For the DEM generation we used TanDEM-X observations acquired between 2010 and 2017. To ensure adequate vertical accuracies, we only used acquisitions with a height of ambiguity smaller than 80 m . The radar incidence angles range from 36 to 44∘. For an accurate orthorectification we used the TanDEM-X 12 m DEM pixel resolution as reference and iteratively updated the look-up table based on the measured deviation . We only studied winter acquisitions because vegetation, wet snow, and standing water during the thaw season induce sizeable errors , whereas in winter we expect the low average monthly temperature to produce a dry snowpack through which radar waves can propagate to the ground without being strongly affected . We followed a standard approach to generate the DEMs , which have a planimetric resolution of about 10 to 12 m and vertical accuracies of about 2 m in areas with high coherences. We did the interferometric processing using the GAMMA Remote Sensing software . More processing details including tilt removal and the correction of misalignments, specifically for DEMs generated from InSAR observations in permafrost regions, can be found in .

In the next step we averaged and mosaicked DEMs corresponding to the same winter. We then used an automated detection algorithm to identify significant elevation changes in the DEM difference images from DEMs that were obtained more than 3 years apart . For each detection we carried out several processing steps. First we assessed the topography and environment using a TanDEM-X DEM and Sentinel-2 multispectral images taken in summer (snow-free). For all study sites at least one Sentinel-2 image during the years 2016 to 2019 was available. The criteria for classifying a detection as an active RTS were the exposure of bare soils, a retreat over time, a location related to a potential sediment removal mechanism, and the presence of a headwall . In uncertain cases we used additional time series of Planet RapidEye optical data to classify the detections .

The error sources and uncertainties that govern the lower RTS detection limit in terms of the headwall height and retreat rate are manifold and difficult to quantify. This is mainly due to the small number of high-resolution, three-dimensional RTS inventories available , where the timescale on which the RTSs are monitored also plays an important role. To obtain an estimate on the lower limit of RTS-induced elevation changes that are detectable, we analysed the smallest RTSs detected in our sample. The 10 smallest RTSs detected have elevation changes in the range of 1.6 to 1.9 m, which is on the same order as the general accuracy of the TanDEM-X DEM and thus represents the smallest RTS headwall heights that are detectable. Similarly, the smallest total area changes of detected RTSs are on the order of 500 to 1000m2, corresponding to about 10 to 12 pixels. Additionally, processes related to the observation properties and InSAR processing further complicate the error estimations. For example, the 40∘ right-looking viewing geometry leads to different pixel resolutions depending on the aspect and slope of the observed area. These error sources and increased uncertainties should be considered in the interpretation and future use of the dataset, especially for small RTSs, in terms of both horizontal and vertical changes.

After the classification step, we generated polygons for each detected RTS, outlining the area with significant elevation changes. Examples of the generated polygons are shown in Figs. S1 to S4 in the Supplement. The polygons outlining the area of elevation change were drawn by a trained student and the first author, as our use of an automated method that implemented a fixed threshold on the elevation change gave unreliable results. We attributed the RTS polygons in terms of location as either “shoreline” (located close to a waterbody) or “hillslope” (located at trenches or riverbeds).

For all calculations we used the polygons, which indicate an area of elevation change and thus a net volume loss. We note that this area can also include a zone of deposition, especially for small and low-relief RTSs or if the time between observations increases. We could not accurately detect areas such as the debris tongues or zones of alluvial deposits, and they are not included. We computed the volumetric and area change as well as the slope and aspect. For parts of the study sites additionally to the winter in 2011/12, observations in 2010/11 and/or 2012/13 were available. To simplify the analysis we normalized the properties to changes per year and took the average if several DEM difference pairs were available. It is important to note that unlike most landslides, RTSs are multi-year features with a strong variability in the erosional intensity as well as a potential change in their morphology over time. In the interpretation of the results and specifically the comparison to landslide studies, the use of the integrated change over several years needs to be considered. We computed the aspect and slope by using the pre-disturbed elevation model and applied Gaussian smoothing with a standard deviation corresponding to 100 m to reduce the influence of random errors . For the aspect distribution we additionally computed the aspect distribution weighted by volume.

