PM radiometry data were the main forcing for the proposed snow density retrieval algorithm. Radiometry data were acquired from the Advanced Microwave Scanning Radiometer Earth Observing System (AMSR-E) Calibrated Enhanced-Resolution Passive Microwave Daily Brightness Temperature Version 2 dataset , resampled to a 12.5 km EASE-Grid 2.0. PM observations spanned eight winter seasons (2003–2011) while the instrument was functional (reference snow density data were not available for the 2002–2003 season). AMSR-E observations for each station were extracted from an adjacent EASE grid cell to the AWS (highlighted in Fig. 1) to minimise water area in observation scene due to their proximity to the coast. Nighttime observations from the descending orbit track (∼01:30 a.m. local time at the equator) were used since snow conditions would be more likely to be cold and dry for optimal microwave retrievals . Radiometry samples were smoothed with a 5 d Gaussian weighted mean filter as described by . The 18.7 and 36.5 GHz vertically-polarised radiometer channels (hereafter 19 and 37 GHz, respectively) were used to estimate ΔTb in the forward model.

Meteorological measurements, acquired from the Environment and Climate Change Canada (ECCC) AWS network were also used for model forcing. The electromagnetic snowpack model was parameterised with AWS data, which required daily measurements of snow depth and air temperature as prior snow conditions. AWS data were the limiting factor in this experiment because the network is sparsely distributed in northern Canada limiting potential sites.

The Snow Microwave Radiative Transfer model SMRT: was used as the forward model in the retrieval algorithm. SMRT was configured with the Improved Born Approximation (IBA) electromagnetic model and microwave grain size microstructure model , which have been demonstrated to be representative of high-Arctic snow conditions . The substrate composition was parameterised to represent cryosolic soil following and atmospheric contributions were estimated as described by .

The physically-based forward modelling approach required the snowpack to be parameterised, so the relevant characteristics needed to be quantified. A two-layer snowpack model was configured to account for the presence of depth hoar underneath a wind slab layer to best represent the microwave signature of tundra snow . Upon initial deposition the snowpack would likely be in a homogenous state, with one layer, but that situation was not considered in this approach. The strong environmental controls present in the tundra contribute to the development of wind slab and depth hoar snow layers quickly after deposition , and algorithm retrievals were performed after 10 cm of snow had accumulated so the pack would be unlikely to be in the initial homogenous state.

Microwave retrieval algorithms have traditionally estimated snow depth using a vertically polarised brightness temperature frequency difference (ΔTb = 19–37 V), because of the sensitivity (insensitivity) of the 37 GHz (19 GHz) channel to snow accumulation, though we believe the same principle could be used to estimate snow density. Generally, ΔTb is thought to increase with snow depth due to increasing volume scattering until a threshold after which the signal is saturated by thermal emission originating in the snowpack . However, that is a simplified explanation of snow microwave interactions (i.e. only considering one layer) and can be complicated by stratification of natural snowpacks. For a tundra snowpack – with characteristic wind slab overlaying depth hoar – volume scattering is dominant for the depth hoar layer and non-scattering emission contributions originate from the wind slab . Thus, it is important to understand how the properties of each snow layer would impact microwave emissions to design an effective snow density retrieval algorithm.

The electromagnetic model (described in Sect. 4.1) was used to simulate microwave emissions from tundra snowpacks to assess its sensitivity to various parameters. The electromagnetic model requires snowpack physical properties to be quantified, including thickness, density, specific surface area (SSA), polydispersity, and temperature of each layer. A series of experiments were designed to illustrate the effects of the various model parameters representative of tundra snow, see and Arctic snow metamorphism; detailed descriptions of each experiment are provided in Table 4.

Snow density is our primary variable of interest, so it is important to understand how it effects microwave emissions. In the IBA model, scattering and absorption coefficients are in part related to snow density. The absorption coefficient increases linearly with snow density because of a greater proportion of ice-to-air in the microstructure representation altering the effective permittivity . On the other hand, the scattering coefficient has a non-linear relationship with snow density because of the interactions between individual scatterers in the snowpack. Volume scattering increases as more scatters are introduced (i.e. increasing density), until the scatterers are close enough in proximity to influence each other and the overall scattering efficiency decreases . Thus, density of the wind slab and depth hoar layers can affect ΔTb in different ways because of their properties that contribute to varying levels of volume scattering and thermal emission.

