To calculate Cx for the entire Antarctic continental coastline, 40 000 points were randomly chosen along the MODIS Mosaic of Antarctica 2008–2009 (MOA2009) coastline dataset (Haran et al., 2014), acquired from the US National Snow and Ice Data Center (NSIDC). The coastal margin is used in the calculation of complexity since the outer margin is more relevant for the processes listed in the introduction here (e.g. ecological habitats, fast-ice formation, polynya location, ice shelf–ocean interaction). Figure 2 illustrates the algorithm for determining Cx. At each random target point x on the merged MOA dataset and for each length scale (of 1, 2, 4, 8, 16, 32, 64, 128, and 256 km), the Euclidean straight-line distance was measured either side of the chosen point to find the corresponding points, a and b, that intersect the coastline. The two vectors xa‾ and xb‾ are vector-summed to give the quantity xc‾, indicating the magnitude of complexity and direction (both relative to north and the local coastline) for the aspect. The maximum distance between successive random points was rarely greater than 1 km, thereby giving a near-uniform and seamless representation of complexity around the continent (Fig. 2).

This approach varies from previous techniques employed to derive Cx, such as the angled measurement technique (AMT) where the length scale is measured forward and backwards of a chosen point on the mapped coastline. In the AMT, the measure of complexity is the supplementary angle (Andrle, 1994, 1996b; Porter-Smith and McKinlay, 2012). The new approach presented here offers a measure not only of complexity (as magnitude) but also of direction. Additionally, the new technique allowed qualification of the chosen section of coastline as either a bay or peninsula for a given length scale (i.e. any angle less than 180∘ would be classed as a bay, and any angle over 180∘ would be classed as a peninsula). An advantage of our technique is that it can be used to quantify coastal complexity at various scales to reflect the multiscale nature of features along the coastline. Additionally, characterizing the orientation (i.e. aspect) of features is useful in that it can be compared to the directions of other potential co-variates, allowing correlations and interactions to be examined.

Given that complexity magnitude varies as length scale changes, the resultant magnitudes of xc‾ were normalized to a range of 0 to 100 to give comparability between length scales. The spectrum of length scales examined was chosen to provide complexity measurements at scales relevant to known oceanic, cryospheric, and geomorphological processes and phenomena at kilometre-to-mesoscale levels, with individual lengths chosen as a series of base 2 powers to minimize the potential problem of spatial autocorrelation (Goodchild, 1986).

Data processing, spatial analysis, and mapping was carried out using the GIS and spatial analysis platforms Arc/Info (ESRI, 1996) and QGIS (Quantum GIS Development Team, 2014). Statistical analysis was carried out using the R language for statistical computing (Ihaka and Gentleman, 1996; R Core Team, 2014) and the R package cluster (Maechler et al., 2018).

Unsupervised classification (clustering) techniques were used to determine how many distinct complexity classes exist around the Antarctic coast. Cluster analysis has a rich history in statistics and machine learning (Hastie et al., 2001; Kaufman and Rousseeuw, 1990). In both fields, it is primarily used as an exploratory technique to identify k groups from n observations, such that observations within groups are more similar to one another in their p multivariate responses than they are compared with those in other groups.

Given the large size of the dataset and the high computational burden of many clustering algorithms, two common and tractable methodologies were selected: k means and partitioning around medoids (k medoids) (Kaufman and Rousseeuw, 1990; Maechler et al., 2018). These centroid-based partitioning methods were applied to the n≈ 40 000 complexity magnitude values for p = 9 length scales (i.e. 1, 2, 4, 8, 16, 32, 64, 128, and 256 km). For both k means and k medoids, length scales were first standardized (0-4116100), and Euclidean distances were used as the metric describing the similarity between observations. The primary difference between these clustering techniques is that while k means attempt to group objects into k clusters based on minimizing the distance of observations to group means (i.e. minimizing the within-cluster sums of squares), k medoids operate by minimizing distances to group medoids, where the latter are data points that are analogous to multivariate medians. Thus, clustering by k medoids can be considered a robust alternative to k means that will be less influenced by outliers and noise in the data. Given the size of the merged MOA coastline dataset, we employ the Clustering LARge Applications (CLARA) implementation of partitioning around medoids, a method that subsets data in order to achieve an optimal solution that is linear (rather than quadratic) in n. The algorithm of Hartigan and Wong (1979) was used for k-means clustering, and optimization was conducted over several random starts to ensure global optimization was achieved.

