Spatio-temporal models for uncertainty quantification in ice-sheet models using Gaussian random fields

Assessing the impact of uncertainties in ice-sheet models is a major and challenging issue that needs to be faced by the ice-sheet community to provide more robust and reliable model-based projections of ice-sheet mass balance. In recent years, uncertainty quantification (UQ) has been increasingly used to characterize and explore uncertainty in ice-sheet models and improve the robustness of their projections. A typical UQ analysis involves first the characterization of the sources of uncertainty followed by the propagation and sensitivity analysis of these sources of uncertainty. Previous UQ analyses in glaciology have usually focused on the last two steps but paid relatively little attention to the preliminary and critical step of the characterization of uncertainty. Sources of uncertainty in ice-sheet models, like uncertainties in ice-sheet geometry or surface mass balance, typically vary in space and possibly in time. For that reason, these sources of uncertainty are more adequately described as spatio(-temporal) random fields, which allow to account for spatial (and temporal) correlation. As a means of improving the characterization of the sources of uncertainties in ice-sheet models, we propose to represent them as a Gaussian random field (GRF) with Matérn covariance function. The class of Matérn covariance functions provides a flexible model that can capture statistical dependence between locations with different degrees of spatial correlation or smoothness properties. Samples from a GRF with Matérn covariance function can be generated efficiently by solving a certain stochastic partial differential equation (SPDE). Discretization of this SPDE by the finite element method results in a sparse approximation known as a Gaussian Markov random field. We solve this equation efficiently using the finite element method within the Ice-sheet and Sea-level System Model (ISSM). In addition, spatio-temporal samples can be generated by combining an autoregressive temporal correlation and the Matérn field. We show the interest of this sampling capability in an illustration concerned with assessing the impact of various sources of uncertainties on the Pine Island Glacier, West Antarctica. We find that both larger spatial and temporal correlations will likely result in increased uncertainty in projections.