To quantify the volumetric change rate density (volumetric change rate per unit area), we first use a simple approach by dividing the summed total of all RTS volumetric changes per year by the study site size. This has a drawback because RTSs often occur heterogeneously and the result strongly depends on the exact outline of the study sites . For example, in the Peel study site only the east-facing part of the mountain range experiences RTS development, but our study site also includes the western part of the range where nearly no RTS activity was detected. To account for this problem, we follow a similar approach to that proposed in and divide our study site into tiles of size 10 km by 10 km, count the number of empty grid cells, and compute a more representative RTS density using only the cells showing RTS activity. It is of note that to interpret the computed density values, the number of empty as well as the number of non-empty grid cells in relation to the total size of the study site should be considered.

To quantify the number of lakes in each study site we used the waterbody mask generated from Sentinel-2 data and computed the area that is covered by the mask . For this computation, we excluded open-water areas.

We investigated the dependency of RTS growth on different terrain controls by computing the aspect, slope, and location (lakeshore or hillslope). For the aspect we identified the most dominant orientation by summing the number of RTSs as well as the volumetric and area change rates in eight aspect bins (N, NE, E, SE, S, SW, W, NW) and used these bins to compute the strength and orientation of the primary direction.

The probability density function (PDF) of the area affected by elevation loss per year corresponding to an RTS inventory can be defined as 1p(ARTS)=1NRTSδNRTSδARTS, where ARTS is the area change affected by elevation loss of an RTS per year, NRTS the total number of RTSs in the inventory, δNRTS the number of RTSs with affected areas between ARTS and ARTS+δARTS, and δARTS the bin width. Equivalently the probability density function p(VRTS) for the volumetric change per year can be defined.

All RTSs in the study show changes per year in the range of 102 to 106m2yr-1 for the area and 102 to 106m3yr-1 for the volume, and we used 30 bins sampled in log space to cover these ranges.

When analysing a landslides PDF three quantities can be used to describe the distribution: the rollover and cutoff points for small events and the coefficient of the power law scaling β for large events. The rollover point is defined as the peak in the PDF and corresponds to the most common occurrence in the distribution. For large RTSs the PDF can be described as a power law function. The point at which the distribution starts to follow a power law is defined as the cutoff point (Fig. ).

To determine how well the data points are described by this model and to estimate the rollover point, we fitted a three-parameter inverse gamma function to the RTS probability density function . To estimate the error of the fit we used the bootstrap method, drawing 1000 random samples with replacement from all data points and computed the R2 value as well as the rollover point for each iteration .

For the computation of the cutoff value and the exponential scaling exponent, we used the method of , which is commonly used in landslide frequency–area analyses . The approach is based on sampling all possible cutoff values and estimating the corresponding exponential scaling coefficients β using a maximum-likelihood fitting method. We then tested the obtained fitting values based on a Kolmogorov–Smirnov statistic and used the values that follow best a true power law distribution as the final cutoff and β value. To quantify the uncertainty we again used a bootstrap algorithm.

One important quantity in comparing landslides of various sizes is the change relation between area and volume. The simplest conversion assumes that an anisotropic scaling exponent, α, relates the area and volume by V≈Aα. Since both variables (area and volume) are affected by measurement errors, we used an orthogonal distance regression model to fit a straight line . To quantify the goodness of the fit, we calculated the RMSE, R2, and p value (in log space).

The estimated PDFs are shown in Fig. a and b. For most areas the quality of fit of the inverse gamma function was good, as indicated by R2 values >0.75. Exceptions were the Noatak and Chukotka study sites with R2 values between 0.6 and 0.7. These two sites also have the lowest number of RTSs in the sample with only 26 (Noatak) and 51 (Chukotka) RTSs (Fig. c and d).

