bulk density. Similar to FYI, the bulk densities of old ice were in the upper range of or even beyond A10. Figure  shows the spatial distribution of the derived sea-ice bulk density when averaged over a typical satellite product grid cell size of 25 km. As expected, lower density values were generally encountered with increasing ice age (Fig. e and f). The lowest sea-ice bulk density values were located 50–150 km northwest off the edge of the landfast ice and the coast of Ellesmere Island and Nansen Sound (Fig. e). Moreover, there was also noticeable along-track variability of bulk density within a single ice type, even FYI where the highest bulk density values were also observed.

We explored the possibility to parameterise sea-ice bulk density using one of the measured sea-ice parameters. Out of the full parameter space, sea-ice freeboard showed the best correlation with the estimated sea-ice bulk density, r=-0.62 (p≪0.001), indicating a significant linear anti-correlation as expected according to Eq. () (ρi∝-hfi). However, the dependence on other sea-ice properties and the fact that keels of ridges with high freeboard contain voids filled with sea water introduced non-linearity to this relationship. To avoid underestimating the density values near the lower and upper ends of the observed freeboard range and to increase the goodness of fit, we therefore fitted an exponential function to the full, along-track averaged sea-ice bulk density data set (N=2246, Fig. ). Least-squares fitting yielded a relationship of 7ρi=a×eb×hfi+c, where the parameters with their respective 1 standard deviation errors are a=72.0±2.4 kgm-3, b=-3.7±0.4 kgm-3, and c=881.8±3.1 kgm-3. The resulting coefficient of determination (R2) was 0.42 and RMSE was 15.2 kgm-3. A bootstrapping approach with 104 random samples of the size N indicated an RMSE range of 14.8–15.7 kgm-3 with 95 % confidence. The fitted curve showed excellent agreement also with the inverse-uncertainty weighted average densities of the 0.05 m freeboard bins resulting in R2=0.89 and RMSE =6.2 kgm-3. Parameterisation split into different ice types or based on a different parameter, such as sea-ice thickness, surface roughness, or snow depth, resulted in poorer fits and values of R2.

Compared to A10, the average sea-ice bulk density estimates derived in this paper are larger by 11.8 kgm-3 (≈1.3 %) for FYI and by 20.4 kgm-3 (≈2.3 %) for MYI but still within the A10 uncertainties, albeit close to the upper limit. In general, our ice-type-averaged bulk density estimates fall within the range of previous studies . Reasons for the comparably high estimates are twofold. First, the A10 FYI density is representative only to level, undeformed ice whereas our estimate also includes deformed FYI. used drill-hole measurements that were carried out in three to five locations, 150–200 m apart around each aircraft landing site on level ice. Sea-water inclusions within deformed and unconsolidated sea ice increase the bulk density. This is a likely reason contributing to the higher FYI bulk density, especially in the 2017 data, given the increased deformation caused by the reversal of the Beaufort Gyre. Second, calculated the MYI density as a weighted average between the layers above and below the waterline based on values from numerous literature sources but used a density of 550 kgm-3 for the upper layer, which is significantly lower than the majority of the literature indicates (720–910 kgm-3 in ; 863–929 kgm-3 in ) and would correspond to an air-volume fraction of up to 40 %. When using the weighted average method in but the density values from instead, we find a MYI density of 909±28 kgm-3 that is closer to our estimate.

The effect of the uncertainties in the measured parameters and assumed sea-water and snow density values on the sea-ice bulk density was studied using Gaussian error propagation in Eq. (). Single point measurements typically resulted in sea-ice bulk density uncertainties of approximately 70 and 35 kgm-3 for FYI and MYI, respectively. Other sources of error arise from the different length scales illustrated in Figs.  and . We are not able to resolve snow depth fully within the total thickness measurement given the comparably large footprint of the EM-Bird, but we calculate it as an average of the snow depth measurements along a chord of the circular EM-Bird footprint. Due to the cross-track movement of the EM-Bird under the aircraft, the ground locations and the number of the snow depth measurements within the total thickness measurement vary slightly along the survey track but generally remain at 8 to 10 measurements close to the centre line. To ensure representative snow depth estimates, we require at least five valid snow depth measurements for each total thickness measurement, which translates into at least 50 % coverage along the chord. Errors may occur locally, e.g. at cross-track transition from a sea-ice floe to young ice in a newly refrozen lead as in the leftmost measurement points in Fig. , but we assume them to occur randomly and not cause systematic bias. However, uncertainties are reduced when averaged along-track over a length scale. Using the 800 m length scale, the resulting sea-ice bulk density uncertainty is generally less than 10 kgm-3 (Eq. ) but remain the highest for thin ice and low sea-ice freeboard where the relative uncertainties of the input parameters are the largest (see the size of the scatter points in Fig. ). In turn, averaging over a length scale simplifies the natural variability of sea ice. Figure  shows how a single 25 km satellite grid cell can already contain several sea-ice types: level, deformed, FYI, SYI, and MYI. However, assigning the sea-ice types is limited by the spatial (12.5 km) and temporal (weekly) resolution of the NSIDC sea-ice age product, which we try to compensate for with the additional thickness-based conditions (see Sect. ). In addition and similar to , the data are seasonally and regionally limited. Our measurements are confined to the western Arctic in early April, and therefore, more measurements across the Arctic and the seasons are needed to evaluate the spatial and temporal variability of sea-ice bulk density.

