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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, E01003, doi:10.1029/2008JE003122, 2009
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Long-term precipitation and late-stage valley network formation:
Landform simulations of Parana Basin, Mars
Charles J. Barnhart,1 Alan D. Howard,2 and Jeffrey M. Moore3
Received 20 February 2008; revised 4 August 2008; accepted 26 August 2008; published 24 January 2009.
[1] We use a computer landform evolution model to show that Noachian-Hesperian-aged,
late-stage valley network formation required numerous and repeated moderate flood
events rather than one or a few continuous, multiyear, deluge-style flows. We introduce a
technique that generates an estimated ‘‘initial conditions’’ digital elevation model (DEM)
of the Parana Valles drainage catchment (PDC) prior to valley network incision. We then
explored how variations in three classes of environmental parameters related to fluvial
processes, and surface material properties evolve the initial conditions DEM. Specifically,
we parameterized discharge scaling, evaporation from ponded water, and the effects of an
indurated surface crust. Each simulation run produced a model output DEM that was
qualitatively and statistically compared to the actual surface DEM. Simulations with an
arid to semiarid climate, moderate evaporation rates, and an indurated surface crust
provide the best match to the actual surface. Simulated valley network formation requires
periods of fluvial activity that last a minimum of 103 –104 years under constant delugestyle conditions. However, craters within the PDC in deluge-style simulations overflow
and generate exit breaches that cut through all crater walls. Longer simulations (105 –
106 years) that modeled repeated, episodic flows with interim evaporation avoid universal
crater breaching. The paucity of crater rim exit breaches in the PDC and the southern
highlands in general implies both that the precipitation was not continuous and that
formation conditions were inconsistent with a few short-lived extreme climate excursions
such as might be induced by large-scale impacts or other cataclysmic events.
Citation: Barnhart, C. J., A. D. Howard, and J. M. Moore (2009), Long-term precipitation and late-stage valley network formation:
Landform simulations of Parana Basin, Mars, J. Geophys. Res., 114, E01003, doi:10.1029/2008JE003122.
1. Introduction
[2] The southern highlands of Mars record a complex
history of erosional processes. The Noachian period was
characterized by extensive, widespread, predominately fluvial erosion of highlands and crater rims [Grant, 1987;
Craddock and Maxwell, 1993; Craddock and Howard,
2002; Hynek and Phillips, 2001; Forsberg-Taylor et al.,
2004; Hartmann, 2005]. Highland and crater rim material
deeply infilled crater floors and intercrater basins, effectively
smoothing the surface and generating vast intercrater plains
[Malin, 1976; Irwin et al., 2005a; Howard, 2007]. However, despite pervasive fluvial reworking of the surface,
masking by localized sediment redistribution caused currently
visible mid-Noachian and earlier drainage networks to be
isolated and sparse [Irwin et al., 2005a].
1
Department of Earth and Planetary Sciences, University of California,
Santa Cruz, California, USA.
2
Department of Environmental Sciences, University of Virginia,
Charlottesville, Virginia, USA.
3
Space Sciences Division, NASA Ames Research Center, Moffett Field,
California, USA.
Copyright 2009 by the American Geophysical Union.
0148-0227/09/2008JE003122$09.00
[3] The mid-Noachian erosional regime contrasts with
subsequent, intensive late-stage erosion that occurred
around the time of the Noachian – Hesperian transition
[Howard et al., 2005; Irwin et al., 2005a] (hereinafter
referred to as late-stage activity). Instead of ubiquitous yet
muted erosion, fluvial incision and deposition were spatially
concentrated and are comparatively well preserved. Moreover, the late-stage erosional regime mainly affected the
equatorial highlands, and so was less widespread than the
earlier type of fluvial degradation. Late-stage fluvial erosion
is characterized by valley networks that sharply incise 50–
350 m into what appear to be earlier, relatively planar
upland surfaces (Figure 1). These earlier surfaces are
interpreted to be Noachian fluvial basin fills and deltaic
sediments deposited where drainage debouched into
enclosed basins [Howard et al., 2005; Irwin et al.,
2005a]. Some regions, including the Parana Valles area,
may have been blanketed episodically by widespread sediment [Grant, 1987; Grant and Schultz, 1990; Grant and
Parker, 2002]. In general, crater rims served as drainage
divides, which blocked incoming drainage from most craters
and are rarely breached by exit valleys [Irwin et al., 2005a;
Howard, 2007; Maxwell et al., 2008; Fassett and Head,
2008]. The abrupt cessation of late-stage fluvial activity
[Fassett and Head, 2008] contributes to the preservation of
the valley networks it formed.
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[4] Two principal hypotheses have been advanced for the
evolution from widespread fluvial activity across much of
Mars during Noachian time to the limited yet focused
fluvial erosion during the Noachian – Hesperian transition.
These hypotheses then explain the ultimate decline and
cessation of fluvial activity throughout the Hesperian period. In one hypothesis, the long-term decline was gradual
and resulted from a reduction of atmospheric temperature
and pressure caused by waning geothermal and volcanic
activity, and by loss of atmospheric components both to
space and to weathering reactions [e.g., Carr, 1996]. In the
second hypothesis, episodic, and catastrophic events including orbital cycles, outflow floods, volcanism, and impactinduced climate optima generated or contributed significantly to short-lived greenhouses and brief dramatic fluvial
episodes [Carr, 1989; Segura et al., 2002]. These two
mechanisms are not exclusive and could have been operating in conjunction, such that short-lived events punctuated a
trend of gradual decline. They should, however, produce
different diagnostic erosional and depositional features, and
different regional signatures. Presumably, large, sustained
discharges create thoroughly integrated networks that
breach crater walls and connect large drainage basins.
Moderate discharges and associated periods of quiescence,
on the other hand, would concentrate valley network
development to areas with large catchments and regional
slopes. Some episodic events, such as addition of shortlived greenhouse gasses (e.g., SO2 from volcanic eruptions)
would probably contribute to a general atmospheric warming that would produce seasonal precipitation cycles similar
to terrestrial ones. Orbital variations might also induce
seasonal precipitation cycles.
[5] This paper examines possible environmental conditions responsible for creating the late-stage valley networks
present in the Parana Valles drainage basin catchment
(PDC) using techniques not previously applied to Mars
hydrology. We use morphometric analysis to create an
approximate preincision version of the PDC and then use
landform evolution simulations to explore controls on
valley network formation. The study focuses on valley
networks because they provide evidence of mass transport
over significant distances, and therefore imply sustained or
repeated surface flow. Furthermore, valley network integration is evidence of substantial modification of the land
surface by erosion. Fluvial processes, such as valley network incision, are strongly sensitive to their climatic environment; incision and deposition often alternate in response
to climatic and geologic factors [e.g., Bull, 1991]. Finally,
valley networks are a landform class that is readily amenable
to quantitative analysis through the use of digital elevation
models (DEMs) [Stepinski and Collier, 2004; Stepinski and
Stepinski, 2005].
[6] The mid-Noachian degradation was areally extensive
but involved primarily local sedimentary redistribution
rather than the development of integrated drainage networks. In contrast, the late-stage incision involved a short
episode of focused incision with regionally integrated
drainage. Environmental conditions responsible for latestage incision remain uncertain. One possibility is the
episodic melting of snowpacks and other ice accumulations
[Howard et al., 2005]. This would produce more runoff
relative to sediment yield than would runoff from local
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convective rainstorms in an arid environment. Another
possibility is the development of a thick indurated surface
crust (ISC) over much of the highlands prior to the latestage fluvial activity [Howard et al., 2005]. Subsequent
channel incision would then be limited to larger streams that
had the power to cut through the resistant crust into weaker
underlying substrate [Howard et al., 2005]. A third possibility is renewed erosion following deposition of a mantling
blanket that smoothed an earlier landscape possibly produced by fluvial erosion [Grant, 2000].
[7] Our study consists of three parts. First, we used data
from the Mars Orbiter Laser Altimeter (MOLA) and Thermal
Emission Imaging System (THEMIS) to define and quantitatively investigate the valley networks debouching into
Parana Basin. Next, we used the Mars Simulation Landform
Model (MSLM) [Howard, 2007] to simulate various scenarios for the erosional processes responsible for valley
incision and evolution. Third, we compared model surfaces
to the actual surface both qualitatively and statistically. Our
efforts concentrate on exploring the effect of three parameter classes on the erosional style of the later, geographically
limited, yet strongly focused, late-stage, fluvial activity that
occurred during the Noachian – Hesperian transition. These
three parameter classes are: discharge scaling; downstream
loss via evaporation and infiltration; and surface crust
induration. Our overall goal is to determine whether discharge responsible for the evolution of late-stage valley
networks within the PDC consisted of one or a few large,
multiyear ‘‘deluge’’ events or episodic, seasonal to semiseasonal floods.
2. Background
2.1. Parana Drainage Catchment, Eastern
Margaritifer, Mars
[8] Together the Samara, Loire, and Parana Valles systems dominate the eastern flank of the Chryse Trough that
trends north through Margaritifer Sinus to Margaritifer
Basin [Grant and Parker, 2002] (Figure 2). Parana Valles
is a watershed-defined drainage network, possessing collecting tributaries and well-defined stem or trunk valleys.
The location and orientation of the valleys are significantly
influenced by the local topographic gradients imposed by a
presumed early to mid-Noachian period impact feature, the
Parana Basin [Grant and Parker, 2002]. This 330 km basin
is highly degraded and smoothed, having an indistinct rim
that grades imperceptibly with surrounding slopes and
plains. The deepest and densest valley dissection occurs
on the basin’s eastern and southeastern rim (the Parana
Valles system). This valley system is deeply entrenched
below the level of a broadly sloping upland surface.
Preserved drainage densities of the valley networks
debouching into Parana Basin, at 0.03 – 0.11 km km – 2,
were recognized from Viking data to be among the highest
on Mars [Carr and Chuang, 1997]. Resurfacing by wind,
mass wasting, and impact gardening [Hartmann and
Neukum, 2001] may obscure even higher drainage densities.
