Size, shape and spatial arrangement of mega-scale glacial lineations from a large and diverse dataset

: Mega-scale glacial lineations (MSGLs) are a characteristic landform on ice stream beds. Solving the puzzle of their formation is key to understanding how ice interacts with its bed and how this, in turn, influences the dynamics of ice streams. However, a comprehensive and detailed characterization of this landform ’ s size, shape and spatial arrangement, which might serve to test and refine formational theories, is largely lacking. This paper presents a detailed morphometric analysis and comparison of 4043 MSGLs from eight palaeo-ice stream settings: three offshore (Norway and Antarctica), four onshore (Canada), and one from under a modern ice stream in West Antarctica. The length of MSGLs is lower than previously suggested (mode 1000 – 2000 m; median 2892m), and they initiate and terminate at various locations on an ice stream bed. Their spatial arrangement reveals a pattern that is characterized by an exceptional parallel conformity (80% of all mapped MSGLs have an azimuth within 5° from the mean values), and a fairly constant lateral spacing (mode 200 – 300m; median 330m), which we interpret as an indication that MSGLs are a spatially self-organized phenomenon. Results show that size, shape and spatial arrangement of MSGLs are consistent both within and also generally between different ice stream beds. We suggest this results from a common mechanism of formation, which is largely insensitive to local factors. Although the elongation of MSGLs (mode 6 – 8; median 12.2) is typically higher than features described as drumlins, these values and those of their width (mode 100 – 200m; median 268m) overlap, which suggests the two landforms are part of a morphological continuum and may share a similar origin. We compare their morphometry with explicit predictions made by the groove-ploughing and rilling instability theories of MSGL formation. Although the latter was most compatible, neither is fully supported by observations. © 2014 The Authors. Earth Surface Processes and Landforms Published by John Wiley & Sons Ltd.


