3.1.1 Exact DPAs (no uncertainty)

• IO: inlets and outlets. Links attached to inlets and outlets are predicted such that flow travels away from inlet nodes and towards outlet nodes.

• PAR: parallel links, Fig. 2a. Parallel links occur when two links begin and end at the same node. To avoid creating a cycle within the graph, all parallel links must flow in the same direction. As a consequence, if the direction of one of a group of parallel links is known, the others are predicted in the same direction.

• CON: continuity, Fig. 2b. Enforcing continuity ensures that no sources or sinks appear within the network other than the inlet and outlet nodes. Continuity is enforced at each node by first identifying nodes for which only one connected link direction is unknown. If the remaining group of known links are all either entering or departing the node, the unknown link is predicted to an orientation opposite the group.

• BDG: bridge links, Fig. 2c. Bridge links are those through which all flow must travel to reach an outlet. Removal of a bridge link from a CN breaks the connectivity of the CN, forming two disconnected CNs. Bridge links are identified in a CN graph via the NetworkX (Hagberg et al., 2008) bridges function and are temporarily removed, creating two subnetworks. Each subnetwork is searched for the presence of inlet and outlet nodes. If either of the subnetworks has either only inlets or only outlets, the flow direction for the bridge link can be predicted as either away from the subnetwork containing the inlets or toward the subnetwork containing the outlets. In some cases, both subnetworks may contain both inlets and outlets; the bridge link direction is thus not predictable.

Figure 2Diagrams showing the direction-predicting algorithms (DPAs). Symbology is further explained in Sect. 3.1. (a) Predicting an unknown parallel link. (b) An example of applying CON to determine the unknown link. (c) Predicting a bridge link with BDG. (d) Using the minimum direction change MDC for predicting the unknown link. (e) A synthetic DEM (SDEM) for the Mackenzie Delta is shown with blue outlets and red inlets. (f) A centerline mesh is shown for the Brahmaputra River with a yellow centerline to demonstrate VD and VA. The box denotes the bounds of the zoom view. The unknown link's endpoints are marked by red points. Dashed lines follow the mesh perpendiculars. (g) Main channels found by MC for the Colville Delta are denoted by blue lines. (h) An example of main channel parallels (PMC). Distances d₁ and d₂ are defined in Sect. 3.1; MC_n refers to the nth main channel node, ordered from upstream to downstream. The dashed link's flow direction is initially unknown.

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3.1.2 Exploitative DPAs

• MDC: minimize direction change, Fig. 2d. MDC is based on the principle that branching angles are more likely to be acute, as observed in both inland (Devauchelle et al., 2012; de Serres and DeRoy, 1990) and deltaic (Coffey and Shaw, 2017) CNs. We extend these observations by hypothesizing that the change in flow directions should be minimized at each node. Candidate links for MDC are identified as unknown links connected to at least one known link. At each end node of a candidate link there may be one or more links flowing into or out of the node. Each of these links, along with the candidate link, is represented by a unit vector whose direction is defined by its endpoint locations (l_u for the unknown candidate link). If multiple links flow into (or out of) the node, their unit vectors are averaged to provide a single direction vector (l_i and l_o for into node and out of node, respectively). The goal is to determine which of l_i or l_o is most parallel to l_u; thus, angles are computed between l_i, l_o and $l_{{\mathrm{u}}_{\mathrm{0}}}$, $l_{{\mathrm{u}}_{\mathrm{1}}}$, where $l_{{\mathrm{u}}_{\mathrm{0}}}$ represents the original position of the unknown link, and $l_{{\mathrm{u}}_{\mathrm{1}}}$ represents its 180^∘ rotation about the node. The minimum of all angles is computed via Eq. (1):

  $$\begin{array}{}\text{(1)}& {\mathit{\alpha }}_{\min }=\left({\mathit{\alpha }}_{u_{\mathrm{o}},l_{\mathrm{i}}},{\mathit{\alpha }}_{u_{\mathrm{o}},l_{\mathrm{o}}},{\mathit{\alpha }}_{u_{\mathrm{i}},l_{\mathrm{i}}},{\mathit{\alpha }}_{u_{\mathrm{i}},l_{\mathrm{o}}}\right),\end{array}$$

  where the subscripts denote the vectors defining the angle. If ${\mathit{\alpha }}_{\min }={\mathit{\alpha }}_{u_{\mathrm{o}},l_{\mathrm{o}}}$ or ${\mathit{\alpha }}_{u_{\mathrm{i}},l_{\mathrm{i}}}$, the unknown link is set to flow out of the node; otherwise, it is set to flow into it. Where possible, this procedure is repeated for both end nodes of l_u, and α_min becomes the minimum of both node minima. The magnitude of α_min provides a measure of certainty of the prediction; α_min closest to 0 represents links whose flow directions are more aligned with at least one of the known connected links. A threshold (ω_ang) may thus be set on α_min to specify the maximum level of direction change allowed before setting the unknown link's direction.

