PUBLICATIONS
Journal of Geophysical Research: Planets
RESEARCH ARTICLE
10.1002/2014JE004698
Key Points:
• Lunar crater degradation can be treated
as a topographic diffusion problem
• Degradation state can be used to
estimate the age of individual craters
• The mean diffusivity of the Moon’s
surface over the last 3 Ga is
2
~5.5 m /Myr
Correspondence to:
C. I. Fassett,
[email protected]
Citation:
Fassett, C. I., and B. J. Thomson (2014),
Crater degradation on the lunar maria:
Topographic diffusion and the rate of
erosion on the Moon, J. Geophys. Res.
Planets, 119, doi:10.1002/2014JE004698.
Received 18 JUL 2014
Accepted 26 SEP 2014
Accepted article online 30 SEP 2014
Crater degradation on the lunar maria: Topographic
diffusion and the rate of erosion on the Moon
Caleb I. Fassett1 and Bradley J. Thomson2
1
Department of Astronomy, Mount Holyoke College, South Hadley, Massachusetts, USA, 2Center for Remote Sensing,
Boston University, Boston, Massachusetts, USA
Abstract Landscape evolution on the Moon is dominated by impact cratering in the post-maria period.
In this study, we mapped 800 m to 5 km diameter craters on >30% of the lunar maria and extracted their
topographic profiles from digital terrain models created using the Kaguya Terrain Camera. We then
characterized the degradation of these craters using a topographic diffusion model. Because craters have a
well-understood initial morphometry, these data provide insight into erosion on the Moon and the topographic
diffusivity of the lunar surface as a function of time. The average diffusivity we calculate over the past 3 Ga is
~5.5 m2/Myr. With this diffusivity, after 3 Ga, a 1 km diameter crater is reduced to approximately ~52% of its
initial depth and a 300 m diameter crater is reduced to only ~7% of its initial depth, and craters smaller than
~200–300 m are degraded beyond recognition. Our results also allow estimation of the age of individual
craters on the basis of their degradation state, provide a constraint on the age of mare units, and enable
modeling of how lunar terrain evolves as a function of its topography.
1. Introduction
The calibration of the lunar cratering rate following the acquisition and radiometric dating of Apollo and Luna
samples has helped establish a geologic history and chronology for the lunar maria [see, e.g., Wilhelms, 1987;
Hiesinger et al., 2000, 2003, 2010, 2011]. Most of the maria were emplaced in the Imbrian period, from ~3.8
to ~3.2 Ga, with less volumetrically significant volcanism both earlier and more recently (the youngest units
are as young as ~1 to 1.2 Ga) [Schultz and Spudis, 1983; Hiesinger et al., 2003]. Following the most recent
volcanism in a given region, it is tempting to think of the Moon’s surface as quasi-static and unchanging, with
only slow, ongoing impacts leading to the accumulation of new fresh craters on the surface. However, such a
perspective is incomplete, since both the lunar regolith and lunar topography evolve with time. For example,
bright ejecta rays from fresh craters fade [e.g., Grier et al., 2001], ejecta deposits become less distinct [Ghent et al.,
2005; Bell et al., 2012], boulders excavated or exposed to the surface are rapidly destroyed [Hörz et al., 1975;
Basilevsky et al., 2013], and topography becomes progressively muted with time [e.g., Craddock and Howard,
2000; Howard, 2007; Kreslavsky et al., 2013].
The final one of these processes—evolution of topography with time—is the focus of this paper.
Specifically, this study derives a new calibration for the rate at which topography of the Moon evolves
by looking at the progressive degradation of craters with diameter D = 800 m to 5 km on the lunar
maria. The results of this calibration provide a way to estimate the age of individual craters from their
degradation state and a measurement of the diffusivity and average erosion rate on the Moon’s surface
over the past ~3.6 Ga.
2. Background
For more than 40 years, crater degradation has been hypothesized to predominantly result from the
cumulative effects of small later impacts [e.g., Ross, 1968; Soderblom, 1970; Soderblom and Lebofsky,
1972]. Soderblom [1970] provides a theoretical basis for recognizing that the form of this degradation
is diffusional [see also Craddock and Howard, 2000; Howard, 2007; Bouley and Baratoux, 2011]. Other
processes that mobilize regolith on the Moon such as thermal expansion and contraction [Molaro and
Byrne, 2012] or seismic shaking [Schultz and Gault, 1976] may contribute to topographic diffusion as
well, and the relative importance of these processes relative to micrometeoritic bombardment is
presently unknown.
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
1
Journal of Geophysical Research: Planets
10.1002/2014JE004698
→
In a diffusive model [see Pelletier, 2008, chapter 2], the volume flux q of material downslope across a contour
is proportional to the slope of topography, so for diffusivity κ and the gradient of topography h:
→
q ¼ ∇h
(1)
Combining this with a statement of volume conservation (or mass conservation if density is constant)
∂h
→
¼ ∇ q
∂t
(2)
∂h
¼ κ∇2 h
∂t
(3)
implies that h evolves as
(Note that throughout this paper, κ is used to denote the diffusivity, rather than D that is often used for
discussing topographic diffusivity. The common use of D to denote crater diameter makes this untenable in a
study characterizing crater topography.)
Along with its application to cratered landscapes, equation (3) has long been applied as a model for the
evolution of terrestrial hillslopes over time [e.g., Culling, 1960; Pelletier, 2008]. This type of diffusive model also
implies that small craters degrade substantially faster than large ones, consistent with past analysis of the
lunar surface [e.g., Basilevsky, 1976]. This means that for craters in the same degradation state—with the same
diffusion age κt—a larger crater will appear substantially fresher than its smaller counterparts. Since the
diffusion timescale grows as the square of the length scale, all else being equal, doubling the crater diameter
increases its effective lifetime by a factor of 4. Even if the diffusivity κ is itself a function of time, it is not
necessary to know the variability of diffusivity with time in advance, because mean diffusivity is sufficient
to characterize the state of a given surface feature [Skianis et al., 2008]. In other words, Skianis et al. show that
if topographic diffusion is operating on a surface, even if the rate of this diffusion is varying with time, the
mean diffusivity that is experienced is what matters for the resulting degradation state.
Not all of the topographic degradation experienced by craters on the Moon is likely to be from diffusive
processes alone. Other processes may contribute to crater degradation as well, including advective mass
wasting processes, such as landslides [e.g., Kumar et al., 2013; Xiao et al., 2013], or depositional processes, such
as deposition of lofted dust [Grün et al., 2011] or distal ejecta [e.g., Li and Mustard, 2005]. However, the
diffusive model provides a useful first-order framework for this type of analysis with few free parameters.
Our measurements, presented below, also suggest that the diffusive model provides reasonable fits to the
observed topographic profiles of craters on the Moon.
2.1. Crater Degradation, DL , and the Age of Geologic Units
One of the most widespread past applications of crater degradation measurements has been to use crater
degradation state to provide a proxy for the relative and absolute age estimates for geologic units. This is
possible because older geologic units should have craters in a more advanced state of degradation than
younger terrains, whose superposed craters have been exposed for a shorter period. A process for taking
advantage of this was developed by Soderblom and Lebofsky [1972], who used Soderblom’s [1970] degradation
model to predict the interior slopes of craters of a given size after a known flux of small impactors. This
allowed Soderblom and Lebofsky to predict which craters would be shadowed versus unshadowed on a
surface for a given solar elevation (based on predicted crater wall slopes), which is possible to measure on
images. They used this to define a reference crater diameter, DL, which is representative of the maximum size
of a crater expected to be reduced to an interior slope of 1° after the net accumulated flux based on their
model. Soderblom and Lebofsky demonstrated that, to first-order, DL correlates with traditional crater density
measurements. In other words, older surfaces have larger DL values, as expected.
This was then applied observationally to essentially all of the near-side maria in a series of papers by Boyce and
colleagues [Boyce and Dial, 1973, 1975; Boyce et al., 1974; Boyce, 1976; Boyce and Johnson, 1978]. Although these
data are valuable and are somewhat consistent with other inferences about age from crater size-frequency
distributions, they are not without some limitations that have been long understood [Soderblom and Lebofsky,
1972]. In particular, all of these measurements use shadowing as a proxy for maximum topographic slope.
