The sampling was done using carbon fiber tubes with sharp edges of 1 m length, 10 cm diameter and 1 mm wall thickness (called “liners”). To start off, the first liner was carefully pushed and hammered into the ground until its top was parallel to the snow surface. Nonetheless in a few cases the snow core was slightly compacted by up to 2 cm in the vertical direction, visible as a reduction of the snow level inside the tube compared to the surroundings. Subsequently a snow pit of 1 m depth was dug next to the tube and the snow was cut off at its bottom using a metal plate or small saw. The tube was removed and its openings were sealed using matching plastic bags. Then the cutting surface was cleaned and the second liner was inserted right below the first one. Finally the pit had to be deepened to 2 m to once again cut off the snow and take the second liner. Theoretically the process described can be iterated up to an arbitrary depth. However, the area of the snow pit required increases significantly with every meter of depth gained. Sampling the upper 2 m took approximately 2 h per site.

The cores were transported to the Alfred Wegener Institute, Bremerhaven, in a frozen condition. All samples were analyzed in the AWI-Ice-CT described in detail in, a unique X-ray computer tomograph (CT) in a cold lab, which allows micrometer-resolution density measurements of whole 1 m core segments in 2-D and 3-D. As part of the measurement procedure a sample holder for liners was constructed, which contains several pieces of pure ice of known geometry for calibration purposes. Amongst others, the effect of the carbon fiber tube being part of the scan was corrected for using empty tube measurements. Thus, the fragile snow cores do not have to be removed from the liners.

As the required measurement time increases with resolution, we chose to do 2-D scans with a pixel size of approximately 0.128 mm. Each of these scans takes about 3 min. However, 15 min m-1 are more realistic when including sample preparation and accurate documentation. Then, the raw measurement data are automatically processed by detecting the calibration unit and directly calculating densities from the CT images. Additionally, for each liner, the mean density is determined from the mass and geometry of the snow as an independent comparison value. Figure  displays an example CT image with a zoomed section showing two melt layers in the snowpack aligned with the respective densities derived from 2-D analysis.

Finally, the snow was gently pushed out of the tubes and cut in samples with a vertical height of 1 cm for the 30 cm right below the surface and 2 cm otherwise. These samples were crushed and sealed in plastic bags. Finally water isotopes were measured using a Picarro L2130-i with a precision of σ = 0.1 ‰ for δ18O.

The snow was dated by determining and counting the maxima (summer) and minima (winter) in the seasonal δ18O signal. Using the density data, accumulation rates at the different sites were calculated from the snow mass for the 3–5 years worth of accumulation contained in the top 2 m of the snowpack. In the present study, we only use winter-to-winter rates (separating years at the δ18O minima) – summer-to-summer values were computed as a reference but do not show different behavior.

In order to efficiently analyze the data sets generated along the traverse, we investigated several ways to automatically detect coherent signals at the different sites. A well-known matching method is maximizing the cross-correlation. However, determining a constant shift in depth between two profiles is not suitable for our case as the accumulation rate, and thus the vertical spacing of layers, is subject to change going eastwards. Under the assumption of constant accumulation over time and no significant compaction in the top 2 m, one would expect a shift which linearly increases with depth and has a slope equal to the ratio of accumulation rates. Then again, local environmental conditions such as wind speed and direction influence the mass accumulated by a certain deposition event . Therefore we aimed to align snow of the same origin and its properties with continuously changing shifts, a problem that has already been worked on at a lower vertical resolution for alpine snow e.g.,.

The dynamic time warping (DTW) method, which was introduced in speech recognition in the 1970s , provides an efficient algorithm for that purpose. It has already been applied in numerous fields, e.g., for the tracking of ice floes in synthetic aperture radar images . For a detailed review of DTW, see .

The basic idea is to discretize the two data sets to be compared with the same step size l (resulting in two vectors S and T of length n and m) and then consecutively assign the values of one to another, whereby each value can be matched with multiple values of the other data set. To find the best fit, one calculates a matrix D¯, where D¯[i, j] indicates the error of the best path that leads to the ith element of the first data set being connected to the jth element of the second one.

The original algorithm starts by calculating the matrix in the upper left corner, fixing the first elements of both data sets to be linked with each other. Then it proceeds through the matrix by taking the path with the minimal error leading to the respective cell and adding the local error, i.e., D¯[i,j]=∞fori<0orj<0∥S[0]-T[0]∥fori=0andj=0∥S[i]-T[j]∥+minD¯[i,j-1],D¯[i-1,j-1],D¯[i-1,j]else.

