The Arctic has been warming at up to four times the global average since 1979 . This rapid atmospheric warming has accelerated the mass loss from the Greenland Ice Sheet (GrIS) over the last decades , contributing to global sea-level rise . Rising sea levels will impact coastal communities worldwide in the coming decades .

Accurate and timely data on ice sheet dynamics are therefore essential for understanding the processes driving mass loss and refining projections of future sea-level rise . Satellite radar altimetry has emerged as a powerful technique for monitoring ice sheet Surface Elevation Change (SEC), which, under certain assumptions, can be converted into estimates of ice sheet mass loss and sea-level contribution . ESA’s CryoSat-2, carrying the SAR Interferometric Radar Altimeter (SIRAL) , was specifically designed to measure SEC in ice-covered regions, regardless of cloud cover or solar illumination, making it particularly valuable for the polar regions. With more than a decade of continuous observations, CryoSat-2 provides an unprecedented record for assessing both long-term trends and shorter-term variability in GrIS SEC .

Capturing seasonal fluctuations and short-term events, such as glacier calving or enhanced meltwater runoff, requires a higher temporal resolution than the traditional SEC products derived from radar altimetry . Conventional approaches, such as the plane-fit algorithm , constructs annual SEC by extracting a mean SEC over a multi-year period, but this limits their ability to resolve seasonal fluctuations and short-term events. As a result, much of the seasonal signal and many episodic anomalies remain hidden in aggregated annual products. To address this, new processing methods have been developed to derive monthly SEC from radar altimetry . These efforts mark a major step forward as they provide a finer temporal detail, which enables more direct comparison and joint analysis with regional climate model outputs and in situ observations.

Recent studies have demonstrated advances in deriving monthly SEC products from altimetry . Different methodological choices are made regarding spatio-temporal regularization and uncertainty treatment. Quantitative comparison against independent laser altimetry, together with inter-comparison against published SEC products, therefore provides a valuable way to place new developments in context. In this study, we evaluate the performance of our state-space framework relative to published SEC products and inter-compare against ICESat-2 ATL15, Operation IceBridge, AWS stations and Surface Elevation Reconstruction And Change detection (SERAC) timeseries.

Here, we address the spatio-temporal reconstruction step of radar altimetry processing by introducing a state space modeling framework that reconstructs monthly gridded SEC fields directly from irregular satellite observations. We integrate state-space models (SSMs) with radar altimetry as a statistical framework to reconstruct spatially and temporally coherent SEC fields from irregular observations, while accounting for measurement uncertainty and spatio-temporal dependence. SSMs have already demonstrated their potential in related Earth Observation applications, for example, in deriving temporally densified water level time series for a river based on multi-mission satellite altimetry . This flexible and data-driven SSM framework can be applied across satellite altimetry missions to generate continuous, gap-filled SEC time series, with seasonal signals emerging directly from the data rather than from predefined models such as in previous studies by e.g. or . In addition, SSMs explicitly separate signal from noise, and are able to handle irregular or missing data, and may provide coherent uncertainty estimates while accounting for spatial and temporal dependencies . Extending this methodology to ice sheets provides a powerful means to generate continuous, gap-filled time series of SEC across a range of spatial and temporal scales, independent of the specific satellite mission.

In this study, we develop our model 3D-ECM to derive monthly surface elevation change (dSEC) over the entire Greenland Ice Sheet from CryoSat-2 radar altimetry data, where the prefix “d” denotes that the model can be configured to any desired temporal resolution; here it is applied at monthly intervals. The derived dataset contains monthly SEC estimates that reflect both multi year trends and intra-annual variations derived from the SSM.

The construction of the dSEC data product is based on CryoSat-2 Level-2 altimetry data and is validated against independent datasets from ICESat-2, Operation IceBridge (OIB), Automatic Weather Stations (AWS), and laser altimetry-based SERAC time series fusion, incorporating firn height change estimation.

To model the ice surface elevation as a function of time and space, we employ our model 3D-ECM, a data-driven SSM designed to estimate monthly elevation change fields. We define a target grid of 5 km resolution. CryoSat-2 POCA ice surface elevations are extracted using the GrIS ice mask from , and use elevations relative to the ArcticDEM Mosaic V 4.1 . The model thus estimates anomalies relative to this DEM rather than absolute surface heights.