Assuming all the other parameters for the conversion of freeboard to thickness remain the same, the average sea-ice bulk density estimates derived in this paper would result in 12.4 % and 16.7 % larger sea-ice thickness values for FYI and MYI, respectively, in comparison to A10. The effect is larger for thicker ice, for which snow depth plays a proportionally less important role. Therefore, improving in particular the MYI bulk density is important to derive accurate time series of sea-ice thickness and volume, as in the past, thicker MYI represented a larger fraction of the Arctic sea-ice cover. recognised the possibility of varying MYI density between the recent younger MYI and older MYI of the previous decades and discussed the impact of MYI density on sea-ice thickness and volume. Moreover, a potentially increasing degree of deformation may lead to a bias in the time series, as satellites underestimate sea-ice draft of deformed ice as shown by and further discussed by . Deformed and unconsolidated sea ice has an increased bulk density due to sea-water inclusions, and thus, using a typical density of consolidated level sea ice for deriving sea-ice thickness from satellite data will eventually lead to an underestimation. Given the thinner and younger sea-ice cover together with observed increase in sea-ice drift speed and deformation , there is a likely premise for systematic underestimation by current parameterisations of sea-ice density in regions where and at times when sea ice is deformed. A full impact assessment of the sea-ice bulk density parameterisation on decadal sea-ice thickness data record is a logical next step but beyond the scope of this paper.

To represent sea-ice bulk density range as a functional relationship to a parameter observable from space rather than fixed values based on sea-ice type classification, we parameterised sea-ice bulk density using sea-ice freeboard and obtained a significant correlation. Opting for an exponential function instead of linear was beneficial for ensuring a better fit, capturing the high bulk density values at low freeboard values and avoiding linear decrease to possibly unrepresentative and unphysical values at high freeboards. Parameterisation in Eq. () sets the limits of bulk density to 953.8 kgm-3 at zero sea-ice freeboard and approaching 881.8 kgm-3 at high freeboards.

Figure  shows the parameterisation in Eq. () applied to the AWI Level-3 Collocated CryoSat-2 Sea Ice Product from the winter 2018/2019, converting the monthly gridded sea-ice freeboard to sea-ice density. The resulting sea-ice density distribution had a smoother transition between ice types compared to the current ice-type-dependent density classification of the retrieval algorithm. Difference between the density parameterised with Eq. () and A10 was positive overall during the winter except in the central Arctic Ocean and locally in the Fram and Bering straits in spring. The density difference was the largest on MYI in proximity to FYI but decreased toward spring.

Only less than 3 % of the airborne data set has a sea-ice freeboard value larger than 0.5 m, with considerable spread in bulk density values and thus introducing uncertainty to the parameterisation at high freeboard values. Constraining the parameterisation at high freeboards would require more data in deformed and multi-year sea-ice environments. However, that needs to coincide with a sufficient number of open leads to ensure accurate conversion of surface elevations to freeboard from the ALS. With the limitations of the current method, it is also not feasible to investigate cases of negative sea-ice freeboard due to the possible presence of liquid water and altered dielectric properties affecting the retrieval of snow depth.

Our parameterisation improves upon the previous formulations of sea-ice density given the significantly larger number of data points, larger areal coverage, the variety of ice types including deformed sea ice, and the choice of predictor variable it is based on. based his analysis on 17 FYI and 4 MYI sea-ice cores from the Beaufort Sea and derived a floe-thickness-based non-linear parameterisation of ρi=936.3-18hi0.5. used drill-hole measurements from 400 m of profile lines in the Beaufort Sea to derive ρi=-194hf′+974, where hf′=hfi+ρ‾sρ‾ihs≤1.05m is effective freeboard and the overbar denotes average density on the floe. The parameters needed for those formulations cannot be directly observed from space, or in the case of effective freeboard not even with a single in situ measurement, which makes them difficult to apply. Demonstrated in Fig. , our freeboard-based parameterisation has potential for future applications in satellite altimetry. While we acknowledge that, due to the variability in snow mass and sea-ice thickness, our parameterisation may not be applicable on sub-kilometre scales as reflected by the scatter in Fig. , we think that a sea-ice density parameterisation is a significant improvement upon a single value or fixed values based on ice type. In this paper, we decided to adopt a single-predictor parameterisation for the sake of simplicity. However, for future studies it could be worthwhile, for example, to apply machine learning algorithms to the full parameter space to discover possible multi-variable relationships. Given the effect of sea-ice deformation on bulk density, including sea-ice surface roughness, a multi-variable approach could explain more of the variability.