Internal isochronic layers in ice sheets sensed by radar show two characteristic relationships to the basal topography: Either they override it, with layers above the crests of rises lying essentially flat, or they drape over it, with the layers following rises and falls in basal topography. A mechanical theory is presented which shows that overriding is the expected behavior when topographic wavelengths are comparable with or less than the ice thickness, while draping occurs at longer wavelengths. This is shown with analytical perturbation solutions for Newtonian fluids, numerical perturbation solutions for nonlinear fluids, and finite element solutions for nonlinear fluids and large-amplitude variations. Bed variation from topography and changes in the basal boundary condition are considered, for fixed bed and sliding beds, as well as three-dimensional flows and thermomechanically coupled flows. In all cases, the dominant effect on draping/overriding is the wavelength of the topography or variation in basal boundary conditions. Results of these full mechanical system calculations are compared with those from the shallow ice approximation and the longitudinal stress approximation. Some calculations are carried out for zero accumulation, where the age of the ice and therefore isochrone geometry is not defined. It is shown that there is a close relationship between isochrones and streamlines, and that they behave similarly when bed wavelength divided by the ice thickness is small compared with the ratio of ice velocity and accumulation rate, which is a useful approximation. Numerical comparisons of isochrones and streamlines show them to be virtually coincident.