Introduction
Outlet glaciers and ice streams are fast flowing (up to 1000 s m per year) corridors of ice within ice sheets (Bennett, 2003). They are the main arteries through which ice sheets lose mass, accounting for up to 90% of Antarctic discharge, for example (Morgan et al., 1982;Rignot et al., 2011). Ice streams are also dynamic and can widen, migrate or shut down on decadal timescales (Conway et al., 1999;Hulbe and Fahnestock, 2004). Thus, their operation is a key control on ice-sheet mass balance at a range of time-scales. A hierarchy of controls are thought to influence the location of ice streams (Winsborrow et al., 2010), but in most cases, their dynamic behaviour is thought to be influenced by a soft sedimentary bed (Alley et al., 1986;Engelhardt and Kamb, 1997;Tulaczyk et al., 2001). Unfortunately, direct access to the ice-bed interface has only been achieved at a handful of sites along the Siple Coast of West Antarctica (Engelhardt and Kamb, 1997) via costly drilling, which while valuable, provides spatially limited 'point' data. Geophysical techniques provide a valid alternative (Smith and Murray, 2009), but the logistics remain challenging and such techniques have been applied only to a few Antarctic sites, with grids necessarily limited to small (10s km 2 ) areas . In contrast, there is an abundance of relatively well preserved palaeo ice stream beds (Winsborrow et al., 2004), which are a useful proxy for the study of ice-bed interactions and which permit easier access to the landforms and sediments created by ice streams. Indeed, there are numerous studies that provide a detailed characterization of palaeo-ice stream beds from onshore and offshore regions (see reviews in Stokes and Clark, 2001;Livingstone et al., 2012).
Mega-scale glacial lineations (MSGLs) are extremely elongated ridges that maintain a parallel conformity over lengths of 10s of km (Clark, 1993). MSGLs have been identified for many decades (Lemke, 1958), but were first formally recognized and named by Clark (1993) from Landsat imagery of Canada. Based primarily on their great length, and parallel conformity, they were hypothesized to have formed beneath fast flowing ice streams or surges (Clark, 1993(Clark, , 1994. Such a proposal gained acceptance after numerous discoveries of MSGLs on the Antarctic continental shelf (Shipp et al., 1999;Canals et al., 2000;Wellner et al., 2001) in positions proximal to present-day ice streams. This association has now been verified by radar profiling of the bed of the modern Rutford Ice Stream, West Antarctica (Smith et al., 2007;King et al., 2009) where MSGLs were imaged and shown to evolve (e.g. evidence of both erosion and deposition) over a 7 year period. As MSGLs have been found in front of and beneath ice streams and evolving during fast ice flow, they are now widely considered to be the key landform signature for identifying palaeo-ice streams Clark, 1999, 2003;Jakobsson et al., 2005;Ottesen et al., 2005b;Stoker and Bradwell, 2005;Przybylski, 2008). The link between MSGLs and ice streams has also helped decipher the history of sediment sequences and palaeo geography of continental shelves that are relevant to petroleum and gas exploration (Sejrup et al., 2005;Nygård et al., 2007). Furthermore, MSGLs have been identified in ancient (440 million years) sandstones in Africa (Moreau et al., 2005) and, more controversially, on Mars (Lucchitta, 2001;Hubbard et al., 2011).
Despite the importance of MSGLs to our understanding of subglacial processes under ice streams, their mechanism of formation is yet to be resolved, although several different ideas have been proposed (Clark, 1993;Tulaczyk et al., 2001;Clark et al., 2003;Schoof and Clarke, 2008;Shaw et al., 2008;Fowler, 2010). Indeed, the formation of MSGLs is likely to be important for answering fundamental questions regarding the mechanisms of ice stream flow, such as the role of basal sliding versus subglacial till deformation and how roughness elements evolve and modulate ice stream motion. It has been hypothesized, for example, that enhanced flow within ice streams might be caused by: (i) sliding across the surface of a soft sedimentary bed (Engelhardt and Kamb, 1997); (ii) deforming soft sediments at depths of a few centimetres to several metres (Alley et al., 1986), or (iii) a combination of the two that includes a component of ploughing (Tulaczyk et al., 2001).
Attempts to convert these hypotheses into physically-based numerical models are ongoing (Schoof, 2002;Bougamont et al., 2011), and this is considered a fundamental requirement for constraining predictions of the consequences of global climate change on ice stream dynamics (IPCC, 2007). Thus, the next generation of numerical models should account for the production of MSGLs, and be validated through their ability to reproduce MSGLs of the correct size and shape. A prerequisite for this, however, is a quantitative characterization of MSGLs based on a large and diverse dataset, similar to that which has recently been undertaken for drumlins Hess and Briner, 2009;Spagnolo et al., 2010Spagnolo et al., , 2011Spagnolo et al., , 2012. This paper presents the results of detailed morphometric analysis of a sizeable dataset (4043) of MSGLs mapped from eight different locations around the world, both onshore and offshore, and both in palaeo and present-day ice stream bed settings. The aim is to provide an inventory of MSGL metrics and a robust dataset to both test and develop theories of MSGL formation.
A Summary of Previous Work on the 'Metrics' of MSGLs MSGLs were initially distinguished and named as a separate type of subglacial landform because of their extraordinary length, well exceeding that of drumlins, flutes and megaflutes, but also because of their straight crestline and repetitive parallel arrangement (Clark, 1993). Using Canadian examples, Clark (1993) described them as typically characterized by a length of 8-70 km, width of 200-1300 m and spacing of 300-5000 m. Since then, numerous studies in different parts of the world have expanded the range of MSGL metrics: lengths of <1-180 km, widths of 39-5000 m, heights of 1-100 m, elongations (length L/width W) of 2-200:1 and across-flow spacings of 50-5104 m (Table I). However, most previously published work is restricted to qualitative or semi-quantitative descriptions, often only citing minimum and maximum values and, more rarely, estimating mean values, typically expressed as a range or a 'lower than' figure (Table I and references therein). Statistical descriptions of large sample sizes (>100 bedforms) are extremely rare (Graham et al., 2009;Livingstone et al., 2013;Stokes et al., 2013), and very few studies compare data from different ice stream beds (Ottesen et al., 2005b).

Methods
The MSGL database was assembled from a total of eight study areas (Table II; Figure 1) that have been previously identified as ice stream beds with abundant MSGLs (Winsborrow et al., 2004(Winsborrow et al., , 2012Graham et al., 2009;King et al., 2009;Brown et al., 2011;Jakobsson et al., 2011Jakobsson et al., , 2012. Although MSGLs have previously been reported from these regions, they have never been systematically mapped or analyzed in detail.

