Constraining the Average Fill Densities of Mars'
Lowlands and Fluvial Erosion of Titan's Polar
Regions.
by
Yodit Tewelde
Submitted to the Department of Earth, Atmospheric and Planetary
Sciences
in partial fulfillment of the requirements for the degree of
MASSACHUSFTS INSTITUTE
OF TECHNOLOGY
Masters of Science in Planetary Science
at the
OCT 0 4 2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
L1RAR E S
September 2013
© Massachusetts Institute of Technology 2013. All rights reserved.
Author ...
Dept'fient of Eardh, A-t-m-ojsperic and Planetary Sciences
August 26, 2013
Certified by. .
..............................
Maria T. Zuber
Vice President for Research at MIT
Thesis Supervisor
Accepted by....
..........
Robert van der Hilst
Schlumberger Professor of Earth Sciences
Head of Dept. of Earth, Atmospheric and Planetary Sciences
Constraining the Average Fill Densities of Mars' Lowlands
and Fluvial Erosion of Titan's Polar Regions.
by
Yodit Tewelde
Submitted to the Department of Earth, Atmospheric and Planetary Sciences
on August 26, 2013, in partial fulfillment of the
requirements for the degree of
Masters of Science in Planetary Science
Abstract
Other than Earth, Mars and Titan are the only bodies in our Solar System where
we have observed widespread fluvial activity. In this thesis I present two approaches
for constraining the extent of multiple resurfacing processes in order to gain insight
into the early history of Mars and Titan. One of the most distinctive features of the
Martian surface is the dichotomy between the heavily cratered southern highlands
and the relatively smooth northern lowlands. The northern lowlands appear smooth
because many of the craters in the north have been partially or completely buried
beneath volcanic and sedimentary fill of unknown relative proportions. In Chapter
1, we use the Mars Orbiter Laser Altimeter (MOLA) topography data, the Mars Reconnaissance Orbiter (MRO) gravity model and a Wiener filter to map these buried
craters and estimate minimum fill thickness and volume as well as maximum fill density. The overall trend observed for the northern lowlands is more sedimentation
near the dichotomy and less sedimentation further north and near the Tharsis region,
which is consistent with the geology of the region. Titan has few impact craters,
suggesting that its surface is geologically young. In Chapter 2 we evaluate whether
fluvial erosion has caused significant resurfacing by estimating the cumulative erosion
around the margins of polar lakes. Images of drowned fluvial features around the lake
margins, where elevated levels of hydrocarbon liquids appear to have partly flooded
fluvial valleys, allow us to map topographic contours that trace the fluvially dissected
topography. We then used a numerical landscape evolution model to calibrate a relationship between contour sinuosity, which reflects the extent of fluvial valley incision,
and cumulative erosion. We find that cumulative fluvial erosion around the margins
of Titan's polar lakes, including Ligeia Mare, Kraken Mare, and Punga Mare in the
north and Ontario Lacus in the south, ranges from 4% to 31% of the initial relief.
Additional model simulations show that this amount of fluvial erosion does not render craters invisible at the resolution of currently available imagery, suggesting that
fluvial erosion is not the only major resurfacing mechanism operating in Titan's polar
regions.
3
Thesis Supervisor: Maria T. Zuber
Title: Vice President for Research at MIT
4
Acknowledgments
I would like to thank my advisors Maria T. Zuber and J. Taylor Perron for all of their
help and guidance with these projects, as well as my other committee member Richard
P. Binzel and Lindy T. Elkins-Tanton. I would also like to thank my co-authors
James Garvin, Junlun Li, Peter Ford, Scott Miller, and Benjamin Black. This work
was supported by my Robert R. Shrock Graduate Fellowship and by NASA Cassini
Data Analysis Program award NNX09AE73G to Peter Ford.
5
6
Contents
1
Determining the Average Fill Densities of Mars' Lowlands.
9
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2
Data Sets ........
1.3
1.4
1.5
.................................
13
1.2.1
Topography . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.2.2
Gravity M odel . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.2.3
Crustal Density Estimate . . . . . . . . . . . . . . . . . . . . .
14
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
. . . . . . . . . . . . . . . . . . . .
15
M ethodology
1.3.1
Estimating Fill Thickness
1.3.2
Isolating the Gravity Anomaly Associated with the Fill Material 16
1.3.3
Solving for Average Fill Densities . . . . . . . . . . . . . . . .
17
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.4.1
Minimum Fill Volume
. . . . . . . . . . . . . . . . . . . . . .
19
1.4.2
Wiener Filter Results . . . . . . . . . . . . . . . . . . . . . . .
21
1.4.3
Maximum Density Values
. . . . . . . . . . . . . . . . . . . .
21
1.4.4
Sensitivity to Fill Thickness and Crustal Density Estimates . .
29
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2 Estimates of Fluvial Erosion on Titan from Sinuosity of Lake Shore31
lines
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.2
Titan's polar lakes
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.3
Landscape Evolution Model . . . . . . . . . . . . . . . . . . . . . . .
37
2.3.1
37
Fluvial Incision . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3.2
Hillslope Erosion . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.3.3
Initial Surface and Simulation Procedure . . . . . . . . . . . .
42
2.3.4
Relating contour sinuosity to cumulative erosion . . . . . . . .
45
2.4
Testing the sinuosity-erosion relationship with terrestrial data
. . . .
48
2.5
Application to Titan's Polar Regions . . . . . . . . . . . . . . . . . .
52
2.5.1
SAR data and study region
. . . . . . . . . . . . . . . . . . .
52
2.5.2
Test of shoreline identification criterion . . . . . . . . . . . . .
53
2.5.3
Shoreline mapping and sinuosity calculation
. . . . . . . . . .
55
2.5.4
Comparison with model-derived erosion proxy . . . . . . . . .
56
2.5.5
Sensitivity to initial conditions and fluvial incision parameters
57
2.6
2.7
D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
2.6.1
Comparison with previous erosion estimates . . . . . . . . . .
60
2.6.2
Geographical trends
. . . . . . . . . . . . . . . . . . . . . . .
60
2.6.3
Implications for erosional resurfacing on Titan . . . . . . . . .
61
2.6.4
Implications for polar sediment and volatile budgets . . . . . .
63
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
8
Chapter 1
Determining the Average Fill
Densities of Mars' Lowlands.
1.1
Introduction
Images from Mariner 9 and the Viking Orbiters in the 1970s revealed one of the most
prominent features of the Martian surface, a near hemispherical dichotomy (Fig. 1-1)
between the heavily-cratered southern highlands and the relatively smooth northern
lowlands [33, 70, 17]. This 3-km difference between the two regions has been proposed
to represent evidence of either a giant impact [117, 5] or degree-1 mantle convection
causing preferential heating and crustal thinning of one hemisphere [119, 123]. Frey
et al. [27] constrained the relative timing of formation of the dichotomy by generating
cumulative frequency curves for the partially-buried impact craters or Quasi-Circular
Depressions (QCDs) of the northern lowlands. The authors conclude that the sizefrequency distribution of the QCDs implies that the basement crust beneath the fill
material is as at least as old as the southern highlands, and therefore the dichotomy,
which must predate the basement crust, is a primordial feature dating from Mars'
early Noachian epoch (4.1 - 3.7 Gyr).
Mariner 9 and the Viking Orbiters also revealed extensive volcanism [60], gradational and fretted terrain boundaries [76], tectonic ridges [18, 112] and seemingly
water-worn outflow channels [17, 6], which changed the previous perception of Mars
9
as a dry, cold and sterile planet. Parker et al. [76] proposed that the gradational and
fretted terrain boundaries are indicative of ancient shorelines and that the northern
lowlands may have contained a global ocean early in Mars' history. But, with only
surface images as a source of information, the mode, timing and long-term cycling of
water on Mars could not be determined.
The Mars Orbiter Laser Altimeter (MOLA) [126] onboard the Mars Global Surveyor (MGS) spacecraft [4, 3] provided high-resolution topography of the Martian
surface (Fig. 1-1) and provided detailed topography, slopes and roughness of the
lowland plains [95, 94, 72]. The topographical data revealed the full extent of the
many tectonic ridges [37], evidence of the flow of water far from the outflow channels
[94] and the previously mentioned Quasi-Circular Depressions [27]. By examining the
detailed features observed in the MOLA topography, Head et al. [37] were able to
propose the following history of the northern lowlands from the Early Hesperian to
the present day (refer to Fig. 1-2 for the estimated period timeline):
1. Early Hesperian-aged volcanic plains form throughout the lowlands
2. Volcanic plains contract and form extensive Tharsis-circumferential and basin
related wrinkle ridges
3. Late Hesperian-aged outflow channels near Chryse Planitia deposit water and
sediment into the northern lowlands
4. Loss of outflow channel water leads to residual sediments deposited on top of
ridged plains (Vastitas Borealis Formation)
5. Amazonian volcanic plains are deposited
6. Late Amazonian polar and circumpolar deposits form
Our study excludes the polar region, so we are not concerned with step 6 in
this sequence. In addition, the extensive denudation of Arabia Terra (Fig. 1-1) and
regions along the dichotomy also suggest that sediments substantially contribute to
10
the resurfacing of the lowlands [41, 23], but the relative proportion of sediments to
volcanic material and the total fill volume is still not well constrained.
Using the MOLA topographic data, it is possible to map these QCDs and measure crater diameters that can be used to estimate the original excavation depths
of individual QCDs. These fresh crater depths can be extended to map fill depths
and to estimate a minimum fill thickness for the northern plains as a whole. The
approximated pre-fill surface acts as a layer of basal relief between the fill and the
Martian crust. This boundary layer is used in combination with the MOLA topography data, the Mars Reconnaissance Orbiter (MRO) gravity model [49] and a Wiener
filter [79, 51] to constrain an average fill density for the northern plains.
11
km
60
A
A
2
11
300
00
fQ
-30*
-60*I!5A
1800
2700
00
900
Figure 1-1: Map of the global topography of Mars from MOLA [94]. Craters included in study are circled in black. The dashed
line represents the dichotomy boundary as mapped by Parker et al. [76]. Areas of interest are labeled.
180
I
4.5
__ I
4
i
3.5
- __ I
3
I
I
I
2.5
2
1.5
-
I -
1
I
0.5
0
Time Before Present (Gyr)
Figure 1-2: The martian geologic epochs based on stratigraphic units on the surface
of Mars (Adapted from Hartmann and Neukum [34]).