• SDEM: synthetic DEM (deltas only), Fig. 2e. As discussed in Sect. 1, DEMs may provide valuable information for discerning flow directions but are often too coarse for use with low-sloped delta CNs. SDEM invokes our conceptualization of long-time, steady-state flow that moves from the apex of a delta to its outlets to construct a synthetic DEM. This procedure creates inlet and outlet DEMs separately, designed such that elevations are higher near inlets and lower near outlets. The final synthetic DEM is simply the sum of the inlet and outlet DEMS.

  For the outlet DEM, an image of the same size and resolution as the input mask is created and filled with values of 1. To estimate the delta's shoreline, the convex hull of the outlet nodes is computed, and the edge of the convex hull connecting the two most distant outlet nodes is removed to provide an ordered set of input nodes. Line segments between each input node are linearly interpolated at 0.1 pixel intervals, and this interpolated shoreline is “burned” into the image of 1 values by lowering their elevations to 0. A distance transform (Jones et al., 2001) of the image returns an image in which each pixel's value represents its distance to the nearest shoreline. This image (I_DEM,o) is normalized on the interval [0, 1] according to

  $$\begin{array}{}\text{(2)}& I_{\mathrm{DEM},\mathrm{o}}={\displaystyle \frac{I_{\mathrm{DEM},\mathrm{o}}-\left(I_{\mathrm{DEM},\mathrm{o}}\right)}{\left(I_{\mathrm{DEM},\mathrm{o}}\right)-\left(I_{\mathrm{DEM},\mathrm{o}}\right)}}.\end{array}$$

  The inlet DEM (I_DEM,i) is constructed similarly but with some exceptions. Only inlet nodes whose associated channel widths are at least 75 % of the widest inlet channel are considered. Before normalization (Eq. 2), I_DEM,i is inverted via

  $$\begin{array}{}\text{(3)}& I_{\mathrm{DEM},\mathrm{i}}=\left(\max \right)\left(I_{\mathrm{DEM},\mathrm{i}}\right)-I_{\mathrm{DEM},\mathrm{i}}\end{array}$$

  to ensure that elevations near the inlets are raised rather than lowered. The final synthetic DEM is simply the sum of I_DEM,o and I_DEM,i. The synthetic DEM for the Mackenzie Delta is shown in Fig. 2e; only one of the inlet nodes contributed to its I_DEM,i.

  The slope of each link may be computed by drawing elevation values from I_DEM, and a prediction of a link's flow direction can be made. Channels often flow perpendicular to the general flow direction dictated by I_DEM, so predictions made by SDEM may be poor. However, the magnitudes of a link's slope and its length serve as measures of certainty; links may be thresholded by length (ω_len), slope (ω_slope), or both to ensure that SDEM only sets the longest, steepest links.

• MC: main channels, Fig. 2f. Typically, but with exceptions, the main channels (i.e., those that transport the largest discharge) of a braided river or delta CN are the widest of the CN. This concept originates in well-studied, quasi-universal hydraulic geometry relationships of the form W=aQ^b for width (W), discharge (Q), and fitted parameters (a, b) (Leopold and Maddock, 1953; Parker et al., 2007). We impose two additional constraints to this relationship to define main channels: they must begin at inlets and end at outlets, and they tend to follow the most direct path.

  Under these conditions, each outlet has a corresponding “shortest and widest” path from each inlet. This path is found by creating a weighted graph of the CN, whereby weights are defined according to

  $$\begin{array}{}\text{(4)}& w{t}_i=l_i\cdot \left(\left(w\right)-w_i\right)\end{array}$$

  for the ith link with length l_i and width w_i. This weighting scheme results in larger weights for longer and narrower channels. The shortest paths of the weighted graph are computed from each inlet to each outlet using Dijkstra's method implemented in NetworkX. The direction of each link along each path may then be predicted according to the ordered list of nodes returned. A CN may contain multiple main channels; if a link's direction has already been predicted by a main channel, it is not re-predicted by other main channels that share it. Therefore, in rare cases in which two main channels might predict opposite flow directions for a link, the link is predicted by only the flow direction of the first. No uncertainty measurements are made for MC.