Thus, they (1) cannot account for topographic irregularities of the impacted surface or measured crater,
(2) are potentially sensitive to the dynamic range, contrast, and viewing geometry of the original image, and
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
2
Journal of Geophysical Research: Planets
10.1002/2014JE004698
Figure 1. Location of craters and study area, superposed on the global LROC WAC mosaic.
(3) only provide information about maximum interior slope rather than full knowledge of crater interior
topography. In addition, a common method for applying the technique was to search for craters in particular
degradation states (i.e., the largest crater with half of their floor covered in shadow), rather than characterize
the full range of crater degradation states on a given unit, which increases the potential uncertainties. If the
craters relied upon are secondaries or simply anomalous, this method could potentially underestimate or
overestimate surface ages. Finally, this method is primarily aimed at constraining the age of surface units,
not individual craters.
2.2. Direct Simulation of Crater Degradation, Topography From Photoclinometry, and Age Derivations
Building on the earlier work by Soderblom, Boyce, and others, Craddock and Howard [2000] developed a
model for diffusive crater degradation that they compared directly to the topography of craters. One way that
Craddock and Howard’s study improved upon the earlier efforts was that their measurements of crater
topography were derived from photoclinometry, which provides more information about the morphometry
of craters than it is possible to attain from the shadow measurements used to determine DL.
Key results of this work were (1) an age estimate (3.85 Gyr) for the mare in Apollo basin, which was challenging
to determine on the basis of crater size-frequency distribution measurements, (2) a characteristic erosion rate
on the Moon of 0.2 ± 0.01 mm/Myr since the Imbrian period, and (3) the suggestion that this erosion rate or
diffusivity has decreased over time. One limitation of this approach is that photoclinometry is time-consuming
and its accuracy is highly dependent on the photometric parameterization of the surface and calibration of
the imaging system. Because it is time-consuming and very dependent on image conditions, Craddock and
Howard were limited to a small portion of the maria in their study.
3. Methodology
Before presenting a more detailed description of the techniques used in this study, we briefly outline the
approach. First, all craters larger than D = 800 m were mapped across a broad region of the maria with Lunar
Reconnaissance Orbiter Camera (LROC) Wide Angle Camera (WAC) data [Robinson et al., 2010]. This portion
of the work is identical to what is typically applied in a crater counting study. Second, we extracted the
topography for each of the mapped craters using digital terrain models (DTMs) produced by the Kaguya
(SELENE) Terrain Camera (TC) team [Haruyama et al., 2012]. Third, a numerical model for the diffusion of crater
topography for D = 600 m to 5 km craters was implemented, which allowed us to establish a database of
crater profiles in various states of diffusive evolution. Fourth, we determined the profile from this database
which best matched each crater’s topography extracted in step two. The result of these steps was an estimate
for each crater’s best fit degradation state, κt, and initial diameter, D0. These estimates of κt for the craters
were then analyzed to better understand crater degradation temporally and spatially on the lunar surface.
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
3
Journal of Geophysical Research: Planets
10.1002/2014JE004698
Figure 2. An example of possible sources of topography for a crater (UniqCratID, 616) in our data set. (left) LOLA data provide
precise profiles across features, but there is sparse sampling of the topography. We therefore primarily rely on the (right)
Kaguya Terrain Camera DTM for the results presented in this study.
3.1. Crater Mapping and Study Area
The extent of our study area and the location of mapped craters are shown in Figure 1. In aggregate, we mapped
~30% of the lunar maria by area total (~1.8 million km2), progressively mapping individual subareas of ~104–105 km2
at a time. The aim in choosing this study area was to have a broad geographic and age sampling of the maria,
excluding areas dominated by obvious clusters or secondary crater chains, and eliminating the ejecta and
near-field secondary craters of large post-mare craters. Our data samples most of the named maria. It would be
useful to extend this data analysis to additional areas in future work, although the regions of the maria that we
have not yet mapped are likely to be more geologically complex and have a greater admixture of secondaries.
To catalog the location and size of craters, we used the LROC WAC global mosaic released to the Planetary Data
System (PDS) on 8 December 2011 with 100 m/pixel image resolution and imported this data set into the ESRI
ArcMap Geographic Information System (GIS) using U.S. Geological Survey (USGS)’s ISIS software (Integrated
Software for Imagers and Spectrometers). After determining an area of interest, a 500 m grid (fishnet in GIS) was
overlaid on the region and progressively removed as craters were identified using the three-point tool in the
CraterTools extension to ArcMap [Kneissl et al., 2011]. CraterTools fits a circle to a crater’s rim and measures its size
in a local map projection regardless of the map projection of the data or the GIS data frame. The grid that is
progressively removed is used to provide a local scale to ensure that all craters larger than the grid are included,
and so that no portions of the study area are missed. Craters with D < 800 m are excluded from all analyses,
though many were mapped to assure a complete catalog of 800 m to 5 km craters.
In total, we mapped 13,657 craters between 800 m and 5 km in our study area.
3.2. Sources and Extractions of Topography
In the initial stages of the study, our strategy was to rely on topography extracted from the Lunar Orbiter Laser
Altimeter (LOLA) profile data. LOLA provides measurements of the lunar surface topography with ~10 cm
precision by making five simultaneous ranging measurements that sample ~5 m spots. The subsequent fivespot pattern is translated along the orbit by ~50 m [see Smith et al., 2010]. For a profile crossing near the center
of a D ~ 1 km crater, LOLA thus typically acquires ~100 samples of the crater’s topography (Figure 2, left). To
date, LOLA has taken more than 6 billion measurements of the lunar surface. Despite this remarkable data set,
for most craters, the sampling of the topography of craters in our size range of interest is fairly sparse. Most
craters do not have profiles passing near their center; as of the LOLA data released to the PDS through 14
December 2012, only 35% of craters with radius R had a sample point within 0.3R.
Because of this sparse sampling, the primary source of topography for this study is the global Kaguya TC stereo
DTM with 10 m/pixel posting. The Kaguya Terrain Camera successfully mapped over ~99% of the Moon in stereo
and is highly complementary to LOLA: its measurements are less accurate (with a 1σ ~ 3–4 m elevation offset
from LOLA and the Kaguya Laser Altimeter [Haruyama et al., 2012; Barker et al., 2014]), but its spatial coverage
of the surface is superior (e.g., Figure 2, right). Intercomparison with LOLA verified that, where sufficient
measurements exist, equivalent crater profiles and degradation states are derived from both instruments.
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
4
Journal of Geophysical Research: Planets
10.1002/2014JE004698
A global mosaic of the TC DTM was released as 3° × 3° tiles to the SELENE data archive (http://l2db.selene.
darts.isas.jaxa.jp/). All of the tiles covering the maria were imported into ArcMap for analysis. A large majority
of craters we measured are covered in this DTM; the main exceptions are in regions where shadows
prevented stereo matching. This problem is most severe at high latitudes, where, in some instances, the
derived stereo data are insufficient to characterize crater topography.
The methodology we used to extract topography for craters after mapping was similar for LOLA and Kaguya.
For LOLA, we maintain a geographically indexed MySQL database of all LOLA shot points that is updated
at every PDS release. This database was queried at measured crater locations, and all valid LOLA measurements
with a location within 2D of the crater’s center were extracted. For the TC data, the ArcMap Extract by Circle
tool was used to save the region within 2D of the mapped crater center. Care was taken to merge data from
multiple tiles for features near or on the edge of the 3° × 3° boundaries, although craters near the edge of
the DTM tiles proved to be somewhat problematic because the boundaries did not internally agree in all
instances. The extracted TC data were bilinearly resampled to 25 m/pixel and converted to a grid of points.
For both the LOLA and TC data, the extracted points for each crater were then projected into a Lambert
Azimuthal Equal Area map projection centered on the crater.