1.2
Data Sets
To isolate the gravity anomaly associated with the fill material and estimate fill
density, it is necessary to incorporate several data sets and make assumptions about
Mars' internal structure. Here we combine the MOLA topography [95, 94], the MRO
gravity model mrol10b2 [49], and Martian meteorite studies.
1.2.1
Topography
The Mars Orbital Laser Altimeter (MOLA) was one of five instruments onboard the
Mars Global Surveyor (MGS) spacecraft. The instrument collected over 671 million
valid measures of the two-way range from MGS to the Martian surface. MOLA sampled the elevation every 300 meters along the spacecraft groundtrack with a surface
laser spot size of 168 meters [95, 94, 127].
Along-track orbit errors were smaller
than the size of the laser footprint; radial orbit errors were reduced by minimizing
the altimetric crossover residuals so the accuracy is typically better than 1 m radially
[72]. A spherical harmonic expansion of topography referenced to the geoid has been
determined to degree and order 1152 [94]. Since we are limited by the resolution of
the gravity field, we truncate the topographic field to degree and order 90.
13
1.2.2
Gravity Model
In 2006, the Mars Reconnaissance Orbiter (MRO) mission [128] began collecting data
for the radio science gravity investigation and Radiometric Doppler data were used
to determine the MRO orbit. To determine precise orbits, it is necessary to model
the non-conservative forces that act on the spacecraft.
The three forces with the
largest impact on orbit determination are atmospheric drag, solar radiation pressure
and angular momentum desaturations of the spacecraft's momentum wheels (AMDs)
[49]. A surface albedo model is also included along with corrections for Mars' solid
tide, factors that affect the DSN ground station positions, radio signal propagation
through the troposphere and third-body perturbations from the sun, moon, all planets
in the solar system, and the Martian moons. The GEODYN/SOLVE programs [78,
77] and DPODP program [68] are used respectively at NASA/GSFC and the Jet
Propulsion Laboratory. Both programs use a Bayesian least-squares approach to
determine the convergence of the spacecraft orbit segments [125]. Once precise orbits
are determined, the gravity field can be computed.
Due to the lower altitude of the MRO spacecraft, 255-km periapse as opposed to
the - 400 km altitude of MGS and Mars Odyssey, it was possible to determine higher-
resolution gravity field models such as mrol10b2 [49]. In this study we use gravity
model mrol10b2, which has been determined to degree and order 110, and because
the sampling of gravity is non-uniform, we expand to degree 90 as recommended by
Konopliv et al. [49].
1.2.3
Crustal Density Estimate
Sources of information concerning the densities of the Marian-crust and mantle include
orbital spectra [69], in situ spectral and chemical observations from the Viking and
Pathfinder landers [63], and geochemical analyses of SNC (shergottite, nakhlite and
chassigny) meteorites, which represent samples from the Martian surface [12, 42].
McGovern et al. [61] found that for much of the Martian surface, a crustal density
value of 2900 kg m- 3 best fit localized admittances, and this value of density that has
14
been adopted in several studies [127, 75, 116]. We utilize this crustal density value as
well.
1.3
Methodology
1.3.1
Estimating Fill Thickness
We use the MOLA topography map (Fig. 1-1; [95, 94, 72]) to identify the location and
diameter of 208 QCDs whose rims have been preserved. Garvin performed a study
on the global scaling relationships for approximately 2000 non-degraded craters on
Mars adequately resolved by MOLA and determined the following relations:
Simple craters: d = 0.25D 0 .65 (D < 7km)
(1.1)
Complex Craters: d = 0.33D 0 5 3 (7 < D < 70km)
(1.2)
Basins: d = 3.5D
0 1 7 (D
> 70km)
(1.3)
where D is the rim crest diameter (in km) and d is the fresh crater depth from rim
crest to floor (in km). Since our study excludes craters with a diameter under 20 km,
only the equations for complex craters and basins were used to determine fresh crater
depths. These relations do not apply to massive impact basins such as Utopia Planitia
(Fig. 1-1), so we use the deepest portion of Hellas Planitia in the southern highlands
to act as a proxy for the original excavation depth of Utopia Planitia. Based on the
fresh crater depths, we linearly interpolate a prefill surface to estimate fill thickness
throughout the northern plains region and treat this surface as a layer of basal relief.
15
1.3.2
Isolating the Gravity Anomaly Associated with the Fill
Material
In order to utilize gravity and topography to map the northern plains fill, it is necessary to remove the gravitational signature of the underlying terrain. To do that we
create a Wiener filter [79]. The filter is constructed by analyzing the residual gravity
anomalies outside of the area of interest, the northern lowlands, which are likely to be
from the crust or mantle. By characterizing the spectral nature of this outside region,
we can filter out the residual gravity signature from the crust and mantle beneath the
fill material in the northern lowlands and isolate the gravity anomaly associated with
the fill material. A similar approach was used to map density anomalies in Mars'
south polar layered terrain [51]. The outside region on Mars needs to have similar
crustal properties as the basement crust under the northern plains but mostly devoid
of fill material. The region that best fits this description is Arabia Terra (Fig. 1-1),
which is along the dichotomy and has a crustal thickness between that of the northern lowlands and the southern highlands. Arabia Terra also has a comparable crater
distribution as the QCDs of the northern lowlands [27].
This approach is based on two assumptions: (1) the gravitational anomalies of
the fill material are spectrally different from those associated with the crust and
mantle; and (2) the crust and mantle heterogeneities are generally similar in both
Arabia Terra and in the northern lowlands. The optimal Wiener filter is found by
minimizing the following term:
< Ig,8 (x, y)
-
g(x, y)h(x, y) I'>
where g, is the desired anomaly from the region in Arabia Terra, g is the gravity
observed in the northern lowlands, a combination of anomalies from crust and mantle
heterogeneities as well as the fill material, and h is the optimum Wiener filter. The
asterisk represents the spatial convolution. The optimal filter can be written in the
wave number domain as:
16
H(u,
H~u, v)
v)
=(.4<
G, (u, v) G* (u, v) >(14
< G(u, v)G*(u, v) >
where the terms represent the Fourier transform of their lower case counter-part and
here the asterisks indicate the complex conjugate of G. If we also assume that the
gravity signal and noise are stationary and ergodic, the filter can be simplified to:
Gs(u, v)|2(15
1G(u, v)|2
H(u, v) ~
(1.5)
We test the success of the Wiener filter to recover the original signal by manufacturing a signal and adding red noise with values up to 40% of the original signal's
strength. Using only the noisy signal (Fig. 1-3b) and the two signals combined (Fig.
1-3c), we are able to create the optimum Wiener filter and remove a majority of
the surrounding noisy signal. We are able to recover a signal with a pattern and
magnitudes that are similar to the original signal (Fig. 1-3d).
I
150
ILI
I50150 W_.
100
200
100
E
-150
100
50 -0
Figure 1-3: Demonstration of the application of the Wiener filter a. Manufactured
original signal b. Randomly-generated red noise that acts as our surrounding signal
(gs) c. The addition of a. and b. which acts as our observed signal (g) d. Recovered
signal after applying the optimum Wiener filter, adequately recovering the original
signal.
1.3.3
Solving for Average Fill Densities
With the estimated fill thickness and isolated gravity anomalies, it is possible to constrain the average fill density. For such a large region it is best to perform the gravity
analysis in the spherical harmonic domain, since the Cartesian domain would result in
large errors due to the curvature of Mars. With the estimation of basal relief and the
17
MRO gravity model, we can use potential theory equations [115, 43] to determine the
density contrast required to produce the observed potential anomalies in the northern
lowlands. Utilizing the orthogonal properties of the spherical harmonic functions in
combination with Newton's law of gravitation yields:
U(r, 0,
=
GMiD
M)
Y
D)- CAYzim(9, #)
(1.6)
ilm
where U is the gravitational potential, G is the universal gravitational constant, M
is the mass of Mars, D is the interface depth, 1 and m are the degree and order of
the field, and Yjm and are the spherical harmonic functions and coefficients. For all
r = RB + max(B), the spherical harmonic coefficients can be expressed as follows:
+
im
47rApD3
him H= 1 (1 + 4 - j)(17)
M(21+1)n= Dnn!
1+3
where Ap is density contrast between the two layers of the interface and hum are the
spherical harmonic coefficients of topography. The height of the geoid (N) at the
surface can be determined using Brun's equation, which is a first-order Taylor series
approximation over the radial coordinate r:
U(U, r + 6r) = U(), r) +
oBr
'
N(Q)
(1.8)
The radial derivative of gravitational potential is the surface gravitational acceleration (g,) which we assume is constant over the surface. We introduce a generic
interface relief B(Q) with density contrast APB at depth DB to address the effect of
the basal relief interface between the fill and the crust of the northern lowlands. By
combining the Equations 1.6 and 1.7 with Brun's equation (Equation 1.8), the coefficients of the gravitational equipotential surface for can be expressed as a function of
radius:
NB
6UB
U1m
Sr
_
4WGRB
9r(21+1)
B+1
1+3 n
pB
(mr)RB
i(m.PB
r
R/nn
18
1( + 4 j)
1+3
(1.9)
where R is the planetary radius and
RB
= R - DB. The magnitude of each higher-
order summation term falls off rapidly as n increases. When the depth of the interface
relief is small, the summation terms for n > 1 are negligible and the geoid contribution
for R > RB can be expressed as follows:
uinear6NB =
1m
3
F4prR
M(21
+ 2) (R )(
B
(1.10)
In this study we assume n > 1 are negligible and use equation 1.10 to solve for the
density contrast APB that best fits the estimated depth of the basal relief interface.
We solve Equation 1.10 using a forward modeling approach over a wide range of APB
values with an interval of 10 kg m-3 .
1.4
Results and Discussion
1.4.1
Minimum Fill Volume
While hundreds of QCD's are visible in the MOLA topography, we appreciate that
there is a significant population that has been completely buried and show no signs
on the surface today. This is why we assume our thickness values to be a minimum
fill thickness. The minimum fill thickness values (Fig. 1-4) were estimated by interpolating fresh crater depth values for QCDs in the northern lowlands. The minimum
fill volume was estimated by integrating between the following two layers in the polar projection: the MOLA topography and the layer of basal relief estimated in this
study. The average fill thickness in the northern plains is approximately 2 kilometers.
We calculated a minimum fill volume of 8.3 t 3.0 x 10 7km 3 , excluding Elysium Rise
and the *polar region. Johnson et al. [45] observe the extent of circumpolar deposits
to be 70± degrees; we conservatively exclude the region above 60' N. If we were to
assume an average fill thickness of 2 km for the polar region above 600 N, that would
add
-
2 x 107km 3 to the minimum fill volume estimate. For reference, 8.3 x 10 7km 3 )
spread over the surface of Mars, would bury the entire planet ~ 0.4 km. The error
bars here are based on the standard deviation associated with the fresh excavation
19
depths of the Garvin study mentioned in section (1.3.1).