Datasets
Offshore, three palaeo ice stream beds were studied: two located beneath the Amundsen Sea, Antarctica (Pine Island, and Getz), and one under the Barents Sea, off northern Norway (Håkerringjdupet). These marine datasets were chosen because of the availability of high resolution 3D terrain data (Table II, Figure 1(A), (B), (C), (E)).
Pine Island trough comprises two distinct sets of MSGLs belonging to distinct ice streaming events and which are separated by a grounding zone wedge . In this paper, the northernmost set has been called 'Pine Island N', and the southernmost 'Pine Island S'. Together, they cover about 40% of the entire Pine Island trough length, and are located in its middle to outer portion. The area is covered by terrain data with horizontal resolution of 20 m and vertical resolution of~2 m, collected using a hull-mounted Kongsberg 12 kHz EM122 multibeam echo-sounder. The Getz area comprises one set of MSGLs from the inner to middle shelf part of the ice stream pathway. The middle-to-outer shelf is either iceberg scoured or unsurveyed. The ice stream trough (from present-day ice-shelf front to the continental shelf break) is 280 km long, and the MSGLs used in this study cover ã 65 km long portion of it (providing data across most of the ice stream width).The area has terrain data which was collected in 2006 by hull-mounted Kongsberg EM120 and Atlas Hydrosweep DS-2 multibeam swath systems. The combined data are gridded at 30 m bin size, well within the capability of both systems at these shelf water depths (~700 m). Depth resolution for the echo-sounders is 0.1 m for both systems and vertical resolution of the gridded output is ≤~2 m. MAREANO 1433 MEGA SCALE GLACIAL LINEATIONS METRICS  (www.mareano.no) multibeam data, collected by the Norwegian Hydrographic Service (horizontal resolution of 5 m, vertical resolution of~5 m), were used in the Håkjerringdjupet area. The area comprises one main set of MSGLs (Winsborrow et al., 2012) covering about 30% of the entire palaeo ice stream.
Four previously recognized palaeo-ice stream beds were studied from onshore settings (Winsborrow et al., 2004;Brown et al., 2011) and selected for mapping due to the excellent preservation of MSGLs in these regions and the lack of cross-cutting landform relationships (Table II; Figure 1(F), (G), (H), (I)). All four are situated in north-west Canada, where ice streaming is thought to have contributed to the low ice surface profile and rapid retreat of the Laurentide Ice Sheet (Beget, 1987;Brown et al., 2011). The areas are covered by Landsat ETM + and SPOT satellite images with a horizontal resolution of 15 and 10 m, respectively. Unlike their onshore counterparts, these palaeo ice streams are not situated in topographic troughs, thus making it difficult to quantify what percentage of the original ice stream bed is covered by the analyzed MSGLs.
MSGLs from the bed of the Rutford Ice Stream in West Antarctica were also mapped , which is the only extant ice stream for which high resolution bed data are available (Table II; Figure 1(D)). Radar data were acquired perpendicular to flow and have an alongtrack spatial resolution of 7.5 m, and a vertical resolution of 3 m. Profiles were spaced at 500 m intervals in the along-flow direction and an interpolated surface was created using a 20-m-perpendicular-to-flow x 200-m-parallel-to-flow grid. This interpolation scheme maintains the continuity of features elongated in the flow direction, while preserving the high spatial sampling of the cross-flow bed profile. The surface representing the bed was then re-sampled at 50 x 50 m spacing to form the digital terrain model used in this analysis. The data covers the lowermost portion of the Rutford Ice Stream, 25% of its total area.

Mapping techniques
MSGLs were mapped from high-resolution bathymetric data (offshore), radar data (extant ice stream), and combined satellite images (onshore). Bathymetric and radar data were converted into digital terrain models and visualized as hill-shaded images with different illumination angles and elevations, as well as various vertical exaggerations, following a standardized method for mapping glacial bedforms (Smith and Clark, 2005). Due to a lack of similar high resolution digital terrain models onshore, MSGLs could only be mapped from satellite images. Improved identification of subtle landforms was achieved by using two different satellite sources and by (locally) stretching the contrast of these images. All landforms were mapped at a scale between 1: 30 000 and 1: 40 000, depending on the resolution of the terrain or satellite data.
Offshore and modern (Rutford) MSGLs are characterized by a continuous series of parallel crests separated by troughs or grooves. No evident breaks of slope are present to help delineate individual landform side-boundaries and MSGLs appear as a continuous rolling or waved surface, i.e. a sinusoidal profile in cross-section. Therefore, they were mapped as lines drawn along their crests (Figure 2 In all onshore settings, satellite images made it possible to identify MSGLs as individual features with distinct boundaries (Figure 2(C)-(D)) highlighted by shadows and by a change in colour which corresponds to a change in vegetation and soil moisture (corresponding to a break in slope). Onshore MSGLs were therefore mapped as both lines along their crests (to compare with the offshore data) and as elongate polygons.

