The net Arctic sea-ice area (SIA) can be estimated from the sea-ice concentration (SIC) by passive microwave measurements from satellites. To be a truly useful metric, for example of the sensitivity of the Arctic sea-ice cover to global warming, we need, however, reliable estimates of its uncertainty. Here we retrieve this uncertainty by taking into account the spatial and temporal error correlations of the underlying local sea-ice concentration products. As 1 example year, we find that in 2015 the average observational uncertainties of the SIA are 306 000 km

In this study, we quantify the uncertainty of total sea-ice area (SIA) and sea-ice extent (SIE) of the Northern Hemisphere. The former is calculated as the sum of the individual sea-ice areas in all Northern Hemisphere grid cells of a gridded product, while the latter is the sum of the grid-cell area of all grid cells with at least 15 % sea-ice concentration. We calculate the uncertainty of these two metrics by taking into account the spatial and temporal error correlations for propagating uncertainties from the local to the Arctic-wide level for the ESA Sea Ice Climate Change Initiative Sea Ice Concentration (CCI SIC) Climate Data Record version 2.1

The local-area fraction covered by sea ice, called sea-ice concentration (SIC), can be inferred at a resolution of the order of tens of kilometres from passive microwave radiometers on board several satellite missions since the 1970s. These SIC estimates do not depend on daylight, have a small sensitivity to atmospheric conditions, and cover most of the polar oceans on a near-daily basis. Several passive microwave SIC products exist, and they are valuable tools for many aspects of climate-related science, including operational weather forecasts

Based on an analysis of these long-term records, we know that the Arctic sea-ice cover is significantly declining in all seasons

Other types of satellite measurements are used to derive the SIC, such as from near-optical sensors and synthetic aperture radar (SAR) sensors. These can, under favourable conditions, be of higher quality than passive microwave SIC products

The uncertainties in SIA and SIE investigated here stem from uncertainties in the underlying SIC fields. Passive microwave SIC estimates in regions of consolidated ice have typically smaller uncertainties (2 % to 8 % SIC) than estimates from low to intermediate SIC areas with uncertainties in the order of 20 % SIC or more

The impact of atmospheric interference and roughening of the ocean from wind exposure near the ice edge has been highlighted in

Microwave emissions of wet snow, wet ice, and melt ponds on top of sea ice resemble the emissions of open water more closely than cold, dry snow or ice, which can lead to misclassifications and, hence, be a major source of uncertainty for summer melt conditions

Smearing effects become important where SIC values vary on scales close to the measurement footprint, for example in the marginal ice zone

Algorithmic uncertainties result from all the decisions taken within the SIC product development, the frequency bands used for a product, the type of algorithm, the corrections for the types of interference (see above), the auxiliary data (e.g. land mask), and the parameter values therein (e.g. thresholds and correction factors). The impact of these potential error sources on the accuracy of the estimated sea-ice extent is part of the investigations made by

For our investigation, we use the CCI SIC product that has generally one of the most advanced uncertainty estimates among available products. This uncertainty estimate attempts to represent the four types of sources of uncertainty described above; however, some additional sources of uncertainty cannot be taken into account. Any physical process leading to a bias in the SIC cannot be adequately taken into account by most of the common uncertainty estimates, including the one from the CCI SIC product. Those underrepresented processes include melt ponds and the influence of weather, despite respective corrections; the underlying land mask; misclassified ice types at the so-called tie points; and a possible unaccounted increase in tie-point emissions from wet snow

The knowledge of inter-SIC-product biases in SIA and SIE is crucial; however, it is not suitable as the sole metric to estimate and communicate SIA and SIE uncertainty. The inter-product differences contain biases, for example from different land masks, which increase the perceived uncertainty and require, in practice, a different treatment than random errors. While biases in SIC products lead to large perceived uncertainties from product inter-comparisons, other uncertainties are not represented at all, such as common algorithmic assumptions or errors in the commonly used passive microwave data sets.

To overcome these limitations, we estimate here the uncertainty of a single SIA product based on the uncertainty of the underlying SIC fields. This approach complements the inter-comparison across various products because it is based directly on the local SIC uncertainty estimates. Our SIA and SIE uncertainty estimates can accompany the whole product lifespan and can evolve with changes in product quality and SIC conditions. By representing temporal error correlations, we can quantify the reduction in uncertainties from temporal averaging.

