Basal slipping at the ice-bed interface is a key parameter in ice sheet modelling because it can have a large impact on the ice thickness. Unfortunately, its effect on surface observations can be hard to distinguish from that of bed undulations. Therefore, inferring the ice thickness from surface measurements is an interesting, useful, and non-trivial inverse problem. This thesis develops methods for simultaneously recovering the ice thickness and the basal slip, using only surface elevation and velocity measurements.

The shallow ice approximation, a time-dependent non-linear partial differential equation for ice thickness evolution, is chosen to model ice flow. Using this model, synthetic surface data is produced for given bedrock and basal slip profiles. To invert the synthesised data, a restriction to unidirectional ice-flow is initially explored. First, a semi-analytical approach is developed and studied. Following its success, an optimisation based approach is implemented. This method requires less data than the first, and its formulation is not dependent on the unidirectional simplification. Finally, the optimisation framework is extended to two-dimensional ice flow. This method recovers a linearised diffusion coefficient which gives the best fit to observations. Combining this recovered diffusion coefficient with observed surface velocity, a simple optimisation is used to recover both the ice thickness and basal slip.

The methods were successful for each test case. In all cases, the errors occurring had a clear source. Causes for error included proximity to ice sheet margins or ice domes, and overestimation of basal slip, which returns a related underestimate in ice thickness. The results for two-dimensional flow, show that while the inverse problem is challenging, it is possible to recover both of these parameters in certain scenarios and hence, the methods presented in this thesis can be useful for real ice flows.