Mathematics – Logic
Scientific paper
Sep 2008
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2008epsc.conf..237k&link_type=abstract
European Planetary Science Congress 2008, Proceedings of the conference held 21-25 September, 2008 in Münster, Germany. Online a
Mathematics
Logic
2
Scientific paper
Introduction The HiRISE camera imaged the Mars surface at scales that had never been studied before. Beside a host of other fascinating features, these images revealed small (diameter D down to 1 m) impact craters. In planetary geology, impact craters and properties of their populations have been used as valuable sources of information about surface history and geological processes. Small craters on Mars can potentially give essential information about young terrains on this planet, resurfacing rates at small scales and the most recent events in the geological history, first of all, the most recent climate changes. Very young crater populations are thought to be unaffected by distal secondary craters, because they are formed after the most recent secondary-forming event. However, extracting this information is not simple or straightforward. Here I illustrate these difficulties and ways of overcoming them using a population of small craters on ejecta of crater Zunil as an example. Population of small craters on Zunil ejecta Terrain I used HiRISE images PSP_001764_1880 and PSP_002397_1880. In these images I outlined an area (totally 52.8 km2) to NE, NW and SW of the crater limited by the toes of the outer walls of Zunil and the image boundaries. Terrain texture within the area is diverse; however, the area is entirely within the proximal ejecta lobes. The ejecta material was obviously emplaced as a result of the Zunil-forming impact and has a uniform age. The morphology of the surface indicates later resurfacing of steep slopes (over a small total area) and minor eolian modification of the terrain; some sub-areas might be modified by the post-impact hydrothermal activity. Crater population I registered diameters and positions of all impact craters in the area, a total of 1025 craters with D > 1.5 m. The largest of them has D = 20 m. Craters usually have no visible ejecta, which indicates some minor (perhaps, eolian) modification of the surface. Almost all craters have flat floors due to infill with loose material (only a few craters have pristine bowl-shaped floors). Thus, the most prominent process of crater modification is deposition of loose wind-transported material (sand and dust). However, the total number of recognisable craters with partly buried rims is small; it looks like the accumulation of sand and dust effectively fills depressions only, while the total accumulation is modest. This suggests that the number of obliterated craters is small, especially among larger craters. Clustering due to atmospheric break-up Some craters in the population form more or less tight clusters. These clusters are formed due to the break-up of projectiles in the atmosphere [1]. The morphology of overlapping craters is perfectly consistent with simultaneous impacts of fragments of the same projectile. The largest cluster contains 44 craters and reaches ~400 m in size, which is noticeably greater than predicted for the atmospheric break-up in [1] (~50 m) and observed for 20 impacts that have occurred during the last decade [2] (<100 m, [1]). The largest cluster(s) can be a superposition of two clusters formed by different projectiles, or the separation of the fragments can be greater due to periods of higher atmospheric pressure in the recent past. For the purposes of age estimates each cluster should be considered as a single impact event. I ran a "clustering" algorithm, which repeatedly searches for the tightest pair of craters and replaces it with an "effective" crater with diameter Deff = (D1 3+D2 3)1/3 located between the original craters. The process was stopped when the separation between craters in the tightest pair reached 40 m. This limit was consistently deduced from: (1) visual comparison of plots of frequency distributions of the nearest-neighbourdistance for the actual population and simulated purely random spatial scattering; (2) application of the "clustering" algorithm to purely random simulations and comparison of the frequency distributions of the nearest-neighbour-distance with the result for the actual population; (3) results of modelling of atmospheric break-up [1]. The "clustering" algorithm resulted in a population of 698 craters and "effective" craters representing clusters. For some clusters the 40 m separation limit is insufficient; for example, the largest cluster after applying the "clustering" algorithm is reduced to 3 "effective" craters and 1 single crater. On the other hand, comparison with the purely random simulations shows that several pairs in the population are merged erroneously (they have a small separation just by chance). The error in the total number of independent impact events, however, is well below 10%. For denser populations of small craters (for older terrains) the overlap of clusters produced by different projectiles would preclude identification of individual impact events; this would bring much greater uncertainty in the age considerations. The majority of the craters after the "clustering" procedure remain single. Among clusters identified by the "clustering" algorithm, pairs dominate. Only 23 formally identified clusters contain 5 or more craters. Among 19 craters with Deff > 10 m, 12 are "effective" craters representing pairs or multiple craters. This proportion is lower than observed for the latest impacts [1]; in the latter case craters smaller than 1.5 m are identifiable [1]; this explains the discrepancy. Spatial randomness To test spatial randomness I compared some statistics of the actual population and a set of simulated purely random populations, all having undergone the "clustering" algorithm. In particular, I used the standard deviation of the nearest neighbour distance and the interquartile amplitude of the adjacent area (see [3] for details). These tests do not reject spatial randomness of the actual population. Size-frequency distribution I applied the technique from [4] to find simultaneously the maximum-likelihood power-law fit for the cumulative size-frequency distribution (SFD) (after "clustering") and its low-diameter cut-off Dmin. This technique gave a rather good fit for Dmin = 4.85 - 4.95 m and power-law exponent α = 3.16 - 3.20. The latter values coincide perfectly with the typical slope of the Neukum production function (NPF) for Mars [5] for the smallest diameters D < 100 m (the NPF has been defined only for D > 10 m). Thus, my observations give grounds for power-law extrapolation of the NPF down to D = 5 m. For D < 5 m the observed SFD is progressively gentler, which can be caused by difficulty in identification of small craters in rough terrains and possible obliteration (burial) of small craters. Age constraints from the crater population The density of craters larger than D N(D) has been widely used to establish stratigraphic relationships between terrains and to estimate absolute ages. Such inferences assume that crater emplacement can be considered as a Poisson process with a known rate R(D) per unit area. The use of N(D), however, is not straightforward; many additional considerations are necessary for meaningful and reliable inferences. Crater obliteration. N(D) gives an estimate of the crater retention age. We can identify this age with the terrain age, if we have reasons to neglect obliteration of craters. A steep SFD is a good reason for such an assumption: the crater obliteration rate is higher for smaller craters, and if the obliteration is significant, one should expect the resulting SFD to be gentler than the production function. For the case of Zunil ejecta, the SFD suggests the use of N(D=5m). Morphological observations (see above) also suggest minor crater obliteration; nevertheless, some crater rims can be buried, and it is probable that N(D=5m) underestimates the terrain age. My subjective guess based on the morphology is that this bias is less than ~20-30%. Formal statistical error. The observed number of craters M(D) = A N(D) in an area A can be used to obtain a confidence interval for the average crater retention age T: 1(1- ; ) < ṡ ṡ < -1( ; +1) Γ - FΓ p M T A R F p M , where R is the cratering rate
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