Judging a Planet by its Cover: Insights into Lunar Crustal
Structure and Martian Climate History from Surface Features
by
MASSACHUSEr rS INTrrlJTE
OF TECHN CLOGY
Michael M. Sori
L I
C
B.S. in Mathematics, B.A. in Physics
Duke University, 2008
LIBRA RIES
Submitted to the Department of Earth, Atmospheric and Planetary Sciences
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Planetary Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2014
2014 Massachusetts Institute of Technology. All rights reserved.
Signature redacted
Signature of Author:
Department of Earth, Atmospheric and Planetary Sciences
August 1, 2014
Certified by:
Signature redacted
Maria T. Zuber
E. A. Griswold Professor of Geophysics & Vice President for Research
Signature redacted
20RE
Thesis Supervisor
Accepted by:
Robert D. van der Hilst
Schlumberger Professor of Earth Sciences
Head, Department of Earth, Atmospheric and Planetary Sciences
2
Judging a Planet by its Cover: Insights into Lunar Crustal
Structure and Martian Climate History from Surface Features
By
Michael M Sori
Submitted to the Department of Earth, Atmospheric and Planetary
Sciences on June 3, 2014, in partial fulfillment of the requirements for
the degree of
Doctor of Philosophy
Abstract
Orbital spacecraft make observations of a planet's surface in the present
day, but careful analyses of these data can yield information about
deeper planetary structure and history. In this thesis, I use data sets
from four orbital robotic spacecraft missions to answer longstanding
questions about the crustal structure of the Moon and the climatic
history of Mars. In chapter 2, I use gravity data from the Gravity
Recovery and Interior Laboratory (GRAIL) mission to constrain the
quantity and location of hidden volcanic deposits on the Moon. In
chapter 3, I combine GRAIL data with elevation measurements from the
Lunar Orbiter Laser Altimeter (LOLA) aboard the Lunar Reconnaissance
Orbiter (LRO) to investigate the nature of isostatic compensation in the
lunar highlands. In chapter 4, I present a new technique for analysis of
the Martian polar layered deposits (PLDs). In chapter 5, I apply that
technique using images of the PLDs from the MOC and HiRISE
instruments aboard the Mars Global Surveyor (MGS) and Mars
Reconnaissance Orbiter (MRO) to constrain their ages and deposition
rates.
Thesis Supervisor: Maria T. Zuber
Title: E. A. Griswold Professor of Geophysics & Vice President for
Research
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4
Acknowledgements
"If you're succeeding at everything you do, you're not thinking big
enough." I'm not sure if my advisor remembers giving me that piece of
advice years ago, but I've thought of it frequently. Of all the people I
have to thank, Maria Zuber clearly tops the list. She expects a lot from
those who work for her, but those expectations are matched by a
genuine faith in her students to accomplish big things. Maria is one of
those people whom you try and soak up as many things as you can learn
when you're the same room as her. I've heard her say before that the
best thing she does is recruit the right people, but I'll politely disagree
and say that the effect she has on her students through her
encouragement of big ideas surpasses that.
Maria is hardly the only professor that has had a large impact on me in
my time at MIT. Taylor Perron has advised my work on Martian polar
caps and has taught me countless lessons about science and not-sciencethings, and shares Maria's faith in his students to take charge of their
own work and succeed. The other members of my thesis committee
also deserve thanks beyond their agreement to read this document: Ben
Weiss has educated me in many ways, most notably by hiring me as his
teaching assistant and entrusting me with the lives of undergraduates
on our field trip to the Himalayas, and Jim Head has been a figure to look
up to for years and a source of advice and inspiration outside of MIT. I
thank Rick Binzel for chairing my generals exam committee and for his
support of my organization of our department's planetary seminar, and
Lindy Elkins-Tanton for being the most influential non-advisor
professor I've had at MIT by teaching me invaluable lessons about the
nature of scientific research and need to be open-minded in science. I
thank Walter Kiefer for inspiring my study of lunar isostasy, and the rest
of the GRAIL science team for their constant feedback and inspiration.
My fellow planetary science students in the department deserve special
thanks. Alex Evans and Peter James have been sources of advice as the
veteran students of the lab during my time here. Frank Centinello and
Yodit Tewelde are close friends who have always been voices of reason.
Anton Ermakov, Zhenliang Tien, and Matthieu Talpe are the younger
students in the group who I jokingly have referred to as my children, but
have taught me as least as much as I've taught them. Sonia Tikoo has
5
been my mentor of sorts from my first week at MIT, despite my
revelation years into school that she's younger than me, and will be a
fantastic scientific colleague for the remainder of my scientific career.
Elizabeth Bailey was my undergraduate research assistant, will always
be remembered as my first "apprentice," and undoubtedly do great
things in her graduate career at Cal Tech and beyond. Jason Soderblom,
Brandon Johnson, and Katarina Miljkovic have been the postdocs in my
group during my time here and have always provided a useful
perspective on how to graduate and take the next steps.
I thank all of my family and friends and teachers from home and college,
but a few people deserve special mention. My parents have provided
constant support and have been surprisingly understanding as my "I'll
visit Florida many times each year" slowly morphed into "I'll come
home for Christmas" over the years. Christine Ryu has almost singlehandedly kept me sane over the past five years, and for two people with
no training in planetary science, she and Robbie Hunter came up with a
pretty awesome and relevant thesis title for me in the middle of a late
night cab ride in New York City.
Additional people I need to thank include: Jon Grabenstatter and
Morgan O'Neill for being my first close friends in the department, Phil
Wolfe for being one of my best friends and roommate for more years
than I'm willing to admit, Kat Thomas for her sometimes unearned
constant faith in me, Roberta Allard and Margaret Lankow for bailing
me out of administrative problems more times than I deserved, Javier
Matamoros and Erin Koksal for their constant friendship from my first
day in Boston, Arthur "Peaches" Olive and Mike Byrne for being my
partners-in-crime in my EAPS class, Paul Richardson for being my lab
brother, Elena Steponaitis for being a sister to me despite her UNC
heritage, Cory Ip for her matzoh ball soup, Alex Toumar for her
therapeutic IHOP sessions, Ben Mandler for keeping my EAPS
orientation events alive, and many, many, many others who I would list
explicitly if I didn't think I already was breaking a record for number of
people mentioned in an acknowledgements page.
6
Table of Contents
Abstract
Acknowledgements
Table of Contents
3
5
7
Chapter 1: Introduction
9
Chapter 2: Gravitational Search for Lunar Cryptomaria
Abstract
2.1. Introduction
2.2. Gravity Maps
2.3. Modeling of Igneous Deposits
2.4. Results and Discussion
2.5. Summary and Conclusions
Acknowledgements
References
Figures and Tables
13
13
13
17
19
21
25
26
26
30
Chapter 3: The Nature of Lunar Isostasy and Implications
for Mantle Structure
Abstract
3.1. Introduction
3.2. Elevation-Density Correlations
3.3. Geoid to Topography Ratios and Spectrally
Weighted Admittances
3.4. Geoid to Topography Ratio Results
3.5. Discussion
3.6. Conclusions
Acknowledgements
References
Figures and Tables
Chapter 4: A Procedure for Testing the Significance of
Orbital Tuning of the Martian Polar Layered Deposits
Abstract
4.1. Introduction
4.2. Polar Layered Deposit Formation Models
4.2.1. Insolation Forcing
4.2.2. Ice and Dust Accumulation
36
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41
43
46
48
51
51
51
55
63
63
63
70
70
72
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4.2.3. Synthetic Stratigraphic Sequences
4.3. Statistical Analysis
4.3.1. Tuning by Dynamic Time Warping
4.3.2. Monte Carlo Procedure
4.4. Results
4.4.1. Qualitative Characteristics of
Synthetic PLD Stratigraphy
4.4.2. Detection of Orbital Signals for
Different Accumulation Models
4.5. Discussion
4.5.1. Feasibility of Identifying a
Orbital Signal through Tuning
4.5.2. Fraction of Time Preserved in the
Polar Cap Stratigraphy
4.5.3. Additional Considerations for
Modeling PLD Formation
4.5.4. Implications for Orbital Tuning
of the Observed PLD Stratigraphy
4.6. Conclusions
Acknowledgements
References
Figures
73
75
75
76
78
Chapter 5: Dynamic Time Warping of the Martian PLDs
Abstract
5.1. Introduction
5.2. Stratigraphy
5.2.1. MOC Images
5.2.2. HiRISE Images
5.3. Dynamic Time Warping
5.4. Results
5.5. Discussion
5.6. Conclusions
Acknowledgements
References
Figures and Tables
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101
102
106
106
107
108
112
113
115
116
116
119
Chapter 6: Conclusions and Future Work
6.1. Moon
6.2. Mars
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129
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87
91
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1. Introduction
Planetary science is the art of attempting to learn a lot with a little. Any one
planetary mission has, at best, a handful of ways to make observations, and missions
are often fleeting in time. But scientists have developed techniques to use what
might appear to be relatively limited data sets taken from a planet's orbit in the
present day and learn a wealth of information about that world's interior structure
and history. In this thesis, I continue that tradition using data from four orbital
spacecraft to study the Moon and Mars: the Gravity Recovery and Interior
Laboratory (GRAIL), the Lunar Reconnaissance Orbiter (LRO), the Mars Global
Surveyor (MGS), and the Mars Reconnaissance Orbiter (MRO).
The Moon preserves its history more completely than any other terrestrial
planet, save, perhaps, Mercury. It is also, of course, closer to Earth than any other
terrestrial planet, and therefore the most accessible terrestrial planetary body. We
can take advantage of this fortunate occurrence and study the Moon to learn about
the solar system and planets in general. Volcanism and isostasy are phenomena that
take place on all terrestrial planets. GRAIL and LRO make the Moon an ideal place to
study these planetary processes, as they have produced the highest-resolved,
highest accuracy gravity and topography maps of any planet. At the time of this
writing, the best GRAIL-produced lunar gravity map is accurate up to spherical
harmonic degree and order 1080, and the Lunar Orbiter Laser Altimeter (LOLA) has
collected -6.8 billion measurements of elevation with a precision of -10 cm and
accuracy of -1 m. Both will continue to improve.
9
Anyone can look up at the night sky and observe the dark surface areas that
are the Moon's volcanic deposits. Despite this visibility, lunar volcanism contains
many fundamental, unanswered questions. Why are the vast majority of the
deposits on the lunar nearside? What is the spatial extent, timing, and volume of
lunar volcanism? Chapter 2 of my thesis contributes to answering these questions
by searching for cryptomaria, which are volcanic deposits that are hidden from
direct view by an overlying bright layer of impact basin ejecta. The basalts that
compose these deposits have higher density than the more widespread anorthositic
part of the lunar crust, and thus should exhibit a positive gravity signature. I use
gravity data to search for such a signature while taking into consideration geological
evidence in the form of dark-halo craters. My results put constraints on the quantity
of not just cryptomaria, but of lunar volcanism as a whole.
When searching for the gravity signature of any particular feature, one must
subtract away the gravity signature of other known sources. While searching for
cryptomaria, I became interested in how to best correct for the gravity of the Moon's
topography, which led me to an investigation of lunar isostasy. Isostasy is a way to
support topographic loads on a planetary surface by balancing out the weight of that
load at some compensating depth. While the Moon's topography has generally
thought to be isostatically compensated at a large scale, the exact mechanism of
compensation has been harder to identify - because all types of isostasy by
definition involve balance of mass, their gravity signatures are more subtle than
simple positive or negative anomalies. In chapter 3, I use a combination of GRAIL
and LOLA observations to search for the "how" of compensation in the lunar
10
highlands. I search for a negative elevation-density correlation that is characteristic
of Pratt isostasy, and use the method of spectrally-weighted degree-dependent
admittances to analyze the Moon's geoid to topography ratios and determine the
plausibility of various Airy isostasy models. Analysis of isostatic compensation
necessarily yields insights into a planet's deep crust; in the case of the Moon, I argue
that it informs us about the mantle as well.
One of the many themes of planetary exploration has been the realization
over time that water ice, like volcanism and isostasy, is common on planets in the
solar system. Mars is a microcosm of this theme, but the subject of this work is the
Martian ice that has been known the longest: the polar caps. Like Earth, Mars has
polar caps made predominantly of water ice. These caps contain stratigraphy of ice
and dust, called the polar layered deposits (PLDs), that are hypothesized to be a
record of past Martian climate. Decoding that record is not trivial, and has been the
subject of many studies. In chapter 4, I adapt a signal processing technique called
dynamic time warping for use on the Martian PLDs, and discuss its relative
strengths compared to techniques used in past study and its capability of detecting a
climate signal in the PLDs. In chapter 5, I apply that technique to Mars, using images
from cameras aboard MGS and MRO. I test whether the patterns observed in the
layering of the PLDs can be connected to changes in the Martian orbit (precession,
obliquity variations, and eccentricity variations) over the past several millions of
years. Doing so yields constraints on the ages and deposition rates of the polar caps,
and has implications for Martian climate and its sensitivity to the Martian orbit.
11
On the Moon, I have used observations taken from orbit to look downward in
space and probe the subsurface lunar crust and upper mantle. On Mars, I have used
dynamic time warping and orbital images to look backwards in time and study ice
deposition and climate at the poles (of course, the studies have implications for
lunar history and the features directly beneath the surface of the Martian polar caps
as well). Both themes, as is usually the case in science, have inspired at least as
many questions as they have provided answers, but with the completion of this
thesis, we inch our way a little closer toward an understanding of the Moon's
structure and Mars' history.
12
Chapter 2: Gravitational Search for Lunar
Cryptomaria
This research was conducted in collaboration with Maria T. Zuber, James W. Head,
and Walter S. Kiefer.
ABSTRACT:
Lunar cryptomaria are subsurface basaltic deposits that are obscured by overlaying higher
albedo material. Knowledge of the volume and extent of cryptomaria is necessary for a
comprehensive understanding of lunar volcanic history, particularly in early (>3.8 Ga)
epochs when the presence of more abundant impact craters and basins favored
obscuration of surface volcanic deposits by lateral emplacement of ejecta. We use
Gravity Recovery and Interior Laboratory (GRAIL) gravity and Lunar Orbiter Laser
Altimeter (LOLA) topography data to construct maps of the Moon's positive Bouguer
and isostatic gravity anomalies, and explore the possibility that these features are due to
mass excesses associated with cryptomaria by cross-referencing the regions with geologic
data such as dark-halo craters. We model the potential cryptomare deposits as buried
high-density rectangular prisms at depth, and find a volume of candidate buried
cryptomaria between 0.42 x 106 km 3 and 2.45 x 106 kM3, depending on assumptions
about cryptomaria density and crustal compensation state. These candidate deposits
correspond to a surface area of between 0.50 x 106 kM 2 and 1.03 x 106 kin 2 , which would
increase the amount of the lunar surface containing volcanic deposits from 16.6% to
between 17.9% and 19.3%. The high-resolution GRAIL and LOLA observations thus
indicate that there does not exist large volumes of non-dike basaltic intrusions trapped in
the lunar crust.
1. Introduction
Volcanism is a ubiquitous phenomenon on large terrestrial planetary bodies,
having occurred on Mercury, Venus, Earth, the Moon, and Mars [Basaltic Volcanism
Study Project, 1981]. Understanding the timing, frequency, and magnitude of this
13
planetary process provides important constraints on a planet's geological and thermal
histories, but this information can be difficult to obtain, particularly early in a planet's
history. The surficial deposits associated with eruptive volcanism can be studied
geophysically or geologically to characterize and quantify past activity. The Moon is a
particularly instructive place to learn about ancient planetary volcanism, given its
accessibility and preservation of surface features due to its lack of plate tectonics [Head
and Solomon, 1981] and atmosphere.
Maria are basaltic extrusions found on the lunar surface, characterized by a low
albedo relative to the anorthositic highlands [e.g., Wilhelms, 1987; Hiesinger and Head,
2006]. The deposits represent secondary crust; i.e., crust resulting from partial melting of
the lunar mantle [Taylor, 1989], and are younger than the highlands [Wilhelms, 1987].
From studies of impact crater size-frequency distribution, morphology, and stratigraphy it
has been inferred that volcanism on the Moon was active from -4.0 Ga to -1.2 Ga, with
most of the volume being emplaced between ~3.7 Ga and ~3.3 Ga [Hiesinger et al., 2000,
2003, 2011]. Age estimates of basaltic materials from radiometric dating are consistent
with these results [Papike et al., 1998]. The maria cover approximately 17% of the lunar
surface (or 6.3 x 106 kM2 ), occurring preferentially in topographic lows on the nearside
[Head, 1975]; volumetrically, they compose -1% of the lunar crust, only rarely
accounting for 10% or more of crustal thickness in any location [Head, 1982; Head and
Wilson, 1992], for a total volume of 5 x 106 kM 3 [Horz, 1978; Budney and Lucey, 1998].
A problem with using mare deposits to infer quantitative information about the
spatial and temporal distribution of basaltic volcanism on the Moon is that some deposits
may be hidden. Cryptomaria are basaltic extrusions that have been overlain by higher
14
albedo material, thereby shielding the unit's characteristic low albedo from direct
observation [Head and Wilson, 1992]. The high-albedo material is impact-sourced
primary ejecta mixed with local material, and is responsible for the Moon's "light plains"
[Oberbeck et al., 1974; Oberbeck, 1975]. However, other pieces of geological evidence
exist to identify units of cryptomare; importantly, sufficiently energetic impacts may
excavate through the overlying high-albedo unit and into underlying low-albedo
cryptomaria. The result is a crater with a ring of low-albedo ejecta (Figure 1); such
structures have been named dark-halo craters (DHCs) and have been used to identify
deposits of cryptomaria [Schultz and Spudis, 1979, 1983; Hawke and Bell, 1981; Bell
and Hawke, 1984; Antonenko et al., 1995]. Constraints on the thickness of the basaltic
unit and high-albedo cover can be inferred from the dimensions of the crater. Variations
in the multispectral signature caused by mixing of preexisting mare material into basin
ejecta have also been used to identify cryptomare deposits [Mustard and Head, 1996].
Studies have analyzed the spectral signature of the lunar surface using near-infrared,
visible, and ultraviolet reflectance properties [Lucey et al., 1991; Hawke et al., 1993;
Blewett et al., 1995; Mustard and Head, 1996], and probed tens of meters into the lunar
subsurface using long-wavelength radar echoes [Campbell and Hawke, 2005].
Identification of cryptomaria is crucial to our understanding of lunar history.