Previous studies have made estimates of the volume of sediments and volcanic
material deposited in the northern lowlands. Hynek and Phillips [41] examined inliers
in Arabia Terra and by assuming that they are representative of the original surface
elevation were able to estimate that 4.5 x 10 6 km 3 of material has been deposited into
the northern lowlands. Evans et al. [23] used gravity, topography and a flexural
model to examine the same region and concluded that the maximum volume that
could have been eroded from Arabia Terra is 3 x 10 7km 3 . Based on the volume of
large outflow channels and other fluvial features near Chryse Planitia, Carr et al. [16]
estimated that 4.2 x 10 6 km 3 of material had been deposited from that region. Head et
al. [37] used flood models and geological indicators such as wrinkle ridges to estimate
the volume of Hesperian-aged volcanic flows in the lowlands to be 3.3 x 10 7 km 3. This
brings the range fill material volumes to a total of approximately 7 x 10 7km 3 . This
rough estimate falls well within our estimated volume range and gives us additional
confidence in our volume results.
IkS mrI
6
5
4
3
2
Figure 1-4: Minimum fill thickness map in kilometers. Estimates are derived by
interpolating a pre-fill surface from the fresh crater depths and subtracting this layer
from the topography. The fresh depth of Utopia Planitia is scaled from Hellas Planitia,
a basin of comparable size in the southern highlands. The Elysium volcanic rise and
the polar region north of 600 latitude have been removed from the area of interest.
The fill thickness values overlay the grayscale image of the topography data [72].
20
1.4.2
Wiener Filter Results
To isolate the gravity anomaly associated with the fill material, we must remove the
anomaly associated with crust and mantle variations and heterogeneities. One way of
doing this is to construct a Wiener filter using the gravity signature of a region with
similar crustal and mantle variations. Arabia Terra is a region along the dichotomy
that is mostly devoid of fill material and has an elevation closer to the northern
lowlands than the southern highlands. It arguably provides the best approximation of
what the crustal variations and underlying mantle look like beneath the fill material.
Fig. 1-5(a-b) is a close up of the Arabia Terra topography and Bouguer anomaly
with degrees 1-12 removed, which partially filters out the long-wavelength gravity
signature that is likely to come from greater depths. We use a region within Arabia
Terra to construct a Wiener filter for the northern lowlands, and the residual gravity
anomaly associated with the fill material is mapped in Fig. 1.5c.
1.4.3
Maximum Density Values
With estimates of the thickness and the gravity anomaly of the fill, we use equation
11 to solve for the density contrast of the material and the crust. If we assume the
average crustal density to be 2900 kg m-3, the resulting fill density values can be seen
in Figure 1-6. Since our fill thickness estimate is a minimum, our density values are
considered a maximum as a thicker layer of material would require a lower density
contrast to produce the same gravity signature.
Density values can give insight into the relative proportion of sedimentary to
volcanic material. It is difficult to discuss values in absolutes without specific density
values of sediments and volcanic material on Mars' surface. While we do have the
SNC meteorites to give some indication of density values, the specific surface locations
of those samples are unknown. Instead we observe overall trends in density values
and compare the results to the mapped geology of the region (Fig. 1-7). The values
range from approximately 2500 - 3100 kg m-3. From the SNC meteorites we expect
pore-free basalt density values to be as high as 3220 - 3390 kg m- 3 or 3040 - 3220
21
[km]
IMOaS]
3
200
100
05
so
.1
.0
.4
-100
(agpisi
ano
60-
ISO
100
301
so
0
.100
-100
180*
270*
00
90*
10*
450
60
3040
20
000
.20
40
.........
-300
-60
-100
270*
00
9W
130
Figure 1-5: a. Arabia Terra MOLA Topography b. Arabia Terra Bouguer anomaly
with degrees 1-12 removed c. Northern lowlands Bouguer gravity anomaly with
degrees 1-12 removed [49]. d. Northern lowlands residual gravity anomaly after
applying the optimum Wiener filter. Both c. and d. overlay the grayscale image of
MOLA topography [72]
22
kg m- 3 if we assume 5% porosity [73], which agrees well with our maximum values.
Our first observation is that regions near the dichotomy, on average, have a lower
density than regions further north. Arabia Terra and fluvial channels along the dichotomy have deposited sediments along the dichotomy, explaining the lower average
density. As we might expect, Chryse Planitia (Fig. 1-1; the flood plain unit [Hchp]
in Fig. 1-7) near the large outflow channels, has a lower density and thus a higher
sediment volume than regions further north where we begin to see the effects of the
Arcadian volcanic flows [Aal]. Also note the higher density values of the region directly north of Alba Patera (Fig. 1-1). If we look at the geologic map in Fig. 1-7, we
can see that the majority of this area is classified as the lower member flow of the Alba
Patera Volcanic Formation [Hal]. These density values are consistent with volcanic
flows. Another area of interest is Utopia Planitia (Fig. 1-1), which we have estimated
to have a fill thickness of over 6 km. The geologic map has designated the majority
of the region as a combination of Vastitas Borealis Formation [Hvg] and Elysium
Formation [Ael3]. The Hesperian-aged Vastitas Borealis Formation is interpreted as
a combination of degraded lava flows and sediment [37] and the Elysium Formation
member as a volcanic flow unit. Our maximum density results for this region range
between 2800 - 2900 kg m-3, which agrees with the suggestion that sediments have
significantly contributed to the burial of the Utopia basin [124].
23
[kg m4]
3100
3000
2900
2800
2700
2600
2500
Figure 1-6: Average fill density map in kg m-3 . Measurements are taken at a spacing of 2 degrees in the latitudinal and
longitudinal directions. The density results overlay the grayscale image of the topography data [72].
Geology
AHa
Aa2
Achp
Ael3
Aoal
Aos
Hal
Hhet
Htm
Hvr
Npl2
b
AHat
Aa3
Achu
AeI4
Aoa2
Apk
Hch
Hpl3
Htu
Nb
NpId
cb
AHpe
Aa4
Ae
Aml
Aoa3
Aps
Hchp
Hr
Hvg
Nf
Nple
cs
AHt3
Aa5
Aell
Amm
Aoa4
As
Hcht
Hs
Hvk
Nm
NpIh
s
Aal
Ach
AeI2
Amu
Aop
HNu
Hf
Ht
Hvm
Npll
NpIr
v
Figure 1-7: Geologic map of Mars' northern lowlands [93] used to compare the results of our density study overlaying the
grayscale image of the topography data [72].
26
Unit Symbol
Unit Name
Aal
Arcadia Formation, member 1
Aa2
Arcadia Formation, member 2
Aa3
Arcadia Formation, member 3
Aa4
Arcadia Formation, member 4
Aa5
Arcadia Formation, member 5
Ach
younger channel material
Achp
younger flood-plain material
Achu
younger channel system material, undivided
Ae
eolian deposits
Aell
Elysium Formation, member 1
Ael2
Elysium Formation, member 2
Ael3
Elysium Formation, member 3
Ael4
Elysium Formation, member 4
AHa
Apollinaris Patera Formation
AHat
Albor Tholus Formation
AHpe
etched plains material
AHt3
Tharsis Montes Formation, member 3
Aml
Medusae Fossae Formation, lower member
Amm
Medusae Fossae Formation, middle member
Amu
Medusae Fossae Formation, upper member
Aoal
Olympus Mons Formation, aureole member 1
Aoa2
Olympus Mons Formation, aureole member 2
Aoa3
Olympus Mons Formation, aureole member 3
Aoa4
Olympus Mons Formation, aureole member 4
Aop
Olympus Mons Formation, plains member
Aos
Olympus Mons Formation, shield member
Apk
knobby plains material
Aps
smooth plains material
As
slide material
b
tear-drop shaped bar or island
27
Unit Symbol
Unit Name
cb
impact crater material, partly buried
cs
impact crater material, superposed
Hal
Alba Patera Formation, lower member
Hch
older channel material
Hchp
older flood-plain material
Hcht
older chaotic material
Hf
younger fractured material
Hhet
Hecates Tholus Formation
HNu
undivided material
Hp13
plateau sequence, smooth unit
Hr
ridged plains material
Hs
Syrtis Major Formation
Htl
Tempe Terra Formation, lower member
Htm
Tempe Terra Formation, middle member
Htu
Tempe Terra Formation, upper member
Hvg
Vastitas Borealis Formation, grooved member
Hvk
Vastitas Borealis Formation, knobby member
Hvm
Vasitas Borealis Formation, mottled member
Hvr
Vastitas Borealis Formation, ridged member
Nb
highly-deformed terrain materials, basement complex
Nf
highly-deformed terrain materials, older fractured material
Nm
mountainous material
Npll
plateau sequence, cratered unit
Np12
plateau sequence, subdued cratered unit
Npld
plateau sequence, dissected unit
Nple
plateau sequence, etched unit
Nplh
plateau sequence, hilly unit
Nplr
plateau sequence, ridged unit
s
impact crater material, smooth floor
v
volcano, relative age unknown
28
1.4.4
Sensitivity to Fill Thickness and Crustal Density Estimates
We test the sensitivity of our results to the fill thickness parameter. Using the same
conditions that we used to constrain our volume estimate, we run our model at the
fill thicknesses that are 0.43 km thicker and thinner than our estimate. This value
comes from the standard deviation associated with complex craters in the Garvin
study mentions in section 1.3.1,which provide the most variability in fresh excavation
depths. We find that changing our thickness values results in a maximum of ±70 kg
m-3 change in our maximum density values. This leads us to conclude that our results
are fairly insensitive to the range of depths we are examining and that the overall
trend in density distribution, which is the focus of our study, remains consistent.
We also consider the effects of changes in the crustal density values. As mentioned
in section 1.2.3, a crustal density of 2900 kg m-
3
has been used in multiple studies.
A more conservative range of values would be 2700 - 3100 kg m-
3
[116]. Since the
parameter that we test in equation 1.10 is the density contrast between the crust
and fill, APR, varying the crustal density value would alter the results by the same
amount. The linear relationship between crustal density and fill density would make
it difficult to constrain a total sediment volume, which is why our observations are
based on overall trends which are not affected.
