Estimating erosional exhumation on Titan from drainage network morphology

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Citation Black, Benjamin A., J. Taylor Perron, Devon M. Burr, and Sarah A. Drummond. “Estimating Erosional Exhumation on Titan from Drainage Network Morphology.” Journal of Geophysical Research 117, no. E8 (2012). Copyright © 2012 American Geophysical Union

As Published http://dx.doi.org/10.1029/2012je004085

Publisher American Geophysical Union (AGU)

Version Final published version

Citable link http://hdl.handle.net/1721.1/85624

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, E08006, doi:10.1029/2012JE004085, 2012

Estimating erosional exhumation on Titan from drainage network morphology Benjamin A. Black,1 J. Taylor Perron,1 Devon M. Burr,2 and Sarah A. Drummond2 Received 20 March 2012; revised 15 June 2012; accepted 26 June 2012; published 15 August 2012.

[1] Drainage networks on Titan, Earth, and Mars provide the only known examples of non-volcanic fluvial activity in our solar system. The drainage networks on Titan are apparently the result of a methane-ethane cycle similar to Earth’s water cycle. The scarcity of impact craters and the uneven distribution of fluvial dissection on Titan suggest that the surface may be relatively young. The purpose of this study is to assess the importance of erosion relative to other plausible mechanisms of resurfacing such as tectonic deformation, cryovolcanism, or deposition of aerosols. We present a new method, based on a measure of drainage network shape known as the width function, to estimate cumulative erosion into an initially rough surface. We calibrate this method with a numerical landscape evolution model, and successfully test the method by applying it to river networks on Earth with different exhumation histories. To estimate erosional exhumation on Titan, we mapped fluvial networks in all Synthetic Aperture Radar swaths obtained by the Cassini spacecraft through T71. Application of our method to the most completely imaged drainage networks indicates that for two of four regions analyzed, Titan’s fluvial networks have produced only minor erosional modification of the surface. For the best-constrained region in the northern high latitudes, we find that fluvial networks reflect spatially averaged erosion of more than 0.4% but less than 9% of the initial topographic relief. This result implies either a recent, non-fluvial resurfacing event or long-term fluvial incision rates that are slow relative to the rate of resurfacing. Citation: Black, B. A., J. T. Perron, D. M. Burr, and S. A. Drummond (2012), Estimating erosional exhumation on Titan from drainage network morphology, J. Geophys. Res., 117, E08006, doi:10.1029/2012JE004085.

1. Introduction [3] As a result of the thick organic-rich haze that shrouds Titan, the actual landscape on the surface of the moon [2] Prior to the arrival of the Cassini-Huygens spacecraft remained mysterious until the first spectacular results from in orbit around Saturn in 2004, the surface of Titan was the the Cassini spacecraft. While Titan does not host a global subject of wide-ranging interest and speculation. Gerard surface ocean, Synthetic Aperture Radar (SAR) images from Kuiper was the first scientist to detect methane in Titan’s the Cassini Titan Radar Mapper [Elachi et al., 2004] show atmosphere [Kuiper, 1944]. Methane is near its triple-point 100 km-scale drainage networks with branching morpholo- (91 K and 0.2 bars) at surface conditions on Titan of 94 K gies (Figure 1). This morphology was revealed in even and 1.47 bars [Fulchignoni et al., 2005]. This observation greater detail and at finer scales for networks imaged by led Sagan and Dermott [1982] to envision methane oceans the Huygens Probe Descent Imager/Spectral Radiometer on Titan hundreds of meters deep. Lunine et al. [1983] [Tomasko et al., 2005]. The negative topographic relief suggested that methane photolysis might instead result in associated with the Huygens networks (that is, they are an ocean dominated by ethane, of even greater depth. Other valleys) [Tomasko et al., 2005; Soderblom et al., 2007] researchers, however, concluded that global methane oceans further suggests that they were created by fluvial erosion seemed unlikely [Eshleman et al., 1983]. [Perron et al., 2006; Moore and Pappalardo, 2011; Langhans et al., 2012]. These networks were most likely carved by a mixture of liquid methane [Kouvaris and Flasar, 1Department of Earth, Atmospheric, and Planetary Sciences, 1991] and possibly ethane [Lunine et al., 1983; Brown et al., Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. 2008] as they eroded “bedrock” composed of water ice 2Earth and Planetary Sciences Department, University of Tennessee, [Griffith et al., 2003; McCord et al., 2006; Rodriguez et al., Knoxville, Tennessee, USA. 2006], perhaps with a sedimentary veneer of organic atmo- Corresponding author: B. A. Black, Department of Earth, Atmospheric, spheric precipitates [Yung et al., 1984; McKay et al., 2001; and Planetary Sciences, Massachusetts Institute of Technology, 77 Lavvas et al., 2008; Krasnopolsky, 2009]. Near the poles, Massachusetts Ave., Bldg. 54-810, Cambridge, MA 02139, USA. river valleys feed into typically RADAR-dark hydrocarbon ([email protected]) lakes; the RADAR-darkness of the lakes most likely corre- ©2012. American Geophysical Union. All Rights Reserved. sponds to lower backscatter associated with a smoother 0148-0227/12/2012JE004085

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Figure 1. Low-latitude branching networks in Cassini’s T13 and T43 swaths (249.5 E, 9.6 S).White arrows point to locations of visible valleys, and are oriented approximately parallel to flow direction. Black arrow points to perpendicular valley segments in a network that has been identified by Burr et al. [2009] as potentially reflecting tectonic influences. surface [Mitri et al., 2007; Stofan et al., 2007]. Some lake- apparently patchy and uneven fluvial dissection [Drummond shaped features are distinctly brighter and are probably dry et al., 2012] is truly representative of the surface, it further lakebeds [Hayes et al., 2008]; possible SAR-bright paleo- supports the hypothesis that Titan’s surface is locally rela- lakes have also been identified in the low-latitude Hotei tively young, and subject to variable amounts of modifica- Regio and Tui Regio basins [Moore and Howard, 2010]. tion. Cryovolcanism [Sotin et al., 2005; Mitri et al., 2008], [4] Titan has relatively few visible impact craters, and tectonic deformation [Radebaugh et al., 2007; Burr et al., these are unevenly distributed geographically. Depending on 2009], deposition of organic aerosols [Khare et al., 1978], the crater production rate, Titan’s globally averaged surface possible migration of observed dunes [Lorenz et al., 2006], age may be between 0.2 to 1 billion years [Lorenz et al., and erosional exhumation all provide plausible mechanisms 2007; Neish and Lorenz, 2012]; tropical dunefields and the for resurfacing, but determining which process has been north polar region display a particular dearth of craters dominant is a key question for deciphering Titan’s geologic [Wood et al., 2010]. SAR swaths currently cover 40% of history. Titan’s surface. These images show abundant fluvial net- [5] The high concentration of methane in Titan’s atmo- works at high northern latitudes (Figure 2), several large sphere provides some clues to the enigma of Titan’s youthful networks and numerous fragmentary ones near the tropical surface, but also underscores an additional puzzle. Titan’s region of Xanadu (e.g., Figure 1) and in the southern high total atmospheric pressure is 1.47 bars at the surface, latitudes, and scattered networks elsewhere. If this composed of roughly 95% nitrogen and 5% methane [Lindal

Figure 2. (left) Branching networks that drain into a hydrocarbon lake near Titan’s North Pole (Swath T28; image centered at 108.5E, 75N). (right) The same region after mapping fluvial networks (white lines).

