TAKING A SECOND LOOK: CORRECTING SEA ICE FORECASTS WITH SPARSE OBSERVATIONS Tianshuo Zhang Xianglei Xing∗ Aowen Yang Jia Gao Wenzhe Zhai ShanShan Liu
arXiv:2609.24591v1 [cs.LG] 21 Sep 2026
College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin 150001, China ABSTRACT Sea ice forecasts are issued several days ahead, allowing errors to accumulate while new, often sparse sea ice concentration (SIC) observations become available. We find that fixedpropagation errors concentrate near structured, high-gradient ice edges, whereas homogeneous interiors require limited propagation, suggesting that propagation distance should be state dependent. We therefore introduce ECHO (Evidenceguided Correction with Heterogeneous prOpagation), where ECHO-Scale adapts propagation distance while preserving correction geometry, and ECHO-Delta learns a bounded residual around fixed propagation. Across all 96 standard evaluation settings spanning diverse priors, observation times, sparsity levels, geometries, and noise conditions, both outperform fixed propagation. ECHO-Delta achieves the best average accuracy, while ECHO-Scale is more robust to geometry shifts. Code is available at https://github. com/yingtian22/TAKING-A-SECOND-LOOK. Index Terms— Sea ice forecasting, Sparse observations, Forecast correction, Spatial propagation, Data assimilation 1. INTRODUCTION Reliable short-term sea ice forecasts are important for maritime operations in the Arctic, where accurate knowledge of the evolving ice cover is required for safe and effective decision making [1, 2]. Operational systems therefore routinely issue sea ice forecasts several days in advance [3]. However, the ice state continues to evolve after a forecast is issued, allowing forecast errors to accumulate before the target day. During this interval, new satellite observations may arrive intermittently with partial spatial coverage, providing an opportunity to revise the issued forecast. In this work, we consider a second-look setting: an open-loop forecast is kept fixed, while sparse intermediate SIC observations are used to revise it before the target time. Fig. 1 provides an overview of the second-look setting and the two ECHO variants. ∗ Corresponding author: [email protected].
Fig. 1. The Framework of ECHO.
Sparse observations must be propagated into unobserved regions to obtain a full-field correction, a central challenge in data assimilation and sparse satellite reconstruction [4, 5, 6]. In our setting, the key question is how each observed innovation should influence its surroundings. A simple deterministic reference, Fixed Propagation, extends the nearestobservation innovation with distance-dependent decay but applies the same rule across heterogeneous ice conditions. This leaves a central question: how far should each innovation propagate? Spatial error analysis shows that Fixed Propagation errors concentrate in structured, high-gradient, high-innovation iceedge regions, while homogeneous ice and ocean interiors require little propagation. This suggests that the main limitation lies in the spatially uniform propagation extent rather than the nearest-observation geometry, motivating state-dependent adaptation of propagation distance. Based on this diagnosis, we introduce ECHO (Evidenceguided Correction with Heterogeneous prOpagation) with two variants: ECHO-Scale preserves nearest-observation geometry while adapting a bounded spatially varying scale, whereas ECHO-Delta learns a bounded residual around Fixed Propagation. Across 96 standard settings, both improve upon Fixed Propagation and compare favorably with representative data-assimilation baselines. ECHO-Delta achieves the best average accuracy, while ECHO-Scale is lighter and more robust to geometry shifts, revealing a trade-off among accuracy, robustness, and complexity. Our contributions are fourfold. First, we formulate a
second-look sea ice forecast correction setting using sparse intermediate SIC observations. Second, spatial error diagnosis identifies uniform propagation extent as the main limitation of Fixed Propagation. Third, we introduce ECHO, with ECHO-Scale adapting propagation distance and ECHODelta adding bounded residual refinement. Fourth, extensive evaluations demonstrate strong overall gains and an accuracy–robustness–complexity trade-off between the two variants.
tion is ( Rt (x) =
Yt (x) − Xtf (x), 0,
x ∈ Ωobs , otherwise.