Because CryoSat-2 POCA elevation anomalies are irregular in space and time, we adopt a framework that jointly models spatial and temporal correlations to reconstruct coherent surface elevation anomaly fields. This is done via a state-space approach, in which the evolution of surface height in space and time is represented as a set of hidden states that are indirectly observed through noisy satellite elevation measurements. In this framework, the hidden states correspond to the true, but unobservable, surface height anomalies on a regular grid. The observations are the CryoSat-2 Level-2 POCA elevation anomalies relative to ArcticDEM. By linking the hidden and observed components through probabilistic relationships, a State-space model provides a statistical framework that separates measurement noise from true elevation changes.

The large amount of satellite data requires a computationally efficient approach to reconstruct the spatial-temporal field of surface elevation anomalies Htrue. To do this, we apply a Gaussian Markov Random Field (GMRF; ). A GMRF is a multivariate Gaussian random field in which the probability of each location is conditionally dependent only on its neighbors, leading to a sparse representation of the precision (inverse covariance) matrix. This is computationally efficient compared to a Gaussian Process (GP) model, in which each location is correlated with every other through a dense covariance matrix that requires costly matrix inversion. Instead, sparsity in the GMRF allows operations to scale almost linearly with the number of grid cells, whereas GP models typically scale cubically with data size. The use of Gaussian Markov Random Fields for spatial modelling follows established geostatistical approaches for representing spatially correlated fields while maintaining computational efficiency . Similar statistical frameworks have been used to reconstruct glacier elevation change from satellite observations, where spatially and temporally correlated signals must be inferred from irregularly sampled measurements .

To model the ice surface elevation anomalies as a function of time and space, we define a 3-dimensional regular grid. In the spatial dimension, we use a target grid with a 5 km resolution in a polar stereographic projection (EPSG:3413) covering the Greenland Ice Sheet. In the temporal dimension, a monthly resolution is selected. The 5 km spatial resolution and monthly temporal resolution represent a compromise between resolving spatial and seasonal variability and ensuring sufficient data density for spatio-temporal reconstruction given the CryoSat-2 sampling that has a 369 d repeat cycle and a 30 d subcycle . For computational efficiency, the model is applied to overlapping spatial tiles that are processed independently. The overlap is introduced through a 15 km padding region, which provides additional observational context near tile boundaries and reduces edge effects in the state-space estimation. Following processing, only the central region of each tile is retained, while results within the padding zone are discarded. Consequently, each 5 km grid cell is represented by a single tile estimate and no averaging between overlapping tiles is required.

Here, we assume that an observation of ice elevation anomaly Hiobs is a function of space (x,y) and time t is represented by the following expression. 1Hiobsx,y,t=Himodelx,y,t+ϵix,y,t,ϵi∼N0,σobs2, where Himodel is the true unobserved elevation anomaly, and ϵi represents the observational error, which here is assumed to be Gaussian with the standard deviation σobs. The modeled elevation anomaly is represented by a Gaussian Markov random field consisting of a spatial and temporal component with a mean and covariance given as 2Hmodelx,y,t∼N(μ,σp2Q1⊗Q2-1).

Here, Q1 and Q2 are the precision matrices that define the neighbor structure related to the spatial and temporal parts of the GMRF. The spatial part of the precision matrix is defined as 3Q1=ϕ⋅Q0+I, where ϕ is a model parameter controlling the spatial correlation, I is the identity matrix, and Q0 defines the neighborhood structure, which is described as 4Q0i,j=number of neighbors of nodei,ifi=j,-1,ifi∼j,0,otherwise. In our framework, the neighborhood structure is built with a 4-nearest-neighbor structure (each grid cell connects to its north, south, east, and west neighbors). The notation i∼j indicates that the grid cells i and j share an edge (are directly adjacent to that 4-neighborhood). The precision matrix Q2 related to the autoregressive process of order 1 is defined as 5Q2=1-ρ-ρ1+ρ2⋱⋱⋱⋱⋱1+ρ2-ρ-ρ1 where ρ is a parameter that controls the temporal correlation assumed to be between -1 and 1 to ensure that the process is stationary. In summary, the full model parameters vector θ=(σobs,σp,μ,ϕ,ρ) and the hidden states Htrue are estimated by the joint likelihood function 6LHobs,Htrue,θ=LHobs,θLHtrue As the hidden states Htrue are unobserved, the joint likelihood cannot be calculated directly. Instead, we form the marginal likelihood function, which is only dependent on the fixed model parameters and the observation. 7LMHobs,θ=∫LHobs,Htrue,θdHtrue. The 3D model of ice elevation anomalies is implemented in the open source software R using the Template Model Builder (RTMB) . In RTMB, the marginal likelihood is derived via the Laplace approximation. In short, this is a second-order Taylor approximation of the logarithm of the function to be integrated around the optimum. In practice, this will change an integral into an optimization problem.