2of17 E08006 BLACK ET AL.: EROSIONAL EXHUMATION ON TITAN E08006 et al., 1983; Fulchignoni et al., 2005; Niemann et al., 2005]. not aware of any extensive valley networks on Earth that This methane abundance creates the potential for methane have formed mostly or entirely through dissolution. saturation, precipitation, and fluvial modification of the Mechanical erosion therefore appears to be the simplest surface [Perron et al., 2006; Turtle et al., 2011; Schneider explanation for Titan’s valley networks. et al., 2012]. However, photochemical breakdown and [7] In this study, we analyze the planform geometries of hydrogen loss to space means that the lifetime of methane in Titan’s drainage networks in order to quantify erosional Titan’s atmosphere is in the range of 10–100 Myr [Niemann exhumation, and thereby describe the evolution of the land- et al., 2005]. Therefore, in order to maintain the observed scape through time. Metrics of drainage network shape— methane pressures on Titan, there must be some source of such as drainage density, the width function, and channel replenishment. Over the course of geologic history, Yung sinuosity—are commonly applied on Earth to understand the et al. [1984] estimate that Titan may have lost a total of way network structure relates to environmental variables 8 bars of methane. This assessment was confirmed by such as precipitation or bedrock lithology [e.g., Rodriguez- Niemann et al. [2005], who find that Titan’s primitive Iturbe and Rinaldo, 1997]. Our measurements of purely atmosphere may have been several times as massive as at planform geometry are unusual by terrestrial standards, present. Mitri et al. [2007] determine that the observed however, because analyses of fluvial incision on Earth typ- atmospheric concentrations of methane could be supported ically also rely on elevation data to measure landscape by evaporation from lakes covering 0.2–2% of Titan’s sur- characteristics such as slope, drainage area, and valley relief. face. Alternatively, the atmospheric concentrations of In the case of Titan the limited available topographic data methane could provide evidence for transfer of subsurface currently inhibit such measurements. Even on Earth, it is materials to the surface and atmosphere of Titan [Lunine and challenging to assess the cumulative erosion that a landscape Lorenz, 2009]. Possible cryovolcanic features are not wide- has experienced without independent knowledge of the ini- spread on Titan’s exterior, however [Lopes et al., 2010; tial conditions. The goal of this paper is to develop and apply Moore and Pappalardo, 2011], suggesting that some of the a method to assess the extent of erosional development of a other processes mentioned above, such as erosional modifi- fluvial landscape, using only features that can be measured cation, are important in the rejuvenation of the surface. in plan view. [6] The history of Titan’s surface is thus critical to [8] We first present a geometrical statistic for quantifying understanding both its interior processes and hydrocarbon the shape of drainage networks, and demonstrate its useful- cycle. While erosion of water-ice bedrock or organic sedi- ness for evaluating the extent of erosional modification of a mentary layers by methane-ethane rivers might seem surface. We calibrate our statistic with a simple landscape inconsistent with the conditions of terrestrial erosion, theo- evolution model, and we validate it through comparisons retical and experimental results imply that the rates and with terrestrial drainage networks with constrained exhu- dominant mechanisms of fluvial erosion may be similar. mation histories. Finally, we map fluvial networks on Titan Collins [2005] investigates scaling relationships between and apply our method to those networks. Our results suggest Titan and Earth. He argues that mechanical erosion pro- that some of Titan’s most prominent drainage networks have cesses can operate on Titan at rates that are not radically caused spatially averaged erosion of only a small fraction different from those for the corresponding processes on (<10%) of the initial topographic relief in the regions Earth. Burr et al. [2006] and Perron et al. [2006] conclude where they have formed. that it may be slightly easier to transport sediments on Titan than on Earth or Mars, due to the greater buoyancy of sed- 2. A Statistic Relating Drainage Network iment and the lower gravitational acceleration. Tomasko Geometries to Erosional Exhumation et al. [2005] report that grains at the Huygens landing site, which are 2–20 cm in diameter [Keller et al., 2008], are 2.1. Evolution of Drainage Networks rounded, suggesting that they have been transported over a [9] With few impact craters and little ground-based or sufficient distance for rounding to occur. Solubility analyses historical data available, Titan’s drainage networks provide suggest that chemical erosion of water-ice bedrock on Titan one of the best sources of information about past erosion and is unlikely [Lorenz and Lunine, 1996], and low surface resurfacing patterns. Over time, we expect the geometry of temperatures and the lack of significant daily temperature these networks to evolve. Topographic data that resolve the variation would impede thermal erosion of water ice [Perron cross-sectional and longitudinal forms of fluvial valleys on et al., 2006], leaving mechanical erosion as the most likely Titan are currently limited. In the absence of widespread incision and particle shaping mechanism for water ice bed- topographic data, as discussed above, we must focus our rock. As noted by Perron et al. [2006], some of the valley analysis on the planform geometry of drainage networks. heads imaged during the Huygens descent are located on Our conceptual model for drainage network development is isolated topographic highs, the networks have many similar to that outlined by Horton [1945]. The basic stages in tributaries, and valleys appear to widen downstream, all of this model are shown schematically in Figure 3, starting with which are most consistent with mechanical erosion by run- initial perturbations in a random surface (Figure 3a). These off. Organic materials that have precipitated from Titan’s initial perturbations develop into channels that compete for atmosphere [Yung et al., 1984; McKay et al., 2001; Lavvas water and therefore drainage area. Because the erosion rate et al., 2008; Krasnopolsky, 2009] may be more soluble in channels is proportional to flow discharge (see section 3), than water-ice bedrock in hydrocarbon liquids [Malaska incipient channels that have smaller drainage areas and et al., 2011]. If significant thicknesses of organic fallout therefore smaller discharge than their neighbors will grow layers do mantle areas of Titan, dissolution erosion may be less rapidly than their faster-eroding neighbors. The more locally important [Malaska et al., 2011]. However, we are rapidly eroding networks will gradually integrate more

