Calculating balance velocities with a membrane stress correction

Abstract

Inverse methods, where surface data are 'inverted' in order to quantify basal properties of ice sheets, play a major role in initializations. The balance-velocity method is a unique linear initialization, in which accumulation, surface elevation and thickness data are used to calculate the velocities and basal conditions required to maintain the observed ice-sheet altimetric signal, resulting in an estimate of the basal sliding viscosity that is guaranteed to be non-negative. We examine the observation that balance velocities based on the shallow-ice approximation (SIA) are extremely dependent on grid size, showing that Antarctic balance velocities on a 1km grid are excessively overchannelized. Incorporating the membrane-stress approximation into balance-velocity calculations and comparing them with a simplified analytical solution shows that numerical error monotonically decreases with grid resolution and over-channelization is eliminated for Newtonian and non-Newtonian rheology. In contrast, for the SIA reducing grid size below the membrane-stress coupling length fails to improve accuracy. However, since this approach is nonlinear, a unique viscosity solution is not guaranteed, and in practice 'sliding viscosity' estimates are noisy. This raises problems of the sensitivity of these estimates to data and model errors, which may mean using inverse or smoothing techniques in association with balance-velocity methods in many, if not all, practical applications.