In this study, we investigate an original approach combining data assimilation and machine learning to correct past model estimations of sea ice thickness using present observations. While ground truth observations offer unparalleled accuracy, they lack global coverage, contrasting with remote sensing observations that, although global, are associated with large uncertainties due to necessary assumptions for estimation. At present, the best estimation is commonly obtained by integrating remote sensing observations into models to reduce their biases. However, this approach relies on the availability of observations and, as a result, cannot help retrieve historical sea ice thickness. Studies focusing on long-term evolution, particularly those oriented toward climate research, demand extensive and accurate time series of sea ice thickness, given the essential role that sea ice plays as the interface between the ocean and the atmosphere.

Arctic sea ice acts as a multifaceted and vital interface between the ocean and the atmosphere, playing a major role in regulating energy exchange, reflecting sunlight, and influencing local weather patterns. Sea ice significantly influences marine ecosystems, providing habitat and migration routes for diverse species . As sea ice melts, it injects freshwater into the ocean, affecting salinity levels and exposing the ocean to the atmosphere. Moreover, as Arctic sea ice extent is declining due to warming , the Arctic is becoming more navigable, opening up new opportunities for maritime transportation and resource exploitation but also raising concerns about environmental impacts and sustainable management of the region's fragile ecosystems . Notably, the thickness of Arctic sea ice stands as a major unknown quantity, as thicker ice, usually older and deformed, resists melting and mechanical stresses better. Its variations are intricately tied to the heat and freshwater budget, the sea ice dynamics, and the ecosystem.

The current deficiency in a comprehensive and accurate climate record for sea ice thickness (SIT) is attributed to the sparse availability of SIT observations and the relatively recent integration of satellite technology. Although SIT observations have been taken in situ and by no fewer than five satellites, they generally suffer from severe representativity issues and high uncertainties and lack both the temporal and spatial continuity that long-term climate studies need. Consequently, model reanalyses of Arctic SIT diverge substantially and lack credibility. An extended reconstruction of Arctic sea ice thickness, along with its uncertainty estimates, is essential to unlocking investigations of the Arctic climate, including heat budgets , freshwater fluxes , and its ecosystem .

Physical-based sea ice models (e.g. ) can simulate reasonable sea ice thickness, yet SIT biases in numerical models remain significant, originating from various factors including external components like atmospheric or ocean fluxes and internal aspects intrinsic to the model itself. Intercomparisons of SIT between state-of-the-art models thus exhibit large deviations from one model to the next in terms of spatial distribution . Similarly, large deviations are observed when comparing satellite products or diverse in situ datasets, mostly due to differences in spatial and temporal coverage and in data processing methods, such as retracking algorithms for satellite altimeters (e.g. ).

Since 2010, the merged remote sensing product CS2SMOS, combining data from Soil Moisture and Ocean Salinity (SMOS) and CryoSat-2 (CS2) for thin and thick ice, respectively, provides continuous SIT every winter , yet longer time series are required to conduct climate studies. Assimilating CS2SMOS data in the coupled ocean–sea ice model TOPAZ corrects a low SIT bias of roughly 16 cm, reducing average root-mean-square errors from 53 to 38 cm and further to 20 cm in March . Can we extrapolate the benefits of data assimilation to past periods without SIT observations? introduced a method to combine data assimilation (DA) with machine learning (ML) to build a hybrid numerical model. The present study applies this approach to rewind a climate record.

Machine learning has advanced to a point where it can effectively address the high dimensionality, complexity, and nonlinearity inherent in dynamical systems , especially when combined with DA . Recent investigations have demonstrated the potential of machine learning for sea ice, focusing on various objectives such as parameterizing subgrid-scale dynamics , emulating sea ice melt ponds , or skilfully predicting DA increments of sea ice concentration across all seasons . In the present study, our assumption is that a suitable compression of the variables at play (e.g. via empirical orthogonal function, EOF) identifies the complex nonlinear relationships between physical variables without altering them .

In the present investigation, we train a machine learning algorithm to learn the systematic SIT errors between two versions of the model TOPAZ4 over 2011–2022, with (TOPAZ4-RA) and without (TOPAZ4-FR) CS2SMOS assimilation. Then, we use the algorithm to predict the SIT error and extend the SIT estimates to periods before 2011 (Fig. ). For this work, the training period (2014–2022) supports algorithm development and includes a validation period (20 % of the training period, in chronological order without randomization) to optimize hyperparameters. The test period (2011–2013) enables us to verify our algorithm performance using the data held specifically for this purpose. The evaluation period (1992–2010) allows us to assess the ML-adjusted SIT, called TOPAZ4-ML, compared to independent datasets.

