1. The Relationship Between Topographic Roughness and Crater-Size-
Frequency-Derived Age on the Surface of Mars
Registration Number: 13066281
Degree Programme: Geography (BSc)
Module Code: GEO356
Submission Year: 2016
Supervisor: Dr. Felix Ng
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Abstract
Here a new method for rapidly assessing the age of surface geological units on Mars is
proposed. In the absence of in-situ radiometric dating, age determination using impact
craters is the primary method used to date the surface of other planets within our Solar
System. Calibrated using samples of Lunar rock returned by the Apollo and Luna
missions, measurements of crater size frequency allow for accurate determination of
surface age based on the changing rate of impactors over time. With adjustments to take
into account the different cratering rates relative to the Moon, crater dating has been
expanded to all rocky surfaces in the Solar System. However, in the absence of suitable
automatic detection techniques, counting craters remains a manual and time-consuming
process. In light of this, a novel method to rapidly date geological units is suggested.
Roughness is a property all natural surfaces share and has previously been linked with
age derived from crater density on the Moon. On Mars the commonly held assumption is
that the highest values of roughness correspond spatially to areas of highest crater density
and therefore age. To test this assumption, and the degree to which roughness can be used
to predict surface age, 125 HRSC images were analysed from 10 regions across Mars.
Age was calculated for each image using measurements of crater size frequency. Age was
then compared to three measures of roughness calculated using topography profiles
derived from the Mars Orbiter Laser Altimeter. Two measures of roughness were
significantly related to age and the relationships bear the signature of a changing cratering
rate over time, a maximum limit on roughness that changes over time was discovered.
However, there was considerable variation within the datasets, inferred to be due to the
effects of partial crater degradation, acting to smooth the topographic expression of
craters without affecting size frequency. Here degradation was qualitatively associated
with relatively smooth surfaces older than 3.2 Ga, and partial degradation events were
dated to known periods of increased degradation rates early in Mars’s history.
Word count: 10,000 words
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Contents
1. Introduction ...................................................................................................................1
2. Background and Rationale ............................................................................................2
2.1. Dating Using the Crater Population .......................................................................2
2.2. Topographic Roughness.........................................................................................5
3. Aims and Hypothesized Outcomes ...............................................................................7
4. Methodology .................................................................................................................9
4.1. Study Site Selection ...............................................................................................9
4.2.1. Deriving Age from Crater Counts.......................................................................9
4.2.2. Uncertainty in Age Determination Using Crater Counts ..................................15
4.3. Calculating Roughness.........................................................................................17
5. Results.........................................................................................................................20
6. Analysis and Discussion..............................................................................................26
7. Conclusion...................................................................................................................36
Acknowledgments...........................................................................................................37
References .......................................................................................................................38
Appendix .........................................................................................................................47
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1. Introduction
Running from 1961-72, NASA’s Apollo missions to the Moon together with the Soviet
Luna programme of unmanned probes represent the only times materials (~382 kg) from
another planetary surface have been successfully collected in-situ, then returned for
radiometric dating (Neukum 1983; Marchi et al. 2009). With NASA unlikely to land
humans on Mars before 2035 at least, there remains, and will remain for some time no
way to radiometrically date any surfaces beyond Earth’s. This represents a problem when
seeking to understand the geomorphological history of other planets.
However, dating planets is possible; and requires the utilisation of craters, the most
ubiquitous landforms within the Solar System. Craters are left behind when a impactor
strikes the surface of a planet (Neukum 1983). On Earth, tectonic and erosional processes
act to erase evidence of cratering and so few examples are preserved (Milton &
Macdonald 2005). However, on other terrestrial planets, much less geologically active
than Earth, craters are more likely to be preserved. Long used as a relative dating
technique, measurements of crater-size frequency (number of craters of size x per unit
area) allow establishment of relative stratigraphic relationships among different areas on
a planet (Hartmann 1977). The principle is that areas of higher crater frequency have been
exposed to cratering for longer (Hartmann 1977). Absolute dating became possible with
the retrieval of lunar material. Corresponding to an area of known crater-size frequency
distribution (CSFD), radiometric samples permitted calibration of crater frequency with
age (Arvidson et al. 1979; Neukum 1983). Measurements of age using CSFD have since
been used in a wide array of settings and to date a range of surface processes (e.g Arvidson
et al. 1979; Neukum 1983; Grant & Schultz 1993; Mustard et al. 2001; Hartmann &
Neukum 2001; Neukum et al. 2001; Kadish & Head 2014; Kadish et al. 2014). However,
dating using the crater population relies on collecting a large enough sample to be
statistically significant (Neukum 1983), and so is a time-expensive endeavour. Automatic
machine learning algorithms have sought to address this issue (e.g Bue & Stepinski 2007;
Ding et al. 2013) but manual methods remain the best way of calculating CSFD for a
given area. The purpose of this study is to test the relationship on Mars between CSFD-
derived age and an inherent, and comparatively much easier to calculate property of
natural surfaces: roughness.
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Roughness is a property all natural surfaces possess, and is simply a measure of how
much any given surfaces deviates about a plane (Shepard et al. 2001). Given its
fundamental link to surface processes, which act to either increase or decrease the
roughness of a surface (Shepard et al. 2001), roughness is a quantity that has been well-
studied in the solar system (e.g., Orosei et al. 2003; Torrence et al. 2010; Rosenburg et
al. 2011; Pommerol et al. 2012; Kreslavsky et al. 2013). Cratering is one such process
that alters the roughness properties of surfaces, and more craters would result in a rougher
surface in absence of any other factors that may modify crater morphology (Appendix 1
& 2). For the Moon this relationship has been explored and roughness has been compared
with the spatial density of large craters, showing a good correlation (Yokota et al. 2014).
However, an investigation of how CSFD and roughness may relate on Mars, which
comparatively has been much more geologically active over time, has never been
undertaken.
2. Background and Rationale
2.1. Dating Using the Crater Population
With the exception of Earth and Venus, which are both the subject of more active
endogenous and exogenous resurfacing (Neukum 1983; Kreslavsky et al. 2015),
terrestrial planets all possess surfaces heavily pitted by craters (Pommerol et al. 2012).
Crater size ranges from a few metres across (Marchi et al. 2013), to many thousands of
kilometres across (Marinova et al. 2008). Large craters less common than small craters.
Formed when an impactor strikes the surface, the crater is the resultant cavity (Guzman
et al. 2015). Craters share broadly the same morphology, impactors leave behind circular
depressions where the target surface has been both compressed and excavated (Marchi et
al. 2011). Differences in crater morphology do arise, and morphologies are usually
categorised into two classes: simple and complex (Milton & Macdonald 2005). The
difference is primarily one of size, but also of shape, simple craters are much smaller and
more bowl-shaped, while complex craters are larger and often exhibit central uplifts due
to underlying crustal rebound and extensive rim slumping (Milton & Macdonald 2005).
An example of a complex crater can be seen in Appendix 3.
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The number of craters on a surface is ultimately determined by the surface age, and
average crater production rate over time. Knowing one allows the other to be constrained
(Neukum 1983). Prior to the Apollo missions, attempts at absolute dating had already
been made. The most accurate attempt came in 1965 when Hartmann constructed a
Terrestrial/Lunar crater production rate. Using dated areas of the Canadian shield, and
then comparing this to the newly discovered CSFD of each area, Hartmann (1965)
calculated the cratering rate of Earth. This was then used to estimate how frequently
craters of a reference diameter (1 km) form over time. On the Moon a crater of 1 km
forms more easily than on Earth due to the Moon’s weaker gravity, meaning they will
occur on average, more frequently over time. The cratering rate can thus be converted
into a mass flux. Hartmann (1965) estimated a Lunar Mare formation age of 3.6 Ga, later
proven to be almost exactly correct (Neukum et al. 2001).
The retrieval of lunar rock from craters marked the first time impact craters on other
planetary bodies had been dated directly (Marchi et al. 2009). However, in order to create
a lunar-wide dating model, both a production and chronology function are needed.
Production functions describe the CSFD observed at the surface, i.e. how many craters of
diameter D1, can be expected compared to diameter D2, and are directly linked to the size
frequency of impacting objects. The calibrated age samples correspond to source units of
known crater densities could be used to construct a calibrated chronology function,
Figure 1- a) chronology function describing the formation rate of 1 km craters. Rate constant until 3 Ga after which it
exponentially increases (Neukum et al. 2001). b) isochron diagram showing the expected cumulative CSFD of surfaces
from 1 Ma to 4 Ga (Neukum et al. 2001)
ba
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describing the changing crater formation rate of a reference diameter (1 km) over time.
