We attempt to statistically model SIT using surface-measurable metrics (e.g., mean and standard deviation of the snow freeboard), in order to see the limitations of this method. To accurately calculate SIT without making assumptions of snow distribution, we need to use combined measurements of ice draft (AUV), snow freeboard (lidar) and snow depth (probe). Here, we primarily use PIPERS data to focus on early-winter Ross Sea floes and also because this is the largest such dataset from one season and region, which is important so that the ridges have consistent morphology.

We use a simple (multi)linear least-squares regression with either one (snow freeboard, F) or two (F and snow depth, D) variables with a constant term, such that T=c1F+c2D+c0.

For the two-variable fit, we do an additional fit with the constant forced to be zero, in order to obtain coefficients that can be used, following Eq. (), to estimate the snow and ice densities.

To measure the fit accuracy, we use the mean relative error (MRE), as this avoids weighting errors from thin or thick ice differently. The Radj2 value, adjusted for a different number of variables, is also reported where possible (it is not defined for a fit forced through the origin). When comparing the generalization of the fits to test data excluded from the fit data, we also report the relative error of predicting the mean survey-wide thickness (REM), as often researchers are interested in the aggregate statistics of a survey. These fit errors in estimating mean SIT are compared to both prior relationships derived from drilling data to highlight uncertainty when used with different ice conditions and to our ConvNet predictions of ice thickness.

In order to motivate more complex methods in subsequent sections, we also use surface roughness (standard deviation, σ) to predict thickness to demonstrate that surface morphological characteristics have some information that can be used to predict thickness.

One advantage of deep-learning techniques is that they are able to learn complex relationships between the input variables and a desired output, even if the relationships are not obvious to a human. Although they are commonly used for image classification purposes, they can also be used for regression e.g.,. We expect a convolutional neural network (ConvNet) to achieve lower errors in estimating SIT, as they are able to learn complex structural metrics, in addition to simplistic roughness metrics like σ. Our input is a windowed lidar scan (snow freeboard) and an output of mean ice thickness. Notably, there is no input of snow depth, or any input of ice or snow densities. This allows the ConvNet to infer these parameters by itself and more importantly to potentially use different density values for different areas.

Our architecture is shown in Fig. . The input consists of 20 m × 20 m (100 pixel × 100 pixel) windows, with three convolutional layers, with a stride of 2 in the first two layers, and two fully connected layers. We used scaled exponential linear units (SELUs) to create nonlinearity . The loss function used was the mean squared error. We also used dropout (p=0.4) and augmentation (random 90∘ rotations, horizontal and vertical flipping) to reduce overfitting . An overview of ConvNet basics and full implementation details are given in the Appendix.

The training–validation set consisted of randomly selected windows from three PIPERS ice stations, each on a different floe. We chose 20 m as the window size by using the range of the semivariogram for the floes (25 m), which we expect to represent the maximum feature length scale. This compares well to an average snow feature size of 23.3 m from early-winter Ross Sea drill lines from . We chose 20 m instead of 25 m windows to balance this with the need for a smaller window size to ensure a larger number of windows (data points) for our analysis. These data were randomly divided into 80 %–20 % to make the training and validation sets. The remaining floe (divided into windows) was kept as a test set, in case the training and validation windows had similar morphology and the validation set was thus not entirely independent of the training set. To prevent cherry-picking, the ConvNet was trained four times, with a different floe used as the test floe each time. Results are shown in Table . Although the training error is directly analogous to the fit error for linear models for some dataset, it is much easier to overfit with a ConvNet as the training error can be made arbitrarily low. As a result, we compare our validation error to the linear fit errors, and we also use our test errors as a test of the generalization of our model. From here onwards, analysis of the ConvNet refers to the one using PIP8 as a test set, though using a different one would yield qualitatively similar analysis.

The (irreducibly) poor generalization of linear fits, likely due to a locally varying proportion of snow and ice amongst different surface types, motivates the use of more complex algorithms that can account for the surface structure. For this, we use a ConvNet with training, validation and test datasets as described in Sect. .

The best validation error was 15 %, corresponding to a training error of 11 % (Fig. a and b). The mean test error (on the excluded floe) was 20 %. Although the linear models have a similar fit error, they do not generalize as well to the test set, and the resulting thickness distribution is visibly different to the real test distribution (Fig. c).

