Icarus 273 (2016) 329–336
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Icarus
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The steepest slopes on the Moon from Lunar Orbiter Laser Altimeter
(LOLA) Data: Spatial Distribution and Correlation with Geologic
Features
Mikhail A. Kreslavsky a,b,∗, James W. Head c
a
b
c
Earth and Planetary Sciences, University of California, Santa Cruz, CA 95064 USA
MExLab, Moscow State University of Geodesy and Cartography (MIIGAiK), Moscow 105064, Russia
Earth, Environmental and Planetary Sciences, Brown University, Providence, RI 02912, USA
a r t i c l e
i n f o
Article history:
Received 4 July 2015
Revised 26 January 2016
Accepted 22 February 2016
Available online 2 March 2016
Keywords:
Moon Surface
Geological processes
Impact processes
a b s t r a c t
We calculated topographic gradients over the surface of the Moon at a 25 m baseline using data obtained by the Lunar Orbiter Laser Altimeter (LOLA) instrument onboard the Lunar Reconnaissance Orbiter
(LRO) spacecraft. The relative spatial distribution of steep slopes can be reliably obtained, although some
technical characteristics of the LOLA dataset preclude statistical studies of slope orientation. The derived
slope-frequency distribution revealed a steep rollover for slopes close to the angle of repose. Slopes significantly steeper than the angle of repose are almost absent on the Moon due to (1) the general absence of cohesion/strength of the fractured and fragmented megaregolith of the lunar highlands, and (2)
the absence of geological processes producing steep-slopes in the recent geological past. The majority of
slopes steeper than 32°–35° are associated with relatively young large impact craters. We demonstrate
that these impact craters progressively lose their steepest slopes. We also found that features of Early
Imbrian and older ages have almost no slopes steeper than 35°. We interpret this to be due to removal
of all steep slopes by the latest basin-forming impact (Orientale), probably by global seismic shaking. The
global spatial distribution of the steepest slopes correlates moderately well with the predicted spatial
distribution of impact rate; however, a significant paucity of steep slopes in the southern farside remains
unexplained.
© 2016 Elsevier Inc. All rights reserved.
1. Introduction
The statistical characterization of topographic slopes is a valuable tool for comparison of planetary landscapes and formulating
an understanding of landscape evolution processes (e.g., Sharpton
and Head, 1986; Kreslavsky and Head, 1999; Aharonson et al.,
2001; Thomas et al., 2002; Rosenburg et al, 2011). The steepest
slopes are of special interest because they are more sensitive to alteration and degradation processes; on the Moon with its generally
old and stable surface, this sensitivity is important. On one hand,
the presence of steep slopes indicates the youthfulness of the related geologic features. Therefore, the steepest slopes are an independent source of information about feature ages. On the other
hand, statistical differences about the occurrence of steep slopes
for geologically similar features of different stratigraphic position
provide information about the nature and peculiarities of slopemodification processes in different geological epochs. In this paper
∗
Corresponding author: Tel.: +1 831 459 5143; fax: +1 831 459 3074.
E-mail address: [email protected] (M.A. Kreslavsky).
http://dx.doi.org/10.1016/j.icarus.2016.02.036
0019-1035/© 2016 Elsevier Inc. All rights reserved.
we analyze the steepest slopes on the Moon and present our primary conclusions.
Slope steepness depends on the baseline at which the slopes
are considered: generally, the shorter the baseline, the steeper the
slopes. Slopes at long baselines, kilometers and longer, are limited
by the total topographic amplitude of associated topographic features and are less useful because in a sense they just duplicate
the information that can be derived from the topographic amplitudes (e.g., crater and basin depths, etc.). In this paper we focus
on the shortest baselines at which we can accurately derive slopes.
