Elsevier

Icarus

Volume 226, Issue 1, September–October 2013, Pages 885-890
Icarus

Planetary surface dating from crater size–frequency distribution measurements: Multiple resurfacing episodes and differential isochron fitting

https://doi.org/10.1016/j.icarus.2013.07.004Get rights and content

Highlights

  • Differential isochron fits are useful for identifying multiple resurfacing events.

  • The range of crater-distribution flattening for a resurfacing episode is limited.

  • A flattened crater distribution covering a range greater than Ω corresponds to an age.

  • The binning bias for differentially/incrementally plotted crater data can be removed.

  • A revised calculation of the martian epoch boundary times is presented.

Abstract

The analysis of crater size–frequency distributions and absolute densities forms the basis of current approaches for estimating the absolute and relative ages of planetary surfaces. Users of the Neukum system of crater dating have conventionally used a cumulative presentation of the data, but because of the recent proliferation of interest in identifying resurfacing ages, it is worth emphasising the utility of the differential presentation of crater data in identifying resurfacing events and, particularly, in distinguishing the signature of short-lived events, such as volcanic flows, from long-acting processes, such as aeolian erosion.

The work describes some additional considerations for making isochron fits to differentially plotted crater populations with respect to the removal of a binning bias for incrementally/differentially plotted data.

The Hartmann approach has not typically employed the fitting of isochrons, but differential fitting would be a natural choice for this system, and is implemented in the Craterstats software.

A revised calculation of the martian epoch boundary times in both chronology systems is provided.

Introduction

In the Neukum system of surface dating (Neukum, 1983, Hartmann and Neukum, 2001, Ivanov, 2001, Neukum et al., 2001), we have conventionally used a reverse cumulative presentation of the crater size–frequency distribution—plotting the number of craters larger than a given diameter D against D—because of its advantage with respect to data noise: with decreasing diameter, the curve rapidly converges to a position on the plot which uniquely corresponds to a model age. This technique is fine when measuring a single age: the formation of the unit. If we are also interested in unravelling the chronology of some post-formation modification of the unit, the picture is more complicated. A resurfacing event causes the removal of craters from the small end of the size–frequency distribution at some time after the unit’s formation. If the data are plotted in a reverse-cumulative style, the location of the large-end tail on the plot still corresponds to the formation age of the unit, but the small-end tail does not correspond directly to the time of the resurfacing event: it is offset (upwards) by the relatively older section earlier in the cumulative calculation. This offset can be compensated by an iterative procedure (Michael and Neukum, 2010), provided that the upper and lower size limits of the resurfaced part of the distribution can be adequately determined. Often, the resurfacing appears as a step between two portions of the distribution whose curves run parallel to plotted isochrons: in such cases, specifying the diameter range representing the part of the crater population corresponding to the resurfacing is reasonably straightforward (Fig. 1).

In other cases, where the resurfacing process was not short-lived, or where multiple resurfacing events have occurred, it can be difficult to identify appropriate diameter ranges for the correction procedure. Here it is better to examine the population in a differential presentation (Arvidson et al., 1979), where portions of the distribution which correspond to isochrons may be seen directly, with abrupt steps between isochrons being indicative of short-lived resurfacing events, and more gradual transitions between isochrons indicative of long-acting, diminishing resurfacing processes.

The differential presentation is similar to the incremental presentation used by Hartmann (Hartmann, 1966, Hartmann, 1977, Hartmann, 2005), the distinction being that the differential plot is normalised according to the choice of bin-width such that the plot is independent of the binning (aside from statistical effects). The use of a differential isochron fit is a natural choice for the presentation of results in Hartmann-style.

Section snippets

How do resurfacing processes change the crater population?

Consider a resurfacing process, either erosional or depositional, which acts on a surface such that the crater population is reduced. The surface, as always in crater dating studies, should have experienced a near-homogeneous geological history, which is to say that, aside from the presence of impact craters, the whole of the surface should have been emplaced at once and that any modifying processes can be reasonably assumed to have affected all parts of the surface in equal measure.

Let us

Choice of bin width

A complexity of dealing with a differential plot is the choice of bin-width: if spaced too finely, a real data plot will show both empty bins and bins containing only a single crater: it is hardly possible, however, to discern a trend which may coincide with an isochron until the bins begin to encompass a larger number of craters. At the same time, the lower expected count of craters per bin increases the point scatter arising from the random nature of the cratering process. On the other hand,

Production functions in differential form

To make use of the differential plot for extracting ages requires having the production functions available in differential form. For a Neukum cumulative polynomial, the conversion equation was given previously (Michael and Neukum, 2010). For the Hartmann system (Hartmann, 2005), the differential form isFbin=HD(21/4-2-1/4)where H are the tabulated incremental values, and D are the geometric bin centres.

Exponential binning bias

A significant additional consideration is that the binning of a decreasing exponential function introduces a bias when plotting against the bin centre, since statistically, there will always be more craters close to the left edge of the bin than the right. The consequence is that points are plotted higher than would be the case for a continuous function. The effect becomes larger as the bin width increases, and makes a difference of the order of a few percent in model ages for 2-binning. The

Hartmann functions

In earlier works, Hartmann gave expressions for production functions in the form of piecewise line fits in contrast to Neukum’s polynomials. Most recently (Hartmann, 2005), he provides the function in tabular form, giving the number of craters in each 2 diameter bin for standard aged surfaces.

To fit the Hartmann function as isochrons, it is possible either to convert the function to cumulative form or to use a differential form. The cumulative form is straightforward: the table values are

Software

These calculations are implemented in the cross-platform software Craterstats, available from http://hrscview.fu-berlin.de/software.html.

When fitting an isochron to some portion of the crater distribution, one may select either to make a ‘cumulative fit’ or a ‘differential fit’. The cumulative fit uses the conventional Neukum technique; the differential fit follows the exact same procedure in the differential data space, while also removing the binning bias. Note that the data space is distinct

Conclusion

The use of a differential presentation of data is suggested for the chronological analysis of surfaces with a complex resurfacing history, and the additional mathematical elements required for performing isochron fitting in differential form with equivalent precision to the previous approach are provided.

The influence of multiple resurfacing episodes is considered, and it is established that an extended flattening of the crater size–frequency distribution, over a diameter range exceeding Ω  2,

Acknowledgments

The work was supported by the German Space Agency (DLR Bonn), Grant 50QM1001 (HRSC on Mars Express), on behalf of the German Federal Ministry of Economics and Technology. I thank Boris Ivanov and an anonymous reviewer for their suggestions, which helped improve the manuscript.

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