This shows better generalization than the linear models (test MREs from 28 % to 47 %). Although the best-performing linear models have only slightly higher test MREs (24 % for the three-variable fit in Sect. ) than our ConvNet (20 %), the range of errors is much greater, with test MREs of 18 %–29 %, whereas the ConvNet has remarkably consistent test MREs of 18 %–20 %. Furthermore, it is important to remember that achieving these comparably low MREs with linear models requires snow depth as a variable, which is generally not available. These fits also typically include a negative constant (Table ), which means T<0 for F=D=0, which is clearly unphysical and limits the application of these models to areas of low snow freeboard. The fits to snow freeboard only, which uses the same input data as the ConvNet, have considerably higher test MREs (23 %–39 %; see Table ). For the sake of comparison to models that use rms error, such as , the validation rms error for our survey-averaged mean thickness values is 2 cm, which is lower than the rms error of 11–15 cm from . Our fit uses three surveys from three different floes as an input, which means the fit is likely lower in error than , which uses 23 floes. However, we would also expect poorer generalization for our test set from using only three surveys. Although our test rms error for the mean survey thickness (3 cm) cannot be directly compared, it is reasonable to surmise that our ConvNet achieves better generalization than a linear fit. Note that the rms error is not linked to the surface rms roughness, which is just the standard deviation of the snow freeboard.

As shown in Fig. c, the ConvNet does seem to be capturing the thickness distribution of the test floe, even if the individual window mean estimates have some scatter. In contrast, the linear models have considerably different thickness distributions (Fig. , red points and lines) despite having similar fit MREs (Table ). The ConvNet also successfully reproduces the spatial variability of the SIT distribution better than the linear fit (Fig. ). Note because of the small size of the dataset, there is significant oversampling in the ConvNet prediction of the floe SIT distribution. The primary difference between the ConvNet and linear fit for this floe is a large overestimation of level ice thickness. This demonstrates the inability of the linear fit to account for variations in effective densities and/or snow / ice freeboard ratios. The ConvNet prediction can have some large local errors, in this case chiefly on the flanks of the ridge, where steep freeboard or thickness gradients may affect performance. Comparisons for other floes (not shown) are qualitatively similar, though the spatial distribution of fit errors varies among floes. The key result of the ConvNet is in the significantly reduced error in the local (20 m scale) mean thickness (MRE of 15 %–20 %), which also gives a low ∼10% error of the average scan-wide thickness. Moreover, this high accuracy also carries over to test sets from the same region and season. In contrast, linear models, which do not generalize well to new datasets, have a considerable bias (Fig. ), despite having an ostensibly good fit. Analysis of why the ConvNet may be performing better than linear fits is given in Sect. .

Our linear regression results for fitting T=c1F+c0 have markedly different coefficients from drill line data from the same region and season . Here we discuss possible reasons for their differences. The first difference is that our value for c1=7.67 (Table ) is much higher. This is almost certainly because our dataset includes much more deformed ice, as we deliberately sampled deformed areas on floes. At one extreme, where the snow load is large such that the snow depth = snow freeboard assumption is approximately valid (set F=D in Eq. ), which for our data occurs for level, thin ice where there is some snow load, Eq. () would simplify to T=2.7F (using density values from ). In contrast, when the topography is sufficiently rough, there is considerable ice freeboard, which may even exceed snow depth. If we assume the snow is negligible (D=0), which may be the case at the sail peak, Eq. () becomes T=9.4F. These values become lower and upper bounds for fitting c1 in T=c1F (without the constant c0). The best fit value for c1 is 5.8 when fitting to the full dataset (Fig. ), which falls between these two extremes of snow-only F and ice-only freeboard F. Our coefficient is also comparable to , who found a coefficient of 5.23 from first-year Weddell ice. Much as in , our dataset includes considerable deformed ice, which has a nonzero ice freeboard, and so the coefficient of F is higher than 2.7. We can estimate the ratio of snow to ice by comparing this with the hydrostatic equation: for example, if we assume typical snow and ice densities of 300 kg m-3 and 920 kg m-3, this implies that snow, on average, comprises 54 % of the measured snow freeboard. Using these values, Eq.  simplifies to T=5.8F, as in Fig. . In further support of this, our dataset has mean snow depths for the four surveys ranging from 16 to 26 cm and mean snow freeboards ranging from 24 to 37 cm, implying considerable nonzero mean ice freeboards.