Measuring MSGLs
The azimuth of all mapped MSGLs (n = 4043) was automatically derived with specific GIS tools as the angle (0-360°) from grid North of each digitized line. Crestline length was evaluated from the total length of each MSGL crest line. Not all of the mapped MSGLs could be mapped for their entire extent (see also following section). These MSGLs were not included in the length analysis, which was therefore limited to a total of 3068 features. Width and elongation could only be extracted from the polygon data from onshore settings (n = 1929). Width was estimated using Euler's approximation for the width of an ellipse . Elongation was computed as length/width.
The across-flow spacing of MSGLs is the distance between adjacent crestlines. MSGLs are relatively long features, and some variation in lateral spacing along their extent has been noted. Therefore, multiple values (on each side) were regularly collected at 1 km intervals along each MSGL, by creating a series of 1 km spaced topographic profiles transverse to crestlines (Figure 3). The cross-profile interval represents a compromise between attempting to take multiple measurements along each individual MSGL and maintaining the time needed to process all data efficiently. With the 1 km cross-profile pacing, mean lateral spacing of MSGLs was obtained by typically averaging eight (four on each side) spacing values measured along individual MSGL. A semi-automated GIS procedure was then applied to determine the distance between adjacent crestlines along each profile (Figure 3). Two spacing measurements (i.e. the distance to each lateral neighbour) were collected at each interception point between a cross-profile and a MSGL crestline. Data were therefore summarized by calculating a mean MSGL spacing from these measurements. Only those MSGLs that had two adjacent neighbours (and that were not separated from these by large erosional features like meltwater channels or a field of iceberg furrows) were included in this analysis (1697 offshore and 1846 onshore). In addition, in order to assess how across-flow spacing varies downstream, a mean spacing was also determined for individual cross-flow profiles in the down-flow direction.
MSGL amplitude was defined as the difference between the elevation of the crestline and the lowest elevation recorded in the trough between adjacent MSGL crestlines on both sides of each MSGL (Figure 3). The same 1 km spaced cross-profiles used to measure spacing were employed for this task. The two amplitude values generated at each cross-profile interception point were averaged in order to produce a single value, an effective way to de-trend the variation in elevation due to the regional-scale topography.
All intersection values were then averaged for each MSGL. Note, however, that amplitude could only be measured for the offshore MSGLs (for which a high resolution terrain model was available) and the same criteria were applied to dictate where spacing was measured (n = 1697). In order to assess how amplitude varies downstream, the mean amplitude was also determined for individual cross-flow profiles in the down-flow direction.

Data fidelity
MSGLs have likely experienced post-glacial modification. Attempts to quantify post-glacial erosion, current winnowing, glacimarine sediment draping, etc., are challenging and extremely rare Finlayson, 2013). However, other modifications are easier to recognize (e.g. iceberg furrows, meltwater channels, etc.) and their influence on the derived metrics could be considerable. Where possible, these have been taken into account. For example, only fully imaged MSGLs (i.e. exclude those running off the edge of image domains) and those without a clear sign of being partly overridden (e.g. by moraine) or eroded (e.g. subsequent meltwater channel) were included in the length analysis. Similarly, spacing and amplitude were determined only for MSGLs showing continuity (i.e. no interruptions due to the presence of iceberg furrows or meltwater channels) between adjacent features.
The frequency distributions of MSGL metrics from different settings were assembled into single histograms (Figure 4). In order to avoid bias associated with settings with a greater number of measurements, the data were normalized according to the relative number of MSGLs of each setting.

Results
The metrics of the MSGL are presented in Table III for each study area and as an aggregated population for all mapped MSGLs. Statistics from each setting and normalized frequency histograms are shown in Figure 4. MSGLs are characterized by lengths of up to 37 km and elongation ratios of up to 134:1 (Table I). However, although these maximum values have been commonly reported in previous work, they are not representative of the vast majority of sampled features. Moreover, these extreme values can strongly affect mean values, which also become unreliable statistical descriptors when frequency distributions are skewed, as is the case here. Consequently, a simpler and better way to describe their metrics is to focus on modal and median values and to provide a most common range, i.e. 10 and 90 percentiles.

