A method to generalize stream flowlines in small-scale maps by a variable flow-based pruning threshold.

Author:Tinker, Michael


Hydrography is a conspicuous and complex natural theme on topographic maps due to its large number of features (Savino et al. 2011). It must be drawn with strict tolerances in order to appear well integrated with the terrain and is very sensitive to changing scales (Buttenfield, Stanislawski, and Brewer 2011). A significant amount of previous scholarly research has explored systematic methods for generalizing hydrography suitable for automation.

Hydrographic generalization methods are often based on stream ordering schemes. Recent methods emphasize on generalizing by stream drainage area, several of which we briefly review in the following section. In this paper, we employ the National Hydrography Data Set (NHDPlus) to explore a novel generalization method that uses variable flow thresholds. NHDPlus is a publicly available, comprehensive vector spatial data set containing hydrography for the United States at 1:100,000 scale, provided by the US Environmental Protection Agency (EPA). Originally conceptualized as a tool for surface-water modeling, the data are used for a variety of applications, including cartography (EPA, USGS, and Horizon Systems Corporations 2010a, 2010b). NHDPlus data are ideal for exploring generalization methods because they include many additional attributes, such as Strahler stream order and mean annual estimates of natural flow and velocity for all streams.

The intent of this generalization method is to improve upon traditional hydrographic presentation to produce a river map that does not under-represent rivers and streams in high precipitation areas or over-represent them in arid areas. This method represents an advancement over previous work by using mean annual precipitation to derive the flow rate pruning thresholds, serving to emphasize on a more accurate rendering of precipitation and runoff regime. While we use NHDPlus to develop and discuss this work, these methods may be applied to other hydrography data sets, as long as they have been enriched with flow rates (see, for example, ESRI 2012 or Global Runoff Data Center 2013).

Related work

Any method of generalization must address two key issues: which features to retain (or delete) and how many features to retain (or delete), which can be represented by the selection and the target. The selection process can address either retention or deletion, but the target is generally a number of features to retain. Topfer's radical law (or principle of selection) has been adapted for use in hydrography. The original formula is: [n.sub.f] = [n.sub.a] x [square root of ([M.sub.a]/[M.sub.f])], where [n.sub.f] is the number of features shown on the derived map; [n.sub.a] is the number of features on the source map; [M.sub.a] is the scale denominator of the source map; and [M.sub.f] is the scale denominator of the derived map (Topfer 1966). Some researchers have used or modified the radical law to establish targets (e.g., Ai, Liu, and Chen 2006; Gardiner 1982; Zhang 2007; Stanislawski 2009; Wilmer and Brewer 2010; Mazur and Castner 1990). While this principle helps to answer the question of how many features to retain when generalizing to smaller scales, it provides no guidance about which features to retain.

Generalizing by stream order

Most hydrographic generalization methods are based on the stream ordering schemes of Horton (1945), Strahler (1957), or Shreve (1966). In Horton's scheme, a first-order stream is a headwater with no tributaries; a second-order stream is represented by the confluence of two first-order streams, and so on. At confluences, the shorter stream or the stream joining the parent stream at a greater angle is assigned a lower order. Stream order increments only when two streams of the same order flow together. Once all confluence-to-confluence segments of the stream network have been ordered, the main stem of each river, from headwater to outlet, is assigned the same order as its outlet. The cartographer must decide which segments comprise the main stem of the river (Mazur and Castner 1990), though Horton's original criteria were to select the "longest and straightest path" (which implies curvilinear continuity rather than the actual straightness of line). Strahler modified Horton's method by abandoning this subjective retracing stage. Shreve then further modified Horton's scheme to account for the number of upstream tributaries. In Shreve's scheme, stream order increments whenever two streams join, regardless of the tributary length or angle of confluence.