Calculations of volcanic activity on the Moon based on surface properties will be an
underestimation if substantial buried volcanic units exist, with the degree of inaccuracy
increasing with the amount of cryptomaria. Cryptomaria may also affect our
understanding of the spatial distribution of volcanism; exposed maria have a strong
asymmetric distribution, with the vast majority of the deposits on the lunar nearside
15
[Head, 1975; Wilhelms, 1987]. Given that the farside crust is thicker than the nearside
[Zuber et al., 1994], the question had been raised whether basalt stalled below or within
the crust at a neutral buoyancy level [Head, 1982; Head and Wilson, 1992]. Additionally,
most cryptomare deposits are old by nature in a geological sense; cryptomaria are
generally believed to be older than 3.8 Ga, so they preferentially inform us about the
early volcanic and thermal history of the Moon [Hiesinger et al., 2011], including mare
basalt petrogenesis [Neal and Taylor, 1992]. The presence of basaltic clasts as old as
4.23 Ga in returned Apollo samples [Taylor et al., 1983] and lunar meteorites [Terada et
al., 2007] are also suggestive of the importance of cryptomaria in earliest lunar history.
In this study, we use gravity data derived from the Gravity Recovery and Interior
Laboratory (GRAIL) mission [Zuber et al., 2013a] in combination with topographic data
from the Lunar Orbiter Laser Altimeter (LOLA) instrument [Smith et al., 2010] aboard
the Lunar Reconnaissance Orbiter (LRO) [Chin et al., 2007] to construct global maps of
the Moon's Bouguer and isostatic gravity anomalies. Cryptomare units are of higher
density than the anorthositic crust, and thus should exhibit positive gravity anomalies
relative to their surroundings consistent with their corresponding to a near-surface or
deeper subsurface mass excess. We identify features in the gravity maps that are
attributable to impact basins [Head et al., 2010; Fassett et al., 2012; Melosh et al., 2013;
Neumann et al., 2014], surface mare deposits [Hiesinger et al., 2011], or igneous vertical
tabular intrusions [Andrews-Hanna et al., 2013], and eliminate them as cryptomaria
candidates. The remaining features exhibiting positive gravity anomalies are considered
as candidates for hidden igneous (extrusive and intrusive) deposits. For each candidate
region, we create models of gravity anomalies due to high-density subsurface material to
16
estimate the thickness and volume needed to produce the observed gravity anomalies.
Finally, we compare our candidate regions with proposed cryptomaria locations from
other (non-gravity based) studies.
We note two items of importance about our method before proceeding. First,
analysis of gravity data alone yields non-unique solutions; a positive anomaly may be due
to a relatively small mass nearby (i.e., shallowly buried in a planetary surface) or a
relatively large mass far away (deeply buried). Thus, other data and/or assumptions must
be integrated into our analysis to produce meaningful results, as will be described below.
Second, our method does not distinguish between basaltic extrusions that have been
subsequently buried by impact ejecta and shallow subsurface intrusions. The term
"cryptomaria" was created in reference to the former, but both features inform us about
partial melting of the lunar mantle and the Moon's thermal history in general.
2. Gravity Maps
GRAIL data have provided the most accurate and highest resolution gravity data
of any planetary body to date from orbit [Zuber et al., 2013b]. The mission uses dual
spacecraft to continuously map the lunar gravity field using spacecraft-to-spacecraft
tracking. Data from both the GRAIL Primary Mission and GRAIL Extended Mission are
combined to model lunar gravity fields up to degree and order 900 [Lemoine et al., 2013;
2014; Konopliv et al., 2013, 2014].
In our study, we expand a gravity field out to degree 660 in order to minimize
high-frequency noise, filtering out the first six degrees to remove long-wavelength
structure in the gravity field. We also use a 30-degree cosine taper to reduce ringing.
17
These gravity data are combined with data from a LOLA topography solution [Smith et
al., 2012] to produce a global map of the Moon's Bouguer anomalies, assuming a
reference density for the crust of 2560 kg/m3 [Wieczorek et al., 2013]. Such a map
reveals the gravity field of the Moon after the effect of topography is subtracted out
(Figure 2).
The positive Bouguer anomalies on the map reveal regions of the Moon with
gravity that must be the result of some excess of mass relative to the reference density,
2560 kg/m 3. Such mass excesses can take the form of higher local crustal density,
thinner crust, or deposits of higher density material. In particular, many of the positive
features on our Bouguer anomaly map are sourced from mantle uplift resulting from large
impacts, exposed basaltic deposits resulting from ancient lunar volcanism, or vertical
tabular igneous intrusions resulting from early thermal expansion of the Moon (Figure 3).
Smaller amplitude features, such as gravity anomalies associated with floor-fractured
craters, could also play a role [Jozwiak et al., 2012]. We identify these features, making
use of a database of lunar craters and basins [Head et al., 2010] and images from the
Lunar Reconnaissance Orbiter Camera (LROC) [Robinson et al., 2010], and eliminate
their corresponding gravity signatures as candidates for cryptomaria or shallow igneous
intrusions. The remaining features on the positive Bouguer anomaly map are considered
as candidates for hidden igneous deposits.
Lunar topography may have isostatically compensating roots [O'Keefe, 1968;
Zuber et al., 1994; Wieczorek and Phillips, 1997], and so we also consider gravity
anomalies that include contributions from crustal roots. We create a map of lunar gravity
anomalies under the condition that topography is compensated by variations in crustal
18
thickness (Airy isostasy) with a mean crustal thickness of 40 km [Wieczorek et al., 2013],
as derived in Turcotte et al. [1981] (Figure 2, bottom row). As with the Bouguer gravity,
we consider the positive anomalies in this isostatic anomaly map as candidate cryptomare
locations. The Bouguer anomaly map (Figure 2, top row) and the isostatic anomaly map
(Figure 2, bottom row) serve as end members with respect to the state of compensation,
and provide constraints on the thicknesses of igneous deposit candidates.
3. Modeling of Igneous Deposits
For each candidate cryptomare region from our positive Bouguer anomaly or
isostatic anomaly maps (Figure 2), we model igneous deposits in an effort to produce the
observed gravity anomaly. As noted previously, modeling with gravity data alone would
yield non-unique results, but a synthesis of gravity and appropriate geological data can
produce more precise constraints. In this study, the relevant non-gravity observations are
measured densities of lunar samples and geometries of dark-halo craters.
The most important factor in modeling the gravity signature of hidden igneous
deposits is the contrast in bulk density between the cryptomare deposit and the overlying
basin ejecta. A recent study using the bead method and helium pycnometry on Apollo
samples and lunar meteorites [Kiefer et al., 2012] found that bulk densities of lunar
basalts typically vary between 3010 and 3270 kg/m 3 . Combining their measurements
with previous work [Horai and Winkler, 1976], the study reports bulk densities for lunar
basin ejecta between 2350 and 2600 kg/m 3 . This is consistent with GRAIL-derived bulk
densities of the highlands crust, which has an average of 2560 kg/M 3 [Wieczorek et al.,
2013]. We therefore consider density contrasts between 450 and 710 kg/m 3 in this study,
19
i.e., the differences between the bulk density constraints for lunar basalts and the average
crustal density. The low-density contrast (450 kg/M 3) provides a maximum constraint on
deposit thicknesses and volumes, and the high-density contrast (710 kg/m 3 ) provides a
minimum constraint.
We must also estimate the thickness of the overlying high-albedo layer in order to
estimate the thickness and volume of the igneous deposit; the deeper in the crust a deposit
is located, the more mass will be needed to produce the same gravity anomaly. Darkhalo impact craters yield information about the location, depth, and thickness of
cryptomaria. In a region where cryptomaria is predicted, the smallest DHC and largest
non-DHC give an estimate of the upper bound of top of the deposit (or, equivalently, the
thickness of the overlying high-albedo layer), while the largest DHC gives an upper
bound of the bottom of the deposit (Figure 1). Thus, a study of DHCs yields estimates
for minimum thicknesses of cryptomare deposits. We use the results of such previous
studies [Antonenko, 1999, as summarized by Shearer et al., 2006] to guide how we model
the depths our deposits, and to compare with our estimates of thicknesses and volumes
from gravity data. It should also be noted that some DHCs on the Moon form as a result
of pyroclastic activity [Schultz and Spudis, 1979], but these can be distinguished from
impact craters based on their elongated shape, alignment with linear rilles or fissures
[Head and Wilson, 1979], and their lack of an elevated rim [Melosh, 1989].
With constraints on the density contrast from sample analysis and depth of
deposits from DHC studies, we model cryptomare deposits as rectangular prisms to
reproduce the observed gravity anomalies in the GRAIL-derived Bouguer and isostatic
anomaly maps. The gravity anomaly, g, measured at the origin of a rectangular prism of
20
density pi in a crust of density pc extending between
and X2, yj and y2, and z, and z2 is
x,
given by [Blakely, 1995]:
I I
i=1 j=1 k=1
R,
k=
x
2
+
Y2
rj
k
.
arctan x'
I
Xi
nRj+y) yj InRjk + X)
ZkRik
/
2 22
g =G(p - pc) y,
J
+ Zk
tijk =
The x and y directions are lateral with respect to the surface, and the z direction is vertical
with respect to the surface.
4. Results and Discussion
Maps of the Moon's positive Bouguer and positive isostatic anomalies are shown
in Figure 4. The major features on these maps that are not obviously attributable to
impact basins, surface maria, vertical tabular igneous intrusions, or the rim of the South
Pole-Aitken basin must be due to some other sort of mass excess. Such features and their
prospects as cryptomare deposits are discussed individually below.
The most prominent feature appearing on our positive Bouguer anomaly map is a
large arc-shaped feature south of Oceanus Procellarum on the near side that stretches
between approximate longitudes 900 W and 90' E and approximate latitudes 30* S and
700 S. The eastern end of the feature corresponds to Mare Australe, the western end
overlaps with the Schiller-Schickard region, and the central portion corresponds to
Maurolycus crater (Figure 5, top).
Mare Australe was first geologically mapped by Wilhelms and El-Baz [1977] as
basaltic lava ponds of middle-to-late Imbrian age within the older Australe basin on the
southeastern limb that was largely destroyed by subsequent impacts prior to volcanic
21
activity [Whitford-Stark, 1979]. Later research dated the Australe basalt deposits
between 3.08 Ga to 3.91 Ga, with 70% of the deposits between 3.5 Ga and 3.8 Ga
[Hiesinger et al., 2000]. Identification of DHCs between the mare patches in Australe
[Schultz and Spudis, 1979] suggests the presence of cryptomaria in the region as well,
with an estimated minimum thickness based on DHC geometry of 500 m and estimated
area of 6.4 x 105 km 2 [Antonenko, 1999, as summarized by Shearer et al., 2006].
The Schiller-Schickard region, located ~1400 km southeast of Orientale basin, is
one of the most well studied candidate areas for cryptomaria. A combination of DHCs
[Schultz and Spudis, 1979; 1983], spectral mixing analyses [Mustard et al., 1992; Head et
al., 1993], and proximity to the Orientale basin and light plains units [Hawke and Bell,
1991] provide strong evidence for the presence of cryptomare deposits. Estimates of the
minimum thickness and areal extent of the deposits based on these data are 400 m and 3.6
x
105 km 2 , respectively.
Maurolycus crater in the south-central highlands has also been proposed as a
region containing cryptomare deposits [Antonenko et al., 1999, as summarized by
Shearer et al., 2006], although other studies [Hawke et al., 2002] have shown that such
deposits are not strictly required. The identification was largely based on an anomalously
high FeO area and the presence of dark-rayed craters inferred to indicate excavation of
mafic minerals [Giguere et al., 1998]. The minimum thickness proposed for this deposit
is 400 m, with an area of 1.6 x 105 km 2 [Antonenko et al., 1999, as summarized by
Shearer et al., 2006]. Since areas of previously proposed cryptomaria align well with the
western end, center, and eastern end of our continuous positive Bouguer gravity feature,
we consider the entire arc a candidate for cryptomare deposits.
22
We model segments as rectangular prisms of high-density material buried at depth
in an attempt to reproduce this gravity feature which we designate the Southern Arc. The
volume of cryptomaria needed depends on the material's density, the material's depth,
and the density of the overlying crust. We do the same type of modeling for the gravity
anomaly that appears in the positive isostatic anomaly map (Figure 5, bottom). The
feature is still present between Maurolycus and Australe, but weaker. Note that a positive
anomaly is associated with Schiller-Zucchius basin in the region of interest in both the
Bouguer and isostatic anomaly maps. Since impact-associated mantle uplift is not
representative of igneous deposits, that anomaly is subtracted out according to an
empirical relationship between basin diameter and gravity anomaly amplitude, as shown
in [Neumann et al., 2014]. A summary of the thickness and volume of cryptomaria
needed as a function of these parameters is given in Table 1 for both the Bouguer map
and the isostatic map.
We have shown in Table 1 that the thicknesses required are strongly dependent
upon the density contrast between the overlying high-albedo material and underlying
low-albedo cryptomaria, and only weakly dependent upon the depth of the deposits. For
the Southern Arc, the Bouguer gravity requires a total volume of cryptomaria between
1.54 x 106 km 3 and 2.45 x 106 kMi 3 , and the isostatic gravity anomaly requires a total
volume of cryptomaria between 4.2 x 105 km 3 and 6.6 x 105 km3 . The areal extent of the
cryptomaria required is 1.03 x 106 kM 2 for the Bouguer anomaly map and 5.0 x 105 km 2
for the isostatic anomaly map.
We find no other regions of the Moon outside of impact basins that present a
convincing case for the presence of a large amount of cryptomaria with a combination of
23
positive gravity anomalies and DHCs. This does not imply that there is no other buried
mafic material or cryptomaria elsewhere in the lunar crust; however, it is likely to be
small in comparison to the volume of visible maria emplaced on the Moon's surface and
the candidate modeled cryptomaria in the Southern Arc feature described above. A
buried basaltic deposit of 105 km 3 would be detectable with our method, even with
conservative estimates for the density contrast between the lunar crust and the deposits;
this corresponds to ~2% of the volume of known maria.
There is a large anomalously positive feature in the Bouguer gravity north of the
farside highlands (Figure 4). It extends from approximately 1750 W to 900 W and from
600 N to 750 N. Bouguer gravity data imply thicknesses between 1.62 km and 2.35 km if
the feature was due to buried high-density deposits, depending on the density contrast
between the deposit and the overlying material. However, because the feature is absent
in the isostatic anomaly map and there exists no geologically based evidence for
cryptomaria or other igneous deposits in the region, we do not consider this area as a
strong candidate. The feature in the Bouguer gravity is likely due to the compensation
state of the region.
There are positive anomalies in the farside highlands in the isostatic anomaly map
but not the Bouguer anomaly map. The crust is thicker here [Wieczorek et al., 2013], and
these anomalies are thus due to our assumption about compensation depth in constructing
the isostatic anomaly map being too low for this region. There is an interesting positive
anomaly southeast of the South Pole-Aitken basin centered at 1000 E, 860 S (Figure 4).
The lack of DHCs in the area [Schultz and Spudis, 1979, 1983] and circular nature of the
feature lends itself to speculation of an erased impact basin; see [Neumann et al., 2014].
24
Though we have excluded gravity anomalies within lunar basins from our
analysis, it is plausible if not highly likely that hidden igneous deposits contribute to the
Bouguer anomaly there, in addition to contributions from impact-induced mantle uplift
and visible mare deposits. The Bouguer anomalies associated with impact basins is the
subject of another study [Neumann et al., 2014], and warrants future consideration in
regards to cryptomaria, particularly for the South Pole-Aitken basin.
5. Summary and conclusions
The areas, thicknesses, and volumes of our proposed cryptomare deposits derived
on the basis of the GRAIL gravity data are summarized in Table 1. The Southern Arc
feature provides the only case with a compelling combination of geophysical and
geological evidence for cryptomaria and includes the previously proposed deposits
associated with Mare Australe, Maurolycus crater, and the Schiller-Schickard region; it
also includes the possibility of new deposits in between these three regions. These
proposed deposits are all on the nearside of the Moon, and thus our analysis does not
change our understanding of where lunar volcanism is prevalent [Wilhelms, 1987; Head
and Wilson, 1992].
We have eliminated the possibility that there are unrevealed large volumes of
basaltic lava trapped within the lunar crust. This includes near-surface extrusions and
intrusions, as both would contain signatures in the GRAIL data. Our analysis of the
Moon's Bouguer gravity anomaly yields a total surface area of candidate cryptomaria
(the Southern Arc region) of 1.03 x 106
kM2.
and a total volume between 1.54 x 106 M3
and 2.45 x 106 kMi 3 , depending on the assumed density contrast. Analysis of the Moon's
25
isostatic gravity anomaly yields a total surface area of 5.0 x 105 km 2 of candidate
cryptomaria (the Southern Arc region), and a total volume between 4.2 x 105 km 3 and 6.6
X
105 Mi 3 , depending on the assumed density contrast. The thicknesses we calculate are
higher than previous estimates for those regions, but those studies have been interpreted
as minimum constraints [Shearer at al., 2006]. The results from the Bouguer gravity
analysis and isostatic gravity analysis can be viewed as end members, with the most
likely answer lying somewhere within their defined range. The Moon's crust has
generally been interpreted to be isostatically compensated [e.g., Wieczorek and Phillips,
1997], which may hint that the true volume of cryptomaria is closer to the lower
quantities derived from the isostatic analysis, but more analysis on the local isostatic state
should be performed using GRAIL data for further understanding.
Acknowledgements
This study was supported by the GRAIL mission, which is part of NASA's Discovery
program and is performed under contract to the Massachusetts Institute of Technology
and the Jet Propulsion Laboratory, California Institute of Technology.
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29
r-,
J_-i_
r__
Highlands crust
Figure 1. Impact craters into lunar crust that contains cryptomaria. (Left) Crustal crosssection of a cryptomaria deposit. The left crater in the diagram shows a crater formed by an
impact event that is only energetic enough such that it will only penetrate into the
overlaying high-albedo basin ejecta; the middle crater represents a sufficiently energetic
impact such that it excavated low-albedo cryptomaria and formed a dark halo with its
ejecta. The right-hand crater penetrated through the cryptomaria and excavated underlying
highlands crust. Using these relationships the depth and thickness of cryptomaria can be
determined. (Right) LROC Narrow Angle Camera image #M 144409490L [Robinson et al.,
20 10] shows an example of a 25 m HC on the ejecta of Censorinus A; note that only the
largest crater in the image has excavated into the cryptomaria.