[9] We established our study domain using spacecraft
image maps and DEMs. A flow-routing program that
operated on MOLA precision experiment data records
(PEDR) identified drainage divides and defined the Parana
drainage catchment. Typically, catchments are defined with
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drains Parana Basin. In general, the depth and density of
fluvial incision correlate with regional gradients, being
greatest in the Parana Valles system at the eastern and
southeastern portions of the PDC.
[11] Parana Basin is ancient and degraded, predating all
the visible craters in the PDC. The plateau at location A
(Figure 3) may be another strongly degraded and infilled
ancient basin. Loire Valles was created by overflow from
Parana Basin [Grant, 1987, 2002]. The work of Howard
[2007] documents two levels of possible depositional
benches along the southeastern margin of Parana Basin that
coincide, respectively, with the level of the overflow from
Parana Basin before incision of Loire Valles, and with the
present sill level of the overflow into Loire Valles.
[12] Early stages of fluvial erosion in the PDC were
episodically interrupted by continued impact cratering,
creating new basins and damming or diverting preexisting
drainage, as discussed for other places on Mars by Irwin
and Howard [2002] and as simulated by Howard [2007].
Apparently, the northward draining valley at location B
(Figure 3) was diverted out of the PDC by the crater at
location C to flow through the narrow canyon at location D.
The original continuation of the valley at location B flowed
northward within the PDC, but was also interrupted by the
Figure 1. A broadly SE –NW sloping plain (23.275°N,
350°E) southeast of Parana Basin records two distinct
erosional regimes. Widespread early to mid-Noachian
erosion smoothed the surface, generating vast intercrater
plains (middle of plot). Conversely, spatially concentrated
late-stage erosion, occurring near the Noachian – Hesperian
transition, resulted in deeply incised valley networks (top
and bottom of plot). The contrast in erosional regimes is
well expressed by the faint, muted hanging valley (arrows)
intersecting a sharply incised valley network (top middle of
plot). Thermal Emission Imaging System (THEMIS) VIS
image V06717003, image width 16.6 km, overlies a
THEMIS daytime IR mosaic with elevation cueing from
Mars Orbiter Laser Altimeter (MOLA) topography.
reference to a narrow outlet. The PDC drains into Loire
Valles, and our study domain focuses on tributaries feeding
the headward end of Loire Valles including Parana Valles
where significant valley incision and integration occurred
(Figure 3). Full resolution (100 m pixel1) daytime
THEMIS infrared (IR) images were assembled into a
mosaic using the U.S. Geological Survey image processing
program, ISIS, and coregistered with MOLA PEDR topographic data. Data from the Viking Orbiter Mars Digital
Image Mosaic (MDIM) filled daytime THEMIS IR coverage
gaps.
2.2. Geomorphic Evolution of the Parana Drainage
Catchment
[10] The PDC resides on a broad regional slope descending from the high divide between the Margaritifer and
Hellas basins (Figure 3). The portion of the PDC southeast
of the Parana Basin is characterized by a strong regional
slope to the NW and is deeply incised by the Parana Valles
system. The northeast corner of the region is more gently
sloped and less incised except for the deep Loire Valles that
Figure 2. Shaded relief topography of Margaritifer Sinus
provides context for the Parana drainage catchment (PDC),
which resides on a SE – NW sloping plain on the eastern
side of Chryse Trough (middle of plot 20°N, 15°W). The
model domain used in our landform evolution simulations
by the Mars Simulation Landform Model (MSLM) is
bounded by the white box (shown in detail by Figure 3).
Elevation cueing and shaded relief from MOLA topography.
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the Parana Valles network strongly imply that the valley
formation occurred over an extended period of time, whether
continuous or episodic. Other midsized and large craters
throughout the PDC possess whole, intact crater rims without
any evidence of breaching.
Figure 3. The Parana drainage catchment is defined by
drainage divides (dashed yellow line) derived from MOLA
elevation. The PDC is captured by a rectangular digital
elevation model (DEM) shown here by a mosaic of
THEMIS daytime IR data and Viking Mars Digital Image
Mosaic (MDIM) with elevation cueing from MOLA
centered at 25.5°S, 12.25°W. The DEM extends 840 km
in the north-south direction and 630 km in the east-west
direction from roughly 18.5° to 32.5°S and 342.5° to
353.0°E and horizontally covers roughly 529,200 km2. The
southeastern portion of the PDC is characterized by a strong
regional slope to the NW and is deeply incised by the
Parana Valles system. In general, the depth and density of
fluvial incision correlates with regional gradients. Near the
end of fluvial incision a row of craters became erosionally
breached, permitting headward erosion of the stubby
canyon at location H. Other midsized and large craters
throughout the PDC possess whole, intact crater rims
without any evidence of breaching. Additional features of
the PDC, labeled A –H, are described in the text.
crater at location E. Another degraded fluvial valley extends
north-south through location F, and apparently was interrupted by the crater at location G as well as beheaded by the
cluster of large craters west and southwest of the crater at
location E. The row of craters at the south end of Parana
Basin may have dammed drainage that formerly extended
northward into Parana Basin, causing deep sedimentation in
the upland plateau at location A. Near the end of fluvial
incision this row of craters became erosionally breached,
permitting headward erosion of the stubby canyon at
location H. These impact interruptions in the evolution of
2.3. Valley Networks
[13] We digitally mapped all resolvable valley networks
within the PDC. Valleys resolved by MOLA topographic
data are mapped in white (Figure 4). We digitized the
principal, central trunk valleys of stream systems in 3-D
space (latitude, longitude, and altitude) on the basis of
points 3 km apart along thalwegs, defined by contour lines.
Following the valley digitalization techniques discussed by
Howard et al. [2005], the elevation of each point is defined
by all MOLA PEDR (shot) data within a 3 km search radius
to determine the maximum, minimum, average and 75th
percentile elevation. A 3 km search radius ensured that
points from the plain surrounding the valley were captured.
The valley floor is assumed to be the minimum point. The
75th percentile captures the elevation of the surrounding
surface while diminishing the influence of local highs.
Although selection of the minimum point within a 3 km
radius introduces a downstream bias, this bias is present
throughout the profile. Accordingly, minima point selection
preserves profile shape and, together with valley shoulders,
yields measurements of incision depth, gradient, and downstream length. Valley profiles measured in Parana Basin
share many characteristics similar to those on the Isidis rim
[Howard et al., 2005]. Profiles are broadly convex, stepped
and irregular (Figure 5). Moreover, there is a positive
relation between incision depth, gradient, and cumulative
downstream length [Howard et al., 2005].
2.4. Mars Surface Landform Model
[14] MSLM simulates long-term landform evolution by
weathering, mass wasting, fluvial, eolian, and lacustrine
processes [Howard, 1994a, 2007; Forsberg-Taylor et al.,
2004; Howard and Moore, 2004]. MSLM is a gravitationally scaled version of the terrestrially based Detachment
Limited Model (DELIM). DELIM has successfully predicted the evolution of terrestrial landscapes [e.g., Howard,
1997]. The models provide explicit simulations of landform
development and thus predict the evolution of the surface
topography and the final landscape. Here, we apply them in
particular to quantify valley network formation.
[15] The model reads in a user-supplied file containing
parameter controls and models evolution of landforms
starting from an initial DEM or matrix of elevation values.
Elevation changes at any given cell within the DEM result
from a linear combination of diffusive mass wasting, which
tends to smooth the topography, and fluvial erosion and
deposition, which operate in channelized flows. The potential for fluvial incision is a positive function of discharge
(parameterized to increase as a power function of contributing area) and local gradient. Mass-wasting processes
dominate at divides and upslope regions. Fluvial processes
dominate downstream. Model specifics are outlined in
Appendix A and described extensively by Howard
[1994a, 1997, 2007].
[16] To first order, MSLM simulates channel incision by
calculating terrestrially equivalent mean annual flood dis-
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Figure 4. Parana Valles (boxed region in Figure 3) drains
to the northwest into Parana Basin. Valley networks
observed in MOLA topographic data are mapped in white.
Valley section A – B is a 270 km segment of the 370 km
central trunk valley of the Parana Valles network. THEMIS
daytime IR mosaic with elevation cueing from MOLA.
charges as a function of contributing area, and subsequently,
by calculating incision rate as a function of discharge. In
terrestrial drainage systems, the majority of geomorphic
work occurs during floods. An annual flood is the maximum discharge peak that occurs during 1 year. Mean annual
floods, or the mean of a series of peak annual stream
discharges, appear to control alluvial channel dimensions
in many terrestrial regions [e.g., Knighton, 1998]. On the
basis of channel width and meander wavelength measurements in preserved Martian channels, the formative discharges within these channels is about the same magnitude
as mean annual floods in terrestrial channels having the
same contributing area [Howard et al., 2005; Moore et al.,
2003; Irwin et al., 2005b].
[17] We assume that the mean annual flood, expressed as
a function of contributing area and any evaporative losses in
lakes, is an appropriate discharge value for estimating the
shear stress responsible for bedrock channel incision. The
bedrock erodibility (Kb in equation (A4)) is estimated by
relating long-term terrestrial erosion rates in weak sedimentary rocks to the corresponding shear stresses within the
channel produced by the mean annual flood. The resulting
rate of erosion is scaled to equivalent rates of erosion in
terrestrial drainage networks in arid to semiarid climates.
The actual erosional time scale on Mars might be different if
the frequency (recurrence interval) of flood discharges is not
the same as on Earth. For example, if erosion occurs only
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during rare climatic optima produced by volcanic greenhouse gas emissions or during favorable orbital configurations then the Martian time scale could be much longer than
the terrestrial one.
[18] The study uses three DEM products: the actual
surface (AS), an ‘‘initial conditions’’ surface (ICS), and
the model output surface (MOS). The AS is defined by the
present-day topography of the PDC. It was generated by the
commercial program Surfer1 from the MOLA PEDR shot
data based upon a natural neighbor interpolated DEM with a
square topographic grid with pixels 1 km on a side. The AS
DEM is centered at 25.5°S, 12.25°W and extends 840 km in
the north-south direction and 630 km in the east-west
direction from roughly 18.5° to 32.5°S and 342.5° to
353.0°E. The AS DEM spans an elevation range from
3134 to 1959 m and comprises an area of roughly
529,200 km2.