Section 2 describes various datasets and the model TOPAZ4. Section 3 further explains the method used to combine DA and ML. Section 4 presents the results and evaluation of the ML algorithm, as well as an assessment of the extended SIT time series with independent datasets, and highlights unprecedented outcomes from this brand-new product. Section 5 discusses the limitations of and uncertainties in this investigation.

Our adjustment method is applied as a post-processing operation during the TOPAZ4 free run, dependent on the state of the sea ice but also the external forcing variables. The approach is based on the empirical orthogonal function (EOF) decomposition to reduce the dimensionality of our problem. This data compression enables us to apply a substantial adjustment that requires minimal computational resources and remains unaffected by static geographic features, such as the coastlines. During the training period (2014–2022), we compute the empirical orthogonal functions (EOFs; spatial component of the statistical patterns) and associated principal components (PCs; temporal evolution of the statistical patterns) of the SIT biases. Applying the method outside of the training period assumes that this EOF decomposition is stable back in time. Consequently, the EOFs of the SIT biases are assumed to be invariant, while the target variables to predict are the PCs of SIT biases back in time (Fig. ). The increments from data assimilation give the best estimates of SIT biases, and we learn to emulate these increments, similarly to what is done in, for example, or . Using the increments rather than the innovations means that the algorithm can be used with irregular observations, while the data assimilation takes care of their interpolation. Likewise, each input feature (listed in Table ) is decomposed independently using either eight EOFs (sea ice thickness and age) or four EOFs (all other variables). At first, 14 a priori relevant features are used as inputs, and then an arbitrary threshold of the variable importance enables the adequate selection of the best-suited variables. The input features are provided to the algorithm at different time lags (in d): t-30, t-7, t0, t+7, and t+30. Considering that using eight components for the EOF decomposition of the SIT bias yields satisfactory results (Supplement Fig. S2), the subsequent results will exclusively focus on this configuration.

Long short-term memory (LSTM) is a recurrent neural network designed to model chronological sequences and store information on a long time range . LSTM estimates the current prediction using data from its own prior prediction and enables the propagation of the bias backward in time, like a nonlinear type of autoregressive process. A unique model is developed for every single PC, as each depends on different input variables and time lag. The architecture is composed of three backward-prediction LSTM layers alternating with dropout layers, which prevent overfitting by randomly deactivating neural connections during training. Hyperparameters such as the number of components of the inputs, the input variables, and their time lags can change between models, while the overall architecture remains the same. Details regarding the differences between each model can be found in Table . Since certain PCs proved more challenging to predict than others, a comprehensive analysis of PC prediction is provided in Appendix  to better understand the performance of each model. Throughout this investigation, we discovered that the input variables have a much greater impact on the prediction than the ML architecture.

The uncertainty associated with the nonlinear estimation is computed by introducing random-walk processes to perturb the inputs of the LSTM. Multiple perturbation instances are employed to compute the ML-adjusted SIT, and the standard deviation of the resulting ensemble of SIT predictions is used as an uncertainty estimate. It is important to note that this uncertainty solely characterizes the sensitivity of the algorithm to its inputs and does not encompass the uncertainty associated with the training process of the ML algorithm. The final uncertainty is computed using 50 members, with a random-walk perturbation of the inputs set to 100 % of the original values scaled between -1 and 1.

To predict SIT biases in the past, our method is the following. We project the values of each input variable onto its principal components. As a result, we obtain a time series of each principal component for each variable. Then, the ML algorithm predicts the PCs of the SIT bias, and thus SIT biases can be retrieved by inverting the EOF projection. Lastly, TOPAZ4-ML SIT is reconstructed by adding SIT biases to TOPAZ4-FR. For a comprehensive assessment, we evaluate the total sea ice volume as the product of the sea ice thickness with the concentration and area of each grid cell.

We introduce a trivial bias correction as a baseline to evaluate the efficiency of our ML adjustment. Monthly biases between TOPAZ4-RA and TOPAZ4-FR are averaged from 2014 to 2022. The daily baseline SIT, called TOPAZ4-BL, is then obtained by adding the monthly biases to the SIT from TOPAZ4-FR at each grid point for the corresponding month. Considering that TOPAZ4 generally has too thin SIT in areas of thick ice, even this simple baseline a priori constitutes a solid benchmark.

After a brief analysis of the SIT biases of TOPAZ4, the sections below follow the standard steps of ML applications: first testing the algorithm on omitted data (2011–2013) and then predicting SIT biases outside of the training and testing windows, in our case extending the SIT data into the past (1992–2010).