30
mGal
-400
-300
-200
-100
0
100
200
300
400
500
600
700
Bouguer (top) or Isostatic (bottom) anomaly, in mGal
800
Figure 2. Stereographic projections of the Moon's Bouguer anomalies (top row) and
isostatic anomalies (bottom row) centered on the near side (left column) and far side (right
column), in mGal. Isostatic anomalies assume Airy compensation with a compensation
depth of 40 km.
31
(b)
700
10N
70N
250
600
020
<65N-15
0400
30010
00
200
105
5
60N
140W
130W
120W
(c)
145E
170E
500
20
20
N
-
0
103
195E
(d)
105
40E
50E
60E
40E
50E
60E
Figure 3. Examples of positive Bouguer gravity features that are not due to cryptomaria,
and need to be accounted for in our search. These include mantle uplift associated with
basin-forming impacts such as in Hertzsprung basin (a), and vertical igneous tabular
intrusions associated with early thermal expansion of the Moon (b) [Andrews-Hanna et al.,
2013]. One can also see surface maria in the gravity field; for example, notice the match
between the positive Bouguer features in (c) and an LROC image [Robinson et al., 2010] of
Mare Crisium and Mare Fecunditatis in (d).
32
WN
I
600
500
E
30S
200
608
100
=5
0
60S
100
0
135W
sow
46W
0
45E
OE
135E
ISO00
9ON
No0
E
30
E
C
M
4-0
1,0
30S200
0
100
60S
13W
9OW
46W
0
46E
90E
136E
180
0
Figure 4. The Moon's positive Bouguer anomalies (top) and positive isostatic anomalies
(bottom) centered on the nearside. Isostatic anomalies assume Airy compensation with a
compensation depth of 40 km. All negative Bouguer and isostatic anomalies are set to zero
to highlight the regions where some mass excess must be present. Simple cylindrical
projection.
33
100
200
300
400
Positive Bouguer (top)/Isostatic (bottom) anomaly, mGal
Figure 5. The positive Bouguer anomalies (top) and positive isostatic anomalies (bottom)
of the area referred to as the Southern Arc, which we consider to be the strongest
cryptomare candidate in this study. The feature in the Bouguer gravity overlaps with three
regions that have been proposed as deposit locations previously on the basis of geological
evidence. The positive feature at 450 W, 60* S is due to mantle uplift from the SchillerZucchius basin and its contribution is subtracted away from our analysis.
34
Bouguer
Anomalies
450s
kg/m 3
4 50d
580s
kg/M 3
580d
kg/M 3
710s
kg/m 3
7 10d
kg/M 3
Segment 1
(2.5 x 10s
3.20 km
3.22 km
2.49 km
2.50 km
2.03 km
2.03 km
1.70 km
1.73 km
1.32 km
1.34 km
1.08 km
1.09 km
2.72 km
2.74 km
2.11 km
2.13 km
1.72 km
1.72 km
2.21 km
2.22 km
1.72 km
1.73 km
1.40 km
1.40 km
2.43 x
2.45 x
1.89 x
1.90 x 106
km 3
1.54 x 106
1.54 x 106
km 3
km 3
7 10d
kg/M 3
)
km 2
Segment 2
(3.7 x 10s
)
km 2
Segment 3
(1.8 x 105
)
km 2
Segment 4
(2.3 x 105
)
km 2
Total Volume
(1.03 x 106
106
km 3
106
km 3
106
km 3
)
km 2
Isostatic
Anomalies
Segment 1
450s
kg/m 3
1.28 km
4 50d
580d
1.29 km
580s
kg/M 3
0.99 km
0.99 km
710s
kg/m 3
0.81 km
1.33 km
1.34 km
1.04 km
1.04 km
0.85 km
0.85 km
Total Volume
6.6 x
6.6 x
5.1 x 105
5.1 x 105
4.2 x 10s
4.2 x 105
(5.0 x 105
105 km 3
10s km 3
km 3
km 3
km 3
km 3
kg/M 3
kg/m 3
kg/M 3
0.81 km
(1.9 x 105
)
km 2
Segment 2
(3.1 x 105
)
km 2
)
km 2
Table 1. Thicknesses and volumes needed to reproduce the observed positive Bouguer
(top) or isostatic (bottom) anomalies. The columns are the density contrast between the
low-density overlying basin ejecta and the high-density cryptomaria layer. Each density
contrast is placed at the surface (subscript s) or at a depth of 1 km (subscript d). The
Bouguer feature is approximated by four buried rectangular prisms (segments 1-4) and the
isostatic feature by two (segments 1,2). Each entry is the thickness required to reproduce
the mean gravity anomaly in that rectangular segment for a given density contrast. The
bottom row in each table is the volume of cryptomaria needed for a given density contrast
to reproduce the entire feature in the Bouguer or isostatic map.
35
Chapter 3: The Nature of Isostasy in the Lunar
Highlands and Implications for Mantle Structure
This research was conducted in collaboration with Maria T. Zuber, Brandon C.
Johnson, and Jason M. Soderblom.
ABSTRACT:
The lunar highlands are known to be isostatically compensated on a large scale, but
the exact mechanism of compensation has been difficult to precisely identify. We
use topographic data from the Lunar Orbiter Laser Altimeter (LOLA) and gravity
data from the Gravity Recovery and Interior Laboratory (GRAIL) to investigate the
support of lunar topography. By analyzing the correlation between crustal density
and elevation on various spatial scales, we show that Pratt isostasy is not an
important mechanism in compensation of the lunar highlands. We use the method
of spectrally-weighted admittances to compare the predicted geoid to topography
ratios (GTRs) of various isostatic models with observed GTRs. We find that GTRs in
the nearside highlands are consistent with Airy isostasy in the crust but a crustal
Airy mechanism is inconsistent with farside GTRs. Instead, we propose a two-layer
Airy mechanism for the farside, in which compensation of farside topography occurs
in both the crust and upper mantle. To match the observed GTRs, the part of the
upper mantle layer involved in compensation needs to be at least 125-km thick with
a density between 3000 and 3180 kg/m 3 . This suggests a composition of pyroxenes,
rather than olivine, for the upper mantle. We have thus used GRAIL data to detect
the lunar mantle and constrained lunar formation models to those that produce a
compositionally heterogeneous and stratified mantle.
1. Introduction
One way planetary topography can be supported is through isostatic
equilibrium, in which the overburden pressure of rock is balanced at some depth of
compensation. At such a compensating depth, differences in mass balance the
variations in weight resulting from topographic relief at the surface. Large
planetary topographic features are generally isostatically compensated [Watts,
36
2001]. Thus, careful study of a planet's topography may yield information about
that planet's crustal or perhaps even mantle structure.
The regions of the Moon that are not associated with basaltic extrusions
(maria) have been deduced to be generally isostatically compensated [O'Keefe,
1968], and regions of higher topographic relief systematically have higher crustal
thickness than regions of lower topographic relief [Wieczorek et al., 2013]. This
observation was initially made with the construction of the first lunar gravity maps
[Muller and Sjogren, 1968], and has held with each subsequently more precise and
accurate data set [e.g., Zuber et al., 1994]. However, isostatic compensation can
undertake different forms, each with different implications for crustal formation. In
this paper, we seek to understand the nature of isostasy in the lunar crust and its
implications for planetary history using the most recent lunar data sets.
The two basic models of isostatic compensation are Airy isostasy [Airy, 1855]
and Pratt isostasy [Pratt, 1855]. In the Airy isostasy model, thicknesses in an upper
layer vary in such a way that overburden pressures are equal at some depth. The
upper layer is of a uniform density that is less than the density of the lower layer. In
the Pratt isostasy model, densities in an upper layer vary laterally in a layer of
uniform thickness (except for topographic relief), again producing constant
pressure at a compensating depth. Note that the upper and lower layers do not
necessarily need to correspond to the crust and mantle; a planetary crust may be
vertically stratified. These two models can be thought of as end members in a
continuum; both mechanisms can be operating on the same planet and even the
same local region. The mechanisms may also operate on more than two layers; for
37
example, one can consider a model for compensation in which a top layer has a
uniform thickness and a middle layer has variable thickness, all above a high density
lower layer (see Figure 1)
Airy and Pratt mechanisms arise under different conditions. Airy isostasy
could be primitive in origin and result from floatation of sections of crust that
crystallized in the lunar magma ocean, or be an effect of lateral redistribution of
material from large impacts [Wieczorek and Phillips, 1997]. Pratt isostasy is
common in the Earth's oceans, where bathymetry is frequently the result of thermal
expansion and contraction of the oceanic lithosphere [Lambeck, 1988]. On the
Moon, a different method to produce chemical heterogeneity would need to be
invoked, such as repeated re-differentiation of some regions by large impacts
[Wetherill, 1975]. Thus, identification of the type of isostatic mechanism operating
on the Moon holds the promise of providing information about lunar crustal
formation.
Solomon [1978] investigated the possibility of a Pratt mechanism being a
significant component of lunar isostasy. His study considered regions of the Moon
in which relevant chemical information (Fe and Mg concentrations and Al/Si and
Mg/Si ratios) was known from Apollo X-ray fluorescence and gamma-ray
spectroscopy experiments [Bielefeld et al., 1976; Bielefeld, 1977]. Using normative
mineralogy, Solomon inferred bulk densities and found a negative correlation with
elevation, characteristic of the Pratt mechanism, providing observational support of
the hypothesis that Pratt isostasy was important on the Moon. However, the study
was necessarily limited to only fourteen data points as a result of the limited spatial
38
extent of the Apollo 15 and 16 experiments, and the 99% confidence interval of the
correlation coefficient between bulk density and elevation was large (-0.04 to -0.92).
Wieczorek and Phillips [1997] revisited the issue using data from the
Clementine mission [Nozette et al., 1994; Zuber et al., 1994]. With more precise and
spatially-distributed data, they did not observe a significant negative correlation
between density and elevation in the lunar highlands, and thus ruled out the
possibility that lateral density variations in the lunar crust play a substantial role in
compensating large crustal regions. The study also introduced a new methodology
for interpreting geoid to topography ratios (GTRs) on a sphere by showing that the
GTR for a given isostatic model could be computed using a sum of spectrallyweighted degree-dependent admittances. They showed that observations of the
highlands were consistent with either a single-layer Airy model of the crust or a
two-layer Airy model with upper crustal thickness variations and a uniform lower
crust. The study, however, only reported results for the nearside highlands, as the
uncertainty in the measured geoid of the farside highlands was much greater than
that of the nearside (owing to the absence of direct farside radio tracking) and could
not provide statistically strong conclusions for that hemisphere.
Two recent spacecraft missions allow us to investigate the isostatic state of
the Moon more accurately than ever before. The Lunar Orbiter Laser Altimeter
(LOLA) [Smith, et al., 2010] aboard the Lunar Reconnaissance Orbiter [Chin et al.,
2007] has provided the most spatially dense and precise topographic map of any
planetary body to date, and the Gravity Recovery and Interior Laboratory (GRAIL)
mission [Zuber et al., 2013a] has provided the highest-resolution gravity data of any
39
planetary body to date, using spacecraft-to-spacecraft tracking [Zuber et al., 2013b].
The gravity models we use from the GRAIL primary and extended missions
[Konopliv et al., 2013, 2014; Lemoine et al., 2013, 2014] are expanded to degree and
order 660, which gives the resulting global gravity maps a half-wavelength surface
resolution of -8 km. This is a large improvement compared to the gravity models
available to Wieczorek and Phillips [1997] from the Clementine mission, which were
expanded to degree and order 70 for a half-wavelength surface resolution of -73
km [Lemoine et al., 1997]. The GRAIL maps represent an improvement in accuracy
of 4-5 orders of magnitude over Clementine maps, depending on the spherical
harmonic degree. Also of note is the difference in accuracy for Clementine geoid
maps between the nearside and farside: the model used by Wieczorek and Phillips
[1997] had formal errors ranging from 2 m in the nearside geoid to 24 m in the
farside geoid [Lemoine et al., 1997]. Solomon [1978] did not use any gravity maps
in his analysis of lunar isostasy. Topographic data have also undergone substantial
improvement from previous studies. Our elevation data comes from a LOLA
topographic map [Smith et al., 2010]; LOLA has collected -6.8 x 109 measurements
of elevation, yielding a topographic model expanded to degree and order 2500 with
precision -10 cm and accuracy -1 m. The topographic map used by Wieczorek and
Phillips [1997] was derived from the Clementine lidar instrument, which collected
-70,000 measurements of elevation, and produced a topographic model expanded
to degree and order 72 with precision -10 m and accuracy -100 m [Smith et al.,
1997]. Elevation data used by Solomon [1978] were taken from measurements from
40
the Apollo 15 and 16 laser altimeters, which only sampled specific orbital tracks and
had precision -100 m and accuracy -400 m [Kaula et al., 1974].
In this paper, we will use the GRAIL and LOLA data in two main ways. First,
we will search for the signature negative correlation between elevation and crustal
density indicative of Pratt isostasy in a more local sense than previous study has
been able to perform. Second, we will use the method of spectrally-weighted
degree-dependent admittances to construct isostatic models that fit the observed
GTRs, with special attention paid to the farside highlands. We will discuss the
results from these procedures as they relate to the structure of the lunar crust and
mantle, and the geophysical evolution of the Moon.
2. Elevation-Density Correlations
Our search for a negative correlation between elevation and density makes
use of four data sets. The elevation data come from a LOLA topographic map [Smith
et al., 2010], and is referenced to the lunar geoid, which is calculated from the first
660 degrees and orders from the GRAIL primary and extended missions [Konopliv
et al., 2013, 2014; Lemoine et al., 2013, 2014]. For crustal density, we consider data
that are derived from Lunar Prospector spectroscopy [Prettyman et al., 2006] and
from GRAIL gravity measurements [Wieczorek et al., 2013]. The spectroscopy
results represent an estimate of the grain density, while the gravity results
represent an estimate of the bulk density. The grain densities in particular rely on
41
the assumption that the composition of the lunar surface is representative of the
underlying crustal column.
We calculate moving windows of a set radius across three maps (elevation
referenced to the geoid, grain density, and bulk density), sampling data points in
these maps in a grid pattern with a spacing of 8 km. We create scatter plots of grain
or bulk density as a function of elevation, and calculate the slope and R 2 value of the
best-fit line determined from a least-squares fit (e.g., Figure 2). A negative slope in
the line indicates that Pratt isostasy may be an important mechanism, with a high R 2
value indicative of a high goodness-of-fit of that line. However, this is not the case
for windows that contain large regions of maria, as these basaltic extrusions are
preferentially emplaced in topographic lows [Head, 1975] and do not compose a
majority of the crustal thickness at any one location, except in the interior of some
large impact basins [Head, 1982; Head and Wilson, 1992]. Therefore, these areas
violate our important assumption that the density we observe near the surface is
representative of the underlying crustal column and should not be interpreted as
places with a strong Pratt mechanism.
We consider three ways to test for significance of the resulting set of R 2
values. The first is hypothesis testing. We perform a t-test, where the null
hypothesis is that the slope of the best-fit line between density and elevation at each
location is equal to zero. The second test, which tests the sensitivity of the model, is
to identify regions that exhibit a positive correlation between elevation and density.
Such a relationship is not predicted by any model of isostasy and, therefore,
provides a false-positive threshold against which we can compare our results. As an
42
additional sensitivity test, we generate random density maps of a planetary surface
that have the same mean and standard deviation of densities as the actual lunar
crustal densities. We perform the moving window procedure using these random
density maps and the real lunar topography. We know any locations with a negative
correlation between density and elevation occurred by chance and thus provide a
measure of the magnitude and extent of R 2 values that are not statistically
significant.
A resulting map of R 2 values is shown in Figure 3 for bulk density and
elevation measured with a 500-km window. Since the R2 values are generally close
to zero in windows that do not contain mare regions or parts of the South PoleAitken basin, we do not find evidence for a strong Pratt mechanism operating in the
lunar highlands on a large scale. This result holds equally for maps that consider
grain density and different spatial filters.
3. Geoid to Topography Ratios and Spectrally-Weighted Admittances
To test whether Airy isostasy is a significant mechanism on the moon, we use
the method of Wieczorek and Phillips [1997], which shows that the GTR of a given
Airy isostatic model on a sphere is equivalent to a sum of spectrally-weighted
degree-dependent admittances. We calculate the GTRs of the Moon by using the
LOLA-derived elevation map and GRAIL-derived geoid map. We take a moving
window of a set radius across these maps, sampling data in a grid pattern within
that window every 8 km. The slope of the best-fit line between geoid and elevation
43
is the GTR for that window. This is repeated across the lunar surface to obtain a
map of GTRs (Figure 4) for the highlands. We consider moving windows of 500,
750, and 1000-km radius, and find that while the resulting distribution of GTRs is
somewhat sensitive to window size, our results will hold for each size. We also
consider GTRs by forcing the best-fit line between geoid and elevation to pass
through the origin and calculating the resulting slope, but find that it does not
change the mean GTR of a region by more than 10%.
The predicted GTR for an Airy isostatic model is a weighted sum of
admittances, ZL, for each spherical harmonic degree.
L
The weighting function WL, is the fractional topographic power at degree L. For a
single-layer Airy model compensated at the crust-mantle interface, the admittance
at degree L is given by [e.g., Lambeck 1988],
Z
-
A4(2L+ 1)
10(2
\(R
)]
(2)
where R is the lunar radius, M is the lunar mass, T is the reference crustal thickness
(i.e., the thickness at zero elevation), and pc is the density of the crust (Table 1). One
can also consider Airy models with two layers in the crust. The admittance
equations are initially shown in Wieczorek and Phillips [1997], but the equations for
two different models were swapped; the corrected equations are shown in Pauer
and Breuer [2008]. For a two-layer Airy model with upper crustal thickness
variations overlying a uniformly thick lower crust, the admittance at degree L is
44
- iU
ZL
4Zp=R
M(2L+1)
[
R -RT
[1 p
Pp
R
T
PIC - PUC R - Tuc
R
R
2]-
)
[-T L 2P c
R
PM - Pic
R-T
(3)
where Tc is the total crustal thickness at zero elevation, Tuc is the upper crustal
thickness at zero elevation, and puc, Plc, and pm are the densities of the upper crust,
lower crust and mantle respectively. Finally, for a two-layer Airy model with lower
crustal thickness variations underlying a uniformly thick upper crust, the
admittance at degree L is
ZL
cR 3 1+
M(2L+1)
4
PUC
(
-
)]
-T
R
+
P""
U)
R-T+Pc 2
R
) j
(4)
These two models can be thought of as end members. Equation 3 represents a twolayer crust where the inter-crustal boundary follows the crust-mantle interface;
equation 4 represents a two-layer crust where the inter-crustal boundary follows
the topography.
