Historically, depth (d) and diameter (D) relations are predominantly described as a power law [4], 3rd order log-domain polynomials [1] or linear fits [3]. Such relationships obtained using the Apollo era data are deterministic (data model pre-decided before fitting) and non-causal. Accordingly, data points can be fitted to multiple deterministic relationships with yet undetermined confidence zones (since topographic data is still being collected, and of better quality). As an alternate method of understanding crater formation and associated morphology we propose probabilistic modeling that does not hypothesize a relation between physical quantities. Reasons for this are : (a) It is not deterministic, so it does not coerce us into believing a relationship that does not hold true with more data;(b) A probabilistic model can be used to infer deterministic relations, which then improve the existing model; (c) The model allows incorporation of additional parameters that allow relating causes and effects and also help determine the more-likely causes (and effects). This ultimately leads to a Bayesian network which operates on the basis of conditional probability.

To illustrate the proposed method, a simple Bayesian network is constructed from an existing impact crater database [2] and relations between depth, diameter and relative age is analyzed. Large craters (30 km <D< 100 km) whose relative age was predetermined from the sources [5,6] were selected. The depth, diameter, relative age information was used to model the joint probability distribution of the data as a 3-component Gaussian mixture model (GMM). Specifically, three non-overlapping ranges were constructed for the d/D ratio : (0 < d/D <0.08), (0.08 < d/D <0.14), (0.14 < d/D <0.2) and three relative age classes were defined : 1- Pre-Nectarian and Nectarian, 2-Lower and upper Imbrian, 3- Eratosthenian and Copernican and a graphical map connected the d/D values to relative age classes. The graph links were marked with the conditional probability values P(Age|d/D) which is the probability of a relative age classification being true given the d/D condition. The probability of an age class is then given by the product of the conditional probability P(Age|d/D) and the probability of a particular d/D value (P(d/D)). For the fixed scenario where the choice of d/D is pre-fixed - P(d/D) = 1 and hence P(Age) = P(Age|d/D). As an example, for the range (0 < d/D <0.08) it is found that there is ~85% probability that craters 30km - 100km are Pre-Nectarian or Nectarian. Similar probabilities were obtained for all the 9 (3x3) graph links.

The described simple Bayesian network that is by no means complete, but illustrates the potential of the proposed method in the context of ever-increasing high resolution lunar data in recent times.

[1] R. Baldwin (1963) [Chicago]University of Chicago Press

[2] Losiak, A., et al. (2009) LPSC 2009, 1532

[3] C. Elachi, et al. (1976)Earth, Moon, and Planets 15(1):119.

[4] R. Pike (1967) Journal of Geophysical Research 72(8):2099.

[5] Wilhelms, D.E., 1987, U.S. Geological Survey Professional Paper 1348.

[6] Wilhelms, D.E., and Byrne, C.J. (2008)