Abstract

The only method of constraining regional surface ages on Mars at present is through use of the impact crater size-frequency distribution system developed by Hartmann and Neukum. The ideal size-frequency distribution follows a power-law distribution of the form log Y=log A+ b log X (where Y is number and X is crater diameter) until saturation occurs (where crater density is so great that one more impact obliterates all traces of an older crater). Ages of surfaces at saturation cannot be characterized any better than "very old." As A declines, the surface is deemed progressively younger. These curves are calibrated against lunar rock and soil samples from the Apollo and Luna missions to generate rough absolute ages.

Actual size-frequency curves, however, typically show two or three inflection points, changes in the steepness of b for a segment. The steepness is greatest for craters with a diameter below ~1 km for all size-frequency distributions or ages. Smaller craters are over-represented. One explanation suggested is secondary cratering: debris from an impact shot outward and upward, impacting some distance from the primary crater. Fresh craters suggest these fall in rays, but ray evidence deteriorates with time, making it hard to identify a given small crater with a given primary crater. This paper proposes a variant on nearest neighbor analysis and regression to detect linearities among size classes of small craters in a region to reconstruct missing ray structures and allow the size-frequency distribution to be adjusted for secondary impact distortions.

Keywords:

Mars, regionalization, crater count

Introduction

The only method of constraining regional surface ages on Mars at present is through use of the impact crater size versus frequency distribution system, developed by Arvidson et al. (1978), Tanaka (1986), Neukum and Ivanov (1994), and Hartmann and Neukum (2001), among others. [ SLIDE 2 ] The ideal size-frequency distribution follows a power-law distribution. The slope, b, is broadly consistent across martian surfaces, averaging about -1.8. As a and, thus, the height of the curve rises, the surface is deemed progressively older, until saturation occurs. Saturation is the steady-state equilibrium reached by extremely old surfaces, where craters are so dense that a new crater necessarily obliterates an older one. [ SLIDE 3 ] These curves are calibrated against lunar rock and soil samples from the Apollo and Luna missions to generate rough absolute ages, on the assumption that all bodies in the inner solar system experienced a similar bombardment in the early history of the system.

Actual size-frequency curves, however, typically show slight inflection points below saturation, changes in the steepness of b for a segment, one or two at each end of the diameter distribution (Hartmann 2005). The steepness increases at the right end of the distribution, turning down at a crater diameter of roughly 64 km, due to limitations on the number of giant impactors available in the relatively narrow temporal window when these were perturbed and sent into the inner solar system. The steepness on the left end of the distribution is greatest for craters with a diameter below ~1 km for all size-frequency distributions or ages. Smaller craters are, thus, over-represented. Over-representation implies that age estimates could be inflated for images with fine spatial resolution.

[ SLIDE 4 ] One explanation frequently proposed for over-representation of most of the sub-kilometer size range craters, and possible age inflation, is secondary cratering, This is debris from an impact falling some distance from the primary crater, creating smaller craters (e.g., Preblich, McEwen, and Studer 2007; Chapman 2008). [ SLIDE 5 ] Fresh impacts detected on Mars and well-preserved craters on the moon suggest these fall in rays. Mars, however, is geologically active, so ray evidence erodes with time, making it hard to link a given small crater with a given primary crater. Previous attempts to deal with secondary cratering have included examining craters for irregularities in shape allowed by the sub-sonic velocities of secondary impactors and visual inspection for alignments and clustering.

This paper proposes a variant on nearest neighbor analysis to detect linearities among small craters in a region. The method can identify candidates for possible missing ray structures and allow the size-frequency distribution to be adjusted for secondary impact distortions. The purpose of this paper is to present and assess the method on a test case image of a badly cratered region on Mars.

Data and Methods

[ SLIDE 6 ] I searched the NASA Photojournal site for an image that satisfied the following criteria: first, moderately high spatial resolution showing the smaller craters likely to be problematic and, second, a site in an ancient and heavily cratered region of Mars, so that a moderate resolution image would provide a large number of craters. [ SLIDE 7 ] I picked a Mars Global Surveyor Mars Orbital Camera Narrow Angle (NASA 2010) image taken in March 2006 of Terra Sabæa, one of the most densely battered landscapes of Mars, dating back to the Noachian Epoch before 3.8 or 3.5 billion years ago.