1.5
Conclusions
We use MOLA topography data and the Quasi-Circular Depressions of the northern
lowlands to estimate a minimum fill thickness for the region as a whole. With the use
of a Wiener filter and the mrol10b2 gravity model, we are able to isolate the residual
gravity associated with the fill and calculate the maximum fill density of the northern
plains.
We see that regions near the dichotomy have a lower average fill density,
which is consistent with more sedimentation in those areas. Regions further north
and specifically near the Alba Patera volcanic region have a higher density value as is
29
expected by regions near large volcanic flows and further from outflow channels and
regions like Arabia Terra that show evidence of extreme erosion. These results are
consistent with the sequence of northern lowlands history that was proposed by Head
et al. [37].
30
Chapter 2
Estimates of Fluvial Erosion on
Titan from Sinuosity of Lake
Shorelines
2.1
Introduction
Images from the Descent Imager/Spectral Radiometer (DISR) on the Huygens Probe
and the Synthetic Aperture Radar (SAR) instrument on the Cassini spacecraft have
revealed extensive fluvial networks in many regions on Titan [104, 97, 57, 14, 15, 50, 1],
some of which drain into lakes near the north and south poles [66, 101]. Titan appears
to be the only body in the Solar System other than Earth where rivers have recently
flowed on the surface. Unlike Earth, however, Titan has rivers of liquid methane
and ethane incising into a surface of water ice and organic sediments [32], and river
valleys near the poles feed into lakes of methane, ethane and other hydrocarbons
[101].
No river channels have been observed directly on Titan due to the coarse
image resolution, but the observed landforms are interpreted to be fluvial valleys
based on their morphology and context, which are similar to features on Earth that
were formed mechanically by surface runoff [84, 1, 15]. In addition, methane comprises
several percent of Titan's atmosphere, and the surface conditions allow methane to
31
exist as a gas or a liquid [74]. All of this suggests that Titan has a methane cycle
that is analogous to Earth's water cycle.
This high concentration of atmospheric methane presents a puzzle, because methane
undergoes photochemical breakdown that limits its atmospheric lifetime to 10-100
Myr [74]. To sustain the observed methane concentration over billions of years, there
must be a source of replenishment [122].
It is possible that a transfer of subsur-
face material to the surface and atmosphere could be responsible for the methane
replenishment [58, 52].
However, the geological processes operating on Titan remain an outstanding problem, in part because of questions regarding the history of Titan's surface. Cassini
observations have revealed only a few scattered impact craters, which indicate a relatively young surface that is estimated to be between a few hundred Myr and 1 Gyr
old [56, 120, 71].
Several plausible resurfacing mechanisms could potentially help
to explain this young surface, including cryovolcanism [99, 54, 52], tectonic deformation [88], organic aerosol deposition [62], dune migration and aeolian deposition
[87], and widespread erosional denudation [11]. Determining which of these processes
are significant is essential to understanding Titan's geologic history, and thus Titan's
interior processes and hydrocarbon cycle.
The purpose of this paper is to assess the extent of resurfacing by fluvial erosion on Titan, particularly in the vicinity of the polar lakes. On Earth, rates and
amounts of fluvial erosion can be estimated by a combination of three-dimensional
topographic measurements, field observations, and geochronologic techniques that
require field sampling [9, 11, 25].
On Titan, we are currently unable to directly
measure rates or amounts of landscape change. The Cassini-Huygens mission was
not designed to perform widespread topographic mapping, but a limited set of topographic measurements has been obtained from a combination of radar altimetry
[22], stereogrammetry of overlapping SAR swaths [47, 46], and a technique known as
SARTopo that uses the overlapping returns of the radar instruments multiple beams
[100]. However, these basic digital elevation models (DEMs) are rare, covering a small
fraction of Titan's surface. Although the acquisition of such data is an impressive
32
technical feat [88, 97, 100], the resolution, precision and coverage are generally not
sufficient for measurements of the width, depth, or cross-sectional form of individual
fluvial valleys or of the surrounding hillslopes. Moreover, precise measurements of
the present topography would be insufficient to measure cumulative erosion without
additional constraints on the initial topography. Thus, it is not clear whether Titan's
landscape is the result of steady, protracted, and significant exhumation of its crust
(over perhaps billions of years), or only minimal, recent fluvial incision and hillslope
adjustment.
In the absence of adequate knowledge of the initial and present topography of
the fluvially dissected landscape, alternative approaches must be employed to estimate how much erosion has occurred. Black et al. [11] developed a procedure for
estimating the spatially averaged cumulative erosion of a surface, as a fraction of the
initial topographic relief, by measuring the map-view geometry of drainage networks.
They studied fluvial networks in SAR images covering two north polar areas, one
equatorial area, and one south polar area. Drainage networks in all four areas are
consistent with minimum erosion of 0.4% of initial relief. Two of the regions yielded
estimates of maximum erosion (9% for one of the north polar areas and 16% for the
south polar area), but these upper bounds were less certain than the lower bounds,
and the remaining two sites did not yield well-defined upper bounds. The study by
Black et al. [11] implies that some areas of Titan's surface have experienced relatively
little erosion since the most recent resurfacing event, but it is not clear how extensive
these areas are. To address some of these questions and extend our understanding of
fluvial processes on Titan, we seek to obtain additional estimates of regional erosion
using independent techniques. Despite the lack of topographic datasets that resolve
individual fluvial valleys, there are some surface features on Titan that delineate fluvial topography and are visible in images. The polar lakes provide a two-dimensional
imprint of Titan's three-dimensional fluvial topography at a level of detail that is
otherwise inaccessible with current data. SAR images reveal elevated levels of hydrocarbon liquids in some of the polar lakes, which have partly flooded fluvial valleys
around the lake margins, similar to ria shorelines on Earth [101, 111]. If each lake level
33
is a surface of constant gravitational potential, the imaged shoreline is an elevation
contour with a shape that reflects the dissection of the surrounding landscape.
Geomorphological studies on Earth (and even on Mars) typically make use of
abundant topographic data, so there is no established procedure for estimating erosion
from a single elevation contour. In this chapter, we describe such a procedure and
use it to obtain estimates of cumulative erosion in the vicinity of Titan's polar lakes.
In section 2.2, we provide an overview of current knowledge about Titan's polar
lakes and the associated fluvial features. In section 2.3, we use a simple numerical
landscape evolution model to calibrate a relationship between the sinuosity of an
elevation contour surrounding a topographic depression and the average amount of
cumulative fluvial erosion of the surrounding landscape. We test this relationship
in section 2.4 by applying it to a terrestrial landscape, a partly dissected plateau
where cumulative erosion and contour sinuosity can both be measured, and confirm
that contour sinuosity provides useful estimates of erosion. In section 2.5, we map
the shorelines around several polar lakes on Titan, measure the sinuosity of these
shorelines, and use the model-derived relationship to estimate the cumulative fluvial
erosion around the margins of Titan's polar lakes.
2.2
Titan's polar lakes
While Titan's current conditions allow for liquid methane anywhere on its surface,
only the highest latitudes have relative humidity levels at which bodies of liquid
methane are expected to persist without evaporating [101].
Stofan et al. [101] re-
ported the first hydrocarbon lakes observed by Cassini RADAR, and as of this writing,
with 55.4% of Titan's surface imaged by SAR, 655 lake features from < 10km 2 to
> 100, 000km 2 have been identified, all of which are poleward of 550 latitude, with
most at high northern latitudes [1]. The formation processes of the original lake depressions are still unknown, but possible mechanisms include impact cratering [1011,
cryovolcanic caldera collapse [53], karst-like dissolution [54, 64], or tectonic opening
of structural basins [110]. Alternatively, hydrocarbon liquids may have simply filled
34
Figure 2-1: SAR image of Kraken Mare shoreline illuminated from the top at approximately 780 N, 600 E. Illumination direction is from the top of the image. Pixel
values represent normalized radar cross-section o in decibels, from -20 dB (black)
to +0 dB (white). The large peninsula or island is Mayda Insula. The sharp contrast
between the dark lake and the rough, lighter surrounding terrain traces out a topographic contour through a fluvially dissected landscape that appears to have been
partially flooded by a rising lake level.
local depressions on a rough surface. Three types of lake shorelines are observed
[101, 35]. Lakes of the first type have sharply defined, rounded shorelines with no
associated channel networks visible. These may be fed only by subsurface flow of
hydrocarbons, though it is impossible to rule out the possibility that surface drainage
networks are present, given the currently available image resolution [15]. Lakes of the
second type have more distinctly polygonal shapes with rough shoreline geometries
and appear to be fed by fluvial networks. Lakes of the third type are similar to the
second type, in that they are clearly associated with fluvial networks, but portions of
the lake margins where fluvial features terminate have a branching shape that suggests that rising lake levels have flooded river valleys that eroded when lake levels
were lower [101, 35, 110, 1, 111]. This third class of lakes is the focus of our study.
A pronounced hemispherical asymmetry has been observed between the northern
35
polar region, where roughly 10% of the imaged surface area is covered by lakes [1],
and southern polar region, which is largely devoid of lakes [2]. Aharonson et al. [2]
suggest that this observation can be explained by a combination of Titan's 29.5-year
seasonal cycle and an asymmetry in the seasons due to Saturn's axial obliquity and
orbital eccentricity, somewhat similar to the effects of Earth's Milankovitch cycles on
Pleistocene glaciations. Currently, lake levels in the northern polar region appear to
be rising, while observations of Ontario Lacus suggest that lake levels in the southern
polar region are falling [7, 67, 110, 36, 109]. Since Cassini's arrival in 2004, Titan
has progressed from southern summer to early southern autumn and from northern
winter to early northern spring [108].
Continued study may reveal whether, and
by how much, lake levels vary over seasonal time scales [58], providing insight into
Titan's methane cycle.
Several previous studies have discussed the morphology of Titan's lake shores and
the processes that appear to be shaping them. Wall et al. [110] reported evidence
of several processes modifying the margins of Ontario Lacus, including wave-driven
erosion and sediment transport, river delta construction, and possibly regional tilting,
but concluded that the overall form suggests a drowned coast that includes flooded
fluvial valleys. Cornet et al. [21], drawing an analogy to lakes in semi-arid regions
on Earth, additionally suggest that the margins of Ontario Lacus could have been
modified by dissolution, though the solubility of Titan's surface materials in hydrocarbons remains uncertain [55]. Sharma and Byrne [91, 92] compared lake shapes
on Titan and Earth, as measured by fractal dimension, shoreline development index
(the ratio of the shoreline perimeter to the circumference of a circle with the same
area), and an elongation factor, in an effort to identify the geological processes that
formed Titan's lakes. Although they determined that certain formation mechanisms
do produce statistically distinct shapes on Earth, they found that Titan's distribution
of lake shapes is consistent with multiple formation mechanisms.