If supplied at all, SIC uncertainties are kept on a grid cell level by the data providers. The analytical propagation of these uncertainties to the aggregated measures of SIA and SIE is challenging due to spatial and temporal correlations and due to the application of thresholds (for SIE) on SIC fields with dependent uncertainties and computational limits when a full correlation matrix needs to be used. This is because even the correlation matrix of 1 month with a 50 km resolution SIC product would have more than 1 trillion entries, which clearly exceeds typical computational memory resources.

To overcome these issues, instead of an analytical uncertainty propagation, we use a Monte Carlo approach here and derive an ensemble of SIC estimates that possess correlated error fields. For this approach, it is crucial to distinguish between errors and uncertainties: an

In the following, we address these criteria, starting in Sect.

The error correlations are assumed to be constant in space and time, with one characteristic length scale in space and one in time. Correlations are therefore assumed to solely depend on the (space–time) distance between two locations.

The estimate of the spatial correlation length scale used here is based exclusively on the data in

The total error correlation length results from the described processing when using the total_standard_error variable (renamed as total_standard_uncertainties in newer versions) of the CCI SIC product. The total error correlation length therefore describes whether the amplitude of uncertainties is correlated but not whether the errors that these uncertainties describe are correlated. An example for this would be a process which creates independent noise on a persistent spatial scale. In this case, the amplitude of the noise would have a typical length scale, but the errors would nevertheless be uncorrelated. The total_standard_error variable is largely based on the maximum SIC minus the minimum SIC of a moving

The sea-ice concentration error correlation length (hereafter the SIC error length), in contrast, results from the described processing when using the raw SIC values themselves, including values outside of the [0 %, 100 %] range. Analysing these untruncated SIC values shows that they regularly reach up to 110 % SIC, indicating that product SIC errors, even for pack ice regions, can be of the order 10 % SIC, since SICs above 100 % are physically impossible. By assuming a symmetric error distribution, it follows that all SIC values above 90 % can originate from fully ice-covered regions, which informed the

Spatial

In contrast to the spatial correlation length scales, we estimate the temporal error correlation length (i.e. the duration) ourselves but largely follow the processing of the spatial SIC error length in

We use the previously identified spatial and temporal error correlations to create a Monte Carlo ensemble in order to propagate the CCI SIC uncertainty estimates to the SIA and SIE estimates. The spread within the SIC ensemble represents its correlated uncertainties. Therefore, the ensemble spread in the resulting SIA or SIE estimates provides an estimate of the propagated uncertainty.

The generation of ensemble members with correlated random errors consists of the following four steps. (1) Independent white noise with zero mean and a standard deviation of 1 is generated in the whole domain by a numerical random generator. The noise is generated for the whole time period and hemisphere at once to avoid discontinuities in the final error fields. (2) A three-dimensional Gaussian low-pass filter with sigma values of 5 d in the time dimension and 288 km in the two space dimensions (compare Fig.

To examine the quality of the generated SIC field, we need to examine two questions describing the two basic quality measures mentioned before. (1) How well is the local spread within the ensemble reflecting the uncertainty estimates of the CCI SIC product? (2) How well does the generated ensemble reproduce the spatial and temporal error correlation characteristics of the original product? If both criteria are met, then we have shown that our synthetic errors are a good approximation for the inherent product errors.

To examine whether the first quality measure is met, we compare the spread in the generated ensemble with the uncertainty estimate by the data providers (Fig.

Example of

Comparison of the error correlation distributions of the CCI SIC product with the statistically generated ensemble. The shown spatial distributions

The spatial and temporal error characteristics of the statistically generated ensemble members are similar to the ones of the original CCI SIC product (Fig.

In summary, we have generated a statistical ensemble of SIC time–space fields, which are centred around the CCI SIC product while the added noise is in excellent agreement with the local CCI SIC uncertainty estimates and the estimated temporal and spatial error correlations. In the following, we can therefore use the generated ensemble to quantify uncertainties in the SIA and SIE.