Horton's method may be considered a hierarchy of rivers, while Strahler and Shreve's methods are hierarchies of segments. Generalization based upon the ordering scheme is well represented in recent literature (Catlow and Du 1984; Mazur and Castner 1990; Richardson 1994; Touya 2007; Zhang 2007; Thomson and Brooks 2000; Savino et al. 2011) and other scholars have since refined or expanded upon these methods.

Mazur and Castner (1990) proposed a method to arbitrarily select the Horton orders and compare the results to other published small-scale maps. Richardson (1994) calculated the number of streams retained after generalizing by both ordering schemes, setting the target number of streams by selecting a percentage of the number of original streams. Thomson and Brooks (2000) described a method of selecting by stroke "a curvilinear segment that can be drawn in one smooth movement and without a dramatic change in style." In their method, once strokes are identified, attributes such as their length or representative class can be derived and sorted. Selection then proceeds by arbitrarily deleting strokes on the basis of their attributes. The authors found that selection by strokes generally produces the "longest and straightest path", which is essentially Horton's original criterion for defining the main fiver stem. Touya (2007), following Thomson and Brooks (2000), used a different set of criteria to determine the stroke, including river name, priority of permanent (perennial) over intermittent streams, priority of artificial path through irrigation zones, fiver length, and angle of confluence. Once strokes are determined, the Horton orders are assigned. Selection retains higher order strokes and lower order strokes are retained only if they meet a minimum length threshold.

Additional generalization methods have been devised by drawing directly from Strahler's work. Catlow and Du (1984) argued that simple deletion of all low-order streams does not result in acceptable cartography because all headwaters are indiscriminately culled and single-river systems are eliminated. They used length, therefore, as a secondary criterion. First-order single-river systems are retained if they are longer than an arbitrary length threshold. Savino et al. (2011) determined the main course of the fiver by starting from streams with the largest Strahler order and then moving upstream. They identified the main stem by evaluating attributes such as total distance to the furthest upstream source, total number of branches uphill, and width. River courses that are shorter than an arbitrary length threshold are pruned, and density of the remaining rivers is also considered. Arbitrary buffers are built around each river course, and selection proceeds recursively by pruning fiver courses below a threshold percentage of overlap.

Zhang (2007) described a method in which stretches of river are numbered, as in the Shreve scheme, but then the main stem of each river is assigned the same order as its outlet, as in the Horton scheme. While both Zhang's and Horton's methods depend on an arbitrary identification of the main stem, Zhang's method also applies arbitrary length thresholds to each river group. In Zhang's method, the target is set with a modified application of the radical law, based on the number of tributaries.

Limitation of Strahler order method to generalize the NHDPlus

The NHDPlus can be generalized by selection of Strahler stream order. The greatest stream order value in the conterminous US is 10, found on the main stern of the lower Mississippi River. In the Pacific Northwest, the greatest value is nine, found on the lower Columbia River, below its confluence with the Snake River (Pierson et al. 2008).


In Figure 1, the US Pacific Northwest (NHDPlus region 17) is shown to be successively generalized by Strahler orders 4, 5, and 6. Figure 1a shows only streams of order 4 or greater, which could work for display at 1:500,000. Figure lb shows only streams of order 5 or greater; while this could work for display at 1:1,000,000, it appears to be missing important coastal streams. Figure 1c, which shows streams of order 6 or greater, has lost so many rivers that it would not be useful for a small-scale map. This serves to demonstrate that the NHDPlus can be generalized by Strahler order, but the resulting maps show a very wide range in scale. Strahler order is not fine enough to generalize the 1:100,000 NHDPlus to more than one or two output scales.

Figure 1b and c also show how generalizing by order may indiscriminately cull important rivers. The coastal region of the US Pacific Northwest is one of the wettest areas in the United States, yet few coastal rivers appear in the generalized maps because they were of lower order. In reality, they are likely to be important rivers with high flow and should remain on the map.

Generalizing by drainage area

Other researchers have devised generalization methods based on drainage area, rather than stream order...

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