Experiments 1 to 3 were designed to simulate microwave emission from isolated wind slab and depth hoar layers, accounting for variations in SSA, polydispersity, and layer thickness, respectively (Figs. 2 to 4). The relationships between snow density and brightness temperatures follow skewed curves with minima at densities of 150 kgm-3 and frequency dependent amplitudes. Snow volume scattering is less sensitive to 19 than 37 GHz, so the frequency difference (ΔTb) is approximately the reflected 37 GHz curve and its magnitude depends on different microstructure properties . Lower (higher) SSA values produce greater (lesser) volume scattering, with minimal dependency on density, effectively translating the ΔTb curves vertically (depth hoar ∼9 K between 10 to 12 m² kg⁻¹ and wind slab ∼3 K between 15 to 25 m² kg⁻¹). Similarly, polydispersity effectively scales SSA, also translating ΔTb curves (depth hoar ∼19 K between 1.2 to 1.8 and wind slab ∼3 K between 0.6 to 0.9). Alternatively, layer thickness amplifies the relationship between snow density and simulated ΔTb, increasing sensitivity to depth hoar density (∼10 K between 150 to 450 kgm-3 at 5 cm vs. ∼28 K at 15 cm) and the wind slab to a lesser extent (∼0.5 K between 150 to 450 kgm-3 at 10 cm vs. ∼3 K at 30 cm). Seasonal snow density is typical above the 150 kgm-3 inflection point (ignoring fresh snow), so we can assume snow density has a negative relationship with ΔTb – with all other parameters equal, greater (lesser) ΔTb would indicate lower (higher) snow density.

While there was minimal model sensitivity to the isolated wind slab in Experiments 1 to 3, the effect of the wind slab on brightness temperature should be more apparent when parameterised over depth hoar. Experiment 4 was designed to demonstrate brightness temperature sensitivity for the two-layer snowpack representation, configured to replicate mid-season wind slab compaction over an established depth hoar layer. Wind slab thickness was parameterised to decrease with compaction (i.e. densification) for SWE to remain constant, and a range of initial SWE values (i.e. thicknesses) were considered (shaded areas in Fig. 4). When introduced over the established depth hoar layer, absorption and thermal emission originating in the wind slab mask ΔTb by several K depending on its SWE (∼2 K for 50 mm vs. ∼3 K for 85 mm). Then, absorption increased linearly with snow density and ΔTb was accordingly masked by the wind slab as it compacted (∼5 K between 250 to 400 kgm-3 for 50 mm vs. ∼8 K for 85 mm). Thus, wind slab formation resulting from compaction or thickening should be apparent in AMSR-E radiometry (i.e. evident from decreasing ΔTb), given its radiometric sensitivity of ±0.6 K. Furthermore, the magnitude of ΔTb masking by the wind slab is enhanced by the snowpack thermal gradient and a relatively colder wind slab compared to the substrate will increase ΔTb (∼2 K between 0 to -10 °C, not shown).

The results from the various experiments in the sensitivity test suggest there should be sufficient sensitivity to estimate snow density conditions from space-based PM radiometry. Further, PM radiometry was shown to be more sensitive to the thickness of depth hoar than the wind slab (and in turn overall snow depth) and, in terms of estimating Arctic snow mass, might be better suited to retrieving snow density rather than depth. PM retrievals of snow density were conducted at each AWS site, where meteorological conditions dictated when retrievals were performed. A minimum snow depth of 10 cm was imposed for algorithm retrievals because of the transparent nature of shallow snow to microwave emissions . Similarly, algorithm retrievals were not conducted when AWS air temperatures were above freezing because of the likelihood of liquid meltwater in the snowpack attenuating microwave emissions . With the AWS observations prescribed to the electromagnetic model an inverse modelling approach was applied to optimise the snow density parameters. The forward model was inverted by minimizing the cost function (J) 1J(ρws,ρdh)=(ΔTb,sim(ρws,ρdh)-ΔTb,obs)2 representing the vertically polarised 19 and 37 GHz spectral difference in the AMSR-E observation (ΔTb,obs) and the simulated SMRT signature at the same channels (ΔTb,sim), given the prescribed wind slab and depth hoar layer densities (ρws and ρdh, respectively).