There does not exist an analogous simple spherical admittance model for
Pratt isostasy, so we use the approximation from Haxby and Turcotte [1978]:
GTR=
2T
'rpCR
M
In this case, the density pc is the reference crustal density (i.e., density at zero
elevation) and T is the reference crustal thickness.
45
We consider other studies as a guide for what constitutes reasonable input
parameters for our isostatic models. GRAIL observations show that the reference
crustal thickness on the Moon is 34-43 km, and the average crustal density in the
highlands is 2550 kg/M 3 [Wieczorek et al., 2013]. A study using the bead method
and helium pycnometry on Apollo samples and lunar meteorites [Kiefer et al., 2012]
found that bulk densities of lunar basalts typically vary between 3010 and 3270
kg/M 3 . Theoretical models of lunar magma ocean solidification [e.g., Elkins-Tanton
et al., 2011] predict an upper mantle density of -3000 kg/M 3 that monotonically
increases with depth, with a discrete jump in density at -700 km depth. A study of
the "effective density" at different spherical harmonic degrees using GRAIL data
shows that crustal density should not increase with depth in the lunar highlands, at
least in the upper -10 km of crust [Besserer et al., 2014].
4. Geoid to Topography Ratio Results
The GTRs for the nearside highlands generally range from 15-25 m/km
(Figure 4d). Our results are in good agreement with Wieczorek and Phillips [1997]
and support their conclusion that the nearside highlands data can be described by a
one or two-layer Airy model with upper crustal thickness variations. We find that
the GTRs on the farside highlands, however, are much higher (Figure 4e), with an
average value of 45 m/km for a moving window size of 500 km (Figure 4b, c) and
require a different model. We consider a number of isostatic models to reproduce
the observed farside highlands GTR values.
46
Figure 5 shows the predicted GTRs for single-layer Pratt and Airy models of
the crust as a function of crustal density and crustal thickness. The GTRs are much
lower than 45 m/km for all considered input parameters. Examples of two-layer
crustal models of Airy isostasy are shown in Figure 6. They too predict GTRs that
are too low for all reasonable input parameters. The observed farside highlands are
inconsistent with either Airy or Pratt mechanisms with compensation in the crust,
unless one invokes unrealistic values for average crustal thickness (> 100 km),
unrealistic values of average crustal density (> 3000 kg/m 3 ), or a density inversion
for which there currently exists no other observational evidence (i.e., a layer of
material underlying the farside highlands that is at least 300 kg/m 3 less dense than
the density observed at the surface).
We consider instead a two-layer Airy model in which some compensation
takes place in the mantle. In this model, the upper layer is the entire crust, which
varies in thickness, and the lower layer is the upper mantle, which has a uniform
thickness and higher density than the crust. Both these layers overlay the rest of the
mantle, which has a density higher than both the crust and the upper mantle layer.
The admittance for this model is similar to equation (3) and is given by
1 (R
pR 3
ZL - M(2L+D
-Tu
L[
Pi,
u(R -T -T
R- T
-PUM- PC
R
L
RT-T)
R
2'.
1+
P
p,,-
(6)
C R - Tc
.R- T -T
Figure 7 shows the predicted GTRs for this model. We assume an average crustal
density of 2550 kg/M 3 and reference crustal thicknesses between 34 and 43 km
47
[Wieczorek et al., 2013]. The parameters we vary are the density of the upper
mantle layer and the thickness of the upper mantle layer. We can match the
observed GTRs with an upper mantle layer that is at least 125 km thick and has a
density in the range of 3000-3180 kg/M 3, when the lower mantle is assumed to be
3400 kg/m 3. This density range of the upper mantle layer is more likely indicative
of a pyroxene composition on the Moon, as opposed to olivine [Elkins-Tanton et al.,
2011].
5. Discussion
An Airy isostasy model that invokes an upper mantle layer that is
intermediate in density between the bulk density of the crust and the mantle and at
least 125 km thick can describe the observed GTRs of the farside highlands. The
important question then becomes: is such a structure geochemically reasonable?
Many studies of the lunar magma ocean predict a compositionally stratified
mantle [e.g., Solomon and Longhi, 1977; Snyder et al., 1992; Elkins-Tanton et al.,
2011]. In particular, Elkins-Tanton et al. [2011] predict a post-overturn vertical
density structure that invokes a 650-km thick layer in the upper mantle with a
density between 3000 and 3200 kg/m 3, above the remaining mantle which has a
density of at least 3400 kg/M 3 (the exact densities depend on the fraction of
interstitial melt in the model). These results are consistent with our prediction for
an upper mantle layer that produces the observed GTRs of the farside highlands
(Figure 7).
48
A compositionally stratified mantle is also consistent with Apollo seismic
results [Nakamura et al., 1973]. Previous studies have argued for discontinuities in
the mantle at depths of 270-750 km, on the basis of observed changes in seismic
wave velocities [e.g., Nakamura et al., 1974, 1976; Dainty et al., 1976; Goins et al.,
1981; Khan and Mosegaard, 2002; Legnonne et al., 2003]. Seismic data must be
interpreted with caution, however, as seismic stations only exist on the nearside and
the data are not sufficient to resolve fine structural differences between layers
[Nakamura, 1983], and models with a chemically uniform mantle can also fit the
observations [Khan et al., 2006].
Though the vertical structure is consistent with other predictions, we require
an explanation for the hemispherical asymmetry. Why does our two-layer mantle
model apply to the farside highlands, but not the nearside highlands? There are two
possibilities: the upper mantle layer we invoke exists on the farside and not the
nearside, or the upper mantle layer is global, but only participates in isostatic
compensation on the farside (Figure 8). The former possibility could simply be the
result of lateral mantle heterogeneity after solidification of the magma ocean; such
heterogeneities are predicted by Elkins-Tanton et al. [2011]. The latter possibility
implies that the nearside upper mantle was able to flow more readily than the
farside upper mantle at the time of formation of the farside highlands. This could be
explained by elevated temperatures in the nearside upper mantle caused by the
high concentration of heat-producing elements in that hemisphere, an idea already
proposed to explain the asymmetric distribution of lunar basin evolution [Solomon
49
et al., 1982], lunar maria [Wieczorek and Phillips, 2000], and lunar basin size and
corresponding excavation depth [Miljkovic et al., 2014].
An alternative hypothesis to mantle compensation would invoke significant
elastic thickness in the farside highlands, an effect that would increase admittances
and GTRs compared to a strict Airy model [e.g., Watts, 2001]. The farside highlands,
however, are thought to have formed very early in lunar history [Wasson and
Warren, 1980; Jolliff et al., 2000], as a result of magma-ocean convective
asymmetries [Loper and Warner, 2002] or spatial variations in tidal heating
[Garrick-Bethel et al., 2010, 2014]. Both of these formation models precede cooling
of the Moon and formation of a thick elastic lithosphere. If, however, the farside
highlands were formed later, as a result of ejecta deposits from the South PoleAitken basin [Zuber et al., 1994] or accretion of a companion moon [Jutzi and
Asphaug, 2011], elastic thickness would be a relevant factor. We favor the mantle
compensation hypothesis because basin ejecta deposits are not predicted to be
emplaced in a degree-2 pattern [Garrick-Bethel et al., 2010; 2014] and GRAIL
studies do not observe an increase in density with depth in the farside highlands
[Besserer et al., 2014] as might be expected from accretion of a second moon [Jutzi
and Asphaug, 2011]. Dynamic compensation of topography can result in high GTRs
compared to a strict Airy model, and has been proposed to be important for Earth
[Ceuleneer et al., 1988] and Venus [Smrekar and Phillips, 1991]. However, we do
not favor a dynamic explanation since vigorous convection in the lunar mantle has
likely not been active recently [e.g., Evans et al., 2014].
50
6. Conclusions
Analysis of the Moon's crustal density and topography shows that Pratt
isostasy is not an important mechanism in compensating the lunar highlands.
Analysis of the Moon's GTRs reveals a fundamental difference in the state of
compensation between the nearside and farside highlands. The nearside highlands
can be compensated with simple Airy isostasy in the crust. The observed farside
highlands GTRs, however, require more complex compensation, and are best fit with
a two-layer mantle structure, in which the upper layer of the mantle is 3000-3180
kg/M 3 in density and at least 125 km thick. This structure is consistent with, though
not demanded by, seismic data and geochemical data and models. We thus argue
that we have detected the lunar mantle using GRAIL data, and have constrained
lunar formation models to those that produce a compositionally stratified mantle.
Acknowledgements
This study was supported by the GRAIL mission, which is part of NASA's Discovery
program and is performed under contract to the Massachusetts Institute of
Technology and the Jet Propulsion Laboratory, California Institute of Technology.
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Wieczorek, M.A., and Phillips, R.J., 1997. The structure and compensation of the
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Wieczorek, M.A., and Phillips, R.J., 2000. The "Procellarum KREEP Terrane":
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20417-20430.
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Zuber, M.T., Smith, D.E., Lemoine, F.G., and Neumann, G.A., 1994. The Shape and
Internal Structure of the Moon from the Clementine Mission. Science 266,
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Zuber, M.T., et al., 2013a. Gravity Recovery and Interior Laboratory (GRAIL):
Mapping the Lunar Interior from Crust to Core. Space Sci. Rev. 178, 3-24.
Zuber, M.T., et al., 2013b. Gravity Field of the Moon from the Gravity Recovery and
Interior Laboratory (GRAIL) Mission. Science 339, 668-671.
54
P2
Pc
3
Pm
Pm
PPuc
Pm7
Figure 1. Basic models of isostatic compensation. Single-layer Airy (upper left) and Pratt
mechanisms (upper right) invoke differences in the thickness or density respectively of a
layer overlying a region of greater density to balance the weight of topography. Dual-layer
models of Airy isostasy invoke compensation in two layers. Variations in thickness can
occur in either the upper layer (lower left) or the lower layer (lower right).
55
2950-
2940-
-.
2920-
*
.
..
2910
..
0) o
F 2900.
.
.0.
-
2930-
*.o0
- 0
ta
sas. t,
0
289
0 .. (
087
2880
287
-4
..
01
:
.' 0- -o.
0
-2
the 0 fasd ihad
.
na0
0
r
-t
0.
o
q
a
t
*.
1 t
0.: ..
0
2
km0
81
radiu widw0nti xmltebs-iiei
negative in slope, but only has an R 2 value of 0.04, and thus does not lend strong evidence in
support of Pratt isostasy operating in the region.
56
C
2_
1.0
10.5
0
0
0.
0
Nearside
Farside
Figure 3. Maps of our check for Pratt isostasy, overlaying shaded topographic maps of the
Moon's surface. In this example, we look for negative correlations between GRAIL-derived
bulk density and LOLA-derived elevation by plotting the R2 value s of the best-fit line
between density and elevation within a 450-kmn moving window centered on a point on the
Moon's surface. In the top row, we plot the R2 value if the slope of that line is negative;
in
the bottom row, we plot the R2 value if the slope of that line is positive. High R2 values
only
appear when the moving window contains mare regions or the South Pole-Aitken (SP-A)
basin. In the highlands, R2 values do not appear more significantly in the negative
slope
case (which would be expected from Pratt isostasy) than in the positive slope case (which
no isostatic model predicts). Therefore, these maps indicate a lack of importance for the
Pratt mechanism in the highlands.
57
(a)
LOLA Topographic Map
10
8
6
4
E
2
0 0
-2
-4
-6
-8
(b)
Farisde GTRs (500 km window)
(c)
Farisde GTR histogram
60
500 km window
750 km window
- 1000 km window
-
$16
-
C
aj 12
30
(>
(U-4
20
60
40
80
GTR (r/km)
GTR (m/km)
50
""'Nearside
7
Farside
U
U
010
65 S
a
5
I
a e
80
-40
(d)
Nearisde GTRs (500 km window)
(e)
-20
(G
0
Ra
20/
m)
40
60
Near- vs farside GTRs
Figure 4. Observed GTRs of the lunar highlands from GRAIL and LOLA data. The locations
of the nearside (dashed box) and farside (solid box) analyzed are in (a). GTRs of the farside
using a 500-km radius moving window are in (b). Comparison of 750-km and 1000-km
radius moving windows are shown in (c). GTRs of the nearside using a 500-km window are
in (d). Comparison of the observed GTR distribution between the nearside (dashed line)
and farside (solid line) for a 500-km radius moving window are in (e).
58
Pratt GTRs
Airy Single Layer GTRs
2900
2900
45
40
2800
280035
02700
E
_&2700
S2600
2600
2500
32500
'D 2400
2400
2
30
20
15
10
10
20
30
40
50
Crustal Thickness (kn)
60
70
5
10
20
30
40
50
60
Crustal Thickness at zero elevation (km)
70
0
Figure 5. GTRs predicted by compensation of a single-layer crust by variable crustal
density (left) and variable crustal thickness (right). In neither model do the predicted GTRs
approach the observed value of 45 m/km of the farside highlands. The color scale applies to
both plots.
59
Airy 2 Layer Crust, 25% Uniform Upper Layer
Airy 2 Layer Crust, 75% Uniform
Upper Layer
02600
26026001
2400
2400
10
20
30
40
50
0
Airy 2 Layer Crust, 25% Uniform Lower Layer
-
10
20
30
40
50
60
70
Airy 2 Layer Crust, 75% Uniform Lower Layer
70
2M0
2900
46
U2800
2800
35
2700
2700
30
U40
25
2600
2600
10
2400
2400
5
10
20
30
40
50
W
70
10
20
30
40
so
"60
70
0
Crustal thickness at zero elevation (kin)
Figure 6. GTRs predicted by compensation of a two-layer crust. In the top row, the upper
crust is a uniformly thick layer and the lower crust has variable thickness; in the bottom
row, the reverse is true. In the left column, the uniformly thick layer is 25% of the total
crust at zero elevation, and in the right column, the uniformly thick layer is 75% of the total
crust at zero elevation. The lower crust is 300 kg/M 3 denser than the upper crust. For
GRAIL-derived values of the average crustal density (2550 kg/M 3) and the reference crustal
thickness (34-43 km) [Wieczorek et al., 2013], the GTRs of the model do not match the high
GTRs observed in the farside highlands. The color scale is the same for all examples.
60
Airy 2 layer mantle
65
3300
60
3250
50
3200
-
55
IE
-
-
3150
0
CL
35
3100
CL
'I)
0.
0.
D
30
3050
25
100
200
300
400
500
600
700
800
Upper mantle thickness (km)
Figure 7. GTRs predicted by Airy compensation with a single-layer crust and a two-layer
mantle. The crustal density is 2550 kg/M 3 and the lower mantle density is 3400 kg/M 3. We
vary the thickness and density of an upper mantle layer. The thick black line represents
where the model predicts a GTR of 45 m/km for a reference crustal thickness of 40 km; the
thinner black lines represent where the GTR would be 45 m/km if instead we used a
reference crustal thickness of 34 (bottom) or 43 (top) km, which are the bounds as derived
by GRAIL [Wieczorek et al., 2013].
61
Farside crust
Upper mantle
F
Mantle
Farside crust
Upper mantle
Lower Mantle
Figure 8. Schematic (not to scale) of two different possibilities for structure of our model.
The upper mantle can be laterally heterogeneous and exist only on the farside (top), or can
exist globally and only participate in isostatic compensation of topography on the farside
(bottom).
Parameter
Lunar radius
Lunar mass
Reference crustal
thickness
Average crustal density
Variable
R
M
T
Value
1737.4 km
7.348 x 1022 kg
34-43 km
Reference
Smith et al., 2010
Zuber et al., 2013b
Wieczorek et al., 2013
Pc
2550 kg/m 3
Wieczorek et al., 2013
Table 1. Parameters used in our spectrally weighted admittance isostatic models, with
references.
62
Chapter 4: A procedure for testing the significance
of orbital tuning of the Martian polar layered
deposits
This research was conducted in collaboration with J. Taylor Perron, Peter Huybers,
and Oded Aharonson.
ABSTRACT:
Layered deposits of dusty ice in the Martian polar caps have been hypothesized to record
climate changes driven by orbitally induced variations in the distribution of incoming
solar radiation. Attempts to identify such an orbital signal by tuning a stratigraphic
sequence of polar layered deposits (PLDs) to match an assumed forcing introduce a risk
of identifying spurious matches between unrelated records. We present an approach for
evaluating the significance of matches obtained by orbital tuning, and investigate the
utility of this approach for identifying orbital signals in the Mars PLDs. Using a set of
simple models for ice and dust accumulation driven by insolation, we generate synthetic
PLD stratigraphic sequences with nonlinear time-depth relationships. We then use a
dynamic time warping algorithm to attempt to identify an orbital signal in the modeled
sequences, and apply a Monte Carlo procedure to determine whether this match is
significantly better than a match to a random sequence that contains no orbital signal.
For simple deposition mechanisms in which dust deposition rate is constant and ice
deposition rate varies linearly with insolation, we find that an orbital signal can be
confidently identified if at least 10% of the accumulation time interval is preserved as
strata. Addition of noise to our models raises this minimum preservation requirement, and
we expect that more complex deposition functions would generally also make
identification more difficult. In light of these results, we consider the prospects for
identifying an orbital signal in the actual PLD stratigraphy, and conclude that this is
feasible even with a strongly nonlinear relationship between stratigraphic depth and time,
provided that a sufficient fraction of time is preserved in the record and that ice and dust
deposition rates vary predictably with insolation. Independent age constraints from other
techniques may be necessary, for example, if an insufficient amount of time is preserved
in the stratigraphy.