The modes of the volume change distributions (rollover points) differ between sites. The two study sites located in the high Arctic (Ellesmere and Banks) show rollover values that are an order of magnitude higher. The range of measured volumetric and area change rates show large variations for the Tuktoyaktuk and Peel study sites, whereas the other study sites show smaller variations.

The PDFs above the cutoff value and the relation between rollover and cutoff as well as the exponential-decay values differ between sites (Fig. ). For the PDF based on the volumetric change, a high rollover value is moderately associated with high cutoff values, indicated by a correlation coefficient of 0.72. By contrast, the PDF based on the area change rate shows a much stronger separation between the high-Arctic sites and the other study sites and consequently also shows a high correlation factor of 0.96. For the power law exponent for RTSs above the cutoff values, no large difference between the areas is visible (β is approximately 2 to 3, and correlation coefficients are <0.64). All correlation coefficients are shown in the Supplement (Fig. S5). It is of note that for the yearly area and volumetric change rates the cutoff value for the Peel study site is relatively small but the distribution continues to high values with yearly area change rates of up to 6×104 m2 yr-1 and 3×105 m3 yr-1 (mega-slumps). The computed values of the rollover, cutoff, and exponential-decay coefficients as well as the fit parameters for the inverse gamma function are reported in Tables S1 and S2 in the Supplement.

The estimated area-to-volume scaling law based on all data points in log–log space shows a clear relationship that spans over 4 orders of magnitude between the area and volumetric change rates (Fig. a). The estimated scaling exponent across all study sites was α=1.15±0.01. The quality of fit was decent, with an R2 value of 0.81, RMSE of 0.21 m3 yr-1, and p value smaller than 10-6, showing a strong dependency between RTS area and volumetric change rates. This is remarkable considering that RTSs in the sample occurred in different topographic and geomorphological settings. Nevertheless we found a moderate inter-region variability in the scaling coefficient. The α coefficients for the individual sites was in the range of 1.05 to 1.25 with the exception of RTSs in the Banks site with a high coefficient of 1.37 (Fig. b). The data points and fitted lines for each study site individually can be seen in the Supplement Fig. S6. The strong association between area and volume change rates can facilitate the estimation of volume changes from multispectral satellite images.

Among the investigated terrain controls, aspect shows the greatest variability between the study sites. RTSs located in Siberia as well as on Ellesmere tend to favour a south-west-facing orientation (Fig. a). The very small number of RTSs in the Noatak study site showed a preferred orientation towards the north-west, and RTSs in Peel have a preferred orientation towards the north-east. For Tuktoyaktuk and Banks no clear trend is visible. To consider the possibility of more than one preferred orientation, we additionally looked at the initial aspect bin distribution (Supplement Fig. S7). Here only the aspect distribution of RTSs in the Noatak Valley shows two preferred orientations, but this could be related to the low number of RTSs in the study site. Additionally to the number of RTSs in each aspect bin, we weighted the aspect by the volumetric change rates. This only slightly alters the preferred orientation, and large RTSs do not occur at different aspects.

The slope of the pre-disturbed area shows some difference between the study sites (Fig. c). In general all RTSs evolve at slopes ranging from 2 to 3∘ up to slopes of 20∘. Interestingly, in the study site of the largest RTSs on Banks, they tend to favour lower slopes with values below 12∘.

We investigated the dependency of RTS locations in terms of their occurrence. We distinguished two types of location, either at a shore (including lake and coastal) or at hillslopes with no large waterbodies close by. Several study sites have mostly one type of RTS location. The RTSs in Ellesmere (99 % hillslope), Peel (96 % hillslope), and Noatak (88 % hillslope) have mostly RTSs at hillslope locations. On the contrary, RTSs in the Tuktoyaktuk site are nearly all located at lakeshores (99 %). All other study sites have a mixture of hillslope and shoreline RTS locations: Banks (66 % hillslope), Chukotka (52 % hillslope), Taymyr (27 % hillslope), and Yamal/Gydan (26 % hillslope). In the study sites with both types of RTS location, no significant difference between the distributions is visible (Fig. c). Furthermore, we did not find a significant correlation between RTS size and the percentage of hillslope or shoreline RTS locations (Supplement Fig. S5).