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Figure 3. (a–c) Schematic drawings illustrating the expected evolution of drainage networks in a landscape with sinks at the top and bottom of the frame. Initially, headward advance dominates. Basins develop roughly uniform width (Figure 3a). Once upslope area is exhausted, lateral tributary growth causes widening (Figure 3b). Interaction among basins concentrates area at intermediate distances from outlet (Figure 3c). (d–f) Three frames from a landscape evolution model run illustrating the evolution predicted in Figures 3a–3c. Time increases by a multiple of five in each successive frame. Colors denote elevation, with blue to red show- ing lowest to highest elevations, respectively. Maximum initial elevations are 400 m, yielding average slopes of 0.008. (g–i) Idealized illustration of the relationship between the cumulative width function (red line) and the DW statistic (shaded area) for the drainage networks shown in Figures 3a–3c. As shown in the figure, DW is proportional to the deviation of the cumulative width function from a uniform distri- bution of network links. Figures 3g–3i are schematic; actual width functions may not be symmetric about their midpoints. drainage area and develop simple tributary channels geological evidence can often be used to constrain the (Figure 3b). At first, the channel networks will expand development of terrestrial networks. mainly through headward growth of the trunk channel, but [11] As the geometric sequence described above proceeds, once they reach the drainage divide, tributaries will grow the drainage networks incise through the initial relief (the laterally and capture additional drainage area. As the sur- initial range of elevations in a landscape over a given hori- viving networks approach a steady form, they develop den- zontal area), gradually overprinting the initial topography dritic drainage networks with evenly spaced trunk channels (Figures 3d–3f). We can define the relative “age” of a fluvial and less-elongated drainage basins (Figure 3c). landscape as the extent to which it has modified an initial [10] By devising a method to quantify these trends, we can surface. All else being equal, this sequence will proceed use planform drainage network morphology to gauge the more slowly at larger spatial scales with higher relief. By extent of erosional modification. Although this assessment normalizing the total vertical erosion E (spatially averaged sounds like a simple task, it has not yet been completed for over the landscape) by the initial relief H0, we obtain a Earth—in part because elevation data and various lines of dimensionless measure of the relative erosional “age” of the landscape. Because landscapes can develop at different

4of17 E08006 BLACK ET AL.: EROSIONAL EXHUMATION ON TITAN E08006 rates, the morphologic age does not correspond to an abso- available for Titan confirms that many of the networks lute calendar age. Henceforth, we will use the qualitative visible in SAR are incised valleys with steep valley walls, terms “immature” and “mature” to refer to networks that and therefore bedrock erosion must have occurred during are in the early stages of fluvial dissection (small E/H0) their formation [Tomasko et al., 2005; Soderblom et al., and networks that have caused extensive erosion of an 2007]. Images from the Huygens probe [Tomasko et al., initial surface (E/H0 approaching unity, or exceeding unity 2005] show branched networks dissecting the landscape if uplift or base level lowering generate additional relief), down to a certain scale, but with channels that appear to respectively. truncate at a scale coarser than the image resolution. This observation implies that at the finest scales, non-fluvial ero- 2.2. The Cumulative Width Function sion mechanisms are more important than fluvial incision. [12] Given these general expectations for how a drainage However, because we are interested in the shapes of net- network should evolve over time, especially the develop- works which span 20 to 500 km, the relative contribution of ment of dendritic tributaries and the trend toward more hillslope processes to network geometry is likely to be small. equant and integrated basins, we can define a morphologic Cross sections through valleys near the Huygens landing site statistic to quantify the relative maturity of a network. show nearly straight hillslope profiles with slopes of roughly Rodriguez-Iturbe and Rinaldo [1997] define the width 30 degrees [Tomasko et al., 2005], suggesting that hillslope function as the number of links in a drainage network erosion may be modeled with a critical angle, where any at a given distance from the drainage basin outlet. The hillslopes that steepen beyond this angle experience rapid cumulative width function, as demonstrated in Figures 3g–3i, mass wasting. is the integral of the width function: the fraction of the net- [17] We therefore employed a simple model that includes work that lies within a given flow distance from the outlet. only bedrock river incision and mass wasting. The mass [13] We predict that the cumulative width functions of less wasting is driven by a slope threshold. When valley side mature drainages (such as Figure 3a) will reflect a uniform slopes exceed a critical gradient of 0.6 (31) we assume that distribution of channel segments, because the drainage will failure will occur, transporting material downslope to the not have developed an equant shape with dendritic tributar- channel and then out of the system. We assume that the ies. Basins like those in Figure 3b–3c, which are further flow’s transport capacity exceeds the supply of landslide- along in their development, will have sigmoidal or concave derived sediment, such that the transport of sediment does cumulative width functions. In order to compare the cumu- not limit the rate of channel incision. Channel incision into lative width functions of various networks, we calculate the bedrock is modeled with a stream power equation [Seidl and absolute value of the area between each network’s cumula- Dietrich, 1992; Whipple and Tucker, 1999], tive width function and the straight line that represents a – ∂z m uniform width function (Figures 3g 3i), and we define j ¼KA jjrz ð1Þ this dimensionless quantity as DW. This statistic, which ∂t channel measures the deviation of the measured width function from a uniform width function, is conceptually similar to the which is derived from the assumption that the rate of inci- Cramér-von Mises criterion, which measures the difference sion is linearly proportional to the rate of energy expenditure between a theoretical cumulative distribution function and a by the flow per unit area of the channel bed, also called the D “unit stream power.” In equation (1), z is elevation, t is time, measured distribution. The W statistic is distinct, however, r from the Kolmogorov-Smirnov statistic, which only considers A is drainage area, m is a constant, and | z| is the channel the maximum difference between a measured cumulative slope. We assume that there is no change in surface elevation distribution and a theoretical one. due to uplift or subsidence. K is the stream incision coeffi- cient, which represents the effects of bedrock erodibility, [14] In summary, DW is a measure of how much the distribution of segments within a fluvial network deviates precipitation rates, drainage basin hydrology, and hydraulic from a uniform distribution. In order to test our hypothesis geometry on fluvial incision rates. We assume that K is that DW should increase as drainage networks dissect a spatially uniform, and therefore that bedrock erodibility, surface, and to quantify any such relationship between DW precipitation, and other components of K are also uniform. and cumulative erosion, we turn to a numerical model of The coefficient m, which typically has a value of approxi- landscape evolution. mately 0.5 [Howard and Kerby, 1983; Whipple and Tucker, 1999], incorporates the relationships between erosion and stream power, channel geometry and discharge, and basin 3. Landscape Evolution Model geometry and discharge [Howard, 1994; Whipple and [15] Numerical modeling provides a clear theoretical Tucker, 1999]. It influences the concavity of the channel framework for both testing and calibrating our predictions for longitudinal profile and the relative balance between drain- how the DW statistic should behave as drainage networks age area and channel slope in shaping the overall geometry evolve. We identify several processes that are likely to be of the network. Both theoretical arguments [Willgoose et al., most important in shaping fluvial landscapes on Titan and 1991; Howard, 1994; Tucker and Bras, 1998] and empirical include them in our evolution model. observations [e.g., Rosenbloom and Anderson, 1994] [16] Drainage basins on Titan, like those on Earth, have support the utility of equation (1) as a simplified general developed through a combination of fluvial incision, which model for bedrock river incision, and it has been used pre- creates networks of incised channels, and hillslope processes viously in many models of landscape evolution [e.g., that govern the response of the surrounding topography to Howard, 1994; Pelletier, 2004; Perron et al., 2008, 2009]. channel incision. The limited stereo topography currently

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Figure 4. Example 25 Myr model run, with one drainage network highlighted in red. We only analyze the portion of the network that has eroded through >20% of the initial relief. The DW increases over time, eventually approaching a steady value for a given drainage network, even though the topography does not reach a steady state.