(1)
The second-look problem is therefore to construct a dense correction field from the sparse innovation Rt , while keeping the original forecast trajectory fixed. For every location x, let j ∗ (x) = arg min ∥x − j∥2 , j∈Ωobs
d(x) = ∥x − j ∗ (x)∥2 (2)
2. RELATED WORK 2.1. Sea Ice Forecasting and Data Assimilation Sea ice prediction spans physics-based forecasting, datadriven modeling, and data assimilation. Operational systems such as TOPAZ4 couple ocean–sea ice models with ensemble assimilation [3], while learning-based methods predict future SIC from historical observations, including PredRNN++ [7], SICNet [8], IceDiff [9], and SIFusion [10]. Classical data-assimilation methods propagate innovations through prescribed statistical structure, including nudging and covariance-based optimal interpolation, whereas EnKF represents forecast uncertainty with ensembles [4, 5]. Learned variational schemes such as 4DVarNet address sparse spatiotemporal reconstruction [6]. Unlike direct forecasting or reconstruction, our setting revises an issued target-day forecast using sparse intermediate observations, focusing on innovation propagation into unobserved regions. 2.2. Learning-based Forecast Correction Learning-based modeling can exploit structured inductive biases [11], while forecast post-processing reduces errors through neural calibration [12], residual correction [13], and supervised refinement of operational SIC forecasts [2]. These methods generally learn direct mappings from available predictors to improved forecasts. ECHO instead considers a second-look setting with a frozen forecast and sparse intermediate SIC observations. To obtain a dense target-day correction, ECHO retains a deterministic propagation reference: ECHO-Scale adapts its spatial influence extent, while ECHO-Delta learns a bounded residual around it, bridging prescribed propagation and learned correction. 3. METHODOLOGY Second-Look Correction Formulation. Let Xtf and XTf denote the frozen open-loop SIC forecasts at an intermediate observation time t and the target time T , respectively. We consider T = 7 and t ∈ {3, 5}. Sparse SIC observations Yt are available only over a subset of grid cells Ωobs ⊂ Ω, where Ω denotes the valid ocean domain. The observation innova-
denote the nearest observed location and its spatial distance, respectively. We define a generic propagation operator d(x) Rt (j ∗ (x)) , (3) PL [Rt ](x) = exp − L(x) where L(x) > 0 controls the propagation extent and Π[0,1] (·) denotes projection onto the valid SIC range. Fixed Propagation as the Reference Operator. Fixed Propagation is the homogeneous case L(x) ≡ L0 : d(x) f fix ∗ X̂T (x) = Π[0,1] XT (x) + exp − Rt (j (x)) . L0 (4) The validation-selected scales are L0 = 8 for the persistence prior and L0 = 5 for the Direct U-Net prior. This formulation is parameter-free at inference and preserves an explicit relationship between observation distance and correction magnitude. Its main restriction, however, is that the same propagation extent is imposed throughout the spatial domain despite strongly heterogeneous local SIC structure. ECHO-Scale: Geometry-Preserving Adaptive Propagation. ECHO-Scale introduces the minimum state-dependent relaxation of Eq. (4): the nearest-observation assignment j ∗ (x) and the innovation geometry are kept unchanged, while only the propagation extent is made spatially adaptive. Specifically, we parameterize Lθ (x) = L0 exp[ln 2 tanh(sθ (x))] ∈ [0.5L0 , 2L0 ].
(5)
and the corrected forecast becomes d(x) f ∗ scale X̂T (x) = Π[0,1] XT (x) + exp − Rt (j (x)) . Lθ (x) (6) This parameterization separates the deterministic propagation geometry from the learned propagation extent. The final scale head is zero-initialized, so sθ (x) = 0 initially and Lθ (x) = L0 . ECHO-Scale therefore starts exactly from Fixed Propagation and learns only a bounded state-dependent deviation in propagation extent. Fig. 2 illustrates the resulting adaptation. ECHO-Delta: Bounded Residual Relaxation. While ECHO-Scale restricts learning to the propagation scale,
ECHO-Scale uses the quadratic loss, whereas ECHO-Delta uses the mixed ℓ1 –ℓ2 loss; predictions are projected onto [0, 1] before loss evaluation.
Fig. 2. Spatial adaptation learned by ECHO-Scale.
Fig. 4. Comparison of Fixed Propagation and ECHO variants.