The necessity of accurate and statistically unbiased measurements
of the steep slopes dictates the use of laser altimeter data. Topographic data derived from stereo images (e.g., Robinson et al.,
2010; Scholten et al., 2012) despite their potentially high resolution, have some shortcomings for the statistical analysis of steep
slopes, because the images used for photogrammetric analysis often have shadows in association with the steepest slopes of certain
orientation, which makes it impossible to derive elevation differences reliably for some slopes. Here we use data from the Lunar
Orbiter Laser Altimeter (LOLA) (Smith et al. 2010a, b) onboard the
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M.A. Kreslavsky, J.W. Head / Icarus 273 (2016) 329–336
Lunar Reconnaissance Orbiter (LRO), and study steep slopes at the
shortest baselines allowed by this instrument, namely, 25 m. Initially, the general statistical results about slopes obtained from the
LOLA data set have been reported by Rosenburg et al. (2011). Here
we specifically focus on the steepest slopes.
In this paper, when we use the word “slope” in a quantitative
sense, we mean the so-called “two-dimensional” slope, also known
as the absolute value of the topographic gradient. For discussions
of the relationships between two-dimensional slopes and onedimensional slopes (also known as slopes along profiles) see, e.g.,
Aharonson and Schorghofer (2006), and Rosenburg et al. (2011).
Useful equations for converting isotropic slope-frequency distributions of one-dimensional slopes to two-dimensional slopes and
vice versa are listed by Kreslavsky et al. (2015).
In this paper we first describe the technical details of slope
calculations and analyze the related biases. Then we consider
the slope-frequency distribution and briefly discuss the paucity of
slopes steeper than the angle of repose on the Moon. Then we analyze the association of the steepest slopes with specific geological
features, on the Moon mostly associated with large impact craters
of different stratigraphic age, and we discuss the observed trends.
Finally, we consider the global distribution of the proportion of the
steepest slopes and compare it with other global data sets.
2. Slope calculation and bias analysis
Normally, every time a LOLA laser shot probes the surface (every “shot”), it obtains range (and inferred elevation) measurements
in 5 small (∼5 m) “spots” that form a specially designed pattern on
the surface: Spot 1 is in the center and Spots 2, 3, 4, and 5 around
it are separated from it by ∼25 m and are located in the vertexes
of a square (Smith et al., 2010a,b). The coordinates of each spot
and the measured elevations are available as LOLA Reduced Data
Records (RDRs) from the NASA Planetary Data System (PDS). For
any three non-collinear points with known coordinates and elevations, straightforward geometric calculations yield the topographic
gradient vector; its absolute value is the (two-dimensional) slope.
For each LOLA shot we calculated four slopes at ∼25 m baseline
using four triplets of spots 1-2-3, 1-3-4, 1-4-5, and 1-5-2. Slopes
were calculated with respect to a reference sphere. For our purposes, the difference between slopes with respect to the geoid and
to the sphere is negligible.
The internal LOLA ranging precision is ∼10 cm; however, for the
steep slopes and rough terrains considered in this paper, the precision is worse due to actual elevation variation within the ∼5 m
LOLA spot. A possible ∼1 m uncertainty within each spot corresponds to ∼4º uncertainty in slope. This uncertainty does not affect
our ability to distinguish between steep (>32º) and typical (∼10º)
slopes, and very steep (> 40º) and steep (∼32º) slopes. The effect of
this uncertainty on the result is also reduced by the large statistics.
We performed the slope calculations for the entire set of
LOLA data obtained during LRO operations in its circular orbit
phase from September 2009 to December 2011 (LRO orbits 1005 –
11403). When LRO was in the non-illuminated portion of the Moon
or close to it, distortion of the alignment of the LOLA laser beams
and the detector optical axes regularly occurred due to the thermal
contraction (see Smith et al. 2010b for details). We excluded all
data affected by this LOLA anomaly. The data set used consisted of
∼6 × 108 successful laser shots, so we calculated ∼ 2.4 × 109 slopes.
