Long-term precipitation and late-stage valley network formation: Landform simulations of Parana Basin, Mars



[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 deluge-style 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.

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].

[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.

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.

[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 impact-induced 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 short-lived 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 late-stage 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 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 late-stage 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.

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.

[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 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 pixel−1) 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.

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.

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 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 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 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.

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].

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.

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.

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 discharges 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 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 Surfer® 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 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 Surfer® 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 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 Surfer® with 1 km grid cells. This method smooths the surface by removing valleys while preserving crests of drainage divides and regional slope gradients.

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.

[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):

equation image

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 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 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.

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.

[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.

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 s−1), is the flow going through a stream and is assumed to be proportional to the contributing area:

equation image

where A is the upstream contributing area (m2) and α 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 α 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 α 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, 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 α near 0.5. (Figure 8).

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 s−1) through a stream, Q, is proportional to contributing area, A. Discharge increases more rapidly downstream for higher values of the discharge exponent α. As a proxy for climate, we test α 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 α value of 0.48.

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 overflow from other basins may contribute to the present basin at a rate VI. A yearly water balance for the basin is thus:

equation image

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:

equation image

All lakes overflow as X → −1, and lakes become indefinitely small as X → ∞. 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.

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 three parameter classes have broad scale effects on landscape evolution (Figure 9) and are discussed in detail (section 7).

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: α, 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, α, 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 α values and low evaporation ratios simulating sustained multiyear deluge-style precipitation events breach nearly all crater rims (top right corner).

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 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.

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 baseline- and model-generated histograms was calculated by employing χ2 statistics (equation (5)).

[40] We employ a chi-square (χ2) 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 (χ2) values are calculated for areas of net erosion as follows:

equation image
equation image

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 χ2 statistics requires that the total number of observed data, equation imageXi, equals the total number of predicted or baseline data, equation imagePi. 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, β, 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 β ranged from 1 to 1.5.

[42] The ICS is an estimation of the preincised surface. An exact χ2 match (equaling zero) would not be expected. Thus, relatively low χ2 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 χ2 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 errors, however, do not invalidate our analysis. We take advantage of χ2 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

[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 χ2 values become smaller as the surface erodes up to some minimum point. Following the minimum χ2 point erosion continues and χ2 values increase. We sampled each run and computed the χ2 value every 500 out of a total 10,000 iterations to determine the minimum χ2 value for each run. Plotting the χ2 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 χ2 values especially when compared to other runs. We use the MOS DEM and discharge data from the run sample with the lowest χ2 value to intercompare model results.

[44] Strong correspondence exists between positive qualitative, visual representation of model performance and low values of the χ2 measure of fit. Surfaces that were severely overeroded or undereroded produced very poor representations of the surface and had extremely large χ2 values. Only simulation runs with minimum χ2 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 χ2 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 χ2 values. Despite complex interplay between parameters during run evolution, correlation between minimum χ2 values and run parameters can be discerned. Table 1 summarizes averaged χ2 dependence on simulation parameters. All tabulated and charted χ2 values were normalized to 14,258, the highest value for runs analyzed in this study. Discharge scaling correlates strongly with minimum χ2 values. Arid discharge regimes (smaller α in equation (2)) have lower χ2 values than more humid ones (Figure 11). Indurated surface crusts of moderate thickness and resistance have low minimum χ2 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 χ2 values (Figure 13). In summary, variations in discharge exponent, α, and indurated surface crust thickness and resistance have a more significant effect on minimum χ2 values than evaporation ratios.

Figure 11.

Discharge scaling exponent α versus normalized χ2 value. Minimum χ2 values strongly correlate with the area exponent, α. Closed circles represent runs with an indurated surface crust. Column averages are marked by a cross. Output DEMs from model runs with lower α values are a proxy for arid climates and produce a superior statistical match to the actual surface.

Figure 12.

ISC product versus normalized χ2 value. Model runs with an ISC product produced the lowest χ2 values. The ISC product was calculated by multiplying the ISC thickness by its relative resistivity. Normalized χ2 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 χ2 values.

