A critique of methods for analysis of the diameter-frequency relation for craters with special application to the Moon

The incremental and cumulative methods of analyzing data on the diameter-frequency relationship for craters are contrasted. The traditional cumulative method is found to lead to serious inconsistencies for two basic reasons: (1) The diameter-frequency relation of the form N = ADB (N the number of craters per square kilometer per kilometer diameter increment, D the diameter in kilometers) is not expected to, and does not in reality, have a constant B; B varies with diameter in a manner to which the cumulative frequency distribution is insensitive; (2) cumulative crater counts do not form a normal distribution of independent points and therefore cannot be fitted by straightforward least-squares methods without leading to spurious values of B because of the propagated error. A much improved least-squares method is developed to determine the incremental frequency distribution. The method includes proper weighting and the proper determination of the average diameter of craters in a given diameter increment. When crater counts subject to observational incompleteness are properly excluded from the fitting, a computer program based on the above least-squares methods yields consistent results for the highlands and for the maria from various independent sets of crater data. It is found that there is a continuing variation with diameter in the value of B (varying from about −4.3 to −2.0 for maria craters and from about −2.2 to −2.9 for highland craters), which makes it imperative that the diameter range be defined when discussing frequency relations and which makes extrapolations into unobserved diameter ranges very questionable.