We propose a novel way to measure and analyze networks of drainage divides from digital elevation models. We developed an algorithm that extracts drainage divides based on the drainage basin boundaries defined by a stream network. In contrast to streams, there is no straightforward approach to order and classify divides, although it is intuitive that some divides are more important than others. A meaningful way of ordering divides is the average distance one would have to travel down on either side of a divide to reach a common stream location. However, because measuring these distances is computationally expensive and prone to edge effects, we instead sort divide segments based on their tree-like network structure, starting from endpoints at river confluences. The sorted nature of the network allows for assigning distances to points along the divides, which can be shown to scale with the average distance downslope to the common stream location. Furthermore, because divide segments tend to have characteristic lengths, an ordering scheme in which divide orders increase by 1 at junctions mimics these distances. We applied our new algorithm to the Big Tujunga catchment in the San Gabriel Mountains of southern California and studied the morphology of the drainage divide network. Our results show that topographic metrics, like the downstream flow distance to a stream and hillslope relief, attain characteristic values that depend on the drainage area threshold used to derive the stream network. Portions along the divide network that have lower than average relief or are closer than average to streams are often distinctly asymmetric in shape, suggesting that these divides are unstable. Our new and automated approach thus helps to objectively extract and analyze divide networks from digital elevation models.

Drainage divides are fundamental elements of the Earth's surface. They define the boundaries of drainage basins and thus form barriers for the transport of solutes and solids by rivers. It has long been recognized that drainage divides are not static through time but that they are mobile and migrate laterally (e.g., Gilbert, 1877). The lateral migration of divides is a consequence of spatial gradients in surface uplift (positive or negative) and stream captures. These frequently accompany tectonic deformation due to shearing, stretching, and rotating stream networks (Bonnet, 2009; Castelltort et al., 2012; Goren et al., 2015; Forte et al., 2015; Guerit et al., 2018), but recent studies have shown that even in tectonically inactive landscapes, drainage divides migrate over prolonged periods of time (Beeson et al., 2017). Such behavior is consistent with the notion that small and local perturbations can trigger nonlocal responses with potentially large effects on drainage form and area (Fehr et al., 2011; O'Hara et al., 2019). At regional scales, mobile divides can lead to profound changes in drainage configurations and subsequent alterations of the base level and the dispersal of sediment to sedimentary basins. For example, Cenozoic building of the eastern Tibetan Plateau margin has been proposed to account for major reorganization of large East Asian river systems and associated changes in sediment delivery to marginal basins (Clark et al., 2004; Clift et al., 2006). Moreover, changes in drainage area that are associated with migrating divides affect river incision rates (Willett et al., 2014) and thus the topographic development of landscapes, which potentially confounds their interpretation in the context of climatic and tectonic changes (Yang et al., 2015).

Recent studies of the causes and effects of mobile drainage divides have focused on topographic differences across several specific, manually selected drainage divides (e.g., Willett et al., 2014; Goren et al., 2015; Whipple et al., 2017; Buscher et al., 2017; Beeson et al., 2017; Gallen, 2018; Guerit et al., 2018; Forte and Whipple, 2018). Even if appropriate in these studies, such a procedure introduces unwanted subjectivity, both in the selection of divides and how any across-divide comparison is done. This choice of procedure may be attributed to the fact that, so far, there has been no straightforward approach to reliably extract the drainage divide network from a digital elevation model (DEM). Functions that classify topographic ridges (the common shape of drainage divides) based on local surface characteristics and a threshold value (e.g., Little and Shi, 2001; Koka et al., 2011) are prone to misclassifications. The gray-weighted skeletonization method by Ranwez and Soille (2002) (homotopic thinning) requires the determination of topographic anchors (e.g., regional maxima), which makes it sensitive to DEM errors. The approach by Lindsay and Seibert (2013), who identified pixels belonging to drainage divides based on confluent flow paths from adjacent DEM pixels and a threshold value, is computationally expensive and sensitive to edge effects that depend on DEM size. Furthermore, drainage divides that coincide with pixel centers are inconsistent with the commonly used D8 flow-routing algorithm (O'Callaghan and Mark, 1984), in which each pixel belongs to a specific drainage basin. A probabilistic approach based on multiple flow directions exists (Schwanghart and Heckmann, 2012), but computation is expensive and thus restricted to a few drainage basin outlets. Finally, all of these approaches merely yield a classified grid but no information about the tree-like network structure of drainage divides, which requires the ordering of the divide pixels into a network (Fig. 1).