Parallel conformity
The vast majority (80%) of all studied MSGLs have an azimuth within μ (mean value) ±5°, confirming the high level of parallel conformity described by many but rarely quantified (Clark, 1993). The narrowest azimuth distributions are found in Pine Island trough and on the Liard palaeo ice stream beds (all three with 80% of the population within just 5°). The broadest azimuth distributions (with a range of 18°and 19°, respectively) are found in the Getz and Håkerringjdupet palaeo ice streams, where MSGLs are not straight but can be observed to bend, following a curving ice flow trajectory. However, even in these settings, the parallel conformity within adjacent MSGLs remains striking (Figure 1(D)).  (Table IV; Figure 5). The most statistically-significant trend is that of the Pine Island N ice stream bed, where MSGL amplitude shows a tendency to increase downstream. Other significant trends (within the 99% confidence interval) are those revealed by the Håkerringjdupet and Getz ice stream beds, the former showing increasing amplitude (and variability), while the latter shows decreasing amplitude downstream. It should also

Metrics
Taken together, the metrics presented here quantitatively confirm MSGLs as kilometre-long, highly elongate features with amplitudes of only a few metres (mode of 1-2 m) and a close and consistent spacing with a high parallel conformity. Their low amplitude and great length explains why MSGLs are typically not visible in the field and can only be fully appreciated and mapped from airborne and spaceborne imagery (Clark, 1993), or shipborne bathymetry.

Consistency
All measured variables show a unimodal distribution within each setting, thus indicating a prominent scaling preference rather than an even spread of values across a wide scale range or a multimodal distribution. Comparing distributions between ice stream settings reveals a strong consistency in the production of MSGLs of similar size, shape and spacing. With a few exceptions where local factors might have had an influence on specific metrics (e.g. Great Bear MSGL width), MSGL metrics were found to be very similar and always within the same order of magnitude. In some cases (e.g. spacing, Figure 5) the statistics are so strikingly close that there is virtually no difference between MSGLs from offshore Antarctica and onshore Arctic Canada. This is particularly remarkable because some of the settings are thousands of kilometres apart, and each was subjected to a specific glacial history (duration of flow, evolving ice thickness and velocity) and specific local topographic, sedimentary and lithological conditions. We interpret this consistency within and across the settings to indicate that MSGLs share a common origin, which is largely insensitive to local factors.

MSGL length
MSGLs are known for being extremely long features. The present study confirms this idea (maximum length 37 km), but also scales down the proportions by indicating that the vast majority (80%) of MSGLs are less than 9 km long (see also Stokes et al., 2013). This figure is almost one order of magnitude lower than reported in previous studies (Table I), although these often describe MSGLs by the maximum length measured within a field of MSGLs. This difference is also largely due to the fact that MSGLs have been mapped here in unprecedented detail, making it possible to identify and separate smaller features. Indeed, what generally appears to be a single feature at a small scale (e.g. 1:70 000), often reveals itself to be the sum of two or more aligned, but clearly-separated (by some tens of metres), landforms at a larger scale (e.g. 1:30 000).
A fundamental consequence of lineations sets extending up to hundreds of kilometres (Mosola and Anderson, 2006) and individual MSGLs being shorter than hitherto supposed is that multiple features exist at different downstream positions throughout the extent of an ice stream. In other words, individual MSGLs are not found to extend continuously throughout an ice stream bed, i.e. from its upstream to the downstream ends. Rather, they terminate and others initiate in multiple locations (Figure 7).

Comparison with drumlins
The length and the elongation of MSGLs are typically higher than drumlins , whereas the amplitude is about half as much as a typical drumlin relief (Spagnolo et al., 2012). However, other metrics, such as the width (compared with Clark et al., 2009) and the spacing are very similar to drumlins. In particular, the similarity in width but larger lengths are supportive of the argument that MSGLs could represent extremely elongate drumlins (Stokes and Clark, 2002) and that a similar mechanism might produce both MSGLs and drumlins, depending on the ice flow velocity (Clark, 1993;Stokes et al., 2013) and the nature and availability of sediment . Indeed, there are now many documented palaeo ice stream settings, both onshore (Stokes and Clark, 2002;Stokes et al., 2013) and offshore (Ó Cofaigh et al., 2002;Heroy and Anderson, 2005;Graham et al., 2009), where drumlins upstream become progressively more elongate and into MSGLs downstream, thus making a distinction between the two landforms very difficult.
Log-normal distributions Like drumlins , the metrics of MSGLs show a unimodal frequency distribution with a positive skew. When length, amplitude or spacing for a population of MSGLs are converted into their natural logarithm and their frequency evaluated for bins of equal intervals (Figure 6), the resulting frequency distribution is log-normal, in common with drumlins (Fowler et al., 2013;Hillier et al., 2013). This type of distribution is frequent in nature, with examples from many disciplines, including geology and mining (e.g. concentration of elements), biology (e.g. Figure 2. Examples illustrating the different mapping of MSGLs in different study areas that was dictated by the availability of imagery. Lines along the crests offshore (Pine Island Bay, A and B on the figure) and polygons onshore (Great Bear Ice Stream, C and D on the figure). Arrows indicate the inferred palaeo direction of ice flow. This figure is available in colour online at wileyonlinelibrary.com/journal/espl airborne contamination by bacteria and fungi), human medicine (e.g. latency period of diseases) and ecology (e.g. species abundance) (Limpert et al., 2001 and references therein). Typically, log-normal distributions emerge from variables that are characterized by an incremental ('multiplicative') growth or fragmentation, but that initiated as a large number of independent events (Limpert et al., 2001). The log-normal distribution implies an element of randomness in the growth/decay of MSGLs, similar to that which has been advocated for ribbed moraines (Dunlop et al., 2008) and argued for drumlins (Fowler et al., 2013;Hillier et al., 2013). In view of this, MSGLs are likely to represent a growing phenomenon for which the growth phases occur randomly, or for random durations, or under variably random physical conditions of the flow parameters (Hillier et al., 2013).