[ SLIDE 8 ] The image has 1.5 m/pixel resolution, covering approximately 3 km from northeast to southwest and 6.6 km from northwest to southeast. It is near 22° S at 21° E. The image is dominated by a relatively fresh crater about 3 km in diameter to the northwest, with what looks like a much older, much degraded crater not quite 2 km in diameter to the southeast. Outside these two larger craters, the landscape is nearly saturated with mostly degraded old craters, except where newer æolian material partly buries the cratered surface. [ SLIDE 9 ] I divided the image with a 150 m x 150 m grid and selected 36 of the resulting quadrats in a 6 by 6 square to the northeast. [ SLIDE 10 ] Each quadrat was given a code based on cell identification number and then the center of each identifiable crater within each quadrat was given a letter code. Its X and Y coördinates were recorded, resulting in a database of 146 craters, a statistically large sample for the process to be applied to them. Each crater's diameter was also measured and entered into the database. [ SLIDE 11 ] I mapped them by creating an X-Y scatterplot in OpenOffice Calc and then putting the image of the 36 quadrat study area behind the scatterplot as the chart wall.

The database, then, consisted of 146 records, each containing a crater ID code, X and Y coördinates in meters from the southwest corner of the test area, and diameter in meters. The X and Y coördinates were used to calculate distance in meters between every crater and every other crater in the database using the Pythagorean theorem. These distances were then ranked for each crater, and the ranks were used to identify for each crater the nearest neighbor and the next nearest neighbors up to the 6th order neighbor.

The X and Y coördinates were also converted into azimuths through use of the arctangent in degrees, each crater serving as the origin and each of its six nearest neighbors as the destination for each azimuth. From the azimuths, another five variables were generated. I started with the azimuth between a given crater and its nearest neighbor. Then, I subtracted the azimuths to each of the other five higher order neighbors from the azimuth of the original pair and created five azimuth differences.

[ SLIDE 12 ] I went through the resulting 730 azimuth differences, looking for differences that fell within a narrow cone of alignment. I experimented with several standards trying to accommodate the look of lunar and martian ray structures. I compared the counts of "aligned" and unaligned azimuth differences, repeating this process for each standard. [ SLIDE 13 ] These counts were then compared to an expected distribution of alignments among the five higher order neighbors generated from the binomial distribution. The significance of the discrepancy between observed and expected counts was calculated using the Chi-square goodness-of-fit test. [ SLIDE 14 ] I plotted the resulting prob-values against the candidate cutoff angles for aligned and unaligned craters to create a scree plot. 15o was the tightest angle to produce significance below 0.05 in collapsed Chi-square tables. So, I used 15° of separation around 0° and 180° as a cutoff criterion for identifying potential alignments.

Having two or more higher order neighbors aligned with the nearest neighbor pair out of the five higher order neighbors available produced 33 groups of neighboring craters that were, thus, deemed promising as linear chains of craters of the type expected from secondary crater and ray structures. The 4-6 crater chains thus identified among the six nearest neighbors of any given crater were then plotted and a correlation and regression analysis was performed on each. Of the 33 chains, 18 had correlations greater than 0.80 (absolute). For any chain with R < 0.80 (absolute), I identified individual outlier craters that degraded R and experimented with removing them. This process identified sixteen craters that damaged the alignments. Removing them removed one entire chain and raised the correlations on all the remaining chains above 0.80 (absolute). These outliers were then removed from the map of the study area.

Results

[ SLIDE 15 ] The resulting map of short alignments of neighboring craters represents potential secondary crater chains. What is interesting and completely unanticipated is how several of these chains are themselves lined up, an alignment beyond the scale of the original analysis confined to the six nearest neighbors of each crater. [ SLIDE 16 ] For example, one striking lineament of chains runs up the western half of the study area. This lineament is notable for its crossing of several distinct terrain subtypes: the flat fill in an old and degraded crater to the south, softened crater rim structures, and a markedly softened terrain to the northwest.

[ SLIDE 17 ] There are another three, much shorter chains in the eastern two columns of quadrats, all roughly oriented northeast-southwest, which were surprising in a different way. [ SLIDE 18 ] These three lines converge to a point about 400 m to the northeast of the study area, [ SLIDE 19 ] as seen in this scatterplot done in Statistica to include 95% confidence bounds as well as regression lines.

[ SLIDE 20 ] Unfortunately, imagery of comparable resolution is not available for the adjacent area. MOC coverage is scattered irregularly along orbital flight paths around the planet. [ SLIDE 21 ] The best imagery is the Mars Digital Image Mosaic, with spatial resolution of 231 m/pixel in the martian "tropics." [ SLIDE 22 ] This is available through Google Earth Mars, which allows plotting of the locations and sizes of higher resolution images by various sensors and craft. It is too coarse, however, to make out what, if anything, might lie at the intersection of these three lines.

[ SLIDE 23 ] Serendipitously, visual inspection of the craters in the study area turned up a fourth line of larger craters. Plotting their trajectory (with an R of 1.00) leads to intersection with the other three lines in the same vicinity! What the meaning is of this radial pattern of small craters with a line of larger craters among them remains unclear.