[18] We modeled the evolution of the landscape by statistic. There is a tradeoff between initial relief, K, hori- solving equation (1) forward in time on a regular grid using a zontal scale, and the time interval over which the landscape finite difference method. We began with random, self-affine erodes. We are primarily interested in how drainage network initial surfaces defined by the variance in elevation and the form changes as a function of how much of the initial relief slope of the surface’s power spectrum. We assume there are has eroded. Because this evolution is independent of the no structures that would introduce strongly oriented initial absolute rate of erosion, our results are valid without refer- topography. The lowest 10% of elevations in the initial sur- ence to the value of K or elapsed time. face were set to a fixed elevation of zero to simulate a base [19] Our model runs employed a drainage area exponent level, as might be provided for high-latitude networks by of m = 0.5 and a slope exponent of one, which implies that Titan’s polar lakes. Initial relief for our model landscapes was erosion rate is linearly proportional to stream power approximately 400 m. The average relief of equatorial [Whipple and Tucker, 1999]. Varying these exponents will mountains reported by Radebaugh et al. [2007] was 400 m shift the concavity and response time of the drainage profiles over horizontal profiles tens of kilometers long. The grid size [Whipple and Tucker, 1999], but exploratory model runs not is 400 by 400 points, with Dx = Dy = 125 m. We selected K = presented here indicate that this has only a minor effect on 0.005 yr1 with m = 0.5, and ran the model for 25 Myr. DW. The 50 km by 50 km scale of the model runs is similar Because the magnitude of K is not known for Titan, these to that of most of the networks we analyzed on Titan, though model results are not tuned to be specifically applicable to the some of the networks are several times larger. In Figure 4, rate of erosion on Titan. Rather, they form a test set for our the imprint of initial topography is clearly visible even after general predictions of landscape evolution and the DW 25 Myr of landscape evolution. The self-affine random

6of17 E08006 BLACK ET AL.: EROSIONAL EXHUMATION ON TITAN E08006 surface that we use to initiate the models is qualitatively networks (drainage area >10% of the total grid) in these runs consistent with available data from Titan, which generally were used to construct a calibration data set for DWasa show gentle slopes over long baselines [Zebker et al., 2009], function of E/H0. We binned these data according to E/H0. though higher resolution data from the Huygens landing site The resulting distributions of DW for various values of E/H0 contain valley side slopes up to 30 degrees over short dis- are plotted in Figure 5, with the mean trend with increasing tances [Tomasko et al., 2005]. E/H0 shown in Figure 6. Because the distributions for low [20] Figures 3d–3f show three stages in a typical run of the and high DW statistics are associated with different E/H0, landscape evolution model. To investigate our hypothesized we can use our results to calculate the statistical likelihood relationship between fluvial network morphology and the of a given erosional exhumation for drainage networks on extent of erosion, we require a criterion for defining fluvial Titan or Earth. networks in our model topography that is comparable to the visibility of fluvial networks in coarsely resolved Cassini 4. Validation With Terrestrial Drainage SAR images. As we discuss in section 5, deeper valleys Networks should be much easier to detect in SAR imagery of Titan, and very shallow valleys may be invisible. Therefore, in our 4.1. Glaciated And Unglaciated Landscapes model runs, we calculated the DW only for those portions of [23] To complement the theoretical predictions of land- the network that have incised a depth greater than 20% of the scape evolution provided by numerical modeling, we sought initial relief of the landscape. 10–25% thresholds all yield observational validation of our hypotheses. We measured the similar trends, but lower thresholds correspond with lower DW for drainage networks in several landscapes in North confidence in our upper limits on erosion. We discuss this America. These terrestrial locations are summarized in sensitivity in greater depth in section 6.2. In order to avoid Table 1. We selected four sites in North America with one of fragmented networks, we included all portions of a network two histories: a long history of fluvial erosion, or a short downstream from a segment that has eroded to this depth. history of fluvial erosion following resurfacing by glacial This approach is comparable to our procedure for mapping processes. The first two sites, in the Allegheny Plateau of discontinuous but clearly related fluvial features. Pennsylvania and the Oregon Coast Range, have not been [21] Four timesteps from a 25 Myr model run are shown in glaciated in Pleistocene time. The other two sites, on the Figures 4a–4d. In Figures 4e–4h, we isolate a single network southern edge of Lake Superior in Michigan and near Hudson and show the portion that we analyzed after applying the Bay in northern Québec, were glaciated during the Last “visibility” criterion described above. These frames show Glacial Maximum [Ehlers and Gibbard, 2004; Kobor and how, for the most deeply eroded fluvial features, the evolu- Roering, 2004]. Because continental ice sheets caused tion of the network generally matches our predictions as extensive erosion of the landscapes that we categorize as shown in Figure 3. Because stream incision rate is propor- recently glaciated, the onset of Northern Hemisphere degla- tional to drainage area, A, in equation (1), there is a positive ciation around 19–20 ka [Clark et al., 2009] serves as the feedback between stream incision and drainage area. The effective surface age for the Michigan and Hudson Bay sites. more drainage area a given network claims, the faster it [24] We also selected four sites that straddle the line incises, which allows it to lengthen or widen, thereby cap- marking the maximum extent of the Wisconsonian glacia- turing more drainage area. Incision rate is also proportional tion. These sites—in Washington’s Puget Sound and north- to land-surface slope, so networks grow more quickly in the western Pennsylvania—provide a range of surface ages upslope direction. As networks begin to form on an initially while holding erodibility and climate generally constant. undissected surface, they will first grow headward, because [25] As shown in Table 1, these eight sites all had rela- headward growth is unimpeded by competition with neigh- tively similar mean DW statistics, compared to the broader boring networks. This growth leads them to become elon- range observed for individual terrestrial drainage networks gated, as in Figures 4a, 4b, 4e, and 4f. Once networks begin and during model simulations. In part, this difference to impinge on one another on opposite sides of a drainage between the terrestrial and model data may be explained by divide, however, they widen and become more equant, as in the absence of an incision threshold for the terrestrial data. Figure 4c, 4d, 4g, and 4h. Because initial relief is not known, we calculated the DWfor [22] These trends are reflected in DW, as shown in all parts of a network, not only for those parts that had incised Figures 4i–4l. On the x axis, instead of time, we express the more than 20% of the initial relief. The eight terrestrial sites do, progressive development of the drainages in terms of E/H0, however, show a systematic pattern wherein recently glaciated the normalized measure of average erosion defined in section landscapes are biased to lower DW statistics and older land- 2.1. Very early in the development of the networks, the DW scapes are biased to higher DW statistics (Figure 7). This is low. It rises quickly to a nearly steady value that varies pattern is consistent with the general prediction that drainage from network to network. The precise final configuration of networks that have eroded a larger fraction of the initial each drainage network depends on initial conditions [Perron topographic relief will have a larger fraction of the network and Fagherazzi, 2012]; the overall shape of the basin will concentrated at intermediate distances from the basin control the shape of the cumulative width function (of which outlet (Figures 3 and 4). Figures 4i–4l show examples), and will therefore also influ- 4.2. Drainage Networks on Kaua’i ence the ultimate value of DW achieved by each network. In order to better estimate the distribution of possible outcomes, [26] In addition to this qualitative confirmation of the D we employed a Monte Carlo procedure in which we per- hypothesized relationship between W and E/H0, we sought formed twenty model runs with twenty different initial sur- to make a quantitative comparison of terrestrial fluvial net- faces that shared similar statistical characteristics. The largest works with the landscape evolution model predictions. The