Wider implications MSGL formation theories
The formation of MSGLs is enigmatic, despite their importance for understanding ice stream dynamics. Different formational hypotheses/ideas have been proposed, some invoking an erosional mechanism (Lemke, 1958) while others suggesting a constructional process (Bluemle et al., 1993), perhaps involving the deformation of subglacial sediment (Clark, 1993). Some advocate a prominent role for ice ploughing through sediment (Tulaczyk et al., 2001;Clark et al., 2003), while others consider water, either in terms of a mega-flood (Shaw et al., 2008) or a smaller film between ice and sediment that breaks down into a series of rills (Fowler, 2010). Currently, only two hypotheses provide testable, quantitative predictions about the morphometry of MSGLs with which we can compare our observations. These are the grooveploughing (Tulaczyk et al., 2001;Clark et al., 2003) and the rilling instability hypotheses (Fowler, 2010). The former suggests that MSGLs are the product of 'groove-ploughing', formed by a series of ice keels at the sole of the glacier. The keels would develop when streaming ice encounters a topographically rough area upstream (i.e. outcropping bedrock) or through lateral compression as the ice stream narrows through a convergent onset zone. The ice keels would then plough into soft, saturated sediments downstream for many kilometres, thereby producing grooves defining the MSGLs (Tulaczyk et al., 2001;Clark et al., 2003). MSGL formation would thus be considered as primarily erosional, although some sediment squeezing and deposition would be likely to occur in the intervening ridges between adjacent keels.
Our work highlights three main issues for the grooveploughing hypothesis to address. With the availability of higher resolution data it is apparent that individual MSGL initiate and terminate in various positions within an ice stream bed (Figure 7). This is instructive, as it implies that an individual MSGL can come into existence anywhere on the bed, contradicting the groove-ploughing expectations of the same ice keels surviving over very long distances (Tulaczyk et al., 2001).
In relation to the previous point, a second issue for the groove-ploughing hypothesis relates to the unimodal distributions of MSGL metrics (Figure 4). With some exceptions (e.g. outcrops of bedding planes), bedrock roughness is usually uneven across an ice stream bed and keels (and MSGLs) should therefore form at variable sizes. Large differences might be expected between outcrops of rocks from different settings. However, results indicate that, for the most part, MSGL size is relatively similar across various ice stream beds with different underlying geology. The same argument applies to MSGL spacing. Uneven outcropping should generate unevenly spaced keels (and MSGLs). However, results show that MSGL spacing varies relatively little within (and between) ice stream beds.
A further testable prediction of the groove-ploughing hypothesis is that ice keels should reduce in size while moving downstream as they melt because of frictional heat. A reduction of the size of the keel would therefore imply that groove depth should shallow and the groove width should widen in a downstream direction, with a corresponding increase in the spacing between adjacent lineations. The studied MSGLs extend for tens of kilometres and cover considerable portions of the original ice stream beds (see section Datasets). However, results indicate that across-flow spacing either shows no statistically significant downstream trend or, with the exception of the Haldane palaeo ice stream bed, Figure 3. Spacing and amplitude determination using Rutford ice stream bed (see red box in Figure 1 for map overview). (A) hillshaded bathymetric data; the purple lines represent cross-sections spaced 1 km apart, while the yellow arrow indicates the direction of ice flow; colours refer to bathymetric scale of Fig. 2(A). (B) MSGLs mapped along their crestlines (black lines); red rectangle represents the area in C. (C) Blue arrow indicates MSGL spacing. Point 1 and 2 refer to the crosssection shown in D. (D) MSGL cross-section 1-2: red arrows indicate the amplitude measured as the difference in elevation between the crestline and adjacent low points in troughs (two values per MSGL). This figure is available in colour online at wileyonlinelibrary.com/journal/espl consistently decreases in a downstream direction (Getz, Pine Island N and Great Bear palaeo ice stream beds). For amplitude, different ice streams reveal different trends or no trend. Those showing MSGL amplitudes decreasing downstream (e.g. parts of the Rutford Ice Stream and the Getz palaeo ice stream bed) are consistent with the groove-ploughing prediction; while those characterized by no evident trend (e.g. Pine Island S) or amplitudes increasing downstream (e.g. Håkerringjdupet and Pine Island N palaeo ice stream beds) are inconsistent. In its current form, therefore, the groove-ploughing hypothesis is mostly unable to account for our observations of MSGLs (see also Ó Cofaigh et al., 2005).
The other hypothesis that provides quantitative predictions of the size and shape of MSGLs is that based on subglacial meltwater rilling (Fowler, 2010), which invokes an instability in an initially uniform water-film (typical value of 2.7 mm thick) flowing between the ice and the deformable subglacial till. Water-flow depth will be larger in an incipient stream, and this would allow faster flow and thus higher erosion and sediment transport. Consequently, the stream would deepen further, and this positive feedback, initiated by an instability, would result in the excavation of grooves, or rills, between ridges. The mathematical simulations suggest that the overlying ice would dampen the growth of the landform's height at relatively length (km) 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 13-14  Our data show that most spacing values are between 200 and 300 m, with lengths between 1 and 2 km and amplitudes from 1 to 2 m. Thus, Fowler's (2010) predictions are overestimating both the length and amplitude of MSGLs, but the theoretical spacing is very close to observations and this hypothesis is yet to explore the full range of parameters. The hypothesis also suggests, implicitly, that MSGLs should be evenly spaced across an ice stream bed and this is indeed supported by the unimodal distribution of the across-flow spacing. A predominant spacing indicates that MSGLs tend to evenly occupy the available surface and could represent a spatially self-organized phenomenon, which is typical of instability-related bedforms (Clark, 2010). However, the data from this paper show that MSGLs initiate and terminate anywhere on the bed and this is not easily reconciled with rills forming where meltwater is expected to flow uninterrupted, unless changes in sediment properties are able to rapidly modify the character and pathway of meltwater flow on an ice stream.