The solution to the two-layer snowpack model presented was imprecise because different layer density combinations could produce the same predicted ΔTb in Eq. (1), resulting in a system with no global minima. The practical impact of this equifinality issue was that the algorithm may be confronted by seemingly equally valid but different layer density combinations, producing the same microwave signature. Without additional information there was no suitable way to identify the optimal layer density combination, so the retrieval algorithm was designed to solve for all microwave-plausible layer density combinations for a given observation scene to address equifinality in the inverse model.

To constrain the modelled layer density estimates to a plausible range, boundary conditions were established to limit the parameter space in which the algorithm could search for solutions to the inverse model. The first boundary condition was defined based on the strong environmental controls present in the tundra that result in a characteristic wind slab snow layer overlaying less dense depth hoar . Logically, the wind slab layer should be denser than the depth hoar layer, so all parameter combinations where ρws<ρdh were discarded, and the lower boundary situated where the densities of the two layers were equal. The second boundary for the model was defined based on the behaviour of microwave interactions in the electromagnetic model. Simulated ΔTb peaks at a snow density of 150 kgm-3 (see Sect. 4.2), and the apparent permittivity in IBA is applicable up to a volume fraction of 50 %, or 458.5 kgm-3 . Thus, the domain of each layer was limited to densities between 150 to 450 kgm-3 to ensure consistent behaviour in the electromagnetic snowpack model, and the upper boundary situated where either layer was at the edge of that domain.

An important aspect of the retrieval algorithm was to exploit how the various minima on the cost surface, defined by Eq. (1), were positioned throughout the parameter space to reduce computational requirements. Figure 6 is an example of how the minima formed a valley transecting the parameter space. The microwave-plausible density range is shown in Fig. 6a as the set of layer density combinations situated along a straight line connecting the solutions at the two established boundary conditions for the inverse model. Wind slab and depth hoar density combinations were mapped to bulk values in Fig. 6b, where the contours of iso-density will pivot clockwise (counterclockwise) around the lower solution when the proportion of depth hoar thickness increases (decreases). It should be noted that under some instances, the valley could intersect the upper boundary related to the minimum depth hoar density (i.e. left axis in Fig. 6a), though the situation shown (intersecting the upper axis) was more common.

The lowest contour level (±0.6 K) in Fig. 6a represents the sensitivity of the AMSR-E radiometer at 19 and 37 GHz and the grid spacing corresponds to algorithm retrieval accuracy (10 kgm-3). In theory, the lower boundary can produce a solution with similar magnitudes of measurement uncertainty and retrieval accuracy (the upper solution is less precise being situated in a broader part of the valley). However, the stated radiometric sensitivity of the AMSR-E instrument is related to measurements of surface emission and additional uncertainty in the retrievals could be introduced because of atmospheric contributions. The atmosphere was configured to represent typical subarctic conditions but the frequency dependent relationship with water vapour content could change observed ΔTb by ∼2 to 3 K from typical conditions. Thus, trends in snow density conditions should be apparent from spaceborne radiometry, with fluctuations from varying atmospheric water vapour content (partially mitigated with temporal filtering of PM radiometry); higher (lower) water vapour concentrations will cause the solution to shift towards the bottom-left (top-right) in Fig. 6a.

The range of microwave-plausible snow densities raised the question of how to evaluate the algorithm estimates against the reference data. A heterogeneity (H) parameter was introduced into the algorithm to estimate densities for the two snow layers and reduce the microwave-plausible snow densities to a single estimate of bulk snow density – ranging from 0 to 1 (i.e. the least and most heterogenous solutions, respectively). Wind slab (ρws) and depth hoar (ρdh) densities were estimated related to H with 2ρws=ρws,lower+H⋅(ρws,upper-ρws,lower)3ρdh=ρdh,lower-H⋅(ρdh,lower-ρdh,upper) where (ρws,lower, ρdh,lower) and (ρws,upper, ρdh,upper) are the lower and upper solutions, respectively, and bulk density (ρbulk) estimated based on the depth hoar thickness divided by the total snow depth (depth hoar fraction, DHF) 4ρbulk=ρws⋅(1-DHF)+ρdh⋅DHF Ultimately, the bulk snow density estimated with H was treated as the final algorithm estimate with uncertainty defined by the microwave-plausible range.