The high scatter of our fit also suggests that the snow / ice ratio varies locally, as can be expected around level and deformed ice. If the proportion of ice to snow were constant, then the best-fit line, for whatever slope, would have no scatter. This is not the case in Fig. , and indeed the standard deviation of ice freeboard across all windows was 7.9 cm (mean: 9.0 cm). This means that assuming a constant snow and ice density or a constant snow–ice proportion is not justified, and hence it is likely that simple statistical models break down when looking at deformation on a small scale or when large-scale snow deposition and ice development conditions vary. This mirrors the conclusions in , who found that linear regressions could not capture locally and regionally varying snow / ice proportions. Even when including regime-dependent fits (Sect. , Fig. ), this does not improve the test errors because this is likely too simplistic (even within a ridge, the ratio of snow to ice is likely varying). An important point regarding σ is that it does not actually account for the surface morphology very well, as any permutation of elevations within the window will give the same σ. This means that the “shape” or “structure” of the surface is not truly accounted for. This motivates more complex metrics for surface roughness (Sect. ).

Unlike our approach, the fits in and use large-scale, survey-averaged data. Their coefficients for c1, 2.4–3.5 and 2.8 for Ross Sea and Bellingshausen Sea data, respectively, are near the theoretical value of 2.7 assuming no ice freeboard. This suggests that at large scales for some seasons and regions, it may be reasonable to assume that the mean ice freeboard is zero, but this is not the case at smaller scales. It is also possible that drill lines have undersampled ridged ice due to sampling constraints or (in our case) sample heavily deformed areas that are not typically sampled in situ. Thus, empirical fits should be used with caution.

The second major difference is that our intercept is negative, whereas those from and are all positive. In our case, it is possible to interpret our negative intercept as a result of fitting a linear model across two roughness regimes. From above, the two regime extremes (no-ice vs. no-snow contribution to snow freeboard) give T=2.7F and T=9.4F as limiting cases. In general, we expect the proportion of ice freeboard to gradually increase as F increases from thinner, level ice to thicker, deformed ice. Although snow also accumulates around deformed ice, there may also be local windows at parts of the ridge with no snow (e.g., the sail). This means that we expect a gradual transition from T=2.7F to T=9.4F as F increases. Fitting one line through these two clusters of points would result in a coefficient for F between 2.7 and 9.4 and a negative intercept, which we find in almost all our cases. The one exception is the fit for the level category, which is essentially a null fit (as over 90 % of the thickness values are clustered around 0.5±0.05 m). In contrast, the coefficients for F from and are all ∼3 because these studies average over multiple floes and have a sufficiently small proportion of deformed surface area to assume a negligible ice freeboard as discussed above. In their case, their intercept would be positive, as their ice thickness estimates would be otherwise underestimated due to some of the snow freeboard being ice instead of snow.

When fitting a linear or ConvNet model to snow freeboard data, we cannot know whether there are negative ice freeboards; as such, these methods account for it only implicitly, with a linear fit effectively assuming that a similar percentage of freeboards will be negative. This may contribute to errors when trying to apply a specific linear fit to a new dataset. A ConvNet could conceivably do better here, in that significant negative freeboard is likely to matter most when there is deep snow, which might have recognizable surface morphology, although this is quite speculative.

The ConvNet performs better than the best linear models in both fit and test MREs. However, the ConvNet trained with our dataset is very limited in applicability to only datasets from the same region and season. When we applied our trained ConvNet to lidar inputs from a different expedition (SIPEX-II; ) from a different season and region, the MRE is 69 %, and the REM is 51 %. This suggests that other seasons and regions may have different relationships between the surface morphology and SIT, which is not surprising given that snow accumulates throughout winter. The SIPEX-II data were collected during spring in coastal East Antarctica in an area of very thick, late-season ice with very deep snow with large snowdrift features of length scales >20 m (which would not be resolved by the ConvNet filters here). It is also possible that datasets from spring, such as SIPEX-II, will not be as easy to train networks on because the significantly higher amounts of snow may obscure the deformed surface. Although this points out a limitation of this method, which restricts any trained ConvNet to a narrow temporal–spatial range, it also adds weight to the idea that the ConvNet is learning relevant morphological features. A ConvNet trained on Arctic data would likely learn different features (e.g., melt ponds and hummocks), although additional filters may be needed to distinguish multiyear and first-year floes.