To estimate the volumetric change rate density of RTSs within the RTS-affected regions of each study site, we gridded them into tiles of size 10 km by 10 km. Figure a shows the volumetric change rates per square kilometre using only the tiles with RTSs present. The volumetric change densities over the total study site strongly depend on the exact outline of study sites, and removing tiles without RTSs present gives a more consistent and comparable volumetric change rate density. To make this more visible, the number of tiles with RTSs present and without can be seen in Fig. b. Here for example, the Chukotka Peninsula has only a small number of tiles with RTSs present.

The computed probability density functions of the yearly area and volumetric change rates follow a characteristic inverse gamma law with first an increase in frequency up to a maximum value with the most abundant thaw slump sizes (rollover) and then a decrease with an exponential-decay tail above a certain cutoff value. Our findings show that the applicability of this universal scaling also applies to permafrost landscapes, despite differences in the governing geomorphic processes with respect to lower-latitude environments. Here we emphasize again that another difference between our analyses and those of common landslide studies is that RTSs are a multi-year phenomena with variable yearly erosion rates. Some variability in the exact form of the distributions should therefore be expected if different time periods are chosen.

To further investigate the distributions we distinguish between two parts: (1) the exponential-decay part for large RTSs and (2) the part that deviates from this exponential decay below the cutoff point. For landslides the exponential-decay part is typically explained in a statistical way by the concept of self-organized criticality, where a constant “input” of a specific landslide size at a random location, together with a merger of landslides that are close to each other, reproduces this distribution . For RTSs this explanation seems plausible since initiation and evolution are strongly linked to soil properties that can promote RTS development in close proximity and also since RTS coalescence is common . In addition to the universal exponential-decay behaviour in all study sites, we found that the largest RTSs in the Peel, Banks, and Ellesmere study sites have growth rates that are orders of magnitude larger (Fig. a and b). A possible explanation is that topographic and geomorphological properties, like the amount of massive ice, overburden thickness, or the steepness of terrain, only allow RTSs to grow to a certain size . For example in the Tuktoyaktuk study site (NRTS=212) where RTSs occur at lakeshores in mainly flat regions, the largest RTSs show growth rates of between 5200 m2 yr-1 and 31800 m3 yr-1 compared to, for example, the Ellesmere site (NRTS=223) with more topographic features and mainly hillslope RTSs, which show maximum growth rates that are 3 to 4 times higher (range is between 23000 and 106400 m2yr-1). This suggests that additionally to the exponential-decay factor, a maximum RTS growth rate is also important to characterize the high end tail of the probability density function.

For the deviation from the exponential decay, two types of explanation have been proposed for landslides in temperate climate : first an under-sampling of small landslides due to limitations in resolution and secondly an explanation that attributes this divergence to physical processes. Investigating our dataset, a divergence due to under-sampling seems unlikely since the PDFs in Peel, Banks, and Ellesmere show this divergence (cutoff point) at high yearly change rates of >104 m2 yr-1 and >3×104 m3 yr-1, which corresponds to area and volumetric changes high above the resolution limit (TanDEM-X DEM resolution – horizontal ≈10 m, vertical 2 to 5 m). The physical origins are likely related to environmental conditions and physical characteristics of ground materials like ground ice content but are outside the scope of this work. Future models for thaw slump initiation and evolution should be able to investigate the drivers and reproduce such distributions.

