Corresponding author: J. D. Pelletier, Department of Geosciences, University of Arizona, 1040 East Fourth St., Tucson, AZ 85721-0077, USA. (email@example.com)
 In this article, a method for drainage network extraction from high-resolution digital elevation models (DEMs; e.g., those derived from airborne laser swath mapping) is presented, which requires just two user-defined parameters and is capable of handling discontinuous valley networks. The accuracy and robustness of the method are illustrated using synthetic valley networks that mimic the complexities of real landscapes and for which the true drainage network is known exactly by construction. The method involves six principal steps: optimal Wiener filtering to remove microtopographic noise, mapping of the contour curvature, identification of valley heads using a user-defined contour-curvature threshold criterion, routing of a unit discharge of water from each valley head using a multiple-flow-direction routing algorithm, removal of discontinuous reaches from the drainage network using a user-defined discharge-per-upstream-valley-head threshold criterion, and thinning of the valley network to a single pixel width. The method yields accurate results using the same user-defined parameters for the two field sites considered in this study, suggesting that for DEMs with resolution of approximately 1 m, the method has the ability to produce accurate results for a variety of landscapes by using the same parameter values used in this study.
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 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].
 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.
 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 , for example, gives the user a choice between using a contributing area threshold and using the method of Pueker and Douglas . The Pueker and Douglas  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 . Recently, two methods have been developed that use a positive threshold for topographic curvature to define valley networks. Molloy and Stepinski  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  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 , applied to a terrestrial example by Luo and Stepinski , 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).
Passalacqua et al. [2010b] developed a method similar in many aspects to that of Molloy and Stepinski . 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 , 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 . The algorithms of Molloy and Stepinski  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.
 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 55 or 3125 cases to be considered, a number sufficiently large that a complete search of the parameter space becomes practically difficult.
 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.
 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.
 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  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, h0 is the valley relief, S0 is the regional slope of the landscape, and Pe is the Peclet number (equal to vR/2D, 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 h0 = 10 m, S0 = 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 m2/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.
 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].
2. Steps 1–3: Smoothing, Contour Curvature Mapping, and Valley Head Identification
 The principal goal of the first three steps of valley head identification is to identify valley heads using map criteria that do not include contributing area. Following the work of Molloy and Stepinski  and Passalacqua et al. [2010a, 2010b], the contour curvature is used for this purpose. The use of contour curvature to map valley heads has roots in the contour-crenulation method used to identify valleys from contour maps in the predigital era [e.g., Melton, 1957].
 Local curvature values are extremely sensitive to microtopographic noise in high-resolution DEMs. With each derivative (from elevation to slope and from slope to curvature), the amplitude of microtopographic noise is greatly enhanced relative to the amplitude of the larger-scale shape of the landscape that the method uses to identify valley heads. Microtopographic noise is ubiquitous in high-resolution DEMs. Such noise can be sourced from errors in the identification of the true ground surface in the presence of vegetation or from the fact that some colluvial transport processes, e.g., bioturbation, result in the roughening of topography (e.g., the development of pit-and-mound topography) at small space and time scales even as they smooth the landscape at larger space and time scales [Jyotsna and Haff, 1997]. Great care must be taken when attempting to suppress microtopographic noise to avoid artificially modifying the large-scale shape of the hillslope-valley system used to define the valley network.
 In this section, the efficacy of three techniques for noise suppression in high-resolution DEMs is tested: (1) diffusive smoothing, (2) optimal Wiener filtering (OWF), and (3) nonlinear diffusive or Perona-Malik (PM) filtering. Diffusive smoothing (also known as a Gaussian filter) involves solving the diffusion equation, i.e.,
for the evolution of the DEM topography forward in time over an interval T. Diffusion of the DEM topography forward in time smooths the small-scale or high-spatial-frequency components of the landscape more rapidly than the large-scale or low-spatial-frequency components. Diffusion can be a simple and effective method for suppressing microtopographic noise without significantly modifying the large-scale shape of the landscape. The amount of filtering in the diffusion method is defined by the product of the diffusivity D and time interval T. The units of DT are m2. The user must determine the value of DT that corresponds to just the right amount of smoothing to suppress the microtopographic noise without artificially changing the large-scale morphology of the valley network.