We also tried different inputs, such as using 10 m × 10 m windows, which had training, validation, and test errors of 9%, 18 %, and 25 % and using 20 m × 20 m inputs with half the resolution (i.e., 0.4 m), which had errors of 7%, 13 % and 25 %. The smaller window case has a slightly higher validation error than the above ConvNet, and the coarser-resolution input has a slightly lower validation error than the above ConvNet, but both cases have slightly higher test errors. Larger windows, which are more likely to capture surface features, are likely to improve the fit, but our dataset is too small to test this as larger window sizes would mean fewer training inputs. However, it is promising that the validation errors are lower at a coarser resolution. This suggests that this method may indeed extend to coarser, larger datasets like those from airborne laser altimetry from OIB. We also tried training for the mean snow depth given the lidar inputs, with training, validation, and test errors of 15%, 17 % and 18 %, which is very similar to the thickness prediction. This is not entirely surprising as, if hydrostatic balance is valid, being able to predict the mean thickness given some snow freeboard measurements naturally gives the mean snow depth via Eq. ().

Although the ConvNet achieved a much lower test error than the linear fits, the inner workings of a ConvNet are not as clear to interpret. We can try to analyze the learned features by passing the full set of lidar windows through the ConvNet to see if the final layer activations resemble any kind of metric. The below analysis of features is very qualitative, as it is inherently very difficult to characterize what a ConvNet is learning.

One helpful way to gain insight into what the ConvNet is learning is to inspect the filters. Filters in early layers tend to detect basic features like edges (analogous to a Gabor filter, for example), with later layers corresponding to more complex features like lines, shapes or objects . We see similar behavior in our filters; typical filters learned in our model are shown in Fig. . Early filters highlight basic features like edges when convolved with the input array, while later filters show more complex features. These complex features are hard to interpret, but are clearly converged and not just random arrays. For example, a “blob” feature could be a snow dune filter, while filters with a clear linear gradient could correspond to the edge of ridges. The filters in the final layer are around ∼8 m in size. This may be too small to resolve the entire width of the ridges in our dataset, but would be enough to identify areas near ridges. With a larger windowed lidar scan, such as those from OIB with a scan width ∼250 m , we expect better feature identification, as the entire width of a ridge can be resolved within a filter.

The learned weights for the final (8×1) hidden layer and their activations (when each input window is fed forward through the ConvNet) are shown in Fig. a, grouped by category (level, ridged, snowy). These should correspond to (unspecified) metrics, which are linearly combined with the weights shown in Fig. b. It is clear that level surfaces are distinguished from ridged and snowy surfaces, but ridged and snowy surfaces show considerable overlap with each other. While it is not possible to determine with full certainty what each of the eight features corresponds to, we can correlate these features to metrics that we may expect to be important for estimating the ice thickness and see which ones match. Performing this analysis, for ridged surfaces, features 0, 3 and 6 had a strong correlation (|R|>0.95) to the mean snow freeboard (Fig. d); for snowy surfaces, these three features had a slightly weaker correlation (0.88<|R|<0.96) to the mean snow freeboard; and for level surfaces, features 1 and 5 had a slight correlation (|R|=0.67 and 0.80, respectively) to the mean snow freeboard (Fig. a). However, features that correlated to the ridged surface mean snow freeboard did not correlate to the level surface mean snow freeboard, and vice versa (Fig. b and c). This suggests that the mean snow freeboard for level surfaces is treated differently (e.g., given a different effective density) than other categories.