1. Introduction
The topographic domes of the north and south polar ice caps on Mars are mostly
composed of kilometers-thick layered sedimentary deposits, the polar layered deposits
63
(PLDs), which are exposed in spiraling troughs cut into the caps [Murray et al., 1972;
Cutts, 1973; Howard et al., 1982; Byrne, 2009], as shown in Figure 1. The PLDs were
initially observed in images from the Mariner 9 spacecraft [Murray et al., 1972], and
were immediately inferred to be composed of atmospherically deposited dust and ice
[Cutts, 1973]. Since then, the PLDs have been more thoroughly characterized. Carbon
dioxide ice and clathrate hydrate have been shown to be compositionally insignificant
based on their effects on thermal properties [Mellon, 1996] and bulk strength [Nye et al.,
2000]. Water ice dominates dust volumetrically; dust volume composition has an upper
limit of 2% in the north polar cap [Picardi et al., 2005] and 10% in the south polar cap
[Plaut et al., 2007] according to MARSIS radar transparency data, and ~15% in the south
polar cap according to gravity anomalies associated with the area [Zuber et al., 2007;
Wieczorek et al., 2008]. Concentrations far smaller than these upper bounds could
produce the observed brightness differences [Cutts, 1973]. MOLA topography
demonstrates that the ice caps are dome-like structures 3-4 km thick [Zuber et al., 1998],
with volumes of 1.14 million km 3 for the northern dome [Smith et al., 2001] and 1.6
million km 3 for the southern dome [Plaut et al., 2007]. The deposits are locally overlain
by seasonal carbon dioxide frost [Smith et al., 2001]. Radar soundings from the
SHARAD instrument [Phillips et al., 2008] have revealed that large-scale stratigraphy is
similar in different parts of the northern ice cap, implying that the PLDs record regional
or global climate phenomena rather than local conditions.
Many authors have attempted to constrain the deposition rates of polar ice or dust
[Pollack et al., 1979; Kieffer, 1990; Herkenhoff and Plaut, 2000], but these estimates
span orders of magnitude. Populations of impact craters on the polar caps provide some
64
constraints, including an estimated mean surface age of 30 to 100 Myr for the southern
PLDs [Koutnik et al., 2002] and an estimated upper limit on the accumulation rate of 3-4
mm/yr for the northern PLDs [Banks et al., 2010]. Despite these efforts, the ages of the
PLDs remain poorly constrained.
It has been proposed that patterns in the thickness and brightness of these layers,
which are thought to result from variable dust concentration in the ice, are controlled by
changes in the distribution of solar radiation due to quasi-periodic variations in the
planet's spin and orbital characteristics over time, specifically climatic precession,
obliquity variation, and eccentricity variation [Murray et al., 1973; Cutts et al., 1976;
Toon et al., 1980; Cutts and Lewis, 1982; Howard et al., 1982; Thomas et al., 1992,
Laskar et al., 2002; Milkovich and Head, 2005; Milkovich et al., 2008; Fishbaugh et al.,
2010; Hvidberg et al., 2012]. In this way, the PLDs may record past Martian climate.
An analogous argument is often made regarding ice cores or marine sediment
cores and Earth's paleoclimate. Some of the variability in marine Pleistocene
paleoclimate proxies has been convincingly linked to orbital changes [Hays et al., 1976].
However, there is debate about how much of the recorded climate variability was
deterministically controlled by Milankovitch cycles [Kominz and Pisias, 1979; Wunsch,
2004]. In theory, the problem on Mars should be more tractable than the analogous
problem on Earth. The Martian atmosphere is orders of magnitude less massive than
Earth's, and Mars has not had a surface ocean in the recent past, two factors that should
make the Martian climate system simpler than the terrestrial one. Mars also experiences
larger obliquity and eccentricity variations than Earth [Ward, 1973; Touma and Wisdom,
65
1993; Laskar et al., 2004], which should make an orbital signal, if present, stronger and
perhaps easier to detect.
Despite the likelihood of a simpler climate on Mars, detection of an orbital signal
in the PLDs is not a trivial task. The relationship between time and stratigraphic depth in
the PLDs is unknown, and is likely nonlinear. There are no absolute ages available for
any part of the deposits. Image brightness may contain noise from image artifacts,
inherent noise in the deposition rates of ice and/or dust, and an indirect relationship
between visible albedo and PLD composition [Tanaka, 2005; Fishbaugh and Hvidberg,
2006; Herkenhoff et al., 2007; Levrard et al., 2007]. Because of these complexities and
uncertainties, detection of an orbital signal in the Martian PLDs using spacecraft
observations poses a considerable challenge [Perron and Huybers, 2009].
The problem of orbital signal detection has been considered almost since the
PLDs were first discovered. Given the lack of an absolute chronology, most efforts to
interpret the PLDs have focused on modeling or analyzing their stratigraphy. The first
study to consider in detail how different PLD formation mechanisms influence the
resulting stratigraphy was that of Cutts and Lewis [1982]. They considered two
deposition models. In their first model, material composing the major constituent of the
PLDs is deposited at a constant rate, and differences between layers are caused by a
minor constituent that is deposited at a constant rate only when the obliquity of the planet
is below a certain threshold value. In their other model, only one type of material is
deposited, but only when the obliquity is below a certain threshold value; layer
boundaries correspond to periods with no deposition. Although these models are highly
simplified, their work revealed the sensitivity of PLD stratigraphy to factors such as ice
66
deposition rates and thresholds, and thus hinted at the difficulty of detecting an orbital
signal. More recently, Levrard et al. [2007] used a global climate model for Mars to study
ice accumulation rates and concluded that formation of PLD layers must indeed be more
complex than originally modeled. Hvidberg et al. [2012] built upon the models of Cutts
and Lewis with physically plausible mechanisms of ice and dust deposition, and showed
that their models could generate synthetic PLD sequences consistent with some
stratigraphy observed in the top 500 m of the PLDs.
Other authors have used time series analysis to search for coherent signals in the
PLD stratigraphy, particularly signals that may be related to orbital forcing. Milkovich
and Head [2005] analyzed spectra of brightness profiles through the north PLDs, and
reported the presence of a signal with a 30 m vertical wavelength in the upper 300 m of
the PLDs, which they interpreted as a signature of the approximately 51 kyr cycle of the
climatic precession. They assumed a linear time-depth relationship, however, and did not
evaluate the statistical significance of the signal they identified. Perron and Huybers
[2009] expanded this analysis, also assuming a linear time-depth relationship on average,
but allowing for local variability ("jitter") in this relationship. They also evaluated the
significance of peaks in the PLD spectra with respect to a noise background. Perron and
Huybers [2009] found that the PLD spectra closely resemble spectra for autocorrelated
random noise, but that many stratigraphic sequences contain intermittent, quasi-periodic
bedding with a vertical wavelength of 1.6 m. Subsequent studies have confirmed and
refined this measurement of 1.6 meter bedding through analyses of higher-resolution
imagery and stereo topography [Fishbaugh, 2010; Limaye et al., 2012].
67
These applications of conventional time series analysis techniques have revealed
signals within the stratigraphy, but have not been able to conclusively identify evidence
of orbital forcing due to the absence of multiple periodic signals with a ratio of
wavelengths that matches the expected ratio of orbital periods [Perron and Huybers,
2009]. They have also been limited by the assumption of a linear time-depth relationship,
a scenario that, while possible, is rare in terrestrial stratigraphic sequences [Sadler, 1981;
Weedon, 2005]. Thus, while the Mars polar caps do appear to record repeating regional
or global climate events, the duration of these events and their relationship to orbitally
forced variations in insolation remain unknown.
In studies of terrestrial paleoclimate records, it is common to address the problem
of unknown time-depth relationships by tuning an observed record - adjusting its time
model nonuniformly by moving points in the record closer together or further apart - to
match an assumed forcing with a known chronology, or by tuning two or more observed
records with unknown chronologies to match each other. There have been limited efforts
to apply tuning procedures to the Mars PLDs. Laskar et al. [2002] compared the PLD
stratigraphy with an insolation time series using an approach in which a portion of the
photometric brightness image was stretched to provide an approximate fit to the
insolation time series. They analyzed only one image, however, and did not evaluate the
goodness of fit statistically. Milkovich et al. [2008] used the signal-matching algorithm of
Lisiecki and Lisiecki [2002] to search for stratigraphic correlations between PLDs in
different regions of the north polar cap, but did not attempt to tune PLD sequences to
match insolation records.
68
The need to assess the statistical significance of proposed tunings is widespread in
the study of terrestrial paleoclimate [Proistosescu et al., 2012] and in other analyses that
seek correlations among time series with uncertain chronologies. The essential problem is
that any effort to tune records to match one another will produce some agreement, but it
is not clear whether this agreement arose by chance, or whether it reveals an underlying
relationship. To address this need, methods have been proposed that estimate the
significance of a tuned fit between records, generally by comparing the fit between
records that are hypothesized to share an underlying relationship with fits to random
records that share no underlying relationship with the observed record. This was the
general approach adopted by Milkovich et al. [2008] in their effort to correlate PLD
stratigraphic sequences with one another.
In this paper, we adapt a statistical procedure for evaluating the significance of
orbital tuning that has been successfully applied to terrestrial paleoclimate records and
has been shown to be applicable to comparisons between any two time-uncertain series
[Haam and Huybers, 2010]. That study considered an application where the time series
were known to be approximately 9000 years in total; for our application to the Martian
PLDs, the total duration is much more uncertain but the same statistical methods apply,
albeit with the expectation that the power of the method will be lower. Of course, one
should also consider any independent age constraints on the PLDs to guide the technique
and determine if a resulting match is physically plausible. We use the procedure to
compare two data series - insolation as a function of time and composition of strata as a
function of depth - and assess the potential for detecting an orbital signal in the Mars
polar layered deposits. Our approach is divided into two main steps. First, we construct
69
simplified models for PLD accumulation and drive these models with a Martian
insolation time series to create synthetic PLD records. We consider three different
models, none of which produces a linear time-depth relationship. In the second step, we
perform a statistical analysis to determine how reliably we can detect the orbital signal in
the synthetic PLD records. The statistical analysis uses a dynamic time warping
algorithm to tune the synthetic PLD records to the insolation time series and a Monte
Carlo procedure that evaluates the statistical significance of that tuning by applying the
same dynamic time warping algorithm to random signals. For each modeled PLD
formation mechanism, this procedure yields an estimated confidence level for detection
of an orbital signal. We then consider the implications of this analysis for the
interpretation of the PLD stratigraphic sequences measured from spacecraft observations,
including the prospects for identifying evidence of orbital forcing. The purpose of our
work is not to definitively identify the accumulation function controlling PLD formation,
but to assess the performance of a technique that can be used to analyze PLD records that
do not have a linear depth-age relationship.
2. Polar Layered Deposit Formation Models
2.1 Insolationforcing
In the models presented here, hypothetical ice and dust deposition rates expressed
as functions of insolation are integrated forward in time to produce synthetic PLD
stratigraphic sequences. Changes in the seasonality and global distribution of insolation
on Mars are controlled mainly by the planet's climatic precession, obliquity variations,
and eccentricity variations [Ward, 1973, 1974, 1992; Touma and Wisdom, 1993; Laskar
70
et al., 2004]. The climatic precession of Mars has a period of approximately 51 kyr. The
obliquity of Mars varies with an average period of 120 kyr due to variation of the spin
axis and is modulated by a 1200 kyr period due to variation of its orbital inclination
[Ward, 1973]. The eccentricity of Mars's orbit varies with periods of 95 kyr, 99 kyr and
2400 kyr [Laskar et al., 2004].
The evolution of Martian orbital parameters over long time intervals is chaotic
[Laskar and Robutel, 1993; Touma and Wisdom, 1993]. Given the precision with which
present-day orbital parameters can be measured, the current solution for insolation over
time [Laskar et al., 2004] is accurate for the last 10-20 Myr. We calculate insolation over
this interval from the orbital solution of Laskar et al. [2004] using methods described by
Berger [1978]. Like previous analyses of the PLDs [Laskar et al., 2002], we use the
average daily insolation at the north pole on the summer solstice (Fig. 2) as a proxy for
the climatic conditions controlling the deposition of polar ice and dust. This assumes that
the effect of the axial precession on the magnitude of ice deposition in a given year is less
important than the effect of obliquity. As noted above, our objective in this study is to
evaluate a procedure for analyzing PLD sequences with nonlinear time-depth
relationships, not to identify the exact relationship between insolation and PLD
formation, so our results do not rely on the correctness of this assumption.
The orbital solution features a significant reduction in mean obliquity, and
therefore summer insolation at the poles, after approximately 5 Ma. Paleoclimate models
suggest that polar ice caps would not have been stable before this time [Jakosky et al.,
1995; Mischna et al., 2003; Forget et al., 2006; Levrard et al., 2007], which would imply
that the PLDs exposed in the upper portions of the ice caps are younger than 5 Ma.
71
However, other studies have estimated the age of the southern PLDs to be an order of
magnitude older than this, which may be related to protective lag deposits [Banks et al.,
2010]. There is an observational constraint from crater counts that yields a maximum age
of ~I Ga on the north polar basal units [Tanaka et al., 2008], and our approach does not
depend on an estimate of the absolute age of the PLDs. In the models presented here, we
only consider the past five million years of Martian insolation history (Fig. 2).
2.2 Ice and dust accumulation
We consider three classes of PLD formation models, which are illustrated
schematically in Figure 3. Although our models are more complicated than those
originally studied by Cutts and Lewis [1982], they are not intended to capture all aspects
of the physical processes controlling ice and dust deposition rates. The key attribute of
our simple, insolation-driven models is that they produce strata with a non-linear timedepth relationship, and therefore provide a useful tool for exploring how insolation
forcing may be recorded in the PLDs. In each model, dust deposition rate fdut [L/T] is
held constant, and ice deposition ratefice [L/T] is expressed as a simple function of
insolation,
# (W/m
2
). In the first model, ice deposition ratefice (#) varies linearly with
insolation. Higher insolation corresponds to slower ice deposition. The insolation value
at which no ice is deposited (flcek() = 0) is chosen to be greater than the maximum
insolation reached in the past five million years, so thatfice(#) is always positive, and the
resulting PLDs contain no hiatuses in accumulation.
The second model is the same as the first model, except that the insolation at
whichfice(#) = 0 is chosen to be less than the maximum insolation reached in the past five
72
million years. For insolation values above this threshold,fice() = 0. Therefore, for
certain time intervals in the past five million years, no ice is deposited, and the resulting
PLDs contain hiatuses in accumulation.
The third model is the same as the second model, except thatfice() maintains its
linear relationship with insolation at all insolation values, which means that
fice(o)
is
negative for insolation values above the threshold. A negative ice deposition rate
corresponds to ablation, which destroys a previously deposited section of the PLD. The
resulting PLDs therefore contain hiatuses, as in the second model, but the hiatuses are not
limited to time intervals when insolation exceeds a threshold value. Figure 3 summarizes
the ice deposition functions for the three models. All three models can have their
parameters adjusted in order to vary the absolute values of their deposition rates. The
units of brightness and depth in the models are arbitrary, so the slopes of the trends
relating deposition rate to insolation in Figure 3 do not affect our tuning procedure.
2.3 Generationof synthetic stratigraphicsequences
For each instance of a model, the insolation time series (Fig. 2) is sampled every
1000 years, for a total of 5000 time steps. At every time step, ice and dust deposition
rates are calculated, an increment of ice is deposited using a forward Euler method, and
the dust concentration of the ice is calculated as the ratio of the dust and ice deposition
rates. This iterative procedure constructs a synthetic PLD stratigraphic sequence
consisting of a series of "beds" of unequal thickness and variable dust concentration.
Figure 4 shows examples of outputs for each model class.
73
The models make a number of simplifications. Dust is assumed to be
volumetrically negligible, on the basis of work that suggests an upper limit for dust
content of 2% by volume for the northern polar cap [Picardi et al., 2005]. Dust is
assumed to blow away during hiatuses in ice deposition, such that dust lags do not
develop in models with hiatuses or ablation. This assumption is consistent with abundant
evidence for eolian sediment transport in the north polar region [Byrne, 2009]. We
neglect topographic differences involving aspect and shadowing that could potentially
cause local variations in deposition rates, based on the observation that large-scale
stratigraphy is consistent across the polar ice caps [Phillips et al., 2008]. In this study, we
have chosen to ignore insolation-induced variations in dust deposition rate, because we
expect ice deposition to be more strongly influenced by insolation [Toon et al., 1980].
Dust deposition rate is likely to be affected by global dust storms, which may correlate
with insolation [Zurek and Martin, 1993], but in the absence of a clear expectation for the
relationship between insolation and dust, and given the evidence that atmospheric
dustiness varies considerably over intervals much shorter than the periods of orbital
changes [Zurek and Martin, 1993], the relation between insolation and ice deposition rate
is a logical starting point. Stratigraphic thickness and dust concentrations are presented
in arbitrary units, because long-term deposition rates of ice and dust are poorly
constrained, with estimates spanning three orders of magnitude [Pollack et al., 1979;
Kieffer, 1990; Herkenhoff and Plaut, 2000]. This does not pose a problem for the tuning
procedure described below, because potential detection of an orbital signal involves
consideration of the relative amplitudes and frequencies of stratigraphic signals in PLD
records rather than the absolute dust concentrations and stratigraphic distances.
74
3. Statistical Analysis
Our statistical analysis consists of two main components: a dynamic time warping
algorithm that tunes a synthetic PLD record in an effort to match the insolation function,
and a Monte Carlo procedure that evaluates the statistical significance of the match.
3.1 Orbitaltuning by dynamic time warping
Dynamic time warping (DTW) allows for the possibility that the PLDs do not
follow a linear time-depth relationship. We use a DTW algorithm proposed by Haam and
Huybers [2010] that tunes a record - stretches or contracts its time dimension
nonuniformly - to find the optimal match between the record and another time series.
The goodness of the match for a given tuning is measured by the covariance between the
tuned record and the other time series, and the optimal tuning is the one that maximizes
this covariance. In this case, the records are the synthetic PLDs, and they are tuned to
match the insolation function.
The DTW algorithm tunes the record to the forcing function by using a cost
matrix, which is constructed by computing the squared differences between each point in
the synthetic record and every point in the insolation function. The resulting matrix of
squared differences represents the costs (penalties) of all possible matches between points
in the two records. The algorithm then finds the path through the cost matrix that incurs
the lowest average cost, starting from an element that corresponds to the top of the PLD
record and the estimated time in the insolation function when the uppermost layer was
deposited, and ending at an element that corresponds to the bottom of the PLD record and
75
the time in the insolation function when the first layer was deposited. The calculated path
represents the tuned record that has the maximum possible covariance with the insolation
function. Figure 5 shows an example of an output of the DTW algorithm with both the
tuned and actual time-depth curves. The least-cost path is not required to terminate with
the earliest time in the insolation function; since most troughs only expose the uppermost
few hundred meters of stratigraphy out of a total of ~2km, it is likely that exposed
deposits only correspond to a fraction of the 5 Myr insolation function. Similarly, the
path is not required to start at the present day, because the uppermost strata may have
formed some time before the present. However, we expect that the age of the bottom of a
PLD sequence is much less certain than the age of the top, so we do not allow the starting
point of the least-cost path to vary as freely as the ending point. This is implemented by
imposing a non-zero cost on the leftmost column of the cost matrix and no cost on the
rightmost column (Fig. 5). We also impose a non-zero cost on the bottom row because a
path traveling along that row would correspond either to the unlikely scenario of a thick
layer of ice deposited instantaneously at the present day or to the unphysical scenario of
strata that are younger than the present.
