On the theoretical aspects of joint inversion for basal slipperiness and viscosity in ice-flow models

When modelling ice flows there are a number of properties of the ice which are poorly constrained by observations, in particular the ice rheology and basal slipperiness at the ice bed. Inversion methods are frequently used to estimate the distribution of these 'hidden' fields in computational ice flow models. These methods use surface measurement data in combination with a forward model of the ice dynamics that relate the hidden fields to the surface fields. In this study we use first-order linear perturbation theory to gain insights into our ability to extract information about the ice viscosity at the same time as basal slipperiness, and understand the theoretical limitations of this approach. We frame the typical inversion problem in terms of a Gaussian maximum a-posteriori estimation with explicitly stated priors for the hidden fields. We illustrate the inversion behaviour with perturbations applied to flow down a laterally confined channel, where both viscosity and slipperiness play a significant role in the ice sheet dynamics. Our results indicate that it is possible to extract information about the viscosity field at the same time as estimating the basal slipperiness, and that explicitly recognising uncertainty in the viscosity field is important.