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Figure 5. Six snapshots of the distribution of DW from an ensemble of 20 model runs. These histograms show that mean DW increases with increasing erosion relative to initial relief, eventually approaching a steady value of DW. western coast of the Hawaiian island of Kaua’i, above the Monte Carlo calculation (Figure 8). In each iteration, we Mana Plain, provides a unique opportunity to measure the randomly selected twenty networks from the numerical DW statistic for a landscape with known amount of cumu- model results (the same as the number of measured Kaua’i lative erosion. This portion of the island is relatively networks) for each of eight logarithmically spaced values of homogeneous, with a classic shield volcano structure that E/H0, and calculated the mean DW for each group of twenty has been partially eroded by drainage networks [Macdonald model networks. This simulation was performed 100 times. et al., 1960]. We identified remnants of the original shield If the mean of the simulated population was separated by at surface surrounding each basin and fit these remnants with least two standard errors from the mean of the Kaua’i popu- spline functions in order to reconstruct the initial surface. lation, then we counted this instance as a statistical difference We then estimated the volume of material that has been in the mean. The number of times (out of 100 simulations) removed through erosion, allowing us to calculate E/H0 for that the mean DW for the model was two standard errors each basin. We calculated DW for 20 adjacent drainage higher or lower than the mean DW for Kaua’i gives an networks (Haeleele, Hanakapiai, Hanakoa, Hikimoe, Hoea, estimate of the likelihood that the Kaua’i drainage basins Honopu, Hukipo, Kaawaloa, Kaaweiki, Kahelunui, Kahoa- have eroded less or more, respectively, than each value of loha, Kalalau, Kapilimao, Kauhao, Kaulaula, Kuapaa, E/H0. Makaha, Milolii, Nahomalu, Niu, Ohaiula, Paua, Wailao, [28] Figure 8 plots these Monte Carlo percentiles for and Waipao), and obtained a relatively low DW = 0.061 Kaua’i, along with the E/H0 value we actually observe there. 0.012. Based on an approximate initial relief of 1000 m in this [27] In order to compare these measured combinations of area of Kaua’i, the Monte Carlo calculation suggests a very E/H0 and DW with data from our model, we performed a high likelihood (95th percentile) that the landscape has

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5.1. Mapping and Analysis [30] We produced detailed maps of Titan’s fluvial networks using SAR images from the Cassini spacecraft (see Figure 2 for an example from Titan’s North Polar region). SAR ima- ges are radar backscatter images. Backscatter intensity is influenced by surface orientation relative to radar incidence and azimuth angles of observation, roughness, dielectric constant, and density of the targeted surface. The resolution of SAR swaths is 350–720 m/pixel at the center of the swaths, and decreases by a factor of 3 at the two ends. Each SAR swath ranges from 120 to 450 km wide, and images roughly 1.1% of Titan’s surface [Elachi et al., 2004, 2005]. To date, approximately 40% of Titan’s surface has SAR coverage. [31] We mapped all fluvial networks visible in swaths Ta through T71 [Drummond et al., 2011, 2012]. We identified networks on the basis of five criteria: light-dark pairings, branching patterns, crosscutting of other radar features (such as dunes or mountains), connectivity, and orientation with Figure 6. Compilation of binned DW as a function of E/H0 respect to suspected topographic sinks (such as lakes). Light- for all twenty model runs. As erosion proceeds, drainages dark pairings are created by differing illumination conditions branch and become more equant, leading to higher values on either side of a valley: the slope that is facing the of DW. After erosion has reached approximately 10% of spacecraft is illuminated by the radar and appears bright, the initial relief, values of DW reach a steady state. Large whereas the other slope will appear dark. The networks we error bars represent one standard deviation; small bold error mapped ranged from first-order networks, which essentially bars show one standard error. consist of only one visible link, up to fourth-order networks with dozens of tributaries. These orders are apparent orders that describe the features that we have mapped, but because experienced more than 4 m of spatially averaged erosion, many fine scale features are probably invisible in the SAR and a moderate likelihood (50th percentile) of less than imagery, the actual orders of the drainage networks may be 160 m. Our best estimate for the actual spatially averaged much higher. Drummond et al. [2011, 2012] provide a more erosion, based on current topography and the spline-fit initial detailed description of the global mapping efforts. surface, is 80 m. The good agreement between the [32] For this paper, we focused on networks from swaths observed erosion and the constraints based on DW suggests T13, T43, T28, T29, and T39, which include some of the that the DW value of a drainage network is a robust indicator most extensive and most completely imaged networks of the approximate amount of cumulative fluvial erosion in found on Titan so far. T13 and T43 are equatorial swaths a landscape. near Xanadu (Figure 1), T28 and T29 cover the lakes around the north pole (Figure 2), and T39 is near the south 5. Application to Fluvial Networks on Titan pole. We also designated a probable sink for each large network—a point through which we expect flow from the [29] Analysis of terrestrial and model networks indicates rest of the network to drain. We identified these sinks the usefulness of DW as a proxy for the cumulative erosion based on flow directions inferred from junction angles and experienced by a landscape. In this section, we discuss the apparent trends in valley width. Starting from the sink in application of this statistic to Titan. each network, we worked upstream to assign a flow direction

Table 1. DW Measurements From Titan and Eartha Number of Current Location Description (Swaths) Coordinates Networks DW Relief Titan NP1 Titan Northern High-Latitude [T28/T29] 78.5E, 80.1N 19 0.07 0.016 >300 m Titan NP2 Titan Northern High-Latitude [T28/T29] 117.0E, 74.0N 21 0.05 0.012 >300 m Titan T13 Titan Low-Latitude [T13] 249.5E, 9.6S 4 0.07 0.035 500 m Titan T39 Titan Southern High-Latitude [T39] 328.0E, 73.2S 8 0.06 0.015 SW Pennsylvania Unglaciated 279.5E, 39.5N 4 0.107 0.041 300 m NW Pennsylvania Glaciated 280.0E, 42.0N 6 0.066 0.016 200 m NW Pennsylvania South of LGM 280.5E, 41.4N 4 0.084 0.030 200 m Washington Glaciated 237.9E, 47.2N 7 0.068 0.012 600 m Washington South of LGM 237.6E, 46.5N 4 0.092 0.023 1000 m Oregon Unglaciated 236.0E, 43.4N 8 0.091 0.026 650 m Michigan Glaciated 271.4E, 46.5N 7 0.082 0.010 350 m Hudson Bay, Canada Glaciated 262.5E, 51.9N 5 0.076 0.019 250 m Western Kaua’i Mana/Na’Pali Coast 200.3E, 22.06N 20 0.061 0.012 1000 m aEstimates of current relief for Titan are based on SAR topography from Stiles et al. [2009]. “Glaciated” means that a given area was glaciated at the Last Glacial Maximum. Means for Titan are weighted by horizontal length of the networks. Uncertainties in DW are stated as 2 standard errors.