In its cumulative form (Figure 1a), this function describes a constant cratering rate from
present until 3.5 Ga after which it exponentially increases (Michael & Neukum 2010).
The exponential increase is caused by the Late Heavy Bombardment; a period of intense
cratering after the initial formation of the Solar System. To find how rate of formation of
other crater diameters has changed over time one multiplies the 1 km by an appropriate
constant derived from the production function (Neukum et al. 2001). Together
chronology and production functions allow the construction of isochron diagrams (Figure
1b), showing the expected cumulative crater densities for surfaces of different ages. To
measure age on the Moon, one plots its cumulative CSFD normalised to a unit area on an
isochron diagram, the closest matching isochron line gives the surface age (Michael &
Neukum 2010).
Subsequent missions to Mercury, Mars and the satellites of Jupiter and Saturn, revealed
other bodies experienced cratering similar to that observed on the Moon (Neukum 1983).
(Hartmann & Neukum 2001). However, dating other planets using the lunar chronology
function requires consideration of cratering rate relative to that on the Moon (Neukum &
Wise 1976). The most important elements that vary between planets are: the relative
impactor flux and its velocity distribution; the crater size as a function of impactor size,
which varies according to velocity distribution, gravity and target properties; and a
normalisation with respect to the planet’s surface area and collisional cross-section
(Michael & Neukum 2010). Crater dating has proved powerful in many fields of planetary
geoscience, permitting the absolute dating of observed landforms ex-situ (e.g Roberts &
Zhong 2007; Berman et al. 2011; Berman et al. 2015), which may then be used to
reconstruct the conditions of landform emplacement and dynamics (e.g Banks et al. 2010;
Berman et al. 2011; Goudge et al. 2012; Hauber et al. 2013; Kadish & Head 2014; Kadish
et al. 2014).
As the only way of dating the surface of other planets, dating using the crater population
is a well-studied field with constant refinements to the production and chronology
functions (Michael & Neukum 2010; Xiao & Strom 2012; Daubar et al. 2013). Improved
remote sensing techniques and data-availability has increased the method’s reliability and
accuracy (Kneissl et al. 2011). However, the process of crater counting remains a time-
consuming and manual process. Even the most comprehensive databases, compiled over
many years of manual crater identification do not include all craters. The database created
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by Robbins and Hynek (2012) comprises of 384,343 craters on Mars, and is statistically
complete, but only includes craters ≥1 km. Given the nature of the production function,
ignoring craters below 1 km means ignoring a large part of the total population, a
population that is especially important when dating young surfaces (Hartmann 2007).
Some have sought to automate the process of crater counting (Salamunićcar et al. 2011).
Most automatic crater detection techniques are based on imagery data, and are therefore
the subject of inherent limitations to imagery data (Bue & Stepinski 2007). Crater
‘visibility’ depends on illumination, surface properties and the atmospheric state at the
time the image was captured. Variation in crater morphology also complicates the process
(Bue & Stepinski 2007). Despite their common circular appearance, craters are often
complex landforms, with some degraded by erosion and barely distinguishable (Grant &
Schultz 1993), others overlap with pre-existing craters (Hartmann 1971). To overcome
the limitations of imaging data and the relative complexity of craters, elaborate multistep
algorithms have been developed, often utilising a combination of supervised and
unsupervised techniques (Bue & Stepinski 2007). Automatic techniques have had limited
success, and their efficiency decreases in proportion to terrain complexity (Bue &
Stepinski 2007; Salamunićcar et al. 2011). Manual techniques remain the most effective
method for documenting craters, especially in older and more complex regions of Mars.
2.2. Topographic Roughness
All natural surfaces exhibit roughness, defined as the vertical expression of topography
at horizontal scales of millimetres to hundreds of meters (Shepard et al. 2001). These are
the scales most familiar to geologists, and as such roughness is a useful parameter in
differentiating between surfaces, particularly on other planets, where in-situ study is
impossible (Shepard et al. 2001). Roughness is particularly useful in the context of radar
remote-sensing, as the scales on which roughness is measured are the same as those which
have the greatest effects on the scattering on microwaves (Orosei et al. 2003; Sultan-
Salem & Tyler 2006; Putzig et al. 2014). However, it is also useful in differentiating
between surfaces (Torrence et al. 2011), the comparison between surfaces based on their
roughness and scale dependence is a powerful tool for interpreting the relationship
between topographic units and how they have changed over time (Torrence et al. 2011).
Studies of Lunar roughness began soon after the commencement of the Apollo
programme (Daniels 1963). However, it was not until the launch of the Mars Global
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Surveyor (MGS) satellite in 1996 that high-resolution studies of the Martian surface
became possible (Smith et al. 2001). MGS carried a laser altimetry tool designated the
Mars Orbiter Laser Altimeter (MOLA) (Zuber et al. 1992). MOLA’s primary function
was to map the global topography of Mars to a level suitable for geophysical, geological,
and atmospheric circulation studies (Smith et al. 2001). To this end, MOLA consisted of
a diode-pumped, neodymium-doped yttrium aluminium garnet laser, a silicon avalanche
photodiode detector, and a time interval unit with a resolution of 10-ns (Zuber et al. 1992).
MOLA operated with a vertical precision of ~1.5m, a spatial accuracy of ~100m, recorded
surface height along tracks at intervals spaced ~300m apart (Appendix 2) (Rosenburg et
al. 2011). MGS circled Mars every two hours, 378 km above the surface in an areocentric
manner (Smith et al. 2001), before failing in 2007. The Lunar Orbiter Laser Altimeter
(LOLA) is a similar instrument currently orbiting the Moon aboard the Lunar
Reconnaissance Orbiter (LRO) (Torrence et al. 2011).
Numerous studies have utilised the high-resolution topographic data returned by the
MOLA instrument particularly studies focussing on constraining the dimensions of
landforms. MOLA data has been especially useful in the study of the Martian polar
deposits. Measurements accurate to within 1m vertically have been vital in constraining
the volume of ice stored at the poles (Zuber et al. 1998; Limaye et al. 2012), and
understanding the system dynamics (Tanaka et al. 2008). MOLA data has been useful in
understanding the potential dynamics of water flow on Mars (Irwin et al. 2005; Penido et
al. 2013; Hobley et al. 2014). Understanding the shape of potential basins such as the
Hellas Impact basin (Diot et al. 2014) and the lower topography North of the North-South
dichotomy (Nikora & Goring 2004; Roberts & Zhong 2007b; Marinova et al. 2008)
permits more accurate estimations of the Martian water budget. Lunar research has
included site-selection for future missions (Smith et al. 2010) and the potential presence
of surface ice sheltered within craters (Eke et al. 2013).
Studies calculating the roughness characteristics of the entire surface of the Moon
(Rosenburg et al. 2011; Kreslavsky et al. 2013; Cao et al. 2015) and Mars (Orosei et al.
2003; Nikora & Goring 2004) have also been carried out. On the Moon LOLA data has
been used effectively to quantify the difference in roughness between the Lunar
Highlands and Mare Plains, the former appearing vastly rougher (Rosenburg et al. 2011).
On Mars, the most obvious global roughness characteristic is the hemispheric difference
between the Northern lowlands and Southern Highlands (Appendix 6), the latter
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appearing rougher (Orosei et al. 2003). Lunar highlands are heavily perturbed by
meteoroid impacts compared to the relatively crater-sparse Mare (Yokota et al. 2014) and
so roughness is spatially associated with CSFD. Previous work has been carried out, using
data from another Lunar laser altimeter, SELENE. Observed roughness from the Lunar
highlands was compared to the spatial density of large craters, they were found to be
strongly correlated (Yokota et al. 2014). No comparable study has been conducted on
Mars, where a similar spatial association can and often is made between the oldest
Highland areas and the areas of densest cratering.
3. Aims and Hypothesized Outcomes
The aim of this study, is to assess the degree to which CSFD derived age correlates with
three measures of roughness on Mars: RMS height, RMS Slope and the Hurst Exponent.