Given the evidence that multiple processes may have modified the shorelines of
some of Titan's lakes, we focus our analysis on a small number of large, well resolved
lakes with shoreline morphologies that appear to be dominated by flooded river val36
leys. Before turning to measurements of lake shorelines on Titan, however, we seek a
general relationship between the shape of a flooded lake shoreline and the extent of
fluvial erosion that occurred along the lake margins prior to the rise in the lake level.
2.3
Landscape Evolution Model
The valley networks that surround Titan's polar lakes have formed by erosion into
the icy surface [15] by two general processes: fluvial incision and hillslope erosion [11].
We use a simple model that incorporates these two processes [82, 83, 85], which has
previously been applied to Titan [11], to investigate the relationship between erosion
of a rough initial surface and the sinuosity of contour lines around the lake margins
that define the landscape's base level.
2.3.1
Fluvial Incision
The governing equation for the model is based on the conservation of mass of the
erodible substrate, conservation of mass of the fluid that runs off of the surface and
flows through the channels, and expressions describing the rate of erosion of the land
surface. Some tropical valleys on Titan appear to be broad and multithreaded, suggesting that they have dominantly alluvial river beds surrounded by floodplains that
may store significant volumes of sediment [28, 15]. Topographic measurements show
that other rivers have steep-sided, narrow-floored valleys inset into the surrounding
terrain [104, 97, 44, 47, 46, 1, 15], which suggests that they have incised into the
surface and transported the resulting sediment downstream. The characteristics of
the polar networks studied here are consistent with the latter category [11, 15], and a
previous study of finer-scale fluvial networks near the Huygens landing site found evidence that channel incision occurs through mechanical erosion driven by runoff [84].
We therefore assume that the rate of fluvial incision in the polar regions is limited by
the rate at which the channelized flows can detach cohesive material from the channel
bed, rather than the rate at which the flows can transport channel bed sediment and
any sediment delivered to the channel from surrounding hillslopes. We also assume
37
that all eroded material is transported downstream and deposited in lakes, rather
than aggrading in channels.
Theoretical analyses have demonstrated that, despite considerable differences in
materials and physical parameters between Titan and Earth, the mechanisms that
drive channel incision and sediment transport on Earth should also be effective on
Titan [20, 13].
Following models for detachment-limited fluvial incision on Earth
[38, 39, 40, 113] and supporting field observations [26], the rate of channel incision in
our model is set to be linearly proportional to the rate of energy expenditure by the
flow, or "stream power" [90, 39, 114]:
(9t
= -KA'I V z
(2.1)
where z is elevation, t is time, A is the drainage area, I V zJ is the magnitude of the
topographic gradient (that is, the dimensionless surface slope), and K is a coefficient
that depends on substrate erodibility, precipitation rates, channel cross-sectional geometry, and runoff efficiency. The exponent m describes how fluid discharge and
channel geometry vary downstream, and therefore influences the concavity of longitudinal channel elevation profiles. This simple model of fluvial incision is similar to
the approach used to describe river incision into bedrock in numerous other models
of landscape evolution [39, 106, 80, 118, 105].
2.3.2
Hillslope Erosion
As channel beds lower, they steepen the adjacent hillslopes, driving erosion of the
surrounding landscape. The response of hillslopes to this steepening depends on the
nature of the surface material. Possible hillslope configurations include a "bedrock"
surface that is mantled with regolith, a surface of exposed bedrock with no regolith
cover, or a thick deposit of granular material. Images from the Huygens landing site
show abundant, rounded granular material covering the ground surface [104], but it is
unknown how thick this granular layer is or whether this is representative of Titan's
surface in other locations. These different hillslope types will respond differently to
38
a
U
1 km
Transt Points
Transec Line
Valley Bottom
-
Drainage Divide
Elevation (i)
-194
58
Figure 2-2: (a) Orthoimage based on Descent Imager/Spectral Radiometer images
of an elevated area near the Huygens Probe landing site. Superimposed maps show
drainage divides and hillslope transects in the portion of the image analyzed here.
(b) Digital elevation model with color representing elevation relative to an arbitrary
datum. Superimposed maps are the same as in (a), but with the addition of valleys
mapped from the orthoimage. Data from [97]
channel incision, and could develop distinct topographic profiles. If the hillslopes are
mantled with or consist entirely of cohesionless, granular material, slopes should be
at the angle of repose or lower. If the hillslopes consist of exposed bedrock or some
other material with significant cohesion, then slope angles may be steeper than the
39
40
a
20
Basin 1
0
C
.2
-20
0U
-40
-60
-80
0
0
20
40
60
80
Distance from Drainage Divide (m)
b
10 )
Basin 2
-50
0
LU
-100
-150
0
50
100
150
Distance from Drainage Divide (m)
20C
8
C
6
0
4
U~
2
0
-60
-40
-20
0
20
40
Hilislope Angle (Degrees)
60
Figure 2-3: Average hillslope profiles extracted from Huygens stereo topography at
the locations shown in Fig. 2-2 for (a) Basin 1 and (b) Basin 2. Points are average
elevations obtained by binning multiple adjacent hillslope profiles, with error bars
denoting the standard error of the mean of each bin. (c) Histogram of individual slope
measurements from all profiles. Negative slopes correspond to sections of profiles that
slope in a direction opposite to that expected from the overall valley shape.
40
angle of repose.
In the absence of detailed observations of the hillslopes surrounding channel networks on Titan, we use the few available hillslope profiles near the Huygens landing
site (HLS) to constrain the hillslope response to channel incision. The DISR instrument onboard the Huygens probe returned the highest-resolution images of Titan's
surface (< 20 m/pixel). A 50 m/pixel digital elevation model (DEM) and corresponding 12.5 m/pixel orthophotos covering an area of approximately 3 by 5 km were
generated from six overlapping DISR stereo pairs [98] (Fig. 2-2). We identified valley
bottoms as linear features that appear dark in the images and ridgelines as intervening topographic highs that lie between valley heads. We then extracted hillslope
profiles in the two drainage basins where the mapped valleys correspond to topographic lows in the DEM and where valley side slopes were long enough for several
elevation measurements to be taken between a drainage divide and valley bottom.
Elevations were interpolated linearly from the four nearest points in the DEM. In
order to extract representative hillslope shapes from the noisy DEM data, multiple
adjacent profiles within each valley were binned to generate average hillslope profiles
Fig. 2-3(a-b). The average profiles Fig. 2-3(a-b) and the distribution of all slope
measurements Fig. 2-3c show that slopes are typically about 30' (median 31', 95%
confidence interval 170 to 51') but vary widely, with several measurements steeper
than 600. Experiments indicate that static friction angles for sand and gravel under
Titan gravity can be up to 5' higher than terrestrial values, and dynamic friction angles up to 10' lower [48], suggesting that friction angles for cohesionless material on
Titan may lie between 200 and 36'. Although the small number of negative slopes in
Fig. 2-3c suggests that there may be significant uncertainties in the DISR elevations,
and hence slope estimates, the large number of steep slopes and the truncation of the
distribution at approximately 60' are consistent with the hypothesis that many of
the hillslopes in the DISR images equal or exceed the friction angles for cohesionless
material. This observation suggests that hillslope surfaces near the Huygens landing site consist of either exposed bedrock or sediment at a critical angle, and that
hillslope erosion keeps pace with fluvial incision through oversteepening that leads
41
to hillslope failure [10]. Given these measurements, we incorporate mass wasting as
the dominant hillslope erosion mechanism in the model. This is implemented as a
simple rule: when slopes surpass a threshold gradient of 0.6, failure occurs and material is eroded (and transported out of the system) until the gradient returns to the
threshold value [107, 80]. At the spatial scales of interest (tens of km), neither the
specific threshold gradient nor the hillslope erosion law in general has a strong effect
on drainage network development [106].
2.3.3
Initial Surface and Simulation Procedure
The initial terrain is a random, autocorrelated surface punctuated by lakes Fig. 24a, intended to mimic Titan's polar landscape. It is generated by constructing a
synthetic two-dimensional Fourier spectrum with a power-law relationship between
Fourier amplitude F and radial frequency
f, F oc f-8,
adding random perturbations
to the amplitude and phase, and taking the inverse Fourier transform [81].
The
lowest 10% of elevations are designated as lakes, consistent with the approximate areal
coverage of lakes in Titan's north polar region [1], and are assigned fixed elevations and
treated as fluid and sediment mass sinks. Autocorrelation of the surface is controlled
by varying
#: a larger # gives a smoother surface with fewer local minima and maxima,
while a smaller
use
#
#
creates a rough surface with many local minima and maxima. We
= 2.0, which best reproduces the distribution of lakes in the north polar region
and is consistent with the power spectral exponent estimated by Sharma and Byrne
[91], but we also test the sensitivity of our results to this parameter (see Section
2.5.5).
The model landscape evolves by solving Equation 2.1 forward in time using an
explicit finite difference method [82, 83, 85]. Drainage area A is calculated at each
iteration with the Doc algorithm of [103], with local minima forced to "overflow",
such that all points eventually drain into a lake. After each iteration, any slopes that
exceed the threshold gradient are eroded until their gradients return to the threshold
value. All the simulations analyzed in this paper were performed on a 50 x 50 km
domain with periodic boundaries and a grid point spacing of 125 m in both the x
42
Elevation [m]
375
250
125
0
Figure 2-4: Shaded relief maps of topographic surfaces at four instants in a representative landscape evolution model simulation. Color indicates relative elevation.
Normalized cumulative erosion amounts E/Ho are (a) 0, (b) 0.10, (c) 0.20, and (d)
0.50.
and y directions. Initial relief, the difference between the maximum elevation and
the lake level, was approximately 400 m. Each simulation was run for 1 Myr with an
adaptive time step that ensured numerical stability and second-order accuracy. This
time interval is not intended to match the actual duration of fluvial erosion on Titan,
which is unknown, nor is the relief intended to match the actual topography of the
polar regions. The initial relief, the run duration, and the magnitudes of K and m in
equation 2.1 trade off in determining the amount of fluvial erosion that occurs over
the course of a run. Neither the duration of fluvial erosion nor the values of K and
m are known for Titan, so we focus our analysis on the amount of cumulative erosion
relative to the initial relief rather than the absolute amount of erosion or the rate
43
Elevation (m]
350
300
250
200
150
100
50
0
Figure 2-5: Illustration of procedure for measuring shoreline sinuosity. Background
image is a coarsened landscape evolution model solution with colors indicating relative
elevations. White line is the elevation contour taken 20 m above lake level. Gray line
is the contour after smoothing by averaging. Yellow points represent the decimated
version of the smoothed contour, which is used as an estimate of the contour shape
prior to fluvial incision.
at which it occurs, and simply choose a combination of run duration (1 Myr) and
parameters (K = 510-6yr
1, m
= 0.5) that produces extensive erosion of the initial
topography over the course of a run. We set the value m = 0.5, which is consistent
with typical values inferred from field observations on Earth [96, 86].