The GMRF captures both the spatial coherence and temporal evolution of elevation changes. This joint formulation allows efficient and realistic reconstruction of SEC over the Greenland Ice Sheet, even in areas with irregularly distributed data. RTMB also reports the model uncertainty, which is approximated via the generalized delta-method, and the variance is expressed as 8varH^true=H2-1+(∇H^trueθ)tH1-1∇H^true Here H2 is the block of full Hessian matrix related to the hidden states, H1 is the Hessian matrix related to the parameters, and ∇H^true are the gradients of the hidden states with respect to the model parameters θ . The generalized delta-method approximates the uncertainty of the reconstructed field by propagating parameter uncertainty through a first-order Taylor expansion around the optimum.

To reduce residual high-frequency variability not fully captured by the statistical model and enhance the spatiotemporal coherence of the SEC dataset, we subsequently apply a two-step postprocessing of the monthly dSEC gridded fields. The trends of both the raw (pre-smoothing) and the smoothed dSEC grids will be evaluated separately in Sect. . First, each monthly elevation map is smoothed with a median spatial filter with a window size of 5 × 5 gridcell to suppress small-scale noise while retaining the larger-scale elevation signals. Secondly, for temporal consistency, a Savitzky-Golay filter is applied along the time axis. For the seasonal component, we use a 17-month window, which is longer than the annual cycle and therefore provides a smoothed and damped representation of the seasonal signal while removing month-scale irregularities. To extract the multi-year trend, we use a 41-month window, which effectively isolates the long-term trends of SEC. This procedure efficiently suppresses noise and outlier-driven seasonal spikes, while preserving the large-scale spatial patterns and long-term elevation-change signals.

The grid-cell uncertainties are first obtained from the state-space model as posterior standard deviations derived from the RTMB Hessian. These uncertainties are then propagated through the full post-processing chain using a Monte Carlo approach, in which Gaussian perturbations consistent with the model estimated variance are added to the elevation fields and passed through the same spatial and temporal post-processing chain. The final uncertainty at each grid cell and epoch is computed as the standard deviation across ensemble realizations of the filtered product.

For regional averages of SEC (e.g., Greenland-wide, interior, and ablation-zone means shown in Fig. ), grid-cell values were combined using a latitude-dependent area-weighted mean, with weights proportional to cos⁡(ϕ), where ϕ is the latitude of each grid cell. The uncertainty of each regional mean was propagated from the individual grid-cell uncertainties using standard error propagation for a weighted mean. Spatial correlations between neighbouring grid cells were not explicitly included in this uncertainty propagation, and the resulting regional uncertainties may therefore underestimate the true uncertainty.

Using the described 3D-ECM model framework, we computed the monthly SECs across the GrIS at a 5 km resolution using radar satellite data from the CryoSat-2 L2 data product. A spatial map of the average SEC from 2011–2025 produced by our 3D-ECM is presented in Fig. , with time series of elevation change at selected locations in Fig. . Widespread thinning is observed along the margins and generally stable or slightly thickening conditions in the ice sheet interior (in the following defined as areas above 1500 m elevation based on ArcticDEM). The average trend in SEC over the entire GrIS is -0.069 ± 0.002 m yr⁻¹, as illustrated in Fig. , whereas the trend in the ablation zone (defined as areas below 1500 m elevation based on ArcticDEM) is -0.314 ± 0.009 m yr⁻¹. The shaded regions in Fig.  represent the uncertainty of the mean elevation change at each timestep, derived from the grid-cell uncertainties estimated by the state-space model and propagated through the spatial averaging. As the time series represent elevation anomalies at each timestep rather than cumulative changes, the uncertainties do not accumulate over time.

