Sensitivity of ice sheet surface velocity and elevation to variations in basal friction and topography in the Full Stokes and Shallow Shelf Approximation frameworks

Abstract. Predictions of future mass loss from ice sheets are afflicted with uncertainty, caused, among others, by insufficient understanding of spatio-temporally variable processes at the inaccessible base of ice sheets for which few direct observations exist and of which basal friction is a prime example. Here, we use an inverse modeling approach and the associated time-dependent adjoint equations, derived in the framework of a Full Stokes model and a Shallow Shelf/Shelfy Stream Approximation model, respectively, to determine the sensitivity of ice sheet surface velocities and elevation to perturbations in basal friction and basal topography. Analytical and numerical examples are presented showing the importance of including the time dependent kinematic free surface equation for the elevation and its adjoint, in particular for observations of the elevation. A closed form of the analytical solutions to the adjoint equations is given for a two dimensional vertical ice in steady state under the Shallow Shelf Approximation. There is a delay in time between a perturbation at the ice base and the observation of the change in elevation. A perturbation at the base in the topography has a direct effect in space at the surface above the perturbation and a perturbation in the friction is propagated directly to the surface in time. Perturbations with long wavelength and low frequency will propagate to the surface while those of short wavelength and high frequency are damped.