The Big Tujunga catchment, San Gabriel Mountains, United States, with the stream network (blue) and drainage divide network (red) draped over hillshade image. The drainage divide network is obtained with the approach developed in this study. The thickness of the stream and divide lines is related to upstream area and divide order, respectively. Divide orders are based on the Topo ordering scheme, which we describe in the main text. The map projection is UTM zone 11. North is up.

Although divide networks might be thought of as mirrors of stream networks, there are fundamental differences between the two. Starting at channel heads, i.e., the tips of stream networks, streams always flow downhill and the upstream area monotonically increases. Stream networks are therefore directed networks that have a tree-like structure and a natural order, which has been quantified in different ways (e.g., Horton, 1945; Strahler, 1954; Shreve, 1966). Divide networks, however, are neither directed nor rooted, and they may even contain cycles. They do not obey any monotonic trends in elevation or other topographic properties that could be easily measured. As a consequence, their ordering is less straightforward. Nevertheless, it is intuitive that some divides (e.g., a continental divide) should have a different order than others. In addition, the structure of divide networks could be important in their susceptibility to drainage captures. For example, higher-order divides may record perturbations longer, as they are farther away from the base level and thus cannot adjust as quickly as lower-order divides. Furthermore, where higher-order divides are close to higher-order streams, drainage-capture events would result in profound changes in drainage area and thus a greater impact on stream discharge and power (e.g., Willett et al., 2014).

In this study, we propose measuring and analyzing networks of drainage divides to address questions like the following. How is the geometry of a divide network related to that of a stream network? Do similar scaling relationships apply? And can the divide network be used to infer catchment–drainage dynamics? Empirically driven answers to these questions require tools to study drainage divides, most efficiently from DEMs. We present our study in two separate papers. In the following, part 1, we present a new approach that allows for the identification and ordering of drainage divides in a DEM. We investigate ways of ordering drainage divide networks and analyze basic statistical and topographic properties with a natural example from the Big Tujunga catchment in the San Gabriel Mountains in southern California. In part 2 of this study (Scherler and Schwanghart, 2020), we present the results from numerical experiments with a landscape evolution model that we conducted to examine the response of drainage divide networks to perturbations.

Drainage divides are the boundaries between adjacent drainage basins, and
thus their determination is based on the definition of drainage basins. In a
gridded DEM, drainage basins are generally defined through the use of flow
direction algorithms. The D8 flow direction algorithm (O'Callaghan and Mark,
1984) assigns flow from each pixel in a DEM to one of its eight neighbors
in the direction of the steepest descent. As a result, each pixel is
associated with a distinct upstream, or uphill, drainage basin. In contrast,
multiple flow direction algorithms, such as the D

For a given point in a channel network, its drainage basin is uniquely defined to be the upstream area of that point. The drainage divide of that basin, however, does not intersect the channel itself. We thus define drainage divides as lines (or graphs) that mark the margin of drainage basins and that do not cross rivers (Fig. 2). When derived from a DEM, these graphs consist of nodes and edges: nodes are located on pixel corners and edges follow pixel boundaries. A meaningful property of divide nodes and edges is that they should not coincide with nodes or edges of the drainage network. When applying the D8 flow-routing algorithm to a gridded DEM with square elevation cells, however, this requirement poses a problem due to the different pixel connectivity of divides and rivers. Whereas divide nodes can be connected to only four cardinal neighbors, river nodes can be connected to eight different neighbors. In consequence, divide nodes may exist that coincide with diagonal edges of drainage networks (Fig. 2). In a gridded DEM this issue could be resolved with a D4 flow direction algorithm; or, more generally, this issue could be avoided if flow is only allowed orthogonal to cell boundaries. In our approach, we nonetheless adopt the D8 flow direction algorithm and allow for spatial congruence of streams and divides. In practice, such issues mainly arise near confluences (Fig. 2).