4. EXPERIMENTS
Fig. 3. Qualitative comparison on a representative test case. ECHO-Delta introduces a second, more expressive relaxation around the same deterministic reference. Let ∆ϕ (x) = δmax tanh(zϕ (x)) ,
δmax = 0.25,
(7)
denote a bounded residual correction. The final prediction is i h (8) X̂Tdelta (x) = Π[0,1] X̂Tfix (x) + ∆ϕ (x) . The bounded parameterization prevents unrestricted departures from the reference correction while allowing errors not captured by an isotropic distance-decay model to be compensated directly. The two variants therefore instantiate two levels of relaxation of the same reference operator: PL −→ PLθ (x) −→ PL0 + ∆ϕ , |{z}0 | {z } | {z } Fixed
ECHO-Scale
(9)
ECHO-Delta
where ECHO-Scale preserves the deterministic correction geometry and adapts only its influence extent, whereas ECHODelta increases expressiveness through bounded residual refinement. This progression tests whether gains arise from adapting propagation extent alone or require residual correction. Fig. 4 summarizes this constraint–expressiveness hierarchy. Learning Objective. Let e(x) = X̂T (x) − XT (x) denote the prediction error over valid ocean pixels. Both variants minimize X 1 L= ρ(e(x)), (10) |Ωvalid | x∈Ωvalid
where the error penalty is chosen according to the degree of model flexibility: e 2 , ECHO-Scale, ρ(e) = λ = 0.2. (11) |e| + λe2 , ECHO-Delta,
Experimental Setup. We use the NOAA/NSIDC Sea Ice Concentration CDR (G02202 v5) [14] from 2014–2020 on a 448 × 304 polar-stereographic grid, with 1819/365/360 train/validation/test samples. Correction is evaluated at lead day 7 using two frozen priors, persistence and a pretrained Direct U-Net. Sparse observations are provided at t ∈ {3, 5} with {5%, 10%, 30%} ocean coverage, four geometries (random, edge-biased, stripe, coarse-grid), and Gaussian noise σ ∈ {0, 0.03}, yielding 96 settings with 360 test cases each. RMSE over valid ocean pixels is the primary metric, with MAE, ice-edge RMSE, MIZ RMSE, and sea-ice extent error also reported. Baselines include residual interpolation, nudging, localized OI, and EnKF-based assimilation; OI uses validation-selected parameters (L = 2, rloc = 4) fixed for all test settings. Both ECHO variants use AdamW with learning rate 5 × 10−4 , batch size 4, 20 epochs, patience 6, and three training seeds. This protocol tests robustness across forecast priors, observation times, sparsity, sampling geometries, and noise. Main Results. Table 1 summarizes performance over all 96 settings, with ECHO reported as mean±std over three training seeds and the remaining baselines using their frozen formal results. ECHO-Delta achieves the lowest average RMSE (0.06249 ± 0.00028), followed by ECHO-Scale (0.06735 ± 0.00003). Using the seed-42 paired protocol, both variants outperform Fixed Propagation in all 96 settings, with mean ∆RMSEs of −0.001634 for ECHO-Scale (95% CI [−0.001745, −0.001523]) and −0.006765 for ECHO-Delta (95% CI [−0.007356, −0.006177]). Against validationtuned Localized OI, ECHO-Scale is better on average despite winning 61 of 96 settings (∆ = −0.000917, 95% CI [−0.001333, −0.000507]), while ECHO-Delta wins all 96 settings (∆ = −0.006048, 95% CI [−0.006777, −0.005338]). Fig. 5 highlights the improvement around the ice edge, while Fig. 3 provides a full-domain qualitative comparison. Global Retuning and Spatial Scale Adaptation. To test whether ECHO-Scale simply compensates for a suboptimal global scale, we re-optimize L on the validation set. The
Table 1. Results over 96 settings. ECHO reports 3-seed means; CIs use seed 42. Method
Edge MIZ RMSE ↓ MAE ↓ RMSE ↓ RMSE ↓
Prior 0.08161 0.01753 0.15568 Residual Interp. 0.07172 0.02065 0.13921 Fixed Prop. 0.06897 0.01850 0.13463 Nudging [4] 0.06950 0.01911 0.13499 Localized OI [15] 0.06825 0.01794 0.13351 EnKF-PertObs [5] 0.07989 0.01934 0.15312 4DVarNet [6] 0.06909 0.02645 0.13581 ECHO-Scale 0.06735 0.01736 0.13209 ECHO-Delta 0.06249 0.01309 0.12477