Fig. 1 presents the histogram (incremental slope-frequency distribution) for this dataset
We selected and listed all ∼6 × 106 slopes steeper than 32º:
such slopes are steeper or comparable to the angle of repose of
dry material and therefore are interesting for the analysis of slope
degradation. Fig. 2 presents the spatial distribution of the steepest slopes on the Moon. Each black dot marks the location of the
Fig. 1. Incremental frequency distribution of two-dimensional slopes at 25 m baseline calculated with 0.1° increment for the entire data set used: (a) linear frequency
axis, (b) logarithmic frequency axis. Shaded area is the range of slopes studied in
this paper.
slopes steeper than the value listed; many points overlap. The data
coverage of the surface is inhomogeneous: a typical distance between orbit tracks at low latitudes is on the order of 0.8 km; however, the distance varies widely, and there are gaps as wide as
∼4 km.
The dataset contains a tiny proportion of spurious points,
wrongly measured elevations not marked as bad data points in the
data release. Such points usually produce spurious steep slopes.
Among ∼5 × 104 slopes steeper than 45º, about 80% are spurious,
and the rest are genuine. This number of spurious slopes is an approximate estimate based on our automated analysis of proximity of steep slopes to each other. For example, if slopes from the
spot triplets 1-2-3 and 1-5-2 are steeper than 45º and have similar orientation, while slopes form two other triplets of this shot as
well as all slopes from neighboring shots are gentler than 32°, it
is likely that spot 2 elevation is spurious. We manually checked
∼10 cases like this using images. The majority of the scattered
black dots in Fig. 2a are spurious, although some of them may be
genuine steep slopes associated with walls of small young craters.
For slopes steeper than 40º (Fig. 2b) the proportion of spurious
points in the scattered slopes is still noticeable; it becomes negligible for gentler slopes. Clusters of black dots in locations of large
young impact craters (Fig. 2a) coincide with steep crater walls and
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331
Fig. 2. Spatial distribution of slopes steeper than (a) 45°, (b) 40°, (c) 35°, (d) 32°. Simple cylindrical projection for the entire Moon centered at the center of the nearside.
Each black dot marks the position of a steep slope; many dots overlap.
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M.A. Kreslavsky, J.W. Head / Icarus 273 (2016) 329–336
therefore are genuine. Among ∼1.4 × 105 slopes steeper than 40°
(Fig. 2b), the percentage of scattered spurious points is still significant, but here they do not dominate. For slopes steeper than 35º
(∼1 × 106 , Fig. 2c) the proportion of spurious points is already negligibly small. In Fig. 1b, the steep-slope tip of the distribution (38–
40°) decreases more gently than the main rollover (34–37°); this is
probably due to the contribution of the spurious data points.
There is a small proportion of missing (or marked as bad) data
points in the LOLA data set. However, on steep terrain, this percentage is significantly higher and can reach about one half on
the steepest slopes. This means that our data significantly underestimate (up to a factor of two) the absolute percentage of the
steepest among all measured slopes. Moreover, the proportion of
missing data points is different for different spots and depends on
slope orientation. This means that the statistics of slope orientation (direction of the topographic gradient vector) is biased. When
we tried to analyze the rose diagrams of steep-slope orientation,
we found that they are obviously biased: the slope orientation
distributions are weirdly anisotropic with characteristic directions
aligned with the five-spot configuration axes. The rose diagrams
calculated separately for each of the four spot triplets used were
different. Thus, the described bias makes it impossible to study the
statistics of slope orientation with the data used.
We analyzed this issue in more detail and found that the proportion of missing data points as a function of slope is a rather
stable quantity for each of the five spots; it does not depend on
latitude and only weakly drifts through the mission. This means
that the relative spatial variations of the proportion of steep slopes
are detected reliably. In other words, both the association of steep
slopes with particular geologic features and global spatial distributions of steep slopes are obtained reliably.
3. Paucity of steep slopes
The global slope-frequency distribution (Fig. 1) has a distinctive structure. The downward roll-over at the zero-slope margin is
usual for two-dimensional slopes of natural surfaces. The most frequent slopes are 1–3° steep, and the slope-frequency distribution
has a heavy steep-slope tail. Preliminary analysis by Kreslavsky
et al. (2015) showed significant differences in the nature of slopefrequency distributions between highlands and maria and the presence of another characteristic slope of ∼6° in highlands. Fig. 1
mixes highlands and maria together; however, it is dominated by
the highlands. The frequency distribution of slopes is an interesting subject of a separate study; here we only focus on the steepest
slopes (>32°, as shown by the shaded domain in Fig. 1). For these
slopes, the frequency distribution sharply drops (Fig. 1b), reflecting
the strong relative paucity of slopes steeper than typical angles of
repose (e.g., Jaeger et al., 1988) of dry materials (28–35º).