In summary, an important implication of our quantitative data, therefore, is that current ideas and theories about MSGL formation are either un-supported or insufficiently developed (see also Stokes et al., 2013). Since MSGLs tend to evolve into a self-organized pattern (unimodal spacing distribution) and a dominant size and shape that is largely insensitive to local factors, it is plausible that some type of instability is governing their evolution, similarly to that which has been suggested for other glacial bedforms (Hindmarsh, 1998(Hindmarsh, , 1999Fowler, 2000). However, it is unclear at this stage if this would be the instability of the coupled ice and deforming bed invoked for drumlins, with MSGLs possibly representing elongate drumlins, or if it is an instability specific to the MSGL, such as the rilling one suggested by Fowler (2010), but which is not fully supported by the results presented in this paper.
MSGLs and ice stream flow mechanisms Hydrological observations beneath the Whillans (B), Kamb (C) and Bindschadler (D) ice streams in Antarctica have suggested    that the basal water pressure is very close to the ice overburden pressure, implying that it is plausible for the ice to flow via sliding over its bed (Engelhardt and Kamb, 1997;Kamb, 2001). The formation of MSGLs is compatible with ice stream sliding if these landforms are primarily formed by erosion, for example through a groove-ploughing mechanism (Tulaczyk et al., 2001;Clark et al., 2003) or in association with megafloods (Shaw et al., 2008). However, with regards to grooveploughing, the results presented here indicate that predictions from this hypothesis are not fully supported by observations. Issues have also been raised in relation to the mega-flood theory , among which that the volume of water required to form features occupying areas of thousands of square kms appears implausibly large Shaw and Young, 2010) and recent investigations have recorded MSGLs being modified without any signs of concomitant megafloods , although strictly, this does not rule out floods having created them. The notion that ice streams predominantly slide over their beds is, perhaps, also hard to reconcile with observations of trough mouth fans and grounding zone wedges deposited at ice stream grounding lines (Vorren and Laberg, 1997;Ó Cofaigh et al., 2003), which have always been assumed to reflect high sediment transport rates and a strong coupling between the base of the ice and the underlying sediment. Unless erosion occurs independently from ice streaming (Christoffersen et al., 2010) or an exceptional amount of (fine) sediment mobilization could be accounted for by the action of meltwater flowing in thin films, it is difficult to reconcile the deposition of large volumes of sediments with ice stream sliding. The alternative to sliding, which was traditionally advocated was that ice streams could largely (or entirely) move via subglacial sediment deformation, as supported by the discovery of several metres of fluted subglacial till with high porosity (i.e. likely to be dilated) beneath Whillans Ice Stream (Alley et al., 1986). Further theoretical and observational constraints have indicated that rapid fluctuations in ice stream velocity (for instance those observed on the Whillans Ice Stream) are indeed associated with variation in shear strain rate of the subglacial till (Tulaczyk, 2006). Under the scenario of ice flow being facilitated by sediment deformation, the ice would be coupled to its bed and the basal till would essentially be transported downstream with the flow of the ice, helping explain the advection of large volumes of sediment towards the grounding line of ice streams. MSGLs could evolve in this scenario either by net erosion or deposition or indeed both (redistribution of sediments). This is supported by observations of erosion and subsequent deposition beneath Rutford Ice Stream, where MSGLs have been seen to evolve Smith and Murray, 2009); and by acoustic profiles that show MSGLs being part of an acoustically semi-transparent unit interpreted as deformation till . A further complication, however, is that recent studies have challenged the idea of a deep, pervasive, viscous shearing of the bed and observations from the Lake Michigan Lobe of the Laurentide Ice Sheet have suggested that the flow occurred there via shallow (decimetres) deformation of till patches (Thomason and Iverson, 2009;Iverson, 2010). Such a scenario is unlikely to promote spatially-continuous formation of MSGLs over hundreds of square kilometres and would require a series of build-up events in order to generate metres-tall features. Furthermore, the internal structure of some MSGLs has shown the presence of intact sedimentary bodies which are also hard to reconcile with pervasive deformation (Shaw et al., 2000;Ó Cofaigh et al., 2013). Taken together, these observations would suggest that MSGL formation takes place through a combination of sliding in the presence of near-overburden pressurized basal meltwater, with some sediment excavated/deformed and advected from in between and along ridges. The rilling instability theory (Fowler, 2010) appears to include most of these elements. However, our observations are only partially supportive of the theory's predictions in its present form.

