Using anomaly correlation coefficients (ACCs) for different target months at different lead times ( Figure 3 ), we examine the persistence of pan-Arctic SIC and total SIE based on the satellite observations during 1979–2020. Persistence of SIC is represented as the regional averaged ACC for SIC on all grid points. It is obvious that the persistence is much longer in September, October, and March, for both the ice concentration and the extent. In contrast, low correlations exist between successive months in the transition season (e.g., May and June) when sea ice is rapidly growing or melting. In this study, we mainly focus on the sea ice situation in summer and early autumn.

Sea ice in September and October is highly correlated with sea ice in the previous 1–3 months, and thus months from June to August show predictive skills in forecasting late summer sea ice. As pointed out by Andersson et al. (2021), the precondition of the sea ice pack is a key predictor, with persistence of sea ice anomalies potentially lasting at seasonal timescales, especially for the ice in late summer. In addition, correlations are extremely low for sea ice between summer and spring for the same year. The weak correlation may be a manifestation of the springtime predictability barrier.

The above-mentioned possible sources of sea ice predictability, including the seasonal cycle, the multi-decadal trend, and the persistence of sea ice anomalies (especially in summer months), are essential for our data-driven forecasting approaches used in this study.

LSTMs is a special type of Recurrent Neural Networks (RNNs) introduced by Hochreiter and Schmidhuber in 1997 (Hochreiter and Schmidhuber, 1997). Like RNNs, LSTMs are networks in the form of a chain of repeating modules, allowing information to persist, so that they are generally used to deal with time series problems. However, when training a classic RNN using back-propagation, the long-term gradients can “vanish” or “explode”, because the computations involved in the process use finite-precision numbers. LSTMs could partially solve the vanishing gradient problem, because LSTM modules or units allow gradients to also flow unchanged, making LSTM capable of handling long-term dependencies (Calin, 2020).

Generally, a single LSTM cell contains four interacting neural network layers acting as four activation functions [three sigmoid (σ) and one hyperbolic tangent (tanh) functions], which are marked by yellow boxes in Figure 4B . This kind of module structure is more complex than that in RNNs, which contains only one layer of tanh. LSTM controls the transfer and loss of information by a memory cell state (the straight black line at the top of Figure 4B ) and three “gates” (marked by red dashed boxes in Figure 4B ). The cell state is responsible for remembering the previous state while the gates are responsible for controlling the amount of memory to be exposed.

Chattopadhyay et al. (2020) have found that deep neural networks including LSTM are able to reproduce both the short-term evolution and the long-term statistics of dynamical systems, which provides a basis for this study to extract the nonlinear relations of sea ice time series and to make sea ice forecasts based on observational data. In contrast to the traditional statistical methods, the nonlinearity introduced by LSTM could effectively contribute to the skill of the sea ice forecast. Based on the finding of Chi and Kim (2017), we use the observed SIC in the preceding 12 months {x_{t -12}, x_{t -1}} as the input data to make 1-month lead forecast at each grid point. These data are fed into LSTM one by one, and the final hidden state h_{t -1} at the last month is used to predict y_t via a fully-connected operator. The whole network is trained by the widely used loss function based on the mean squared error (MSE) between the predicted y_t and observed SIC x_t , which can be described as 1 n ∑ i = 1 n ( x ( i ) − y ( i ) ) 2 Then, we keep the network structure unchanged, and only add the mean absolute error (MAE) into the original loss function, resulting in a combined loss function of MSE and MAE, i.e., 1 n ∑ i = 1 n ( x ( i ) − y ( i ) ) 2 + | x ( i ) − y ( i ) | Such a loss function could effectively reduce the impacts of outliers, thus expecting to achieve better predictions. Using this loss function, we finally develop an improved LSTM model.