[19] MSLM simulation runs begin with an ICS DEM,
whose generation is discussed later. The ICS DEM spans
the same latitudinal and longitudinal extent as the AS DEM.
However, computational efficiency considerations limited
the spatial resolution of simulation-run DEMs. Therefore,
we degraded the resolution of the ICS to 3 km per pixel.
Reduced model resolution introduces a notable caveat:
valleys and channels smaller than 3 km were not directly
simulated. As a MSLM simulation proceeds, MOS DEMs
are generated at user specified intervals by MSLM evolving
from the ICS DEM. Accordingly, they share the same
resolution (3 km per pixel side) and longitudinal and
latitudinal range as the ICS DEM. Data for quantitative
analysis and shaded relief maps for qualitative analysis are
derived from MOS DEMs.
2.5. Model Caveats and Assumptions
[20] Domain boundaries significantly affect the simulated
regional evolution. The domain was chosen to capture the
Figure 5. The valley profile of Parana Valles from point A
to B marked in Figure 4 is stepped and irregular. The valley
floor (solid line) was obtained from the minimum point
MOLA precision experiment data records in a 3 km radius
from the valley center. The shoulders were defined by the
75th percentile elevation in the same 3 km radius. In
general, valley depth correlated with gradient and downstream reach.
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PDC in its entirety. Because of the PDC’s irregular shape,
other drainage basins, particularly the Samara drainage
basin in the southwest part of the domain, were subjected
to the same MSLM flow routing and erosive controls as the
PDC but without their full aerial extent and corresponding
discharges. Other drainage basins were likely underincised
when compared to the actual surface. For this reason, we
focused our qualitative analysis on valley networks in the
PDC and not on other drainage basins only partially
captured by the model domain.
[21] Although verified terrestrially and scaled to Martian
gravity, MSLM simulates landform evolution based on
terrestrially derived empirical hydrologic relations [Howard,
2007]. Surface properties, loss rates due to evaporation and
infiltration, and general climatic and geologic properties
were possibly different on Mars. For this reason a wide
range of parameters was explored. Comparing runs to one
another and using the actual surface as reference best serves
hypothesis testing. Simulation results and interpretation
allow exclusion of hypotheses but cannot determine exact
parameter values.
3. Generating an Initial Conditions Surface
[22] As a necessary preliminary to running model simulations of the fluvial incision present in the PDC, we derived
an initial conditions surface from the extant topography. The
initial topography has a strong influence on the evolution of
topography in the simulations, primarily through determining the overall layout of main channels and divides. It has a
weaker influence on details of individual slopes and only a
modest influence on averaged morphometric properties
such as drainage density, average stream gradients, and
moments of slope gradient or divergence [Howard,
1994b]. Previous studies show clear and significant fluvial
reworking of the surface [Grant and Parker, 2002]. Geomorphic work includes downslope transport of sediment,
valley incision, and slack water infilling with alluvium. The
actual preincised ICS cannot be reproduced, but the following method generates a reasonable approximation. Our goal
has been to reproduce the topography that existed just prior
to the late-stage fluvial incision but subsequent to the major
fluvial diversions due to impact cratering discussed in
section 2.2.
[23] ICS generation is a new technique in modeling of
Mars landform evolution. The following describes the steps
we took to generate an ICS for the PDC. The valleys
throughout the PDC were removed first. Following widespread fluvial activity in the mid to late Noachian, intercrater plains were assumed to be largely ungullied, and large
(>200 km) crater basins were assumed to be graded and
shallow. This smoothed surface was recreated by systematic
infilling of extant valleys. We produced a detailed contour
map using the commercial program Surfer1 from the AS
DEM. We examined this topographic map for broad divides
between valleys, and we collected a region-wide network of
digitized divide points. These digitized divide points were
assumed to define the broad, preincision surface. We did not
include narrow intervalley divides because slope processes
may have lowered them. In addition, degraded craters larger
than 10 km in diameter within the region were defined by
digitizing their rim crests, the bases of the interior rims, and
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points on the crater floors. The resulting irregular grid of
XYZ (latitude, longitude, elevation) points (Figure 6) was
converted into a square-gridded DEM using natural neighbor interpolation in Surfer1 with 1 km grid cells. This
method smooths the surface by removing valleys while
preserving crests of drainage divides and regional slope
gradients.
[24] This preliminary ICS grid was then modified by
reconstructing craters larger than 30 km in diameter to fresh
crater geometry using simulated impacts as described by
Forsberg-Taylor et al. [2004] and Howard [2007]. Scaling
statistics obtained from morphometric studies of fresh
Martian craters determined simulated crater shapes [Garvin
et al., 2000]. Crater reconstruction was conducted for two
reasons. The first is that craters reconstructed solely from
digitized points did not have well-defined circular rims of
nearly constant relative relief. Low points on the crater rim
(generated artificially from interpolation) would allow
drainage to enter or exit where it actually did not occur.
The second reason is that many of the intermediate-sized
craters (30 – 80 km in diameter) would presumably have had
higher rims and lower floors. These features would allow
greater water and sediment storage capacity than was the
case at the end of late-stage incision. The reproduced craters
have nearly circular rims although some of the actual craters
are noticeably elliptic.
[25] A second modification to the preliminary ICS DEM
reshaped Parana Basin to compensate for late-stage fluvial
and lacustrine infilling. The basin is the principle sink for
alluvium in the Parana catchment and was therefore assumed to be significantly infilled during the late-stage
erosional epoch. The volume of fill within Parana Basin,
calculated using an initial crater volume minus the volume
remaining unfilled, is estimated to be 23,300 km3 [Grant,
2000]. In ICS DEM construction, 22,650 km3 of material
was removed from the ICS grid using a radially symmetric
cosine function rotated about an axis centered at Parana
Basin (22.5°S, 12.5°W):
zðrÞ ¼
i
z0 h rp
cos
þ 1 ; for 0 r R
2
R
ð1Þ
where z is the depth of removal in kilometers as a function
of radius, r (km), from the basin center, z0 (km) is depth of
material removed from the center of the basin, and R is the
radial extent of the excavation. We chose an arbitrary but
reasonable central removal depth of 1 km and a radial extent
of 120 km to match volumetric removal estimates of
23,000 km3. A cosine function was selected to minimize
both an abrupt gradient change or lip and gradient influence
on model valley network incision near the artificial
excavation of Parana Basin. We did not attempt to
reconstruct a fresh crater rim for Parana Basin because its
advanced stage of degradation suggests that fluvial activity
and/or possible mantling prior to the late-stage fluvial
activity had already obliterated the rim.
[26] To provide tractable boundary conditions, all cells on
the edges of the simulation domain are assumed to be
unerodible in the MSLM simulations. However, two
boundary DEM cells were set to present topographic levels
where Loire and Samara Valles exit the DEM domain to the
northwest and west, respectively. These cells effectively act
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then becomes the basis for estimating late-stage fluvial
incision depths from valley depths below the initial surface.
As discussed in section 2.5, we reduced the ICS grid
resolution to square 3 km grid cells, for computational
efficiency.
[29] Our methodology, specifically the construction of the
ICS, and the use of MSLM, involves several assumptions.
In particular, we assume that the ICS approximates the
landscape in the region just before the end of the Noachian
period, and that it lacked deeply incised channels. The
smoothness of the surface is assumed to represent some
combination of (1) local fluvial reworking without much
regional drainage integration, (2) a hyperarid climate, and
perhaps, (3) regional mantling. Modeled scenarios intend to
determine what combination of processes and surface material properties reproduce, in a general way, the observed
pattern of subsequent channel incision. The actual processes
responsible for late-stage incision were probably both
temporally and spatially unique. They likely combined
continuous, episodic, and high-intensity processes. However,
for the sake of hypothesis testing and amenability to numerical simulations, we assume that a constant climate and
regionally uniform material properties extending over a finite
duration can represent the late-stage erosional regime.
Figure 6. An initial conditions surface (ICS) was created
as the input DEM for all simulation runs. Here digitized
points defined by MOLA derived latitude, longitude, and
elevation used in the construction of the ICS are superimposed on a shaded relief map of the Parana Valles region
of the PDC. An ICS DEM was generated by interpolating
points (black circles) thought to represent a surface that
postdated the mid-Noachian regime but predated late-stage
fluvial incision. Selection focused on broad plains and
interfluvial divides as denoted by the asterisk symbol. The
rims and floors of craters smaller than 30 km in diameter
(i.e., C and D) where heavily sampled in order to accurately
reproduce their shape. Craters larger than 30 km (i.e., A and
B) were reconstructed to fresh crater geometry.
as drainage sinks. The fixed points allow fluid and alluvium
to exit the domain. These exit points encourage headward
migration of knickpoints and motivate incision, acting as a
proxy for regional topography external to the simulation
domain in which the Loire and Samara Valles systems both
exhibit downstream control.
[27] Although exit points for Loire and Samara Valles are
artificially low, they do not affect the evolution of Parana
Valles. The Parana Basin presents a significant topographic
minimum between Parana Valles and Loire Valles. Parana
Valles enters the SE Parana Basin whereas Loire Valles
drains the Parana Basin from the NW at a present-day
elevation that is 500 m higher than the basin floor. Samara
Valles drains an entirely different drainage catchment to the
southwest of the PDC.
[28] In summary, although the actual preincised ICS is
impossible to reconstruct, this technique results in a DEM
that preserves regional gradients, drainage divides, and large
craters, while removing valley incision (Figure 7). This ICS
4. Hypotheses to Test
[30] We explore how variations in three classes of environmental parameters related to fluvial processes and surface material properties affect patterns of valley network
incision. The three parameter classes determine incision
patterns by (1) discharge scaling; (2) the relation between
precipitation and evaporation; and (3) the presence of an
indurated surface crust. Each parameter class is tested
against actual spatial distribution and depth of incised
valleys within the Parana Valles region.