For any given application, clustering should be carried out for the spatial extent and at spatial scales relevant to the phenomena under investigation. As the present study seeks a synoptic, Antarctic-wide summary of complexity, we first consider all data (Antarctic-wide, all length scales) in a single analysis. In this case, all length scales are afforded equal weight in the analysis. However, it is likely that many local- to regional-scale phenomena impacting oceanic and cryosphere processes may be relatively unaffected by smaller-scale complexity. For this reason, cluster analyses were repeated on complexity data restricted to length scales ≥ 8 km and results compared with those derived from analyses of all length scales considered simultaneously.

A common problem when conducting unsupervised classification is that often the true number of groups, say k∗, is unknown and must be estimated from the data. Estimating k∗ is a difficult and somewhat ill-defined problem since there is no universal definition of what should constitute a group, and this has led to a wide variety of approaches for estimating k∗ under different clustering scenarios (Charrad et al., 2014; Milligan and Cooper, 1985). The gap statistic, which can be used in conjunction with many clustering techniques, is one of the more useful approaches to objectively determining k∗ (Tibshirani et al., 2001). While it is known to perform imperfectly in a limited set of circumstances (Mohajer et al., 2011), Tibshirani et al. (2001) use simulation experiments and analyses of real data to demonstrate that the technique outperforms a wide range of alternate established methods. The technique determines the optimal number of groups by examining the within-cluster dispersion Wk as a function of the number of clusters k. Obtaining separate clustering solutions for k∈{1,2,…,kmax⁡}, along with corresponding Wk values {W1,W2,…,Wkmax⁡}, shows that by itself Wk is uninformative since it always decreases with increasing k, even for independent data with no structure. The gap statistic overcomes this problem by defining 1Gapn(k)=En{log⁡(Wk)}-log⁡(Wk)∀k,k=1,2,…,kmax⁡, where En denotes the expectation under a sample size of n from a reference (H0) distribution. The latter is determined by resampling from a uniform distribution on the p hypercube determined by the ranges of the data after first centring and rotating them to align with their principal axes. The optimal cluster number k∗ is estimated as the value maximizing Gapn(k) after considering sampling variability associated with determining the reference distribution. In practice, this is achieved by choosing k∗ to provide the maximum gap statistic that is within 1 standard error (Breiman et al., 1984) of the first local maximum over the range of k (Tibshirani et al., 2001). For the present study, En{log⁡(Wk)} was estimated by an average of 100 separate Monte Carlo samples of the reference distribution. For both k means and k medoids, Gapn(k) was assessed over the range k=1 to 20. The gap statistic can be calculated for a range of clustering algorithms, which allows the similarity in clustering solutions to be compared between methods.

The total length of the outer merged MOA coastline is 39 593 km. The length of ice shelf and grounded ice coastline around the continent are 21 269 and 18 324 km, respectively and roughly proportional in western and eastern Antarctica. There is a strong positive skew in the distribution of Cx at all length scales, and this skew is especially pronounced at shorter length scales, i.e. complexity is not normally distributed, indicating that the Antarctic coastal margin has a tendency to be straighter rather than highly complex (Fig. 3).

A notable difference between the western and eastern sectors of Antarctica (-180–0 and 0–180∘) is the orientation of both bays and peninsulas. In East Antarctica, these features generally face directly offshore across all length scales (Fig. 4), with the higher Cx magnitude generally facing directly offshore, i.e. a normal distribution of magnitudes and their orientations. In West Antarctica, on the other hand, both bays and peninsulas have a general skew toward the west-of-offshore direction. This becomes particularly dominant at length scales of > 16 km. This bias in the bay and peninsula feature orientation may have implications for key physical processes (e.g. formation and persistence of fast ice) and biological processes highlighted in the Introduction. These variances could be used to examine and differentiate between regional and local areas and with other co-variates to analyse specific phenomena.

Analysis of the gap statistics shows that omission of smaller length scales (≤ 8 km) produces a pronounced local maximum at k=3. This suggests that the optimal number of complexity groups is three, as shown by the “elbow” in the gap statistic plots (see Fig. 5).