A characteristic vertical scale associated with the steep slopes is
comparable to the slope baseline, that is, about tens of meters in
our case. At this vertical scale the lunar highlands are composed of
megaregolith (e.g., Hartmann, 1973), a layer of impact-fragmented,
disintegrated material. Slopes in such material are maintained by
dry friction between the fragments; therefore, the occurrence of
steep slopes in the megaregolith is limited by the angle of repose.
This explains the paucity of slopes steeper than the angle of repose
in the highlands.
Another, complementary (rather than alternative) explanation
of this observation is the absence of geological processes capable of producing new steep slopes on the Moon. On the Earth,
at the scale of tens of meters, nearly vertical cliffs are constantly
produced by water erosion (streams, rivers, waves), a process absent on the Moon. On Mars, the majority of slopes steeper than
45° at 300 m baseline are associated with the huge walls of Vallis Marineris and walls of young Cerberus Fossae (Kreslavsky and
Head, 2003a,b); these steep linear walls are created by tectonics.
On the Moon, morphological evidence of geologically recent tectonic deformation has been reported (e.g., Watters et al., 2012);
however, the associated features are minor in their distribution
and magnitude and no steep slopes at a 25 m baseline are produced. Steep cliffs on the Earth and Mars are also associated with
collapse features, however, such features ultimately originate from
water erosion, absent on the Moon, and volcanic, processes generally absent on the Moon in the recent geological past (Copernican
period and later part of Eratosthenian periods, 1–2 Ga) (Hiesinger
et al., 2011). Braden et al. (2014) have recently described features
interpreted to represent basaltic volcanism occurring in the last
100 million years, but the majority of these features are too small
to be reliably detected in our data. Volcanic and tectonic processes
capable of producing steep slopes (Head and Wilson, 2016) generally date back to the Imbrian and earlier periods (∼3 Ga and older)
(Hiesinger et al., 2011).
The only widespread process that does effectively produce geologically young steep slopes is impact cratering. In the next subsection we see that the majority of the steepest slopes on the Moon
are associated with impact craters. Slopes in fresh craters, however, rarely exceed ∼45°. This occurs because during crater formation, there is a significant concentration of seismic/acoustic energy,
which mobilizes material; this, in turn, makes steep crater walls
unstable.
In summary, on the Moon, we observe a virtual absence of
slopes significantly steeper than the angle of repose. We explain
this by a combination of two factors: (1) an absence of cohesion/strength of the fractured and fragmented megaregolith of the
lunar highlands, making slopes steeper than the angle of repose
unstable, and (2) the absence of active geological processes that
could produce slopes steeper than the angle of repose (e.g., volcanic vents, collapse features, lava flow channels) in the competent
material of the lunar maria in the recent geological past.
4. Association with geologic features
4.1. Large impact craters
As we noted in Section 2, the majority of inferred slopes steeper
than 45° are actually artifacts caused by unidentified spurious
points in the LOLA data set. However, some slopes steeper than 45°
are present on the Moon. Almost all of them are associated with
large Copernican-aged impact craters. In Fig. 2a, dense clusters of
black dots indicating the locations of steep slopes clearly mark all
of the largest Copernican-age craters (except Taruntius, an impact
crater that will be discussed later). For progressively gentler slopes
(40°, 35°, 32°, Fig. 2b–d), large craters of Eratosthenian and Late
Imbrian age appear in the maps. Usually, craters of Early Imbrian
age and older are not seen as distinctive clusters of steep slopes in
the maps (Fig. 2).