4.1. Discharge Scaling
[31] The first approach is to explore various discharge
scaling relations with respect to contributing area as a proxy
for different precipitation climates and substrate hydrologic
properties. Discharge, Q (m3 s1), is the flow going through
a stream and is assumed to be proportional to the contributing area:
Q ¼ kAa ;
ð2Þ
where A is the upstream contributing area (m2) and a scales
the dependence of discharge on contributing area. In
terrestrial settings, discharges of magnitude approximating
the mean annual flood perform the majority of annual
geomorphic work [e.g., Bull and Kirkby, 2002]. The
assumed default value of constant, k, scales discharge to
the mean annual flood for terrestrially semiarid conditions.
An a value of 1 means that the mean annual flood is
directly proportional to the contributing area. Lower values
of 0.5 or 0.3 mean that discharge does not increase as
rapidly downstream. For drainage networks on Earth, an a
value of 0.7 would be typical of a humid environment;
whereas lower values are characteristic of semiarid to arid
environments [Bull and Kirkby, 2002]. This study tests
alpha values of 0.3, 0.5, and 0.7. The discharge constant, k,
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overflow from other basins may contribute to the present
basin at a rate VI. A yearly water balance for the basin is
thus:
V0 ¼ VI þ ðAT AL ÞPRB þ AL P EAL :
ð3Þ
The work of Howard [2007] details an iterative procedure
for calculating basin inflows and outflows in a network of
interconnected channels and basins. For Mars we do not
know within wide limits the actual values of precipitation
and evaporation, but the degree to which basins overflow is
determined by the ratio, X, of net evaporation in lake basins
to runoff depth on uplands:
X ¼
ð E PÞ
:
PRB
ð4Þ
All lakes overflow as X ! 1, and lakes become
indefinitely small as X ! 1. For the present simulations
we assume that P, RB, and E are spatially invariable and we
assume values for X of 1.0, 5.0, and 10.0.
Figure 7. The ICS DEM was used as the starting point for
all model simulations. It is assumed to approximate the
landscape of the Parana drainage catchment just prior to the
late-stage fluvial activity. The ICS was extrapolated from
the actual surface through (1) systematic infilling of extant
valleys by interpolating digitized interfluvial divide points,
(2) geometric reconstruction of highly degraded craters, and
(3) removal of material in Parana Basin.
is adjusted so that the discharge, Q, generated by a moderate
contributing area of 7.5 km2 is the same for all simulation
runs and scaled to terrestrial semiarid conditions. For
example, the Colorado Plateau, a semiarid setting on Earth,
exhibits a strong correlation between mean annual flood
discharge, Q, and contributing area with an a near 0.5.
(Figure 8).
4.2. Runoff Evaporation Scaling
[32] Discharge is further limited by evaporative losses
within depressions, mostly craters. Little direct evidence
exists of the actual early Mars precipitation rates and their
dependence on time, latitude, and elevation. However, the
degree of basin overflow, and hence drainage network
integration, depends largely on the ratio of evaporation to
runoff. The works of Howard [2007] and Matsubara and
Howard [2006] developed a flow-routing model that
accounts for evaporative losses. These models successfully
predict the lake size and distribution of lakes in the Basin
and Range during both Pleistocene and Holocene times.
Consider an enclosed drainage basin of total area AT with an
included lake of area AL. We treat a multiyear water balance
with the average precipitation rate P (depth per year). On
the uplands the fractional runoff yield is RB. Yearly evaporation rate on the lake is E. With sufficient precipitation the
lake may overflow at a yearly volumetric rate V0, and
4.3. Incision Through a Late Noachian Indurated
Surface Crust
[33] An ISC or layer may have developed on the late
Noachian landscape prior to the late-stage incision [Howard
et al., 2005], perhaps gradually or during an episodic
climate favoring its development [Dixon, 1994]. Furthermore, the presumed absence of vegetation and bioturbation
on Mars may arguably lead to thicker, more cohesive
indurated surface crusts. On the other hand, impact gardening, if the atmospheric density at the time permitted it to
occur, might have inhibited crust formation. MSLM simulates an ISC by dividing the erosion rate by an erosive
resistivity value, r, for the topmost layer of a given
thickness, Hd, that blankets the model domain (see equations (A1) and (A4) in Appendix A). Also, the intrinsic rock
weathering rate, W0 (equation (A1) in Appendix A), is
divided by the factor r. An ISC layer would reduce sediment
yields from upland areas and focus erosion within larger
channels. In addition, an impermeable ISC might increase
upland runoff, although we do not simulate this. ISC
thicknesses of 1, 2, and 10 m with enhanced erosive
resistivities of 10x, 20x, and 30x were simulated.
4.4. Simulation Procedures
[34] Starting from the ICS topographic grid, the model
simulates landform evolution under a set of assumed model
parameters. The simulations are grouped into families of
runs in which one parameter is varied while the others are
held constant. Combinations of discharge scaling, evaporation ratio, and ISC parameters constitute 108 possible
environmental combinations. We ran 72 simulations that
covered end-member scenarios while iteratively focusing on
combinations that yielded superior chi-square values (see
section 5). Simulations are run to the point that the surface
is eroded well beyond its actual state. This insures that
simulations were not underrun. Run data and MOS DEMs
are collected at the time of optimal statistical fitness
between the MOS DEM and the AS DEM (discussed in
section 5). Simulations discussed in this work are listed in
Table A1 in Appendix A. Qualitatively, variations of the
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Figure 8. Mean annual flood values for 37 streams and
rivers in Arizona, Colorado, Utah, and New Mexico (the
largest being the Colorado River at Grand Canyon before the
building of Glen Canyon Dam) as a function of contributing
area. Equation (2) states that flow (m3 s1) through a
stream, Q, is proportional to contributing area, A. Discharge
increases more rapidly downstream for higher values of the
discharge exponent a. As a proxy for climate, we test a
values of 0.3, 0.5, and 0.7, which are typical of arid to
semiarid environments on Earth. The regression trend for
these 37 streams and rivers generates an a value of 0.48.
three parameter classes have broad scale effects on landscape evolution (Figure 9) and are discussed in detail
(section 7).
5. Quantitative Analyses
[35] The use of high-resolution DEMs and MSLM allows
detailed statistical and quantitative analysis of model simulations and comparisons with the actual pattern of valley
incision. A variety of statistics was used to guide hypothesis
making and testing, and to test correlation and goodness of
fit between the actual surface and modeled landscapes
evolved under specific processes. Additionally, valley network formation lifetimes and minimum required discharges
were calculated for runs with the best statistical fit.
[36] The principal statistical tool we used in this study
was an elevation difference histogram. As a simulation
proceeds, the DEM continues to evolve by diffusive and
fluvial transport processes. Material erodes from higher
elevations, and valleys become incised, so that material is
deposited at lower elevations and in local minima. Snapshot
MOS DEMs are captured throughout a simulation. Elevation difference DEMs are produced by subtracting the
‘‘initial,’’ preincision DEM (the ICS), node-by-node, from
MOS DEMs generated by the simulation.
[37] Though illustrative, elevation difference DEMs are
not directly useful for comparing suites of model runs with
the actual valley pattern. Valley network formation is a
complex, evolutionary process and we cannot be assured
that our initial surface exactly represents a prior geomorphic
surface. Although similar network morphology may form, it
will not form in exactly the same place in different simulations or in the same detailed pattern as the actual valley
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network. Rather than being interested in exactly where
geomorphic work was performed, we concentrate on the
cumulative pattern of erosion and deposition and on variations in degree of incision.
[38] To produce an elevation difference histogram integrated over the entire simulation domain, cells of each
elevation difference DEM were binned into 5 m intervals.
Negative values or areas of the simulated surface that are
lower than the initial surface are assumed to indicate areas
of erosion. Positive values (conceptually, areas of deposition) are not considered in this study for two reasons: (1) the
technique used to generate the ‘‘original’’ surface added
material somewhat arbitrarily to the extant DEM of the
Parana catchment, and (2) the geomorphic process we are
exploring, valley network incision, is an erosive process.
Additionally, to focus on valley network incision on the
intercrater slopes and plains, crater basins, including Parana
Basin, which underwent sediment filling, were masked and
systematically removed from calculations that generated the
elevation difference histogram.
[39] By subtracting the ICS DEM from the AS DEM and
then binning the cell-by-cell differences we generated a
baseline difference histogram to which simulation runs were
compared (Figure 10). Of particular interest is the strongly
negative end of the histogram. This indicates deep, concentrated erosion that we assume was generated by late-stage
valley incision. Each model simulation generates numerous
elevation histograms as the run progresses. Runs that are
overly erosive overshoot the baseline, whereas runs that
produce a histogram line that is under baseline fail to
generate deep-focused erosion.
[40] We employ a chi-square (c2) distribution to quantify
differences between the baseline histogram and histograms
generated from model runs. This has two purposes: (1) to
statistically compare elevation difference histograms between model simulations, and (2) to compare multiple
elevation difference histograms from a particular simulation
in order to determine when that run came to its closest
statistical match to the inferred pattern of late-stage incision.
Chi-square (c2) values are calculated for areas of net
erosion as follows:
!
N
X
ðbXi Pi Þ2
;
c ¼
Pi
i¼1
2
N
X
b¼
i¼1
N
X
ð5Þ
Pi
;
ð6Þ
Xi
i¼1
where i is the elevation range bin number, N is the total
number of bins, Xi is the number of MOS DEM cells whose
elevation difference from the ICS DEM fall within bin i, and
Pi is the number of baseline, or DEM cells from the AS
elevation difference histogram that fall within bin i.
[41] Traditional use of c2 statistics requires that the total
N
P
Xi, equals the total number
number of observed data,
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Figure 9. A sample of nine model output surface (MOS) DEMs cover the parameter space explored in
this study. Each row demonstrates the effect of a particular parameter: a, evaporation ratio, and indurated
surface crust (ISC). Unless varying with their particular row, the parameters are held fixed and set to 0.5,
0, and 0 m, respectively. Each run is labeled with its variable parameter value (bottom). The contributing
area exponent, a, controls how rapidly discharge increases downstream (see equation (1) and section 4.1).