In order to derive SIA and SIE uncertainties, we repeat the calculation of daily SIA and SIE values for each ensemble member individually (Fig.

Daily Arctic SIA

In a next step, we derive the weekly mean and monthly mean SIA and SIE estimates from the daily time series and afterwards calculate the corresponding ensemble SD (Fig.

This example shows that weekly averages have nearly the same uncertainty as daily estimates, which is due to the typical temporal error correlation of 5 d. In general, the uncertainty of daily SIA and SIE values strongly depends on the spatial error correlation, while the amount of uncertainty reduction by temporal averages is determined by the temporal correlation (see the Appendix, Fig.

We address the sensitivity of our uncertainty estimates on the SIC error correlation length scales by repeating the ensemble generation for realistic lower-end and upper-end correlation length scales. The difference in daily uncertainty between these two setups is about 80 000 km

Of particular interest both scientifically and in the public discussion is the trend in September SIA and SIE because September is the month that typically contains the yearly sea-ice minimum and is one of the months with the fastest observed decline in sea ice

The uncertainties studied here can be understood as an improved representation of the total_standard_error variable provided with the ESA CCI sea-ice concentration product. We fully rely on the experience and extensive validation efforts of the data providers to quantify the local uncertainties

The assumption of homogeneous and purely radial error correlations is of course a simplification: some sources of errors are expected to play a stronger role in specific conditions. This includes the land spill over, which originates from a strong contrast between microwave signatures from the land and the ocean, while the signatures of the land and sea ice are very similar. The passive microwave sensors permit a blurring of this sharp contrast, leading to a contamination of measurements near the coast with land signatures

Additional error sources with likely non-circular error correlations are tie-point errors. Errors in tie points are expected to create errors at all locations with conditions similar to the tie-point conditions. Therefore, error correlations are expected to be higher between locations with high sea-ice concentrations and between locations with low sea-ice concentrations. In other words, since the ocean and sea-ice tie points are defined independently of each other, one would expect the errors in an ocean measurement to be less correlated with errors in sea-ice measurements, everything else being equal.

Despite these caveats, we use a circular correlation pattern in this study, based purely on the distance between two locations. The existence of a non-circular correlation pattern is further supported, for example, by Fig. 5 in

Another assumption we rely on is that the error characteristics derived from nearly 100 % SIC are applicable for all ice conditions. A similar analysis for intermediate SIC is not possible because variations in real SIC and SIC errors cannot be distinguished. For conditions close to 0 % SIC at locations close to the ice edge, the approach used here and in

We compare two trend uncertainty estimates of different natures: the traditional standard error in the trend, which is often provided with linear regressions, is based exclusively on the SIA values and is driven by the length of the time series and measurement-to-measurement variability. Our estimate is representing the measurement uncertainty, based on the propagation of the total_standard_error variable and is therefore not representing the influence of inter-annual variability. We realize that it can be confusing to use the same notation for different kinds of trend uncertainties and propose the use of the “measurement trend uncertainty” for the type of uncertainty produced here and the “standard error in the trend” or, more generally, the “trend fitting uncertainty” for the type of uncertainties based exclusively on the SIA values. Our measurement trend uncertainty here is less sensitive to the decision to fit the trend to daily or monthly mean SIA values (9151 vs. 9042 km

For a specific SIC data set, the work presented here looks at the year 2015 for a continuous time series and data from 2002–2017 for the September trend analysis (CCI SIC at 50 km resolution). It demonstrates how error estimates can be supplied for SIE and SIA estimates and for their trends. In a next step, our method can readily be implemented for sea-ice indicators on a daily basis by operational services such as the EUMETSAT OSI SAF. The implementation of our method by data providers could allow them to provide error estimates not only to daily and monthly mean SIE and SIA time series but also to set confidence intervals for other widely used metrics. Such metrics include the trends in monthly SIE and SIA (typically September and March) as well as rank values such as record low/high and earliest/latest sea-ice extremes. Such an implementation could, thus, increase the maturity of these key climate indicators. We further hope that this work will inspire the development of more sophisticated SIC error correlation estimates to refine SIA and SIE uncertainty estimates and improve the SIC ensemble from different SIC algorithms and new applications. If regional error characteristics are sufficiently well represented, then the SIC ensemble could be used directly in regional coupled models to investigate the impact of correlated SIC uncertainties in oceanic and atmospheric surface fluxes.