Eratosthenian and Late Imbrian craters appear as empty circles
in the maps (Fig. 2c and d), sometimes with a small cluster of dots
in the center, indicating that the steep slopes are associated with
crater walls and sometimes with the central peak(s). Many Copernican craters appear as filled circles, indicating that steep slopes
are also associated with terraces or are caused by the presence of
meters-sized blocks on the floor.
All these observations are readily explained by the progressive
degradation of steep slopes in impact craters (Head, 1975). Freshly
formed large craters have some steep slopes and meter(s)-sized
blocks on the crater floors are apparently responsible for some
steep slope data point retrievals. The blocks disappear geologically
rapidly, at time scales on the order of 100 Ma (Basilevsky et al.
2013, 2015). The slopes also degrade, and talus deposits accumulate at the base of the slopes. The volume of talus material needed
M.A. Kreslavsky, J.W. Head / Icarus 273 (2016) 329–336
to hide a steep wall is proportional to the square of wall height,
while the rate of talus accumulation is approximately proportional
to the height of exposed part of the wall; therefore taller walls preserve their steepness for longer times. Short steep slopes on/near
crater floors disappear first, and tall outer parts of crater interior
walls keep their steepness for longer periods. Meters-size blocks
on the crater floors apparently responsible for some steep slope
data point retrievals also disappear rather quickly, at time scales
on the order of 100 Ma (Basilevsky et al. 2013, 2015). The talus accumulation rate also strongly depends on slope: steeper slopes degrade more rapidly. Thus, progressive degradation of steep slopes
and the accumulation of talus appears to be a very plausible mechanism to explain the absence of steeper slopes in relatively older
craters.
The sharp difference in steep slope occurrences between Late
Imbrian and Early Imbrian craters is readily explained by the hypothesis that basin-forming impacts can cause global resurfacing. On the basis of the kilometer-scale topographic roughness
signature, Kreslavsky and Head (2012) argued that the Orientale
basin-forming impact event, separating the Early and Late Imbrian
epochs, was the last large basing-forming impact, and thus was the
last event producing global slope resurfacing at the kilometer spatial scale. Naturally, this global resurfacing event is also the latest event that instantly removed all preexisting steep slopes globally. Kreslavsky and Head (2012) suggested that global seismic effects produced by basin-forming impacts (Schultz and Gault, 1975)
are the physical mechanism responsible for such global resurfacing. In this interpretation (Kreslavsky and Head, 2012), global seismic shaking immediately following the initial impact caused pervasive mobilization of surface material and failure of pre-existing
steep slopes, which effectively erased almost all pre-existing steep
slopes. In this scenario, each large basin-forming impact event
erased all preexisting steep slopes; therefore all currently observed
steep slopes are predicted to have been created after the latest of
these large impact events, the impact that formed the Orientale
basin. We found that large (∼10 km) distal secondary craters associated with the Orientale impact are characterized by abundant
32°–35° steep slopes on their walls, similar to typical Late Imbrian
primary craters. This is consistent with global seismic shaking, because the seismic wave travels rapidly and precedes the arrival of
the secondary-forming projectiles (Schultz and Gault, 1975).
There are some exceptions to the prominent and wellunderstood degradation trend described above. Below we consider
two of the most obvious examples.
The crater Taruntius (56 km in diameter) has been mapped as
Copernican in age by Wilhelms (1987) on the basis of its associated bright rays; however it has no slopes steeper than 32° except
those associated with superposed small craters. The morphology
of this crater seen in the high-resolution LROC NAC images is also
inconsistent with its Copernican age. The identification of Tarunitus as Copernican in age is probably incorrect. The ray system may
be a visual effect due to the superposition of faint rays from distal craters, and/or includes a compositional aspect of bright material that acts to subvert the simple regolith mixing-rate interpretation of the crater-ray degradation time scale (Hawke et al.,
2004). However, Taruntius clearly postdates the Mare Fecunditatis
volcanic plains of Late Imbrian age; it is therefore Late Imbrian
or younger in age and should have steep slopes according to its
age. Tauruntius is also a floor-fractured crater (Schultz, 1976; Wichmann and Schultz, 1996; Jozwiak et al. 2012, 2015) and the process
of intrusion and uplift could have caused tilting and local seismic
shaking sufficient to modify the slopes.