 OWF involves distinguishing a “signal” (the large-scale morphology of the hillslope-valley system) from noise in frequency space and filtering the DEM to enhance the signal relative to the noise. Figure 3a illustrates the OWF technique using the analytic valley head case (Figure 2b) as an example. Figure 3a plots the radially averaged power spectrum of this DEM, S(ν), as a function of the spatial frequency, ν, i.e., the inverse of the wavelength. The power spectrum is plotted only up to the Nyquist frequency, defined as one over twice the pixel size, because the power spectrum is aliased at frequencies higher than that frequency. The large-scale shape of the analytic valley head is characterized by a power-law radially averaged power spectrum denoted by |H(ν)|2 in Figure 3a. The white noise component has, by definition, a flat power spectrum with equal power at all spatial frequencies (labeled |N(ν)|2 in Figure 3b). The idea of the OWF is that at large scales, the noise contributes little to the power spectrum, whereas at higher frequencies, the opposite is true. As such, the power spectrum of the signal (without noise) can be estimated using the trend of the power spectrum of the input DEM at low frequencies (which often has a power-law relationship with spatial frequency), whereas the power spectrum of the noise can be modeled using a fit to the power spectrum of the input data at high spatial frequencies (i.e., a constant fit to the mean amplitude of the power spectrum just below the Nyquist frequency). Wiener  showed that the best possible linear filter works in frequency space with the following transfer function:
For relatively low frequencies, |H(ν)|2 ≫ |N(ν)|2, equation (4) is nearly equal to 1, and the transfer function does not modify the input data at all. At high frequencies, |H(ν)|2 ≪ |N(ν)|2, and the amplitude of the noisy signal is reduced by a factor equal to the ratio of the noise to the estimated amplitude of the true signal. A key advantage of the OWF is that it involves no free parameters. Unlike diffusion, in which the user must decide how much diffusion is “enough,” the OWF uses the structure of the power spectrum to decide how much filtering should be performed.
 Figure 3b plots the transfer function (equation (4)) for the analytic valley head case and compares it with the filtering of the diffusion equation for three representative values of DT. Solving the diffusion equation is equivalent to filtering in frequency space with a transfer function equal to exp(−(2πν2)DT). Figure 3b shows that for the analytic valley head case, the value of DT that most closely matches the transfer function of the OWF is DT = 0.5 m2.
 Figures 3c and 3d illustrate the radially averaged power spectra and OWF functions |H(ν)|2 and |N(ν)|2 corresponding to the two synthetic landscapes (Figure 3c) and the two real landscapes (Figure 3d) considered in this study. In all cases, the power spectra are characterized by a power-law function of ν for the lowest spatial frequencies and a constant (flat) spectrum for the highest spatial frequencies. Figure 4 illustrates the results of OWF on the four test cases. The results show more smoothing in the synthetic cases when compared with the real cases. This is appropriate because the magnitude of the noise is relatively large in both of those test cases when compared with the real-world landscapes, and hence, the OWF responds with filtering over a larger range of spatial frequencies to suppress the noise.
 Once filtered, the valley-head-identification component of the method computes the contour curvature using equation (1) and then searches the landscape, from the highest elevation to the lowest elevation in rank order, identifying locations where the contour curvature, κ, is greater than a user-prescribed threshold value, κt. Once a valley head is found, the method “flags” all pixels located downstream (along any downslope flow pathway, not just the direction of steepest descent), and the algorithm removes those downstream points from consideration as valley heads regardless of their value of κ. This procedure requires that the DEM be hydrologically conditioned after the filtering step but prior to the flagging step. In this study, the fillinpitsandflats routine of Pelletier  is used for this purpose. This routine increases the local elevation of any pit or flat in the DEM by a small amount (1 cm) and then recursively operates on all neighboring pixels, automatically stopping when the DEM drains properly. This method can be adjusted to remove only those pits and flats that are below a certain threshold area or it can be otherwise adjusted to minimize the side effects of hydrologic conditioning.