3.2 Monte Carloprocedure
The DTW algorithm gives the maximum covariance between a tuned synthetic
PLD and the insolation time series, but does not assign a statistical significance to that
covariance. The procedure therefore requires an additional step that quantitatively
evaluates the null hypothesis that the PLD record is a random time series uncorrelated
with insolation, and that the maximum covariance between the PLD and insolation is no
76
better than that obtained by chance. We evaluate this null hypothesis through a Monte
Carlo procedure in which random records with statistical characteristics similar to those
of the synthetic PLDs are tuned to match the insolation function. For each synthetic
PLD, 1000 random records with the same mean, variance, and lag-I autocorrelation as
the synthetic PLD are generated. The DTW procedure then tunes each random record to
the insolation record using the same procedure applied to the synthetic PLD, yielding a
maximum covariance for each random record. A comparison of the resulting distribution
of 1000 maximum covariances with the maximum covariance between the insolation and
synthetic PLD provides a way of gauging the likelihood that the match is not spurious,
and therefore the confidence level at which the null hypothesis can be rejected. An
example is shown in Figure 6. We express this confidence level as the percentage of
random Monte Carlo records, PMc, that yield a smaller maximum covariance than the
synthetic record. If PMc = 100%, then the synthetic PLD matches insolation better than
all random records, and the orbital signal is detected in the synthetic PLD with an
extremely high degree of confidence. If PMc = 50%, then the orbital signal is so obscured
by the PLD formation mechanism that the tuned match between the PLD and insolation is
no better than the median match between a random time series and insolation, and thus
there is little confidence that the modeled stratigraphy is related to insolation. Between
these two extremes is a range of confidence levels for detection of an orbital signal. This
approach provides a way of quantifying the feasibility of detecting an orbital signal given
a hypothesized PLD formation mechanism, as well as a way of quantifying the
significance of orbital tuning applied to real PLD records, for which the formation
mechanism is unknown. Figure 7 compares dynamic time warping analyses of synthetic
77
PLD models and random time series for one case in which the covariance between
insolation and the tuned PLD is substantially higher than for the tuned random time series
(Fig. 7a,b) and another in which the PLD and random time series yield comparable
covariances (Fig. 7c,d).
4. Results
4.1 Qualitativecharacteristicsof synthetic PLD stratigraphy
Model outputs of synthetic PLD records yield noteworthy trends, even before
application of the DTW algorithm and Monte Carlo procedure. In the no-hiatus case,
where ice deposition rate varies linearly with insolation and is always positive (Fig. 4a,b),
varying the coefficient relating ice deposition rate to insolation changes the absolute
values of dust concentration in the resultant stratigraphic sequences, but not the relative
frequencies of bedding. The outcome of this simple formation model is therefore
qualitatively independent of model parameters.
The relative frequencies of bedding in models that allow hiatuses are also
insensitive to changes in the coefficient relating ice deposition rate to insolation (Fig. 4cf). However, adjusting the threshold insolation value in these models does change the
stratigraphy qualitatively, because it influences the fraction of time that is preserved.
Figure 4g,h shows two instances of the model with hiatuses but no ablation, with
different thresholds for ice deposition. Note that adjustment of this threshold changes not
only the values of dust concentration, but the number of bright peaks as well.
4.2 Detection of orbitalsignalsfor different accumulationmodels
78
As mentioned in section 3, a maximum covariance was calculated for each
synthetic PLD and was then compared to the maximum covariances obtained for 1000
randomly generated records that shared several statistical properties with the synthetic
PLD. For models with no ablation and no hiatuses (Fig. 4a,b), the maximum covariance
is close to 1 and is always greater than the maximum covariances for all randomly
generated records (PMC = 100%). Thus, for this simple formation function, we can
confidently identify an orbital signal in all cases, despite a nonlinear time-depth
relationship that would complicate or preclude detection with conventional time series
analysis methods. This result illustrates one of the main benefits of the tuning procedure,
and suggests that tuning analyses of the PLDs, combined with an appropriate statistical
test, could reveal underlying structure that conventional time series analyses have missed.
For the more complicated models that produce hiatuses (Fig. 4c-f), Pmc generally
scales with the insolation threshold for ice deposition (Fig. 8a), because higher thresholds
result in shorter hiatuses. That is, when less of the insolation time series produces strata
that are preserved, the match between the PLDs and insolation is worse, and is less likely
to be better than the match to a random record. For sufficiently high insolation thresholds
(> 225 W/m 2 for the model with hiatuses but no ablation, and > 270 W/m 2 for the model
with ablation), the maximum covariance for the model output is greater than all
maximum covariances for random records (PMc = 100%), despite incomplete
preservation of the modeled time interval (Fig. 8a). Below those threshold insolation
values, PMc decreases as the threshold is lowered. For models without ablation but with
ice deposition stopping above a threshold insolation value of 222 W/m 2 , an orbital signal
can be detected with a 95% degree of confidence. For a threshold insolation value of 174
79
W/m 2 or lower, PMC is not significantly higher than 50%, and thus the model output can
not be tuned to an orbital signal better than a random record; detection of an orbital signal
is infeasible. For models with ablation above a threshold insolation value of 269 W/m 2
an orbital signal can be detected with a 95% degree of confidence. For a threshold
insolation value of 243 W/m 2 or lower, PMc is not significantly higher than 50%, and thus
the model output cannot be tuned to an orbital signal better than a random record;
detection of the signal is infeasible. For a threshold value of 210 W/m 2 or lower, no PLD
record exists - it is all ablated away.
We find that this relationship can be generalized by plotting PMC as a function of
the fraction of time preserved in the stratigraphy (Fig. 8b). For the formation models
investigated here, the modeled PLDs can be distinguished from random time series (PMc
> 50%) even if only a few percent of the modeled time interval is preserved in the
stratigraphy, and can be confidently distinguished (PMC > 90%) if approximately 8-10%
of the time interval is preserved. Between these extremes, Pmc increases approximately
linearly with the fraction of time preserved.
We also examined the influence of the total duration of PLD accumulation on the
ease of identifying an orbital signal. In addition to the insolation time series for the past 5
Myr (Fig. 2), we drove the model that allows ablation with the insolation for the past 3
Myr and the past 1 Myr, and performed the same statistical analysis on the model
outputs. The results in Fig. 7 demonstrate that, in addition to the dependence on
insolation threshold, PMC is higher when the total accumulation interval is longer:
depositing the PLDs over a longer period of time makes it easier to detect an orbital
influence.
80
5. Discussion
5.1 Feasibilityof identifying an orbitalsignal through tuning
In general, our results imply that detection of an orbital influence on PLD
formation is feasible (though not trivial), even if the relationship between depth and time
in the stratigraphy is strongly nonlinear. Indeed, we find that PLD sequences formed by
ice and dust deposition models that include no hiatuses in deposition can be distinguished
from stochastic time series 100% of the time. While such a deposition model is probably
overly simple (see section 5.3), this result nonetheless emphasizes that a nonlinear timedepth relationship is not an insurmountable complication.
In the more likely scenario that the PLD stratigraphy contains gaps, our analysis
provides a framework for determining whether the accumulated record contains enough
information to reliably identify orbital influence. Features such as unconformities and
crosscutting troughs suggest that the accumulation of the polar stratigraphic record was
punctuated by periods of no ice deposition [Tanaka et al., 2008]. In models with hiatuses
or ablation, the ability to detect orbital signals is a function of the threshold insolation at
which ice deposition stops. This result makes intuitive sense: when more of a PLD
record is ablated away, it is more difficult to detect the underlying forcing that drove PLD
formation. Our procedure identifies a clear, roughly linear relationship between the ease
of identifying an orbital influence, as measured by PMc, and the insolation threshold for
ice deposition in each model (Fig. 8a). However, these particular values of the insolation
threshold should not be interpreted as absolute, because the true relationships between
insolation and ice and dust deposition rates are unknown. Instead, we emphasize that the
fraction of time preserved in the stratigraphy is the more relevant quantity for
81
determining whether an orbital signal can be confidently detected. The clearest
demonstration of this point is that the trends in PMC for the different models collapse to a
more uniform trend when plotted against fraction of time preserved (Fig. 8b) rather than
the threshold insolation (Fig. 8a).
The other main factor that influences the ease of detecting orbital influence is the
total duration of PLD formation. In general, the shorter the time period over which the
PLDs form, the more difficult it is to detect an orbital signal in the stratigraphy (Fig. 9).
This too makes intuitive sense: a stratigraphic sequence that preserves 50% of five
million years contains more information than a sequence that preserves 50% of one
million years, and the additional information makes it easier to distinguish the orbitally
driven record from a random record.
5.2 Fractionof time preserved in the polarcap stratigraphy
Although the northern polar cap of Mars is thought to have experienced net
accumulation of ice over the past few Myr [Pollack, et al., 1979; Kieffer, 1990; Laskar et
al., 2002], it is unclear whether the cap is presently in a state of net accumulation or net
ablation. If we assume that Mars is in a state of net ablation today, then our models
suggest that the current PLDs represent only a small fraction (< 10%) of the total record
deposited over time. The current insolation at the Martian north pole during the summer
solstice, 265 W/m 2, is near the mean insolation for the past 5 Myr of Martian history (Fig.
2). Thus, if the PLDs are ablating today, it is likely that they have ablated more often than
they have accumulated, and their strata may only record a small fraction of the past 5
Myr. It should be noted, however, that these models assume ablation occurs at a similar
82
rate to ice deposition. If ablation is much slower than ice deposition (which might be the
case if, for example, ablation forms a dust lag that inhibits further ablation), the PLDs
could record a larger portion of recent Martian history, even if the caps are experiencing
net ablation today.
5.3 Additional considerationsfor modeling PLDformation
The objective of this study is to identify the main factors that influence the
viability of orbital tuning applied to the PLDs. We therefore have not attempted to
formulate a model for PLD accumulation that incorporates all the factors that influence
the appearance of the stratigraphy, nor have we attempted an absolute calibration of rate
parameters. Nonetheless, given the finding that orbital tuning may indeed be a viable
means of identifying the cause of paleoclimate signals preserved in the PLDs, it is
important to consider the limitations of, and possible improvements to, the simple models
presented here.
Several improvements could be implemented to make the PLD formation models
more realistic. In particular, both ice and dust deposition rates could be expressed in
terms of a fuller complement of physical variables. Ice deposition rates could take
humidity into account. Dust deposition rates could consider the occurrence of global dust
storms, which historical observations [Pollack et al., 1979; Toon et al., 1980; Haberle,
1896; Zurek and Martin, 1993] suggest produce a high frequency signal, but which may
also include long-term trends related to insolation [Fernandez, 1998]. These additional
complexities will almost certainly make detection of an orbital signal more difficult, and
83
thus the confidence in detection abilities presented in this study should be interpreted as
an upper limit.
Other potential complications are the possibility of stochastic variability in
deposition processes and the imperfect relationship between PLD composition and
appearance. To explore how these factors influence the orbital tuning procedure, we
performed an additional analysis in which the modeled ice deposition rate includes a
stochastic component. Specifically, we added red noise (a random signal in which
spectral power P declines with frequencyf according to P oc f
2
) to the amount of ice
deposited in a given time step in our models to generate synthetic PLDs that are not
constructed with the assumption of a deterministic relationship between ice deposition
rate and insolation. Starting with a model that forms hiatuses when the insolation is 300
W/m2 or greater, we varied the amplitude of the noise and produced 100 random
realizations of the PLD strata for each value of noise amplitude. We then used the DTW
algorithm to calculate the maximum covariance between each modeled stratigraphic
sequence and the insolation time series. Figure 10 shows how the maximum covariance
depends on the amplitude of the noise. The addition of red noise to the ice deposition
rate changes the maximum covariance in a gradual fashion, suggesting that a nondeterministic relationship between insolation and PLD accumulation does not necessarily
prevent the DTW method from identifying an orbital signal.
5.4 Implicationsfor orbitaltuning of the observed PLD stratigraphy
Given the probable influence of insolation on the deposition or ablation of water
ice, the major constituent of the PLDs, it is likely that the relationship between time and
84
depth in the PLDs is nonlinear, as our simple models predict. One of the main
implications of our results is that it may nonetheless be possible to identify evidence of
quasi-periodic insolation forcing by applying a tuning procedure like the one described
here. Such an analysis could reveal coherent signals in the PLD stratigraphy that would
not be detected by conventional time series analysis procedures that assume a linear or
nearly linear time-depth relationship [Perron and Huybers, 2009].
The appropriate future direction of this study is to apply the statistical analysis
described here to actual images of the Martian PLDs. Images obtained by the Mars
Orbiter Camera (MOC) on the Mars Global Surveyor spacecraft and the High Resolution
Imaging Science Experiment (HiRISE) aboard the Mars Reconnaissance Orbiter can be
converted to sequences of brightness vs. depth that can be analyzed with the same
procedure as the synthetic sequences of dust concentration studied here [Milkovich and
Head, 2005; Milkovich et al., 2008; Perron and Huybers, 2009; Fishbaugh, 2010; Limaye
et al., 2012]. These sequences should be compared to insolation records of varying time
spans, so that we do not assume a certain total age for the sequences from the outset, and
with the important consideration that, in the best-case scenario, a match would still only
determine the age of the exposed sequence. Such an age would be younger than that of
the entire PLDs, which would need to be extrapolated. This procedure can determine if a
time-uncertain PLD sequence matches an insolation time series better than random
records, but it cannot confirm that such a match reveals the true PLD chronology. In
particular, if a PLD sequence containing quasi-periodic signals [Perron and Huybers,
2009; Limaye et al., 2012] is tuned to match an insolation record composed of multiple
quasi-periodic signals, there is a possibility that the periods in the sequence will be tuned
85
to match the wrong periods in the forcing. In practice, a procedure such as ours should
be applied to PLD records in concert with other lines of evidence [Hinnov, 2013],
including climate models [Levrard et al., 2007], polar cap and trough formation models
[Smith et al., 2013], and regional stratigraphic analyses, to identify a PLD chronology
that is statistically probably and takes into account the relevant geological constraints. It
should be noted that conversion of images to brightness-depth sequences introduces an
additional source of noise that must be considered [Tanaka, 2005], but recent efforts to
quantify these uncertainties have found them to be modest [Limaye et al., 2012]. The
dynamic time warping procedure we have applied to brightness records can in principle
be applied to other proxies for PLD composition, such as sequences of slope or roughness
vs. depth, or composite records incorporating both brightness and topographic
information. Thus, for any possible identification of an orbital signal in the PLDs, the
statistical procedure presented here can yield a quantitative estimate of the likelihood of a
spurious match. If the PLDs preserve a sizeable fraction of the total accumulation time,
and the deposition rates of ice and dust are sufficiently deterministic, it may well be
possible to detect an orbital signal, if one is present.
6. Conclusions
We use a statistical procedure that evaluates the significance of time series tuning
to examine the feasibility of detecting an influence of orbital variations on the polar
stratigraphy of Mars. We apply the procedure to synthetic stratigraphic sequences
generated by simple formation models for the Martian polar layered deposits, and find
that detection of an orbital signal in the resulting stratigraphy is feasible, though not
86
trivial. Models in which ice deposition rate varies linearly with insolation produce
stratigraphy in which orbital signals are easily detected with the tuning procedure, despite
a nonlinear relationship between depth and time that can foil conventional time series
analysis methods. For more complicated models of ice deposition, detection ability
depends strongly on the threshold insolation at which ice deposition stops or an ablation
episode begins, and more generally, on the fraction of total formation time preserved in
the strata. Improved constraints on ice and dust deposition rates on Mars would permit a
more definitive assessment of whether detection of an orbital signal in the PLDs is
feasible, but our analysis does not reveal the problem to be necessarily intractable at the
current state of knowledge. HiRISE images should be adequate to identify evidence of
an orbital influence if PLD formation is controlled by a sufficiently simple mechanism
and sufficient time preserved. We find that when too little time is preserved in the
stratigraphy, confident identification of an orbital signal may be impossible without
absolute ages, even given simple formation scenarios and no matter the quality of the
spacecraft images.
Acknowledgments
This study was supported by the NASA Mars Data Analysis Program, award 65P1089493. We thank Shane Byrne and an anonymous referee for their suggestions.
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299-303.
90
100 m
Figure 1. Mars Orbiter Camera (MOC) Image #MO001754 of a PLD stratigraphic sequence,
corrected for topography. The vertical scale corresponds to vertical depth within the PLD
sequence, and the horizontal scale corresponds to distance along the outcrop.
91
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Time before present (kyr)
Figure 2. Martian insolation over the past five million years at the north pole on the summer
solstice, calculated from the orbital solution of Laskar et al. [2004].
92
C
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Insolation (W/m2
Figure 3. Ice deposition rate (arbitrary units) as a function of insolation for the three models
considered. Model 1 (solid black line) is a simple linear dependence of deposition rate on
insolation, with no hiatuses in deposition. Model 2 (dotted line) allows ice deposition rate to drop
to zero at high insolation values, creating hiatuses. Model 3 (solid gray line) allows ice
deposition rate to become negative at high insolation values, causing alabation of existing layers.
93
Model 1
a
c
b
4-
Model 2
d
CL
Brightness (arbitrary units)
Brightness (arbitrary units)
e
Mode 3
f
Model 2, thresh. 200 W/m 2
g
Model 2, thresh. 250 W/m 2
h
C
.0
-C
0.
-c
CB
0
Brightness (arbitrary units)
Brightness (arbitrary units)
Brightness (arbitrary units)
Figure 4. Examples of synthetic PLD stratigraphic sequences produced by the three model
classes. Plots in (a,c,e) show dust concentration in arbitrary units as a function of depth in
arbitrary units. Images in (b,d,f) show simulated images of the stratigraphy (compare with Fig. 1)
created by assuming that brightness scales inversely with dust concentration and adding Gaussian
noise. The third model class (e,f), which includes ablation, produces synthetic PLDs most
visually similar to actual images. Plots in (g,h) were both produced by the model with hiatuses
and no ablation, but with different values of the threshold insolation for ice accumulation: 200
W/m 2 in (g), 250 W/m 2 in (h).