The crater Tsiolkovskiy (180 km in diameter) has been mapped
as Late Imbrian in age by Wilhelms (1987). This age is constrained
on the younger side by the Late Imbrian age of mare infill of the
crater interior, which in turn was obtained from counts of 100 m–
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1 km craters (Tyrie, 1988), although counting of craters larger
than 500 m on the mare infill yielded an age close to the Imbrian/Eratosthenian boundary (Williams et al., 2013). Tsiolkovskiy’s
walls and central peak have a number of sites with slopes steeper
than 45º and clearly appear in the 40° steep slope map (Fig. 2b).
According to this steep slope signature it is predicted to be on
the younger side of the Eratosthenian age. Tsiolkovskiy’s rim is
also known for its anomalously high rock abundance (Greenhagen
et al., 2013) inferred from night-time thermal infrared emission
(Bandfield et al. 2011). The observed rock abundance is typical of
Copernican-aged craters only, and abundant rocks together with
other morphological characteristics of Copernican-age craters indeed are seen in high-resolution LROC NAC images (Neish et al.
2013). Anomalously steep slopes explain the anomalously high
rock abundance: rapid regolith downwasting on steep slopes exposes rocks. The presence of steep slopes, however, remains unexplained. A possible explanation is that both Tsiolkovskiy and
its mare infill are close to the Copernican/Eratosthenian boundary in their age, while the high density of superposed craters on
mare infill is due to an anomalous contamination with secondaries.
The latter is consistent with an anomalously high hectometer-scale
topographic roughness displayed by the Tsiolkovskiy mare infill
(Kreslavsky et al., 2013).
4.2. Impact basins
The Orientale basin, the youngest impact basin on the Moon
(Wilhelms, 1987; Whitten et al., 2011) is apparent on the steep
slope maps (Fig. 2b–d). The basin rings (Montes Rook and Montes
Cordillera) have steep slopes, including those steeper than 40°
(Fig. 2b). The hypothesis of global small-scale resurfacing by basinforming impacts (Kreslavsky and Head, 2012) predicts that all
other, older basins would not have steep slopes because these
slopes would be removed by the seismic effects of the latest Orientale impact. In general, this is correct: no other basin shows as pronounced a signature as Orientale. There are, however, some qualifications.
The outer ring of the Schrödinger basin does not show any
enhancement in steep slopes. This basin is of Early Imbrian age,
meaning that it predates Orientale, therefore the absence of steep
slopes here is consistent with steep slope removal as a consequence of the Orientale-forming impact. There are, however, some
steep slopes in the inner ring. These steep slopes are actually associated with tectonic graben cutting the ring and floor and therefore
may postdate the Orientale impact. This interpretation is strengthened by the close association of floor-fracturing with the era of
mare volcanism (Jozwiak et al. 2012, 2015), and the location of
volcanic vents in association with these graben (Kramer et al.,
2013). Furthermore, the abundance of boulders on the floor of the
Schrödinger basin and their association with the floor graben and
scarps (Kumar et al., 2016) supports a post-Orientale basin origin
for these steep slopes.
A part of the Imbrium basin outer ring, Montes Apenninus, has
some moderately steep (32°–40°) slopes (Fig. 2c and d), while all
other exposed parts of this basin ring do not have steep slopes.
It is not clear, however, what is special about Montes Apenninus
that produces this anomaly. It is possible that some tectonic or
other processes (for example the formation of the adjacent 83 km
diameter Archimedes crater) triggered several large landslides in
this particular region after the Orientale impact and before emplacement of the latest Mare Imbrium volcanic material embaying
Montes Apenninus.
