Computers & Geosciences 28 (2002) 1051–1060

Algorithms for using a DEM for mapping catchment areasof stream sediment samples$

Richard Jones*

1289 Tate Drive, Reno, NV 89523, USA

Received 29 May 2001; received in revised form 5 April 2002; accepted 10 April 2002

Abstract

Algorithms are presented which use a digital elevation model (DEM) to create maps of catchment areas of stream

sediment sample points, and to locate optimal sample point locations. Drainage flow directions are determined by the

popular ‘‘D8’’ algorithm, which assigns the drainage from one point on the DEM grid to one of its eight adjacent

neighbors. A new ‘‘drainage enforcement’’ algorithm is presented that insures drainage continuity through flat areas

and out of depressions. This algorithm is based on the priority-first-search weighted-graph algorithm. Relatively simple

methods for defining internal basins and incorporating digitized stream data are also discussed. All of the selected

methods are robust, computationally efficient, and avoid many of the problems associated with other methods. In order

to support this claim, the literature is reviewed and comparisons are made between the chosen methods and other

methods. A full implementation has been created, and the Pascal source-code and Win95 executable program may be

downloaded from the IAMG web site. r 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Algorithms; Digital elevation model; Drainage network; Catchment area; Geochemistry

1. Introduction

Digital elevation models (DEMs) are readily available

data sets that represent the elevation of the earth’s

surface at discrete points in a regular, rectangular grid.

This paper presents computer algorithms that use as

data input a DEM, and output maps showing drainage

patterns and the catchment areas of stream sediment

sample sites.

The examples and illustrations that are presented in

this article are based on a ‘‘1-degree’’ DEM that was

provided by the US Geological Survey (USGS).1 A

1-degree DEM covers an area of 11 in longitude by 11 in

latitude. The DEM contains 1201 rows by 1201 columns

of elevation data, which corresponds to a grid spacing of

3 s of arc. Other DEM data sets are available for the US

and for other countries.

2. The basic algorithm for defining a drainage network

The algorithm that is used here for modeling a

drainage network was first presented by O’Callaghan

and Mark (1984) and Mark (1984).

The basic concept of the algorithm is to model the

drainage network in terms of catchment area. A

catchment area is that area of a watershed that lies

upstream from a point. As the flow of water is traced

downhill from a point, a counter is incremented for all

the downstream points through which the water flows.

The drainage network is defined by the relative counts.

A threshold is chosen which determines when a node

becomes a point on a ‘‘drainage’’. Since larger drainages

have bigger catchment areas than smaller drainages,

they can be ranked by size.

$Code available form server at http://www.iamg.org/CGE-

ditor/index.htm.

*Tel.:+1-775-746-9765.

E-mail address: rjones@pyramid.net (R. Jones).1http://edcftp.cr.usgs.gov/pub/data/DEM/250/

0098-3004/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 9 8 - 3 0 0 4 ( 0 2 ) 0 0 0 2 2 - 5

A separate grid-sized matrix (ca) is initialized to count

the units of catchment area that lie upstream from each

grid node. There are two ways to count ca. The first is to

follow the flow path from each node downhill until it

leaves the grid or ends in a ‘‘pit’’. (A pit is a point that

terminates a flow path in the downhill direction. Water

cannot flow past a pit because none of the adjacent

points around it have a lower elevation.) If there are n

nodes (n ¼ nx � ny), then the calculation requires about

n3=2 to n2 operations because most paths visit many

nodes. The second technique reduces the running time to

O(n log n) by presorting the nodes by elevation. In this

case, each node is visited only once in order of

decreasing elevation, and the downstream counter is

incremented by the total count of the upstream counter.

The downstream direction from a node is determined

by the direction of steepest decent (most negative

gradient). Using only the adjacent, orthogonal nodes

to determine the downstream direction results in a

streaky drainage pattern, but using the eight adjacent

nodes works well. When two or more nodes tie for being

the lowest, one is arbitrarily chosen. The algorithm is

often called the ‘‘D8’’ algorithm (Fairfield and Leymar-

ie, 1991).

2.1. Other methods

The basic D8 algorithm is probably the most popular

method for automated drainage recognition and catch-

ment area determinations (Martz and Garbrecht, 1998).

However, it has a number of deficiencies. The continuity

of drainages does not extend across flat areas or

anywhere that there is a ‘‘pit’’ in the DEM. This subject

is explored in detail in the next section of this paper. The

D8 technique of using only one of eight possible outflow

directions for a node is another problem that has been

addressed in several papers.

In modeling hydrologic flow using a DEM, Quinn

et al. (1991) do not assume that all of the water flowing

out of one unit area goes entirely into one neighboring

cell. In particular, if groundwater also interacts with

surface runoff, then the flow paths may be quite broad

and diffuse. Their method is to partition flow to all

downslope, neighboring cells. However, the distinction

between hydrologic flow units (e.g. water volume, mass

of dissolved contaminant) and catchment area is not

made clear in the article. Whereas certain flow units may

be dispersed, the catchment area from where the flow

originates probably should not be partitioned in the

same way (Tarboton, 1997). In particular, multiple

outflow paths create diffuse and overlapping catch-

ment area boundaries, which conflicts with the basic

definition of catchment area. Also, for mapping catch-

ment areas for stream sediment sampling, sharply

defined catchment area boundaries simplify the final

map product.

The D8 method limits the direction of flow between

two adjacent nodes to only eight possibilities. For most

drainage lines, the individual segments are linked in a

zigzag pattern, which over the length of a particular

drainage approximates the true stream orientation.