 Figure 5 illustrates a test of valley head identification in the analytic valley head test case. The diffusion filter is used instead of the OWF for this test because by using the diffusion filter and varying the value of DT, the robustness of the method with respect to the degree of smoothing, DT, and the value of the threshold contour curvature, κt, can be documented. Figure 5a illustrates the error between the actual valley head and the predictions of the valley head identification for a range of values of DT and κt. The value of κt is multiplied by DT to define the horizontal axis of Figure 5a because greater smoothing (higher DT values) generally necessitates smaller κt values. As a consequence, it is convenient to multiply κt by DT for plotting purposes. The error visualized in the map of Figure 5a has units of meters and is equal to the distance between the predicted valley head and the actual head. If the algorithm predicts multiple valley heads, the displacement errors are summed. In this way, the error penalizes the algorithm for predicting valley heads in the wrong place and for predicting more valley heads than actually exist. As Figure 5a shows, the method works poorly, i.e., the error is greater than 20 m (shown as white), for values of DT lower than approximately 0.3 m2. Without sufficient smoothing, the valley head cannot be identified within the noise of the DEM regardless of the value of κt. For values of DT in the range of approximately 0.3 to 1.0 m2, the method correctly identifies the valley head (i.e., errors are <1 m, shown as dark red or red; areas with exactly zero error are in black) for a range of κtDT values. The best result, i.e., zero error, is achieved for the widest range of κtDT values for DT = 0.5 m2, which corresponds to the value of DT most similar to the OWF. For values of DT greater than approximately 0.5 m2, the range of κt values for which the method achieves errors less than 20 m (shown as various colors) increases; however, the method also has less average precision within that range (there are more areas indicative of high error than black areas along transects of constant DT for larger DT values). For relatively large values of DT, the noise is more effectively suppressed, resulting in a broader range of κt values that yield reasonable results; however, the valley head is also more distorted in shape as a consequence of the more aggressive noise suppression, which results in lower precision.
 Figures 5b–5e illustrate two ways in which the method can fail to accurately identify the true valley head. Figure 5b illustrates the contour curvature map with a relatively small amount of smoothing (DT = 0.2 m2), and Figure 5c illustrates the contour curvature map with a relatively large amount of smoothing (DT = 0.9 m2). Figure 5d illustrates the predicted valley head map using the smoothing of Figure 5b and a κt value of 0.75 m−1. In this case, the method falsely identifies many valley heads because the microtopographic noise has only been partially filtered out. Figures 5c and 5e illustrate the opposite case of too much smoothing. In this case, the DEM has been smoothed to the point where the valley head has been significantly distorted. In this case, the method identifies three valley heads (using κt = 0.15 m−1, i.e., nearly the same κtDT value as in Figure 5d) in the immediate vicinity of the one true valley head (Figure 5e). The advantage of using the OWF method for smoothing is that an appropriate value of smoothing is chosen automatically by the analysis, allowing the user to focus on choosing an appropriate value of the threshold contour curvature, rather than having to select the combination of multiple parameters that yields the most accurate result.
 Figures 5f and 5g illustrate the difficulty of using contour curvature alone to identify valley networks. Figure 5 also illustrate areas where the contour curvature is greater than κt = 0.05 m−1 (Figure 5f) and 0.1 m−1 (Figure 5g) for the case of the analytic valley head smoothed with the OWF. If the threshold contour curvature value, κt, is chosen to be large enough to avoid false valley identification on the hillslopes, then the method identifies a discontinuous valley (Figure 5g). If the value of κt is lowered in an attempt to allow the contour curvature method to identify a continuous valley, the method identifies many false valleys on the hillslope (Figure 5f). Therefore, even in this case of a single valley, using contour curvature to identity the valley fails. This result is consistent with the fact that Molloy and Stepinski  and Passalacqua et al. [2010b] added filtering and reconnection steps using contributing area to obtain reasonable results using contour curvature alone to identify valleys.
