## 1. Introduction

[2] Distinguishing valleys from hillslopes is a fundamental step in many hydrological and geomorphic analyses. In hydrology, for example, many models require that different values of hydraulic roughness, infiltration, or other model parameters be applied to valley bottoms and hillslopes for realistic results to be obtained [*Julien et al*., 1995]. In geomorphology, the computation of valley or drainage density, defined as the ratio of the total length of valley bottoms divided by the basin area, requires that the valley or drainage network first be defined. Drainage density, which varies over approximately 2 orders of magnitude in fluvially dominated terrain on Earth, i.e., from approximately 3 to 300 km^{−1} [*Melton*, 1957; *Madduma Bandara*, 1974], is important because it defines the transition from predominantly colluvial to predominantly fluvial sediment transport in landscapes [*Perron et al*., 2008].

[3] Most existing methods for drainage network extraction rely on contributing area or length (and user-defined contributing area or length thresholds that define the transition from hillslope to valley) to distinguish valley bottoms from hillslopes in digital elevation models (DEMs). Most early methods of drainage network extraction from DEMs used contributing area only [e.g., *O'Callaghan and Mark*, 1984] or a combination of contributing area, length, and slope [e.g., *Tribe*, 1991, 1992; *Montgomery and Dietrich*, 1992; *Giannoni et al*., 2005; *Hancock and Evans*, 2006; *Tarolli and Fontana*, 2009]. Using contributing area or length as a criterion for drainage network extraction is problematic for two reasons. First, as drainage density is inversely related to the average contributing area upstream from valley bottoms, using contributing area as a criterion for drainage network extraction results in circular reasoning—the drainage density resulting from the analysis depends strongly on the contributing area threshold (or similar contributing-area-dependent parameter) used to identify the transition from hillslopes to valleys in the extraction procedure. Second, using contributing area as a mapping criterion is problematic because valley bottoms are areas of localized, confined flow, not simply areas with large contributing area. As such, the topographic curvature or V shapedness should be used to define valley bottoms. In many landscapes, discontinuous valley networks exist in which well-defined V-shaped valleys with high topographic curvature grade into low-relief zones of distributary sheetflow that are better classified as hillslopes for the purposes of hydrologic and geomorphic analysis, despite the fact that they might have a relatively large contributing area.

[4] To mitigate the problems associated with using contributing area or length as a mapping criterion for drainage networks, methods based on topographic curvature have also been developed. The method of *Tarboton and Ames* [2001], for example, gives the user a choice between using a contributing area threshold and using the method of *Pueker and Douglas* [1975]. The *Pueker and Douglas* [1975] method identifies areas of positive curvature as valley networks. Given that hillslope segments close to valley heads can be zones of positive curvature and convergent flow but nonetheless be dominated by overland flow, distinguishing hillslopes and valleys for the purposes of hydrologic modeling often requires using a small positive threshold curvature rather than a threshold curvature equal to zero as in *Pueker and Douglas* [1975]. Recently, two methods have been developed that use a positive threshold for topographic curvature to define valley networks. *Molloy and Stepinski* [2007] developed a method for drainage network extraction that involves (1) DEM smoothing (using a two-dimensional (2-D) second-order polynomial fit to each 5 × 5 pixel moving window), (2) calculation of the contour curvature, defined as [*Mitasova and Hofferka*, 1993],

where *h*(*x,y*) is elevation and the subscripts denote derivatives, (3) identification of the valley network as the areas with contour curvature values larger than a user-defined threshold, (4) filtering of this “draft” map to remove circular and large components using circularity and contributing-area-based thresholds, (5) thinning of the map to a single pixel width, and (6) reconnection of the disconnected pieces into connected networks. The heart of the *Molloy and Stepinski* [2007] algorithm is the use of the contour curvature to map the valley network. However, because the contour curvature only partly correlates with where valley bottoms occur, the method requires additional filtering and reconnection steps (labeled 4 and 6 above) involving contributing area to obtain a reasonable result. The method of *Molloy and Stepinski* [2007], applied to a terrestrial example by *Luo and Stepinski* [2008], requires six site-specific user-defined input parameters, including the width of the smoothing filter (which must be allowed to vary for the method to work with DEMs of variable resolution).