An analysis of errors in the CCI passive microwave SIC product indicates typical error correlations of around 300 km in space

The Arctic September SIA trend for the period from 2002 to 2017 is estimated to be

Using a simple (spatial- and temporal-)distance-based correlation model to propagate the uncertainties in the underlying SIC fields to uncertainties in the key climate indicators of SIA and SIE and their trends, we have been able to improve our understanding and the quantification of these uncertainties. We expect this quantification of observational uncertainties to be essential for our understanding of the ongoing Arctic climate change, both as a means in themselves and in providing a more robust basis for model evaluation studies.

The SIA and SIE uncertainty estimates are strongly dependent on the error correlation length scales. While we attempt to constrain the error correlation length as well as possible, there is some ambiguity in the best representation of these correlations. To investigate the impact of uncertainties in the error correlations, we test the sensitivities with regards to several aspects of the processing.

Left: daily Arctic SIA and SIE ensemble of 20 SIC ensemble members with 1 member highlighted (red) and the mean (black). Right: standard deviations of SIA and SIE derived from an ensemble of 100 SIC ensemble members with only the (top) spatial error correlation and (bottom) temporal correlation.

First, we set the spatial (temporal) error correlation to zero and use our best estimate for the temporal (spatial) error correlation. In this way, we separate the impacts of the spatial and temporal correlations from each other (Fig.

Second, we define error correlations on the lower and upper end of consistency with observations. Based on Fig.

Left: daily Arctic SIA and SIE ensemble of 20 SIC ensemble members with 1 member highlighted (red) and the mean (black). Right: standard deviations of SIA and SIE derived from an ensemble of 100 SIC ensemble members with (top) lower-end and (bottom) upper-end spatial and temporal error correlations.

Uncertainty estimates (1 SD) of 100 ensemble members for the year 2015 based on different processing types. The spatial (Sp) and temporal (Tmp) error correlations are nominal values and do not necessarily correspond to the averaged analysed error correlations (see text).

To assess the sensitivity to the filter type, we use two alternative filters to create the noise ensemble: a fast Fourier transform (FFT) and a wavelet filter

Left: daily Arctic SIA and SIE ensemble of 20 SIC ensemble members with 1 member highlighted (red) and the mean (black). Right: standard deviations of SIA and SIE derived from an ensemble of 100 SIC ensemble members based (top) on a wavelet filter and (bottom) on a fast Fourier transform filter.

The same as Fig.

A wavelet transformation is a multi-resolution decomposition of an

The filtering steps are the same for FFT and wavelet filters: first, the three-dimensional white noise field is decomposed into its frequency components. Then, frequency components/wavelet coefficients outside of a manually defined window are set to zero, followed by a reverse transformation/recomposition into the original space–time domain. For the FFT bandpass filter, the range is set from 238 to 338 km and from 4 to 6 d. The wavelet filter decomposes the noise field into four levels using a discrete wavelet transform with Daubechies-12 wavelets (function

The original SIC data are available from the ESA CCI website and the CEDA Archive

AW designed the study, conducted the analysis, and wrote the first version of the manuscript. DN advised on the study design, SK helped, in particular, with the use of spatial correlation estimates and processed data specifically for this study. TL advised on the CCI uncertainty estimates. All authors engaged in discussions, which where absolutely essential for this project, and in manuscript revisions.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors are thankful to the two anonymous reviewers.

This work was supported by the ESA Climate Change Initiative CMUG project. Andreas Wernecke, Dirk Notz, and Thomas Lavergne also acknowledge the support given by the ESA Climate Change Initiative Sea Ice project (contract no. 4000126449/19/I-NB), and Dirk Notz acknowledges the funding by the Deutsche Forschungsgemeinschaft (DFG) under Germany's Excellence Strategy – EXC 2037 “CLICCS – Climate, Climatic Change, and Society” (project no. 390683824). The article processing charges for this open-access publication were covered by the Max Planck Society.

This paper was edited by John Yackel and reviewed by two anonymous referees.