We have analyzed topography tracks over the Mars northern polar cap (composed of H2O ice, CO2 ice, and sediments) and the Antarctic ice sheet on Earth using a Mexican hat wavelet transform. The great utility of the wavelet transform is that it gives both spatial and spectral resolution. We have used the variance of the wavelet output in order to quantify the spectral content of the topography tracks. At short wavelengths, for both the Mars northern polar cap and the Antarctic ice sheet, we find a power law dependence of the wavelet transform variance on wavelength, with a power law exponent of β ≈ 3.5-3.7. This compares with a power law exponent of ~ 2.0 (Brownian motion), typical of topography on both Mars and Earth. There is a power law smoothing of the ice topography at short wavelengths on both planetary bodies. At long wavelengths we infer a similar power law dependence with β ≈ 1.5-2.0, typical of planetary topography. A transition is observed between the smooth short-wavelength behavior (high β) to rougher long-wavelength behavior (low β). This fairly sharp transition occurs at a wavelength of ~ 24 km for Mars and ~ 11 km for Antarctica. The transitions appear to scale with ice thicknesses and suggest a relaxation of short-wavelength topography by ice flows or surface processes.