To this end, imagery of the Martian surface will be collected, CSFD established, and then
compared to roughness values corresponding to the image location. One can hypothesize
several forms this relationship may take (Figure 2). As CSFD and therefore derived age
is linked to impactor flux, any relationship between age and roughness is hypothesised to
change after ~3 Ga. This change is expected to reflect the change in impactor flux from
constant to exponentially increasing before this time. A correlation between CSFD and
roughness has already been demonstrated on the Moon (Yokota et al. 2014). However, in
comparison, Mars has been considerably more geologically active over its history, which
may affect any relationship between roughness and age. If roughness is correlated to age,
the statistical model of the relationship(s) may represent a useful tool for rapidly assessing
the age of geological units.
One of the primary processes that may alter any relationship between roughness and age
on Mars is degradation. By reducing the depth of craters, degradation could reduce the
roughness characteristics of a crater, without decreasing its rim diameter (Appendix 4).
This process of depth reduction has been observed in MOC imagery of Terra Sirenum
(Malin & Edgett 2001) (Appendix 5) and likely involved subaerial transportation of
granular material. Degradation rates are known to have been higher early in Mars’s
history (Weiss & Head 2015). Spatial and temporal variation in the crater degradation
rates could result in areas of similar CSFD with contrasting roughness quantities, which
may result in the relationship breakdown observable in (Figure 2a). By preferentially
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eroding smaller craters (Michael & Neukum 2010), degradation may be visible as a kink
in the shape of an CSFD curve (Michael & Neukum 2010).
Using roughness to constrain age is based on the assumption that cratering is the dominant
expression of surface topography on Mars, an assumption often stated in the context of
roughness variations over Mars’s surface. The degree to which this assumption is true
will affect any relationship between roughness and age. In areas where cratering is not
the dominant expression of surface topography, any relationship between age and
roughness may break down, dependent on how the formation of other surface units is
related to time. While cratering has decreased over time at a known rate (Hartmann 2005),
the formation of other surface features may not be related to time in the same manner.
The formation of glacial features for example is thought to be linked to Mars’s obliquity
(Forget et al. 2006), which is known to be highly dynamic over time (Laskar et al. 2004).
Other processes, such as those linked to volcanism are much more stochastic.
Figure 2- Four hypothetical relationships between roughness and age. a) How the relationship may look if
degradation events early in Mars’s history are important, with a breakdown in relationship as areas of the same age
have different roughness characteristics. b) Cratering is the dominant expression of roughness, and varies according
to the chronology function, with a near-constant increase before ~3 Gyr increasing exponentially thereafter. c)
roughness varies over time according to the formation of another feature, such as those linked to fluvial erosion. d)
change in term of chronology function means cratering becomes dominant only after ~3 Ga
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4. Methodology
4.1. Study Site Selection
To understand the relationship between age and roughness on Mars, 10 sample sites were
selected (Table 1 & Appendix 6). The sample site selection sought to address the major
geological features of Mars, mainly the North-South Dichotomy1
between the expansive
and sparsely cratered Northern Plains; and the heavily cratered Southern Highlands. Also
studied are the large impact basins of Argyre and Isidis Planitia. Each sample site falls
within one region of Mars, and are ~20° by 30°, with the exception of Utopia Planitia,
which was enlarged to collect enough imagery. Sample site dimensions are used to define
search criteria for the HRSC imagery used in this study (Table 1). Regions poleward of
65°N/S are not included due to the relatively limited HRSC coverage in these regions.
4.2.1. Deriving Age from Crater Counts
In this study High Resolution Stereo Colour (HRSC) imagery was used to count craters,
available from the Planetary Data System Geosciences Node Orbital Data Explorer (PDS
ODE). HRSC imagery allows craters over a large area (~3°x10°) to be precisely counted.
With a nadir resolution of up to 25m/pixel (Murray et al. 2005; Williams et al. 2008) the
resolution of HRSC is high enough to resolve smaller (<1 km) impact craters, without
needing to resort to the computationally-expensive process of loading multiple higher
resolution images for comparable areal coverage. These smaller craters are important as
crater frequency increases as size decreases (Neukum 1983). Craters <1 km represent a
large portion of the total crater population, and are especially useful in dating young
surfaces, as small craters are the only ones likely to be found (Michael & Neukum 2010).
Before any model of surface age can be applied, both the diameter of craters and am area
of geological homogeneity (i.e. an area in which the same processes formed/modified the
geological units) need to be precisely recorded (Kneissl et al. 2011). Geological
homogeneity is important to try and ensure the crater population has undergone the same
post-formation modification (Marchi et al. 2011; Michael et al. 2012).
1
The North-South Dichotomy is the largest geological division on Mars, dividing the North and Southern
Hemispheres The division is most obvious in MOLA global mosaics as the difference in elevation
between the Northern Lowlands and Southern Highlands (Appendix 6)
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Study Site Northernmost
Extent
Southernmost
Extent
Westernmost
extent
Easternmost
extent
HRSC Imagery used
Noachis
Terra
-19.3° -51.0° 349.9° 11.6° h0430_0000, h0551_0000, h1910_0000, h1932_0000, h2412_0000, h2467_0000,
h2478_0001, h4295_0000, h4328_0000, h4372_0000, h6631_0000, h8525_0000,
h8599_0000, h8708_0000, ha637_0000
Terra
Sirenum
-20.1° -58.1° 205.9° 227.9° h2527_0000, h2538_0000, h2681_0000, h2692_0000, h4073_0000, h6486_0000,
h6511_0000, h7127_0000, h8537_0000, h8734_0000, ha677_0000, ha758_0000,
hc685_0000, hc766_0000_
Tyrrhena
Terra
-0.1° -30.1 84.6° 104.1° h0561_0000, h4272_0000, h7171_0000, h7196_0000, h8401_0000, h8408_0000,
h8429_0000, ha460_0000, ha481_0000, ha541_0000, ha548_0000, ha562_0000,
hb462_0000, hb469_0000, hb483_0000, hc598_0017
Utopia
Planitia
67.5° 18.5° 103.2° 167.3° h1240_0000, h1266_0001, h1317_0000, h1354_0000, h5151_0000, h5302_0000,
h5320_0000, h5431_0000, h8253_0000, h9733_0000, ha305_0000, ha312_0000
Table 1- Dimensions and location of each of the 10 regions used in this study. Included are all of the HRSC images in which crater counts could be made. Coordinates are reported with respect
to the areocentric grid
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Homogeneity maximises the likelihood that the derived age represents the true base age
(age at which craters began forming) (Marchi et al. 2011). Within each HRSC image,
areas of homogenous geology were identified by eye (Figure 3a,b). Identifying geological
homogeneity meant mapping the outline of a region that was interpreted to have
undergone spatially uniform geological processes (similar surface features, texture etc.),
and excluding any interruptions to homogeneity (Michael & Neukum 2010). Images with
a complex mosaic of geology that resulted in small geologically homogenous areas were
rejected. This is to ensure at least 15 MOLA profiles 30 km long (~1500 individual
topographic heights) could be collected for roughness calculations. However, several
images with geological divisions that still allowed >15 30 km MOLA profiles were
partitioned into areas according to this division, and then separate age and roughness
calculations were made for each area.
Determining geological homogeneity by eye inevitably introduces an element of
subjectivity (Michael et al. 2012). In order to reduce this subjectivity, after CSFD was
recorded for an area, CSFD was tested for clustering (Figure 3c). Clustering was tested
using the mean second nearest neighbour (M2CND) procedure detailed by Michael et al
(2012). Cratering is a spatially random process (Arvidson et al. 1979), and so any
deviation from this distribution is often due to apparent clustering within areas of differing
geology (Michael et al. 2012). Randomness can also indicate that the CSFD includes
secondary craters (Michael et al. 2012), which are clustered around their parent-primary
crater. This randomness analysis was carried out to identify crater counts that either
needed to be recounted or rejected.
To obtain CSFD the procedural recommendations of the Crater Analysis Techniques
Working Group were followed (Arvidson et al. 1979). The diameter of all craters whose
centre falls within the measurement area was recorded. Care had to be taken to avoid
recording common features that appear morphologically similar to craters, like collapse
pits, sublimation pits and volcanic calderas (Michael & Neukum 2010). Due to the
resolution limit of HSRC images it is not possible to record all craters below a certain
diameter, (Figure 4a), leading to a characteristic roll-off in crater size frequency. In this
study roll-off occurred at around 600m, due in part to the resolution of the imagery, but
also for practical purposes, as measuring smaller craters increases measurement time
considerably.