Definition of drainage divides in a digital elevation model. Note the point where a drainage divide (red) coincides with a river channel (blue). See text for details.

Analogous to streams, drainage divides are typically organized into
tree-like networks (Fig. 1), although cycles that
correspond to internally drained basins may exist. Because of the directed
flow of water, stream networks can be regarded as directed graphs that start
at channel heads (the leaves of the tree) and end at an outlet or a river
mouth (the root of the tree). Flow directions in stream networks can be
easily derived from node elevations (e.g., O'Callaghan and Mark, 1984), and
the hierarchy of streams can be related to their upstream area, for example.
In contrast, drainage divides have no inherent direction, and there is
no terrain property, like elevation, that could be used to assign a
direction to them. A meaningful metric for ordering divides may be the
average

Ordering of divides based on the average distance to a common
stream location,

Instead, we suggest that directions can be derived from the tree-like
structure of drainage divide networks. Analogous to a parcel of water that
travels down a river from its source to its mouth, we propose starting at
the leaves of the tree, which we call the

Iterative sorting and ordering of the divide network. The divide network is assembled starting with divide segments that contain endpoints (green) and which are then removed from the collection of divide segments. Former junctions (red) that have only one segment remaining become endpoints, and the iteration continues until no more endpoints exist.

We implemented the above-described way of extracting and ordering drainage
divides from a DEM in the TopoToolbox v2 (Schwanghart and Scherler, 2014), a
MATLAB-based software for topographic analysis. Figure 5 shows the workflow of our approach, which
consists of the following steps.

For a given DEM, we first define a stream network based on the D8 flow direction algorithm and a threshold drainage area (Fig. 5a, b). The lower the threshold, the more detailed the stream and divide networks will be.

We extract drainage divides based on drainage basin boundaries that we obtained for drainage areas at tributary junctions and drainage outlets (Fig. 5c). Initially, each drainage basin boundary is composed of one divide segment that connects two endpoints, and junctions do not yet exist. These divide segments do not cross any rivers but their nodes may coincide with stream edges (Fig. 2). We remove redundant divide segments in the collection of divides, which arise from nested and adjoining drainage basins. As a result, we are left with a set of unique divide segments, which, however, may be continuous across junctions or terminate where they should be continuous (Fig. 6).

We next organize the collection of divide segments into a drainage divide network (Fig. 5d). This is the core of the algorithm, in which we identify endpoints and junctions, merge broken divide segments, and split divide segments at junctions (Fig. 6). Our algorithm distinguishes between junctions, endpoints, and broken divide segments by computing for each node of the divide network the number of edges linked to it, the number of segment termini linked to it, and the existence and direction of a diagonal flow direction. For example, most nodes with two edges and two segment termini correspond to a broken segment and need to be merged, unless they coincide with a stream and merging them would make the resulting divide cross that stream (Fig. 6). See the Appendix for more details.

Finally, we sort the drainage divide segments within the network (Fig. 5e). The algorithm iteratively identifies segments that are connected to endpoints and removes them from the list of unsorted divide segments until no divide segments are left (Fig. 4). This step assigns a direction to each divide segment and transforms the divide network into a directed acyclic graph. For the sorted divide network, we then compute the divide distance, i.e., the maximum distance from an endpoint along the sorted divide network (Fig. 5f).

Workflow of identifying and ordering drainage divides in digital
elevation models.