Evaporation ratio controls how rapidly water is lost from ponds and lakes (black regions) and indirectly
controls network integration. An ISC channelizes and focuses flow, generating deep, concentrated
incision. Many simulations here fail to match the pattern of late-stage valley incision and occur at the
extreme range of simulation parameter values. They either fail by forming premature valley networks or
by overincising the surface and downcutting crater rims. Elevation difference histograms for the top left
and top right runs are plotted in Figure 11 as examples of too little and too much valley incision with
respect to the actual surface. Note that runs with high a values and low evaporation ratios simulating
sustained multiyear deluge-style precipitation events breach nearly all crater rims (top right corner).
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errors, however, do not invalidate our analysis. We take
advantage of c2 statistics as a numerical technique that
objectively and quantitatively compares an observed distribution of elevation differences to a theoretical one: something the geomorphologist’s mind does intuitively but
qualitatively and subjectively.
6. Quantitative Results
Figure 10. Simulation runs were compared to a baseline
elevation difference histogram (solid line) as described in
section 5.1. Of particular interest is the negative end of the
histogram. This indicates deep, concentrated erosion that we
assume was generated by late-stage valley incision. Each
model simulation generates numerous elevation histograms
as the run progresses. Runs that are overly erosive
overshoot the baseline (e.g., run 37, dashed line), whereas
runs that produce a histogram line that is under baseline
(e.g., run 01, dotted line) fail to generate deep-focused
erosion. A measure of goodness of fit between the baselineand model-generated histograms was calculated by employing c2 statistics (equation (5)).
of predicted or baseline data,
N
P
i¼1
Pi. Because elevation
difference histograms were calculated only for model cells
where erosion is represented by a negative change in
elevation, the total number of observed data varied throughout a simulation. The normalization factor, b, maintains
population equivalency between elevation difference histograms from separate runs as well as histograms generated by
the same run at different time steps (equation (6)). Values
for b ranged from 1 to 1.5.
[42] The ICS is an estimation of the preincised surface.
An exact c2 match (equaling zero) would not be expected.
Thus, relatively low c2 values indicate a close statistical fit
between the range of incision depths in the simulation and
the inferred range of actual incision depths based on modern
topography and on our ICS preincision topographic reconstruction. Note that this technique compares only statistics
of total net erosion and is insensitive to the actual locations
of the simulated or actual stream network. Also, we are not
using the c2 statistics in the traditional sense of formally
testing the fit of an observed distribution to a theoretically
derived population because the baseline frequency observations, Pi, utilize the statistics of differences between the
modern and our empirically reconstructed topography and
thus may involve both systematic and random errors. These
[43] Each simulation evolved under a suite of parameters.
The parameters that varied between runs were discharge
regime, evaporation ratio, and ISC thickness and relative
resistance. These parameters create a complex interplay
throughout model evolution producing widely different
surfaces and valley network patterns. As MSLM progresses
c2 values become smaller as the surface erodes up to some
minimum point. Following the minimum c2 point erosion
continues and c2 values increase. We sampled each run and
computed the c2 value every 500 out of a total 10,000
iterations to determine the minimum c2 value for each run.
Plotting the c2 values at subsequent iterations produces a
parabolic shape with a minimum value at a particular
iteration. Higher-resolution sampling had a negligible effect
upon minimum c2 values especially when compared to
other runs. We use the MOS DEM and discharge data from
the run sample with the lowest c2 value to intercompare
model results.
[44] Strong correspondence exists between positive qualitative, visual representation of model performance and low
values of the c2 measure of fit. Surfaces that were severely
overeroded or undereroded produced very poor representations of the surface and had extremely large c2 values. Only
simulation runs with minimum c2 value in the lowest 15%
are analyzed here. There were not any simulations that had
average or good qualitative representation of the PDC with
a minimum c2 value above the lowest 15%. Table A1 (see
Appendix A) lists the simulation runs used in our analysis
and discussion.
[45] Variations in run parameters strongly affected minimum c2 values. Despite complex interplay between parameters during run evolution, correlation between minimum c2
values and run parameters can be discerned. Table 1
Table 1. Average c2 Values for Parameters Used to Simulate
Valley Network Formation
Simulation
Parameter
Discharge
exponent (a)
Evaporation
ratio (X)
ISC product
(thickness resistance)
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Parameter
Value
Average
Normalized
c2 Value
0.3
0.24
0.5
0.7
0
0.42
0.64
0.46
1
5
10
0
0.42
0.42
0.45
0.68
100
200
0.21
0.44
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BARNHART ET AL.: LATE-STAGE VALLEY NETWORK SIMULATIONS
summarizes averaged c2 dependence on simulation parameters. All tabulated and charted c2 values were normalized
to 14,258, the highest value for runs analyzed in this study.
Discharge scaling correlates strongly with minimum c2
values. Arid discharge regimes (smaller a in equation (2))
have lower c2 values than more humid ones (Figure 11).
Indurated surface crusts of moderate thickness and resistance
have low minimum c2 values as compared to simulations
with no crusts or very strong and thick crusts (Figure 12). In
nearly all cases, runs with an indurated surface crust
produce systematically better fits. Evaporation ratios did
not significantly affect the minimum c2 values (Figure 13).
In summary, variations in discharge exponent, a, and
indurated surface crust thickness and resistance have a more
significant effect on minimum c2 values than evaporation
ratios.
6.1. Discharge Estimates
[46] Discharge estimates were obtained for every cell in
the model domain throughout the simulation. The model
records the maximum and minimum discharge throughout
the domain at each time step and scales discharges in each
cell to an eight-bit value (0 to 255) that is used to generate a
normalized gray scale image of discharges throughout the
domain. Because channels form in different cells in each
simulation, we manually select points from the normalized
discharge images that functionally fulfill identical roles in
the valley network. Finally, we scale the eight-bit value into
an actual discharge. In particular, we focused on the
discharge at three functionally identical points common to
every simulation: (1) Loire Valles model domain exit; (2)
Samara Valles model domain exit; and (3) the point where
Parana Valles debouches into the lake occupying Parana
Basin. In simulation runs 24, 58, and 62 the lake in Parana
Basin breached to the north and significant discharges
exited the top of the domain. Simulated top exit breaches
drain the PDC instead of Loire Valles. When calculating
discharges, we substituted the top exit breaches for Loire
Valles in runs 24, 58, and 62. All other simulation runs
developed Loire Valles in a geomorphically similar way to
the actual surface.
[47] Simulated discharges for Loire Valles, Samara
Valles, and Central Parana Valles were strongly controlled
by the discharge exponent, a (equation (2)), and modestly
so by the evaporation ratio (equation (4)). Discharges
flowing out of Parana Basin into Loire Valles ranged from
125 to 4,682 m3 s1. Loire Valles discharge for five of the
six best qualitative runs were 600 m3 s1. As expected,
higher evaporation ratios correlated with lower stream
discharges (see Table 2, runs 72 and 62). Discharge values
for the ‘‘parameter space example’’ runs (Figure 9) and the
six runs that were most similar, qualitatively, to the actual
surface (Figure 14) are listed in Table 2.
[48] The stream discharges we obtained using MSLM
compare well with previous studies that calculated discharges by the considering hydraulic geometry of Martian
valleys and channels [Grant and Parker, 2002; Irwin et al.,
2005b]. For example, conservative discharge estimates
have been obtained by applying terrestrially derived, but
gravitationally scaled, empirical relations between channel
width and discharge to channels within Martian valleys.
Discharges for Loire Valles, assuming sand bed channels
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with sand banks, were estimated to be between 300 and
3000 m3 s1, [Irwin et al., 2005b]. Stream discharges of the
magnitude calculated by MSLM and estimated by Irwin et
al. [2005b] are equivalent to channel-forming floods in
terrestrial drainage systems of equivalent size and relief.
6.2. Formation Time Scales
[49] We estimate the time required to form the valleys in
the PDC for each simulation run using MSLM. As mentioned in section 2.4, simulation ‘‘time’’ progresses one
mean annual flood at a time. Therefore, we can use MSLM
as a chronometer only if the valleys in the PDC were carved
by discharges equivalent to terrestrial mean annual floods.
Mathematical analyses have shown that the mean annual
flood has a recurrence interval of 2.33 years; that is, once
every 2.33 years, on average, the highest flow of the year
will equal or exceed the mean annual flood [Leopold et al.,
1995]. Discharge values calculated by MSLM in this study
are, by nature of the MSLM model, equivalent to mean
annual floods. Empirical measurements of stream discharges mentioned in section 6.1 support our assumption
that mean annual flood level flows are responsible for the
bulk of the erosion of valley networks within the PDC.
[50] The total number of mean annual flood magnitude
flows that occurred in a particular simulation provides a
rough estimate on the duration and frequency of channelforming flows responsible for creating the valley networks.
However, duration and frequency are necessarily coupled
and, only together, estimate the temporal extent of the latestage epoch. Sustained and frequent flows at mean annual
flood stage would form the valley networks within the PDC
much more quickly than short-lived, sparse, and infrequent
flows. At minimum chi-square, runs with the best qualitative and statistical match to the actual surface require
500,000 to 700,000 mean annual flood size flows. Accordingly, the Parana drainage catchment subjected to an arid to
semiarid environment with terrestrial weather patterns
requires 105 – 106 Earth years for formation. On Earth,
flood stage is sustained for a week, or roughly 2% of a year.
Therefore, a continuous flow at mean annual flood stage
sustained for 103 – 104 Earth years could have formed the
valley networks at Parana Basin. On the other hand,
infrequent or episodic discharge at flood stage spaced by
long periods of quiescence would increase formation time
scales to 107 years. Larger formation time scales are limited
only by the cessation of valley development in southern
highlands during the early Hesperian period. The shortest
runs had a high discharge exponent. The shortest run
(Figure 9, top left; Table 2, Run 37) reached a minimum
chi-square fit at 300,000 mean annual floods which is
equivalent to 6000 years of continuous and sustained
mean annual flood stage discharge levels. Finally, a Martian
year is 1.88 Earth years and would, hypothetically, have
longer seasons in past climate optima. We discuss the
implications of these formation time scale estimates in
section 8.1.