All existing retrieval algorithms have considered a single snow layer, so a new scheme was needed to parameterise the two layer snowpack model over the course of a season. Arctic snowpacks have been studied in detail during field campaigns see, though they are mostly restricted to end-of-season conditions around March-to-April and much less is known about Arctic snowpack composition early in the season. There have been some studies that focused on early season conditions , though they mainly provide qualitative descriptions of the temporal evolutions of Arctic snowpacks. Thus, our approach started with end-of-season conditions and worked backwards to parameterise the snowpack over the full season, with some parameters informed from available literature where possible and others calibrated.

Our temporal parameterization of snowpack properties was based on identifying trends in satellite PM and AWS observations, which we assumed to indicate different stages of snowpack evolution. Generally, two different behaviours were identified in the forcing datasets which we attributed to normal and restricted conditions for depth hoar development. In normal cases, ΔTb increased rapidly over a short period in the fall immediately after the first snowfall, coinciding with an extended early season zero-curtain period producing extreme vertical temperature gradients for rapid depth hoar metamorphism . In restricted cases, ΔTb increased gradually over longer periods of the season, consistent with high density layers slowly metamorphizing slowly into depth hoar . Later in the season ΔTb would plateau, attributed to a halt in depth hoar formation, before temperatures increase at end-of-season and ΔTb drops rapidly.

In total 4 different stages of snowpack evolution were identified, presented in Table 5. The proposed stages are numbered in the expected order of occurrence, but in practice their order can vary with some exceptions. Stage 0 is a special circumstance (i.e. does not happen every season) and must occur at the beginning of the season when temperatures are still around freezing. Then, the snowpack can alternate between Stages 1 and 2 throughout the season, owing to fluctuations in air temperature that change the thermal regime of the snowpack and snowfall events, prior to reaching equilibrium in State 2 towards end-of-season. Finally, the snowpack begins to warm in Stage 3 at the end-of-season with increasing air temperatures inverting the temperature gradient before ripening and final melt. The relevant state variables (i.e. layer thickness, thermal regime, and microstructure) were estimated dynamically considering the identified stage of snowpack evolution.

Some algorithm parameters could not be based on observations and instead needed to be determined through a calibration procedure. The calibration procedure consisted of two stages and ran each year from 15 March onwards, assuming snowpack properties would be mostly stable then. Calibrating for end-of-season conditions also allowed for parameters to be compared to those measured during field campaigns. First, wind slab SSA, depth hoar SSA, and depth hoar thickness were adjusted to produce the greatest overlap between the range of microwave-plausible snow density estimates and the in situ reference samples, with an overlap metric: 8overlap=1n∑t=1n|{ρest(t)}∩{ρobs(t)}||{ρest(t)}| where {ρest(t)} is the set of microwave-plausible estimated snow densities and {ρobs(t)} the corresponding CanSWE density sample with a ±10 % uncertainty range, at time t. Thus, the overlap metric described the proportion of the microwave-plausible snow density range that intersected the uncertainty range of the in situ samples, averaged over n time steps. Second, H was calibrated by converting the microwave-plausible algorithm estimates, from the first step, into discrete values to minimise mean absolute percentage error (MAPE). MAPE was chosen for this purpose, rather than absolute or squared error, because of the heteroscedastic nature of the uncertainty in the reference dataset.

At each site, algorithm snow density estimates were evaluated with CanSWE bulk density samples using the same metrics as in the calibration stage (i.e. overlap and MAPE); bias, root mean square error (RMSE), and correlation were also reported as indicators of algorithm performance. MAPE was treated as the primary measure of absolute accuracy of algorithm estimates – if MAPE was within the uncertainty range of the in situ samples (±10 %) then snow density estimates from the algorithm could be comparable to those collected with snow courses.

Calibrating the two layer snowpack model with bulk density measurements (i.e. CanSWE) introduced some uncertainty into the algorithm configuration parameters. As demonstrated by the sensitivity test, depth hoar SSA and thickness have complementary effects on simulated ΔTb – i.e. lower (higher) SSA can compensate if the depth hoar is too thin (thick) – so various SSA and thickness combinations could produce similar microwave emissions. At each site, SSA parameters were kept constant over all seasons because inter-season variations in SSA should be relatively low , but DHF was free to account for varying environmental conditions. End-of-season H values were also kept constant for each site due to the lack of stratigraphic data to conduct a meaningful calibration and in an effort to reduce the number of free parameters in the calibration procedure. In the future, extensive stratigraphic data from multiple sites should be used for calibration to increase confidence in specific algorithm parameters.

