Variational inference at glacier scale

We characterize the joint Bayesian posterior distribution over spatially-varying basal traction and ice rheology of an ice sheet model from observations of surface speed using stochastic variational inference, the first application of such methods to large-scale fluid simulations subject to real-world observations. Assuming a low-rank Gaussian process posterior, we use natural gradient descent to minimize the Kullback-Leibler divergence between this assumed distribution and the true posterior. By also placing a Gaussian process prior over traction and rheology, and by casting the problem in terms of eigenfunctions of a kernel, we gain substantial control over prior assumptions on parameter smoothness and length scale, while also rendering the inference tractable. In a synthetic example, we find that this method recovers known parameters and accounts for situations of parameter indeterminacy. We also apply the method to Helheim Glacier in Southeast Greenland and show that the proposed method is computationally scalable to catchment-sized problems. We find that observations of fast flow provide substantial information gain relative to a prior distribution, however even precise observations offer little information in slow-flowing regions. The approach described here is a road-map towards robust and scalable Bayesian inference in a wide array of physics-informed problems.