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model runs, E/H0 = 0.4% corresponds to erosion within the valleys of 25–30% of the initial relief, or 100–120 m. [36] For two of our study areas on Titan, the measured DW was statistically indistinguishable from the upper values of DW obtained during the model runs (Figure 9), and therefore no upper limit on erosion could be determined. For the other two study areas, we were able to estimate upper limits on erosion, though with less confidence than for our estimates of the lower limits. Our constraints for an area of Titan’s northern lakes region are most robust: we estimate an 85% probability that fluvial erosion in this region is less than 9% of the initial relief. Once again, this value is an average over the entire landscape; within the valleys, the incision may be much deeper.

6. Discussion 6.1. Implications for Resurfacing and Long-Term Erosion Rates on Titan [37] In section 1, we reviewed several potential mechan- Figure 7. DW for regions on Titan and Earth. DW for isms for resurfacing on Titan, including cryovolcanism, Titan is averaged over all four study regions, with a total tectonic deformation, deposition of organic aerosols, possi- of 52 networks. Uncertainties represent two standard errors ble migration of observed dunes, and erosional exhumation. of the mean. On the basis of the results presented in section 5.2, we can evaluate the potential effectiveness of fluvial erosion for resurfacing. The maximum relief in Titan’s landscapes at to each pixel along the flow path, according to an eight- low latitudes is approximately 500 to 2000 m along SAR- direction routing scheme. based topographic profiles 5 to 20 km long [Radebaugh [33] We calculated DW for 52 fluvial networks. These et al., 2007]; much greater relief would trigger flow at the networks were located in four general regions, as shown in base of Titan’s dominantly water ice crust, resulting in oro- Figure 9 and in the auxiliary material, and we performed our genic collapse [Perron and de Pater, 2004]. Elachi et al. analyses separately for each population.1 The mean DW [2006] also estimate maximum relief of 1000 m. Assum- statistics for each region are weighted according to the ing that relief in landscapes on Titan does not generally lengths of the individual networks; larger networks are exceed 1000 m, we can calculate an order-of-magnitude therefore weighted more strongly. The results are summa- rized in Table 1. Below we compare these results with DW statistics from numerically simulated landscapes in order to estimate the magnitude of fluvial erosion on Titan. 5.2. Estimating Erosion on Titan [34] In order to place constraints on the amount of erosion that has occurred on Titan, we performed a series of Monte Carlo calculations to compare the distributions of DW derived from the landscape evolution model (Figure 5) against the distributions of DW measured for Titan. These Monte Carlo calculations followed the same procedure we applied to terrestrial networks on Kaua’i, as described in section 4.2, except that the number of randomly selected networks in each Monte Carlo iteration was chosen to match the number of mapped networks in each Titan region. Table 1 lists the number of networks we analyzed within each study area on Titan. [35] Figure 9 shows percentiles for the upper and lower bounds on E/H0 for each region, overlain on a map of Titan. Based on our Monte Carlo simulations, we estimate lower Figure 8. Monte Carlo simulations provide constraints on limits on erosion on Titan of 0.4% to 0.6% of initial relief in the likely lower (dashed red line) and upper (solid blue line) all our study regions. These numbers correspond to the bounds on erosion relative to initial relief. For Kaua’i, the regionally averaged erosion; erosion in fluvial valleys is percentiles based on a measured DW of 0.061 0.012 for much larger. As an analogy, for our landscape evolution twenty drainage networks are consistent with the actual E/ H0, shown by the gray bar. The width of the gray bar repre- 1Auxiliary materials are available with the HTML. doi:10.1029/ sents E/H0 values within two standard errors of the mean 2012JE004085. erosion among our 20 Kaua’i basins.

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Figure 9. Available SAR swaths and probabilistic estimates of erosional exhumation for mapped regions of Titan. North Pole Area 1 is outlined in light green, North Pole Area 2 is blue, the Equatorial region is red, and T39 in the southern high latitudes is purple. Ligeia Mare, one of Titan’s largest lakes, is visible next to the north polar networks. Four exterior panels show results of Monte Carlo calculations for each area. Dashed red curves show probability that E/H0 for a particular region on Titan is greater than a given value on the x axis, and solid blue curves show the probability that E/H0 for Titan is less than a given value. estimate of erosion. For NP2 drainages near Ligeia Mare the [38] Impactors smaller than 1 km in diameter may fail mean DW is 0.05 0.012 (Table 1). This relatively uniform to penetrate Titan’s thick atmosphere, producing a dearth distribution of network segments, in conjunction with of microcraters [Artemieva and Lunine, 2003]. Erosional numerical model results (Figures 4 and 9) and an assumed exhumation of the magnitude that we predict might be H0 of 1000 m, implies an upper limit on areally averaged sufficient to erase some of the small craters that are created, erosion of 100 m. Fluvial erosion of <9% of the initial especially those that have also experienced viscous relaxation relief (our estimate for the region of the NP2 drainages), may [Parmentier and Head, 1981; Artemieva and Lunine, 2003]. have produced a significant modification of the initial land- However, as is evident from the persistence of topographi- scape, in particular in valleys where incision may be much cally prominent features [e.g., Radebaugh et al., 2007], the deeper than 9% of the initial topography.

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Figure 10. Apparent inconsistencies in drainage network flow paths located at 95E, 75N on the shores of Titan’s Ligeia Mare (T28). Possible link between adjacent valleys via diffuse dark band could indicate stream capture (arrow A). Junction angles suggest that the large network entering the image from the lower left does not drain into Ligeia Mare (arrow B). magnitude of fluvial erosion has been insufficient to entirely frequency of those events, would significantly improve our level Titan’s surface. knowledge of likely runoff discharge [Perron et al., 2006]. 8 [39] If crater counting surface ages for Titan of 10 – If mechanical erosion also occurs through fallout deposits of 109 years [Lorenz et al., 2007; Wood et al., 2010; Neish and organic aerosols from the atmosphere, then additional data Lorenz, 2012] are accurate, we can use our constraints on on the strength of such materials would be important for maximum exhumation to estimate a long-term erosion rate. understanding their erosional behavior. 8 A maximum of order 100 m of erosion in 10 years provides [42] A further possible explanation is that the surface age an upper limit on fluvial erosion rates of 106 myr1 for for some regions is in fact much younger than crater counting the NP2 area of Titan’s northern high latitudes. The calcu- seems to suggest. Surface ages on Titan may be heteroge- lated long-term erosion rates are even slower if we consider neous; in Figure 2 there are both mottled, slightly darker an initial relief in the landscape of less than order 103 meters, areas and smooth, light-toned areas, each incised by drainage or surface ages of more than 108 years. In contrast, the typical networks. These different surface appearances may corre- range of fluvial erosion rates on Earth’s continents is 105 spond to different ages of resurfacing. If the northern polar to 103 myr1 [Portenga and Bierman, 2011]. In other region is younger than 108 years, then our estimates for ero- words, in order to be consistent with our results and with the sion rates on Titan would increase proportionally. surface age predicted from crater counting, erosion rates for [43] Finally, some fluvial networks may be relics, either some areas on Titan would have to be far slower than typical from an earlier period with higher atmospheric methane terrestrial rates, despite the fact that rapid surface changes concentrations or from a previous topographic state that has provide signs of present-day methane precipitation at low been altered tectonically or resurfaced. Methane photolysis latitudes on Titan [Turtle et al., 2011]. [Yung et al., 1984] could engender gradual decreases in [40] There are several possible explanations for the methane concentration and precipitation over time, despite apparently slow erosion rates on Titan. Collins [2005] has reports of present-day precipitation [Turtle et al., 2011]. previously suggested that during bedrock incision on Titan, Examination of SAR imagery provides some localized the lower kinetic energy of saltating grains is offset by the indications that changes in flow paths appear to have out- lower abrasion resistance of water ice at Titan surface tem- paced the rate at which fluvial erosion erases previous peratures. Assuming similar discharge rates, the erosion drainage patterns. Figure 10 shows possible inconsistencies rates on Titan and Earth would then be comparable [Collins, in the flow paths of fluvial features near Ligeia Mare that 2005]. However, recent experimental evidence shows that may reflect relict earlier drainage patterns. Junction angles ice on Titan may in fact be much stronger than originally are ambiguous, raising the possibility that the drainages in thought [Collins et al., 2012], with a tensile strength similar Figure 10 do not terminate into the lake, which we assume is to welded tuff or quartzite, among the strongest of terrestrial currently a local topographic minimum. The diffuse dark continental materials [Sklar and Dietrich, 2001]. As a result, band in Figure 10, which does connect to the lake, seems to river incision on Titan could be slower than on Earth, given crosscut the narrower branching networks. similar discharge rates. [44] Depending on the dominant tectonic style on Titan, [41] On the other hand, typical discharge rates and the our upper bounds on areally averaged erosion hint at more frequency of high flow events on Titan may be very different speculative implications for Titan’s surface geology. In the from those on Earth. Additional data on the intensity and case where uplift of bedrock is driven purely by isostatic duration of precipitation events, and also on the long-term response to erosional exhumation, that uplift of bedrock