As shown in Figure 4 , our final model incorporates an attention module into the original LSTM network. Different from the traditional LSTM models which only consider the contribution of the last hidden state h_{t -1} to the final prediction, the attention module is able to deal with all of the hidden states {h_{t -12}, h_{t -1}} in the preceding 12 months, and automatically recognize the importance of them. To achieve this goal, we construct the module by two fully-connected layers, and the last one outputs the importance of the weights {w_{t -12}, w_{t -1}} for 12 months. After that, the hidden states are multiplied by the corresponding weights to get the final hidden value, which can be formulated as h f = ∑ i = 1 12 [ w t − i × h t − i ] Same as traditional LSTM models, hf is finally fed into a fully-connected layer to get y_t . In addition, during the training phase, we employ a combination loss of MSE and MAE to better optimize the proposed network. For LSTM in the network, there is 1 hidden layer with 24 neurons. For the attention module, the number of neurons in the first and the second fully connected layer is 96 and 12, respectively. We choose Adam as the optimizer to optimizing the weights. The learning rate and the batch size are set to 0.001 and 128, respectively. For simplicity, the three forecasting models are abbreviated as LSTM_MSE (original LSTM with MSE-based loss function), LSTM_MAE (LSTM with MSE and MAE combined loss function), and LSTM_Attention (attention-based LSTM with MSE and MAE combined loss function), respectively.

For sea ice forecast at longer lead times (2–6 months), we use a recursive approach. The predicted values at shorter lead time are treated as new inputs into the same forecasting model to make predictions at longer lead time. Take the forecast of September 2019 SIC as an example. For the 1-month lead time forecast, SIC observation from September 2018 to August 2019 is used as inputs to LSTM network. Then to generate a 2-month lead time forecast, we use SIC observation from September 2018 to July 2019 and SIC forecast of August 2019 as inputs. Monthly SIC of the passive microwave observations (see the section “Datasets”) from 1979 to 2014 (36 years, 432 months) are used to construct the training set for LSTM network, while data from 2015 to 2020 (6 years, 72 months) are used to construct the test set to verify the skills of the prediction models. To generate the training set, the 432-month data needs to be divided into input-output data pairs. The input consists of preceding 12-month data, while the output is the subsequent 1-month data. At each month, all of the grid points could be used. Thus, more than 10 million of training samples were generated to train the proposed model. Similar methods are used to the 72-month test data, and more than 3 million of test samples were produced. The LSTM related models are implemented on a personal computer with an Intel core i7-4790, 3.60-GHz processor, 32-GB RAM, and a GTX TITAN X graphic card. The training time for LSTM_MSE, LSTM_MAE, and LSTM_Attention are 69.07, 69.08, and 69.31 minutes, respectively, while the testing time for them are 0.90, 0.91, and 1.21 seconds, respectively.

Referring to the work of Liu et al. (2021), we also assess our LSTM mos against two empirical benchmark models using sea ice persistence and climatology. The persistence is defined as the SIC anomaly at lead time step 0 added to the SIC climatology of the target forecast month, while the climatology of SIC in each month is defined as the average of SIC in a 10-yr sliding window preceding the respective forecast target time. For example, the 1-month lead persistence forecast of SIC in September 2019 is the anomaly of August 2019 plus the climatology of September SIC in 2009–2018.

To compare LSTM with numerical model-based forecasts, sea ice prediction results from two numerical climate models are also analyzed. Previous studies have indicated that the numerical systems from the European Centre for Medium-Range Weather Forecasts (ECMWF) and the U.K. Met Office (UKMO) provide the state-of-the-art Arctic sea ice prediction skills (Zampieri et al., 2018; Wayand et al., 2019). In this study, the seasonal ensemble forecast products from these two systems are compared with forecasts of our deep learning methods. The two model systems (hereafter, abbreviated as ECMWF-C3S and UKMO-C3S) are contributing to the Copernicus Climate Change Service (C3S). The dataset includes forecasts created in real-time (since 2017) and retrospective forecasts (hindcasts) from 1993 to 2016. The brief information of these two models is given in Table 1 . These models are chosen because of their time range (January 2015–December 2020 for ECMWF, and January 2015–December 2016 for UKMO) available for the intercomparing in this study and the popularity of their sea ice models. Forecasts of SIC from the two models are re-gridded from their latitude-longitude grid to the NSIDC equal area grid using bilinear interpolation method. The interpolation is performed after converting the latitude-longitude coordinate to distance coordinate.