7. Qualitative Evaluation
[51] Shaded relief and contour maps were evaluated
qualitatively for each simulation. As a simulation progresses
the surface erodes and reaches a time step that has the
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Figure 11. Discharge scaling exponent a versus normalized c2 value. Minimum c2 values strongly correlate with
the area exponent, a. Closed circles represent runs with an
indurated surface crust. Column averages are marked by a
cross. Output DEMs from model runs with lower a values
are a proxy for arid climates and produce a superior
statistical match to the actual surface.
lowest c2 value for that particular run. Further evolution of
the surface produces higher c2 values. The shaded relief
maps were generated from DEMs corresponding to the time
step with the best statistical fit assuming that this would be
the best qualitative fit as well. The shaded relief maps were
qualitatively evaluated with regard to the spatial pattern,
depth, and density of valley development as compared to
the extant network. To remove potential bias, each author
evaluated each run independently without knowledge of run
parameters or c2 value. We visually evaluated simulation
DEMs and compared them to the actual PDC shaded relief.
Evaluations were based on the following criteria:
[52] 1. General drainage pattern and density: Are valley
networks long, well integrated and continuous or short and
disparate? What fraction of the surface has been incised as
compared to the actual valley pattern?
[53] 2. Replication of the strong dissection of the east end
of the simulation domain and of the entrenched dissection
southeast of Parana Basin: The valley networks southeast of
Parana Basin are the most integrated, have the highest
drainage densities, and extend to the eastern divide. Simulations were evaluated on how well they recreated the
pattern in this critical area.
[54] 3. The dissection or lack thereof on the intercrater
plains. In particular the vast plain south of Parana Basin
lacks dissection. Simulations that produced deeply incised
valley networks with high drainage densities in this area are
a poor representation of the present-day surface.
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[55] 4. Crater lake formation, overflow, and rim breach.
Are several crater rims cut by rim breaches due to lake
overflow?
[56] 5. Development of Loire Valles: Did Loire Valles
develop in the northwest of the model domain and if so,
how deep was its incision? This particular criterion was
given low weight in the qualitative assessment because back
cutting and headward migration of knickpoints from downstream of our model domain could have realistically contributed to the formation of Loire Valles. Initial incision of
Loire Valles might also have occurred from a short, atypically wet, episode in which runoff filled the basin to
overflowing.
[57] 6. Incision of valleys in the simulation domain
external to the Parana Valles watershed. Here the development of Samara Valles in the southern part of the model
domain was evaluated. We also considered smaller valleys
and gullies forming on crater walls particularly just to the
east of Parana Basin.
[58] Just as with the quantitative analysis, our qualitative
evaluation did not consider the interiors of craters, alluvial
and fluvial deposits, sedimentation, and deposition patterns
in the PDC. Parana Basin’s principle role in the simulations
was to act as a sink for alluvium. However, we did evaluate
whether valleys developed from crater basin overflow and
crater wall breaching in the model as compared to their
occurrence in the actual landscape. Runs were evaluated on
the basis of the above criteria, and the six best qualitative
matches were selected for further analysis.
Figure 12. ISC product versus normalized c2 value.
Model runs with an ISC product produced the lowest c2
values. The ISC product was calculated by multiplying the
ISC thickness by its relative resistivity. Normalized c2
values were lowest for modest ISC products of 100X. These
runs had ISC thicknesses of 2 or 10 m and resistivities of 50
or 10, respectively. Simulation runs that did not have an ISC
generally had poor normalized c2 values.
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sediment transport. On the other hand, high values of a
result in deep incision throughout the region due to high
discharges downstream and low alluvial channel gradients.
Low evaporation ratios (Figure 9, middle left) lead to
overflowing crater lakes that breach and downcut most
crater rims. High evaporation ratios (Figure 9, middle right)
mean little overflow from ponded drainages. In particular,
high evaporation ratios prevent Parana Basin from overflowing and inhibit the incision of Loire Valles. An absent,
thin, or weak indurated surface crust (Figure 9, bottom left)
leads to a high density of valleys and fails to preserve much
of the initial surface. In contrast, a thick or resistant
duricrust limits incision, producing a few, deeply incised
valleys (Figure 9, bottom right) that, in number, total less
than the observed valleys in the Parana Valles region.
8. Discussion
Figure 13. Evaporation ratio versus normalized c2 value.
Normalized c2 values do not correlate well with evaporation ratio (column averages are marked by a cross).
Although higher evaporation ratios are marginally better
than low or zero evaporation ratios, the correlation is
dominated by other model parameters.
[59] The simulation runs generated wide ranges of valley
morphologies. A simulation run’s MOS DEM was assumed
to most closely resemble the AS at whichever iteration
minimized the c2 fit between the MOS DEM and the AS
DEM. Many runs contained either too little or too much
valley incision. Figure 9 gives examples of runs morphologically dissimilar to the actual surface. In contrast, a small
suite of runs generated valley morphologies similar to the
actual surface, particularly in the region of Parana Valles,
southeast of Parana Basin (Figure 14). Moreover, qualitatively similar runs have relatively low normalized c2 values
(Figure 15). These runs had an a value of 0.5, which in a
terrestrial setting, would be indicative of a semiarid environment. Modest indurated surface crusts produced better
qualitative similarity to the actual surface whereas nonzero
evaporation ratios, although present in all six runs, seemed
to have little effect on normalized c2 values. None of the
six, qualitatively superior runs had exit breaches cutting
across the rims of craters throughout the domain other than
the Parana Basin.
[60] Simulations that fail to match the pattern of late stage
valley incision in the Parana Valles region generally occur at
the extremes of the range of simulation parameter values.
Simulations with very low values of the discharge exponent,
a (Figure 9, top left) have very shallow channel incision
because the lower parts of the drainages become steep
alluvial valleys and alluvial fan deltas. This occurs because
low discharges in the downstream portion of the drainage
network result in steep gradients required for alluvial
[61] Arid to semiarid environments generate MOS DEMs
that best fit the actual surface. The following is an interpretation of the evolution of the PDC that was motivated by
the simulation results. As a surrogate for climate, including
humidity, precipitation, and infiltration, simulations explored a range of assumptions regarding the proportionality
between stream discharge and contributing area (equation
(2), section 4.1). The a parameter controlled the proportionality between discharge and contributing area. Runs
with a low a value of 0.3 (typical of arid environments)
provided the best statistical match to the actual surface.
Under these environmental conditions, discharge increases
slowly with increasing contributing area. Drainage densities
are reduced and network integration is frustrated.
[62] Evaporation ratios control ponding, network integration, and, indirectly, groundwater infiltration. In general,
evaporation controls had the least effect on minimum chisquare fit. However, qualitatively, runs without any evaporative loss from ponded water created networks that were
too deeply dissected and too well integrated. In addition,
most craters developed obvious valleys cutting the rim (exit
breaches) that do not occur in the actual landscape. Thus,
relatively high evaporation ratios, consistent with semiarid
to arid climate, best replicate the actual landscape. The
paucity of crater exit breaches argues against a climatic
scenario of one or more episodes of nearly constant, delugestyle, precipitation for a period of several years as has been
suggested might occur after major basin-forming impacts
[e.g., Carr, 1989; Segura et al., 2002].
[63] The presence of an ICS generates a good qualitative
and statistical match to the actual surface. A thick, chemically indurated crust may have developed over much of the
highlands during the Noachian period. Episodic climate
favoring its development may have come in the form of
low-intensity but frequent precipitation as snow or rain
[Howard et al., 2005]. A change in climate with rapid
snowmelt or high precipitation rates would generate flow
rates with enough intensity to erode through the ISC.
Locations of ISC removal and exposure of more erodable
substrates where exposed would deepen rapidly and concentrate flow. The ISC would channelize such flows and
inhibit delivery of sediment to the channel system.
[64] Indurated surface crusts are found in many arid
terrestrial environments [Cooke et al., 1993; Dixon, 1994]
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Table 2. Simulated Discharge Rates in Loire, Central Parana, and
Samara Channels
Run
1
19
20
21
22
25
37
56
72
23
24
55
57
58
62
Alpha
ISC
Producta
0.3
0.5
0.5
0.5
0.5
0.5
0.7
0.5
0.5
0
0
0
0
100
200
0
500
0
0.5
0.5
0.5
0.5
0.5
0.5
100
100
100
200
100
100
Evaporation
Ratio
Simulated Discharges (m3 s1)
Loire Central Parana Samara
Example Runsb
0
125
0
626
1
592
5
522
0
627
0
604
0
4682
0
653
10
362
Best Qualitative Runsc
1
601
5
660
1
609
1
614
5
670
10
276
39
198
218
229
219
217
1416
227
216
64
440
410
430
433
452
2915
434
356
227
200
221
222
202
195
421
357
434
421
357
362
a
Indurated surface crust.
Compare Figure 10.
c
Compare Figure 15.
b
and their presence is argued for on Mars [Jakosky and
Christensen, 1986]. Findings in nonpolar regions by the
Viking landers [Moore et al., 1987], Pathfinder [Moore et
al., 1999] and the Mars Exploration Rovers [Squyres et al.,
2004] of soils with various concentrations of chlorine,
sulfur, and silica support the presence of ISCs on Mars.
E01003
An ISC may be visible in hyperspectral imaging such as
CRISM and OMEGA. Probable morphological evidence of
an ISC is found just north of Loire Valles near its head at
Parana Basin (Figure 16). Formation of indurated crusts by
surface leaching and reprecipitation at shallow depth are
favored by aeolian deposition of fine, easily weathered dust
[e.g., Dixon, 1994, and references therein]. The inferred
early Martian conditions of strong winds, arid environment,
and abundant volcanic activity producing fine, chemically
reactive dust might have strongly favored development of
surface crusts.