Finally, on arrival at cell D¯[n, m] it backtraces the path of minimal errors to D¯[0, 0], obtaining the best fit of the complete data sets in the given norm ∥⋅∥.

For our application – matching measurements of the upper 2 m of the snowpack – we do not aim to fit complete data sets, but rather allow for different offsets at the top and bottom. The former may be caused by variations of the snow surface due to current conditions, the latter by different accumulation rates, leading to data at the bottom of the liners not having any physical relation apart from being the deepest snow analyzed at the given location. To accomplish that, we expand the idea of , introducing maximal surface and bottom index offsets s and b. Then we initialize D¯ by D¯[0,j]=∥S[0]-T[j]∥for0≤j≤s and D¯[i,0]=∥S[i]-T[0]∥for0<i≤s before proceeding through the matrix. Finally instead of backtracing simply from D¯[n, m], we end our fitting path at minD¯[i,j]|(i=nandm-b≤j≤mor(j=mandn-b≤i≤n) and search a trace back to any of the initialized elements. Thereby we find the best matching of subsets of S and T with a maximal shift of s ⋅ l at the top and b ⋅ l at the bottom. In between, we verify that a linearly increasing maximal shift is not exceeded.

The simple way we proceed through the matrix so far, often referred to as “stepping pattern”, is unrealistic for our case as a single value of one data set could be fit to arbitrarily many values of the other. Along the traverse we find the maximal ratio of the respective accumulation rates between two sites to be a little smaller than 2. Therefore, we apply constrained stepping as presented by such that each value of one data set can be fit to at most two values of the other. This is obtained by D¯[i,j]=∥S[i]-T[j]∥+min(D¯[i,j-1],D¯[i-1,j-1],D¯[i-1,j])fori=1orj=1∥S[i]-T[j]∥+min∥S[i-1]-T[j]∥+D¯[i-2,j-1]D¯[i-1,j-1]∥S[i]-T[j-1]∥+D¯[i-1,j-2]else.

Figure  illustrates the different patterns for proceeding through the matrix. Here, usage of cell [i, j] refers to S[i] being assigned to T[j]. In the aftermath, the backtracing has to occur according to the implemented stepping.

Finally, we do not only want to fit one type of data (e.g., densities) but combine the available information in the profiles to gain a robust picture of the developing stratigraphy along the traverse. In a first step, we match the δ18O signal, which shows a clear seasonal behavior but almost no small-scale variations, as the high-frequency component is lost by diffusion. Then, we use the obtained depth assignment of the two different sites to resample the measured densities to a common depth scale. In a second step, we apply the algorithm to these densities at a much higher resolution to fine-tune our depth alignment according to small-scale stratigraphic features. As a norm we use the Euclidean distance divided by the path length (i.e., the root mean square error), which means that we have to keep track of the path lengths in a second matrix. Table  summarizes the final set of parameters. The maximum allowed offsets for the coarse fitting have been chosen according to the measured height of variations in the snow surface (e.g., dunes) and the maximum ratio of estimated accumulation rates. In the second step we allow for fine-tuning up to the maximum remaining shift, which was manually identified by aligning the vertical centers of the 2012 melt layers.

This method does not only allow us to compare data from two sites, but also to obtain a moving depth alignment by fitting the profiles to the first data set one by one. The result, a continuous image of the snow layering, can be compared with other indicators such as the melt layer positions. In addition, being able to align densities and stratigraphic features all along the traverse enables us to provide a representative density profile for the region. For its construction, we first use the continuous layering to transform all density curves to the first depth scale (NEEM) and average them. This, however, is not yet a representative density profile as all profiles now replicate the layering at NEEM; e.g., a layer that is very thin there but thicker at most sites would be considered thin. To overcome this, we calculate the mean shifts applied to the values that were aligned and thus averaged. On average, i.e., for constant accumulation rates, we would expect these shifts to go linear with depth for the layering to be representative. Thus we calculate a linear least squares regression and correct the depth accordingly.

Nonetheless, the depth scale still represents the accumulation rate at NEEM. To transfer the average profile to any location X in the sampling area of known accumulation (not necessarily one of the N2E sites), we need to calculate a linear rescaling factor fX for the depth dX that fulfills dNEEM=dX⋅fX. We expect fX to be determined by the accumulation rate, or rather its ratio to the one at NEEM.