 Figure 6 illustrates the results of valley head identification on the synthetic landscape with second-order drainage basins. Figure 6a illustrates the contour curvature of the landscape, after OWF has been applied to the initial DEM. Figure 6b illustrates the error map (analogous to Figure 5a) for a range of values of DT and κt. Note that as in Figure 5a, the error map is shown using diffusive smoothing, despite the fact that the use of OWF is recommended, because the diffusive filter is similar to the OWF for an optimal value of DT and because by varying the value of DT, it is possible to illustrate the robustness of the results with respect to variations in the amount of smoothing and the chosen value of the threshold of contour curvature. The shape of the error map for this case is comparable with that in Figure 5a, i.e., significant regions of the parameter space have zero error once a certain amount of smoothing has been performed, suggesting that the method can work well in cases with multiple valleys and different magnitudes of noise.
Passalacqua et al. [2010b] proposed using the nonlinear diffusion or PM filter to minimize the effects of DEM noise in drainage network extraction. The PM filter [Perona and Malik, 1990] was originally developed for image processing and has been shown to be very effective at minimizing noise while maintaining discontinuities in an image. The PM filter solves the diffusion equation with a diffusivity that varies spatially according to slope, i.e.,
Equations (5) and (6) result in normal diffusive smoothing of portions of the landscape that have slopes comparable with or less than λ, whereas portions of the landscape that have slopes significantly greater than λ are not affected because the effective diffusivity (a combination of D and c) goes to zero as the slope becomes much larger than λ.
 Figures 7a–7c illustrate the application of the PM filter to a landscape with a central “plateau” oriented in the north-south (NS) direction. This DEM has a pixel size of 0.5 m, a central portion that is 3 m higher than the surrounding landscape, and a Gaussian noise with a standard deviation of 1. Results of the PM filter are shown using λ = 1 for DT = 0 (Figure 7a), DT = 1 m2 (Figure 7b), and DT = 3 m2 (Figure 7c). The slope on the edges of the plateau in this DEM is much greater than λ (i.e., 3 m/0.5 m or 6 times larger than λ), and hence, the nonlinearity in equation (6) works to maintain the steep slope in this portion of the landscape while diffusing noise in the rest of the landscape. Although the PM filter works well in cases with changes in elevation between adjacent pixels that are much larger than the noise present in the rest of the DEM, the method may work less well in some cases that mimic the actual morphology of valley heads, where discontinuities in elevation are less pronounced than in Figures 7a–7c. Figures 7d–7f illustrate the output of the PM filter applied to the analytic valley head case using λ = 1 and the same values of DT as in Figures 7a–7c. If the value of λ is set too high, the PM filter reduces to simple diffusion, and the results are indistinguishable from those in Figure 5. However, if the value of λ is set low enough that nonlinear diffusion actually occurs, the method works to preserve some of the noise present on the hillslopes (Figure 7e). Figure 7f illustrates the results of the error analysis for valley head identification using the PM filter on the analytic valley head case for a range of values of DT and κt with λ = 1. Figure 7f shows that the PM filter achieves results that are worse than the simple diffusion filter for this test case. Specifically, there are no values of DT and κt that yield zero error in Figure 7f, whereas the simple diffusion filter (Figure 5a) yields results with zero error for a significant portion of the parameter space. The PM filter can be shown to work better (i.e., yield more accurate results over a wider range of κt values) than the simple diffusion filter for the analytic valley head case for some DT values if the amplitude of the noise is sufficiently small, e.g., 0.1 m instead of 0.5 m. Nevertheless, the results of this test case indicate that in at least some cases, the PM filter works less well than simpler methods while also having the disadvantage that it requires the user to specify an additional site-specific parameter.