All Nectarian-age and older basins (those predating the Imbrium impact) (Fassett et al., 2012) do not appear as circles with
abundant steep slopes in the map of slopes steeper than 35º
(Fig. 2c). Only one basin, Crisium, becomes visible in the map of
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Fig. 3. Global distributions smoothed down to spherical harmonics of degree ≤ 2 (a, e, g) and ≤ 3 (b, f, h). Simple cylindrical projection for the whole Moon centered at the
center of the nearside. Brighter shades denote higher values. (a,b) proportion of slopes steeper than 35°; (c,d) predicted impact rate according to Le Feuvre and Wieczorek
(2003); (e,f) decameter-scale topographic roughness according to Kreslavsky et al (2013); (g,h) topography with respect to a reference sphere.
slopes above 32º (Fig. 2d) due to a slightly higher density of scattered steep slopes within its proximal ejecta. It is presently not
clear what particular factors are responsible for this increase in the
scattered steep slope abundance.
4.3. Other features
Walls of several sinuous rilles (see Hurwitz et al. 2013, for
the global distribution, characteristics and modes of origin of sinuous rilles) often have slopes steeper than 35º due to the exposure of bedrock outcrop layers in the upper parts of the rille
walls. The most prominent examples are seen in Vallis Schröteri
in the Aristarchus Plateau. Some linear rilles of tectonic origin
(graben) also have consistently steep walls, for example, the graben
in Schrödinger previously mentioned, a system of graben on the
Humboldt crater floor, Rima Hyginus and Rima Ariadaeus. These
features postdate the Orientale basin impact; thus, the preservation of their steep slopes is consistent with global seismic shaking as a small-scale resurfacing mechanism for earlier steep slopes,
while steep slopes that formed post-large impact basin by tectonic
(linear rille) and thermal erosion (sinuous rilles) processes remain
well preserved.
Scattered steep slopes in Fig. 2b–d are often related to small
young craters. Some scattered slopes in the 32°–35° range are at
the walls of old craters (Early Imbrian or possibly older); however,
they are not grouped in clusters or chains. Fig. 2d gives the impression that scattered steep slopes are everywhere, but this is
a visual effect caused by size of the dot used to make the figure: as seen from Fig. 1; slopes steeper than 32º are still rare:
they comprise only ∼0.15% of all measured slopes and due to lost
data points (see Section 2), only about ∼0.3% of the total surface
area.
5. Global distribution
As discussed in Section 2, although the absolute proportion of
steep slopes is strongly biased, the relative spatial variations are
reliable. The apparent paucity of steepest slopes in the polar area
seen in Fig. 2 is related to distortion due to the projection. To study
the global variations of steep slope abundances we need to mitigate the inhomogeneity of surface sampling with LRO orbit tracks.
To do this we first mapped the proportion of the steepest slopes in
the simple cylindrical projection at 8 pixels per degree sampling.
For each pixel of the map we counted the number of good slope
measurements, the number of those steeper than 35°, and then divided the latter by the former. In this way we excluded any spatial
inhomogeneity in the measurements. Since the absolute value of
the proportion of steep slopes is biased, we normalized the map
by the mean percentage of the steep slopes. The majority of pixels
have zero values. To study the global variations and remove the effects of the huge concentration of steep slopes in specific individual large young craters, we smoothed the percentage map down
to the lowest spherical harmonics. Fig. 3a and b shows the resulting global distribution smoothed down to harmonics of degree two
and three, respectively. It is seen that there is some concentration
of the steepest slopes toward the equator. There is also a relative
paucity of steepest slopes in the southern farside (approximately
consistent with the South Pole–Aitken basin) and a maximum in
the central–eastern farside. Fig. 4 shows separately the latitudinal
dependence of the percentage of the steepest slopes averaged over
all longitudes.
From the analysis in Section 4.1 we know that the steepest
slopes are predominantly produced by impacts and that these
degrade with geological time. Therefore it is reasonable to expect
that the number of steepest slopes is proportional to cratering rate,
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335
Correlation with topography is even more puzzling. Kreslavsky
et al. (2013) noted and discussed the correlation between roughness and topography. They suggested the following explanation:
On the basis of geodynamical reasoning, the equator is close to topographic highs, and relative roughening of the equatorial region
might be caused by more frequent impacts. This explains the generally higher roughness at low latitudes and therefore some part
of the correlation with topography. After that, the similarity between other global-scale details of the distributions might be coincidental. The same logic can be applied one-to-one to the steepest
slopes.