In an ideal situation, the resulting (x,y, z) point files would then lie in a coordinate system with an origin
corresponding to each crater’s center. Determining radial profiles from these data would then simply involve
calculating the radial distance of each point from the origin. However, in practice, neither the LOLA measurements
nor the TC DTMs were perfectly co-registered to the LROC WAC data set, and there was inherent uncertainty in
the mapped craters’ centers. Thus, we developed a procedure to improve this coregistration using Generic
Mapping Tools (GMT) [Wessel and Smith, 1998] to spatially shift the (x,y, z) points until a location with maximal
radial symmetry is identified.
Specifically, this procedure is a grid search for the center location with the minimum misfit of a degree five
polynomial to the data with GMT’s trend1d fitting routine, with the assumption that this misfit is minimized
when the points are closest to being collapsed onto a single profile (radially symmetric around the crater
center). The search domain was initially defined as ±500 m from the measured crater center, and the step size
for shifting was refined as the search proceeded, with a final step size of 10 m. In some instances, human
intervention was required to increase the size of the search domain or manually pick the crater center when
this algorithm failed.
The resulting points following this registration step were then saved as radial distance and elevation (r,z)
pairs. The median registration offset between the LROC WAC and TC data sets we found was 320 m and
the shifts required were correlated locally (usually similar on a single portion of a given DTM tile). For
LOLA, the average offset was ~100 m (one LROC pixel). Since the geodetic accuracy of the LROC WAC
mosaic has been subsequently improved [e.g., Speyerer et al., 2014] and efforts are actively underway to
update and merge the TC DTMs with the LOLA [e.g., Barker et al., 2014], the co-registration of these data
sets is likely to be substantially improved in the near future or already has been improved subsequent
to our measurements.
3.3. Fresh Crater Morphometry and Modeling Crater Evolution
Our model for quantitatively fitting the measured crater with a degradation profile is similar to that of Craddock
and Howard [2000] and Pelletier [2008]. A MATLAB program is used to solve equation (3) (the diffusion equation)
numerically in two dimensions (h(x,y)) using an explicit, forward-time, center-space approach. The grid is first
initialized with the morphometry of a fresh, simple crater of the diameter of interest (Figure 3). The model domain
had grid spacing of 10 m for craters with D < 2.5 km and grid spacing of 20 m for craters larger than this size.
The fresh crater model is based on a degree 3 polynomial approximation to measurements of rayed simple
craters from LOLA data identified by Werner and Medvedev [2010] and S.C. Werner (personal communication,
2012), and has initial crater topography as defined in equation (4):
8
0:181
>
>
>
< 0:229 þ 0:228r þ 0:083r 2 0:0393
hðr Þ=D ¼
>
0:188 0:187r þ 0:018r 2 þ 0:015r 3
>
>
:
0
FASSETT AND THOMSON
for r ≤ 0:2R
Central Flat Floor
for 0:2R < r < 0:98R
Interior
for 0:98R < r < 1:5R
Rim and Exterior
for r ≥ 1:5R
©2014. American Geophysical Union. All Rights Reserved.
(4)
Farfield
5
Journal of Geophysical Research: Planets
10.1002/2014JE004698
This has a flat floor within 20% of the crater’s
center, consistent with LOLA observations
and earlier results [e.g., Schultz, 1976; Wood
and Anderson, 1978], and a depth/diameter
ratio (d/D) of 0.218 from the top of the rim
to the flat crater floor, which is close to, but
slightly deeper than, that of Pike [1977].
Figure 3. The model (green) we used for fresh craters in this
study (see equation (4)). LOLA data were extracted where it
crossed the center of six of the freshest rayed craters on the
Moon in our size range.
Figure 4. The model evolution of three craters of different sizes
from 0 to 3 Ga. κt values are selected to be equivalent to ~500 Myr
intervals (κt ~ 25, 5750, 8750, 11275, 13500, 14975, and 16450).
After 3 Gyr of evolution, a 300 m crater is almost imperceptible,
the 1 km crater is muted but still has significant relief, and the
3 km crater has only seen significant changes near its rim and
floor. Note the significant widening of the rim as topographic
diffusion is advanced, which is particularly evident here for the
300 m and 1 km craters.
FASSETT AND THOMSON
The fact that such a model fits a range of
crater sizes is consistent with earlier results
that imply that measured depths and
diameters of the freshest craters are highly
correlated (with d/D ≈ 0.2, R2 = 0.99); [Pike,
1974, 1977, 1980]. Similar morphometric
relationships exist for the rim height of
fresh craters and for the decay of their ejecta
[e.g., Pike, 1977]; these relationships exist
because most fresh, primary, simple craters in
the size range considered here have consistent
morphology [e.g., Pike, 1974]. At smaller sizes,
with diameter less than 100 to 300 m,
substrate effects can become more important,
resulting in craters with concentric terraces
[Oberbeck and Quaide, 1967; Quaide and
Oberbeck, 1968; Bart, 2014]. Note that only
the most oblique impact angles (θ < ~15°)
lead to craters are non-circular in planform
(~5% of the total population) [Gault and
Wedekind, 1978; Andrews-Hanna and Zuber,
2010]; we exclude highly elliptical craters
from our degradation analysis (though they
are included in our crater statistics). The selfsimilarity of simple craters in the size range
we analyze makes it possible to parameterize
the initial crater morphometry over a range of
sizes with a single function (equation (4)).
Following this initialization, the numerical
model allowed derivation of model crater
profiles at a range of diffusion age (κt) values
for each initial crater size, D0. The results
from this modeling were stored in a database
of profiles of κt from 0 to 100,000 m2 and
D0 = 600 m to 5 km. The reason that the
craters smaller than those of our main size
range of interest are necessary to consider in
this database is that the apparent diameter
of craters increases as they degrade (see
Figure 4). There is also an inherent uncertainty
in our diameter measurements of ~10%
[Robbins et al., 2014].
3.4. Profile Fitting and Analysis
For each crater, the extracted (r, z) profiles
were compared with the model profiles,
©2014. American Geophysical Union. All Rights Reserved.
6
Journal of Geophysical Research: Planets
10.1002/2014JE004698
varying two parameters related to the crater, D0 and κt. A third free parameter, z0, was also varied; this is
the offset of the extracted crater profiles from zero elevation. In practice, z0 was estimated during profile
extraction from the topography surrounding the crater by taking the average elevation of all points in the
R to 2R region; this estimate was then refined during profile fitting. Our model implicitly assumes that all of
the craters formed on an initially flat slope, which is a reasonable assumption for the maria as a whole and our
data set in particular, since we excluded areas with significant background topography.
D0, κt, and z0 were obtained using a grid search, with final increments of 10 m, 25 m2, and 1 m, respectively.
The search range for D0 was 0.8–1D (for measured crater diameter, D), from 0 to 100,000 m2 for κt, and from
0 to 85 m for z0.
A best fit model was initially determined by using a least squared fit to the data within 2R of the crater center,
minimizing the weighted model misfit:
X
(5)
w r ðzmodel zmeasure Þ2
where wr = (r/R)2 for the radius r of the data point and the summation is computed over all of the
observations. This weighting compensates for the fact that the number of measurements at a given range
increases as r 2.
All resulting fits were manually inspected. For about 5–10% of the craters, the automatic best fit was
found to be unsatisfactory. Three typical sources of poor fits were (1) in instances where the manually
measured crater diameter was too large, the fitting algorithm sometimes converged on too large a D0,
which was obvious because the rim position of the degraded model crater badly misfit the location
of the measured profile’s crater rim, (2) in cases where the manually measured crater diameter was too
small, the range occasionally needed to be adjusted to allow for a larger D0 (only an issue for fresh
craters), and (3) on occasion, the fitting routine misfit both the rim and crater floor because the search
range for z0 was too limited or because the crater’s profile was heavily modified by later impacts or
preexisting topography. One reason that these qualitative misfits arise is that D0 and κt are not completely
independent of each other: a more degraded, larger crater (higher κt, higher D0) can have a similar general
shape to a less degraded, smaller crater (lower κt, lower D0), although in detail the profiles are fairly
different, especially near the crater rim. In these instances, the search range for D0, κt, and z0 values were
manually restricted to find a profile that best fit the depth of the crater, its interior wall slope, and its rim
location and curvature.