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a
Figure 3- Disruption to geological homogeneity. a) part of HRSC image h2538_0000 (Terra Sirenum) showing two distinct
areas of geology, Resurfacing is evident in the North; several craters that straddle the boundary display breached or totally
obliterated rims. b) total distribution and size of craters in image h2538_0000. Red box shows the extent of a). Clear
differences are observable in distribution across the boundary shown in a) with denser cratering south of the boundary
especially larger diameter craters. c) CSFD cluster analysis of the entire image, without taking into account the geological
divide. Each triangle represents a bin, and this graph shows the degree to which crater distribution is clustered, between 1
to -1 on the y-axis classified as random. Confirming visual observation, most bins are non-random, suggesting the area
being counted is not one of geological homogeneity.
c
b
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Absolute dating using the cratering record requires the use of map-projected imagery in
order to accurately constrain areas of homogenous geology and record crater diameter
(Michael & Neukum 2010). However, this can lead to distortion, especially at higher
latitudes where craters with a circular footprint appear oblate according to the axis of
distortion (Kneissl et al. 2011). Distortion makes the accurate recording of crater diameter
difficult. To overcome this issue, an extension for ESRI’s ArcGIS called ‘cratertools’ was
utilised. Developed by Kneissl et al (2011), cratertools allows for CSFD determination
independent of map-projection distortion (See Appendix 8 for detailed discussion).
With a complete list of crater diameters, and the total area of the geologically homogenous
area in which craters were counted, age could be determined. First the crater diameters
were binned in a psuedo-log manner as reccomended by Hartmann & Neukum (2001).
Then a log Ncum vs. log D plot was constructed (Figure 4), where Ncum was the cumulative
ffrequency, and D, crater diameter. Here the isochron method of calculating age was used
(Michael & Neukum 2010), and this required that a fitting range be selected (Figure 4b).
In most cases, a range of data points in the mid-diameter bins lay across a single isochron
before falling off an isochron at smaller (Figure 4a) and larger diameters. The point at
which the data points began falling off the isochron line informed the fitting range used,
and bins outside this diamater range were removed. The isochron which the resultant
range lay on, corresponded to the surface age (Figrue 4b). Isochrons were constructed
a b
Figure 4- log Ncum vs log D plot of CSFD in HRSC image h4118_0000 (Hesperia Planum). Each point represents one
diameter bin, and the grey lines represent isochron lines. a) Prior to fitting, the cumulative distribution closely follows one
isochron before rolling off at around 700m. b) Having selected a fitting range accounting for the roll-off at 700m, the
cumulative distribution now matches the 3.35 Ga isochron.
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using the 2004 Iteration of Hartmann’s chronology and production functions (Hartmann
2005).
4.2.2. Uncertainty in Age Determination Using Crater Counts
Without radiometrically dated samples from Mars, it is necessary to extrapolate from the
calibrated Lunar chronology function (Hartmann 1977; Ivanov 2001). However, the
transfer of the Lunar chronology model to Mars may introduce a systematic error in flux
uncertainty (Neukum et al. 2010). Systematic errors affect all values in the same way and
are related to the absolute model calibration, therefore affecting how closely the age
relates to the true Solar-System time scale (Michael et al. 2012). The error for individual
age determinations is by comparison much smaller; in the region of 10-30%, and is
independent of the chronology model (Neukum et al. 2010). As a consequence of this,
the systematic uncertainty for crater models could be as large as a factor 2 for ages <3.5
Ga when impactor flux is constant, whereas it is only ±100Ma for ages >3.5 Ga (Hartmann
& Neukum 2001). Despite this, extensive testing of the technique has shown that
modelled ages using the extrapolated chronology model results in ages for volcanic
surfaces and the formation of basins which are in good agreement with crystallisation
ages of Martian Meteorites recovered on Earth (Neukum et al. 2010). Specific error bars
for the modelled age of each geologically homogenous area refer to the statistical
uncertainty of the individual measurement, and not the systematic error discussed above.
Statistical uncertainty determines how precisely two ages could be related to each other.
Key to this uncertainty is the size of the area within which craters were counted. Typically
used to date younger surfaces, smaller craters often occur with a higher areal density,
which may allow a smaller area to be counted, using higher resolution imagery. For older
surfaces, typically dated using larger craters, the opposite is true, and lower resolution
imagery may be adequate (Michael & Neukum 2010). The existence of independently
formed craters on an accumulation surface can be described by a Poisson distribution.
The probability of observing k craters is given by the equation:
𝑝 =
𝜆 𝑘
𝑒−𝜆
𝑘!
(1)
where λ is the expected number of craters during time interval t’ (measured surface age).
Observation of k is already a measurement of the most probable value of λ, but the
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equation determines the likelihood of λ taking any other value, i.e. the probability of a
surface containing more craters than expected for the true surface age, t’ (Michael &
Neukum 2010). Variations in λ increase proportionately with Ncum and by assuming all
impactors are goverened by the same λ (not strictly true as λ is a function of crater
diameter) plots of proability density vs t’ can be created. Figure 5 shows the increasingly
Gaussian distribution with increasing values of k. The progression itself increases
proportionatley with age during the constant phase of the chronology function, markedly
more craters need to be counted to achieve comaparable values of precision between
young and old surfaces (Michael & Neukum 2010). To achieve a probability density of 2
at t=0.5 Ga requires k=8 craters, while at t=2 Ga, k=89 is required. After 3 Ga the
exponential term of the chronology function becomes important, and the value of k
required to reach a comparable level of precision lowers once more (Michael & Neukum
2010).
Crater counts where k was less than that which gave a probability density of 2 for a given
t (Michael & Neukum 2010) were rejected, owing to the increased likelihood of a surface
displaying a different number of craters other than that which would be expected for its
true age, t’. The difference between production and chronology function here and those
used by Michael and Neukum (2010) was ignored, as Hartmann’s 2004 iteration is only
minimally different at extremely small crater diameters (Hartmann 2005)
Figure 5- Statistical age uncertainty
curves for ages, t=0.5,2,3,3.6 Ga,
with varying number of craters, k,
using the Hartmann and Neukum
(2001) Mars chronology function.
Curves show the probability of a
surface having any other age but the
measured value of age. The curves
show the increasing numbers of
crater required to achieve a
comparable level of precision with
age. Figure reprinted from Michael
and Neukum (2010) with
permission. Copyright 2010 Elsevier
B.V.
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Uncertainty of a more vague nature is that invoving secondary cratering (Appendix 7) .
Formed by other impacts, secondary craters contaminates the age record if counted. A
controversial issue (Hartmann 2005; Mcewen & Bierhaus 2006; Dundas & McEwen
2007; Hartmann et al. 2007; Platz et al. 2013) the error caused by secondary crater is
uncertain as they are difficult to identify by morphology alone. However steps can be,
and were taken to avoid clear chains or clusters of craters (Appendix 7) to minimise the
potenial error.
4.3. Calculating Roughness
Roughness describes how much topography deviates about a flat plane. For modelling it
was first treated as the product of a random process, and can as such be characterised
using statistical techniques (Shepard et al. 2001; Smith et al. 2001; Orosei et al. 2003;
Torrence et al. 2011; Pommerol et al. 2012; Kreslavsky et al. 2014). Two parameters
were used in this study to quantify the roughness properties of topography, the first was
the root-mean squared (RMS) height of a discrete profile. When modelling topographic
height, it can be described most simply as a Gaussian-distributed stationary variable,
making standard deviation the most important moment of distribution (Orosei et al.
2003). This can be estimated by calculating RMS height, σ, given by:
𝜎 = {
1
𝑛 − 1
∑[𝑧(𝑥𝑖) − 𝑧̅]2
𝑛
𝑖=1
}
1
2
(2)
where n is the number of sample points, z(xi) is the height of the surface at point xi , and
𝑧̅ is the mean height of the profile over all xi . However, σ does not take into account the
distance over which the measurement was taken, so surfaces of contrasting appearance
but different length can possess the same σ (Orosei et al. 2003).
Therefore, it is important to take into account the length over which the profiles occurred,
RMS slope is one such description of roughness that takes this into account, and is the
second parameter used in this study. Slope is simply the difference in height between two
points divided by their distance. Like σ it can be modelled as a stationary random variable.