In section
2.5.5 we quantify the sensitivity of our results to the value of m and the statistical
properties of the initial surface. As noted in section 2.3.2, the value of the threshold
hillslope angle does not substantially influence the fluvial networks generated by the
model.
44
In the early stages of each simulation, small incipient valleys with few tributaries
form at the lake margins (Fig. 2-4b). As these valleys incise deeper, they propagate
upslope and develop more extensive tributary networks (Fig. 2-4c).
Eventually,
fluvial valleys fill the entire landscape, and drainage divides that began as broad
local maxima are intensely dissected (Fig. 2-4d).
As fluvial incision produces a
landscape of branching valleys interspersed with sharp ridgelines, elevation contours
become progressively more sinuous. In section 2.3.4, we show how increasing contour
sinuosity can be used as a proxy for cumulative erosion.
2.3.4
Relating contour sinuosity to cumulative erosion
As noted in section 2.3.3, neither the relief nor the erosion rates are precisely known
for Titan's polar regions, and therefore the model results are not intended to match
a specific time interval, erosion rate, or spatial scale. Instead, we seek a general,
dimensionless relationship between relative erosion and the shapes of topographic
contours around the lake margins.
Several measures of contour shape correlate'with the extent of fluvial erosion. In
their comparison of lake shapes on Titan and Earth, Sharma and Byrne [92] use
the shorelines' fractal dimension [59] and the shoreline development index, the ratio
of a shoreline's length to the circumference of a circle with the same area. Both of
these are useful general measures of lake shape that are influenced by the increase in
local curvature as fluvial valleys dissect the lake margins (Fig. 2-4), but both have
limitations when used as a proxy for erosion. For example, the shoreline development
index is influenced by lake characteristics that are not related to fluvial processes,
such as the overall non-circularity of the lake shape. We investigated the influence of
fluvial erosion on the fractal dimensions of modeled shorelines and found the changes
to be subtle relative to differences associated with initial topography, particularly
when coarse spatial resolution makes it difficult to obtain a precise estimate of the
spectral slope at short wavelengths. In contrast, we found that a simple measure
of contour sinuosity, described below, is a more direct and reliable proxy for fluvial
erosion.
45
a
6
5
5
OP
.09
P
3
.0
.0
.0
0
2
LieWO
2
1
0
0.1
0.2
0.3
0.4
1
0
0.5
Normalized Cumulative Erosion, E/HO (m/mi
0.1
ar
.9
0.2
0.3
0.4
0.5
Normalized Cumulative Erosion, E/Ho [m/mi
d
C 6
5
6
5
go
4
goI
Kraken Mare
2
*
-.
.
,
Kraken
Mare
3 go1
,
do
2
..- *
ntario Lacus
'..-*Kraken
Mare 1
1
10
0.1
0.2
0.3
0.4
0.5
Normalized Cumulative Erosion, E/Ho (m/mi
0
0.1
0.2
0.3
0.4
0.5
Normalized Cumulative Erosion, E/HO [m/mi
Figure 2-6: Calibration curves relating sinuosity to spatially averaged cumulative
erosion for different shorelines on Titan. Separate calibration curves are required for
different shorelines because SAR image resolution, which varies among shorelines,
influences the measured sinuosity. Spatial resolution (the average spacing between
mapped points) in kilometers is (a) 0.71, (b) 0.62, (c) 0.48, (d) 0.54. In each panel,
the solid curve represents the mean trend and dotted lines represent 95% confidence
envelopes. Horizontal solid bars are placed at the measured sinuosity value of each
shoreline. Intersections of the horizontal bars with the calibration curves and the
dashed envelopes respectively represent the cumulative erosion estimates and the
95% confidence intervals. Confidence limits are estimated by binning the model data,
calculating a 95% confidence ellipse for each bin to reflect uncertainty in both sinuosity
and erosion, and fitting polynomial curves to the boundaries of the error ellipses.
We ran the landscape evolution model for 20 different initial surfaces to average
over the random variability in the resulting final surfaces. The final topographic surfaces were resampled by linear interpolation to the same resolution as the image used
to map each contour or shoreline (see sections 2.4 and 2.5). We selected a contour 20
46
meters above the lake level (5% of the total relief of the landscape) to simulate the
flooded landscape of Titan, and calculated the total contour length. The lengths of
contours 15 meters and 25 meters above the lake surface differed by approximately
4%, indicating that the results are not very sensitive to the sampled elevation. This
contour was then smoothed by averaging the coordinates within a distance increment
that is approximately equal to the width of the largest visible fluvial incision features around the lake margin. The smoothed contour is used as an estimate of the
background shape of the lake. This smoothed contour is still artificially long because
it contains the same number of points as the original unsmoothed contour, so the
points are subsampled. We constructed the subsampled contour by retaining every
20th point, which we determined to be the minimum number of anchor points required
to represent the shape of the smoothed contour. This procedure is illustrated in 2-5.
The ratio of the mapped contour length to the length of the smoothed, downsampled
contour is the contour sinuosity, S. This measure of sinuosity has the disadvantage
of being resolution-dependent, but we account for this in our procedure, as described
below.
The spatially averaged cumulative erosion, E, is calculated by taking the difference
between the elevation field at any time during the simulation and the initial elevation
field, and averaging over the grid. To obtain a dimensionless measure of erosion,
we then divide E by the relief of the initial elevation field, HO [11]. We calculated
the sinuosity S at 20 instants in each model run, which corresponded to 20 values of
E/Ho, to calibrate the relationship between these two quantities. We interpolated the
data into 20 evenly spaced bins according to E/HO, and calculated the mean S and
95% confidence interval within each bin. The means define the calibrated relationship
between sinuosity and E/HO, and the boundaries of the confidence envelope denote
the uncertainty in this relationship.
As noted above, a contour sinuosity measurement depends on the resolution of the
data used to map the contour. This must be accounted for in our calibration, because
the resolution of Cassini SAR images varies, and is not necessarily the same as the grid
resolution of our model. We therefore constructed a separate model-derived sinuosity
47
vs. E/HO curve for each Titan lake shoreline Fig. 2-6 by downsampling the model
topography to the corresponding SAR resolution. These coarsened model solutions
are based on the average spacing of the measured points of each shoreline on Titan.
The moderate differences among the curves in Fig. 2-6 illustrate the magnitude of
this resolution dependence. In section 2.5.5, we test the sensitivity of the sinuosityerosion relationship to the fluvial incision parameters and the characteristics of the
initial topography.
2.4
Testing the sinuosity-erosion relationship with
terrestrial data
To test our procedure for estimating cumulative erosion, we compared our modelcalibrated relationship between contour sinuosity and cumulative erosion with a terrestrial landscape where both quantities can be measured. We sought a study site
where fluvial incision has produced significant drainage network development, but
the initial surface could easily be reconstructed. We selected an area surrounding the
Minnesota River in southern Minnesota, USA, which experienced a sudden drop in
base level roughly 13 ka when outburst floods from glacial Lake Agassiz caused tens of
meters of incision in the Minnesota River valley [19]. This base level drop triggered
a wave of incision that propagated up several tributaries, including the Le Sueur,
Maple and Blue Earth Rivers, and into the surrounding plateau of glacial till and
outwash (Fig. 2-7). Many of the tills are overconsolidated and display mechanical
properties more typical of rock than of sediment, such as brittle fracturing and high
cohesion [31]. The transiently incising channels show characteristics of detachmentlimited fluvial incision, including propagation of a knickpoint roughly 40 km up the
tributaries, which has created numerous strath terraces [8, 29, 31]. Gran et al. [31]
considered the most appropriate model for describing the long-term incision, and
found that a transport-limited model (in which erosion rate depends on the downstream divergence of sediment flux) cannot preserve the slope break associated with
48
748.12
Figure 2-7: Elevation map of the study area adjacent to the Minnesota River Valley
(wide blue feature at top of map) in southern Minnesota, USA, based on laser altimetry acquired and processed by the National Center for Airborne Laser Mapping.
The major tributaries are the Blue Earth River, Maple River, and Le Sueur River.
Boxes indicate areas analyzed to generate the points plotted in Fig. 9. White box
represents area shown in Fig. 8.
the knickpoint.
These observations suggest that the main stage of fluvial incision
can be described with the model presented in section 2.3.1. The nearly horizontal
remnants of the plateau surface and the acquisition of a high-resolution laser altimetry map Fig. 2-7 make it possible to measure both the volume eroded by the wave
of fluvial incision [30, 8, 29, 31] and the sinuosity of topographic contours through
49
Elevation [m]
600
500
400
300
200
100
0
Figure 2-8: Coarsened DEM of Minnesota River study area outlined with white box
in Fig. 7, with colors indicating relative elevation. White line is the elevation contour
20 m above the lowest elevation in the area. Gray points represent the estimated
original plateau margin shape reconstructed with the procedure described in Section
2.3.4
the dissected plateau, creating an opportunity to test the model-derived relationship
between erosion and sinuosity.
We divided the margin of the plateau into five sections Fig. 2-7 and coarsened the
topography in each section using the procedure described in section 2.3.4 to yield the
same relative resolution (defined as the average horizontal spacing of elevation points
divided by the length scale of the area of interest) as the SAR images of Titan's polar
lakes. We used a relative resolution of 0.012, based on the values for the Titan study
50
6
5.5
5 --
-
4.5
-
E~
-
4-
0
1
0
0.1
0.2
0.3
0.4
Normalized Cumulative Erosion, E/Ho [m/m]
0.5
Figure 2-9: Calibration curve relating sinuosity to spatially averaged cumulative erosion, based on 20 landscape evolution model simulations, compared with topographic
measurements for the study site adjacent to the Minnesota River (Fig. 2-7). Solid
and dashed lines were calculated as in Fig. 2-6. Small circles correspond to the areas
indicated in Fig. 2-7. The large circle indicates the measurement for the entire study
site.
sites (see Section 2.5). In each section, we measured the sinuosity of an elevation
contour 20 meters above the minimum elevation of the section. These contours trace
out the fluvially dissected topography Fig. 2-8. We then approximated the initial
plateau surface with a horizontal plane at the highest elevation in each section and
estimated the cumulative erosion by measuring the depth of the modern surface beneath this plane. We also made this measurement for the entire area shown in Fig.