Any alignment method will increase the covariance between records even if they are not related . Therefore, to test the statistical significance of our density alignment, we generate sets of surrogate density profiles with similar statistical properties independently for each site and process them the same way as the original data. Alongside the artificial density profiles, the real δ18O signals are used for the coarse fitting step.

The complexity of the density signal consisting of slow variations, sharp property changes as well as strong melt layer and wind-crust-related density spikes, inhibits the use of simple surrogate construction methods such as autoregressive processes. Instead we propose the following algorithm.

For each site, as a base curve, we identify the δ18O component of the density signal by linear regression, using the same step size llow as for the coarse (δ18O-based) fitting step. This can be done because we rely on δ18O to follow a seasonal cycle – otherwise water isotope dating would be impossible. Let ρbase be the base density from δ18O, rlow the autocorrelation and σbase the standard deviation of the fluctuations of the measured density (averaged to resolution llow) around ρbase for lag llow. We start generating an artificial low-resolution density profile ρlow by ρlowzi=ρbasezi+εiεi=ν0fori=0rlow⋅εi-1+νielseν∼N0,σbase, where zi=z0+i⋅llow.

Here ν ∼ N(0, σbase) implies that the νi are distributed normally with mean zero and standard deviation σbase. In the following, U(0, 1) will represent a continuous uniform distribution for the interval [0, 1]. The inclusion of higher autocorrelation lengths is straightforward. rlow has to be replaced by the autocorrelation matrix, which is multiplied by a vector of the preceding εi. Second, on the fine scale (step size lhigh), we have a look at the differences between the interpolated low-resolution density and the high-resolution density values from the measurements. As we find the distribution to be trimodal, we split the differences into three components – low amplitude variations within snow of similar properties (henceforth denoted “noise” even though they might partly have physical origin), fast and moderate amplitude changes in the density due to layering or wind crusts (“shocks”) and rapid high amplitude changes at melt layers (“melt”). Again, we compute the autocorrelation factor rhigh for lag lhigh. Nonetheless, this time, the standard deviations σnoise, σshocks and σmelt and the means μshocks and μmelt have to be calculated separately. Furthermore we need to estimate the probabilities Pshocks and Pmelt of beginning a shock or a melt layer at a specific position. For this purpose, we determine the number of melt layers Nmelt, the number of shocks Nshocks and the average distance to the previous shock davg. In addition, we denote the total number of data points by N and the distance to the last shock at a given position i by di. Finally, the basic model to generate a random density profile ρhigh is ρhighzi=ρlowzi+κiκi=ϕifori=0orP>Pmelt+PshocksNμshocks,σshocksfori≠0andPmelt<P≤Pmelt+PshocksNμmelt,σmeltfori≠0andP≤PmeltPmelt=NmeltNPshocks=didavg⋅NshocksNP∼U(0,1)ϕi=ν0fori=0rhigh⋅ϕi-1+νielseν∼N0,σbase, where zi=z0+i⋅lhigh.

The same approach as before can be used to expand to higher autocorrelation lengths. However, we use the model in the presented form as it already provides realistic density surrogates.

As an example of the matching process, we present a fit of data from N2E_11 to the first site (NEEM) in Fig. . The distance between the two locations is about 240 km, i.e., a little more than half of the total traverse length. First the δ18O profiles are matched, yielding an approximately linearly increasing coarse shift. In the second step the densities are fine-tuned, which results in small shifts fluctuating around zero and never reaching the allowed maximum of 0.1 m. To provide an overview of the changing snow structure, we fitted all combinations of profiles from two sites and plotted the matrix of the root mean square errors (RMSEs) in Fig. . A remarkable change in the pattern of the fitting errors occurs between the fourth and fifth site along the traverse.

Figure  shows the continuous depth alignment obtained by fitting all liners along the traverse to the first site (NEEM). There were no notable differences when another location (e.g., EGRIP) was chosen as the reference or the fitting was done consecutively. For comparison, the melt layer positions detected during the CT measurements (cf. Table ) have been included. In addition, selected density profiles are displayed. Using the previously calculated depth alignment, density records were stacked to obtain a representative density profile (Fig. ). The gray area indicates a 1 standard deviation error band. Comparing the necessary rescaling factors (known from the construction of the stacked profile) to the ratio of accumulation rates, we apply linear least squares to find fX=0.325+0.665⋅a˙NEEMa˙X, where a˙X denotes the mean annual accumulation rate at site X. The coefficient of determination is R2 = 0.82.