Standard deviation can be estimated by RMS slope s, defined as the RMS difference in
height between two points divided by a lag (Figure 6), or step, ∆x (Orosei et al. 2003):
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𝑠(Δ𝑥) =
1
Δ𝑥
{
1
𝑛
∑[𝑧(𝑥𝑖) − 𝑧(𝑥𝑖 + Δ𝑥)]2
𝑛
𝑖=1
}
1
2
(3)
Related to this, the RMS deviation v is given by (Shepard et al. 2001):
𝑠(Δ𝑥) =
𝑣(Δ𝑥)
Δ𝑥
(4)
These variables are often calculated to statistically analyse roughness throughout the solar
system (Orosei et al. 2003; Pommerol et al. 2012; Kreslavsky et al. 2013; Kreslavsky et
al. 2014).
However, statistical descriptions of roughness as a stationary variable do not account for
scale dependence (Orosei et al. 2003), a commonly observed property of natural surfaces.
σ changes in different profiles of the same area with different lengths, and s changes
according to ∆x. This behaviour can be described by power laws (Shepard et al. 2001):
𝜎(𝐿) = 𝜎0(
𝐿
𝐿0
) 𝐻
(5)
where L is the profile length, and 𝜎0 is the RMS height of the profile computed at 𝐿0, and;
𝑠(Δ𝑥) = 𝑠0 (
Δ𝑥
Δ𝑥0
)
𝐻−1
(6)
where 𝑠0 is the RMS slope at a distance Δ𝑥0 between two points. H is the Hurst Exponent
(0 < H < 1) and is approximately the same in both (5) and (6) for real surfaces (Orosei et
al. 2003). In response to this observation, roughness has been modelled as a nonstationary
random variable according to self-affine fractals (Shepard et al. 2001). Self-affinity
describes the behaviour that governs the scaling of topographic surfaces. Increases in the
scale of x and y axes by a factor r must also change in the z direction by a factor governed
by the Hurst Exponent rH
for the surface to remain identical in the statistical sense (Orosei
et al. 2003).
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To calculate roughness, MOLA data from the PDS ODE was used. MOLA tracks
consisting of a series of distances between the Martian surface and the MGS spacecraft
were used to calculate roughness. Tracks were converted to topographic heights using a
model of the areoid, producing profiles of topographic heights, z (vertical accuracy to
within 1m), with each point separated horizontally by ~300m (due to the time interval
between subsequent laser pulses). Tracks were converted to profiles according to the orbit
number (Frey et al. 1998). Three MOLA profiles for each HRSC image’s spatial footprint
were extracted, and then in accordance with the methodology of Orosei et al (2003) were
divided into continuous non-overlapping 101 point profiles of ~30 km length, yielding
~45 profiles for each image. Each profile was detrended to remove any background trend
and to highlight the variation in small-scale features (Shepard et al. 2001).
For each 30 km profile σ, s & H were calculated. σ and s were calculated using equations
(2) and (3) respectively. For s, ∆x was limited to 10% of the profile length (300 m ≤ ∆x ≥
3000 m) in order to ensure a large enough sample size (Shepard et al. 2001). Deriving H
was more complex and required v to be computed for different values of ∆x. For surfaces
that are self-affine fractals log v(∆x) should be proportional to log ∆x with H being the
proportionality constant. Thus H could be derived from the slope of the line of best fit
(Shepard et al. 2001; Orosei et al. 2003). Overall uncertainty in roughness is difficult to
quantify, especially with regards to the Hurst Exponent, but with a vertical accuracy of
1m, uncertainty due to the MOLA data source is likely to be small.
Figure 6- one MOLA profile, modelled at different intervals of x. Each point corresponds to the elevation recorded by the
MOLA instrument at that location, and the lines between points represent slope. Changing ∆x alters the interval between
horizontal points, which in turn alters the horizontal distance over which slope is recorded.
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5. Results
125 HRSC images were studied, from 10 regions of Mars, in total >60,000 craters were
measured. Several images were split according to geology, resulting in a total of 132 data
points. Figure 7 shows age against H, σ, and s (∆x=300 m & 3000 m), all averaged per
region (Appendix 9). Figure 8 and Appendix 10 show the datasets without regional
averaging, each point corresponds to one crater count. Histograms showing the frequency
of σ, s (∆x=300 m) and H are shown in Figure 9. All results are displayed with the measure
of roughness treated as the independent variable and age as the dependent.
a b
c d
Figure 7- regionally averaged age and a) Hurst Exponent, b) RMS Height, c) RMS Slope (∆x=300m), and d) RMS
Slope (∆x=3000m). On all graphs a linear model fit is applied, and R2
values are also reported. Symbolism is
according to region.
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a
d
b
c
Figure 8- all 132 data points plotted individually with symbolism according to regional sites. Linear model fit applied to all datasets and R2
reported. Age and a) Hurst Exponent, b) RMS
Height, c) RMS Slope (∆x=300m), and d) RMS Slope (∆x=3000m).
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σ and s show strong agreement with each other, both when averaged regionally (Figure
7b, 7c, & 7d) and plotted individually (Figure 8b, 8c &8d), maintaining similar
distribution patterns and containing similar clusters of data. When averaged for each of
the ten regions, H (Figure 7a) shows similarities in distribution with σ and s, although not
as clearly. However, when each of the 132 points are treated independently, there is no
clear similarity between H and σ & s. There are no clear trends in H, with data appearing
to be randomly scattered with any clustering predominantly due to the higher frequency
of older surfaces in this study. When averaged regionally (Figure 7), the relationships
between roughness and age are statistically insignificant (ρ>0.05) and show little
correlation, characterised by very low values of R2
. A maximum of 18.9% of the variation
observed in surface age is linked to s, 8.9% in the case of σ and just 7.6% for H. Appendix
9 shows the considerable variation within regions.
ba
c
Figure 9- frequency of roughness
characteristics. a) Hurst Exponent, binned in 50
equally spaced bins. b) and c) lognormal RMS
Height and Slope (∆x=300m) respectively.
Lognormal and binned in 100 equally spaced
bins.
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When each of the 131 images are treated individually (Figure 8), the trend is markedly
different. Linear regression analysis was carried out; both σ and s showed a higher and
statistically significant R2
value than the data that had been regionally averaged. There
appears to be no relationship between a surface’s H and its age, with a low and statistically
insignificant linear R2
of 0.043. Data transformations according to a Log10 function were
applied, which doubled the R2
of age and both σ and s (∆x=300 m and ∆x=3000 m), but
only modestly increased the still non-significant correlation between H and age. This
prompted the use of non-linear curve estimation to test if the data could be better
represented.
Best approximated by an s-curve equation2
, Figure 11a shows a modelled steep increase
in age with a small increase in σ. After ̴ 3.2 Ga, gradient rapidly falls before plateauing.
Figures 11b and 11c differ and there was considerably higher variation in s with age when
∆x =300m, and log transformation remains the best way of modelling this data. However,
2
Model whose equation is: 𝑌 = 𝑒(𝑏0+𝑏1×𝑡))
𝑜𝑟 𝐼𝑛(𝑌) = 𝑏0 + (𝑏1 × 𝑡) where the independent variable (in
this case roughness) = t, b0 and b1 are model parameter estimates and constants.
a b
c d
Figure 10- log 10 transformation of all 132 data points. Linear model fit applied to all datasets and R2
reported. Age and a)
Hurst Exponent, b) RMS Height, c) RMS Slope (∆x=300m), and d) RMS Slope (∆x=3000m).
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when ∆x is higher (3000m), s does appear to match σ more closely and s-curve fitting
yields a higher R2
than log transformation. The modelled plateauing occurs at almost the
same age for both σ and s. One of the key contrasts between the datasets is their behaviour
for ages <3 Ga, s, appears much more horizontally distributed, especially when
considering each value of s as a percentage of the maximum s value. This is most
prominent when ∆x=300 m, however, for ∆x=3000 m the data distribution appears to
mirror σ more closely, although a similar R2
to s when (∆x=300 m) does not reflect this
(Figure 11d). For both σ and s there is still large deviations from the non-linear fit both
prior to, and after 3 Ga despite higher R2
values suggesting higher correlation than any
linear modelling.
