Simulated Crater Degradation Based on Chebyshev Coefficients

Introduction: Impact craters on the Moon (and other bodies) form and degrade over time resulting in a change in crater shape and hence an overall evolution in lunar topography. Modeling of crater erosion (e.g. [1, 2, 3]) enables the tracking of crater shape evolution with time and can be used to estimate the relative age of a particular crater. Intuitively, crater degradation results from the cumulative effect of micrometeorite bombardment such that the overall nature of the process is diffusional. Theoretical basis of a diffusional model [2] for crater degradation relies on the assumption of the impactor size being smaller than the scale of topography affected, leading to steady erosion of a surface. Topographic diffusion modeling [2, 3, 4] is based on the application of two-dimensional diffusion to a 2-D polynomial representation of crater topography. Diffusion is applied iteratively (each iteration corresponds to one diffusion cycle/ evolution time period) and the modified crater topographic profile is compared to true elevation profiles (taken from craters over a broad range of degradation states) to estimate the diffusion based age. Thus, crater evolution can be tracked as a sequence of elevation profiles (forward modeling) but given a crater elevation profile at a particular degradation stage, the origin (starting state) for a crater is obtained from a dictionary of crater shape profiles. Direct reverse modeling is difficult since the modeling process is unconstrained (application of an accumulation/integral function to a 2D surface). To predictively model crater shapes (both forward and backward), a standardized framework is required to represent crater shapes and such a representation must support the application of topographic diffusion. In this work we show that the shape change of lunar craters due to systematic degradation can be represented as a systematic and predictable state change, when the morphological state is represented by Chebyshev coefficients. The topography of nearly all lunar craters can be represented by a relatively small set of Chebyshev polynomials [5]. In this representation, the individual polynomial functions are scaled and summed to approximate the actual crater elevation profile, and the scaling factors are the Chebyshev coefficients. Thus a crater is represented as a set of numbers. When diffusion is applied, the resulting degraded crater can be represented with another set of numbers. Since each Chebyshev coefficient is associated with a unique Chebyshev polynomial function, the contribution of that Chebyshev polynomial function can be tracked via the corresponding Figure 1: Simulated degradation by Chebyshev coefficients