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Figure 11. Comparison of SAR images and topography for terrestrial drainage networks. (a) SAR image and (b) shaded relief map of the region southwest of L’Anse Bay, Lake Superior, on Michigan’s upper peninsula, centered at 46.57N, 88.67W. (c) SAR image and (d) shaded relief map of a section of the Allegheny Plateau east of the Ohio River in northern West Virginia and southwest Pennsylvania, centered at 39.67N, 80.70W. SAR images are from the C-band antenna on board the Shuttle RADAR Topography Mission [Farr et al., 2007]. Topography is from the USGS National Elevation Data set. Map projections are transverse Mercator, with a pixel size of approximately 30 m. cannot exceed the areally averaged erosion [England and even when downstream of existing fluvial networks (e.g., Molnar, 1990]. Assuming <100 m of areally averaged ero- Figure 2). This observation suggests that sediment deposition sion in the NP2 area, we conclude that fluvial erosion alone during current lake levels has been insufficient to allow sig- has probably been insufficient to exhume material from nificant delta progradation into the standing liquid that fills deeper in Titan’s crust and to expose it at the surface, at least the drowned lower reaches of the networks. A digital eleva- during the formation of the current landscape. Such tion model for the western edge of Kraken Mare, the largest unroofing would require tectonic processes. of the northern polar lakes, does reveal a topographic bench [45] Given the apparent lack of waves on Titan’s lakes that may have been produced by depositional processes [Stofan et al., 2007; Wye et al., 2009; Lorenz et al., 2010], [Aharonson et al., 2012]. Aharonson et al. [2012] estimate the rarity of obvious deltas where fluvial systems drain into that the volume of material required to form this bench hydrocarbon lakes may provide additional evidence that translates to 1 m of areally averaged erosion within Mayda erosion and sediment transport are relatively slow compared Insula. This value is consistent with the areally averaged to changes in lake level. Over geologic timescales, sediment erosion of between 0.4% and 9% of initial relief that we infer transport and deposition will gradually fill in topographic in this study for the areas near to Kraken Mare (Figure 11). sinks. On the southwestern edge of Ontario Lacus, one fea- [46] The few available topographic profiles based on stereo ture has been interpreted as a delta [Wall et al., 2010]. While Descent Imager/Spectral Radiometer imagery also confirm comparisons are problematic between terrestrial river widths that our erosion constraints are reasonable. In the highlands and the widths of valley features visible in SAR data north of the Huygens landing site, topographic profiles show [Burr et al., 2012], Wall et al. [2010] cite the small relative valleys 90 m deep incised into a landscape with 240 m of size of the Ontario Lacus delta as evidence for low sediment relief [Tomasko et al., 2005; Soderblom et al., 2007]. discharge. The shorelines of many of the lakes near the north Assuming that the ratio of valley depths to areally averaged pole evoke drowned drainage networks [Stofan et al., 2007], erosion is comparable to that in our model runs, this translates

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Figure 12. Sensitivity of DW and erosion estimates to resolution of mapped fluvial networks. (a) As the finest network links are excluded, the overall trend in measurements of DW is consistent. Curves are extracted from a set of four trial model runs. The effect on the steady value of DW at large cumulative erosion values is small but significant. Curves are labeled with the minimum contributing area (as a fraction of the total area of the grid) used as a threshold for delineating the drainage network. This is illustrated in the inset with an example channel network. The green dashed curve denotes the value used elsewhere in this study. (b) Monte Carlo calculations for Titan’s North Polar Area 2 using the synthetic data set in Figure 12a. Including or excluding the finest branches of the networks has a negligible effect on the predicted lower limits on exhumation. The magnitudes of the predicted upper limits on erosion are also relatively stable, but the percentiles shift, reflecting uncertainty in the specific percentiles for those estimates.

to an average erosion of 1.5–2% of the initial relief, con- result, the percentiles for the upper limits on E/H0 also shift sistent with the range of erosion shown in Figure 9 for other (Figure 12b). We interpret this shift to mean that the true areas of Titan. certainty with which we know the upper limits may be moderately lower than the stated percentile values. The 6.2. Effects of SAR Resolution lower limits on erosion are robust and independent of [47] Although SAR resolution constitutes one of the major resolution. sources of uncertainty for mapping individual branches [49] As discussed in section 3, we apply a threshold to the of Titan’s drainage networks, small omissions do not signif- model topography such that only relatively deeply eroded icantly affect the measurement because DW is a measure of valleys are analyzed. This threshold is important because drainage shape, not density. The Huygens Descent Imager/ otherwise our analysis would include fluvial networks that Spectral Radiometer revealed fluvial features much smaller extend all the way to the drainage divides, whereas on Titan than the best resolution of the orbital SAR imagery [Tomasko we have only mapped the networks that are visible in SAR et al., 2005; Soderblom et al., 2007], indicating that the finest imagery. Comparison of topography and SAR images scale features on Titan’s surface are not resolved in SAR data. (Figure 11) supports the use of a relief threshold for identi- A comparison between terrestrial SAR and topographic data fying the portion of the fluvial network to be analyzed. As (Figure 11) also underscores the limitations of SAR-based mentioned above, Figure 11 clearly shows that many ter- mapping. Any highly dissected areas on Titan that resemble restrial fluvial features that are visible in the digital elevation the fine-scale valley networks in Figures 11c and 11d may be models are invisible in the SAR images. We hypothesize difficult to delineate. Fluvial features on Titan’s surface may that one factor that contributes to this discrepancy is the be far more extensive than can be detected on the basis of the relief across the valleys. In southwestern Pennsylvania, the SAR images [Burr et al., 2012]. Future missions to Titan may deepest and most easily visible valleys are 150–200 m therefore reveal landscapes that are even more diverse than deeper than the surrounding ridges, or >50% of the total those we can see currently. relief in the landscape (Table 1). In Michigan, many of the [48] It is possible that all regions of Titan imaged with features that disappear in the SAR imagery have relief of SAR have unresolved fine-scale fluvial dissection similar to 20–50 m, or 5–15% of the relief in the landscape. This that at the Huygens probe landing site [Tomasko et al., effect is the basis for our relief filter for the model data; we 2005]. To evaluate the effects of mapping uncertainty analyze only the portions of the networks that we think related to radar instrument resolution, we experimented on a would be visible in Titan’s SAR imagery. small data set of four model runs. We tested the sensitivity of [50] The effects on network shape that result from our results to image resolution by varying the drainage area changing the relief filter are very similar to the effects of threshold that we use to define a stream link when calcu- changing resolution (as described in the previous para- lating DW (Figure 11a). Even for large variations in the fine graph), and so it is unsurprising that these two experi- structure of the network, the overall trend in DW remains ments produce similar results on the modeled DW. consistent, as do DW values for low values of E/H0. How- Changing the sensitivity of this relief filter produces ever, for higher values of E/H0, the inclusion of the lowest- modest shifts in the model calibration curve for DW(a order branches can shift the equilibrium value of DW. As a reduction of 25% in the relief threshold produces a shift of