Argyre Planitia appears anomalously rougher than other regions both when plotted
regionally and as single crater counts. Argyre Planitia’s sample points have notably
higher values of σ and s than other areas of comparable age. Three of the five highest σ
and s values come from this region. Average σ in Argyre Planitia is almost three times
the average σ of any other region, despite average age being just 2.5 Ga. This large σ
value means that when it is included in statistical analysis, Argyre Planitia has a large
influence on all statistical models. This influence is most evident when the data is
regionally averaged (Figure 7). The influence of the Argyre Planitia regions is lower when
modelling all of the individual data points (Figures 8 & 10) but without the Argyre region
the correlation between age and σ in particular, increases prior to 3.2 Ga.
Figures 9b and 9c can be well approximated by a lognormal distribution when divided
into 100 bins of equal size, with similar characteristics to those of Orosei et al (2003).
Mean σ is 102.117 m, Standard deviation is 153.967 m, with a range of 1.8 km. Mean s
(∆x=300 m) is 0.0029, standard deviation is 0.0054 and the range is 0.0618. There is no
clear distribution characteristic with H dataset (Figure 9a), except an overall positive
trend with more frequent higher values of H. Mean H is 0.567, standard deviation is 0.276,
and the range is 0.987.
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d Model Summary Parameter Estimates
Equation Independent
Variable
Dependent R2
Sig. b0 b1
S-Curve RMS Height Age 0.639 0.000 1.247 -8.691
S-curve RMS Slope
(∆x=300m)
Age 0.496 0.000 1.284 0.000
S-Curve RMS Slope
(∆x=3000m)
Age 0.494 0.000 1.180 .0000
Figure 11- non-linear modelling of age and a) RMS Height, b) RMS Slope (∆x=300m) c) RMS Slope
(∆x=3000m). d) model summary and parameter estimates.
a
c
b
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6. Analysis and Discussion
The Late Heavy Bombardment (LHB) was a period of intense planetary bombardment by
debris left over from the accretion phase of planetary formation, a period in which
impactor flux was much higher than at present (Tera et al. 1974; Gomes et al. 2005; Strom
et al. 2005) and the formation rate of craters was at its highest (Robbins et al. 2013).
Although it is thought this ended at around 3.8 Ga during the Noachian Epoch, crater
production did not suddenly drop to current levels. According to the Martian chronology
function of Hartmann (2005) (Figure 12), the rate of crater formation decreased
exponentially from the beginning of Mars’s existence until around 3.24 Ga, which marked
the point at which the chronology function transitions to one describing a constant rate of
crater formation. This transition occurs at almost the same time as the transition in
relationship that is most clearly seen between σ, and age (Figure 11a) and to a lesser extent
s (Figure 11b,c). However, rather than a transition to an exponentially increasing function
as with the chronology function, the rate change effectively plateaus, with areas of almost
the same age possessing vastly different roughness quantities.
Figure 12- Chronology function (Hartmann 2004) describing the changing rate of
crater formation (D>1 km) over time. Martian epochs according to Michael (2013)
are marked. The change from an exponentially increasing crater formation rate to a
constant one occurs marks the boundary between the Hesperian and Amazonian.
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The large variations in roughness with little difference in age, especially for surfaces older
than ~3.2 Ga, may be due to degradational processes, which act to reduce the roughness
quantities while not affecting derived age. Both RMS quantities are sensitive to large
differences in height, as would be the case in the transition from rim edge to floor and
back. Deeper craters produce greater roughness values. Unlike most geomorphological
features, craters form instantaneously (Grant & Schultz 1993), with their initial
morphology linked only to the size of impactor, angle of incidence, velocity and target
properties (Boyce et al. 2006). Crater properties such as diameter and depth are closely
related in fresh craters (Mouginis-Mark & Hayashi 1993; Weiss & Head 2015); depth
ranges from 0.05-0.2 times the crater diameter (Mouginis-Mark & Hayashi 1993).
However, erosional and depositional processes (degradation) can diminish this ratio.
Degradation rates vary spatially on Mars (Grant & Schultz 1993) with differences in
sediment budget, aeolian activity and even fluvial activity all contributing to differing
degradation rates (Malin & Edgett 2001). When compared to fresh craters, partially
degraded craters have much shallower floors, subdued rims, superimposed channels and
rim breaches (Craddock 2002; Weiss & Head 2015). Degradation can mean that
roughness characteristics of the areas in which craters are emplaced can vary considerably
with variation in degradation rates. Areas of fresher and deeper craters would possess
comparatively higher values of σ and s. However, so long as the rim remains visible and
without large breaches (partial degradation), degradation has minimal impact on crater
diameter, which is the key morphology characteristic in age determination. The
consequence of this is that regions of comparable CSFD and therefore age, could possess
vastly different roughness quantities due to differences in degradation rates.
The process of crater formation and subsequent degradation on Mars is one of extreme
complexity, and varies not only spatially but also temporally (Weiss & Head 2015; Levy
et al. 2016). Noachian-aged craters for example may appear significantly more degraded
than younger craters, with erosion rates of up to 1-100m/Myr inferred from large
Noachian aged craters (Levy et al. 2016). Erosion rates over the past 400 Myr are thought
to be just 1.2-2.3m/Myr, inferred from recent measurements of landslide-driven scarp
retreat in Valles Marineris (Grindrod & Warner 2014; Levy et al. 2016). Early
degradation is a phenomena that is thought to be attributable to either fluvial run-off in a
wet and warm early Mars (Craddock 2002), or as predicted by climate models (e.g Forget
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et al. 2013; Wordsworth et al. 2013) surface modification by snow and ice on a cold and
icy early Mars (Weiss & Head 2015).
Encompassing one area of Terra Sirenum; HRSC image h8734_0000 (Figure 13) contains
the only early Noachian aged terrain in this study (3.94 Ga). Many of the craters evident
exhibit clear signs of partial degradation as noted by previous studies, including flat and
shallow crater floors (Figure 13), superimposed channel systems (Figure 13a), lack of
clear ejecta facies (Figure 13), absence of secondary cratering and no visible central raise
as evident in many complex mid-Amazonian age craters (Appendix 3) (Weiss & Head
2015). The roughness characteristics of this area reflect this degradation with an RMS
Height of 121.58m, only 3.4% higher than the average RMS height for Terra Sirenum of
117.13m, and only 19m higher than the average σ of the entire study. This is despite the
surface being exposed to an exponentially increasing crater formation rate for 400 Myr
longer than any other surface in this study. With only one sample point of Noachian age,
degradation rates during this epoch are not responsible for the modelled transition in the
relationship between σ, s and age that occurs at around 3.2 Ga (Figure 11).
Figure 13- degraded craters, Terra Sirenum (3.94 Ga). All
have flattened topography, and display differing levels of
rim breaching. a) two craters (D=20 km). Superimposed
channel system can be seen breaching the eastern crater’s
rim in several places b) Severely breached crater, almost
obliterated. Several of the superimposed craters are
themselves degraded, suggesting multiple degradation
events since 3.94 Gyr. c) crater of similar degradation
state to b). Differing levels of degradation may represent
different degradational episodes, with the respective
craters forming at different times. All images from HRSC
h8734_0000
a b
c
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Previous work on the degradation characteristics of craters in southern Ismenius Lacus
has found that most partial crater degradation events can be linked to not only
geologically brief periods within the Noachian epoch, but also mid-Hesperian and
Amazonian Epochs. Degradation during these periods is more likely to have been due to
localised aqueous mobilisation from the cryosphere, more episodic and widely separated
in space than Noachain degradation (Hauber et al. 2013). Most degraded craters in the
region exhibit signs of fluvial degradation (Grant & Schultz 1993; Levy et al. 2016).
Further work studying crater morphology in two highlands north of Hellas Planitia, found
that craters can be classified into three categories according to the magnitude of
degradation (Mangold et al. 2012). Type 1 craters were degraded during the Noachian
period (~4.0 Ga - ~3.7 Ga) and possess limited ejecta, display signs of heavy degradation,
and show fluvial landforms. Type 2 craters are less degraded, but still possess fluvial
landforms, some preserved ejecta, and were formed between the Early Hesperian and
Early Amazonian ( ̴ 3.7- ̴ 3.3 Ga). Finally type 1 craters are fresh craters with ejecta and
no signs of fluvial degradation (Mangold et al. 2012). Many areas here were dated to the
formation period of type 2 craters (Figure 11). Detailed quantitative assessment of crater
degradation is well beyond the scope of this study. However, qualitatively there does
appear to be a link between degradation and σ in particular, with numerous signs of partial
crater degradation fitting the type 2 description (Mangold et al. 2012) visible in images
with low σ dated between 3.7-3.3 Ga (Figure 14, Appendix 11 & 12).
