Theoretical limitations to englacial velocity calculations

To study the dynamics of ice sheets and glaciers, velocities at the bed of a glacier must be measured directly or calculated using data gathered from boreholes and surface surveys. Boreholes to the bed are expensive and time-consuming to drill, so the determination of basal velocity is almost exclusively by numerical inversion of velocities observed at the surface. For non-linearly viscous glaciers, a perturbation analysis demonstrates that inversions for englacial velocities will magnify measurement errors at an exponential rate with depth. The rate at which calculation errors grow is proportional to a Lyapunov exponent, a measure of “information loss” which is shown to be a simple linear function of spatial frequency with a coefficient depending on Glen’s flow-law exponent, n. The coefficient decreases with increasing non-linearity, demonstrating that inversions with non-linearly viscous ice have smaller calculation errors than inversions with linearly viscous ice. In both the linear and nonlinear cases, the Lyapunov exponent (and rate of error growth) increases with decreasing wavelength, which limits velocity calculations at the bed to wavelengths on the order of one ice thickness or greater. This limitation is theoretical and cannot be countered by more accurate survey data or special numerical techniques.