A principled stopping criterion for the reconstruction of basal properties in ice sheets

A crucial assumption in all ice sheet models concerns the nature and parametrization of the basal boundary condition. Direct observations on large spatial scales are not possible, but inverse methods can be used to determine the distribution of a basal friction parameter from surface measurements. Such reconstructions are not unique -- a range of substantially different basal friction parameter distributions will lead to surface data that match observation to within measurement and model error levels. Additional information must be used to reconstruct a reasonable distribution. We present iterative methods that aim to determine parametrizations that fit observed surface velocities to within a given tolerance. We compare steepest descent, nonlinear conjugate gradient, and incomplete Gauss-Newton algorithms. These iterative techniques have the common property that they tend to correct large-scale features first, and thereby generate minimally featured distributions that are consistent with surface observations. The additional information needed by these methods is an initial estimate for the friction parameter and the desired tolerance for matching observation. We test our methods by creating synthetic data sets and running them through a forward model, the Shallow Shelf Equations. The resulting surface velocity is then perturbed and the inverse methods applied to the perturbed fields. We show that the use of an appropriate and principled stopping criterion avoids fitting noise in the data and leads to a smooth parametrization of basal friction. We also demonstrate the effect of different initial guesses.