All forms of degradational processes are more likely to obliterate rather than partially
degrade the smaller diameter crater population (Michael & Neukum 2010; Williams et
al. 2014). Smaller diameter craters have less topographic expression, and so are less
displaced relative to an even surface, making them more susceptible to obliteration.
(Michael & Neukum 2010). This preferential obliteration of smaller craters can be seen
in cluster analysis (Michael et al. 2012), where crater distribution is compared to a
simulated random distribution (Michael et al. 2012). Figure 15 shows the comparison of
two images with similar age, but different roughness characteristics. While over the fitting
range, all bins were random, craters that fall into 710 m and 1 km diameter bins were
more clustered relative to the random simulation in the image that display signs of
degradation (Figure 15a-c). This clustering indicates that the random distribution of
craters has been disrupted by a partial resurfacing event, with craters that would otherwise
have been present obliterated. In the area that appears rougher but with a similar age
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(Figure 15d-f), craters in 710m and 1 km diameter bins appear more random, closer in
line with the simulated random distribution of craters.
c
b
a
Figure 14- several heavily degraded craters. a) HRSC image ha637_0000 (σ=84m) showing infilled craters in
Noachis Terra. Superposition of smaller fresher craters suggests degradation process has long ceased. b) HRSC
image h7222_0000 showing infilled craters in Hesperia Planum (σ=63m). c) HRSC image hc444_0000 showing a
near obliterated crater in Arabia Terra. (σ=36m)
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c
a
b
d
e
f
Figure 15- cluster analysis
outputs. Left (a,b,c) are
results from h7222_0000
(Hesperia Planum). Right
(d,e,f) are results from
h0551_0000 (Noachis
Terra). a) and d) depict
randomness of each
diameter bin, d) is more
random at lower D than a).
b) and e) show clustering
relative to a random
simulation for 710m D bin,
c) and f) show the same for
1 km D bin. Grey line and
percentage shows Standard
deviations above/below
mean of random
simulation. Both images
are random within their
respective fitting ranges
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For several areas, it was possible to date inferred degradation. Most plots of cumulative CSFD
follow one isochron before rolling off at the resolution limit. However, in some cases a kink in
the size frequency plot is visible (Figure 16, Appendix 13-17). Whatever the process of partial
resurfacing, smaller craters below a threshold diameter are the ones lost, accumulating again
after the degradation event has ceased. In cases with a kink, it was possible to obtain two ages,
one corresponding to the original surface, the other to the time after partial resurfacing event
when craters began re-accumulating (Michael & Neukum 2010). Such a kink can be observed
in the crater count shown in Figure 16a. On a cumulative plot, each bin includes the frequency
of all the bins in which D is larger. By plotting the cumulative frequency of the diameter ranges
either side of the kink independent of each other, in effect plotting two crater counts, it is
possible to shift the resultant cumulative frequencies to two separate isochrons (Figure 16b,
Appendix 13-17).
The younger of the two CSFD curves (Figure 16b and Appendix 13-17) all correspond to ages
within the period (3.3-3.7 Ga) during which degradation rates are thought to have been higher
than current (Grant & Schultz 1993; Mangold et al. 2012; Levy et al. 2016). Application of
this technique to the Noachian aged terrain of h8734_0000 yields a resurfacing age of only
3.35 Ga (Appendix 16). However, this may represent the last major degradation event to have
a b
Figure 16- cumulative CSFD (h7222_0000, Hesperia Planum). a) curve initially follows one isochron, before kinking and
then re-steepening at a lower diameter to follow a younger isochron. b) CSFD split at kink, curves match two distinct
isochrons
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resurfaced the area. The kink observable occurs above the diameter threshold of other terrain
resurfaced (Appendix 16), and may bear the signature of a larger event able to obliterate larger
diameter craters.
This association between degradation state and roughness is clearest seen in images that can
be dated to the Early Hesperian epoch through to the Early Amazonian epoch. As a
consequence of this, during this time-frame and beyond, the measures of roughness discussed
here are not on their own enough to distinguish between areas of different age, as there appears
to be no clear relationship between the variables. Rather it appears that degradation levels exert
a much clearer influence on the roughness of an area than its crater density. Using roughness
to assess age beyond 3.2 Ga may still be possible, but would require a quantitative assessment
of degradation to assess the degree to which degradation has decreased roughness. Doing so
would be a time-intensive procedure, and introduce new uncertainty into age values, which
would represent no improvement over the current method of dating terrestrial surfaces.
As previously stated, roughness is an inherent property of all natural surfaces, and so it is
unreasonable to expect the entire Mars’s surface to display a homogeneity disrupted solely by
the process of impact cratering. Instead the surface of Mars is one of great variety across all
scales, ranging from small dunes to the largest volcano in the Solar System. Each of these
features alters the roughness of an area, increasing it beyond that of a simple flat plane.
Distribution of surface features and their associated processes varies immensely on Mars as it
does here on Earth, with the associated influence on roughness also varying. It was assumed
that cratering was the dominant process on Mars and as such, roughness would be linked
primarily to age through the variable frequency of impactors over time. However, all R2
values
(ignoring H) calculated (Figure 10) suggest that while age does appear to correlate positively
with roughness, and indeed the positive trend is statistically significant in both cases, the
correlation is not strong enough to suggest that statistical models could adequately predict age
using roughness. If roughness were to be dominated by impact cratering, one would expect
roughness to correlate strongly with age, on the basis that crater production changes temporally
at a known rate (Hartmann 2005). It was hypothesized that during the period the chronology
function describes an exponentially increasing crater production rate; roughness would
increase with age at a similar rate. Instead the low R2
values imply other processes with
considerably different temporal distributions to impact cratering, such as glacial and fluvial
processes, as well as processes of degradation are acting to affect the relationship, and are
therefore also important in affecting roughness. While the R2
value for the non-linear models
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are much higher appearing to suggest that roughness is strongly correlated with age, R2
is an
inadequate measure for assessing how well a non-linear model fits data (Appendix 18) (Spiess
& Neumeyer 2010).
This distribution of surface features beyond impact cratering may explain the data points that
appear rougher than the statistical modelling of age and roughness suggest they should. It may
also explain why there is no significant correlation between age and the Hurst Exponent. It has
previously been shown that the H as a measure of roughness is of little use when trying to
distinguish between different natural terrain types (Shepard et al. 2001). However, it was
included in this study on the assumption that impact cratering is the dominant landform. It was
hypothesised that H could be useful in distinguishing between roughness characteristics of
craters of different age, the lack of correlation observed may be indicative of the importance of
other terrain types.
Three of the five highest σ and s values including the highest value, correspond to calculated
surface ages of less than 3 Ga. Given the lower flux of large impactors striking Mars’s surface
after 3.24 Ga with the transition from an exponential to constant chronology function it is
unlikely cratering alone explains the high σ and s values. The highest four values of both σ and
s all come from one sample region of Mars; Argyre Planitia. One of the largest impact features
on Mars, Argyre is located in the southern Highlands in Mars’s western hemisphere (Hiesinger
& Head 2002). The plain encompassed by this impact feature is known as Argyre Planitia.
Initially the area was mapped using Mariner 9 imagery as a smooth plain, consisting largely of
eolian deposits. Branching channels draining into Argyre from the south rim suggested a part
fluvial influence (Hodges 1980). With the deployment of increasingly high-resolution imaging
systems, Argyre Planitia has since been revealed to be a region heavily modified by a complex
array of geological processes. Chief among these are glacial and fluvial/lacustrine processes
(Hiesinger & Head 2002; Dohm et al. 2015). The basin is at one time hypothesised to have
contained a Mediterranean-sized sea in which, as it developed over time and complimented by
the interaction of primordial crustal and mantle, life on Mars may have developed (Dohm et
al. 2015). Argyre Planitia is an extremely complex environment, and all HRSC imagery
collected of the region show this (Figure 17). In each image clear signs of undulating
topography unrelated to cratering are visible, with distinct ridges, scarps and mountains (Figure
17). It is clear that the dominant influence on roughness is not impact cratering but rather the
other terrain present.