Algorithm configurations were calibrated to represent end-of-season conditions, for each site some parameters were kept static over all seasons (Table 6) and depth hoar thicknesses varied each season (Table 7). The sensitivity test demonstrated the model was most sensitive to depth hoar parameters, so depth hoar SSA varied between 10.0 to 13.0 at increments of 0.2 m² kg⁻¹, and fewer options considered for the wind slab of 15.0, 17.5, or 20.0 m² kg⁻¹. For the polar desert sites (Alert, Eureka, and Resolute Bay) the calibration routine produced configurations that were fairly similar and in line with those expected in the polar desert, with depth hoar SSA around 10 to 11 m² kg⁻¹ and average DHF of approximately one third . On the other hand, the configuration for Cambridge Bay was different, with higher than expected depth hoar SSA and DHF for tundra type snow .

Snow survey data from were used to evaluate the calibrated model configuration for the Eureka site in greater detail. The model was originally configured to replicate bulk density measurements (i.e. CanSWE) making it difficult to evaluate individual parameters without stratigraphic information. For example, simulated depth hoar thickness and SSA could compensate for one another without discernible differences in bulk density. Although SSA was not measured in the survey protocol, calibrated SSA values were evaluated by forcing the retrieval algorithm with measured layer thicknesses and AMSR-E L2A observations at 25 km , and the output compared to measured bulk, wind slab, and depth hoar densities (Fig. 7). Algorithm estimates showed good agreement with the measured values, though with slight overestimation for depth hoar and underestimation for wind slab densities. Interestingly, the valley of algorithm solutions for three gird-cells (1, 2, and 4) aligned with regions of iso-density in the parameter space (Fig. 6b) so H could increase slightly to reduce underestimation of wind slab density without affecting overall bulk density. While we cannot conclude from this limited sample size that the algorithm is perfect, the similarity of the algorithm estimates and layer densities to independent snow surveys suggest the parametrization of SSA was effective for Eureka.

Snow depth and DHF can be variable in the tundra , so parameterizing the snowpack model with static parameters could lead to uncertainty. Algorithm performance with calibrated depth hoar thicknesses were compared to those using generalised representations (i.e. seasonal thicknesses, average thickness, and average DHF from Table 6). Parameterizing the depth hoar layer with average thicknesses for each site improved algorithm estimates slightly compared to average DHF but the seasonal parameterization performed considerable better than either (Table 8). Further, seasonal depth hoar thicknesses were the only to bring algorithm estimates within the uncertainty range of the reference dataset at all sites (±10 %).

Calibrated seasonal depth hoar thicknesses were plotted against end-of-season DHI from Eq. (5). to identify a relationship to estimate dynamic depth hoar thicknesses (Fig. 8). Model configurations for each site should be equivalent (specifically depth hoar SSA) for a robust comparison of depth hoar thicknesses, so the configuration from Eureka was applied to the other sites since it seems representative of in situ conditions (see Sect. 5.2). Calibrated depth hoar thicknesses had a very strong relationship with DHI at Alert (R2=0.94, p<0.01), moderate relationships for Eureka (R2=0.68, p=0.023) and Resolute Bay (R2=0.64, p=0.20), and virtually no relationship for Cambridge Bay (R2=0.01, p=0.82). There was considerable spread in plotted values for Cambridge Bay and, when removed, the polar desert sites together have a very strong relationship (R2=0.93, p<0.01) fitted with a linear model 9TEOS=0.349⋅DHI15Mar-3.75 allowing end-of-season depth hoar thickness (TEOS) to be estimated in cm from DHI on 15 March (DHI15Mar).

The temporal parameterization (described in Sect. 4.4) was used to force algorithm retrievals over full winter seasons. The calibrated configuration for Eureka was used for all sites and dynamic depth hoar thickness (DT) estimated in cm with 10DT(t)=max⁡TEOS⋅DHI(t)DHI15Mar,TEOS where DHI at time step t was from Eq. (5) and TEOS from Eq. (9), and the maximum operator did not allow for growth after 15 March. Algorithm runs over all seasons at each site were aggregated to calculated performance metrics, presented in Table 9. Results for the three polar desert sites were similar with moderate MAPE (<20 %), weak-to-moderate positive correlations, and low magnitudes of bias, whereas, Cambridge Bay had higher MAPE, larger positive bias, and a weak negative correlation.