At the base resolution of 0.1 cm we find a mean shared variance of R2 = 0.56 between the average and the individual density profiles. It can be increased by smoothing and obtains a maximum of R2 = 0.71 when using a 4.3 cm moving average. In comparison, for 1000 randomly generated density data sets (e.g., Fig. ), the respective stacked profiles share an average of R2 = 0.44 with their components at base resolution. The maximum is R2 = 0.61. We determine a p value (probability of finding such high R2 by chance) of 0.015 for the measured profiles within the distribution; i.e., the high shared variance of the measured profiles is statistically significant.

For all sites we find at least two melt layers in the snow isotopically dating back to the summer of 2012. In addition, some liners show melt layers which are surrounded by snow dating to winter 2011/2012 or summer 2011. For an overview of all melt layers, see Table  or Fig. . From the raw density profiles, we obtain Fig. , which shows the average densities of the top meter and decimeter, which do not contain any prominent melt layers. The density in the top meter tends to decrease from the maximum of 332 kg m-3 at NEEM down to a minimum of 297 kg m-3 roughly 150 km from EGRIP before slightly increasing again. For 15 out of 18 sites the surface density is higher; nonetheless both parameters evolve similarly along the traverse.

Table  displays the mean annual accumulation rates along the traverse. Starting with a maximum of 225 kg m-2 a-1 at NEEM, the values steadily decrease down to the minimum of 115 kg m-2 a-1 about 100 km from EGRIP before increasing again to 140 kg m-2 a-1 at EGRIP. Comparing average values for the different years, there is neither a trend nor considerable variations in the accumulation rate (cf. Table ). However, we observe much higher differences between successive years within the same core (average change 34.67 kg m-2 a-1), where we mainly see alternating behavior of high- and low-accumulation years.

Of the 5 years contained in our data, 2012 had the isotopically warmest summer for 83 % of the sites. At the three remaining locations (N2E_11, N2E_16 and EGRIP), the highest δ18O values occur in 2014. For the winters, 2014/2015 was isotopically coldest in 51 % of the cases, 2011/2012 in 19 % and 2010/2011 in 30 %. Regarding annual δ18O averages of all available sites (Table ), we also find the highest δ18O values for 2012.

Comparing the annual average δ18O values with the accumulation rates we obtain Fig. . Positive linear relations were fit to the data of 2012, 2013 and 2014 respectively, showing that within 1 year higher temperatures coincide with higher accumulation. The coefficient of determination is highest for 2012, while we have larger spreads for the other 2 years, in particular 2013.

To relate the density with the seasonal, low-frequency δ18O signal at NEEM, we applied a 10 cm running mean to the stacked high-resolution density profile in Fig. . On average, snow with a high δ18O value (considered summer snow) has a low density and the other way around. The only exception is the summer of 2012, where we find high-density values in summer, too.

The liner technique allows us to retrieve non-disturbed snow samples from the field and thereby conduct lab-based analysis (such as high-resolution density measurements) to gain further insight in the development of physical snow properties over large distances. This is a major improvement compared to previous methods, e.g., for measuring snow density, which was so far mainly done by weighing a known volume of snow where we have a trade-off of accuracy (bulk density) and resolution (density cutters). Both horizontal resolution and vertical depth can be adjusted to fit the needs of the respective study.

Figure  illustrates that we are able to align δ18O and density data down to small stratigraphic features very well over a distance of over 200 km. Along the traverse, one observes a clear change in the RMSE (cf. Fig. ) and thereby the snow structure at the fourth site, indicated by significantly different fitting errors. This coincides with the location where the ice divide was left eastwards and thereby the traverse entered a different accumulation regime in agreement with the drainage systems given by .

Furthermore the continuous depth alignment agrees very well with the melt layer positions detected during the CT measurements (Fig. ). Stratigraphic features are still well aligned over the complete traverse distance of almost 450 km. We obtain a clear picture of the layering of the snowpack along the traverse. In comparison to radar measurements, which are limited to centimeter vertical resolution but can resolve annual layers down to 12 m , we can give a much more precise picture and observe small-scale structures like wind crusts. In exchange we are limited to shallower depths – the maximum we plan to access in the near future is 6 m in a trench at the EGRIP drilling site.