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Both RMS height and slope datasets when plotted against age show an increasing maximum
roughness over time. A previously undiscovered relationship, it suggests that age, while not
necessarily well correlated with roughness itself, does impose an upper limit on roughness. The
shape of the data implies that the highest values of σ and s can only exist in a terrain older than
the transition in chronology function. After this transition the crater formation rate was much
higher than at present. Argyre Planitia appears to not obey this relationship, with higher
roughness than would otherwise be permitted. Argyre Planitia is a region in which impact
cratering is not the dominant expression of topography, and so may represent an exception that
proves the rule. This discovery implies that roughness is indeed related to age as originally
hypothesised, but not in a manner that has practical applications for the derivation of age from
roughness in place of the established but extremely time-intensive crater counting method.
Figure 17- Mosaic of HRSC imagery from Argyre Planitia, showing the
prevalence of terrain not associated with cratering
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7. Conclusion
There is a statistically significant relationship between age and two of the three measures of
roughness reported here; RMS Height and RMS Slope. However, the relationship is not one
suitable for the rapid assessment of surface age on geological units. The relationship bears the
signature of the changing cratering rate over time; areas dated to <3.2 Ga are typically
characterised by lower σ and s, in contrast, older areas generally appear rougher with higher σ
and s. This reflects the shift in chronology function from one that describes a constant cratering
rate over time, to one that describes an exponentially increasing cratering rate with time.
However, beyond 3.2 Ga, areas of similar age exist along a continuum of roughness, and there
is no trend between roughness and age. It is likely variations in crater degradation rates are
partly responsible, and here an association between roughness and degradation was made.
Areas older than 3.2 Ga but with lower σ and s than other areas of similar age also displayed
clear and numerous signs of crater degradation with flattened floors and breached rims
prevalent. Signatures of degradation in smoother areas were also found using randomness
analysis and the isochron graphs used to date profiles themselves. With the latter it was possible
to date inferred degradation events, and all degradation events inferred correspond to a time
during the Early Hesperian-Amazonian during which degradation rates are known to have been
higher. Without quantitative assessments of degradation rates, roughness is not a useful tool
for quickly assessing the age of geological units. However, despite this, in areas where cratering
is the dominant influence on topographic roughness, as is the case in 9 of 10 regions tested
here, age imposes a previously undiscovered limit on roughness, in particular σ. Future work
may address the relative lack of samples dated at the extreme ends of the age spectrum, and
seek to understand what effect changing baseline lengths (30 km was used here) may have on
any observed relationship between roughness and age.
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Acknowledgments
The author would like to thank Dr Felix Ng for his guidance, assistance, and encouragement
throughout the dissertation process, and Harry Luckhurst for his assistance in troubleshooting
the code used to calculate roughness. The author would also like to thank NASA/Washington
University for providing access to MOLA and HRSC imagery data via the Planetary Data
System Geosciences Orbital Data Explorer Node (PDS ODE) at:
http://ode.rsl.wustl.edu/
Programmes used in this study included Craterstats 2.0; for plotting crater counts and
determining ages, as well as randomness calculations, and CraterTools for map-projection-
independent CSFD measurements. Both of which are available freely from the Freie
Universität Berlin Department of Earth Sciences/ Institute of Geological Sciences at:
http://www.geo.fu-berlin.de/en/geol/fachrichtungen/planet/software/index.html
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J., 1992. The Mars-Observer Laser Altimeter Investigation. Journal of Geophysical
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Appendix
Appendix 2 - cross-section of a crater constructed using MOLA elevation data. Each point is horizontally spaced 300m
apart. Data detrended to remove overall slope.
Appendix 1- three surface profiles with different roughness characteristics. a) roughest surface with a high crater density,
this surface has been exposed to cratering for the longest and is therefore the oldest. b) Mostly smooth plane, with two craters
the only deviations from this. Intermediate age and roughness. c) Smooth and pristine surface, with no deviation from the
plane. Youngest of the three surfaces
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b
a
Appendix 3- Unnamed complex crater in Utopia Planitia, centre 117.37°E, 43.38°N a) HRSC image h1266_0001 b) 3D
rendering of same crater, using directionally aligned MOLA profiles as ‘ribs’ of the mesh, vertical exaggeration is x8 and the
model is aligned northwards from left to right along the length axis. Shading is according to the height and depth of the
crater. Note the central uplift not clearly observable in 1a, as well as the breach in the western rim.
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Topography
Profile length
Pristine Degraded
Appendix 4- Conceptual model of how the 2-dimensional crater cross-section may be altered by degradation. The dashed
horizontal line represents the craters original form, with a high rim, and deep bowl-shaped depression. A central raise is also
present, indicating the crater is a complex one. A combination of depositional and erosional processes may act to
simultaneously erode the rim, lowering its height relative to the flat plain, and infill its depression, leaving a flattened floor
devoid of complex central raise. Vertical dashed lines represent the change in topography. The combination of lowering the
rim and raising the depression means the topographic roughness at scale of the entire crater appears smoother. However, the
rims peak has remained in almost the same place, and so D will be almost unchanged.
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Appendix 5- MOC image AB1-00406, showing eolian bedforms and impact craters on a ridged plain in Terra Sirenum
(172.8°W, 30.8°S). Some craters are deep and have no bedforms; other shallower with bedforms. A period in which windblown
sand infilled all craters present at the time is inferred. Other craters postdate the depositional event. a) non-map-projected
original MOC image. b) map-projected image with north direction and scale indicated. c) Deep crater with shadow and no
eolian bedforms, to compare to d) example of similar diameter crater that is near-filled and contains bedforms. Illumination is
from the left. Figure reprinted from Malin & Edgett 2001 with permission- Copyright 2001 by the American Geophysical
Union
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Appendix 6- Global colourised hill-shade derived from MOLA elevation measurements. Simple cylindrical projection, with areocentric grid. Clearly visible in this map of elevation is the North-
South Dichotomy separating the northern plains and southern highlands. Also visible are the Hellas impact basin centred at 60° W, 40° S, and the Tharsis bulge centred at 210°W, 0°N. Highlighted
are the location and approximate size of the 10 regions used as sample sites in this study: a) Amazonis Planitia, b) Arabia Terra, c) Argyre Planitia, d) Hesperia Planum, e) Isidis Planita, f) Lunae
Planum, g) Noachis Terra, h) Terra Sirenum, i) Tyrrhena Terra, and j) Utopia Planitia. The size of the Utopia Planitia sample site is considerably larger than the other study sites as there were
less HRSC images available per unit area compared to other regions used.
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Appendix 7- HRSC image H5151_0000 showing two large craters South of Utopia Planitia (Image centre: 104°E,
23°N). Extending radially outwards from either side of the largest crater are two areas of denser crater clustering
compared to the surrounding plain (highlighted in red). Given cratering is a random process it is unlikely these
crater clusters were formed by primary impacts, and are inferred to be a secondary crater swarm. Crater swarms
should not be counted in crater counts, and their areas excluded.
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Appendix 8- Cratertools software
The cratertools software allows for CSFD determination independent of map-projection.
Each crater in a HRSC image is assigned a digitisation file, which can then internally be
projected to a stereographic map projection. Using a sinusoidal map projection with a
centre longitude that matches the crater’s central location, the diameter can then be
measured along the true-scale meridian, without any distortion. Finally the crater
digitisation files are re-projected according to the original map projection and the output
is stored as native (.scc) vector-geometry file (Kneissl et al. 2011). This vector geometry
file recorded the crater diameters, the location of their centres, and the total area of
homogenous geology (Kneissl et al. 2011).
Region Average
age (Ga)
Average Hurst
Exponent
Average RMS
Height (m)
Average RMS
Slope (∆x=300m)
Average RMS
Slope (∆x=3000m)
Amazonis
Planitia
1.61 0.57 27.99 0.00149 0.00184
Arabia Terra 3.34 0.6 118.16 0.0039 0.00686
Argyre
Planitia
2.58 0.67 307.9 0.00584 0.01519
Hesperia
Planum
3.37 0.6 83.07 0.0027 0.00561
Isidis
Planum
2.66 0.57 38.91 0.00247 0.00246
Lunae
Planum
3.27 0.58 75.6 0.00242 0.0045
Noachis
Terra
3.47 0.59 128.1 0.00279 0.00621
Terra
Sirenum
3.29 0.63 117.13 0.00338 0.00668
Tyrrhena
Terra
3.54 0.59 133.72 0.00375 0.00795
Utopia
Planitia
1.79 0.57 27.99 0.00125 0.00172
Appendix 9- Roughness and age averaged to each region. Note the anomalously high values of RMS Height and
RMS Slope in the Argyre Planitia region