Transformation of a collection of drainage divide segments

After the sorting, we also assign orders to divide segments based on the
ordering of stream networks, first introduced by Horton (1945). We adopted
both the Strahler (1954) and Shreve (1966) rules of stream ordering and
added a third rule that we call Topo. All ordering schemes start with a value of
1 at endpoints and progressively update divide orders at junctions based
on the following rules:

As previously mentioned, our divide algorithm currently does not handle internally
drained basins. Whereas the divides of internally drained basins are easy to
identify, they are not easily sorted in a meaningful manner. In fact, the
distance to a common stream location (

We investigated basic characteristics of drainage divide networks using a
30 m resolution DEM from the 1 arcsec Shuttle Radar Topography Mission
data set (Farr et al., 2007). We focused on the catchment of the Big Tujunga
River in the San Gabriel Mountains, USA. The catchment is a good example of
a transient landscape with active drainage basin reorganization and
landscape rejuvenation as the river incises into a relict pre-uplift
landscape (DiBiase et al., 2015). We preprocessed the DEM by carving through
local sinks (Soille et al., 2003) to avoid artificial internally drained
basins, and we obtained a stream network based on a minimum upstream area of
0.1 km

We analyzed the divide network, its planform geometry, and its relation to
topography. Planform geometry is studied using statistical analysis of the
number and length of divide segments of different orders. Topographic
analyses are based on metrics that we determined for the entire DEM and that
we subsequently associated, or mapped, to divide edges and entire divide
segments. As topographic metrics, we focus on hillslope relief (HR) and
horizontal flow distance to the stream network (FD). HR was defined to be the
elevation difference between a point on the divide and the point on the
river that it flows to. To quantify the morphologic asymmetry of a divide,
we propose using the across-divide difference in hillslope relief (

We applied our divide algorithm to the Big Tujunga catchment, and the
resulting divide network for different ordering schemes is shown in
Fig. 7. Because the Shreve and Topo ordering schemes yield
larger ranges in divide orders, their visualization allows for greater
differentiation compared to the Strahler ordering scheme. Differences in the visual
appearance of the divide network due to the ordering scheme are also
apparent at the root node, i.e., the junction that is encountered last in
the ordering process (black arrows in Fig. 7). In
the Topo ordering scheme, divide orders increase by 1 during each sorting
cycle so that the last divide segments will have orders that are different
by not more than 1. In contrast, the ordering rules of the Strahler and Shreve schemes
(see Eqs. 1 and 2) may
yield unequal orders during the sorting so that the divide orders of the
last divide segments may be different by more than 1. In the Big Tujunga
catchment, the basin area, and thus the number of divide segments, is larger
north of the Big Tujunga River compared to south of it. As a consequence,
both the Strahler and Shreve divide orders increase more rapidly along the northern perimeter
compared to the southern, and the junction encountered last during the
sorting process (at the root of the tree) opposes divide segments with
orders of 7 and 6 for Strahler and 1463 and 772 for Shreve in the north and south,
respectively (Fig. 7). For the Strahler ordering scheme,
the frequency distribution of divide segments decreases exponentially with
divide order (

Divide network of the Big Tujunga catchment in the western San
Gabriel Mountains, California, USA. Panels

We computed divide-segment lengths for different drainage area thresholds
(

Divide-segment statistics.

We quantified the average branch length, i.e., the average distance one would have to
travel down on either side of a divide to reach a common stream location
(

Distance to common stream location by

We next studied the morphology of the drainage divide network from the Big
Tujunga catchment. Because the divide morphology consists of parts that lie
within the catchment and parts that lie outside it, we analyzed
the entire drainage divide network from the DEM as shown in
Fig. 5. Although the drainage divide network is
truncated along the DEM edges, the following analysis is insensitive to this
issue. Figure 10 shows the drainage divide
morphology of the Big Tujunga catchment based on a stream network that was
derived from a drainage area threshold of 1 km

Drainage divide morphology of the Big Tujunga catchment based on
a stream network that was derived from a drainage area threshold of 1 km

Based on the observation of characteristic values of

Minimum hillslope relief

Anomalous divides in the Big Tujunga catchment.

Our new approach allows for routinely extracting drainage divides from any DEM
without internally drained basins. We have shown that the maximum divide
distance

The proposed sorting procedure (Fig. 4) recovers the tree-like structure of the divide network and allows for the derivation of divide orders, analogous to the well-known stream orders. Because divide segments have similar mean lengths across all divide orders (Fig. 8), divide orders derived with the Topo ordering scheme can serve a similar purpose as divide distance. Shreve (1969) studied link lengths in stream networks and concluded that their distribution is better described with a gamma distribution compared to an exponential or lognormal distribution. Results from the Big Tujunga catchment support this conclusion with respect to divide-segment lengths, although systematic deviations can be observed (Fig. 8a). It needs to be tested with more observations whether these deviations are inherent to drainage divide networks in general and whether they could hold clues about the dynamic state of a landscape.