3. Steps 4–6: Flow Routing from Valley Heads, Identification of Discontinuous Reaches, and Network Thinning
 Given the challenge of identifying valleys using contour curvature alone, the method illustrated in this study divides drainage network extraction into two sets of steps: (1) valley head identification (steps 1–3) and (2) routing of a unit of discharge from each valley head by the multiple flow direction (MFD) method of Freeman  (steps 4–6, including the postprocessing steps related to that routing). The second set of steps has an advantage over using contour curvature alone to identify valleys because the continuity of the resulting valley network is assured, i.e., any hydrologically conditioned DEM has continuous pathways of flow from every point to the boundaries of the grid or, if properly accounted for in the hydrologic conditioning, bodies of standing water within the DEM. In nature, however, many valley networks are discontinuous. In arid environments, for example, drainage networks in upland terrain often drain onto undissected alluvial fans that lack valleys or channels. Therefore, it is important for any method of drainage network extraction to include a component for identifying discontinuous valley segments.
 Discontinuous valleys occur when localized, confined flow transitions into distributary sheetflow. Distributary sheetflow is characterized by a low discharge per pixel, i.e., when flow becomes distributed over a wide zone of unconfined flow, the discharge per pixel in that area decreases significantly below the discharge per pixel in confined valleys immediately upstream. In this study, discontinuous valleys were identified as those in which the discharge per pixel divided by the number of valley heads upstream falls below a user-prescribed threshold value, Rt. For example, at a pixel that has 10 valley heads upstream, the discharge to that pixel will be 10 times the unit of discharge input into each valley head if the landscape is 100% tributary. However, if the discharge is only one unit at that same pixel, nine units of discharge must be located in nearby pixels within an area of distributary or unconfined flow. In this case, the discharge per pixel divided by the number of valley heads upstream is equal to 0.1, and hence, the area will not be classified as a valley segment provided Rt ≥ 0.1.
 In valleys that are significantly wider than a single pixel, the flow routing procedure results in a map with some valleys that are wider than a single pixel. For some applications (e.g., estimating valley length for input to the computation of drainage density), it is desirable to map the valley as a single pixel wide at the location of the valley centerline. Therefore, as an optional step 6, the valley map may be “thinned” to a single pixel. In this study, the method of Rosenfeld and Kak  is used for this purpose.
 In routing flow from each valley head, the user can choose to use the original, hydrologically conditioned but unfiltered DEM or the filtered DEM used to identify the valley heads. As noted in section 2, noise can be sourced from errors in the identification of the true ground surface in the presence of vegetation or from the fact that some colluvial transport processes result in the roughening of topography. If the noise in the DEM is primarily caused by interpolation artifacts, then the smoothed/filtered DEM should be used for flow routing. If the noise represents real changes in valley bottom elevation and/or the presence of boulders, then whether the filtered or unfiltered DEM should be used for the flow routing procedure depends on whether the user wants a map of valleys that includes small contortions of the flow around large boulders and other small-scale irregularities in the valley bottom or whether the user wants a map of the valley network with such small-scale variability removed. In the remaining examples of this article, assuming that users prefer a valley map with small-scale variability removed, DEMs that include filtering are used. However, the flow routing can be performed on either filtered or unfiltered DEMs, depending on the user's preference, once the valley heads have been identified.
 Figure 8 illustrates the results of the flow routing on the analytic valley case using two magnitudes of smoothing. Figure 8a illustrates the results for the case with relatively little smoothing (diffusion filter with DT = 0.1 m2), and Figure 8b illustrates the results using OWF. In both cases, one unit of discharge is introduced at the true valley head (i.e., steps 1–3 for identifying the valley head are assumed to be performed using parameters that lead to an accurate result), and discharge is routed downstream using the MFD method. The true network in this case is (assuming that the small-scale variability represents artifacts or other unwanted noise) a single straight line in the middle of the domain oriented N-S direction. Under minor smoothing (Figure 8a), the valley takes a tortuous path and results in a high error. Under OWF (Figure 8b), discharge runs down the central column of pixels along the true valley pathway. When this map is thinned to a single pixel using the Rosenfeld and Kak  algorithm, the predicted valley matches the true valley exactly except for just one incorrect pixel located near the lower boundary of the grid.