To assess regional variability and temporal evolution, we extracted time series from six locations across the Greenland Ice Sheet (Panels A–F in Fig. ). These sites are representative of dynamically active outlet glaciers: Petermann, Nioghalvfjerdsbræ (79° N Glacier; hereafter 79° N), Zachariae, Sermeq Kujalleq, and Helheim, as well as a site representative of the relatively stable interior region at Summit Station.

The time series are extracted from single grid cells, with shaded areas in Fig. representing the uncertainty estimated from the 3D-ECM model. Derived from the smoothed CryoSat-2 dSEC product, they reveal spatially varying elevation change patterns from 2011 to 2025. The Summit site, located within the interior zone (> 1500 m elevation), shows a modest thickening trend of 0.017 ± 0.001 m yr⁻¹ over the 14-year period, close to the negligible average trend for the interior ice sheet of -0.014 ± 0.002 m yr⁻¹.

All marginal sites exhibit pronounced thinning. Sermeq Kujalleq experiences the strongest thinning of -1.7 ± 0.074 m yr⁻¹, exceeding the mean trend of the ablation zone (-0.314 ± 0.009 m yr⁻¹). The maximum thinning at Sermeq Kujalleq occurs during the early part of the record, between 2011 and 2018.

The CryoSat-2 Level-2 altimetry data contain a spatial data gap over the fast-flowing part of Sermeq Kujalleq, where steep topography and complex radar scattering hinder reliable surface elevation retrievals, as seen in Fig. , which provides a detailed example for the Sermeq Kujalleq, showing dSEC for three grid cells located inside and outside the persistent data gap for CryoSat-2 data. Grid cells B and C are located within the persistent CryoSat-2 data gap, while A is situated further inland in an area with higher data density. The associated uncertainties differ among the three points, reflecting the varying data density, with higher uncertainties in areas of sparse CryoSat-2 coverage and lower uncertainties inland where data are more abundant. Because CryoSat-2 data density in this region is relatively consistent over time per grid cell, and the posterior variances are propagated through the same spatial and temporal filtering as the model output, the resulting uncertainties remain relatively constant between months. These time series reveal distinct seasonal cycles and interannual variability, demonstrating the ability of this approach to resolve fine-scale temporal changes in rapidly evolving glacier systems.

Our results illustrate how our data-driven 3D-ECM framework can advance the monitoring of ice-sheet surface elevation from satellite radar altimetry. By integrating radar altimetry observations with a spatio-temporal statistical model, we obtain continuous, monthly estimates of dSEC over the Greenland Ice Sheet, including regions where satellite sampling is sparse or irregular. The combination of a GMRF representation of spatial dependence with an autoregressive temporal component allows the model to isolate meaningful elevation-change signals, while additional post-processing of the radar data further reduces high-frequency noise.

This capability has important implications for interpreting both seasonal variability and interannual trends in ice sheet surface elevation change. It also highlights the potential to unify data from different altimetry missions and sensor types. Although here we focus on CryoSat-2, the same framework could be applied to past and future radar altimeters, enabling more consistent long-term monitoring across satellite generations. The framework can also be applied to laser altimetry missions such as ICESat-2, although in this study ICESat-2 is used solely as an independent validation dataset.

Uncertainty patterns in the dSEC product highlight the dependence of model performance on data density. Figure  shows how the uncertainty estimates of the dSEC product are higher in grid cells B (7.1 m ± 1.3 m) and C (7.6 m ± 0.34 m), located within the CryoSat-2 data gap, compared to A (4.8 ± 1.6 m), which is situated in an area with a higher data density. Here, the first value denotes the average uncertainty across the time series, while the ± value represents the temporal variability (standard deviation) of the uncertainty estimate. As expected, model uncertainty, and therefore also propagated uncertainties, increases with distance to available data points; nevertheless, the 3D-ECM reconstructs coherent temporal patterns even in poorly sampled regions. For example, cells B and C capture a pronounced thinning from 2011–2016, consistent with previous studies , followed by reduced thinning, and reproduce the surface build-up observed between 2019 and 2021 in the trough of the Sermeq Kujalleq Ice Stream , despite lying within an almost persistent data gap.