[5] *Passalacqua et al*. [2010b] developed a method similar in many aspects to that of *Molloy and Stepinski* [2007]. *Passalacqua et al*. [2010b] smoothed the input DEM using a nonlinear diffusion filter and then computed a draft valley network map based on domains with contour curvature values larger than a prescribed threshold value. As in *Molloy and Stepinski* [2007], the draft map obtained by thresholding the contour curvature map typically results in many disconnected valley segments. To improve the draft map, *Passalacqua et al*. [2010b] developed a framework based on geodesic pathways. In their approach, the valley network is defined using connected pathways that minimize a cost function inversely proportional to contributing area and threshold curvature. The resulting algorithm requires five site-specific user-defined parameters, one less than the method of *Molloy and Stepinski* [2007]. The algorithms of *Molloy and Stepinski* [2007] and *Passalacqua et al*. [2010b] both require the use of contributing area to achieve reasonable results, and hence, the drawbacks of contributing-area-based methods noted above apply to these methods.

[6] The method illustrated in this study divides the problem of drainage network extraction into two parts: (1) identifying valley heads and (2) identifying valley bottoms downstream from valley heads. Valley heads differ from hillslopes in that they are zones of positive topographic curvature where flow transitions from unconfined sheetflow to localized, confined flow. Valley bottoms are portions of the landscape located downstream from valley heads where localized, confined flow is maintained. In this method, these two principles are used to map drainage networks by using just two user-defined parameters, one a threshold for contour curvature required to define a valley head and another a threshold of flow confinement required for a valley to be maintained downstream from the valley head. The fact that the algorithm given in this study reduces the number of free parameters relative to alternative methods is highly significant. In a method with two free parameters, the parameter space can be searched relatively quickly to identify the best results for a particular case. If each parameter is sampled within a range of five values, for example, just 25 cases define the full parameter space of a method with just two free parameters. In a model with five free parameters, in contrast, a search of the parameter space requires 5^{5} or 3125 cases to be considered, a number sufficiently large that a complete search of the parameter space becomes practically difficult.

[7] The analysis of this study is also novel in that the proposed method is rigorously tested on synthetic valley networks, i.e., those in which the valley network is known exactly by construction, prior to its application to real-world landscapes. It is necessary to test any proposed method for drainage network extraction on synthetic valley networks, in addition to real-world networks, because the “true” valley network of natural landscapes often cannot be established precisely in the field or using high-resolution DEMs. This is because nearly all landscapes exhibit a continuous gradation in form from weakly convergent hillslope segments to well-defined V-shaped valleys. Figure 1 illustrates this problem using a first-order valley in the Santa Catalina Mountains (SCM) of southern Arizona as an example. In this landscape, a weakly convergent hillslope grades continuously into a well-defined V-shaped valley. Precisely locating where the hillslope ends and the valley bottom begins is difficult and subject to considerable subjectivity in this case. One approach to this problem would be to explicitly recognize and map transitional zones between well-defined hillslopes and valley bottoms. For most purposes, however, e.g., hydrologic modeling, it is still necessary to make a clear, binary distinction between hillslopes and valley bottoms. As a result of gradual nature of the hillslope-valley transition in many real-world landscapes, it is necessary to test the efficacy of any proposed method for drainage network extraction on synthetic landscapes in which the valley network is known exactly by construction.