 Figures 8c and 8d illustrate the application of the method to a synthetic landscape with a discontinuous valley. This landscape is identical to the analytic valley head case, except that the bottom third of the DEM is replaced with a low-relief planar piedmont that dips gently toward the bottom of the image. This modification of the analytic valley head case was made to illustrate the operation of the method used in this study to landscapes with discontinuous valleys. Figure 8d illustrates the flow map obtained by routing a unit of discharge from the valley head using the MFD method. When flow enters the undissected piedmont, it becomes unconfined, and the discharge per pixel decreases substantially with increasing distance downslope. The method illustrated in this study terminates the valley where the discharge falls below 0.1 discharge units. In practice, Rt = 0.1 is found to be a reasonable default value—it is both small enough that it does not result in many “false” discontinuities being mapped just upstream from tributary junctions (where tributary valleys may widen locally), and it is large enough that it does not fail to identify many discontinuities associated with the transition from a confined valley to an undissected piedmont.
 Figure 9 illustrates the corresponding results for the case of the synthetic second-order drainage basins. Figure 9a illustrates the results of (1) applying the OWF to the initial DEM, (2) computing the contour curvature of the filtered DEM, (3) identifying valley heads as the first downstream pixels (searching the grid from highest to lowest elevation in rank order) with a contour curvature greater than κt = 0.1 m−1, (4) routing one unit of discharge from each valley head, (5) identifying discontinuous reaches as those in which the ratio of discharge to the number of valley heads upstream falls below Rt = 0.1 of the discharge input to each valley head, and (6) thinning the valley network to a single pixel. The method does a good job of mapping three second-order valleys that are very similar in shape to the actual valleys in the original DEM; however, the error map (which shows pixels that should be mapped as valley pixels but are not, as well as pixels that are incorrectly mapped as valley pixels) illustrates that, in detail, the OWF and thinning components of the method can have the effect of shifting valleys as much as one pixel from their true location. This shifting affects approximately 5% of the pixels in this example.
 Figure 10 illustrates the results of the method on a portion of Marshall Gulch in the SCM, Arizona. Figure 10a is a color map of the discharge obtained by routing one unit of discharge from each valley head using the MFD method. This map was created by (1) applying the OWF to the 2 m/pixel airborne lidar DEM of the area obtained from the Pima County Flood Control District, (2) computing the contour curvature of the filtered DEM, (3) identifying valley heads as the first downstream pixels (searching the grid from highest to lowest elevation in rank order) with contour curvature greater than κt = 0.1 m−1, and (4) routing one unit of discharge from each valley head. Figure 10b is a map of the number of valley heads upstream from each pixel. Figure 10c illustrates the flow map with discontinuous valleys (i.e., those in which the discharge divided by the number of valley heads upstream is less than Rt = 0.1 times the unit of discharge input into each valley head) removed. Figure 10c was obtained by computing the ratio of the maps in Figures 10a and 10b and then removing any valley segments with values less than 0.1. There is little difference between Figures 10a and 10c because Marshall Gulch is predominantly a tributary drainage basin. Figure 10d is the valley network obtained by thinning the flow map of Figure 10c using the Rosenfeld and Kak  algorithm. Figure 10e is the valley mask obtained using Rt = 0 (or, equivalently, skipping the steps related to identifying discontinuous valley segments). As expected, this map is essentially the same as Figure 10c, as in cases of 100% tributary drainage, the series of steps designed to identify discontinuous valley segments have little or no effect.