6. Conclusions
Fig. 4. Latitudinal dependence of the proportion of slopes steeper than 35° (bold
curve) smoothed down to zonal harmonic of degree 5, and impact rate according
to Le Feuvre and Wieczorek (2003) (thin curve).
although some deflection from this proportionality might be
caused by variations in mechanical properties of surface materials,
etc. Dynamic considerations and modeling (Morota et al., 2008,
Gallant et al., 2009, Ito and Malhorta, 2010; Le Feuvre and Wieczorek, 2011) predict somewhat higher (tens of percent) effective
impact crater formation rates in the equator than at the poles,
and somewhat higher rates at the apex (the center of western
hemisphere) than at antapex. The quantitative prediction of impact
rate by Le Feuvre and Wieczorek (2011) is shown in Fig. 3c and
duplicated in Fig. 3d. Fig. 4 shows the latitudinal part of this
dependence.
Comparison of the curves in Fig. 4 shows that the concentration of steep slopes toward the equator has the same order of
magnitude as the concentration of impacts, but a slightly different
shape. Comparison of the spatial distributions of the steep slope
proportion and impact rate (Fig. 3a and b vs. Fig.3c and d) shows,
however, significant discrepancies: the steep slope proportion minimum in the southern farside is absent in the impact rate distribution, and the observed maximum in steep slopes is shifted
westward from the apex. The smoothed distribution of the steepest slope proportion (Fig. 3a and b) correlates somewhat better
with the similarly smoothed distribution of decameter-scale topographic roughness (Fig. 3e and f) (Kreslavsky et al., 2013) and topography (Fig. 3g and h). The reason for this correlation is poorly
understood. Decameter-scale topographic roughness as it is defined
by Kreslavsky et al. (2013) describes typical terrains and is not affected by the fresh craters; this scale of roughness is related to vertical scales of meters and typical slopes of a few degrees. The tiny
area of steep slopes had no effect on this measure of roughness.
The decameter-scale roughness is controlled by regolith gardening
(Kreslavsky and Head, 2013). On the other hand, the rare steepest
slopes are related to specific young craters; they are related to vertical scales of tens of meters, and bedrock/megaregolith rather than
the thin surficial layer of regolith. Thus, roughness and the steepest
slopes are physically different things. It is possible that the common control by impact rate causes some similarity between roughness and steepest slopes, and after that their better correlation to
each other is coincidental.
Our analysis of the steepest slopes on the Moon shows that
their occurrence is controlled by a limited set of processes. The
steepest slopes are formed by impacts, and in the geological past
by several tectonic and volcanic processes; these slopes slowly degrade through geological time and can be instantly removed as a
consequence of large scale basin-forming impacts. The latter effect
probably occurs due to intensive global seismic shaking (Schultz
and Gault, 1975; Kreslavsky and Head, 2012). Due to the simplicity of this set of processes, steep slopes can be effectively used
in geologic studies to assess ages. Global removal of the steepest
slopes by the Orientale impact, the latest basin-forming impact, is
a good global stratigraphic marker. For craters postdating the Orientale impact, slopes correlate with crater age. Although it does
not seem possible to accurately calibrate slopes of the crater walls
in terms of absolute ages, slopes do give an independent constraint
on the stratigraphic age of craters.
Our observations revealed several puzzling facts related to the
steepest slopes. The most prominent of these outstanding questions is the significant relative paucity of steepest slopes in the
southern farside, in the South Pole–Aitken basin region.
Acknowledgments
We acknowledge financial support from the NASA Lunar Reconnaissance Orbiter Lunar Orbiter Laser Altimeter (LOLA) experiment
(NNX09AM54G and NNX11AK29G to JWH) and the NASA Solar System Exploration Research Virtual Institute (SSERVI) grant for Evolution and Environment of Exploration Destinations under cooperative Agreement No. NNA14AB01A at Brown University. The technical work on data analysis was carried out by MAK at MIIGAiK
under Russian Science Foundation support, project 14-22-00197.
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