A collection of notable algorithm simulations was included in Fig. 9 – some as examples of when the algorithm performed very well and others to demonstrate limitations – all simulations are included in Appendix B. Seasonal performance at Eureka was mixed, where three seasons had low MAPE (i.e. <10 %, e.g. Fig. 9a), 3 had moderate MAPE (i.e. <20 %, e.g. Fig. 9b), and one high MAPE (i.e. >20 %, Fig. 9c). The algorithm performed similarly at Alert, where three seasons had low MAPE (e.g. Fig. 9d), one moderate MAPE (not shown), and one high MAPE (Fig. 9e). Alternatively, algorithm performance at Resolute Bay was worse overall, where only one season had relatively low MAPE (not shown) and the other three had higher MAPE (not shown). Results for Cambridge Bay were more nuanced and the relatively high overall MAPE did not tell the whole story. In all but one algorithm run simulated density started considerably higher than reference samples in the early season but matched in situ samples very closely from February onwards (e.g. Fig. 9f).

In the following subsections, key parameters (i.e. SSA and depth hoar thickness) of the calibrated end-of-season configurations were compared against measured values from various field campaigns.

Estimation skill over the full season (Table 9) was lower than during the calibration stage (Table 8), though that was expected because the configuration for Eureka was used for all sites and depth hoar thickness was parameterised with Eq. (10) (rather than calibrated values for each site). In some cases the temporal parameterization produced excellent estimates of snow density over the whole season (e.g. Fig. 9a and d) but in other cases struggled to reproduce the observed densification trajectory (e.g. Fig. 9c and e). Yet, algorithm estimation skill at each site consistently improved over the course of a winter season and most algorithm estimates were close to the in situ references samples later on. To quantify this behaviour the reference dataset was partitioned into three temporal sets – October–November–December (OND), January–February–March (JFM), and April–May–June (AMJ) – and overlap, MAPE, and bias calculated for each in Table 10. There were substantial improvements in all metrics at all sites between OND to JFM and JFM to AMJ, and AMJ MAPE for the polar desert sites were within, or approaching, 10 % indicating the snow density estimation uncertainty was similar to the in situ samples. Temporal results for Cambridge Bay were slightly different than polar desert sites as there was a substantial improvement in all metrics from OND to JFM (most notably the reduction in bias) but MAPE increased in AMJ. Possible explanations for these temporal behaviours in algorithm estimates are discussed below.

The most likely reason for improved algorithm performance towards the end-of-season during most simulations is that the snow metamorphic state was captured effectively by model dynamics that align with our understanding of snowpack metamorphism. Prior knowledge from available literature increased confidence in end-of-season algorithm configuration, though much less was ready for the early-to-mid season introducing uncertainty into the temporal parameterization. Specifically, some properties were effectively quantified with physical models over time (e.g. SSA) while others were not because model representation is simply not developed (e.g. depth hoar thickness).

From the point of view of algorithm development, the most difficult element to parameterise over time was depth hoar thickness. The depth hoar model was generalised to not overfit to any specific forcing data, but edge cases were identified where there were issues. In some cases identified as standard development, and with thicker initial snow depth, Eq. (10) appeared to underestimate early-season depth hoar thickness causing simulated bulk density to pin at the bottom of the range to maximise volume scattering (e.g. Fig. 9c). That early-season underestimation could be related to how depth hoar was parameterised to grow vertically in thickness, which would be logical for indurated development (growing at the expense of wind slab thickness) but normal depth hoar should form from early layers morphing simultaneously. On the other hand, under the most restrictive conditions identified for depth hoar metamorphism Eq. (10) overestimated depth hoar thickness throughout the whole season casing algorithm estimates to pin at the upper limit of the density range to minimise volume scattering (e.g. Fig. 9e), despite very similar simulated (2.0 cm) and calibrated (0.8 cm) end-of-season thicknesses. Thus, our depth hoar model could be improved to consider specific situations – for example, initiating thickness with early-season snow depth measurements during Stage 0 (assuming the entire layer would shortly become depth hoar) and using a fixed thickness (∼1 cm) when very restrictive conditions are identified.