Note: This alignment among alignments is the odder for the progression of crater sizes in this fourth line of larger craters, increasing away from the convergence point of the four lineations, a point brought up in the discussion after this presentation by Stephen Tooth of the University of Wales, Aberystwyth. Smaller fragments created by primary cratering attain higher velocity and, thus, can fall farther from the primary. There should, therefore, be some degree of inverse size sorting with distance from the source crater.

Discussion

Lineations identified this way are by no means proof of a secondary impact ray structure, but they do identify candidates for such structures. These require closer geomorphic examination against their contexts to eliminate alternative explanations for such lineation. Alternative hypotheses could include, variously, a catena of sinkholes opening up along weaknesses in a region subject to extensional stresses, a chain of rootless cones of phreatomagmatic interactions, or a random alignment of craters of such different ages that they could not physically be linked to a single larger impact.

[ BACK TO SLIDE 21 ] In this particular area, the first alternative is inconsistent with the presence of wrinkle ridges, which indicate a past or present compressional stress field in the region. The second alternative would be surprising here, as rootless cones are mainly found in the Northern Lowlands of Mars (Fagents, Lanagan, and Greeley 2002; Lanagan et al. 2002). [ RETURN TO SLIDE 23 ] The third alternative remains in the running here, since secondary crater chains usually begin at least one diameter away from the originating primary crater and often farther than that: The convergence point seems a little too close here.

There are shortcomings to the method, which the presence of the fourth converging chain of larger craters implies. The method starts with the nearest and five next nearest neighbors, and the nearest neighbors to a large crater are apt to be smaller ones, not similarly sized larger craters, so alignments of these larger craters drop off the radar. In future iterations of this project, I would like to try constraining it by size class to see if the method works when the search for nearest neighbors stays within size ranges.

Another shortcoming, not of the method, but of my implementation, became apparent in hindsight: I wish I had noted, not only the easting and northing coördinates and diameter of each crater, but its condition as well. This is an important detail, as condition of the crater is related to its age. I would like to go back over the alignments found here to see if all the craters in each possible chain are in roughly the same condition, which would reïnforce the possibility that they were laid down simultaneously and, if so, could plausibly be attributed to the same secondary cratering event.

Conclusion

[ SLIDE 24 ] Even as is, this method, however, does quickly identify candidate chains and thereby focusses the search for alternative explanations. It also helps bound the estimation of the prevalence of secondary cratering at the smaller end of the crater frequency and size relationship. In the current case study, more than half the craters were found in lineations suggesting secondary cratering! The next step is to move this project out of Open Office and Statistica into GIS for development of a routinized and efficient system for secondary crater prospecting. It should be tried out at several scales, perhaps using pre-existing crater databases at certain of these, such as the Barlow crater catalogue and others available at the USGS PIGWAD server (Mars Crater Consortium 2009).

References

• Arvidson, R.; Boyce, J.; Chapman, C.; Cintala, M.; Fulchignoni, M.; Moore, H.; Neukum, G.; Schultz, P.; Soderblom, L.; Strom, R.; Woronow, A.; and Young, R. 1978. Standard Techniques for Presentation and Analysis of Crater Size- Frequency Data. Crater Analysis Working Group, Office of Space Science, National Aeronautics and Space Administration. Washington, DC: NASA Scientific and Technical Information Office.

• Chapman, Clark R. 2008. Mars cratering issues: Secondary cratering and end- Noachian degradation. Paper presented at the Second Conference on Early Mars: 8025.

• Fagents, S.A.; Pace, K.; and Greeley, R. 2002. Origins of small volcanic cones on Mars. Lunar and Planetary Science 33: 1594.

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• Lanagan, P.D.; Kesztheli, L.P.; Milazzo, M.P.; and McEwen, A.S. 2002. Water vapor diffusion and implications for shallow martian phreatomagmatic explosions. Lunar and Planetary Science 33: 1694.

• Mars Crater Consortium. 2009. Mars crater catalogs. U.S. Geological Survey Planetary Interactive GIS -on-the-Web Analyzable Database (PIGWAD). Available at <http://webgis.wr.usgs.gov/pigwad/down/mars_crater_consortium.htm>.

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• Neukum, G., and Ivanov, B.A. 1994. Crater size distributions and impact probabilities on Earth from lunar, terrestrial-planet, and asteroid cratering data. In Hazards Due to Comets and Asteroids, ed. Tom Gehrels, pp. 359-416. Tucson: University of Arizona Press.

• Preblich, Brandon S.; McEwen, Alfred S.; and Studer, Daniel M. 2007. Mapping rays and secondary craters from the Martian crater Zunil. Journal of Geophysical Research 112, E05006.

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