We found a power law relationship (V≈Aα) between the area and the volumetric change rates with a scaling coefficients α of 1.15 for the total dataset and ranging between 1.05 and 1.37 for the individual study sites. Such relationships are known from landslides in temperate climates with typically values of 1 to 1.5 . For RTSs only one study by , investigating RTSs on the Peel Plateau and Richardson Mountains, has estimated this relationship, and it found a scaling coefficient of 1.42, which is relatively high compared to our values (Peel – 1.27, Tuktoyaktuk – 1.17) but inside the estimated error.

Comparing the coefficients between study sites, we found that lower scaling coefficients are not correlated with smaller RTSs. For example the scaling law coefficient in the Tuktoyaktuk site with relatively small RTSs is the same as for the RTSs in the Ellesmere site with the largest RTSs in our dataset. On the other hand, for RTSs in the Peel study site there is little confining topography and deep layers of ice-rich tills that allow the headwalls to grow to large sizes and consequently yield a steeper regression curve . The diversity in landform characteristics also contributes to the variation in the area-to-volume scaling coefficient. At the study sites Banks and Noatak, shallow detachments are dominant in the small-area range. These may promote larger scaling coefficients when combined with older, deeper RTSs . Furthermore, most RTSs initiate as shallow active layer detachments. The gradual increase in headwall heights following the initiation event could lead to a temporal change in the scaling coefficient. Further investigations relating the scaling coefficients to additional RTS and area characteristics (e.g. soil properties, climatic history, age of the RTSs) are needed.

With the available data we could determine several terrain controls, namely the orientation of RTS growth and the slope of the pre-disturbed area the RTS grew into as well as the location in terms of hillslope and shoreline RTSs. Our findings in terms of the preferred orientation of RTSs are mostly consistent with past regional studies: a preferred south-west orientation for RTSs in the Siberian study sites and Ellesmere , towards the north-east for the Peel study site , and north-facing RTSs in Noatak . For RTSs in the Tuktoyaktuk study site we found no preferred orientation, which is consistent with but in contradiction to other studies that found RTSs orientations that favour north-facing slopes . The association with aspect hints at inter-regional differences in the governing geomorphic drivers and controls. A south-west-facing orientation is considered to be related to higher initiation and growth rates of RTSs due to the higher energy available from solar radiation . This would suggest that solar radiation is an important factor in RTS growth and initiation for the study sites in Ellesmere and Siberia. Past studies have shown that a high ground ice content is a necessary condition for RTS development . During the Holocene thermal maximum, the regions in north-west Canada experienced warmer summer temperatures than other Arctic regions and could have lost ground ice on south-facing slopes . Thus the differences in RTS aspect distributions could be related to the climatic history. For example the dominant north-facing exposure on the Peel Plateau could reflect such anisotropic abundance of ground ice.

We did not find a significant relation between RTS size (area and volumetric change rates) and aspect as well as slope or location (hillslope, shoreline). This finding affirms previous studies that have highlighted the complexity of the processes and controls governing RTS expansion.

The scaling relations we quantified are critical for modelling and predicting RTS activity and the impacts on biogeochemical cycling. The regional variability in scaling behaviour needs to be considered when upscaling field observations to estimate large-scale nutrient, sediment, and carbon budgets. Because Earth system models strive to capture the variability in these processes from regional to global scales, our results can be used to calibrate and validate global models. Possible changes in the scaling relations could be important indicators to predict future RTS evolution and impacts.

Our observations of variable RTS development rates and regimes highlight the need for continual pan-Arctic monitoring and further satellite missions to derive high-resolution DEMs. The TanDEM-X data availability only allowed us to compute elevation changes in a 5-year time window. To investigate changes in RTS activity related to climate change, a higher temporal resolution is needed. Here additional observations from the TanDEM-X satellite as well as data from the ArcticDEM could add additional data points . Furthermore, with the derived area-to-volume scaling laws it is potentially possible to use optical satellite images which are available at a higher temporal resolution to estimate the volumetric change.