 Figure 11 illustrates the corresponding results for the south side of Redondo Mountain, New Mexico. In this area, several of the valleys draining from the mountain terminate in undissected piedmonts. As such, this is a good location to test the ability of the method to identify discontinuous valleys. Figure 11d illustrates the valley network map predicted by the method using κt = 0.1 m−1 and Rt = 0.1. This map identifies several of the valleys in the southern portion of the study area as discontinuous as they transition from valley to piedmont. Precisely where the method identifies the transition from valley to piedmont depends on the specific value of Rt. However, for a broad range of values, i.e., from Rt = 0.03 to 0.3, the method recognizes a transition from valley to piedmont close to where they are visually apparent based on the shaded relief image in Figure 2f. The valley mask obtained using Rt = 0 (Figure 11e) illustrates how poorly the method would perform if it was unable to identify discontinuous valleys where confined flow drains onto undissected piedmonts. In this case, the thinning algorithm, as applied to zones of unconfined flow, predicts complex braided and anastomosing patterns in the valley network. There may be cases where a map of such complex areas of distributary flow is of benefit. However, for the purposes of mapping well-defined valleys, the results shown in Figure 11d with Rt = 0.1 are more accurate. The method does a good job of identifying valleys in this area using the same input parameters used in the analysis of Marshall Gulch (Figure 10), despite the fact that the two sites are quite different. This is significant because it suggests that the method can work well with default parameter values, i.e., without precise site-specific calibration. The maps of field-surveyed valley heads for the two field sites considered here are not presented because it is difficult in many cases to objectively and precisely determine where valley heads are located. Because there is a continuum of forms between weakly convergent hillslopes and well-defined V-shaped valleys (especially in the Redondo Mountain study area), it is difficult to produce a precise map of the “true” valley network. For this reason, in this article, the focus is on testing the proposed method using synthetic valley networks in which the correct answer is known. Nonetheless, the valley network maps produced by the method for the two real-world study sites do a good job of identifying where well-defined V-shaped valleys occur based on a visual comparison of the valley masks in Figures 10d and 11d with the shaded relief images in Figures 2e and 2f.
4. Discussion and Conclusions
 The method illustrated in this study requires just two parameters: a threshold contour curvature, κt, and a threshold value of flow confinement, i.e., the discharge relative to that of a confined tributary valley, Rt. The method works with high precision in the synthetic landscapes. Visually accurate results were also obtained for the two real-world study sites using the same two parameter values, i.e., κt = 0.1 m−1 and Rt = 0.1. To better interpret the meaning of this particular value of κt, it is helpful to think of curvature in terms of the radius of curvature of a circle fit to a contour located close to the valley head. Valley heads with κ > 0.1 m−1 have contours with local radii of curvature less than 5 m. As such, the method illustrated in this study suggests that, in DEMs with resolutions of approximately 1 m, valley heads can be identified as areas of the landscape with contours that have radii of curvature less than approximately 5 m, without the need for more precise site-specific calibration in many cases. Of course, obtaining the best results will require that the values of κt and Rt be calibrated to each study site. However, the results presented here suggest that reasonably accurate results can be obtained using κt = 0.1 m−1 and Rt = 0.1 when high-resolution DEMs derived from airborne lidar data are used as input.
 More research is needed to identify which smoothing method is optimal for most landscapes. All three of the methods described in section 2 can provide effective means of smoothing high-resolution DEMs to suppress noise. The OWF has the significant advantage that it requires no user-defined parameters. Depending on the magnitude of the noise relative to the steepest slopes in the landscape, the PM filter can perform better or worse than simple diffusion; however, in either case, it has the disadvantage of requiring two site-specific parameters.
 In this article, a method of drainage network extraction is presented, which is simpler and more robust than alternative methods proposed in the literature. The method involves six principal steps: OWF to remove microtopographic noise, mapping of contour curvature, identification of valley heads using a user-defined contour-curvature criterion, routing of a unit discharge of water from each valley head, removal of discontinuous reaches from the valley network using a user-defined threshold criterion based on discharge per upstream valley head, and thinning of the valley network to a single pixel width. OWF performs well for the topography characteristic of low-order fluvial valleys, and it has the additional advantage that it requires no free parameters. Flow routing from each valley head guarantees the establishment of a continuous valley network. The method identifies discontinuous valley segments, where appropriate, using a threshold criterion that relates to the discharge relative to that expected for a confined valley. The model was successfully tested on synthetic valley networks that mimic the complexities of real landscapes (i.e., microtopographic variability and discontinuous valley networks) and for which the true valley network is known exactly by construction. The method also yields visually accurate results for the same user-defined parameters in the two field sites considered here, suggesting that it can produce accurate results using default parameter values.
 This study was funded by National Science Foundation award 0724958 and Department of Energy award DE-SC0006773. The author thanks David Tarboton and two anonymous reviewers for their helpful comments that improved the manuscript.