[8] The goal of drainage network extraction is to identify locations in the landscape where a transition occurs from undissected hillslopes (where overland flow and rill wash are dominant) to valley bottoms (where concentrated surface water flow occurs, whether or not a channel exists). Valley heads do not necessarily coincide with channels heads due to migration of channel heads with storms and climate change [*Dietrich and Dunne*, 1993; *Tucker et al*., 2001]. As such, many valley heads are colluvial hollows that only become channels (defined as fluvial pathways with well-defined banks) some distance downstream from the valley head. In this article, the terms “valley” and “drainage” are used interchangeably. This usage is consistent with the previous studies of drainage density that have used a morphological criterion (e.g., the contour-crenulation method) to compute drainage density [e.g., *Melton*, 1957; *Abrahams and Ponczynski*, 1984; *Madduma Bandara*, 1974]. Therefore, the focus of this study is on valley density, i.e., the morphological transition between relatively smooth, undissected hillslopes and valley bottoms with concentrated water flow. This is important because drainage density has been defined in the literature in a number of ways, e.g., as the density of channels and of perennial streams.

[9] Figure 2 illustrates the four test cases considered in this study: two synthetic valley networks and two real-world landscapes. The two real-world cases were chosen from the University of Arizona Jemez River Basin-SCM (JRB-SCM) Critical Zone Observatory. The first synthetic network (Figures 1a and 1b) corresponds to an analytic solution, developed and validated by *Pelletier and Perron* [2012] using real-world examples from southern Arizona, of a valley head migrating steadily headward into a gently dipping plateau. The mathematical form of this landscape is given as follows:

where *h* is elevation, *x* and *y* are the cross-valley and along-valley distances from the origin, *h*_{0} is the valley relief, *S*_{0} is the regional slope of the landscape, and *Pe* is the Peclet number (equal to *vR*/2*D*, where *v* is the valley head migration rate, *R* is the planform radius of curvature of the valley head, and *D* is the diffusivity). To create the synthetic DEM used in this study, equation (2) with *h*_{0} = 10 m, *S*_{0} = 0.05, *R* = 0.002 m, and *Pe* = 0.0001 was sampled with a pixel size of 0.5 m (Figure 2a), and a Gaussian white noise with a standard deviation of 0.5 m was superimposed to represent microtopographic variability (Figure 2b). The second synthetic landscape consists of three second-order valleys, each with a different distance from the divide, resulting in a variable drainage density across the landscape. The DEM for this synthetic landscape was constructed by hand drawing a “mask” grid of valley bottoms using a graphics program and then inputting that mask into a landscape evolution model that uplifts and diffuses all nonvalley-bottom areas with an uplift rate of 1 mm/year and a diffusivity of 5 m^{2}/kyr for 100 kyr. The landscape was then tilted 5% toward the downslope edge of the model domain so that the valleys would drain (Figure 2c). Finally, a Gaussian white noise with a standard deviation of 0.2 m was superimposed on the topography (Figure 2d). A lower magnitude of noise was used in the second synthetic landscape when compared with the first simply to test the ability of the method to handle microtopographic noise of different magnitudes.

[10] Figure 2e illustrates a portion of Marshall Gulch in the SCM near Tucson, Arizona, the first of two real-world test cases considered. The DEM illustrated in Figure 2e is a 2 m/pixel lidar-derived bare-ground DEM obtained from the Pima County Flood Control District. The second site is a portion of the south side of Redondo Mountain in the JRB of northern New Mexico. The DEM for this study site is a 1 m/pixel bare-ground DEM provided by the National Center for Airborne Laser Mapping. Both sites are located 2500 m above sea level and hence are predominantly—covered by forest. The Redondo Mountain site is characterized by lower drainage density when compared with the Marshall Gulch study site, most likely as a result of the thick soils (and hence higher infiltration capacities) characteristic of this highly fractured rhyolitic resurgent dome. These two sites were chosen because of the availability of DEMs derived from airborne lidar data, the contrast in drainage density between them, the existence of discontinuous valleys in one of the two sites (i.e., Redondo Mountain), and the author's familiarity with both sites based on previous work [e.g., *Pelletier and Rasmussen*, 2009; *Pelletier et al*., 2011].