The monthly dSEC product is based on satellite radar altimetry and inherits its sensitivity to changes in surface scattering properties, i.e., the way the radar signal interacts with the surface before it is reflected to the satellite. For instance, an increase in surface roughness will broaden and flatten the returned signal (the waveform), making the identification of the surface reflection more difficult in that waveform . Over polar snow and firn (cold, dry conditions), CryoSat-2’s Ku-band (13.6 GHz) radar typically penetrates up to about 1–2 m . Variations in volume scattering within that depth range (changes of snow density or grain size) can affect the returned waveform and distort surface elevation estimates . Likewise, the appearance of an ice layer within that penetration depth may cause a waveform peak at that horizon, which is then interpreted as the surface . Such effects introduce elevation biases that cannot be fully eliminated by statistical smoothing. Consequently, interpretation of dSEC trends must consider changing surface conditions, particularly in regions experiencing melt, refreezing, or firn densification changes.

Independent validation against ICESat-2 ATL15 and OIB shows that post-processing improves quantitative agreement. Compared to the raw reconstruction, the smoothed dSEC product reduces the annual standard deviation of differences by 40 %–45 % and lowers the median absolute deviation, indicating more stable and spatially coherent elevation-change estimates.

The intercomparison with published SEC products further contextualizes these results. Relative to existing monthly SEC products, the 3D-ECM framework shows reduced spread of differences against ICESat-2 ATL15, suggesting that the joint spatio-temporal modeling provides a more coherent reconstruction of elevation change fields. Additional spatial and temporal comparisons presented in Figs. S2–S4 support this interpretation. The largest trend differences are concentrated in dynamically active outlet-glacier regions, whereas agreement is generally stronger across the interior ice sheet.

Station-specific comparisons with AWS records and the altimetry-based SERAC product further illustrate how local glaciological conditions affect dSEC interpretation. At JAR (close to Sermeq Kujalleq), dSEC correlates moderately with AWS (r= 0.58) and more strongly with SERAC (r= 0.76). The weaker agreement with AWS likely reflects dynamic ice-thickness changes near the fast-flowing outlet glacier Sermeq Kujalleq that are not captured by surface instruments, consistent with observations of strong velocity variability in this region . In contrast, KPC_U is located on an ice lobe north of the Northeast Greenland Ice Stream where seasonal dynamics are minimal. Here dSEC, AWS, and SERAC show strong agreement (all r> 0.72; AWS-SERAC r= 0.92). KAN_M, in the southwest percolation zone, exhibits high correlation between dSEC and both AWS and SERAC but a lower correlation between AWS and SERAC themselves, consistent with the complex interplay of firn compaction, meltwater infiltration, retention, and refreezing in this region. Notably, both dSEC and SERAC record a shift toward more positive trends around 2020 at KAN_M (Fig. ), suggesting that both datasets capture the same seasonal evolution despite these challenging surface processes.

Taken together, these comparisons show that the dSEC product agrees well with both the AWS records and the altimetry-based SERAC product, indicating that the 3D-ECM framework captures physically meaningful elevation change signals consistent with independent observations. However, these examples also emphasize that while the 3D-ECM framework successfully extracts coherent elevation change patterns, interpretation at the local scale must account for the diverse physical processes that govern different climatic and glaciological regimes across Greenland.

The 3D-ECM framework integrates radar altimetry observations within a coherent spatio-temporal reconstruction, enabling the generation of spatially and temporally consistent, gap-filled SEC time series across the Greenland Ice Sheet. An advantage of the 3D-ECM framework is its flexibility, which is applicable across multiple satellite missions and enables the production of continuous, gap-filled SEC time series at varying spatial and temporal scales. Importantly, the framework is entirely data-driven: seasonal variability emerges directly from the observations through the joint spatio-temporal estimation, rather than being prescribed using a predefined harmonic (sine-cosine) formulation. In several existing SEC approaches, seasonality is represented using fixed parametric functions that implicitly assume a stationary and symmetric seasonal cycle with constant amplitude and phase, as in and . In contrast, the present framework does not impose a parametric seasonal model, the 3D-ECM framework allows for non-sinusoidal seasonal shapes as well as temporal changes in amplitude and phase. This flexibility is particularly relevant for Greenland, where seasonal elevation changes vary regionally and interannually in response to evolving melt and accumulation conditions. A potential development for future enhancement of the spatial resolution of the dSEC product is the use of swath-processed CryoSat-2 data instead of the Level-2 POCA dataset, due to the higher data density of the swath-processed CryoSat-2 data.