2-7. In Figure 2-9, we compare these six paired measurements of contour sinuosity
and erosion with the calibration curve generated from the landscape evolution model
at the same relative resolution. All but one of the measurements fall within the 95%
51
confidence intervals, and the measurement for the site as a whole lies very close to
the mean trend. This comparison indicates that contour sinuosity is a useful proxy
for spatially averaged fluvial erosion.
2.5
Application to Titan's Polar Regions
The general relationship presented in section 2.3.4 provides a framework for estimating cumulative fluvial erosion in Titan's north polar landscape based on topographic
contours from fluvially dissected lake margins. Since there are currently no digital
elevation models with sufficient spatial coverage, resolution or precision to make detailed measurements of the fluvially dissected topography around the-polar lakes, we
use SAR images to map lake shorelines that trace contours through drowned fluvial
features.
2.5.1
SAR data and study region
The RADAR onboard the Cassini spacecraft operates at a wavelength of 2.17 cm and
collects surface data on each Titan flyby while orbiting Saturn [22]. The RADAR
has four operational modes, of which Synthetic Aperture Radar (SAR) produces
the highest resolution images (300-1500 m/pixel). SAR brightness depends on the
roughness, topographic slope, dielectric constant and subsurface scattering of the
terrain [102, 24]. In general, elevated terrain that is rough at the scale of the RADAR
wavelength or contains surfaces with sub-resolution elements that face toward and
away from the spacecraft appears radar bright, whereas smooth, nearly horizontal lake
surfaces appear radar dark, making SAR images suitable for mapping lake shorelines.
Strip-shaped SAR images, or swaths, taken during flybys have captured partial or
complete shorelines of several large hydrocarbon lakes in the polar regions: Ligeia
Mare, Kraken Mare, and Punga Mare in the north and Ontario Lacus in the south
(Figures 2-10 and 2-11).
52
Figure 2-10: Stereographic projection of SAR image mosaic covering Titan's three
major northern polar lakes (maria). Pixel values represent normalized radar crosssection o0 in decibels, from -20 dB (black) to +0 dB (white). White lines are the
mapped shorelines used in this study. Numbers in Kraken Mare label the three
shoreline segments that were analyzed separately.
2.5.2
Test of shoreline identification criterion
Elevated levels of liquid hydrocarbons appear to have partially flooded valleys around
the margins of these four large polar lakes [101, 110, 1]. Assuming that each raised
53
2574.6
2574.5
2574.4 2574.3
.
2574.2
0
2574.1
**
0
2574
e
A
A',
1: 0
50 km
1700 E
1800
Figure 2-11: South polar stereographic projection of SAR images showing Ontario
Lacus with altimetry profile plotted above. Pixel values represent normalized radar
cross-section o-o in decibels, from -20 dB (black) to +4 dB (white). The straight black
line crossing the map indicates the orbit track, and the data points in the upper panel
indicate the radius of Titan at the corresponding points on the surface. A and A' mark
the first and last locations where the altimetry track crosses the mapped shoreline.
lake level defines a surface of constant gravitational potential, these elevated shorelines
trace out contours through the fluvially dissected topography. We test the assumption
that the bright-dark contrast visible in SAR images represents a fluid shoreline by
examining altimetry profiles across lakes. One of the other Cassini RADAR modes
is altimetry, which cannot operate at the same time as SAR, but one altimetry track
does pass over Ontario Lacus and crosses the apparent shoreline at multiple points
54
(Fig. 2-11). We extracted the two nearest elevation points on either side of the first
and last locations where the altimetry profile crosses the shoreline (Fig. 2-11). The
two groups of four points have mean elevations and standard deviations of 2574082±1
m and 2574085 ± 4 m, and further analysis of the altimeter signal revealed a smooth
surface with an RMS height variation of less than 3 mm across each 100 m coherent
resolution cell [121]. These very consistent elevations, combined with the generally
level altimetry profile across Ontario Lacus, suggests that the bright-dark border
visible in SAR images does represent an elevation contour.
2.5.3
Shoreline mapping and sinuosity calculation
Lake shorelines (Figs. 2-10 and 2-11) were mapped manually in SAR images in a
polar stereographic projection by digitizing points along the sharp boundary between
light and dark areas. The spacing between adjacent points was typically 500 m,
comparable to but slightly larger than the SAR image pixel size, but was wider in
areas where the boundary was not as clear. In those areas, we placed shoreline points
only where we were confident of the shoreline location, which required wider point
spacing. We also compared our mapping with adjacent SAR swaths in which the
contact was more sharply resolved, and compared multiple overlapping swaths where
they were available. Table 2.1 lists the average resolution at which each shoreline
was mapped. In the north polar region, we restricted our analysis to the maria (Fig.
2-10). We mapped a single path around Ligeia Mare and the imaged section of Punga
Mare. A closed contour of Kraken Mare has not yet been imaged at the time of this
writing, so we mapped the three imaged sections of the main body of the lake (Fig.
2-10) and analyzed these three contour segments separately. The section of Kraken
Mare to the east of the section labeled 3 in Fig. 2-10 was not analyzed because its
shape appears to be so significantly influenced by fluvial topography that it is difficult
to estimate a background shape.
We calculated each shoreline's sinuosity in the same manner as described in section
2.3.4 for the landscape evolution model. Table 2.1 lists the resulting values. The
sinuosity is between 2 and 3 for all the northern lakes, whereas Ontario Lacus in the
55
south is considerably less sinuous, with S = 1.77.
2.5.4
Comparison with model-derived erosion proxy
Using the measured sinuosity values S of the Titan shorelines Table 2.1 and the modelgenerated calibration curves Figure 2-6, we estimated the cumulative fluvial erosion
in the areas surrounding the Titan shorelines. In Figure 2-6, we plot a horizontal bar
corresponding to each measured Titan shoreline sinuosity. The intersection of each bar
with the calibration curve for the corresponding spatial resolution yields an estimate
of E/HO, and the intersection of the bar with the 95% confidence bounds defines
the confidence interval for the cumulative erosion estimate. The E/HO estimates and
95% confidence intervals are summarized in Table 2.1. In the north, we estimate
that fluvial networks around Ligeia Mare, Punga Mare, and section 3 of Kraken Mare
have eroded through roughly 30% of the initial relief, with a 95% confidence interval
of roughly 20% to < 50%. The shorter sections of Kraken Mare's shoreline yield
somewhat smaller estimates of E/Ho: 17% (9% - 33%) for Kraken Mare 1 and 20%
(12% - 33%) for Kraken Mare 2. In the south polar region, the less sinuous shoreline
of Ontario Lacus implies an E/HO of only 4% (0% - 14%). Thus, for the shorelines
examined in this study, the best estimate of spatially averaged cumulative erosion
falls between 17% and 31% for the north polar region but is only 4% for the one lake
analyzed in the south polar region.
Table 2.1: Sinuosity and cumulative erosion estimates for all shorelines in the study.
**The average spacing between mapped points and the value used to coarsen the landscape evolution
model output
Shoreline
Sinuosity
E/Ho
**Resolution
Kraken 1
2.28
0.17+0: 0
0.54
Kraken 2
2.54
0.20+0: 0
0.48
Kraken 3
2.99
0.31t8
Ontario
1.77
0.04t:g0
0.54
0.48
Ligeia
2.79
0.31-009
0.62
Punga
2.16
0.30+0:09
0.71
56
2.5.5
Sensitivity to initial conditions and fluvial incision parameters
As noted in section 2.3.3, the / value (the negative slope of the synthetic Fourier spectrum) used to generate the initial topographic surface controls the relative smoothness
of the surface, and therefore the distribution of local minima. The distribution of minima in turn determines the size distribution of lakes in the simulation. All the model
calculations presented above used / = 2.0. We tested the sensitivity of our results
to initial conditions by constructing sinuosity-erosion calibration curves for / = 1.7
and 2.5, the extremes of the range of / values that produce a distribution of minima
comparable to the size distribution of lakes on Titan. / values below 1.7 produce
many small, irregularly shaped topographic minima scattered throughout the landscape, whereas values of / higher than 2.5 produce very smooth terrain with a single
large, roughly circular sink, neither of which is consistent with the intermediate-sized,
moderately irregularly shaped lakes that cover roughly 10% of the land area in Titan's north polar region [35, 1] (e.g., Figure 2-10). The results of this sensitivity test
are presented in Figure 2-12. While the slope of the mean trend and the extent of
the 95% confidence envelope do show some sensitivity to
/,
the mean trend is very
consistent for E/HO ~ 0.3, the value estimated for most of the northern lakes analyzed here. The larger range of sinuosity for small values of E/HO implies that our
estimate of the magnitude of erosion around Ontario Lacus is somewhat less certain,
but either of the extreme values of / imply an E/Ho of less than about 12%, which
is still much smaller than the value for the northern lakes. We therefore conclude
that these results (the magnitude of erosion in the north and the different extents
of erosion between north and south) are relatively insensitive to the choice of initial
topography.
We also tested the sensitivity of our results to the parameter m in Equation 2.1,
which describes how fluid discharge and channel geometry vary downstream, and
which is the only tunable parameter in the fluvial incision model. For our study we
set m = 0.5, a typical value for detachment-limited terrestrial channels [96, 114]. We
57
6
P-=2.5
5.5 -
P=2.0
P=1.7
5 -4.5 -
.*
1* .5
2.5
0
,,.
0.1
0.2
0.3
0.4
0.5
Normalized Cumulative Erosion, E/H 0 [rn/in]
Figure 2-12: Sensitivity of the sinuosity-erosion calibration curve to /3, the negative
slope of the synthetic Fourier spectrum used to generate the random initial topography, at the resolution of Kraken Mare 1 and Ontario Lacus. Lines represent mean
sinuosities at different values of /3. Uncertainties are of comparable magnitude to
those in Fig. 2-6
repeated our analysis with m
=
0.4 and m =0.6. A larger value of m indicates that
flow discharge increases more rapidly with increasing drainage area or that channel
width increases less rapidly with increasing discharge. Larger values of m produce
more concave river longitudinal profiles. Comparison of the calibration curves and
confidence envelopes for the three values of m (Figure 2-12) shows close agreement
between m
=
0.4 and m
contour sinuosity with m
=
=
0.5, but somewhat higher values of E/H0 for large
0.6. The mean trends for E/H0 < 0.2, on the other
hand, are very similar for all three values of m. The maximum sinuosity we measured
is approximately 3 (for Kraken Mare section 3), which corresponds to E/H0 ~ 0.3
58
6
----
5.5
5
-
-
m=0.6
m=0.5
m=0.4
-
-
4.5
2:1
3.5
0
S3
0 --0
-
2.5
2F
1.5
1
0
0.1
0.2
0.3
0.4
0.5
Normalized Cumulative Erosion, E/HO [m/m]
Figure 2-13: Sensitivity of the sinuosity-erosion calibration curve to m, the exponent on drainage area, A, in the fluvial incision term, at the resolution of Kraken
Mare 1 and Ontario Lacus. Lines represent mean sinuosities at different values of m.