94
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Figure 5. Output from the dynamic time warping algorithm comparing (a) the last five million
years of Martian insolation history to (b) a synthetic PLD sequence. Both time series are
normalized to unit variance. In the model, ice deposition stops (but without ablation) above a
threshold insolation of 350 W/m2 . The square region (c) corresponds to the cost matrix. The black
line in (c) shows the path through the cost matrix that incurs the lowest average cost, and
represents the tuned synthetic PLD. The colors represent cost, with warm colors indicating areas
of higher cost and cool colors indicating areas of lower cost. The dashed line in (c) is simply the
diagonal of the cost matrix, which represents a linear time-depth relationship. The gray line in (c)
represents the true time-depth relationship for this synthetic PLD. The covariance for this tuning
is 0.963 despite the hiatuses in deposition.
95
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Maximum Covariance
Figure 6. Histogram showing the distribution of maximum covariances for random records
generated from a synthetic PLD where ablation occurs at a threshold insolation value of 270
W/m2. The maximum covariance for the synthetic PLD tuned to the insolation record is 0.368
(shown here as the vertical black line), which is greater than 97.2% of the maximum covariances
of the random records. We consider this a confident detection of the orbital signal.
96
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Figure 7. Cost matrices, as shown in Fig. 5, for four different dynamic time warping analyses.
Plot (a) shows a synthetic PLD formed over 5 Myr where ablation occurs at a threshold insolation
value of 350 W/m2 tuned to a 5 Myr insolation signal. Plot (b) shows a corresponding random
PLD tuned to the same signal. Plot (c) shows a synthetic PLD formed over 5 Myr where ablation
occurs at a threshold insolation value of 250 W/m2 tuned to a 5 Myr insolation signal. Plot (d)
shows a corresponding random PLD tuned to the same signal. Note that the tuning in plot (a) is
significantly better than that in plot (b), but there is no significant difference between plots (c)
and (d). Solid black lines are the tuned time-depth relationships, gray lines at the true time-depth
relationships of the synthetic PLDs, and colors represent higher (warm colors) and lower (cool
colors) costs, as in Fig. 5.
97
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Figure 8. Percentage of randomly generated time series, PMC, with insolation covariance that is
smaller than the insolation covariance of a modeled PLD sequence, as a function of (a) the
threshold insolation, and (b) the fraction of the 5 Myr time interval that is preserved in the
modeled stratigraphy. Trends for the model with hiatuses but no ablation and the model with
hiatuses that do include ablation differ when PMC is compared with the magnitude of the
insolation threshold for ice accumulation (a), but overlap when PMC is compared with the fraction
of time preserved (b).
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Figure 9. Percentage of randomly generated time series, PMC, with insolation covariance that is
smaller than the insolation covariance of a modeled PLD sequence, as a function of the insolation
threshold for ice accumulation in the model that allows ablation. Different symbols correspond to
models in which PLDs are deposited over the past 5, 3, or 1 million years of Martian history.
99
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Magnitude of red noise
Figure 10. Effect of adding red noise to the modeled ice deposition rate on the dynamic time
warping algorithm's ability to tune the resulting synthetic PLD to the insolation signal. The
magnitude of the noise added is the ratio of the variance of the noise to the variance of the ice
deposition rate. Each point represents the average maximum covariance of 100 different tunings
of a synthetic PLD that contains hiatuses when the insolation reaches a value of 300 W/m2 or
greater. Error bars are one standard deviation.
100
Chapter 5: Dynamic Time Warping of the Martian Polar
Layered Deposits
This research was conducted in collaboration with J. Taylor Perron, Elizabeth A.
Bailey, Peter Huybers, Oded Aharonson, and Ajay B. S. Limaye.
ABSTRACT:
Sedimentary deposits of ice and dust at the north and south poles of Mars have been
hypothesized to record climate change driven by variations in the distribution of
insolation resulting from changes in the planet's orbit over time. Previous studies
have analyzed images of these polar layered deposits (PLDs) from orbital spacecraft
in an effort to identify such an orbital signal, but estimates of their accumulation
rates have spanned orders of magnitude. Here, we use a method we previously
presented called dynamic time warping to study images taken from the Mars
Orbiter Camera (MOC) and High Resolution Imaging Science Experiment (HiRISE)
and compare them to the insolation history of the Martian poles. We obtain
estimates of ages and accumulation rates of the deposits. Analysis of MOC images
yields a mean accumulation rate of 1.4 1.1 mm/yr for the north PLDs. Analysis of
HiRISE images yields a mean accumulation rate of 0.47 0.12 mm/yr for the north
PLDs and 0.18 0.02 mm/yr for the south PLDs. If these rates are representative of
the entire PLD sequences, then the north PLDs are -4 Ma and the south PLDs are
-17 Ma. However, caution must be exercised in extrapolation of accumulation
rates, as the variability we find between different images suggests that the link
between insolation and stratigraphy is complex, and that PLD formation and
modification may also include other mechanisms.
101
1. Introduction
The poles of Mars are covered with kilometers-thick domes of ice, which
largely consist of sedimentary deposits formed from atmospherically deposited
water ice and dust [Murray et al., 1972; Cutts, 1973; Howard et al., 1982; Byrne,
2009]. These polar layered deposits (PLDs) can be seen in exposures caused by
spiraling troughs that cut into the polar caps. It has been hypothesized by many
authors that the PLDs record past Martian climate. Specifically, most propose that
the variability in composition (i.e., relative weight of dust versus ice) between layers
is controlled by changes in the Martian orbit over time [Murray et al., 1973; Cutts et
al., 1976; Toon et al., 1980; Cutts and Lewis, 1982; Howard et al., 1982; Thomas et
al., 1992, Laskar et al., 2002; Milkovich and Head, 2005; Milkovich et al., 2008;
Fishbaugh et al., 2010b; Hvidberg et al., 2012]. Precession and quasi-periodic
variations in obliquity and eccentricity affect the magnitude of insolation at the
Martian poles [Ward, 1973; Touma and Wisdom, 1993 Laskar et al., 2004],
hypothetically changing the deposition rate of ice or dust in a given year. Similar
stratigraphy of ice and dust is seen on a large-scale throughout the northern ice cap,
implying that the PLDs do record a global or regional climate signal rather than a
local one [Phillips et al., 2008].
Analysis of the Martian PLDs first requires extraction of stratigraphy data
from orbital spacecraft images. The deposits were first seen from the Visual
Imaging System on the Mariner 9 spacecraft [Murray et al., 1972], and later the
Viking orbiters. More recently, the Mars Orbiter Camera (MOC) [Malin et al., 2010]
aboard the Mars Global Surveyor (MGS) and the High Resolution Imaging Science
102
Experiment (HiRISE) [McEwen et al., 2007] aboard the Mars Reconnaissance Orbiter
(MRO) have provided hundreds of images of the poles, including the PLDs. The
degree to which a PLD image is accurately representative of the corresponding
stratigraphy is dependent upon the resolution of and errors associated with the
imaging system.
Data from MOC images [Malin et al., 2010] have typically been combined with
topographic information from the Mars Orbiter Laser Altimeter (MOLA) [Zuber et
al., 1998] to produce elevation-corrected stratigraphic profiles of the PLDs. Many
previous studies have analyzed such profiles in attempts to identify periodicities
that might be the signature of orbitally-induced climate change. Laskar et al. [2002]
tuned a brightness profile of a MOC image in the north polar cap to insolation
history at the Martian pole and inferred a deposition rate of 0.5 mm/yr for the
upper 250 m of PLDs, but did not evaluate the statistical significance of their result
and only analyzed a single profile. Milkovich and Head [2005] analyzed multiple
brightness profiles of the north PLDs (NPLDs) and reported the presence of a 30 m
wavelength signal, interpreting it as being sourced from the -51 kyr cycle of
precession, but they assumed a linear time-depth relationship. Perron and Huybers
[2009] also assessed periodicity in the NPLDs using spectral techniques and found
only a -1.6 m wavelength signal, and raised the important question of whether the
PLD brightness spectra differed significantly from that of red noise. Analysis of the
south PLDs (SPLDs) has been less common, but Bryne and Ivanov [2004] measured
beds in Australe Mensa and Milkovich et al. [2008] reported a 35 m wavelength
signal at one SPLD location.
103
Images from HiRISE [McEwen et al., 2007] typically have resolution of -25
cm/pixel, and can be combined in stereo pairs to generate digital elevation maps
(DEMs) with vertical accuracy -30 cm [Kirk et al., 2008; Lewis et al., 2008;
Fishbaugh et al., 2010a; Limaye et al., 2012]. HiRISE-generated finely resolved
stratigraphic columns have been published by Fishbaugh et al. [2010b] and Limaye
et al. [2012]. Fishbaugh et al. [2010b] found thin layers with average separation of
1.6 m, but concluded that any relationship between layer formation and orbitallyinduced climate forcing must be complex. Limaye et al. [2012] analyzed bed
thicknesses in both the NPLDs and the SPLDs, finding a statistically significant
degree of regularity in the north but not the south. Hvidberg et al. [2012] modeled
NPLD formation, showing that their synthetic PLD sequences were consistent with
the HiRISE-generated column of Fishbaugh et al. [2010b].
A major goal of PLD analysis is to constrain the ages of the deposits, but
previous estimates of average deposition rates have spanned orders of magnitude
[Pollack et al., 1979; Kieffer et al., 1990; Herkenhoff and Plaut, 2000; Laskar et al.,
2002]. Analysis of impact crater populations imply an upper limit on accumulation
rate of 4 mm/yr on the NPLDs [Banks et al. 2010], and Martian climate models
suggest that the deposits would be unstable before 5 Ma as a result of higher mean
obliquity [Jakosky et al., 1995; Mischna et al., 2003; Forget et al., 2006; Levrard et al.,
2007]. However, caution is needed when considering these constraints, as the
SPLDs appear to be an order of magnitude older than 5 Ma on the basis of crater
counting [Levrard et al., 2007; Banks et al., 2010]. Protective lag deposits may
shield ice from ablation during periods of high insolation [Banks et al., 2010], a
104
phenomenon that has also been proposed for other planets [Spencer and Denk,
2010].
A potential major obstacle has been the assumption of a linear time-depth
relationship in the stratigraphy. This is known to be a rare occurrence in terrestrial
paleoclimate records [Sadler, 1981; Weedon, 2003], and indeed ice deposition rates
vary significantly within single PLD sequences [Fishbaugh and Hvidberg, 2006].
Haam and Huybers [2010] presented a technique called dynamic time warping for
comparing any two time-uncertain series and applied it to terrestrial paleoclimate
records. Sori et al. [2014] adapted that technique for study of the Martian PLDs,
showing that it had the capability to identify an orbital signal in the deposits even
when the time-depth relationship was nonlinear, as long as enough time was still
preserved in the stratigraphy.
In this paper, we use that technique to analyze the Martian PLDs. Our study
has three major components. First, we extract stratigraphic profiles of brightness as
a function of depth from MOC and HiRISE images of the PLDs. Second, we use
dynamic time warping to tune those profiles to the past few Myr of insolation
history at the Marian poles. Third, we use dynamic time warping to also tune
randomly generated stratigraphic records to the insolation history in order to
evaluate the statistical significance of our results. We discuss the implications of our
results as they relate to the age and deposition rates of the PLDs.
105
2. Stratigraphy
In our analysis we consider stratigraphic brightness profiles generated from
both MOC and HiRISE images. MOC images correspond to topographic profiles from
MOLA for a large number of locations, while HiRISE images offer superior
resolution. Note that for both HiRISE and MOC, the PLDs visible in the image only
represent a fraction of the stratigraphy of the entire PLD sequence. Typically,
exposed PLDs are hundreds of meters in depth, while the entire sequence is 2 km
thick in the north polar cap and 3 km thick in the south polar cap.
2.1 MOC Images
MOC consists of three instruments: a narrow angle camera, a wide angle
camera with a red filter, and a wide angle camera with a blue filter. We use images
from the narrow angle camera, which typically have a spatial resolution between 1.5
an 12.0 m per pixel [Malin et al., 2010].
We have collected 30 MOC images that span a large area of the polar cap and
reveal the NPLDs in gentle slopes eroded into the surface by spiraling troughs. We
use MOLA topographic profiles to correct each of these images for topography,
which yields a vertical profile of image brightness. This technique has been used in
several earlier studies of MOC images [Laskar et al., 2002; Milkovich and Head,
2005; Perron and Huybers, 2009]. The slope of the trough walls is generally only a
few degrees; this translates the 300 m spacing between MOLA shots to an
equivalent -23 m vertically.
106
Of the 30 images, we eliminate 7 from consideration on the basis of
inadequate image quality or severe curving of the deposits. The remaining 23
images form our dataset. For each image, we extract brightness profiles as a
function of vertical depth from the stratigraphic top of each image to its bottom,
tracing a profile perpendicular to the layering. An example of one of these 23
images is shown in Figure 1.
2.2 HiRISE Images
HiRISE is a high resolution camera capable of resolutions of -25 cm per pixel
[McEwen et al., 2007]. Where there exist two HiRISE images of the same surface
coverage from different viewing angles, they can be combined into a DEM with
vertical precision in the tens of centimeters [Kirk et al., 2008].
Our dynamic time warping technique does not require a linear time-depth
relationship [Sori et al., 2014], so it is not imperative to correct HiRISE images for
topography. Elevation correction creates a uniformly-spaced vertical stratigraphic
sequence, but also introduces error in interpolating for brightness values at
elevation for which there exist no direct measurements. In the interest of
minimizing that error and maximizing the number of HiRISE images available for
analysis, we choose not to topographically correct our brightness profiles that are
extracted from HiRISE images. We tested the analysis for one image with both
options (elevation corrected, and not), and found that it did not alter the final
results significantly. It is, however, still desirable to use HiRISE images with
107
associated DEMs where possible in order to obtain a more accurate measurement of
the stratigraphic height of the entire profile extracted.
We have found four HiRISE images of the PLDs that are suitable for analysis.
Three of these are in the north polar cap, and one is in the south. For each image, we
extract five parallel brightness profiles as a function of vertical depth from the
stratigraphic top of each image to its bottom, tracing a profile perpendicular to the
layering. The HiRISE images we analyzed are shown in Figure 2.
3. Dynamic Time Warping
The dynamic time warping method is described as a general application in
Haam and Huybers [2010] and as a specific application to the Mars PLDs in Sori et
al. [2014], but we summarize the important ideas here. The method tunes one time
series by stretching or contracting its time dimension non-uniformly to find the
optimal match with a second time series. In our case, the two times series are
profiles of brightness as a function of depth and the insolation history of Mars,
respectively. The optimal match is determined by construction of a "cost matrix,"
which is a matrix of the squared differences between each point in the normalized
PLD record and each point in the normalized insolation record. The dynamic time
warping algorithm then finds the path through the cost matrix which incurs the
lowest cost, starting at a point (the lower left corner) that corresponds to the top of
a stratigraphic profile and the present day value of insolation and ending at a point
(the upper right corner) that corresponds to either the bottom of that profile and
the oldest value of insolation in the interval. This path corresponds exactly to a
108
specific time-depth relationship in the stratigraphy and represents the optimal
tuning between the two time series. See Sori et al. [2014], Figure 5, for more details.
The insolation records we use are constructed in thousand year time steps.
Each point corresponds to the insolation at summer solstice for that year, at a
latitude of 90 degrees when analyzing the NPLDs and -90 degrees when analyzing
the SPLDs. This choice implies that obliquity variations are more important in
controlling deposition of the PLDs than axial precession. Our results are not
sensitive to latitude, as insolation does not vary greatly in the range of latitudes at
which polar ice exists. The records are calculated from the orbital solution of Laskar
et al. [2004] using the methods of Berger [1978].
For a given extracted brightness profile and a given insolation record, we
construct a cost matrix. Since the age of any given brightness profile (and the age of
the entire PLD sequence) is unknown, we alter the cost matrix as described above in
two important ways: we set the top row of costs and the rightmost column of costs
equal to zero. Thus, the lowest cost path is effectively allowed to terminate at the
top row (which corresponds to the oldest value of insolation) or the rightmost
column (which corresponds to the lowest piece of stratigraphy).
As an example, consider a piece of stratigraphy in which the youngest layer
was deposited in the present day and the oldest layer was deposited 4 Myr ago. If
an extracted brightness profile is a good proxy for composition, and the stratigraphy
was formed as the result of orbitally-driven insolation changes, then when one uses
dynamic time warping to tune that brightness profile to the last 4 Myr of insolation
the optimal path will terminate in exactly the upper right corner of the cost matrix,
109
even with the cost entries in the top row and rightmost column set to zero. If one
instead tunes the brightness profile to the last 2 Myr of insolation, then the optimal
path will terminate somewhere along the top row; the algorithm is matching a piece
of stratigraphy that is not at the bottom to the oldest year considered. Similarly, if
one tunes the profile to the last 6 Myr of insolation, then the path will terminate
along the rightmost column; the algorithm is matching the bottom piece of
stratigraphy to a year that is younger than the oldest year considered. In essence,
when the optimal path terminates at exactly the corner, the method is stating that
the insolation interval considered was a good fit. When the path terminates along
the top row or rightmost column, the insolation interval was too short or too long,
respectively.
In practice, brightness is not a perfect proxy for composition, and thus the
optimal path might not end precisely at the upper right corner of the cost matrix
even if the true insolation interval is considered. We adopt the following metric to
quantify tunings of brightness profiles with insolation. For a given tuning, we
consider the "fractional path displacement" (FPD), which we define as the fractional
distance along the top row or rightmost column that the optimal path terminates
away from the corner (if the path terminates at the upper right corner, FPD = 0; if
the path terminates halfway along the top row or rightmost column, FPD = 0.5, etc.).
For a single PLD brightness profile, we tune to insolation time series of various
durations. The insolation interval that yields the strongest match (lowest FPD) is
considered the best fit to the stratigraphic sequence, and thus gives an age for that
sequence, if shorter and longer insolation intervals yield progressively weaker
110
matches (greater FPDs). An example illustrating the function of cost matrices and
FPDs is shown in Figure 3 using a synthetically generated PLD sequence using the
procedure described in [Sori et al., 2014].
Using an insolation interval that begins in the present day assumes that the
uppermost layer in the PLDs corresponds to the present day, or very near the
present day. However, if a polar cap is currently experiencing net ablation, the
uppermost layer will have been deposited at some point in the past. Thus, we also
consider the FPDs of records of insolation in which the youngest year is some
thousands or millions of years ago.
