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Floating ice shelves that fringe the coast of Antarctica resist the flow of grounded ice into the ocean. One of the key factors governing the amount of flow resistance an ice shelf provides is the rigidity (related to viscosity) of the ice that constitutes it. Ice rigidity is highly heterogeneous and must be calibrated from spatially continuous surface observations assimilated into an ice-flow model. Realistic uncertainties in calibrated rigidity values are needed to quantify uncertainties in ice sheet and sea-level forecasts. Here, we present a physics-informed machine learning framework for inferring the full probability distribution of rigidity values for a given ice shelf, conditioned on ice surface velocity and thickness fields derived from remote-sensing data. We employ variational inference to jointly train neural networks and a variational Gaussian Process to reconstruct surface observations, rigidity values and uncertainties. Applying the framework to synthetic and large ice shelves in Antarctica demonstrates that rigidity is well-constrained where ice deformation is measurable within the noise level of the observations. Further reduction in uncertainties can be achieved by complementing variational inference with conventional inversion methods. Our results demonstrate a path forward for continuously updated calibrations of ice flow parameters from remote-sensing observations. 

Abstract Accurate representation of the viscous flow of ice is fundamental to understanding glacier dynamics and projecting sea-level rise. Ice viscosity is often described by a simple but largely untested and uncalibrated constitutive relation, Glen’s Flow Law, wherein the rate of deformation is proportional to stress raised to the power n . The value n  = 3 is commonly prescribed in ice-flow models, though observations and experiments support a range of values across stresses and temperatures found on Earth. Here, we leverage recent remotely-sensed observations of Antarctic ice shelves to show that Glen’s Flow Law approximates the viscous flow of ice with n  = 4.1 ± 0.4 in fast-flowing areas. The viscosity and flow rate of ice are therefore more sensitive to changes in stress than most ice-flow models allow. By calibrating the governing equation of ice deformation, our result is a pathway towards improving projections of future glacier change.