A projection of random point scores onto the first two principal component axes, accounting for 41 % of the total variation (Fig. 6), shows the three groups in relation to projections of the complexity length classes. As might be expected, arrows representing the complexity length classes appear in approximate order, in a fan shape, indicating that adjacent classes are most closely correlated with one another. Variances look approximately the same (i.e. arrows are approximately the same length) across length classes. In the two-dimensional approximation, the three groups show considerable overlap.

Figure 7 shows violin plots of Cx magnitude (for ≥ 8 km length scales), by the three-group structure determined by k-medoid clustering. The red line joins adjacent medians. This plot reveals the multiscale complexity of each group: group 1 represents coastline with little complexity (i.e. relatively smooth) at all length scales; group 2 represents coastline with more small-scale (≤ 32 km) complexity; and group 3 represents coastline with more large-scale (≥ 64 km) complexity.

The Cx dataset (Porter-Smith et al., 2019) presented here allows spatially resolved characterization of normalized complexity as a function of longitude for each length scale. This is shown in Fig. 8 as a polar plot. For simplicity, we show only the normalized complexity for 16 km (representing the “class 2” short length-scale cluster) and 128 km (representing the “class 3” long length-scale cluster). For both bays and peninsulas, the 16 km Cx is both larger and more homogenous as a function as longitude (bays: mean Cx = 24.8 ± 16.7; peninsulas: mean Cx = 23.9 ± 15.0), whereas the 128 km Cx is more heterogeneous or episodic in nature (bays: mean Cx = 13.0 ± 17.3; peninsulas: mean Cx = 15.1 ± 17.2).

Figure 9 shows the mix of coastline groups contained within a 64 km sliding window (chosen to allow as many data points as possible while still representing reasonably short length-scale variability) for the entire coastal margin. Although groups or typologies are observed to occur heterogeneously around the entire coastline, certain classes tend to dominate at specific scales and locations around the continent. To derive the dominant group within the heterogeneity, each of the three groups were totalled within the sliding window and proportionately normalized to 255. The dominance and heterogeneity could then be expressed and represented as a value within the RGB colour model.

As expected, the coastal margins of the Ronne, Ross, and Larsen C ice shelves are predominantly group 1. This reflects the very smooth nature of these ice shelf fronts, which tend to calve large, tabular icebergs. There are also several other ice shelves exhibiting group 1 dominance but which do not calve large tabular icebergs, including the Larsen D ice shelf on the eastern side of the peninsula, the Venable and Abbots ice shelves on the western side of the peninsula, and the ice shelves of the Sabrina Coast of East Antarctica. Several East Antarctic regions of grounded ice margin also exhibit group 1 dominance, including the Prince Olav, Mawson, Ingrid Christensen, Wilhelm II, Knox, Wilkes Land, and Adélie Land coasts.

Regions dominated by group 2 (indicating high Cx at small length scales) include the grounded ice coastal margin on the northern part of the western Antarctic Peninsula (between Cape Roquemaurel at 63.5∘ S, 58.9∘ W and Cape Jeremy at 69.4∘ S, 68.8∘ W), a mountainous stretch of Victoria Land on the coast of the western Ross Sea that is punctuated by glacier tongues of length 15 to 25 km (between Cape Washington at 74.7∘ S, 165.5∘ E and Coulman Island at 73.3∘ S, 169.7∘ E), and the Sulzberger Ice Shelf region (at 77∘ S, 150∘ W). The latter is characterized by a highly crevassed and rough (on a 25 km scale) ice shelf margin resulting from severe dynamical constraints on outflowing glacial ice.

Regions exhibiting group 3 dominance, on the other hand, occur mainly at major coastal inflection points. Notable locations are where the Transantarctic Mountains meet the McMurdo Ice Shelf, at the tip of the Antarctic Peninsula, and along the coastline of Alexander Island and the Wilkins Ice Shelf, where coastal undulations occur on the large spatial scale captured by group 3 (64 to 256 km).

Enlargements of Fig. 9 around Enderby Land and Victoria Land are presented in Figs. 10 and 11, respectively. These enlargements highlight regions of complex heterogeneity in Cx.