• VD: valley line distance (braided rivers only), Fig. 2g. Rivers typically flow through corridors of some sort, referred to here as valleys. Valleys feature the lowest elevations in a landscape and contain river floodplains. Multi-scale analyses of river valleys indicate that significant information about the local (link) scale is shared with the valley scale (Gutierrez and Abad, 2014; Vermeulen et al., 2016). VD attempts to impart flow direction information from the river valley scale to the link scale.

  A river corridor centerline is created by filling the holes in the CN mask, skeletonizing it, and smoothing. A mesh is generated over the CN by drawing perpendicular line segments along the centerline. This mesh-generation technique was introduced by Schwenk et al. (2017) and adapted to a Python implementation here. Knowledge of the inlet and outlet node locations allows for an ordering of the polygons and perpendiculars comprising the mesh.

  A prediction for each link is made by finding the two perpendiculars that encompass the link's endpoints (dotted white lines, Fig. 2g). The link's upstream node is predicted as the one closer to the upstream perpendicular. Similarly to deltas, channels of a braided river may flow approximately perpendicularly to the centerline, resulting in an uncertain prediction. To account for the certainty of VD, the number of perpendiculars required to encompass a link (N_perps) is also computed. Links passing through more perpendiculars carry a greater prediction certainty, and a threshold (ω_perps) may be applied to predict only the most certain links.

• VA: centerline angle (braided rivers only), Fig. 2g. The logic behind VA exactly follows that of VD, but instead of considering down-valley distance, we consider the local angle of the valley centerline. Flow direction can be predicted by comparing a link's angles with the nearby centerline angle. The endpoints of the link are mapped to the nearest perpendicular, and the centerline angle between these two perpendiculars (α_cl) is computed (Fig. 2g). The angles of the link computed from the vector defined by its endpoints (α₀) and its 180^∘ rotated version (α₁) are also computed. The link's direction is predicted as the orientation whose angle is closest to α_cl. The difference between α_cl and the closer of α₀ and α₁ provides a measure of certainty of VA, with smaller differences corresponding to higher certainties. This difference may be thresholded (ω_cla) to specify the level of parallelism between the link and the valley centerline required to make a prediction.

3.1.3 Heuristic DPAs

• SP: shortest path. SP stemmed from the observation that in most cases in which flow direction is unknown, the true flow path corresponded to the shortest distance to its outlet. The SP implementation is identical to MC except the links are unweighted. In cases in which the shortest path between inlets and outlets results in opposite predictions of flow direction for a link, the mode is selected as the prediction. SP may fail when the macro-morphology of the CN, e.g., the change in the Brahmaputra valley direction from west to south (Fig. 1), imposes a low-frequency direction change.

• PMC: parallel to main channel, Fig. 2h. Similarly to how VA and VD transfer information from the valley line to predict individual links, the links of main channels contain information on local flow directions that may be exploited to predict nearby, approximately parallel links whose flow directions are unknown. For each link that is not part of a main channel, the nearest (Euclidean distance) main channel node is found. Each of the endpoint nodes of the unknown link is mapped to its nearest main channel nodes (for example, d₁ and d₂ in Fig. 2h). If the endpoint nodes map to the same main channel node, no prediction can be made for the link. In all other cases, a prediction can be made that aligns the flow direction of the unknown link with the main channel nodes to which its end nodes were mapped. The strength of this prediction (ω_nodes) is captured by the difference of mapped-to-node positions along the main channel. In Fig. 2h, for example, this number is 1 (ω_nodes = MC₂ − MC₁).

• MMA: multiple methods agree. If DPAs disagree about the flow direction of a link, MMA simply chooses the most common prediction. A minimum number of agreeing DPAs may be enforced (ω_agree) to ensure greater certainty of the predictions made by MMA.

4.2.1 Ambiguous links

A total of 16 of the 42 link direction errors were attributable to ambiguous links for which morphology alone cannot provide certainty of flow direction. Generally, ambiguous links flow perpendicularly to the local (or overall) flow direction (Fig. 5a–d). Flow directions through ambiguous links can reasonably be argued to go both directions, and in many cases bidirectional flow may be reality (e.g., Fig. 5a). In these cases, MDC cannot be applied with certainty due to the high junction angles, nor can SDEM be applied with certainty because ambiguous links are typically short and not parallel to the main flow direction. MAA, which employs shortest-path methods, sets many of these links, but we found the shortest path to be unreliable for CNs with large-scale morphologic variability, e.g., the ∼90^∘ bend in the Brahmaputra CN. In the case of Fig. 5c, neither VA nor VD could set the erroneous links because of their perpendicular orientation with respect to the centerline. Figure 5d shows an unusual ambiguous link created by the formation of an oxbow lake; the expert judgement was based on the flow direction before the oxbow lake was cut off from the main channel, but the modern topology suggests that flow could travel in the opposite direction. We were unable to design DPAs that set ambiguous links with certainty; however, ambiguous links were the last ones (i.e., least certain) to be set by our recipes, which limited the influence that their potentially erroneous flow directions propagated to other links in the network. Although not strictly true, ambiguous links typically play unimportant roles in overall CN routing.

