Uncertainty propagation from Sentinel-2A/B-derived velocity to glacier strain rates: a first-principles perspective

Satellite remote sensing has become a cornerstone for monitoring glacier surface velocity and strain rates, with normalized cross-correlation (NCC) techniques widely adopted due to their efficiency and robustness. However, current approaches to quantifying strain-rate uncertainty exhibit substantial limitations. They rely primarily on empirical statistics or classical error-propagation theory, which implicitly assumes spatially independent Gaussian velocity errors and fails to account for the numerical effects introduced by NCC algorithmic parameters and environmental conditions. As a result, strain-rate errors derived from NCC-based velocity fields are poorly characterised at sub-monthly timescales over rapidly evolving glaciers, and technical uncertainties cannot be effectively separated from systematic velocity-field mismatches, limiting the applicability of remote-sensing products in glacier dynamics studies.

 

We develop a first-principles uncertainty theory by explicitly modelling the fundamental error mechanisms underlying NCC-derived velocity measurements. Building upon the classical error-propagation framework, we combine ordinary differential equations and stochastic process theory to rigorously derive analytical error expressions for two commonly used strain-rate formulations applied to NCC-derived velocity fields: nominal strain rate and logarithmic strain rate. The theory demonstrates that, although the nominal strain-rate error shares a similar mathematical structure with classical error propagation, its coefficients are substantially smaller than those predicted by traditional formulations. In contrast, the logarithmic strain rate (based on the Nye model and Alley grid-based implementation) converges to the true strain rate under normal circumstances, while degenerating to the nominal strain-rate solution in the worst case.

 

We validate the theoretical predictions using Helheim Glacier, Greenland, as a test case. Surface velocities are extracted from 616 Sentinel-2A/B image pairs with time baselines from 1 to 32 days, followed by statistical analysis of strain-rate errors. Under controlled NCC failure rates, the theoretical model achieves a goodness of fit exceeding R > 0.8, confirming the robustness of the proposed framework.

 

Our results further reveal a strong dependence of strain-rate error on temporal baseline and pixel distinguishing capacity. For longer baselines (Δt > 18 days over Helheim Glacier), high-strain environments such as shear margins lead to a loss of image similarity, increasing NCC failure rates and inducing systematic velocity-field errors that cause strain-rate overestimation. For shorter baselines (Δt < 10 days), nominal strain rates are strongly limited by pixel distinguishing capacity, producing random non-zero velocity artefacts over stable terrain. Owing to the error attenuation behaviour of logarithmic strain rate, the effective lower bound of usable time baselines is reduced to approximately 3 days, enabling high-temporal-resolution monitoring. Based on the derived error equations, we propose a practical time-baseline selection guideline that constrains random strain-rate errors induced by technical uncertainty, while facilitating the separation of systematic velocity-field errors.

 

Overall, this work provides an end-to-end uncertainty quantification framework linking remote-sensing techniques to glacier strain-rate products, offering a theoretical foundation for quality control, uncertainty assessment, and data assimilation in next-generation glacier strain-rate monitoring.