Even with the help of existing models there were challenges with the parameterization of SSA. Most notably, there is practically no distinction in the literature between standard and indurate depth hoar microstructure in terms of SSA and polydispersity, so we did not distinguish between their prescribed microstructure properties. While physical grain size of standard and indurated depth hoar are similar , non-metamorphosed wind slab grains can be present in indurated depth hoar ; possibly leading to higher SSA or lower polydispersity compared to standard depth hoar, necessitating thicker simulated indurated layers. Further, our snowpack representation did not account for deposits of fresh snow, which have low density and high SSA, and, therefore, should be radiometrically negligible . However, new snow was immediately incorporated into the simulated wind slab layer affecting simulated, but not observed, brightness temperatures – for example, mid-season snowfall events at Eureka in January 2011 (Fig. 9b) caused measured bulk density to decrease but simulated bulk density increased. Identifying depth hoar type with the proposed stages of snowpack evolution would not only aid in parameterizing algorithm retrievals (should their microstructure properties prove to be sufficiently different) but could also support applications where snow hardness and thermal conductivity are relevant – for example, permafrost thermal regimes and conditions for subnivean life .

Algorithm estimates generally followed expected densification trajectories (i.e. increasing density over time) in the polar desert (e.g. Fig. 9b) but exhibited different behaviour at Cambridge Bay. Early season density estimates were too high in all, but one, simulations at Cambridge Bay and decreased over time to move closer to in situ measurements (e.g. Fig. 9f). Early season overestimation could be explained by penetration depth at 19 GHz exceeding lake ice thickness , which reduced observed ΔTb causing simulated density to pin at the upper limit to minimise volume scattering. Then, estimates improved over the mid-season when lake ice thickness should exceed the penetration depth at 19 GHz, before thinning ice thickness reintroduced uncertainty in observed brightness temperatures at the end-of-season . Additionally, the radiometric influence of water bodies made it more difficult to interpret the stages of snowpack evolution at Cambridge Bay – Stage 0 was only identified during a couple seasons, despite tundra conditions being generally favourable for depth hoar development . Furthermore, unfrozen water bodies around Cambridge Bay caused pre-snow ΔTb to be very low (i.e. negative ∼10 K) artificially modifying DHI values, likely contributing to the spread of points in Fig. 8. After February, when ice thickness should exceed penetration depth , algorithm performance for Cambridge Bay was comparable to the polar dessert sites (MAPE = 13.2 % and overlap = 43.6 %).

The ultimate goal of this research is to develop a pan-Arctic snow density retrieval algorithm, though the algorithm would need to be modified for that purpose. The current retrieval design is predicated on a two-layer snowpack with distinct properties (i.e. found in the tundra/polar desert) and would need to be modified to consider other Arctic snow types (e.g. taiga). Traditional ecological knowledge of snow conditions e.g. could help to identify important snowpack parameters across various environments to be generalised for the electromagnetic model. Additionally, water bodies could impede retrievals using a ΔTb approach (as described for Cambridge Bay) and a single channel retrieval using only 37 GHz might be more appropriate across the pan-Arctic . Also, the dynamic depth hoar parameterization required PM observations from snow-on to 15 March limiting its use to retrospective analyses, though the relatively long PM observation record would allow for climatological analysis.

After necessary modifications, additional datasets would be required to expand the spatial extent of algorithm retrievals. Snow depth data are the most important to force the algorithm (after radiometry) and the sparse distribution of AWS across the pan-Arctic render them unsuitable for extensive model forcing. Spatially continuous snow depth estimates could be derived from reanalysis models, even as a first order effect, despite their uncertainty in high latitude areas where data are sparse . Assimilation of reanalysis snow depth estimates with AWS data for bias correction might be a promising way forward. Similarly, bias corrected ground temperature estimates from reanalysis products could replace our simple model based on AWS air temperature. Additionally, auxiliary wind speed and soil moisture data could aid with parameterizing the depth hoar layer (i.e. quantifying the potential for development) as they restrict and promote development, respectively . Finally, a pan-Arctic snow density product would require extensive reference data to support algorithm calibration and evaluation which will need to be curated, specifically regarding extensive datasets of snow stratigraphy.