Because the degradation rate is evolving as a nonlinear function of crater size and age, the uncertainty in κt is
difficult to formally parameterize. The experience with manual fitting suggests that the derived κt value is
generally constrained to a range of ~ ±30% for a D = 1 km, moderately degraded crater.
4. Results
4.1. Crater Density on the Maria
One of the by-products of this study is new map of crater densities on a large portion of the lunar maria.
Essentially all of the area we mapped has been analyzed in detail in past studies [Hiesinger et al., 2000,
2003, 2010, 2011; Salamunićcar et al., 2014], and this data set mainly supports and replicates this past work.
Nonetheless, this data set is useful since LROC WAC provides an excellent basemap for determining crater
densities. In addition, a technique of looking at crater density [e.g., Ostrach and Robinson, 2014] without
any a priori assumptions about the division of geologic units gives a different way of analyzing crater density
information than has been analyzed in most past work.
Figure 5a shows the frequency of craters with diameter 800 m to 5 km per unit area, n(800 m to 5 km), in our
study area, and Figure 6a shows a histogram of the ages derived from these crater frequencies. There is
reasonably good qualitative agreement between this map and earlier results, with only local-scale to regionalscale exceptions. Our data also appear to be systematically slightly younger than earlier studies (compare
Figure 6a to Hiesinger et al. [2012, Figure 18]). These differences are fairly minor. They are likely attributable
to subtle differences in measurement areas compared to earlier studies and different assumptions about
secondary craters, which creates subjectivity about what craters to exclude.
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
7
Journal of Geophysical Research: Planets
10.1002/2014JE004698
Figure 5. (a) Map of the crater density in our data set, in moving neighborhoods of 50 km radius. (b) Map of the median κt
value in a 50 km radius neighborhood, for all neighborhoods with at least five craters. Note that the median κt is converted
to the unit age, not the model age of the median crater, which would be considerably younger (since the median crater
formed at ~50% of the frequency of the unit as a whole).
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
8
Journal of Geophysical Research: Planets
10.1002/2014JE004698
Figure 6. Histogram of the age of the maria units we sampled, calculated by (a) converting the n(0.8–5 km) incremental crater frequency to age using the Neukum
Production Function and (b) by using the median crater degradation state to determine the age of units using equation (6).
A detailed comparison of our crater density determinations with earlier results in northwestern Oceanus
Procellarum is shown in Figure 7. Hiesinger et al.’s [2003] P9, P10, and P13 spectral units all have ages between
~3.4 and 3.5 Ga (with model ages of 3.47 Ga, 3.44 Ga, and 3.40 Ga, respectively). Our data in the areas sampled
by Hiesinger et al. are fairly consistent
with these results: we find ages of
3.44 Ga, 3.54 Ga, and 3.34 Ga for P9,
P10, and P13, respectively. The slight
quantitative differences between our
results and Hiesinger et al. are likely
a consequence of the fact that our
count areas do not perfectly match
(especially in P10, where the difference
is biggest) and the crater sizes being
used to infer age are slightly different.
In unit P21, the differences in ages
are clearly attributable to sampling.
Some of the area Hiesinger et al.
counted we excluded as being probable
secondariesbased on morphology
and clustering.
FASSETT AND THOMSON
Figure 7. Detail of western Oceanus Procellerum, comparing (a) Hiesinger et al.’s
[2003] data with (b) the crater densities mapped in this study. (Note that the age
scale here is reversed from Figure 5 to match Hiesinger et al.’s colors; younger is
redder.) The crater density map here (Figure 7b), and as a whole (Figure 5a),
agrees reasonably well with Hiesinger et al.’s measurements where they sampled
(polygons outlined in black). Where differences exist between the Hiesinger
et al. measurements and our density map, they suggest greater variation in
crater densities (and ages) in the maria, not fully captured in earlier work.
An example of a more pronounced
difference is that our crater density
map suggests internal crater density
variations exist within P10 from
northeast to southwest, which were
not recognized in earlier work. These
variations suggest that P10 actually
might be divisible into more than one
unit. This sort of analysis illustrates
how density differences alone [Ostrach
and Robinson, 2014] may be a useful
tool to allow identification of units of
different age that may otherwise be
indistinguishable on the basis of
geology or spectral characteristics.
©2014. American Geophysical Union. All Rights Reserved.
9
Journal of Geophysical Research: Planets
10.1002/2014JE004698
Figure 8. Examples of Terrain Camera derived radial profiles for six individual craters with measured diameter D of 1 km,
2
in varying model degradation states, κt (units, m ). From least to most degraded, the location of these are (9.606°E,
22.936°N), (59.280°E, 39.311°N), (13.114°E, 35.615°N), (19.083°E, 20.010°N), (57.037°E, 50.643°N), and (7.489°E,
26.365°N). Note that due to a variety of factors, not every example has as good a fit as these examples (see text).
4.2. Observed Degradation States
In total, we derived the degradation states for 13,514 craters in our study area. (This is less than the number
of craters mapped because some craters had insufficient stereo topography, or were highly elliptical or
overlapping.) Detailed information for each crater, including their topographic profiles and plots of their best
fit models, are available from the authors upon request. Examples of crater topography and their model
fits are shown in Figure 8.
A few systematic issues appear to affect the quality of our degradation state determinations. First, the fit of
the diffusion model to the topography is generally worse for the larger craters we examine (> ~2 km) than
for smaller craters (800 m to ~2 km). One way this issue manifests itself is that the larger craters’ rims do not
appear to be removed as quickly as we would predict given their reduced depths. One possible explanation
for this is that this effect is real, and the physical breakdown of these rims is unable to keep up with diffusive
transport, so that the transport rate is limited by physical weathering and the formation of regolith. This
explanation is consistent with the observation that many of these larger craters on the maria have rocks or
boulders exposed on their steep slopes, even if they are not particularly fresh. Alternatively, our fresh crater form
—particularly the rim height—may be inaccurate for craters of this size. In addition, a fundamental limitation
of fitting these larger craters in the first place is that their topography does not change as rapidly as a function
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
10
Journal of Geophysical Research: Planets
10.1002/2014JE004698
of time as smaller craters (Figure 4).
Nonetheless, since craters 2–5 km in
diameter range only make up ~4%
of our data, the comparatively worse fit
to larger craters should not have a
controlling influence on our results.
A second issue is that the TC DTMs
became systematically and noticeably
noisier at high latitudes (especially
poleward of ~50°) where imaging
conditions made stereo matching more
challenging. This increases the uncertainty
of our measurements in these regions.
As is the case with the crater density,
we examined the degradation states for
each crater spatially. Figure 5b shows
the median degradation state in moving
Figure 9. The cumulative frequency distribution of crater degradation states
neighborhoods of radius 50 km in our
in our data set.
measurement area (requiring a minimum
of five craters in the neighborhood and clipping the data to the measurement area). The agreement between
Figures 5a and 5b is striking considering that these two maps plot-independent parameters: Figure 5a is
only a function of how many craters are found in a given area, and Figure 5b is only a function of crater
topography and degradation within the local crater population. This agreement also supports the view that
the degradation measurements we derive here provide a good proxy for age.
A cumulative frequency distribution for the derived degradation states (κt) is shown in Figure 9. A few broad
observations from this data are worthwhile before we describe how we link degradation state to age in the
following section. First, the observed degradation states in Figure 9 increase smoothly, at least for craters with
κt < 24,000. The steady rate of increase and lack of clustering around particular values is consistent with what we
would expect based on the slow and stochastic accumulation of primary craters on the Moon and a relatively
steady diffusion rate. Second, the subset of the most degraded craters (κt > ~24,000) reach much higher inferred
degradation states. Some of these are likely to be degraded, post-mare primaries on the oldest portions of the
maria we measured (~3.75 Ga), which are rare partly because these regions are a small subset of the total area we
considered (see Figure 6a; <5% of the surfaces we examined had ages >3.6 Ga). The other most degraded
craters—particularly those on younger terrains—are plausibly secondary craters which started abnormally shallow.