This study evaluates the forecasting model performance based on the standard deterministic accuracy metrics: the root-mean-square error (RMSE), mean absolute error (MAE) and anomaly correlation. Taking the evaluation of SIC prediction as example, the RMSE and MAE of SIC are defined in formulae (1) and (2). S I C i f is the forecasted value of SIC, S I C i o is the observed SIC, and N is the total number of grid points. The errors are spatially averaged after masking the land, open ocean, and the observational “pole hole” grids.

In addition, the assessment of the skills in predicting the Arctic sea ice edge is based on the Integrated Ice Edge Error (IIEE, Goessling et al., 2016). The IIEE is defined as the sum of ocean areas where the presence of sea ice (defined with a 15% SIC threshold) is overestimated and underestimated with respect to the observations. In other words, this verification metric describes the area where the forecasts and observations disagree on the presence of sea ice with concentration being above or below 15%. In this study, the two components of IIEE, i.e., the overestimated error and underestimated error of sea ice edge, are abbreviated as OE and UE for simplicity.

where c=0 when the SIC is higher than 15% and c=1 when SIC is less than 15%. The subscripts f and o denote the forecast and observation. A denotes the area of interest.

The IIEE has been widely used in previous studies focusing on the sea ice forecast (e.g., Blockley and Peterson, 2018; Roach et al., 2018; Allard et al., 2020; Batté et al., 2020; Liu et al., 2021). By taking into account possible error compensations between the overestimation and underestimation of the presence of sea ice, the IIEE could present a better estimate of the ability of forecast models to predict SIC in different Arctic basins.

It is noteworthy that only the mean forecast results of the dynamical models are concentrated on in this study, thus the evaluation metrics used here are relatively simple and can only partly reflect the skills of the forecasting model. Besides, considering that the reliability of a forecast model comes from its ensemble forecasts while our preliminary LSTM-based approach has no ensemble component at present, this approach used here is not yet able to report on the reliability of the sea ice forecast.

At first, we evaluate the hindcast or re-forecast skills of LSTM models for pan-Arctic SIE derived from SIC forecasts at 1-month lead time. We examine the month-to-month and inter-annual variability of the predicted pan-Arctic SIE during January 2015–December 2020. Figure 5 shows the monthly SIE anomaly obtained by three LSTM models and three benchmark models (ECMWF-C3S, Climatology, and Persistence), as well as the absolute errors of their SIE forecasts compared to the satellite observation. When calculating the anomaly, the common reference period is the 30-yr climatology from 1981 to 2010, which is also used in this study. In general, all the forecasts from three LSTM models are consistent with observation in the seasonal variations of SIE anomaly. Greater anomaly occurs in summer to early autumn compared to the rest of the year ( Figure 5A ). Obviously, LSTM_Attention and LSTM_MAE perform better than LSTM_MSE in reproducing the extreme low SIEs in October of 2016, 2019, and 2020. Monthly SIE anomaly from LSTM_Attention forecast shows a significant positive correlation with observation during our test period 2015–2020, with the correlation coefficient reaching about 0.8. The absolute errors of SIE between LSTM forecasts and observation ( Figure 5B ) also show that, in most time of the test period, predicted SIEs from LSTM_Attention and LSTM_MAE are closer to the observation compared to that from LSTM_MSE. This result confirms that the optimization of the deep learning network could improve the forecasting skills by considering previous sea ice conditions as much as possible. It is nevertheless noteworthy that all the three models predict a false minimum SIE in October 2017. This loss of skill reflects the dependence of deep learning-based forecasts on the training data, or their weak capacity reflecting the impacts from strong, synoptic-scale processes on sea ice development.

In comparison, forecast from ECMWF-C3S demonstrates slightly higher absolute error against observation ( Figures 5C, D ). During the test period of 2015–2020, mean absolute errors of monthly SIE are 0.23 and 0.25 million km² for LSTM_Attention and ECMWF-C3S, respectively. Consistent with LSTM_Attention, ECMWF-C3S also underestimates total Arctic SIE significantly in October 2017, which is quite interesting and worthy of further study. The two empirical models of Climatology and Persistence show much higher SIE errors compared to forecasts from other methods.