The novelty of the present study lies in the combination of ML and DA to adjust sea ice thickness backward in time over a long period, longer than the training period. Since 1990, the sea ice thickness distribution in the Arctic has shifted drastically towards thinner sea ice , as documented by both satellite and in situ data. With our adjusted dataset (TOPAZ4-ML), the mean sea ice thickness in May (October) 1992 is 2.16 m (1.08 m), while in 2022, it is 1.54 m (0.57 m), resulting in a decrease of 29 % (47 %). Using independent data in the Arctic Basin, found that the annual mean SIT over the period of 2000–2012 declined from 2.12 to 1.41 m (34 %), while September thickness declined from 1.41 to 0.71 m (50 %). When including all the marginal seas up to the 15 % isoline of concentration, we find that the annual SIT is generally lower, but the trends are compatible, reducing from 1.51 m in 2000 to 1.01 m in 2012 (33 %), while September thickness declined from 1.42 m to 0.81 m (43 %). In our estimation, the annual mean sea ice thickness is lower compared to due primarily to differences in the area considered. These disparities diminish in September as the residual sea ice shrinks toward the central Arctic. reported losses of 2870 km³ per decade in winter (February–March) for 15 years of satellite records (2004–2018) from the non-overlapping ICESat and CryoSat-2 periods. For the same period, the TOPAZ4-ML data indicate losses of 2941 km³ per decade (Fig. ), which falls well within the uncertainties caused by the lack of snow depth data . PIOMAS, another data assimilation model, shows trends of -2.7 and 3.2 ± 1 × 10³ km³ per decade for April and September, respectively, from 1979 to 2018 Fig. 5.24. In comparison, over the period of 1992–2022, PIOMAS indicates -3.0 and -3.8 ± 1 × 10³ km³ per decade, while TOPAZ4-ML shows trends of -3120 and -2960 km³ per decade for April and September, respectively. Although the two datasets align well for April, a notable discrepancy emerges in September, with PIOMAS indicating a more pronounced downward trend. Drawing from the range of available data, the ML-adjusted trends correspond closely to those documented in the existing literature. Although TOPAZ4-FR and TOPAZ4-ML differ significantly in the total SIT, their respective trends are close.

By training our algorithm over the latest decade to predict the past, we assumed the following: the EOFs obtained from the SIT bias between 2011 and 2022 are representative of the statistical behaviour of the errors made by the model over a longer time period, including a dramatic regime shift. To probe the robustness of this assumption, we extracted the EOFs over two subperiods of our dataset: the training period with and without the test period. We only found differences in the least important components (from the sixth and further), while showing similar patterns overall (Supplement Fig. S5). The time series of the differences only shows unstructured noise.

Moreover, since we lack summer SIT observations, we assess the differences in SIT between two versions of TOPAZ (with and without assimilation) and not the SIT directly, so the data assimilation residuals may also have caused some loss of signal for the ML. However, the ML algorithm can adjust the thickness even of the thickest sea ice (>6 m) with less than 20 % error (Fig. ), which explains its performance in an earlier period dominated by multiyear ice.

Our approach based on EOF decomposition enables a drastic reduction in dimensions, leading to fewer parameters in the ML algorithm and thus reducing the costs required to train and apply the algorithm. This method is fast to implement and execute (around 1 h on a personal laptop), requiring minimal computational resources. Given its effectiveness, it demonstrates a strong ability to correct a large number of the biases. In comparison, approaches relying on more intricate 2D neural-network layers produced comparable outcomes but at a higher cost (at least 12 h to train) and in a more complex setup. Additionally, it is possible that with higher-dimensional features, the training set would be too small, increasing the risk of overfitting.

Multiple ML models (LSTM, convolutional neural network, dense, extreme gradient boosting, and random forest) were tested, yielding small local variations but no visible advantages in overall performance between models. The decision to select LSTM was thus driven by its robust time series prediction capabilities and its slightly better results. Throughout this study, the ML architecture (i.e. number of layers and hyperparameters) only played a minor role in achieving the optimal prediction; instead, the prediction accuracy is considerably dependent on the input variables, i.e. the choice of variables and associated time lags.

Three distinct sources of errors were identified when predicting SIT before 2011: ML reconstruction error, errors in the free run of TOPAZ4, and errors induced by regime shifts in sea ice conditions. Since the latter two are impossible to obtain within the scope of this study, we can only provide uncertainty estimates related to the ML method itself. Note that the uncertainty obtained here only characterizes the sensitivity of the algorithm to its inputs (details in Sect. ). The areas exhibiting the highest uncertainty encompass the Fram Strait; the Canadian Archipelago; the Beaufort Gyre; and, with a lower degree of uncertainty, the East Siberian Sea (Supplement Fig. S6). Upon examining the temporal evolution of uncertainty (Supplement Fig. S7), it appears that uncertainty diminishes during both the growth and melt phases of sea ice, likely attributable to the strong sea ice thickness seasonal cycle. Moreover, higher uncertainty is noted during the peak of the winter and summer seasons, when sea ice thickness is less affected by predominant freezing or melting, potentially leading to divergence among individual members.