0.28302 0.24391 0.24043 0.23936 0.23841 0.27226 0.22662 0.23732 0.23085
Table 3. RMSE under different observation coverages. Coverage Fixed ↓ ECHO-Scale ↓ ECHO-Delta ↓ 5% 10% 30%
0.07049 0.06890 0.06751
0.06862 0.06720 0.06623
0.06400 0.06226 0.06120
Under sensor-inspired geometries, ECHO-Scale is never worse than Fixed (88 wins, 8 ties), while ECHO-Delta retains the lowest average RMSE but shows 36 reversals, all under the Direct U-Net prior, and wins all 48 persistence settings. Localized OI performs best with dense, locally distributed observations but weakens under stripe sampling and at 5% coverage. Overall, ECHO-Delta favors average accuracy, whereas ECHO-Scale provides stronger geometry-shift consistency. Ablation Study. Table 4 ablates the input guidance of ECHODelta. Removing either the distance-decay confidence or the Fixed increment degrades performance, while jointly removing the Fixed-guidance features causes the largest RMSE increase. The confidence ablation degrades most under stripe masks, where observations are spatially clustered.
Fig. 5. Qualitative comparison at the ice edge. Table 4. ECHO-Delta ablation study with training seed 42. optimum remains L = 8 for persistence and shifts from 5 to 3.75 for Direct U-Net. This retuning improves Fixed Propagation RMSE only from 0.068966 to 0.068936, explaining just 1.8% of the gap to ECHO-Scale. ECHO-Scale still wins all 96 settings (mean ∆RMSE = −0.001604, 95% CI [−0.001719, −0.001488]). Independent scale sweeps further show that higher-gradient regions favor larger propagation extents, consistent with the learned spatial scales. Table 2. Validation results by prior-SIC gradient regime. Regime
Fixed ECHO-Scale Best Learned RMSE ↓ RMSE ↓ L/L0 L(x)/L0
Low (25%) 0.0151 Medium (47%) 0.0393 High (28%) 0.1185
0.0135 0.0376 0.1160
0.25 0.50 1.25
0.51 0.54 0.73
Sensitivity to Observation Sparsity. Performance improves as observation coverage increases from 5% to 30%. Importantly, at only 5% coverage, both ECHO variants still outperform Fixed Propagation in all 32 matched settings at training/evaluation seed 42. ECHO-Scale benefits most from the sparsest observations, while ECHO-Delta maintains a larger gain across all coverage levels. RMSE by coverage (ECHO: 3-seed mean). Robustness and Generalization. Across three mask and noise realizations, both ECHO variants outperform Fixed Propagation in all 96 settings with negligible RMSE variation; date-matched filtering preserves the paired rankings.
Variant
Edge MIZ RMSE ↓ ∆RMSE RMSE ↓ RMSE ↓
Full ECHO-Delta 0.06220 – 0.12432 w/o Confidence 0.06294 +0.00073 0.12606 w/o Base-Delta 0.06290 +0.00069 0.12583 w/o Fixed-Guidance 0.06315 +0.00095 0.12644
0.23202 0.23679 0.23424 0.23725
Efficiency and Complexity. ECHO-Scale uses 76.9K parameters and 136 MiB peak GPU memory, versus 7.85M and 397 MiB for ECHO-Delta. Mean end-to-end latency is 14.43, 42.24, and 39.42 ms/case for Fixed, ECHO-Scale, and ECHO-Delta, respectively. ECHO-Scale is therefore substantially lighter in parameters and memory, while the two learned variants have comparable latency in the current implementation. 5. CONCLUSION We studied second-look sea ice forecast correction from sparse intermediate observations and identified uniform propagation extent as a key limitation of Fixed Propagation. ECHO-Scale adapts this extent, while ECHO-Delta adds bounded residual refinement. Both outperform Fixed Propagation across all 96 standard settings; ECHO-Delta provides the best average accuracy, whereas ECHO-Scale offers stronger geometry-shift consistency at much lower complexity. Future work will extend the framework to broader forecast horizons and systems.
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