8.1. Implications
[65] Our study argues for long-term climates present
during the Noachian – Hesperian transition that allowed for
sustained presence of liquid water on the surface. These
climates, lasting 104 years or longer, were capable of
sustaining episodic (perhaps seasonal) precipitation with
sufficiently long repose times (dry seasons) during which
evaporation and groundwater intake occurred. The geomorphology strongly implies that (1) seasonal or seasonally
driven episodic floods controlled valley formation and that
(2) significant evaporation, groundwater infiltration, or a
combination of the two occurred sequentially with the
flooding episodes. Among scenarios proposed for such
climates are those that invoke sustained levels of SO2 in
the atmosphere. The atmospheric SO2 would have permitted
a thick, warm, precipitation conducive CO2 atmosphere via
carbonate inhibition [Moore, 2004; Bullock and Moore,
2007; Halevy et al., 2007].
Figure 14. These six runs were ranked to be the most qualitatively similar to the actual surface.
Minimum c2 values for each run are shown in the parentheses. Shaded relief maps and contour plots were
judged independently by each author on the basis of (1) general drainage pattern and density,
(2) replication of the strong dissection of the east end of the simulation domain, and (3) the dissection, or
lack thereof, on the intercrater plains. None of these runs have exit breaches cutting across the rims of
impact craters.
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Figure 15. Normalized c2 values for qualitatively superior
runs. Runs that have the best qualitative match to the actual
surface all have relatively low normalized c2 values. All six
runs had an a value of 0.5, which in a terrestrial setting
would be indicative of a semiarid environment. Modest
indurated surface crusts produced better qualitative similarity to the actual surface whereas nonzero evaporation ratios
seemed to have little effect on normalized c2 values. Runs
with no evaporation (an evaporation ratio of zero) were
qualitatively poor. They generated overflowing crater lakes
that cut exit breaches across all crater rims.
[66] The climate responsible for valley network formation
generated flood stage discharge without generalized filling
and breaching craters. Channel discharges need to be
significant enough to form the valley networks, but these
discharges could not have been continuous to the point that
all craters would flood and breach. Among hypothetical
climate scenarios compatible with the geomorphology
would be those that allowed the build up of a seasonal or
episodic snowpack that then melted quickly in a rain-onsnowpack precipitation event. While other hypothetical
climates can be considered, it is certain that the valley
networks in the Parana drainage catchment were not formed
in a few several-year-long massive deluges as has been
proposed by other workers [e.g., Carr, 1989; Segura et al.,
2002]. Irrespective of a groundwater- or precipitation-based
source, the amount of water required to transport enough
sediment to form the valley networks in few continuous
events would overwhelm evaporation and inundation rates;
filling craters with lakes that would breach and downcut
crater rims.
8.2. Future Work
[67] Future work could include more geomorphic studies
and combined landform evolution models and climate
models. On Earth, broad regional slopes and mountain
ranges act as potential barriers that moist wind systems
E01003
have to surmount. This leads to the dumping of rain or snow
on prevailing windward slopes even in dry climates [Leeder,
1999]. The Parana drainage catchment resides on the eastern
trough of Margaritifer Sinus, which resides on a broad slope
that increases in elevation to the southeast. Locally, the
entire eastern flank of the PDB is surrounded by higher
elevations with Meridiani to the north wrapping around
clockwise to Noachis in the south. Hypothetical westward
winds from the Tharsis bulge and Sinai Planum may have
contributed to an orographic effect that concentrated precipitation throughout Margaritifer and in particular on its
eastern slopes. Applying Mars general circulation models in
combination with landform evolution models is a promising
area of future research that would better constrain precipitation amounts necessary for valley network-forming discharges.
[68] The use of landform evolution models, such as
MSLM on Mars is a nascent science. Future work should
include extensive exploration of parameters and dynamic or
variable controls on climate rather than constant parameterization throughout the simulation. By varying surrogate
climate parameters such as discharge dependence on contributing area and evaporative controls one could simulate
episodic or declining climates. Investigations using landform evolution models applied to other valley network
systems on Mars will aid in determining whether or not
formation mechanisms were similar throughout the equatorial highlands or regionally specific.
9. Conclusions
[69] The early climate of Mars remains an area of
considerable interest and research. This study focused on
geomorphic controls responsible for deep valley incision
associated with the concentrated and intense period of latestage fluvial activity near the Noachian – Hesperian transition. The relative influence and effect of discharge regime,
evaporation, and indurated surface crusts were simulated on
a hypothetical representation of the preincision Parana
drainage catchment using the landform evolution model
MSLM. Statistical analysis and qualitative evaluation demonstrate that simulations that model an arid to semiarid
climate over hundreds of thousands of years, moderate
evaporation ratios, and a modestly indurated surface crust
provide the best match to the actual surface of present-day
Parana Basin.
[70] Valley network formation time is calculated from the
number of mean annual flood size (or magnitude) flows
required to perform the geomorphic work. Under delugelike conditions (conditions during which mean annual flood
level discharges run continuously, not just 2% of the year
(valley formation requires a minimum of 103 –104 years).
However, observations of impact interruptions of network
formation concurrent with valley incision strongly imply
that valley formation occurred over a more extended period
of time. Most significantly, a paucity of crater rim exit
breaches in the Parana Drainage Catchment and the southern highlands in general implies that precipitation was not
deluge-style and continuous but rather moderate and episodic with periods of evaporation. This implies that latestage channel erosion did not form as a consequence of
16 of 21
BARNHART ET AL.: LATE-STAGE VALLEY NETWORK SIMULATIONS
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E01003
Figure 16. An example of a probable indurated surface crust or hardpan occurring just to the northeast
of Loire Valles near its head on the west side of Parana Basin (see Figure 3 for location). The image
shows a plateau dissected by gullies draining to Loire Valles with a prominent light-toned layer at the
edge of the plateau. The arrow points to the inset at the top right that shows the indurated nature of this
layer. This plateau is approximately at the level of the pre-Loire-breaching divide of the Parana Basin. An
ISC layer would reduce sediment yields from upland areas and focus erosion within larger channels.
Model runs with a modest 10 m thick ISC better matched the actual surface both statistically and
qualitatively. Note that dissection is most intense on the left side where the ISC layer is breached. CTX
image P07_003696_1592_XN_20S014W.
giant impact-induced short-lived climate excursions alone.
Therefore, if a few large impact events did perturb the
climate toward periods of precipitation, these periods would
have to be long-lived (approximately hundreds of thousands
of years) and seasonally or semiseasonally cyclic, with
evaporation interplaying significantly with precipitation
and runoff.
Appendix A:
[72] It is assumed that the materials below the surface
(lava, sediments, ejecta, etc., collectively termed ‘‘bedrock’’) may be indurated, but can be weathered at a finite
rate by physical or chemical processes to form colluvium.
The bedrock weathering rate, z_ b (m a1), decreases as a
negative exponential function of regolith thickness, H (m):
z_ b ¼
Mars Simulation Landform Model
[71 ] The landscape model, used in the simulations
reported here (see Table A1), is essentially the DELIM
model as reported by Howard [1994a, 1997, 2007] and
Forsberg-Taylor et al. [2004] with components modeling
physical or chemical weathering of rocks to form transportable colluvium, mass wasting by nonlinear creep, fluvial
detachment, and fluvial transport and deposition. Parameters used for these simulations are based upon terrestrial
values in semiarid or arid landscapes except for correcting
for the difference in gravity between Mars and Earth. We
briefly outline the model below, and additional background
and model details can be found by Howard [1994a, 1997,
2007].
W0 wH
;
e
r
ðA1Þ
where W0 and w control the rate and depth of weathering,
respectively. We assume w = 0.03 m1 and W0 = 0.0001 m a1.
Note that z_ b is the rate of lowering of the colluvial bedrock
contact, and when weathering is isovolumetric, as is
assumed here; it does not change the land surface elevation.
In simulations with an indurated surface crust W0 is reduced
by the resistivity factor r through the first Hd meters beneath
the initial surface elevation. At all other elevations the
resistivity factor is set to a value of one.
[73] The potential rate of erosion by mass wasting, z_ m, is
proportional to the spatial divergence of colluvial mass flux,
qm:
*
z_ m ¼ r qm ;
17 of 21
ðA2Þ
BARNHART ET AL.: LATE-STAGE VALLEY NETWORK SIMULATIONS
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Table A1. List of Simulation Runs Selected for Analysisa
Run c2 Value
1
2
3
4
5
6
7
8
9
19
20
21
22
23
24
25
26
27
37
38
39
40
41
42
43
44
45
55
57
58
61
62
63
71
72
73
0.41
0.42
0.37
0.11
0.12
0.14
0.13
0.13
0.21
1.00
0.98
0.77
0.18
0.19
0.19
0.33
0.32
0.26
0.68
0.74
0.73
0.32
0.29
0.41
0.97
0.90
0.90
0.21
0.30
0.18
0.32
0.19
0.14
0.80
0.82
0.40
Discharge
ISC
ISC
ISC Evaporation
Exponent, a Thickness Resistance Product
Ratio
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.5
0.5
0.5
0.7
0.5
0.3
0.7
0.5
0.3
0
0
0
2
2
2
2
2
2
0
0
0
2
2
2
2
2
2
0
0
0
2
2
2
2
2
2
10
10
10
10
10
10
0
0
0
0
0
0
50
50
50
100
100
100
0
0
0
50
50
50
100
100
100
0
0
0
50
50
50
100
100
100
10
20
10
10
10
10
0
0
0
0
0
0
100
100
100
200
200
200
0
0
0
100
100
100
200
200
200
0
0
0
100
100
100
200
200
200
100
200
100
100
100
100
0
0
0
0
1
5
0
1
5
0
1
5
0
1
5
0
1
5
0
1
5
0
1
5
0
1
5
0
1
5
1
1
5
10
10
10
10
10
10
E01003
[74] Because of the large cell size in the simulations in
this study (300 300 m) mass transport by linear creep (Ks
in equation (A3)) and the shape of small slopes is not well
characterized. Longer slopes in rapidly eroding locations
(e.g., on crater rims), however, tend to be close to the
threshold gradient for regolith (0.8).