Uncertainties are of comparable magnitude to those in Fig. 2-6
for m = 0.4 and m = 0.5, compared with E/HO ~ 0.38 for m = 0.6. We therefore
conclude that our results are only moderately sensitive to m. This assumes that m
on Titan falls in the typical range of terrestrial values, which is likely if fluvial erosion
on Titan is driven by open-channel flow fed by runoff [84, 1, 15].
59
2.6
2.6.1
Discussion
Comparison with previous erosion estimates
The range of E/Ho values estimated here for the northern lakes is somewhat larger
than a previous estimate for the average fluvial erosion over a larger area in the north
polar region. Black et al. [11] used a relationship between fluvial network geometry
and cumulative erosion to estimate E/HO for two north polar areas. They based their
approach on the same landscape evolution model used here, so any differences between
their erosion estimates and ours are not due to different assumed erosion laws. Black
et al. [11] estimated 0.4% < E/HO < 9% for fluvial networks south of Ligeia Mare
and > 0.6%(with no reliable upper bound) for fluvial networks between Ligeia Mare
and Kraken Mare. Our measurement approach, which provides a localized estimate of
erosion in areas immediately surrounding the lakes, complements the approach of [11],
who estimated erosion from planform drainage network shapes covering larger areas in
the north polar region. The difference between the two estimates suggests that fluvial
dissection in Titan's north polar region is more extensive near the lake margins than in
areas farther from the lakes, which is consistent with the interpretation that drainage
networks have propagated upslope from the lake margins as fluvial erosion has acted
on a rough initial surface. This is the expected sequence for transient, detachmentlimited fluvial incision (Fig. 2-4). Our estimate for cumulative erosion around Ontario
Lacus is also consistent with a previous estimate of 0.5% < E/HO < 16% in a nearby
region imaged in the T39 swath [11].
2.6.2
Geographical trends
There is a clear difference between the amounts of estimated erosion around the
northern lakes compared with Ontario Lacus in the south polar region (Figure 2-11,
Table 2.1). We estimate that rivers in the north have eroded through 17% - 31% of the
initial landscape, whereas the estimate for Ontario Lacus is only 4%. This estimate
is consistent with the visual impression that the fluvial features near Ontario are less
60
developed and fewer in number, such that the shoreline appears significantly smoother
than the north polar lakes. However, it is also possible that this north-south difference
is partly a consequence of less extensive flooding of fluvial topography in the south:
Cassini has performed multiple flybys over Ontario Lacus and has observed receding
lake levels [7, 67, 110, 36, 109], which raises the possibility that more extensive fluvial
dissection exists around the Ontario margin, but is less apparent because the more
dissected topography is not currently flooded.
Within the northern polar region, the lakes analyzed here have similar estimated
erosional values. Ligeia Mare, Punga Mare and Kraken Mare 3 have E/H0 values of
31%, 30% and 31% respectively, while Kraken Mare 1 (17%) and Kraken Mare 2 (20%)
are somewhat lower. It is not clear if this is an artifact of the analysis on shorter
sections of shoreline, or if different parts of the Kraken margin have experienced
different amounts of erosion. Even if there are systematic errors in the magnitudes of
our erosion estimates, the relative differences in shoreline sinuosity should still provide
an estimate of relative fluvial erosion. We therefore expect that the observation of
similar overall extents of fluvial erosion around different major lakes in the north
polar region is a robust result.
2.6.3
Implications for erosional resurfacing on Titan
Most of the polar fluvial features terminate at lake margins, implying that most
of the sediment produced by fluvial erosion is eventually deposited in the lakes. If
the lake levels are set by the height of a regional subsurface hydrocarbon reservoir
[35, 65], sediment accumulation in the lakes should have a minimal long-term effect
on lake levels. If, on the other hand, lake levels are controlled by direct exchange of
hydrocarbons between the surface and atmosphere, and if the sediment is insoluble in
the liquid that fills the lakes [55], the fluid displacement that occurs when sediment is
deposited in the lakes would raise lake levels. The magnitude of the effect depends on
the amount of erosion that has occurred and the ratio of the lakes area to its drainage
basin area. For example, if the erosion has occurred over a basin about the same size
as the lake itself, all eroded sediment was deposited in a steep-sided lake, and the
61
topography had an initial relief of a few hundred meters, our erosion estimates of
E/HO ~ 0.2 to 0.3 would imply that the lake level has risen approximately 100 m.
a
7
100 kmb
d
f
Figure 2-14: Landscape evolution model simulation of an eroding surface with three
impact craters 50, 100, and 150 km in diameter. Elevation is shown as (a-c) color, with
warmer colors representing higher elevations, and (d-f) shaded relief at a resolution
of 1500 m/pixel, roughly equivalent to the coarsest Cassini SAR resolution. All
panels are 400 x 400 km. Initial elevation range is 400 m. Topography is shown at
normalized cumulative erosion values, E/Ho, of (a,d) 0, (b,e) 0.31, and (c,f) 0.60.
If this effect is as significant as we estimate, it provides a possible non-climatic
explanation for the apparent rise in lake levels that has flooded fluvial topography in
both the north and south polar regions. Estimates of lake volume changes based on
observed lake level changes would need to take sediment displacement into account.
Thus, fluvial erosion has implications for volatile exchange between Titan's surface,
subsurface and atmosphere that extend beyond signatures of atmospheric precipitation. This progressive rise in lake levels would be superimposed on any fluctuations
associated with orbital variability [2].
It is interesting to note that Ontario Lacus, where we estimate the least fluvial
erosion has occurred, is observed to be experiencing a drop in lake level, whereas the
north polar lakes where more erosion is inferred appear to have experienced more
62
flooding around the lake margins. One possible interpretation is that more rapid and
extensive fluvial erosion and lacustrine sediment deposition in the north polar region
have raised lake levels faster than in the south, where slower erosion and sediment
displacement have not kept pace with other factors that are slowly draining Ontario
Lacus. However, the extent of flooding of lake margins may influence our erosion
estimates by producing more sinuous shorelines when lake levels are higher, making
it difficult to disentangle cause and effect. Thus, another possible interpretation of the
hemispheric difference in shoreline sinuosities is that orbital variations [2] have lowered
Ontario's surface to a level where it inundates a less incised and more alluviated
portion of the lake margin, whereas higher levels in north polar lakes have flooded
more incised valleys in which little aggradation has occurred.
2.6.4
Implications for polar sediment and volatile budgets
One of our objectives in estimating the cumulative fluvial erosion of Titan's surface is
to determine whether fluvial erosion might have been extensive enough to explain Titan's young surface age. Determining whether a topographic feature will be obscured
by fluvial erosion is not as simple as comparing our spatially averaged estimates of
E/HO with the relief of the feature, because erosion is concentrated in valleys and
can also vary in the downstream direction. More detailed numerical experiments are
therefore required.
As a qualitative indicator of the potential for fluvial resurfacing, we performed a
set of additional landscape evolution simulations in which the otherwise random initial
surfaces included several impact craters with a range of diameters. We incorporate
craters from the surface of Mars, mapped by the Mars Orbiter Laser Altimeter [95, 94,
72], and scale them to the depths of similarly sized craters on Ganymede. Ganymede
is a good proxy for fresh craters on Titan because both moons have similar gravity
and surface composition, but since Ganymede does not have a thick atmosphere
the craters are relatively well preserved [89, 71]. We then examined the degree to
which the craters remained visually recognizable after different amounts of cumulative
erosion. Our estimates of fluvial resurfacing of these synthetic landscapes should be
63
considered somewhat conservative: no sediment aggradation occurs in the model, so
features such as craters are erased only by eroding them away, not by filling them in.
Figure 2-14 shows the model elevations (Fig. 2-14(a-c)) as well as shaded relief maps based on model topography that has been coarsened to the approximate
resolution of SAR images (Fig. 2-14(d-f)), for different stages in a representative
simulation. We use shaded relief maps as a rough approximation for the appearance
of topography in a SAR image [11, 15], because incidence angle is a major control on
SAR backscatter. At E/HO = 0.31, the largest average amount of cumulative erosion
inferred from our study of the polar lakes (Table 2.1), all three craters are still clearly
visible even in the coarsened shaded relief (Fig. 2-14e). Only once E/Ho exceeds
approximately 0.60 do smaller craters become difficult to recognize in the coarsened
data (Fig. 2-14f). Again, we emphasize that sediment aggradation on crater floors
or in other closed depressions could make craters more difficult to recognize than
Fig. 2-14 suggests. Nonetheless, the observation that crater rims are far from being
obliterated after the amount of erosion we have estimated around Titan's polar lakes
leads us to suggest that fluvial erosion alone may not have been sufficient to create
Titan's geologically young surface, at least in the polar regions studied here.
2.7
Conclusions
We estimated the cumulative fluvial erosion around several lake margins in the polar regions of Titan by comparing the sinuosity of topographic contours defined by
flooded landscapes surrounding Titan's polar lakes to contours produced by a simple landscape evolution model. We find that the north polar lakes in our analysis
have eroded through approximately 17% to 31% of the initial relief of the landscape,
whereas the single south polar lake in our analysis has eroded through approximately
4% of the initial relief. These estimates complement previous studies of erosion over
larger areas in the north polar region, and suggest that the headward propagation
of drainage networks has left lake margins more extensively dissected than highlands
further away from the lakes. These erosion estimates also provide a basis for assessing
64
whether fluvial erosion of topographic features such as impact craters could explain
the young apparent age of Titan's surface. Synthetic impact craters in landscape evolution simulations are still visible at SAR resolution even after 30% of the initial relief
has been eroded, suggesting that, while fluvial erosion has clearly modified Titan's
surface, additional erosional and depositional processes may be required to explain
the extent of resurfacing on Titan.
65
66
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