For long-term (i.e., the test period of 2015–2020) averaged SIE in different months among the year, LSTM_Attention demonstrates higher skill than LSTM_MSE and LSTM_MAE in most months of the year. The RMSE and MAE ( Figure 6A ) of forecasted SIE are the largest in early autumn (October to November). The relatively low skill is probably due to the higher sea ice mobility and thus the larger uncertainty of forecasted ice edge location during this season of year. This is also a common characteristic for sea ice forecasts from the dynamical model ECMWF-C3S and two empirical benchmark models ( Figure 6B ). Except for February and July, LSTM_Attention shows comparable RMSE and MAE with ECMWF-C3S, especially in late summer and early autumn. Combined with their absolute errors in Figure 5 , this result suggests that LSTM_Attention is comparable to dynamical models in predicting ice extent, especially in the season when marine operations peak.

Since 2008, the Sea Ice Outlook (SIO) initiative (Stroeve et al., 2014), which is now under the auspices of the Sea Ice Prediction Network (SIPN, http://www.arcus.org/sipn/sea-ice-outlook), has collected several sources of seasonal forecasts (statistical, dynamical and heuristic) for the September pan-Arctic SIE at up to three month lead times. To further evaluate the skill of LSTM_Attention, we compare our results with the SIO median and the best SIO estimates for the 2015–2020 September SIE as shown in Table 2 . The SIO forecasts are based on the August reports (i.e., with 1-month lead time) submitted by all the contributors each year. Although showing overestimation and underestimation of SIE in the first two years, the prediction of LSTM_Attention is slightly closer to the observations than the SIO median since 2017; this demonstrates the potential of deep learning approaches in operational monthly to sub-seasonal sea ice forecasting. Meanwhile, there is still a gap between LSTM_Attention forecasts and the best SIO approaches, indicating that more work is needed to improve the skills of our LSTM-based method.

To explore the possible explanations of SIE forecasting skills from LSTM models, we examined the performance of LSTM_Attention in representing the spatial distribution of sea ice cover. Examples of the September SIC forecasts from attention-based LSTM at 1-month lead time are shown in Figures 7 , 8 . We examine the extreme years with record minimum September SIE, including 2015, 2016, 2019, and 2020 (the second-lowest ice extent on record). Prediction results from the UKMO-C3S are only available in 2015 and 2016 during our test period, and thus the inter-comparisons including this model are performed only in these years.

As shown in Figures 7 , 8 , the attention-based LSTM successfully reproduces the spatial pattern or the regional variation of summer sea ice cover in extreme years that recorded unforeseen SIE minimums. In the first two years, LSTM_Attention outperforms UKMO-C3S and is comparable to ECMWF-C3S in predicting the location of sea ice edge. Yet, in September 2016, the concentration underestimation in central Arctic from LSTM_Attention is obvious and is probably the cause of total SIE underestimation in this September.

The spatial distribution of year-round mean MAE of ice concentration in January 2015–December 2020 forecasted by LSTM_Attention is shown in Figure 9 . For all 12 months, concentration errors are the largest in marginal ice zones. Physically, marginal ice zones are characterized by extremely active ice motions and air-sea-ice interactions (e.g., Zippel and Thomson, 2016; Boutin et al., 2020), making it more challenging for sea ice prediction. For the whole Arctic, RMSE of ice concentration forecasted by LSTM_Attention is lower than the two empirical models and ECMWF-C3S in each month ( Figure 10B ). Also shown in Figure 10B is that RMSE of forecasted SIC from LSTM_Attention is relatively low in late summer (August and September) but higher in October, similar to the situation of pan-Arctic SIE ( Figure 6B ). ECMWF-C3S also has higher forecasting skill than persistence during this time, but it does not outperform the 10-yr climatology forecast.