An advantage of characterizing the divide network by distance instead of orders is that the divide distance is invariant with respect to the chosen drainage area threshold, whereas divide orders are not because they depend on the total number of divide segments and junctions. Further differences are apparent at the root node, which may oppose divide segments with orders that differ by more than 1 (Fig. 7). In the case of the Big Tujunga catchment, Strahler orders are not that different across the root node, but in a different landscape that could well be the case. This issue is more prevalent in the case of Shreve ordering, but it is avoided with the Topo ordering scheme. Furthermore, the nonuniform distribution of divide-segment lengths (Fig. 8) influences how similar or dissimilar the divide distances of the meeting divide segments are at the root node. If the average divide-segment length of trees that meet at the root node are different, divide distances will make a jump, even if divide orders are similar. In the Big Tujunga catchment, the divide distance jump at the root node is 5400 m.

Divide orders derived with the Strahler ordering scheme can be used to investigate
how the divide network conforms to the Horton (1945) laws of network
composition. In the Big Tujunga catchment, for example, the bifurcation
ratio of the divide network (

Based on the observation of characteristic values of minimum hillslope
relief (300–500 m) and minimum flow distance (1000–1800 m), we identified
drainage divides in the Big Tujunga catchment that are anomalously low,
close to a stream, and asymmetric (Figs. 11,
12). These geometric properties suggest the
existence of wind gaps, hillslope undercutting by rivers, and spatial
anomalies in erosion rates, which are diagnostic for past or ongoing
mobility of drainage divides. Anomalous drainage divides are particularly
frequent along the eastern edge of the catchment, where an area of low
hillslope angles and local relief (Fig. 12), the
so-called Chilao Flats, is bordering a steep catchment to the south and east
of it. This high-elevation low-relief area is thought to represent a relict
peneplain surface that was uplifted during the growth of the San Gabriel
Mountains and is currently being destroyed by the headward incision of rivers
(Spotila et al., 2002; DiBiase et al., 2015). Cosmogenic

We identified another stretch of anomalous divides along the southern margin
of the Big Tujunga catchment (Fig. 12a, d), part
of which is coincident with the trace of the San Gabriel Fault, which
follows the orientation of the valley (Morton and Miller, 2006). Reduced
relief in a

In this study, we presented an approach to objectively extract and analyze
drainage divides from DEMs. We argued that divides can be ordered in a
meaningful way based on the average distance one would have to travel down
on either side of a divide to reach a common stream location, and we have
shown that this distance can be well approximated by the maximum
along-divide distance from endpoints of the divide network, which we termed
the divide distance. We have also shown that the tree-like structure of
divide networks lends itself to topological analysis similar to stream
networks, and we introduced an ordering scheme (Topo), in which divide orders
increase by 1 at divide junctions. Because divide segments tend to have
characteristic lengths, the Topo ordering scheme mimics the divide distance.
Topographic analysis of the drainage divide network of the Big Tujunga
catchment yielded characteristic values of flow distance and hillslope
relief that can be shown to depend on the drainage area threshold, with
which the stream network was derived. Based on these characteristic values
and a minimum divide distance of

Once the drainage divides are defined based on the outline of drainage
basins and redundant divide segments are removed, they compose a network

Divide node classification matrix.

The divide algorithm developed in this study has been implemented in the
TopoToolbox v2 (Schwanghart and Scherler, 2014). The codes will be made
available with the next TopoToolbox release and shall be accessible at

DS developed the algorithm and led the writing of the paper. Both authors contributed to discussions, editing, and revising the paper.

The authors declare that they have no conflict of interest.

We thank two anonymous reviewers for constructive comments that helped improve the paper.

This research has been supported by the Deutsche Forschungsgemeinschaft (DFG; grant no. SCHE 1676/4-1). The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Sebastien Castelltort and reviewed by two anonymous referees.