The procedure described above yields an estimate for the age of a
stratigraphic sequence if insolation forcing forms that sequence, but it does not
attach a statistical significance to that age. There is a chance that any tuning
procedure, including dynamic time warping, may produce a spurious match
between two time series that are in truth unrelated; this risk has been explicitly
documented in the case of terrestrial climate records [Proistosescu et al., 2012]. As
a way of considering the null model that the PLDs and insolation are unrelated, we
compare tunings of a stratigraphic sequence to insolation with tunings of randomly
generated sequences to insolation. Such an approach was suggested by Sori et al.
[2014], and a similar method was used to correlate different PLD sequences with
one another by Milkovich et al. [2008] to test variability of sequences throughout
the cap.
We perform a Monte Carlo procedure to evaluate the null hypothesis
described above. For each brightness profile that yields an estimated age and
111
deposition rate when tuned to insolation, 1000 random records with the same
mean, variance, and lag-1 autocorrelation are generated. The dynamic time warping
procedure tunes each of these random records to the insolation interval that
produced the lowest FPD. A comparison of the distribution of maximum
covariances of the random records with the maximum covariance between the
actual sequence and the insolation record provides a way to gauge the likelihood
that the match found was not spurious. We express this likelihood as the
percentage of random records, PMc, that yield a smaller maximum covariance than
the actual stratigraphic sequence. If Pmc= 100%, then the orbital signal was
detected with an extremely high degree of confidence; if PMc = 50%, then the orbital
signal was detected with very little confidence, as the match between the sequence
and insolation is not stronger than the match between the average random signal
and insolation.
4. Results
We tuned each of the 23 MOC images to successively longer durations of
insolation starting at the present day, and 17 yielded fits to insolation time intervals
that were better than the fits of 80% of the random records. These fits yield ages
between 0.1 and 1.2 Myr, representing stratigraphy between 120 m deep and 571 m
deep. The average accumulation rate was 1.4 mm/yr. A histogram of the
distribution of all the resulting accumulation rates is shown in Figure 4, and the data
for each MOC image are shown in Table 1.
112
We tuned five profiles from each of the four HiRISE images (Figure 2) to
successively longer durations of insolation starting at the present day. An example
of the fitting of one such profile is shown in Figure 5. Data for these profiles
showing the ages, depths, and accumulation rates associated with each profile is
shown in Table 2. Errors in the inferred ages are the uncertainties in where the
minimum value occurs in the FPD plots (Figure 5).
We also considered insolation intervals that do not begin at the present day,
which is equivalent to assuming that the uppermost layer of the PLD was not
formed in the past thousand years. An example of tuning of a HiRISE profile to such
intervals is shown in Figure 6.
5. Discussion
Analysis of the MOC images yields a mean accumulation rate of 1.4 t 1.1
mm/yr for the NPLDs. Analysis of HiRISE images yields a mean accumulation rate of
0.47 t 0.12 mm/yr for the NPLDs and 0.18 t 0.02 mm/yr for the SPLDs, where the
uncertainties are one standard deviation of the accumulation rates for different
profiles analyzed. We emphasize that these accumulation rates only apply to the
upper hundreds of meters that were analyzed. However, if we assume that the
accumulation rates are representative of the entire sequence, then extrapolating the
rates to the entire PLD sequence yields a total age of ~ 4 Myr for the NPLDs and -17
Myr for the SPLDs. Our analysis is thus consistent with previous studies that have
shown the SPLDs are much older than the NPLDs [e.g., Herkenhoff and Plaut, 2005].
113
Though our accumulation rates are of the same order of magnitude for the
NPLDs, an obvious question to ask is why they vary at all from site to site. Three
possibilities are that (1) whereas the PLDs are controlled by orbitally-induced
insolation forcing, that is not the only formation mechanism involved, (2) that
brightness is not a perfect proxy for composition and (3) there are errors associated
with the images that prevent a perfect extraction of brightness from the data. It is
likely that all three are true to some degree. Figure 7 shows the difference in
accumulation rate between each of the 17 MOC images. There is no correlation with
distance; closer PLD sequences do not systematically have more similar
accumulation rates. This implies that sequences are not representative of a smaller
climate signal that operates in regions 100s of km in size.
MOC images yield accumulation rates for the NPLDs ranging from 0.4 mm/yr
to 4.2 mm/yr, while HiRISE images yield accumulation rates for the NPLDs ranging
from 0.3 mm/yr to 0.7 mm/yr. HiRISE results fall within MOC results, but a T-test
reveals that the two results yield different mean accumulation rates with a 95%
confidence level. One possibility for this is that MOC images have enough resolution
to observe most, but not all, of the layers in the PLDs that HiRISE observes. Another
is simply an issue of sample size; that the few MOC images that yield high
accumulation rates are anomalous (note, in Figure 4, that the distribution is not
normal).
Our analysis has yielded estimates of ages and accumulation rates for 37 PLD
sequences. However, these numbers vary if we relax the assumption that the top
layer of the PLDs represents the present day. In the example shown in Figure 6, we
114
find the sequence represents 470 kyr is we assume the top if the present day,
between 300 and 600 kyr if we assume the top is no older than 0.5 Ma, and between
300 and 1200 kyr if we make no assumptions about the age of the top. Analysis on
other HiRISE profiles yields similar variability. Other studies that can determine if
the PLDs are experiencing net deposition or ablation in the present day would be
helpful in elucidating this issue.
6. Conclusions
We have applied a dynamic time warping procedure to 23 MOC images and
20 profiles from 4 HiRISE images of the PLDs. Our method results in an
accumulation rate of 1.4
1.1 mm/yr for the NPLDs based on MOC images, 0.47 L
0.12 mm/yr for the NPLDs based on HiRISE images, 0.18
0.02 mm/yr for the SPLD
based on a single HiRISE image. We conclude that the orbitally-induced insolation
forcing is partly responsible for formation of the PLDs, but variability in
accumulation rates between images, and even different sequences extracted from
the same image, implies that the relationship between insolation and ice and dust
deposition is complicated, and that there are factors other than insolation in play.
Furthermore, brightness is likely not a perfect proxy for composition, which also
introduces error into our result. Future work estimating the present day
accumulation rate of the sequences in both the north and the south would be a
useful constraint.
115
Acknowledgments
This study was supported by the NASA Mars Data Analysis Program, award 65P1089493.
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118
A
E
0
Lr)
Figure 1. MOC Image #M0002072 of an NPLD stratigraphic sequence, corrected for
topography. The vertical depth of the sequence is 350 m.
119
Figure 2. HiRISE images analyzed: images #PSP_009969_2630 (NP1), #ESP_018910_2625
(NP2), #PSP_001580_2630 (NP3), and #ESP_013896_1075 (SP1). NP1, NP2, NP3 are all
from the north polar cap and have associated HiRISE generated DEMs; SP1 is from the south
polar cap and does not have a HiRISE generated DEM. The dashed lines represent the
brightness profiles we extracted to analyze which are each -2km in length; data for these
profiles are listed in Table 2.
120
W-
E
'~Z' \\
a
I
/
0.4
C
b
4-'
C
.2
4-0 0.2
It
U_
1 Myr 2 Myr 3 Myr 4 Myr 5 Myr
Insolation Interval
Depth (arbitrary units)
4
2
E
a
b
)
1=
C
Figure 3. Demonstration of our dynamic time warping and fractional path displacement
method. In the upper left is a simple model PLD with a non-linear time-depth relationship
generated from 4 Myr of Martian insolation history, in which we kept dust deposition
constant, varied ice deposition inversely with respect to insolation, and created hiatuses in
deposition whenever the insolation was above 300 W/m 2 . In the upper right is the result of
our FPD analysis; each data point represents a tuning of the synthetic PLD to an insolation
record of a given interval, beginning in the present day. On the bottom are three cost
matrices, representing a tuning of the synthetic PLD to 2 Myr (a), 4 Myr (b), and 6 Myr (c) of
insolation. A linear time-depth relationship is represented as the white dashed line and the
optimal path found by the algorithm is represented by the solid black line. In each cost
matrix, warmer colors represent higher costs and cooler colors represent lower costs. Note
that the algorithm successfully picks out the correct age, as the tuning to 4 Myr has the
optimal path ending very close to the upper right corner of the cost matrix.
121
3
E
U
0
F
I
21F
0
E
z
1
0.1
10.0
4.0
2.0
1.0
0.5
n Accumulation Rate (mm/yr)
0.2
Figure 4. Histogram showing the mean accumulation rates that were inferred from
dynamic time warping analysis of 17 MOC images.
0*
00
C
.
E
W
00
*
0
*
o00
Oe
00
o.4
0
U-
0.2k
01.
.0*
*
00 0
1.5
0.5
Insolation Interval (Myr)
Figure 5. Tuning brightness profile 1 at site NP1 (see Figure 2) to increasing durations of
insolation history that begin in the present day. In this case, dynamic time warping yields
an estimate of the age of this stratigraphic sequence to be 470 kyr.
122
1.6
0)1.
4-J
M0.
E
0.
Wi
V
]
T
0.4
0.5
1.0
1.5
2.0
2.5
Assumed Age of Top Layer (Ma)
Figure 6. Tuning brightness profile 1 at site NP1 (the same profile analyzed in Figure 5) to
insolation intervals that do not begin at the present day. Each point represents the best age
estimate for tuning the profile to insolation intervals in steps of 104 years.
123
4--
0
3.5-
0
0
C
0
0
3-
2.5
0
-
E
E
0
75.
0
E
0
20
0
1.5
0
*
C
0
0
I
0
0
00
.
0
.
0
0
0
0
0@*
9
-0
*
0
.
0
.
-
0
0
0.5
0
0
0
100
200
300
400
500
600
700
800
900
Great Circle Distance (km)
Figure 7. Differences in accumulation rates, in mm/yr, between the 17 MOC images as a
function of the distance in km between any two images. Images taken of stratigraphy in
close proximity are not more likely to yield more similar accumulation rates.
124
Mean
Stratigraphic Inferred
Age from accumulation
Height (m)
tuning
rate (mm/yr)
(Myr)
0.7
0.7
523
292.18
85.03
E0200078
1.2
0.4
449
166.66
86.11
E0201540
0.6
120
0.2
136.09
E0300016
84.41
1.2
347
0.3
301.57
E0300417
87.07
0.2
2.0
86.44
82.57
391
E0302206
3.3
326
0.1
254.1
M0001646a 84.48
3.0
303
0.1
M0001714
84.08
108.95
545
0.6
0.9
M0001733
86.80
165.68
0.6
0.4
86.55
78.08
268
M0001754
M0002072
85.96
101.11
349
0.5
0.7
1.9
389
0.2
M0204286
82.70
268.58
0.6
0.5
300
17.53
M0303244a 79.52
0.8
566
0.7
M0303244b 79.52
17.53
M0303530
81.02
30.92
572
0.6
1.0
4.2
426
0.1
M2100236
87.05
100.82
M2302039
87.05
96.8
461
0.4
1.2
0.3
1.1
254.1
342
M0001646b 84.48
Table 1. Results for 17 MOC images showing their image numbers, locations, stratigraphic
height extracted from MOLA data, and age resulting from our dynamic time warping
method. Mean accumulation rate is simply the stratigraphic height divided by the age.
Image
Number
Latitude
Longitude
(E)
125
Site and
Profile
Number
Latitude
Longitude
(E)
NP1_1
NP1_2
NP1_3
NP1_4
NP1-5
NP2_1
NP2_2
NP2_3
82.93
82.93
82.93
82.93
82.93
82.36
82.36
82.36
40.87
40.87
40.87
40.87
40.87
34.06
34.06
34.06
NP2_4
82.36
NP2_5
NP3_1
NP3_2
NP3_3
NP3_4
Stratigraphic Inferred
Age from
Height (m)
tuning
(Myr)
0.47 .02
280 10
0.50 .02
280 10
0.50 .02
280 10
Mean
accumulation
rate (mm/yr)
0.60
0.56
0.56
.05
.05
.05
280
280
270
270
270
10
10
10
10
10
0.49
0.49
0.50
0.74
0.61
.02
.02
.04
.03
.03
0.57
0.57
0.54
0.36
0.44
.05
.05
.06
.03
.04
34.06
270
10
0.66
.03
0.41
.04
82.36
83.02
83.02
83.02
83.02
34.06
94.82
94.82
94.82
94.82
270
260
260
260
260
10
10
10
10
10
0.40
0.71
0.82
0.76
0.78
.05
.04
.04
.04
.04
0.68
0.37
0.32
0.34
0.33
.11
.04
.03
.04
.04
NP3_5
SPi_1
83.02
-72.55
94.82
147.36
260
300
10
25
0.78
1.88
.04
.15
0.33
0.16
.04
.03
SP1_2
-72.55
147.36
300
25
1.48
.10
0.20
.03
SP1_3
SP1_4
-72.55
-72.55
147.36
147.36
300
300
25
25
1.78
1.46
.16
.11
0.17
0.20
.03
.03
SPi_5
-72.55
147.36
300 25
1.56 .12
0.19 .03
Table 2. Results for 20 profiles from 5 HiRISE images, showing the same data as in Table 1.
126
Chapter 6: Conclusions and Future Work
My study of the Moon and Mars has yielded insights into the structures and
histories of those planets. It has also raised many new and interesting questions
that are ripe for future study, both with currently available data and potential future
spacecraft missions to those bodies. Those conclusions and ideas for future work
are summarized below.
1. Moon
I have led an investigation in search of cryptomaria using GRAIL gravity and
LOLA elevation data, which also establishes constraints on lunar basaltic volcanism
in general. I find a volume of potential cryptomaria between 0.42
2.45
x 106
x
106 km 3 and
kM 3 , depending upon assumptions about the isostatic state of the lunar
crust. These candidate deposits correspond to a surface area between 0.50
km 2 and 1.03
x
x 106
106 km 2, which would increase the amount of the lunar surface
containing basaltic volcanic deposits from 16.6% to between 17.9% and 19.3%. I
thus find that there do not exist large volumes of non-dike basaltic intrusions or
subsequently buried extrusions trapped within the lunar crust. Also noteworthy is
that the volumes of potential cryptomaria that I do identify are on the nearside, and
therefore strengthens the evidence for the hemispheric asymmetry observed in
lunar volcanism.
There are further possible avenues to better constrain lunar volcanism. Most
importantly, a study of possible cryptomare deposits in lunar impact basins is
warranted. A necessary precursor to that work would be a detailed inventory and
127
analysis of the gravity anomalies in lunar basins using GRAIL data, and such a study
is currently being undertaken by other members of the GRAIL science team.
Another useful study would be a detailed analysis of the lunar maria in general
using GRAIL data; such a study would provide useful additional constraints on the
bulk densities and thicknesses one might expect in cryptomaria.
I have led an investigation into the nature of isostatic compensation in the
lunar highlands. I have shown that Pratt isostasy is not an important mechanism in
compensating the highlands by an analysis of GRAIL-derived crustal density maps
and LOLA-derived topography maps. I have also shown that there is a dichotomy in
isostasy between the nearside and farside highlands. Specifically, one cannot
compensate the farside highlands using simple Airy or Pratt models of isostasy.
Instead, I have shown that the farside highlands are consistent with compensation
partially occurring in the upper mantle. This idea implies an upper mantle of
thickness >125 km and density 3000-3180 kg/M 3 , which would either exist
preferentially on the farside or exist globally but only participate in compensation
beneath the farside highlands.
There are a few ways one could further test this hypothesis. First, additional
seismic data from future missions would generate direct insight into lunar interior
structure. Second, the continued search for exposed mantle material on the lunar
surface will provide useful constraints on mantle composition if it is able to confirm
findings. Third, theoretical models of the lunar magma ocean that are able to
improve geochemical constraints will lend support either for or against this idea.
Additionally, a detailed GRAIL-based study of the gravity anomalies in the South
128
Pole-Aitken basin (another analysis currently being undertaken by members of the
GRAIL science team) would provide interesting insights into the nature of isostasy
there, a region not currently considered in my study.
Although GRAIL is already providing the best gravity maps for any planet,
extended mission data is still being processed at the time of this writing, and higher
resolved lunar gravity maps will be produced in the future. Improved data is, of
course, welcome in my analysis and could provide stronger support of my
conclusions in either study.
Both my studies of cryptomaria and isostatic compensation have required
careful thought as to other lunar features that might cause observed gravity
anomalies. Thus, I have tangentially encountered many other interesting problems
during my analyses that would be elucidated by a GRAIL-based study. In particular,
hypotheses regarding the origin of lunar magnetic anomalies and quantities of ice at
the lunar poles are just two areas of interest to me that are ripe for future study
using GRAIL.
2. Mars
I have adapted a technique generally used to compare two time-uncertain
series and applied it for use on the Martian PLDs. I have shown that it is feasible to
recover an orbital signal from the PLDs, and therefore obtain estimates for the age
and mean accumulation rate of a given stratigraphic sequence, even when the timedepth relationship of a piece of stratigraphy is strongly nonlinear, as long as
129
deposition of ice and dust occurs in a predictable fashion with respect to insolation
at the poles.
I have led an investigation into applying that technique to MOC and HiRISE
images of the PLDS. I have analyzed 23 MOC images and 20 profiles of brightness as
a function of depth from 4 HiRSE images. I find that analysis of the MOC images
yields a mean accumulation rate of 1.4
1.1 Mm/yr for the NPLDs, and analysis of
the HiRISE images yields a mean accumulation rate of 0.47 t 0.12 mm/yr for the
NPLDs and 0.18
0.02 mm/yr for the SPLDs. These rates apply for the upper -300
m of stratigraphy; extrapolation of these rates would yield an approximate age of 4
Myr for the NPLD sequence and 17 Myr for the SPLD sequence. Variability of results
implies that factors other than insolation are affecting PLD formation and
modification and/or that brightness is not a perfect proxy for composition of the
PLDs.
This analysis could be strengthened in a number of ways. Additional HiRISE
images for both the NPLDs and SPLDs could increase the confidence of the results.
In particular, HiRISE images of sequences for which there also exist MOC images
would be fruitful as points of comparison. Images of deep troughs are especially
useful to understand how the accumulation rate varies with depth; these would
elucidate if applying mean rates of the upper 300 m to the whole PLD sequences is a
reasonable extrapolation. Studies of the present day polar caps that can determine
the current accumulation rate - or even just the sign of that accumulation rate - of
the PLDs would provide useful constraints on our work. Finally, better
130
understanding in a general sense of the differences between the north and south
polar caps of Mars is necessary for a complete interpretation of our results.
131