An estimator of the underlying size distribution of overlapping impact-craters |
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Authors: | WW Mullins |
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Institution: | Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA |
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Abstract: | The stochastic model of lunar type impact-crater formation which assumes (a) random impacts, (b) circular craters, each obliterating any portions of earlier craters lying within, and (c) a probability Pi(t) that a newly formed crater (primary or secondary) has an area ai is analyzed to develop a method of estimating Pi from the final overlapping pattern. It is found that if each crater is weighted by the fraction of the rim which is visible and which lies in an observation area A, then the expected value of the weighted sum of craters of area ai is simply proportional to Pi for any degree of coverage under several conditions, including (a) constant Pi for all i, and (b) Pi stepping from a constant early value to zero (for some i's) with otherwise arbitrary bombardment. Furthermore, in the general case, the expected value of the contribution produced during t0 ± Δt/2 is found to be proportional to Pi(t0). Thus measurement of in the first two cases, or of if crater age data is available in the last case, provides an estimate of the desired Pi. Therefore the introduce the correct weighting factors that just compensate for the effect of overlap.Expressions for the variances of are derived from which it is shown that under the above conditions, are consistent estimators of Pi. Formal evaluation of the variances is carried out in the special case of constant Pi and no secondary cratering. A criterion for the degree of coverage is given; in particular it is shown that the expectation of at saturation is just A. |
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