We can also calculate the diffusivity of the lunar surface from Figure 9 with very few assumptions (the more
extensive calculation in the next section gives more detail but is also more model dependent). First, based
on crater statistics (Figure 6a) [see also Hiesinger et al., 2012, Figure 18] and the lunar sample collection, most of the
maria were formed ~3–3.8 Ga [Basilevsky et al., 2011]. This is concordant with our crater statistics as well: the
average crater density we derive for our study area as a whole, n(0.8–5.0 km) is 0.0074 km2, consistent with an
average age of 3.33 Ga using the Neukum et al. [2001] calibration of the lunar impact flux.
Second, after the major period of maria emplacement, the lunar impact flux is generally believed to have been
within a factor of a few of steady [e.g., Hiesinger et al., 2012], with some suggesting it is modestly decreasing
[Craddock and Howard, 2000; Hartmann et al., 2007; Minton and Malhotra, 2010] or increasing [McEwen et al., 1997;
Culler et al., 2000]. Because of this nearly steady post-mare crater flux, the median-aged crater on this surface is
likely to be approximately half of this ~3–3.8 Ga age, ~1.5–1.9 Ga. (Actually, the Neukum model actually implies
that the median-aged crater on a 3.33 Ga surface should be ~2 Ga old, because the exponentially declining
portion of his chronology function is still an influence on the integrated flux from ~2.8–3.3 Ga.) From Figure 9,
the median degradation state for our measured craters is κt =12,400 m2. Dividing this by tmedian = 1.5–2 Ga
implies an average diffusivity for the lunar surface κ ~ 6–8 m2/Myr for craters over the last ~1.5–2 billion years.
4.3. Linking Degradation States to Age and the Lunar Impact Flux
Using the Neukum et al. [2001] model of the crater accumulation rate more extensively provides a way to
directly link the observed crater degradation states to age. We first extracted the local crater density of each
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
11
Journal of Geophysical Research: Planets
Figure 10. A box-and-whisker plot of crater degradation state versus the
local crater density of the surface that a given crater is superposed upon.
The thick black line is the median κt, the wide bar ranges from the 25th to
75th percentile, and the thin bar ranges from the 10th to 90th percentiles.
The 90th percentile values are probably anomalous, as discussed in the
text. Note that along with the medians increasing as a function of increasing
crater density/age (see also Figure 5), the range of degradation states is
increasing with crater density/age.
10.1002/2014JE004698
crater using Figure 5a. Then, we sorted
the craters by the crater density of the
surface they were superposed upon,
and subsetted them into bins of similar
crater frequency. For each bin, the 10th,
25th, 50th, 75th, and 90th percentiles of
the degradation state distribution were
determined (Figure 10). As is clear in
Figure 10, the median κt (50th percentile
crater degradation state at a given crater
density) and the interquartile range of
crater degradation states increases as
crater density (age) increases, as expected.
We then applied the Neukum chronology
to translate the measured crater
frequencies into age (in this case, binning
the data into 131 sets of 100 craters
each). For each sample, we made the
assumption that the crater with the 10th percentile degradation state for its surface formed at an age equivalent
to 10% of the region’s frequency; in other words, after 90% of the other craters in that area’s population had
already been accumulated (and likewise for the 25th percentile, 50th percentile, etc.). On any given terrain, this
assumption is logical if crater degradation monotonically increases from least degraded to most degraded. For the
linear part of the crater chronology curve
this would imply that the crater in the
median degradation state has an age
which is 50% of the age of the maria it is
superposed upon. (Note, again, however,
that this is not the case for the part of
the curve where the flux is significantly
nonlinear: Neukum’s model implies that
the median-aged crater on a 3.7 Ga old
surface is 3.5 Ga.) The resulting calibration
curve is shown in Figure 11a. This data
can be fit with a fifth-order polynomial
(R2 = 0.91), which gives the expected κt
for a given crater as a function of its age, t:
κt ðtÞ ¼ 435:83ðtÞ5 3621:2ðtÞ4
þ11; 204ðtÞ3 16; 811ðtÞ2
þ17; 546ðt Þ
Figure 11. (a) A calibration of crater degradation state to age, using the
Neukum production function to convert Figure 10 to ages. The curve fit
to this is given in equation (6). (b) The effective diffusivity experienced by
a crater of a given age (the derivative of equation (6)).
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
(6)
The 90th percentile points are not
included in this calibration as they
systematically fall above this best fit
curve. The easiest way to understand
this is if the most degraded craters
are commonly contaminated (at the
~10–15% level) by secondaries or by
incompletely buried pre-mare craters.
The degree to which these craters are
outliers can be illustrated by considering
that, if this calibration of the degradation
rate is correct, the most degraded crater
12
Journal of Geophysical Research: Planets
10.1002/2014JE004698
on a ~3 Ga portion of the mare should have κt ≤ 16,440 m2. The observed 90th percentile κt on terrain of this
age is ~21,000 m2 (see Figure 10, with n = 5.6 craters per 1000 km2). Even taking into account the uncertainty
in κt, the most degraded craters are thus more degraded than expected on all surfaces we study. Although
the 90th percentile degradation state does increase as a function of crater density (Figure 10), its quantitative
inconsistency with the other parts of the crater degradation distribution suggests that considering only the
most degraded crater on a given surface to understand unit chronology may be less reliable than examining
the full range of crater degradation states.
Because Figure 11a shows κt as a function of t, its slope is the effective diffusivity κ implied by the crater
degradation measurements (Figure 11b; the derivative of equation (6)).
5. Discussion and Implications
5.1. Evolution of κ With Time
Three observations stand out when considering the evolution of diffusivity with time implied by Figure 11.
First, the most recent ~0.7 Ga apparently has higher κ (κ ~ 6–16 m2/Myr) than the period from ~0.7 to ~3.1 Ga.
In this middle era, ~0.7 to ~3.1 Ga, κ appears approximately constant (κ ~ 4 ± 2 m2/Myr). The period from ~3.1 to
3.6 Ga has an increasing and higher diffusivity (κ ~ 6–20 m2/Myr).
The interpretation of this diffusivity evolution is not straightforward for several reasons. First, by definition,
the recent epoch of apparent higher diffusivity arises only from observations of the freshest craters. Since
these craters start with steep slopes (close to the angle of repose), the early modification of these craters
may be partially nondiffusive, with advective processes like landsliding and dry granular flows playing an
important role [e.g., Kumar et al., 2013; Basilevsky et al., 2014]. The topographic diffusivity of typical slopes on
the Moon might therefore be no different in this epoch from earlier in lunar history, but, on steeper slopes,
the effective diffusivity might be enhanced. In other words, a better model for the topographic evolution
of the Moon may require alteration of equation (1) so that the flux is nonlinear and increases as the slope
increases toward a critical slope [Andrews and Bucknam, 1987; Howard, 1997; Roering et al., 1999; Pelletier, 2008;
Basilevsky et al., 2014].
A second, less favored alternative that might explain the recent enhanced diffusivity era in Figure 11 is that it is
symptomatic of a change in the impact flux in the ~700 Ma, since the form of Figure 11 is dependent on the
model impact flux. However, complicating this idea is that small impacts are presumed to be the controlling
factor for the diffusivity. If the impactors doing the diffusive work form at a rate proportional to the rate at that
the larger craters we measured form, it is hard to understand how a change in the recent impact flux can be
made compatible with Figure 11. In other words, if we wish to explain the recent increase in diffusivity by
arguing that the model impact flux should be higher, a problem arises, which is that the reason the recent era
has higher diffusivity in the first place is that there are fewer craters with κt ≤ ~5000 m2 than expected (i.e., κt
increases more rapidly for the freshest craters; Figure 9).