[75] In the present modeling effort, potential erosion by
fluvial detachment, z_ f in bedrock or regolith-floored channels and on steep slopes where the flow is carrying less than
a capacity load is assumed to be proportional to the shear
stress, t, exerted by flowing water:
z_ f ¼ Kb
ðt t c Þ;
r
ðA4Þ
where Kb is a parameter taking the value of 0.0003 m2 a kg1.
Similar to W0 in the bedrock weathering case, Kb is reduced
by the resistivity factor, r, of an indurated surface crust
specified by a surface crust thickness, Hd. The critical shear
stress, t c, is assumed to be zero in the present simulations.
Assuming that the reference shear stress is that which
corresponds to the mean annual flood, the value of Kb that
we assume corresponds to terrestrial rates of erosion in
weak sedimentary rocks.
[76] Flow of water is assumed to be channelized and
originating from runoff. Shear stress can be related to
channel gradient and drainage area using equations of
hydraulic geometry and steady, uniform flow as discussed
by Howard [1994a, 2007]:
a
Runs in bold are the best qualitative match to the actual surface. Chisquare values are scaled relative to the largest value in the set, 14,258,
which is among the lowest 15% simulations tested in the study.
V ¼
t ¼ rf gRS;
ðA5Þ
Kn g1=2 R2=3 S 1=2
;
N
ðA6Þ
Q ¼ RWV ;
ðA7Þ
Colluvial flux is given by a nonlinear relationship:
qm ¼ Ks jS j þ Kf
1
1
s;
1 fjS j=St ga
Q ¼ kAa ;
ðA8; equationð2Þ of main textÞ
ðA3Þ
W ¼ Kw Qb ;
where jSj is the absolute value of local slope, s is the unit
vector in the downslope direction, g is gravitational
acceleration, St is a threshold gradient at which the rate of
regolith mass wasting becomes infinite (i.e., landsliding)
(assumed to be 0.8), and Ks is creep diffusivity which is
assumed to be 0.0005 m2 a1. The exponent, a, is assumed
to be 3.0, and Kf takes a value, 0.05, that provides for a
smooth but rapid approach to threshold slopes for rapid
rates of erosion. Erosion of bare bedrock slopes (exposed
when rates of erosion are greater than the maximum
weathering rate given by equation (A1)) follows equation
(A3), but with Ks set to zero and a steeper critical gradient,
St, of 2.7. Erosion of bedrock slopes involves a wide variety
of processes and resultant forms [e.g., Howard and Selby,
1994], and the assumed critical gradient (about 70°) is
chosen to represent bedrock slopes in rapidly incising
canyons.
ðA9Þ
where R is hydraulic radius, S is channel gradient, V is mean
velocity, N is Manning’s resistance coefficient, rf is a
specific runoff yield (depth per unit area per unit time), Q is
an effective discharge, W is channel width, A is drainage
area, and Kn, Kp, Ka, and Kw are coefficients. Channel
width, as parameterized in equation (A9), is generally much
less than the size of an individual grid cell, and following
Howard [1994a], each grid cell is assumed to host a single
channel that carries the total discharge through that cell. The
coefficients and exponents in equations (A5) –(A9) are
assumed temporally and spatially invariant. The following
parameter values are assumed: N = 0.03, Kn = 0.3 (for
metric units); a varies in this study between 0.3 and 0.7, k
varies with a to give the same discharge at a drainage area
of 7.5 km2. Specifically, k = 1.27 103 m s1 for a = 0.5.
Also, b = 0.5, and Kw = 5.0 s0.5 m0.5. As discussed by
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BARNHART ET AL.: LATE-STAGE VALLEY NETWORK SIMULATIONS
Howard [1994a], equations (A4) –(A9) can be combined to
express bedrock channel erosion rate, z_ , as a function of
contributing area and local channel gradient:
z_ ¼ Ke Am S n t c ;
ðA10Þ
where Ke, m, and n are functions of the parameters in
equations (A4) – (A9).
[77] In a landscape evolution model the most crucial
process is that of fluvial erosion, because stream incision
transmits the effects of erosion driven by relief relative to
base level throughout the drainage basin and the created
local slopes drive mass wasting processes and sediment
transport and deposition through the fluvial network
[Howard et al., 1994]. Stream channels are divided into
alluvial and bedrock channels [Gilbert, 1877; Howard,
1980]. The former, floored by transported sediment, generally have low gradients and incision or aggradation occurs
in response to the spatial divergence of sediment flux.
Bedrock channels, with thin or absent sediment cover, are
steeper, and erosion rates are determined by the balance of
bedrock strength and fluvial erosional processes. Because
the underlying bedrock must be eroded for channel networks to incise, the characterization of bedrock channel
erosion is thus the most important component of landscape
evolution models. A variety of processes can contribute to
bedrock channel erosion, including chemical and physical
weathering of the bed, hydraulic plucking, and abrasion by
sediment in transport [e.g., Hancock et al., 1998; Howard,
1998; Whipple et al., 2000; Whipple, 2004]. Several process
models have been developed for bedrock erosion, including
the ‘‘stream power’’ model that assumes that incision rate is
proportional to a measure of flow intensity such as shear
stress used here in equation (A4) [e.g., Howard and Kerby,
1983; Howard, 1994a; Whipple and Tucker, 1999], models
of abrasion by transported bed load [Sklar and Dietrich,
1998, 2001, 2004; Whipple and Tucker, 2002; Gasparini et
al., 2006, 2007; Turowski et al., 2007], and erosion by
debris flows in mountainous headwaters [Howard, 1998;
Stock et al., 2005; Stock and Dietrich, 2006]. In addition to
possible different incision mechanisms, the pattern and rate
of bedrock erosion can be affected by changes in channel
width in response to spatiotemporal variations in the rate of
downcutting or in rock resistance [Lave and Avouac, 2001;
Duvall et al., 2004; Finnegan et al., 2005; Stark, 2006;
Wobus et al., 2006a; Amos and Burbank, 2007; Finnegan et
al., 2007; Wobus et al., 2008] and the requirement for flows
to exceed thresholds for plucking or for sediment transport
coupled with temporal variability of flow strength [Davy
and Crave, 2000; Tucker and Bras, 2000; Snyder et al.,
2003; Tucker, 2004; Lague et al., 2005; Molnar et al.,
2006]. A number of studies have attempted to infer rate
constants for bedrock incision (e.g., Ke, m, n, and t c in
equation (A10)) based upon correlating channel profiles and
contributing drainage areas with measured uplift or incision
rates [Howard and Kerby, 1983; Stock and Montgomery,
1999; Snyder et al., 2001; Snyder et al., 2003; van der Beek
and Bishop, 2003; Duvall et al., 2004; Bishop et al., 2005;
Whipple and Meade, 2006; Wobus et al., 2006b; Crosby et
al., 2007], not always with concordant results. Despite the
uncertainties with regard to appropriate models and param-
E01003
eter values for bedrock channel incision, the stream power
model (used in the present simulations) is the most widely
employed for predicting or analyzing erosional response to
tectonic deformation, base level variations, and climate
change, in part because of the few model parameters and
in part because it generally performs well in predicting rates
and patterns of incision [e.g., Lague et al., 2005; Anderson
et al., 2006; Brocard and van der Beek, 2006; Roe et al.,
2006; Berlin and Anderson, 2007; Miller et al., 2007; Oskin
and Burbank, 2007; Riihimaki et al., 2007; Stolar et al.,
2007; Finnegan et al., 2008].
[78] Regolith is assumed to be more erodible than the
bedrock by a factor M = 10.0, which is assumed to influence
the bed erodibility and the threshold of erosion; thus, the
potential rate of fluvial erosion of channels flowing on
regolith, z_ r, is calculated from equation (A4) by multiplying
Kf by M and dividing t c by M.
[79] When the flux of sediment transported as bed and
suspended load reaches or exceeds the transporting capacity
of the flow (an alluvial channel as opposed to a bedrock
channel), the rate of erosion or deposition, z_ f, is proportional
to the spatial divergence of transport flux ~
qs (volume per
unit time per unit width):
*
z_ f ¼ r qs :
ðA11Þ
Sediment transport flux is estimated using a bed load
transport formula that is expressed as the relationship
between a dimensionless transport rate, F, and a dimensionless shear stress, t*:
p
F ¼ Ke t * t c* ;
ðA12Þ
where
F¼
qsb ð1 mÞ
g1=2 d 3=2 ðS
1=2
s
1Þ
and t * ¼
t
:
rf gðSs 1Þd
ðA13Þ
In these equations t c* is the value of t* at the threshold of
motion, qsb is bed sediment transport rate in bulk volume of
sediment per unit time per unit channel width, Ss is the
specific gravity of the sediment, g is gravitational acceleration, rf is the fluid density, d is the sediment grain size, and m
is alluvium porosity. We assume a fine gravel bed with d =
0.02 m is assumed, with Ke = 8.0, and p = 1.5. For all
simulations t c* = 0.05, Ss = 2.65 and m = 0.5. The shear stress
is estimated from equations (A5) – (A9), with the dominant
discharge for sediment transport assumed to be 0.6 of the
mean annual flood, flowing 2% of the year. Rivers vary from
those transporting dominantly suspended load to those
carrying primarily bed load [e.g., Schumm, 1977]. In the
absence of information for Martian channels, bed sediment
load is assumed to constitute 20% of sediment eroded from
slopes.
[80] Acknowledgments. This paper greatly benefited from Taylor
Perron’s thorough review and Don Wilhelms’ careful read of the manuscript. We thank Francis Nimmo for the suggestion to calculate discharge
rates and Erik Asphaug for generously providing computing resources.
NASA’s Mars Data Analysis Program and Graduate Student Researchers
Program supported this study.
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C. J. Barnhart, Department of Earth and Planetary Sciences, University of
California, 1156 High Street, Santa Cruz, CA 95064, USA. (barnhart@
es.ucsc.edu)
A. D. Howard, Department of Environmental Sciences, University of
Virginia, Box 400123, Charlottesville, VA 22904, USA. (ah6p@virgina.
edu)
J. M. Moore, Space Sciences Division, NASA Ames Research Center,
MS 245-3, Moffett Field, CA 94035, USA. ([email protected])
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