Conclusions
The ice-bed interface is a key control on fast ice flow, with associated impacts on ice sheet mass balance and sea level. This is where mega-scale glacial lineations are formed and their study is likely to improve our understanding of ice stream dynamics. A number of hypotheses have been developed to explain the formation of MSGLs, but there is little agreement as to which might be correct, if any. In part, this reflects a lack of knowledge on their size, shape and spatial arrangement. This paper presents the first compilation of MSGL morphometries, allowing comparison between different ice stream beds, including both onshore and offshore, and palaeo and modern settings. Results suggest that MSGLs can be described as subglacial landforms characterized by a combination of considerable size (i.e. kilometres), especially length, with very subtle amplitudes (i.e. metres). Arguably, the most important and distinctive characteristic is their spatial arrangement, i.e. their consistent spacing and striking parallel conformity. Morphometric similarities within and between various settings are a clear indication that MSGLs are formed by the same mechanism, which is relatively insensitive to local conditions (topography, bedrock properties, etc.). Of the ideas pertaining to the formation of MSGLs, only a few provide explicit predictions of their shape and size. The results presented here are difficult to reconcile with some aspects of the groove-ploughing theory (Tulaczyk et al., 2001;Clark et al., 2003). Rather, MSGLs appear to reflect a self-organized (consistent spacing), probably growing (log-normal distribution of their metrics) phenomenon that tends towards a dominant size and shape, which hints at some sort of instability among the most likely ingredients for the formation of MSGLs. Whether this is the rilling instability invoked by Fowler (2010), whose predictions are only partially compatible with our observations, or the till instability invoked for the formation of drumlins (Hindmarsh, 1999;Fowler, 2000) remains to be seen. However, the present work represents a quantitative foundation which can serve as a test for any future development of theories of MSGL formation and, more widely, on the nature and behaviour of the ice-bed interface.