Planetary impact crater analysis with eigenfunction expansion

High‐resolution topography has recently become available for a number of planetary bodies such as the Moon, Venus, Mercury and particularly Mars, with the 32nd‐degree global Martian topography from the Mars Orbital Laser Altimeter on Mars Global Surveyor . Gridded digital elevation models (DEMs) of impact craters can be extracted from these data, and provide an extensive record of planetary impact crater morphologies. It may not be immediately obvious, however, how crater DEMs can be compared, particularly if there are many thousands of individual measurements. Comparison is greatly simplified if the measurements are reduced to an eigenfunction expansion, using the coefficients of expansion for quantitative shape comparison. Four eigenvalue expansions are compared for their suitability: a one‐dimensional Fourier sine expansion of a crater cross‐section; a two‐dimensional Fourier sine expansion; the eigenfunctions of a vibrating circular membrane; and the Zernike polynomials. All are found to be suitable except the two‐dimensional Fourier expansion, which fails to converge well on the data because of inappropriate geometry. Expansion spectra of four Martian impact craters, each representing a different class of planetary crater morphology, are calculated with the three suitable methods. The relevance of symmetry (about the crater centre for cross‐sections and radial symmetry for two‐dimensional expansions) is discussed. Finally, a preliminary survey of Martian impact crater shapes is made, using eigenfunction expansion, which shows three distinct clusters of Martian crater morphology.