For ridged surfaces, in addition to the mean snow freeboard, the rms roughness was also important, with features 2 and 4 weakly correlating (|R|=0.61) to the standard deviation of the window. The standard deviation had a slightly weaker correlation (|R|=0.55) for level surfaces and virtually none at all for snowy surfaces (|R|<0.20). Another measure of roughness is the rugosity (the ratio of “true” surface area over geometric surface area; see ). This was most important for the snowy category, with |R|=0.57 for feature 7, compared to |R|=0.53 for feature 6 for ridged surfaces and |R|=0.22 for feature 2 for level surfaces. As we found before, these features were much more strongly correlated to the mean elevation and standard deviation, respectively for their respective surface category. This was not the case for feature 7 for snowy surfaces, which had a similar correlation (|R|=0.54) to the mean elevation and a much weaker correlation (|R|=0.35) to the surface σ. To summarize, for all categories, the mean snow freeboard is important (though weighted differently, as different filters activate for different categories). For both level and ridged surfaces, the rms roughness is important, and for snowy surfaces, the rugosity is also important. The above analysis suggests that there are important regime differences for estimating SIT. It should be noted that these statistical metrics suggested above, with the exception of rugosity, do not account for structure (any permutation of the same numbers has the same mean/σ), which limits the usefulness of this approach to interpreting the ConvNet.

This is by no means an exhaustive list, but it suggests that the ConvNet is learning useful differences between different surface types. However, as suggested by the considerable overlap in the distributions in Fig. , these categories may also not be the most relevant classifications. Alternatively, a t-distributed stochastic neighbor embedding (see ), which is an effective cluster visualization tool, shows that ridged and level surfaces are clearly distinguishable, but there is considerable overlap between the snowy and ridged categories (Fig. ). However, the ridged category is quite dispersed, and may even consist of different classes of deformation which should not be grouped all together. Nevertheless, it is apparent that at the very least, the level and non-level categories are meaningfully distinguished. With more data and larger scan sizes (e.g., from OIB), a deep-learning neural network suitable for unsupervised clustering (e.g., an auto-encoder) could identify natural clusterings with their associated features .

To emphasize the importance of the mean elevation, we also tried training the same ConvNet architecture with demeaned elevation as the input. Our ConvNet architecture is able to achieve a lowest validation error of 25 % (training error 10 %), but the test MRE is relatively high (40 %). The test error is worse than the linear model and has twice the test MRE of our ConvNet with snow freeboard (test MRE: 20 %).

We also trained the ConvNet to predict the mean snow depth, with comparable training, validation, and test errors of 15%, 17 %, and 18 % when using raw lidar input and errors of 15%, 22 %, and 45 % when using demeaned lidar input, which suggests the same analyses hold for snow depth prediction. As the snow depth is largely correlated with the snow freeboard e.g.,, with the exception of ridged areas, it is not surprising that the demeaned input is not as good a predictor of the snow depth. However, when metrics obtained from the demeaned snow freeboard (such as roughness) are combined with the mean snow freeboard, snow depth estimates (as well as SIT estimates) are improved. This may mean that aside from the mean snow freeboard, surface lidar scans may contain other information (e.g., morphology) capable of improving both SIT and snow depth predictions. This is promising for applications to larger datasets such as OIB or ICESat-2.

Another approach to analyze these learned weights is to look at the sign of the weight and the typical values of the activations in Fig. . Feature 0 has a negative weight for which the ridged category (and to a lesser extent snowy) has the largest (most negative) feature values; this leads to adding extra thickness, primarily for the ridged ice category. This perhaps accounts for a higher percentage of ice freeboard in the snow freeboard measurement than for the level and snowy categories. Indeed, most of the level category has values near 0 for this feature. This could therefore be interpreted as a “deformation correction” of some sort, or increasing the effective density of the ridged surface (perhaps due to a higher proportion of ice). This is also the case for features 3 and 6, which is not surprising as these three features all had strong correlations to the mean elevation for the ridged and snowy categories.

Features 5 and 7 both show some distinguishing of the different surface types, although the weights are so small for these features (Fig. b) that they likely do not significantly change the SIT estimate and we do not speculate what these may account for.

The inner workings of ConvNets are not easily interpreted, but the analysis here suggests that the ConvNet responds in physically realistic ways to the surface morphology. It may be possible to use these physical metrics to construct an analytical approximation to the model, but due to the nonlinearities in the ConvNet as well as the considerable scatter between the features and our guessed metrics, this will not be as accurate as simply passing the input through the ConvNet.