The higher diffusivities observed in the era prior to ~3.1 Ga are more plausibly related to the enhanced
and exponentially increasing crater flux in this period, which is well established from the calibration of the
impact crater record using the dated samples. Beyond the period constrained by our data, the rate of crater
degradation was probably even more rapid [Head, 1975]. During the late heavy bombardment era, the
flux was enhanced by a factor of at least 4× and potentially by 25× [e.g., Fassett and Minton, 2013] over what it
was at the beginning of the period we measured here (post ~ 3.7 Ga), which was likely sufficient to erase a
substantial number of D ≤ 5 km, pre-Imbrian craters.
5.2. Translating Diffusivity to Characteristic Erosion Rate
From equation (3), if we know the curvature, ∇2h, and the diffusivity, κ, we can calculate the characteristic
erosion rate on the Moon. (More precisely, what we can calculate is the characteristic magnitude of changes in
surface topography as a function of time, both positive and negative.) For the Moon as a whole, using the LOLA
128 ppd (236.9 m/pixel) gridded data, the median absolute value of the curvature |∇2h| = 7.26 × 105 m1.
These values imply a characteristic erosion (or deposition) rate over the last 3 Ga, using κ = 5.5 m2/Myr, of
0.4 mm/Myr. This is somewhat larger than the characteristic lunar erosion rate reported by Craddock and
Howard [2000] (0.2 mm/Myr).
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
13
Journal of Geophysical Research: Planets
10.1002/2014JE004698
Interpreting these erosion rates requires some caution. First, the curvature and roughness of the Moon is
scale-dependent [Rosenburg et al., 2011; Kreslavsky et al., 2013]. At meter-scale, topographic roughness is
rapidly created by small impactors and then destroyed by micrometeoritic bombardment, so the upper
portion of the surface is relatively dynamic. The net change in topography resulting from the roughness at
these scales over long periods of time is close to zero, however. The average erosion rates we report
above are at a scale where the topography is controlled by features of ~500 m and larger, which are typically
able to survive on the lunar surface for billions of years.
Second, the actual erosion rate in any given region of the Moon can be substantially greater or less than
the characteristic rate we quote. On flat-lying plains, the net change of topography over long periods is
effectively zero. On the portions of the Moon with the highest curvature, the rate of change of topography
can be much higher than the characteristic value quoted. The 90th percentile curvature of the 128 ppd LOLA
gridded data set—defined by calculating where 90% of the Moon has lower curvature—is 5.50 × 104 m1,
consistent with an erosion rate of 3 mm/Myr, or nearly an order of magnitude faster than is typical for the
median curvature observed at this ~240 m baseline.
Third, the values for the curvature we calculate include both the maria and the highlands, whereas the
diffusivity we use is derived only from craters in the maria. Although there is no reason to believe that the
effective topographic diffusivity should be different in the highlands (for areas of equivalent topographic
roughness and relief), additional assessment is needed to examine whether crater degradation and the
diffusivity of the highlands differs from the maria.
5.3. Crater Age Estimation From Degradation State
By determining κt values for a given crater, we can estimate its age given the calibration curve in Figure 11.
Applying this technique and linking it to other approaches that allow bootstrapping the age of individual
craters from other observational characteristics, such as optical or radar properties [e.g., Grier et al., 2001;
Ghent et al., 2005; Bell et al., 2012], has the potential to be of substantial value for both planning future
exploration and interpreting lunar geomorphology.
As a consequence of the form of this calibration function, the uncertainty in the age derived using the
degradation state we determine is highly dependent on the age of the crater. If we assume a ±30% uncertainty
in the determined κt, the implied age for a crater with κt = 4000 is 0.30 Ga (+0.13, 0.11 Gyr). For a crater that is
close to the median degradation state, with κt = 12,000 is 1.65 Ga (+1.11 Gyr, 0.71 Gyr). This larger relative
uncertainty in age for middle-aged craters is what should be expected from Figure 11 and is also a simple
consequence of the underlying process. Degraded craters evolve more slowly than fresh craters, and the
diffusivity in this era was lower than during the early, mare-forming period when the flux was higher. For a
crater formed in this earlier period, with highly degraded topography, κt = 24,000, the uncertainty in its age is
<20%: its age is predicted to be 3.63 Ga (+0.23 Gyr, 0.56 Gyr).
5.4. Diffusion, Crater Lifetimes, Equilibrium, and Limitation of Crater Counting With Small Craters
on the Moons
With an estimate for the diffusivity of the lunar surface, we can predict the lifetime over which we can
recognize craters of different sizes on the surface of the Moon as well. If we assume craters become
impossible to recognize when they reach 1% of their initial depth and assume κ = 5 m2/Myr, the lifetime of a
100 m crater on the Moon’s surface is ~1.7 Gyr, the lifetime of a 50 m crater is ~400 Myr, and 20 m craters
survive ~70 Myr. These lifetime estimates are likely to be upper limits since experience suggests that
recognizing craters becomes impossible well before they have this little relief (recall that 1% of the initial
depth of a D = 100 m crater is only 20 cm).
Because these diffusional lifetimes of small craters are relatively short, the crater degradation process is
intimately linked to the equilibrium population of craters on the lunar surface [e.g., Shoemaker, 1965;
Gault, 1970]. As Shoemaker [1965] noted, in equilibrium, “below some limiting diameter….there will be a
steady number of craters of any given size, no matter how long the cratering continues, and craters of
a given size will exhibit a complete range of shape [degradation states] from fresh…to barely discernible.”
The model presented here would suggest that the density of craters at any given size in this equilibrium
condition is a function of how fast craters are removed, which is controlled directly by κ and topography,
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
14
Journal of Geophysical Research: Planets
10.1002/2014JE004698
and is independent of the age of the surface. In other words, once a surface reaches equilibrium, you cannot
determine its formation age beyond knowing that it is greater than the time period required to reach equilibrium.
If topographic diffusion is what controlling the equilibrium density of craters, a clear consequence is that
there is no single equilibrium function that can be defined for the whole Moon. At the very least, the
equilibrium density in any given place is a function of surface topography, or, more specifically, the Laplacian
of topography (the topographic curvature). Given the substantial differences in topography between the
highlands and maria, this may lead to different equilibrium densities in these two provinces. The equilibrium
density is also likely to be affected by when craters of different sizes become unrecognizable, which might
not be at same level of relative relief change for all sizes. Second-order factors that may influence κ, such
as regolith thickness, could also play an important role. Fully delineating how crater degradation is linked to
equilibrium thus requires measurements beyond the scope of this work.
Regardless, most of the lunar surface—with the exception of regions resurfaced by young craters—is relatively
old (billions of years). Our data support earlier results [Shoemaker, 1965; Gault, 1970] that imply that statistics
of lunar craters with D <100 m should be avoided, unless one is interested in understanding crater retention
rather than an age of a surface or a geologic event.
6. Conclusions
In this study, we demonstrate that the evolution of a crater’s topography as it degrades can be treated as a
topographic diffusion problem, as suggested by earlier work [Soderblom, 1970; Craddock and Howard, 2000].
Note that this does not necessarily mean that all the topographic degradation experienced by craters is a
result of linear diffusive processes alone; nondiffusive processes such as landslides or deposition of distal
ejecta likely contribute to the crater degradation as well. However, the reasonable fit of a diffusive model for
topographic profiles of craters with a wide range of degradation states and ages suggest that the diffusive
model is a useful first-order framework for this type of analysis.
Other landforms on the Moon should experience similar evolution of their topography as well. However, since
craters have consistent initial morphometry and form at a well-understood rate, they can be used to calibrate
the rate of topographic diffusion. Over the last 3 Ga, we calculate that the average diffusivity experienced by a
crater on the Moon is κ = 5.5 m2/Myr, which is consistent with a median erosion rate on the Moon of ~0.4 mm/Myr.
The initial topographic diffusion of a crater is quicker than this average diffusivity represents, and the diffusion
rate early in lunar history (> ~ 3.1 Ga) was likely faster as well. This calibration of the rate of crater degradation
provides one strategy for estimating the age of individual craters on the lunar surface, as well as an independent
technique that can help constrain the age of geologic units.