Figure 13.

Evaporation ratio versus normalized χ2 value. Normalized χ2 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.

Table 1. Average χ2 Values for Parameters Used to Simulate Valley Network Formation
Simulation ParameterParameter ValueAverage Normalized χ2 Value
Discharge exponent (α)0.30.24
Evaporation ratio (X)00.46
ISC product (thickness × resistance)00.68

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, α (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 s−1. Loire Valles discharge for five of the six best qualitative runs were ∼600 m3 s−1. 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.

Figure 14.

These six runs were ranked to be the most qualitatively similar to the actual surface. Minimum χ2 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.

Table 2. Simulated Discharge Rates in Loire, Central Parana, and Samara Channels
RunAlphaISCProductaEvaporation RatioSimulated Discharges (m3 s−1)
LoireCentral ParanaSamara
Example Runsb
Best Qualitative Runsc

[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 with sand banks, were estimated to be between 300 and 3000 m3 s−1, [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 channel-forming flows responsible for creating the valley networks. However, duration and frequency are necessarily coupled and, only together, estimate the temporal extent of the late-stage 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 lowest χ2 value for that particular run. Further evolution of the surface produces higher χ2 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 χ2 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.

[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.

[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 χ2 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 χ2 values (Figure 15). These runs had an α 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 χ2 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.

Figure 15.

Normalized χ2 values for qualitatively superior runs. Runs that have the best qualitative match to the actual surface all have relatively low normalized χ2 values. All six runs had an α 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 χ2 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.

[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, α (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 sediment transport. On the other hand, high values of α 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

[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 α parameter controlled the proportionality between discharge and contributing area. Runs with a low α 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 chi-square 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, deluge-style, 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] 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. 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.

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.

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].

[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-on-snowpack 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 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 late-stage 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 deluge-like 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 late-stage channel erosion did not form as a consequence of 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:: 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].

Table A1. List of Simulation Runs Selected for Analysisa
Runχ2 ValueDischarge Exponent, αISC ThicknessISC ResistanceISC ProductEvaporation Ratio
  • a

    Runs in bold are the best qualitative match to the actual surface. Chi-square values are scaled relative to the largest value in the set, 14,258, which is among the lowest 15% simulations tested in the study.


[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, equation imageb (m a−1), decreases as a negative exponential function of regolith thickness, H (m):

equation image

where W0 and ω control the rate and depth of weathering, respectively. We assume ω = 0.03 m−1 and W0 = 0.0001 m a−1. Note that equation imageb 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, equation imagem, is proportional to the spatial divergence of colluvial mass flux, qm:

equation image

Colluvial flux is given by a nonlinear relationship:

equation image

where ∣S∣ 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 a−1. 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.

[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, equation imagef 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, τ, exerted by flowing water:

equation image

where Kb is a parameter taking the value of 0.0003 m2 a kg−1. 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, τc, 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]:

equation image
equation image
equation image
equation image
equation image

where R is hydraulic radius, S is channel gradient, V is mean velocity, N is Manning's resistance coefficient, ρf 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); α varies in this study between 0.3 and 0.7, k varies with α to give the same discharge at a drainage area of 7.5 km2. Specifically, k = 1.27 × 10−3 m s−1 for α = 0.5. Also, b = 0.5, and Kw = 5.0 s0.5 m−0.5. As discussed by Howard [1994a], equations (A4)(A9) can be combined to express bedrock channel erosion rate, equation image, as a function of contributing area and local channel gradient:

equation image

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 τc 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 parameter 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, equation imager, is calculated from equation (A4) by multiplying Kf by M and dividing τc 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, equation imagef, is proportional to the spatial divergence of transport flux equation images (volume per unit time per unit width):

equation image

Sediment transport flux is estimated using a bed load transport formula that is expressed as the relationship between a dimensionless transport rate, Φ, and a dimensionless shear stress, τ*:

equation image


equation image

In these equations τc* is the value of τ* 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, ρf is the fluid density, d is the sediment grain size, and μ 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 τc* = 0.05, Ss = 2.65 and μ = 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] 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.