To place the reconstructed dSEC fields in the context of existing community datasets, we compared the product against two published community SEC products using ICESat-2 ATL15 as a reference over the common spatio-temporal domain (Table ). The comparison shows that the dSEC product yields a lower RMSE relative to ATL15 than the evaluated community products and exhibits a reduced spread of differences compared to . The relatively small MAD values for all products suggest that the central tendencies of the distributions are similar, while differences in RMSE and standard deviation mainly reflect variations in the spatial and temporal coherence of the reconstructed SEC fields. Overall, this comparison indicates that the spatio-temporal state-space framework produces SEC estimates that are consistent with independent laser altimetry and comparable to, or slightly improved relative to, existing community SEC products.

The 3D-ECM framework relies on several modeling assumptions that should be considered when interpreting the dSEC product. The spatio-temporal structure combines a Gaussian Markov Random Field in space with a first-order autoregressive process in time, imposing local spatial dependence and exponentially decaying temporal correlations. While computationally efficient for ice-sheet-scale reconstruction, this formulation may not fully capture long-range spatial correlations or complex glacier dynamics. Observational errors are assumed to be Gaussian and uncorrelated, although radar altimetry can exhibit spatially and temporally correlated errors that are not explicitly modeled; uncertainties therefore represent posterior uncertainty under the assumed model structure.

The state-space model is designed to reconstruct the dominant spatial and temporal evolution of surface elevation change, but it does not explicitly separate high-frequency variability from the underlying signal. Consequently, short-term fluctuations associated with measurement noise, irregular sampling, or transient processes may remain present in the reconstructed fields. To derive the final monthly dSEC product, an additional post-processing step consisting of spatial median filtering and temporal Savitzky-Golay filtering is therefore applied to suppress high-frequency noise and improve spatial coherence. While this substantially improves agreement with independent validation datasets, it may also smooth some short-term variability.

In the revised framework, uncertainties are propagated through this post-processing using a Monte Carlo approach, such that the reported 1σ uncertainty reflects the fully filtered product. For computational efficiency, the model is implemented in overlapping spatial tiles, which may introduce subtle edge effects. Despite these limitations, the 3D-ECM provides a flexible and computationally efficient framework for reconstructing coherent monthly elevation-change fields from irregularly sampled altimetry observations, balancing model complexity, interpretability, and computational feasibility at ice-sheet scale.

We have derived monthly dSEC across the Greenland Ice Sheet for the period 2011–2025 from CryoSat-2 radar altimetry. The state-space model combines a Gaussian Markov Random Field for spatial dependencies with an autoregressive temporal process, allowing seasonality, long-term trends, and abrupt changes to emerge directly from the data without prescribing functional forms. The reference surface was taken from ArcticDEM, and the analysis was restricted to the Greenland Ice Sheet mask defined within the IMBIE framework , excluding peripheral glaciers and ice caps not dynamically connected to the main ice sheet.

The resulting dataset captures both broad spatial patterns and local outlet-glacier variability, while providing consistent point-based time series across the ice sheet. Independent validation confirms the robustness of the approach as comparisons with ICESat-2 ATL15 and NASA Operation IceBridge show strong agreement in long-term trends, with spatio-temporal smoothing reducing spread while maintaining minimal bias. Seasonal amplitudes from AWS and SERAC time series also align closely with the linearly detrended dSEC product, demonstrating that the method resolves both interannual and seasonal variability.

Intercomparison with two published community SEC products further shows that the dSEC reconstruction achieves comparable or lower misfit relative to ICESat-2 ATL15, indicating that the spatio-temporal state-space framework provides consistent SEC estimates relative to existing community datasets.

Beyond the Greenland and CryoSat-2 application, the modeling framework is directly transferable to other altimetry missions and multi-sensor combinations. Its computational efficiency enables long-term monitoring of polar ice sheets and provides a pathway for integrating upcoming missions such as CRISTAL into multi-decadal elevation-change records.

In summary, the presented dSEC dataset delivers monthly, high-resolution, and uncertainty-quantified estimates of SEC across the Greenland ice sheet. The approach provides temporally consistent monthly fields from irregularly sampled radar altimetry observations, support future assessments of ice-sheet mass balance and dynamics and provide a foundation for continuous, multi-mission monitoring of the cryosphere.