Acknowledgments
We would particularly like to acknowledge
Jacquelyn Combellick and Waad Kahouli,
who contributed to the measurements
and data analysis that helped make this
study possible. Mikhail Kreslavsky and
an anonymous reviewer provided helpful comments. We thank the SELENE
(KAGUYA) TC team and the SELENE Data
Archive for acquiring and providing
access to their data, as well as the LROC
and LOLA teams for their excellent work.
This research made use of the USGS
Integrated Software for Imaging and
Spectrometers (ISIS) and the Generic
Mapping Tools (GMT). The data used for
this paper are available on request from
the corresponding author.
FASSETT AND THOMSON
References
Andrews, D. J., and R. C. Bucknam (1987), Fitting degradation of shoreline scarps by a nonlinear diffusion model, J. Geophys. Res., 92(B12),
12,857–12,867, doi:10.1029/JB092iB12p12857.
Andrews-Hanna, J. C., and M. T. Zuber (2010), Elliptical craters and basins on the terrestrial planets, Geol. Soc. Am. Spec. Pap., 465, 1–13.
Barker, M. K., E. Mazarico, G. A. Neumann, D. E. Smith, and M. T. Zuber (2014), Merging digital elevation models from the Lunar Orbiter Laser
Altimeter and Kaguya Terrain Camera, in Lunar and Planetary Science Conference, 45, Abstract no. 1635.
Bart, G. D. (2014), The quantitative relationship between small impact crater morphology and regolith depth, Icarus, 235, 130–135,
doi:10.1016/j.icarus.2014.03.020.
Basilevsky, A. T. (1976), On the evolution rate of small lunar craters, Proc. Lunar Planet. Sci. Conf., 7th, 1005–1020.
Basilevsky, A. T., G. Neukum, and L. Nyquist (2011), The spatial and temporal distribution of lunar mare basalts as deduced from analysis of
data for lunar meteorites, Planet. Space Sci., 58, 1900–1905, doi:10.1016/j.pss.2010.08.020.
Basilevsky, A. T., J. W. Head, and F. Hörz (2013), Survival times of meter-sized boulders on the surface of the Moon, Planet. Space Sci., 89, 118–126,
doi:10.1016/j.pss.2013.07.011.
Basilevsky, A. T., M. A. Kreslavsky, I. P. Karachevtseva, and E. N. Gusakova (2014), Morphometry of small impact craters in the Lunokhod-1 and
Lunokhod-2 study areas, Planet. Space Sci., 92, 77–87.
Bell, S. W., B. J. Thomson, M. D. Dyar, C. D. Neish, J. T. S. Cahill, and D. B. J. Bussey (2012), Dating small fresh lunar craters with Mini-RF radar
observations of ejecta blankets, J. Geophys. Res., 117, E00H30, doi:10.1029/2011JE004007.
Bouley, S., and D. Baratoux (2011), Variation of small crater degradation on the Moon, 42nd Lunar and Planetary Science Conference, LPI
Contribution No.,1608, Abstract no. 1388, Woodlands, Texas, 7–11 March.
Boyce, J. M. (1976), Age of flow units in the lunar nearside maria based on Lunar Orbiter IV photographs, 7th Proc. Lunar Sci. Conf., 2717–2728.
Boyce, J. M., and A. L. Dial Jr. (1973), Relative ages of some nearside mare units based on Apollo 17 metric photographs, Apollo 17 Prelim Sci. Rep.,
NASA SP-330, 29–26 – 29–28.
Boyce, J. M., and A. L. Dial Jr. (1975), Relative ages of flow units in Mare Imbrium and Sinus Iridum, Proc. Lunar Sci. Conf., 6th, 2585–2595.
Boyce, J. M., and D. A. Johnson (1978), Ages of flow units in the far eastern maria and implications for basin-filling history, 9th Proc. Lunar
Planet. Sci. Conf., 3275– 3283.
©2014. American Geophysical Union. All Rights Reserved.
15
Journal of Geophysical Research: Planets
10.1002/2014JE004698
Boyce, J. M., A. L. Dial, and L. A. Soderblom (1974), Ages of the lunar nearside light plains and maria, Proc. Lunar Planet. Sci. Conf., 1st, 11–23.
Craddock, R. A., and A. D. Howard (2000), Simulated degradation of lunar impact craters and a new method for age dating farside mare
deposits, J. Geophys. Res., 105, 20,387–20,401, doi:10.1029/1999JE001099.
40
39
Culler, T. S., T. A. Becker, R. A. Muller, and P. R. Renne (2000), Lunar impact history from Ar/ Ar dating of glass spherules, Science, 287, 1785–1788.
Culling, W. E. H. (1960), Analytical theory of erosion, J. Geol., 68, 336–344.
Fassett, C. I., and D. A. Minton (2013), Impact bombardment of the terrestrial planets and the early history of the solar system, Nat. Geosci., 6,
520–524, doi:10.1038/ngeo1841.
Gault, D. E. (1970), Saturation and equilibrium conditions for impact cratering on the lunar surface: Criteria and implications, Radio Sci., 5(2),
273–291, doi:10.1029/RS005i002p00273.
Gault, D. E., and J. A. Wedekind (1978), Experimental studies of oblique impact, Proc. Lunar Planet. Sci. Conf., 9th, 3843–3875.
Ghent, R. R., D. W. Leverington, B. A. Campbell, B. R. Hawke, and D. B. Campbell (2005), Earth-based observations of radar-dark crater haloes
on the Moon: Implications for regolith properties, J. Geophys. Res., 110, E02005, doi:10.1029/2004JE002366.
Grier, J. A., A. S. McEwen, P. G. Lucey, M. Milazzo, and R. G. Strom (2001), Optical maturity of ejecta from large rayed lunar craters, J. Geophys. Res.,
106, 32,847–32,862, doi:10.1029/1999JE001160.
Grün, E., M. Horanyi, and Z. Sternovsky (2011), The lunar dust environment, Planet. Space Sci., 59, 1672–1680.
Hartmann, W. K., C. Quantin, and N. Mangold (2007), Possible long-term decline in impact rates: 2. Lunar impact-melt data regarding impact
history, Icarus, 186, 11–23, doi:10.1016/j.icarus.2006.09.009.
Haruyama, J., et al. (2012), Lunar global digital terrain model dataset produced from Selene (Kaguya) Terrain Camera stereo observations,
43rd Lunar Planet. Sci. Conf., 1200.
Head, J. W. (1975), Processes of lunar crater degradation: Changes in style with geologic time, Moon, 12, 299–329.
Hiesinger, H., R. Jaumann, G. Neukum, and J. W. Head III (2000), Ages of mare basalts on the lunar nearside, J. Geophys. Res., 105(E12),
29,239–29,275, doi:10.1029/2000JE001244.
Hiesinger, H., J. W. Head III, U. Wolf, R. Jaumann, and G. Neukum (2003), Ages and stratigraphy of mare basalts in Oceanus Procellarum,
Mare Nubium, Mare Cognitum, and Mare Insularum, J. Geophys. Res., 108(E7), 5065, doi:10.1029/2002JE001985.
Hiesinger, H., J. W. Head III, U. Wolf, R. Jaumann, and G. Neukum (2010), Ages and stratigraphy of lunar mare basalts in Mare Frigoris and other
nearside maria based on crater size-frequency distribution measurements, J. Geophys. Res., 115, E03003, doi:10.1029/2009JE003380.
Hiesinger, H., J. W. Head III, U. Wolf, R. Jaumann, and G. Neukum (2011), Ages and stratigraphy of lunar mare basalts: A synthesis, Geol. Soc.
Am. Spec. Pap., 477, 1–51, doi:10.1130/2011.2477(01).
Hiesinger, H., C. H. van der Bogert, J. H. Pasckert, L. Funcke, L. Giacomini, L. R. Ostrach, and M. S. Robinson (2012), How old are young
lunar craters?, J. Geophys. Res., 117, E00H10, doi:10.1029/2011JE003935.
Hörz, F., E. Schneider, D. E. Gault, J. B. Hartung, and D. E. Brownlee (1975), Catastrophic rupture of lunar rocks: A Monte-Carlo simulation,
Moon, 13, 235–238.
