fenics_ice 1.0: a framework for quantifying initialization uncertainty for time-dependent ice sheet models

Mass loss due to dynamic changes in ice sheets is a significant contributor to sea level rise, and this contribution is expectedto increase in the future. Numerical codes simulating the evolution of ice sheets can potentially quantify this future contribution.However, the uncertainty inherent in these models propagates into projections of sea level rise, and hence is crucialto understand. Key variables of ice sheet models, such as basal drag or ice stiffness, are typically initialized using inversionmethodologies to ensure that models match present observations. Such inversions often involve tens or hundreds of thousandsof parameters, with unknown uncertainties and dependencies. The computationally intensive nature of inversions along withtheir high number of parameters mean traditional methods such as Monte Carlo are expensive for uncertainty quantification.Here we develop a framework to estimate the posterior uncertainty of inversions, and project them onto sea level change projections over the decadal timescale. The framework treats parametric uncertainty as multivariate Gaussian, and exploits theequivalence between the Hessian of the model and the inverse covariance of the parameter set. The former is computed efficientlyvia algorithmic differentiation, and the posterior covariance is propagated in time using a time-dependent model adjointto produce projection error bars. This work represents an important step in quantifying the internal uncertainty of projectionsof ice-sheet models.