Gaussian Process Decorrelation for Spatiotemporal Deep Learning-Based Snow Water Equivalent Prediction
Authors: Colin Fenster, Adrienne Marshall, Soutir Bandyopadhyay, Daniel McKenzie
Organizations: Department of Applied Mathematics and Statistics, Colorado School of Mines · Hydrologic Science and Engineering Program, Colorado School of Mines
In the Western United States, snowmelt is essential to the agricultural industry in addition to being a key source of municipal drinking water. Consequently, accurate snowpack forecasting is critical for water policy and management. Automated Snow Telemetry (SNOTEL) stations provide accurate daily measurements of snow water equivalent (SWE) that exhibit strong correlations in space and in time. We tackle the problem of predicting future SWE values across the SNOTEL network. Specifically, we use a Gaussian Process-based linear transformation to remove spatial correlations before training a long short-term memory (LSTM) neural network on the decorrelated SWE data. This approach allows the LSTM to learn a clean temporal signal at each station. We show that this separation of spatial and temporal components yields better predictive success than multiple baseline models. Furthermore, we incorporate conformal prediction to quantify uncertainty in the resulting SWE forecasts, providing a distribution-free approach to illustrate a potential framework for establishing predictive intervals for spatiotemporal data. Together, accurate point forecasts and distribution-free uncertainty quantification provide a framework for SWE accumulation forecasting on subseasonal scales or projecting SWE with future data while motivating and supporting future work in predicting a large-scale, spatiotemporally complete SWE map.
Table 1: Comparison of overall forecasting performance and station-level NSE distributions between the spatiotemporal model and the climatological baseline.
Figure 2: Observed (blue) and predicted (orange) SWE time series for four representative SNOTEL stations during the held-out test period. The spatiotemporal whitening framework accurately reproduces seasonal accumulation and melt dynamics across multiple snow years.
Figure 3: Comparison of station-level forecasting performance between the implemented spatially whitened framework and an otherwise identical LSTM trained on the original spatially correlated data. Panel (A) shows the number of stations within each NSE range, while Panels (B) and (C) summarize the distribution and cumulative distribution of station-level NSE values, respectively.
Figure 4: Individual SNOTEL Station NSE Results
Metric
Value
NSE
−6.30
Mean SWE
0.903 in
Dataset Mean SWE
7.92 in
Mean Distance to 10 Nearest Neighbors
362 km
Neighbor Distance Rank
323/323
Table 2: Summary Statistics for the Signal Peak SNOTEL station.
Figure 5: Map of 323 SNOTEL stations with inset containing observed (blue) and predicted (orange) SWE time series for Signal Peak, NM.
Figure 6: Relative conformal half-width by snow day, computed by dividing the conformal half-width q1−α(t) by the mean SWE for each specific day.
Figure 7: Nominal 90% conformal prediction intervals for four representative SNOTEL stations during the held-out test period. The black line shows the observed SWE, the shaded region shows the conformal prediction interval, and the red dotted lines show the lower and upper interval boundaries.
Figure 8: Nominal 90% conformal prediction interval for Copper Camp during the held-out test period.
Metric
Value
Overall coverage
85.89%
Mean snow-day coverage
85.73%
Median snow-day coverage
86.38%
Minimum snow-day coverage
76.55%
Maximum snow-day coverage
92.95%
Mean interval half-width
3.89 in.
Table 3: PICP results for snow-day-specific conformal prediction intervals.
Reliable forecasting of several interrelated environmental variables - such as regional precipitation and temperature, or other correlated geophysical fields - across many locations calls for accurate predictions accompanied by trustworthy statements of their uncertainty. Modern deep-learning models forecast such variables accurately but usually report no uncertainty, and forcing them to output uncertainty through maximum likelihood tends to degrade their accuracy, especially when the variables are strongly correlated. Motivated by this tension, we develop TSCoNet, a two-stage convolutional-recurrent model coupled with a Gaussian copula that jointly forecasts multiple variables over space and time while quantifying predictive uncertainty. The method first learns accurate mean forecasts and then, holding the mean fixed, refines a shared representation to estimate the predictive variance, yielding calibrated prediction intervals after a standard recalibration, so that uncertainty is added without sacrificing point accuracy. We study the approach on simulated non-stationary spatial fields on the sphere and on a real dataset of monthly precipitation and temperature for fifty cities over 2000-2020. The model matches the accuracy of a strong deterministic forecaster while supplying calibrated prediction intervals that the deterministic model cannot, giving a single tool that provides both accurate point forecasts and reliable uncertainty for multivariate spatio-temporal data.
Jongwook Kim, Jong-Min Kim
Statistics Discipline, Division of Science and Mathematics, University of Minnesota-Morris
Managing water resources in mountainous regions depends heavily on reliable Snow Water Equivalent (SWE) and Snow Height (HS) data, yet these variables remain difficult to track at scale. This study evaluates three machine learning architectures (XGBoost, U-Net and SegFormer) for the joint estimation of SWE and HS variations from Sentinel-1 InSAR data over the Italian Alps, using the IT-SNOW reanalysis as reference. SegFormer achieves the best results on both targets, with an MAE of 10.391 cm for HS and 27.113 mm w.e. for SWE and the lowest variability across initializations. A feature sensitivity analysis shows that including all available features does not guarantee the lowest error, with model- and task-specific sensitivities. Spatial metrics (R2, Pearson's r) separate the three architectures far more clearly than mean error (MAE, RMSE) does, and decomposing the error per window attributes most of it to a systematic offset in the estimated mean variation rather than to the spatial pattern.
Luca Barco, Lorenzo Innocenti, Bianca Bartoli +3
Politecnico di Torino, Dipartimento di Automatica e Informatica, Torino, Italy · Fondazione LINKS, AI Data and Space, Italy
Residual error propagation remains a fundamental problem in recurrent models, where small prediction inaccuracies compound over time and degrade long-horizon performance. Accurately modeling the correlation structure of such residuals is critical for reliable uncertainty quantification in probabilistic multivariate timeseries forecasting. While recent time-series deep models efficiently parametrize time-varying contemporaneous correlations, they often assume temporal independence of errors and neglect spatial correlation across the observed network. In this paper, we introduce Teger, a structured uncertainty module that overcomes the spa- tial and temporal limitations of error-correlated autoregressive forecasting. Teger proposes a spatial curvature-aware graph rewiring mechanism explicitly strengthening information-bottleneck edges identified by discrete Forman curvature. The component is integrated into a low-rank-plus-diagonal covariance head, preserving tractable inference via the Woodbury identity. Teger is backbone-agnostic, requiring only the latent state produced by any autoregressive encoder. We provide theoretical evidence of Teger, and experimentally evaluate it on LSTM, Transformer, and xLSTM backbones across four real-world spatio-temporal datasets, showing consistent improvement in Continuous Ranked Probability Score (CRPS). We further provide a formal theoretical analysis connecting curvature-aware rewiring to (i) oversquashing alleviation, (ii) improved spectral connectivity, (iii) reduced effective resistance, and (iv) improved covariance calibration bounds
Seyed Mohamad Moghadas, Esther Rodrigo Bonet, Bruno Cornelis +1