Howard, A. D. (1997), Badland morphology and evolution: Interpretation using a simulation model, Earth Surf. Processes Landforms, 22, 211–227.
Howard, A. D. (2007), Simulating the development of Martian highland landscapes through the interaction of impact cratering, fluvial erosion,
and variable hydrologic forcing, Geomorphology, 91, 332–363.
Kneissl, T., S. van Gasselt, and G. Neukum (2011), Map-projection-independent crater size-frequency determination in GIS environments—
New software tool for ArcGIS, Planet. Space Sci., 59, 1243–1254, doi:10.1016/j.pss.2010.03.015.
Kreslavsky, M. A., J. W. Head, G. A. Neumann, M. A. Rosenburg, O. Aharonson, D. E. Smith, and M. T. Zuber (2013), Lunar topographic
roughness maps from Lunar Orbiter Laser Altimeter (LOLA) data: Scale dependence and correlation with geologic features and units,
Icarus, 226, 52–66.
Kumar, P. S., V. Keerthi, A. Senthil Kumar, J. Mustard, B. Gopala Krishna, Amitabh, L. R. Ostrach, D. A. Kring, A. S. Kiran Kumar, and J. N. Goswami (2013),
Gullies and landslides on the Moon: Evidence for dry-granular flows, J. Geophys. Res. Planets, 118, 206–223, doi:10.1002/jgre.20043.
Li, L., and J. F. Mustard (2005), On lateral mixing efficiency of lunar regolith, J. Geophys. Res., 110, E11002, doi:10.1029/2004JE002295.
McEwen, A. S., J. M. Moore, and E. M. Shoemaker (1997), The Phanerozoic impact cratering rate: Evidence from the farside of the Moon,
J. Geophys. Res., 102(E4), 9231–9242, doi:10.1029/97JE00114.
Minton, D. A., and R. Malhotra (2010), Dynamical erosion of the asteroid belt and implications for large impacts in the inner Solar System,
Icarus, 207, 744–757.
Molaro, J., and S. Byrne (2012), Rates of temperature change of airless landscapes and implications for thermal stress weathering, J. Geophys. Res.,
117, E10011, doi:10.1029/2012JE004138.
Neukum, G., B. A. Ivanov, and W. K. Hartmann (2001), Cratering records in the inner solar system in relation to the lunar reference system,
Space Sci. Rev., 96, 55–86.
Oberbeck, V. R., and W. L. Quaide (1967), Estimated thickness of a fragmental surface layer of Oceanus Procellarum, J. Geophys. Res., 72(18),
4697–4704, doi:10.1029/JZ072i018p04697.
Ostrach, L. R., and M. S. Robinson (2014), Areal crater density analysis of volcanic smooth plains: Mare Imbrium, A revised approach, in Lunar and
Planetary Science Conference, 45, Abstract no. 1266.
Pelletier, J. D. (2008), Quantitative Modeling of Earth Surface Processes, 295 pp., Cambridge Univ. Press, Cambridge, U. K.
Pike, R. J. (1974), Depth/diameter relations of fresh lunar craters: Revision from spacecraft data, Geophys. Res. Lett., 1, 291–294, doi:10.1029/
GL001i007p00291.
Pike, R. J. (1977), Size-dependence in the shape of fresh impact craters on the moon, in Impact and Explosion Cratering, pp. 489–507,
Pergamon Press, New York.
Pike, R. J. (1980), Control of crater morphology by gravity and target type: Mars, Earth, Moon, Proc. Lunar Planet. Sci. Conf., 11th, 2159–2189.
Quaide, W. L., and V. R. Oberbeck (1968), Thickness determinations of the lunar surface layer from lunar impact craters, J. Geophys. Res., 73(16),
5247–5270, doi:10.1029/JB073i016p05247.
Robbins, S. J., et al. (2014), The variability of crater identification among expert and community crater analysts, Icarus, 234, 109–131.
Robinson, M. S., et al. (2010), Lunar Reconnaissance Orbiter Camera (LROC) instrument overview, Space Sci. Rev., 150, 81–124.
Roering, J. J., J. W. Kirchner, and W. E. Dietrich (1999), Evidence for nonlinear, diffusive sediment transport on hillslopes and implications for
landscape morphology, Water Resour. Res., 35(3), 853–870, doi:10.1029/1998WR900090.
Rosenburg, M. A., O. Aharonson, J. W. Head, M. A. Kreslavsky, E. Mazarico, G. A. Neumann, D. E. Smith, M. H. Torrence, and M. T. Zuber (2011), Global
surface slopes and roughness of the moon from the lunar orbiter laser altimeter, J. Geophys. Res., 116, E02001, doi:10.1029/2010JE003716.
Ross, H. P. (1968), A simplified mathematical model for lunar crater erosion, J. Geophys. Res., 73, 1343–1354, doi:10.1029/JB073i004p01343.
Salamunićcar, G., S. Lončarić, A. Grumpe, and C. Wöhler (2014), Hybrid method for crater detection based on topography reconstruction from
optical images and the new LU78287GT catalogue of Lunar impact craters, Adv. Space Res., 53, 1783–1797, doi:10.1016/j.asr.2013.06.024.
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
16
Journal of Geophysical Research: Planets
10.1002/2014JE004698
Schultz, P. H. (1976), Moon Morphology, 626 pp., Univ. of Texas Press, Austin.
Schultz, P. H., and D. E. Gault (1976), Seismically induced modification of lunar surface features, Proc. Lunar Sci. Conf., 6th, 2845–2862.
Schultz, P. H., and P. D. Spudis (1983), Beginning and end of lunar mare volcanism, Nature, 302, 233–236, doi:10.1038/302233a0.
Shoemaker, E. M. (1965), Preliminary analysis of the fine structure of the lunar surface in Mare Cognitum, in Ranger VII—Part II.
Experimenters’ Analyses and Interpretations, Tech. Rep., 32–700, pp. 75–134, Jet Propul. Lab., Pasadena, Calif.
Skianis, G. A., D. Vaiopoulos, and N. Evelpidou (2008), Solution of the linear diffusion equation for modelling erosion processes with a time
varying diffusion coefficient, Earth Surf. Processes Landforms, 33, 1491–1501.
Smith, D. E., et al. (2010), Initial observations from the Lunar Orbiter Laser Altimeter (LOLA), Geophys. Res. Lett., 37, L18204, doi:10.1029/
2010GL043751.
Soderblom, L. A. (1970), A model for small-impact erosion applied to the lunar surface, J. Geophys. Res., 75, 2655–2661, doi:10.1029/
JB075i014p02655.
Soderblom, L. A., and L. A. Lebofsky (1972), Technique for rapid determination of relative ages for lunar areas from orbital photography,
J. Geophys. Res., 77, 279–296, doi:10.1029/JB077i002p00279.
Speyerer, E. J., R. V. Wagner, A. Licht, M. S. Robinson, K. J. Becker, and J. A. Anderson (2014), New SPICE to improve the geodetic accuracy of
LROC NAC and WAC images, 45th Lunar Planet Sci. Conf., Abstract no. 2421.
Werner, S. C., and S. Medvedev (2010), The Lunar rayed-crater population—Characteristics of the spatial distribution and ray retention,
Earth Planet. Sci. Lett., 295, 147–158.
Wessel, P., and W. H. F. Smith (1998), New, improved version of Generic Mapping Tools released, Eos Trans. AGU, 79, 579, doi:10.1029/98EO00426.
Wilhelms, D. E. (1987), Geologic History of the Moon, U.S. Geol. Surv. Prof. Pap., 1348, 302 pp.
Wood, C. A., and L. Anderson (1978), New morphometric data for fresh lunar craters, Proc. Lunar Planet. Sci. Conf., 9th, 3669–3689.
Xiao, Z., Z. Zeng, N. Ding, and J. Molaro (2013), Mass wasting features on the Moon—How active is the lunar surface?, Earth Planet. Sci. Lett.,
376, 1–11.
FASSETT AND THOMSON
©2014. American Geophysical Union. All Rights Reserved.
17