For rescaling the stacked profile to any location in the area with known annual accumulation, we obtain a linear relation of the depth factor with the ratio of accumulation rates. This is plausible because, on average, we find linearly increasing shifts for the matching. Furthermore we do not observe significant densification in the upper 2 m of the snowpack, and therefore the depth of snow from the same deposition event is primarily determined by the accumulation rate. In addition, the relation has a high coefficient of determination for the applied linear least squares.

As the stratigraphy does not seem to change remarkably along the traverse apart from the effect of the decreasing accumulation rate, we consider the profile in Fig.  to be representative of the whole traverse region, potentially even most of North Greenland. For the given error band, there is an overlap of uncertainty in the depth alignment (x direction) with the uncertainty in density (y direction). The former is mainly caused by the variability of the snow mass accumulated from a single deposition event. Regarding the latter, the average density of the snowpack greatly varies as can be seen in Fig. . Thus, for the second meter, even though it is contained in the uncertainty band, we do not expect a straight line, but rather an alternation of high- and low-density layers similar to the upper meter.

A statistical test using surrogate density profiles shows that the high shared variance of the measured profiles is statistically significant (p = 0.015), even though the actual difference in numbers is quite small. This underlines that the density alignment provides additional information as we tried to use the most realistic surrogates (original δ18O signal, seasonal cycle, three component stratigraphy model). Furthermore, a coefficient of determination of R2 = 0.56 between the stacked and the individual profiles shows how much of the layering does reappear. Smoothing increases R2 as it steadily transforms the profile to the low-resolution density curve that shows seasonal behavior (see Fig. ) while smaller local variations vanish.

The vast majority of melt layers are found in snow dating back to the very warm summer of 2012 . Moreover, above most of the melt layers within older snow, we find clear signs of percolation (cf. Fig. ). Therefore we assume that 2012 was the only year in the period 2010–2015 with significant melt occurring in the observed area. From Fig.  we can infer that on the one hand, the average density of the snow in the top 2 m at a certain location can already be deduced from the surface density. On the other hand, the surface snow in May is among the denser ones within the year, thereby rather representing a spring or even winter signal than a summer one (compare Fig. ). Furthermore we are able to visually identify many layers of homogeneous density, often clearly separated by wind crusts, which seem to contain snow from single deposition events.

For the accumulation rate (see Table ), the 1964–2005 average of 220 kg m-2 a-1 determined from the NEEM ice core agrees very well with the 225 kg m-2 a-1 that we obtain from the corresponding snow liner. In addition, both accumulation maps from field measurements and regional climate models show the same behavior towards the east. While Table  shows no significant interannual changes in the average accumulation rate for the study area, we observe high fluctuations in the local annual values, a feature consistent with the strong influence of stratigraphic noise in single profiles . These can be explained by the accumulation of every year compensating previous local variations in the snow surface before new structures are introduced by wind-induced drift and dunes. Nonetheless, they also might partly originate from the uncertainty of separating the years only according to the δ18O extrema.

In the majority of cases we find the highest isotopic summer temperatures and average δ18O values for 2012, underlining the exceptional warmth of this year. The values for 2014 indicate that it was still warmer than the other contained years, in particular 2010, which was formerly regarded as very warm . The picture for the winters is less clear. Indeed, we assume that the isotopic signal of the fresh snow from winter 2014/2015 might still change.

We find a positive linear relationship of annual mean δ18O and accumulation rate (Fig. ) with similar slopes for 2012 and 2014. This relation might partly originate from the changing surrounding conditions (e.g., elevation) along the traverse. The offset between the years could potentially be caused by the very high temperatures and the consequential surface melting in 2012 as we find the relation for 2013 to be a lot closer to 2014 than 2012. The dependence of the offset on the annual mean temperature (which is quite similar along the traverse) could explain why previous attempts to link both parameters by averaging data from several years e.g., show results that are less clear.

We observe a clear anticorrelation of low-resolution density and δ18O in Fig. . This agrees with the widely accepted conceptual model of , which states that snow has lower densities in summer and higher ones in winter. The main causes given are the increased packing due to stronger winds in winter and the larger size of precipitation particles in summer. For the summer of 2012, the high average densities are caused by the prominent melt layers, superimposing the original signal of the snow.