Figure 5Errors of the recipes. (a–d) Ambiguous links that were erroneously set for the Mackenzie (a, d), Indus (b), and Brahmaputra (c) CNs. (e) The Niger CN features tidal channels whose inlets were not considered. A main channel is shown to an outlet node that should be fed by the missing inlets. (f) A problematic main channel (white) is shown for the Lena CN. A zoom view of the shaded area is shown in (g). (h) Synthetic DEM for the Lena CN with a ridge of the synthetic DEM marked.

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4.2.2 The Niger CN

At 9.5 %, the Niger CN contained the highest fraction of erroneous links (Table 1). However, we found that all four erroneous links shared the same source of error. The Niger Delta features a number of tidal channels that are typically wider at their outlets and eventually fade away toward the delta's apex. Some of these tidal channels are connected to the CN, while others terminate on the delta plain without a surface connection (Fig. 5e). The erroneous links of the Niger were feeder links from the main CN to a tidal channel that, while connected, likely receives very little flow from the main subnetwork. In other words, fluxes to the outlet of this tidal channel should originate at the tidal channel inlets, but these inlets were not considered to be inlets of the CN. Their absence resulted necessarily in a main channel from the CN inlet to the tidal channel outlet, which in turn forced flows rightward towards the tidal channel and resulted in erroneous links. We verified that placing a single inlet at the source of the tidal channel resolved these erroneous links, resulting in a 0 % error for the tested links of the Niger CN.

Figure 6DPA effectiveness. (a) Fraction of links set by each DPA for each study CN. DPAs are defined as follows. VAD – valley angle and distance, VA – valley angle, VD – valley distance, MAA – multiple DPAs agree, SDEM – synthetic DEM, MDC – minimize direction change, BDG – bridge links, MC – main channels, PMC – parallels to main channels, CON – continuity, and IO – inlet/outlet links. (b) Fraction of each DPA for all CNs; red (and blue) bars sum to 1.

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4.2.3 The Lena CN

The Lena CN had a total of 15 identified erroneous links and an unresolved cycle. Nine of these links and the cycle are attributed to the Lena CN's unusual structure. The Lena CN features two clusters of outlets; a long, continuous shoreline on its upper right side contains the majority of the outlets, but a separate subnetwork delivers fluxes to the left side (directions with respect to the orientation in Fig. 5f–h). Fluxes entering its inlet node are either immediately routed to the left subnetwork or flow upwards to a pseudo-apex (Fig. 5f). While the majority of flow through the pseudo-apex heads toward the right shoreline, some is routed through smaller channels to the left shoreline. Recall that MC finds the shortest, widest path from inlets to outlets as a main channel. The main channel from the inlet to the outlet denoted in Fig. 5f is incorrect, as flow to that outlet node should travel through the pseudo-apex. However, because of the narrowness of the channels connecting the pseudo-apex to the outlet, the shortest, widest path bypasses the pseudo-apex and flow to the outlet approaches from the wrong side. MC thus erroneously sets a number of links, including one particularly critical link (Fig. 5g). This link is critical because it bridges two subnetworks; incorrectly setting its flow direction prevents flow from the upper subnetwork from reaching the leftmost outlets. As MC is applied early in the DR, its incorrect direction more readily infects nearby links as evidenced by the numerous erroneous links surrounding it. Its incorrect direction also created the unresolvable cycle in the Lena; this cycle was not present when we preset the critical link to the correct flow direction and then applied the DR.

A total of 4 of the Lena CN's 15 identified erroneous links were attributable to the difficulty of creating a representative synthetic DEM. Elevations in the synthetic DEM are proportional to their distance from the outlet nodes and inversely proportional to the distance from the inlet nodes; this scheme created a ridge in the synthetic DEM (Fig. 5h) that divided the inlet subnetwork from the rest of the delta, effectively forcing the links of the inlet subnetwork to flow uphill and resulting in the four erroneous links. Because elevations near the inlet are raised, the slopes of the inlet subnetwork links were relatively smaller, allowing the DR to pass over them (i.e., not set their directions) in the early stages of the recipe. Other DPAs were thus employed to set the vast majority of the inlet subnetwork links, preventing the ridge from adversely affecting their directions. Interestingly, the end of the ridge coincides with the location of the pseudo-apex due to the radial layout of the Lena's outlets.