Despite the baseline yielding good average results, the trivial bias correction displays strong regional biases and mediocre performance during outlier years. In addition, we expect the performance of the baseline to decrease even further as we extrapolate back in time. Indeed, the correction of the baseline is applied once and relies solely on the patterns of mean bias observed during 2014–2022, with no ability to accommodate different environmental conditions. On the contrary, our ML adjustment method proves more adaptable when predicting back in time since it takes into account the past state of environmental variables and the varying relative importance of each component (as independent errors identified by the EOF decomposition).

A distinct feature appears in the SIT averaged (Fig. ) at the onset of the melting season: a second peak, brief compared to the first, occurs shortly after the SIT maximum. It is observed almost yearly in TOPAZ4-FR but only twice (2017 and 2020) in TOPAZ4-RA. The phenomenon can be explained as follows: the relatively thin sea ice melts first, decreasing the area faster than the thickness, thus increasing the average SIT as a case of survivorship bias. This survivorship bias may intensify in cases of erroneously thinner sea ice in the central Arctic. Such instances can arise from either thinner sea ice in the central Arctic (TOPAZ4-BL) or misplaced thick sea ice in the Beaufort Gyre (TOPAZ4-FR). To prevent this artefact, many studies prefer to use the total volume or a geographic restriction to an area of perennial ice in the central Arctic.

Comparing TOPAZ4 to in situ datasets is challenging, primarily due to representation errors. Knowing the true sea ice thickness remains a major issue for evaluation, particularly when considering historical data from older satellite missions such as ICESat-1 and Envisat. This issue becomes more pronounced as we delve further into the past. Large uncertainties linked to in situ observations and the model ultimately lead to differences in SIT and difficulties evaluating our product. Adding to this point, the limited availability of global datasets over extended periods in the Arctic restricts the scope of possible comparisons. One notable advantage of our methodology is its capacity to bridge data gaps when mooring observations are unavailable.

In this investigation, we demonstrated that machine learning (ML) can be combined with data assimilation (DA) to predict sea ice thickness (SIT) errors backwards in time to 1992, using the ice–ocean model TOPAZ4 and atmospheric variables from the ERA5 reanalysis. The SIT biases are the results of accumulated increments from the assimilation of sea ice thickness data from CS2SMOS every 7 d between 2011 and 2022 during the ice growth period (October–May). Then, we reduced the dimensionality of the DA increments using empirical orthogonal functions (EOFs). The LSTM learned to predict SIT biases using principal components (PCs) of various sea ice, ocean, and atmospheric variables as inputs. This study demonstrates that our PC-based approach is effective at providing a major sea ice thickness adjustment.

Our approach significantly reduced sea ice thickness biases throughout the test period (2011–2013) from a low year-round bias of -10.0 to 1.4 cm. Significant improvements are noted during the melting period, likely attributable to substantial errors in TOPAZ4 with assimilation, as sea ice thickness data assimilation is unavailable during summer. Applying our algorithm before 2011, the evaluation with independent mooring data indicates improvement in the central Arctic compared to the TOPAZ4 free run, while results in the Beaufort Gyre show poorer agreement. However, the scarcity of in situ datasets and the often limited continuity of observations restrict the comparison to only a few locations. Remote sensing data from Envisat and ICESat-1 were primarily utilized for qualitative assessment due to their inherent high uncertainties and temporal–spatial discontinuities. Our approach demonstrates a general improvement in SIT despite the challenge of selecting a reliable truth for validation.

Furthermore, this prolonged time series gives new insights into various aspects of SIT, including distribution, spatial patterns, and changes through time. The estimated May (October) mean sea ice thickness in 1992 was 2.16 m (1.08 m), whereas it was 1.54 m (0.57 m) in 2022, resulting in a 29 % (47 %) decline. This amounts to a decrease in total sea ice volume from 19 690 to 12 700 km³, with a corresponding trend of -3153 km³ per decade, corroborating previous model estimates. A decrease in the thickest sea ice has been observed throughout the years, with the proportion of sea ice thicker than 2.5 m going from 28 % in 1992 to 7 % in 2022. In the ML-adjusted data, the transition in 2007 is, however, less abrupt than that deduced from moored observations from the Fram Strait.

The ML-adjustment technique can be implemented for other variables, as long as equivalent resources are available: two model runs with and without assimilation of the target variable and some auxiliary data related to the target variable but in complex ways. Further work is required to compare our SIT time series to the novel year-round processing of CS2 , especially regarding the summer sea ice thickness. The ML-adjustment method was originally introduced within the framework of an iterative method combining the DA and ML techniques . In a subsequent investigation, a second iteration of DA using the reconstructed SIT and its uncertainty will be performed with TOPAZ4, improving the initial conditions of SIT from the latter decade.