Technical Summary: Uncertainty Quantification for EO Regression Tasks

Problem Statement

Earth Observation (EO) regression tasks, specifically building height, tree canopy height, and above-ground biomass (AGB) estimation, are critical for urban planning, forest monitoring, and climate policy. However, standard deep learning models typically yield deterministic predictions (single scalar values per pixel) without indicating the reliability or confidence of those predictions. This lack of uncertainty quantification is particularly problematic in EO due to inherent challenges such as heterogeneous land surfaces, skewed target distributions, sensor noise, atmospheric interference, and signal saturation at high target values.

While uncertainty estimation is well-established in computer vision (often using Bayesian approximations or ensembles), its systematic application to EO regression remains limited. Furthermore, regression tasks present unique difficulties compared to classification: continuous target spaces with skewed or heavy-tailed distributions make defining calibrated uncertainty measures challenging. Standard Gaussian assumptions may fail to capture asymmetric uncertainties common in EO data, necessitating alternative approaches.

Methodology

The authors propose modeling aleatoric uncertainty (irreducible noise in data) using year-long time series of Sentinel-1 SAR and Sentinel-2 MSI imagery (12 months, 10m resolution). They introduce two complementary deep learning approaches sharing a common backbone but differing in their uncertainty headers:

1. Network Architecture

• Backbone: A shared CNN encoder based on ResNet50 processes the 12-month time series (T=12, C=9 channels). Features are mapped to a latent space and processed by two consecutive Swin Transformer blocks using shifted window attention to capture spatiotemporal dependencies.
• Decoder: A multi-level decoder (L1, L2, L3) enhances features with temporal attention and upsamples them to generate pixel-wise predictions.

2. Gaussian Uncertainty (Gaussian UC)

• Formulation: A parametric approach assuming the target follows a Gaussian distribution. The model jointly predicts the mean ($\mu$) and the log standard deviation ($\eta = \log \sigma$).
• Loss Function: Trained using a weighted sum of:
  • Negative Log-Likelihood (NLL): The primary loss, prioritizing nonzero ground-truth pixels.
  • Sobel Edge Loss: Preserves boundaries and suppresses blur.
  • RMSE: Acts as a regularizer for the mean prediction.

3. Quantile Uncertainty (Quantile UC)

• Formulation: A non-parametric approach estimating the 10th, 50th, and 90th conditional quantiles ($Q_{10}, Q_{50}, Q_{90}$). This allows for asymmetric prediction intervals, better suited for skewed distributions.
• Loss Function: Trained using a weighted sum of:
  • Pinball Loss (Quantile Loss): Minimizes error across the three quantiles.
  • Regularization Terms:
    • Coverage Shortfall: Encourages the empirical coverage of $[Q_{10}, Q_{90}]$ to approach a nominal 80%.
    • Non-Crossing: Enforces $Q_{10} \leq Q_{50} \leq Q_{90}$.
    • Confidence-Adaptive Spread: Shrinks interval width when the median prediction is accurate.
  • Sobel and RMSE: Applied to the median ($Q_{50}$) to preserve edges and stabilize estimates.

4. Evaluation Metrics

• Accuracy: RMSE, $R^2$, MAPE, nMAD (Normalized Median Absolute Deviation), and IoU.
• Uncertainty Calibration:
  • ErrorCoverage: For Gaussian UC, measures the percentage of residuals falling within $k\sigma$ ($k=1, 2, 3$).
  • DataCoverage: For Quantile UC, measures the percentage of ground truth values falling within the $[Q_{10}, Q_{90}]$ interval.

Key Contributions

1. Comparative Framework: The paper systematically compares Gaussian and Quantile formulations for aleatoric uncertainty in EO regression.
2. Multi-Task Evaluation: Both approaches are evaluated on three high-impact tasks: Building Height Estimation (BHE), Canopy Height Estimation (CHE), and Above-Ground Biomass (AGB) estimation, utilizing fused SAR and multispectral time series.
3. Calibration Metrics: The authors demonstrate the use of "ErrorCoverage" and "DataCoverage" to rigorously evaluate the calibration of predicted uncertainty intervals.
4. Performance Benchmarking: The study shows that uncertainty-aware models can achieve accuracy comparable to deterministic baselines while providing interpretable confidence estimates.

Results

The models were tested on datasets covering Estonia, Finland, Germany, the Netherlands, and Switzerland.

• Accuracy: Both models achieved competitive results.
  • Canopy Height & Biomass: The Gaussian UC model performed best, with CHE achieving an RMSE of 2.33m and AGB an RMSE of 30.14 tons/pixel. $R^2$ scores were high (0.80–0.90), indicating efficient learning of vegetation variations.
  • Building Height: The Quantile UC model slightly outperformed the Gaussian model (RMSE 2.37m vs. 2.56m, $R^2$ 0.50 vs. 0.42). The authors attribute this to the heterogeneity of urban environments, where asymmetric intervals better capture localized biases and non-Gaussian distributions.
• Calibration:
  • Gaussian UC: Error coverage rates were close to nominal Gaussian expectations (e.g., ~68% within $\pm1\sigma$, ~95% within $\pm2\sigma$). However, in some high-value regions, the model inflated uncertainty to compensate for tail misfits.
  • Quantile UC: Data coverage for the $[Q_{10}, Q_{90}]$ interval was approximately 80% across all tasks (0.79–0.82), indicating well-calibrated asymmetric intervals.
• Distribution Characteristics: The results highlighted that building height and biomass errors often exhibit right-skewed distributions (systematic underestimation at high values), which the Quantile model captured effectively via wider upper intervals ($Q_{50}$–$Q_{90}$) compared to lower intervals.

Significance and Claims

The paper claims that explicit modeling of aleatoric uncertainty is indispensable for reliable EO inference, particularly given the noise and saturation inherent in satellite data. The authors assert that:

• Uncertainty estimation models can match or surpass deterministic benchmarks in accuracy while providing operationally useful confidence estimates.
• The choice between Gaussian and Quantile formulations depends on the target domain: Gaussian assumptions work well for smoother vegetation targets, while Quantile regression is superior for complex, skewed urban environments.
• The proposed models outperform the current 10m state-of-the-art uncertainty-aware model specifically for canopy height estimation.
• The work addresses a critical gap in EO by moving beyond single-value predictions to provide calibrated, interpretable uncertainty maps essential for decision-making in climate policy and urban planning.

The implementation is made available to the community to facilitate further research in this domain.