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3–4% in the mean DW). The sensitivity of our erosion [53] Some drainage networks on Titan, including drainages estimates to such shifts mirrors the response to variable res- from swaths T13 and T28 that are included in this study, have olution. Our estimates for the lower limits on Titan’s erosion been classified as rectangular [Burr et al., 2009; Drummond remain robust, but the percentiles at which we ascertain the et al., 2012]. This geometry may suggest that these regions upper limits on erosion may decrease from the values shown have been tectonically influenced [Burr et al., 2009; in Figure 8. Radebaugh et al., 2011; Drummond et al., 2012]. However, D unless this tectonic control is extensive enough to alter the 6.3. Tectonics, Spatial Variability in Erosion and W shape of the entire drainage basin, it is unlikely to strongly [51] The similarity in our erosion estimates for equatorial, affect measurements of DW or our estimates of cumulative northern high-latitude, and southern high-latitude networks erosion. Sub-basins in which tributaries have preferred on Titan is interesting given the asymmetric distribution of orientations will not necessarily impart an elongated shape to lakes on Titan. Aharonson et al. [2009] suggest that the the entire basin. Drummond et al. [2012] tabulate the geo- preponderance of lakes in the northern hemisphere may metric classification of all fluvial networks mapped through result from nonlinear climate responses to the eccentricity of Cassini swath T71. Around half of the networks with more Saturn’s orbit around the sun. Over timescales of 45 kyr, than 10 links in swaths T28/T29 and T39 are rectangular; all this asymmetry should reverse [Aharonson et al., 2009]. The of the networks of that size in swath T13 are rectangular lack of asymmetry in our data may imply that the networks [Drummond et al., 2012]. Given that the highest measured have developed over times much longer than 45 kyr, which DW is for the rectangular T13 networks (Table 1), it is dif- is reasonable given the constraints on Titan’s surface age ficult to account for Titan’s overall low DW values as a result [Wood et al., 2010; Neish and Lorenz, 2012]. Although of tectonic influences. Limited erosion provides a simpler clouds have been observed most frequently at high latitudes explanation. on Titan [e.g., Rannou et al., 2006], Schaller et al. [2009] [54] Several approaches could provide further insight into and Turtle et al. [2011] report storms in Titan’s tropics. Titan’s erosion rates, surface ages, and resurfacing history. The networks in swath T13 are hundreds of kilometers Better constraints on precipitation rates during rain events, across and are well developed (Figure 1), with high DW. and the distribution and frequency of rain events, would The similar lower limits on E/H0 suggest that these large permit better estimates of long-term erosion rates. Studies of drainages near Xanadu may have eroded as deeply as net- non-fluvial resurfacing mechanisms such as aerosol deposi- works close to the poles. tion and tectonics would clarify the geologic controls on [52] Within our study regions, the most distal branches of surface topography. Improved crater counting estimates some networks appear to originate in mottled light-and-dark would also contribute to our understanding of the chronol- “crenelated” terrains that evoke the appearance of highly ogy of landscape change on Titan. dissected regions in SAR imagery for Earth (Figure 11). This observation does not necessarily conflict with our interpre- 7. Summary and Conclusions tation that in some areas there has been relatively little are- ally averaged erosion. Our DW populations for Titan span [55] We develop a statistic that relates the planform vast areas. If spatially non-uniform erosion has occurred geometry of a fluvial network as it erodes into bedrock to the within those areas, we may be averaging the DW statistics of amount of cumulative erosion. We calibrate this statistic networks with disparate histories. It is also possible to gen- using a simple landscape evolution model, and show that the erate a spatially extensive imprint of fluvial erosion while width function of a fluvial network (the frequency distribu- only eroding through limited relief. For example, in our tion of links in the network with respect to flow distance numerical simulations, there is no bedrock uplift or lowering from the basin outlet) becomes systematically less uniform of sinks, implying that the maximum vertical erosion is as fluvial erosion progresses. Comparisons of recently gla- simply the initial relief. Yet the model drainage networks ciated landscapes versus landscapes with a longer history of propagate throughout the landscape well before the relief is fluvial erosion on Earth are qualitatively consistent with our completely eroded. Finally, our results do not exclude the theoretical prediction. Using fluvial networks on western ’ possibility that some regions on Titan may have experienced Kaua i, where an initial topographic surface with a known extensive erosion. Indeed, for two of our four study sites, age can be reconstructed, we demonstrate that the width including Xanadu, where there is abundant “crenelated” function can be used to probabilistically assess upper and terrain, the DW statistic does not produce a reliable upper lower bounds on the ratio of cumulative erosion to initial bound on erosion. Figure 6 demonstrates that the DW sta- relief. tistic is only sensitive to the early stages of landscape evo- [56] We apply these methods to fluvial networks mapped ’ lution, when drainage networks are invading previously in four regions on Titan, and find that Titan s north polar – undissected terrain. After most of the drainage area has been region has undergone a minimum of 0.4 0.6% of spatially captured, basin shapes no longer change appreciably averaged erosion of an initial surface, and for a subprovince although erosion continues, and DW ceases to provide a with better constraints we suggest that there has been at most ’ unique upper limit on cumulative erosion. For two regions 9% erosion. Titan s Xanadu region has experienced at least where we do provide upper bounds on erosion, our conclu- 0.4% erosion of its initial relief, and the southern high lati- sion based on measurements of DW is that drainage net- tudes show evidence for at least 0.5% erosion of the initial ’ works in these areas appear to have caused relatively little surface. Our upper limit on cumulative erosion in Titan s ’ exhumation. north polar region implies that, if estimates of Titan s surface age based on crater counting are correct, long-term erosion

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