  The velocity field and the height at the surface of a dynamic ice sheet are
observed. The ice sheets are modeled by the full Stokes equations and shallow
shelf/shelfy stream approximations. Time dependence is introduced by a
kinematic free surface equation which updates the surface elevation using the
velocity solution. The sensitivity of the observed quantities at the ice
surface to parameters in the models, for example the basal topography and
friction coefficients, is analyzed by first deriving the time dependent adjoint
equations. Using the adjoint solutions, the effect of a perturbation in a
parameter is obtained showing the importance of including the time dependence,
in particular when the height is observed. The adjoint equations are solved
analytically and numerically and the sensitivity of the desired parameters is
determined in several examples in two dimensions. A closed form of the
analytical solutions to the adjoint equations is given for a two dimensional
grounding line migration problem in steady state under the shallow shelf
approximation.