We also show the July-August-September (JAS) RMSE of lead-time-dependent SIC forecasts by different approaches ( Figure 10A ) to examine the sea ice predictive skills of LSTM_Attention. In general, RMSE increases with the increased lead time except for the climatology forecast. LSTM_Attention outperforms both the persistence and the ECMWF-C3S forecast for lead times from 1 month to 6 months. In particular, the RMSE over time of SIC from LSTM_Attention is lower than the persistence with lead times up to 2 months, providing the basis for its skill in total ice extent prediction. At lead time of 2 months, the skill of LSTM_Attention is comparable to the climatology and thus it is vanished.

The above-mentioned assessments confirmed the potential of LSTM_Attention to predict summer concentration fields. In practice, it is important to know whether a region is ice free or not for the increasingly frequent shipping and related activities in the Arctic Ocean. The Integrated Ice Edge Error (IIEE), which is first introduced by Goessling et al. (2016), has been proved to be an appropriate objective metric. This metric could well evaluate the model’s forecasting skill in the position of the ice edge (with the concentration threshold of 15%) and will be analyzed in the following section.

Figure 11 shows the monthly IIEE and its two components, OE and UE, during the test period 2015–2020 from the three LSTM models. Similar to the absolute error in total ice extent (recall Figure 5B ), IIEE of LSTM_Attention is much lower than LSTM_MSE and LSTM_MAE. The seasonal cycle ( Figure 12 ) of IIEE shows that the forecasting ice edge error is relatively larger in early autumn (October) than in other months, which is consistent with LSTM_Attention’s low skill in forecasting October ice extent (recall Figure 6 ). Figure 10 also demonstrates that the improvement of IIEE from the attention-based LSTM is mainly contributed by the decrease of OE in the forecast fields of LSTM_Attention. As for the UE in LSTM_Attention forecast, it gets worse instead of better compared to the previous approaches ( Figure 11C ). Future effort is warranted to explain and reduce this underestimated component of IIEE in LSTM model.

The extremely high IIEE and its two components in October and November ( Figure 12 ) is associated with the high MAE of total ice extent forecasted by LSTM_Attention ( Figure 6A ). Moreover, the discrepancy between IIEE and MAE also highlights the limitations of conventional metric for ice extent evaluations like MAE: the IIEE contains the “misplacement error”, reflecting too much ice in one area and too little in another, which is not included in MAE of the total ice extent.

Figure 13 shows the IIEEs of summer (July–August–September, JAS) sea ice forecasts over 2015-2016 versus lead time. Forecasts are from LSTM_Attention and the two C3S dynamical models, respectively. It is not surprising that the IIEE increases with the increasing lead time. LSTM_Attention is comparable to ECMWF-C3S and UKMO-C3S in 1-month lead time forecast, while it outperforms the two dynamical models at lead time of 2 months. IIEEs of forecasts from LSTM_Attention and ECMWF-C3S in 2015–2020 are similar to those for 2015–2016 and thus are not shown here. Besides, LSTM_Attention obviously differs from the dynamical models in the allocation of the two components OE and UE. LSTM_Attention forecast provides comparable amounts of OE and UE, while in the two dynamical models’ forecasts, UE dominants the total IIEE error.

We take September 2016 as an example to examine the spatial structures of the gridded two components of IIEE ( Figure 14 ). For all the three forecast fields, the overestimated and underestimated error components are mainly located in the Pacific sector of the Arctic Ocean. One possible reason is that sea ice in the Atlantic sector (including the Kara Sea and Barents Sea) has almost completely melted in September, pushing the ice edge northward to the central Arctic Ocean. The larger ice edge error in the Pacific sector reflects the great inter-annual variation of sea ice cover due to strong advection and melting processes in this region (e.g., Bi et al., 2019). Actually, in addition to the most obvious decreasing trend ( Figure 2B ) since 1979, September ice concentration in the Pacific sector also shows the largest inter-annual variation in contrast to other Arctic regions (figure not shown).

Compared to UKMO-C3S, LSTM_Attention and ECMWF-C3S mainly capture the observed sea ice edge especially in the Atlantic sector, indicating that our improved LSTM network is reliable in monthly to sub-seasonal Arctic sea ice edge forecast in the melting season. This confirms the potential of our LSTM network in providing a monthly to sub-seasonal outlook of ice edge changes, which could help planning the routes in the Arctic Ocean 1 to 1.5 months in advance.