However, there will be a discrepancy between the

lengths of the drainages as calculated by summing the

individual segment lengths as compared to some other

method which manages to somehow produce smooth,

curvilinear drainage lines.

The D8 method fails to model the average flow

direction for laterally extensive, planar slopes. Flow

lines on these slopes are parallel, and should be oriented

down-dip. However, if the slope is featureless, the

absence of ‘‘banks’’ means that the DEM will not

provide the necessary information to jog the drainages

into a zigzag pattern. Hence, the D8 algorithm produces

drainages that are not, in general, oriented down-dip,

but are instead oriented in one of the eight principal

directions. Fairfield and Leymarie (1991) propose a

stochastic function that is applied to the gradient

determinations. The random component in this function

provides the necessary ‘‘jog’’ to cause the flow lines to

zigzag even on featureless slopes, thus producing

drainage lines that are more-or-less parallel to the

slope’s dip direction. Tarboton (1997) proposes using

two outflow paths to overcome the problem. By a rather

complicated formula, the flow direction from a node is

assigned a direction over the range of 0–3601. However,

the discrete grid spacing precludes using this directional

angle in a direct manner. Therefore, the catchment area

is partitioned between the two nodes that lie on either

side of the ideal flow direction.

Both the methods of Fairfield and Leymarie (1991)

and Tarboton (1997) solve the directional problem of

the D8 algorithm for a planar, long slope. However, the

solution may not be worth the effort. First, the slope

must be featureless, otherwise the D8 method would

work. By any method the individual ‘‘drainages’’ are

arbitrarily located on a featureless slope. It may not be

important to have properly oriented drainage lines when

the drainages do not even exist in the real world.

Secondly, highly planar slopes are unusual in most

terrain. Hence, the special methods that are proposed

for these areas make the overall calculation less efficient

and can introduce undesirable features in other areas

where the DEM is well conditioned for the basic D8

algorithm to work.

O’Callaghan and Mark (1984) specify that some

threshold is used to identify when a pixel is considered

to be on a drainage line. Regardless of how the threshold

is chosen, the drainage lines provide important visual

information about flow directions. The method recom-

mended here is to determine the threshold visually by

trial and error. However, some research has been

applied towards more definitive criteria. Tarboton et al.

R. Jones / Computers & Geosciences 28 (2002) 1051–10601052

(1991) provides statistical methods for determining the

threshold. Tribe (1992) suggests that the criteria for

the drainage network should be a close match between

the DEM-derived network and those ‘‘blue lines’’ that

are shown on published maps to represent rivers and

creeks. Tribe found that a threshold that works well for

drainages in upland areas is generally too small for low-

relief areas. She was able to get a better match with the

blue lines when the DEM-based drainages ended at

‘‘valley heads’’, which are determined by the elevation

gradients in the DEM.

3. Problems with DEM data

The basic D8 algorithm generally works well for small

drainages in areas of high relief. However, the major

drainages are not delineated (Fig. 1). There are some

inherent problems with DEM data with respect to

drainage analysis.

Major streams occur in alluvial valleys with low

stream gradients. If the topographic gradient is less than

the precision of the DEM elevation data, then flats will

invariably occur such that there is no immediately

adjacent downhill direction from certain nodes. For

example, the USGS 1-degree DEM data has elevations

rounded to the nearest meter, and the x (longitude) and

y (latitude) spacing translates to about 70 and 90m,

respectively, in the horizontal plane. Thus, if a stream

has a gradient of 1% or less, then the DEM will show

adjacent points along the course of the stream with no

gradient at all. In the D8 algorithm, a drainage

terminates at a point where none of the adjacent points

are downhill.

Precision is not the only problem with the raw DEM

data, a point apparently missed by most writers on the

subject (e.g., 2 Martz and Garbrecht, 1998; Mark, 1984).

Lakes are truly flat, and unless they are in closed basins,

they are part of drainage networks (Turcotte et al.,

2001). Even in areas of relief, spurious pits may occur.

Consider, for example, an area where the rivers occur in

incised channels. Suppose a stream flows NNE from a

node located at (x; y) and passes between the nodes at

(x; yþ 1) and (xþ 1; yþ 1). If these latter points occur

on the banks of the stream they probably will have

elevations greater than the elevation at (x; y). Further-more, the fact that the stream actually flows NNE from

the point (x; y) is not recorded in the DEM data. Hence,

Fig. 1. Basic algorithm for calculating catchment area grid (ca) generally fails to produce desired result on raw DEM data. Here,

drainage pattern is shown using threshold of 1/8 sq. miles. Larger rivers are not shown, and even smaller streams located in upland

areas of higher topographic relief are disrupted by ‘‘pits’’. Map is based on northern half of Wells-west 1-degree DEM, which contains

721,801 points. Of these, 101,042 points are pits without drainage to any one of eight adjacent points. Map contains 2994 drainages,

largest of which has a catchment area of only 2.81 sq. miles.

2Saunders, W., 1999. Preparation of DEMs for use in

environmental modeling analysis. http://www.esri.com/library/

userconf/proc99/proceed/papers/pap802/p802.htm.

R. Jones / Computers & Geosciences 28 (2002) 1051–1060 1053

the drainage calculation needs to predict the true

direction of stream flow at points where the DEM has

spurious pits.

The problems with flats and spurious pits are

compounded by the potential to have real basins located

within the boundaries of the grid. Basins may include

the margin of an ocean, a closed lake basin, or a playa.

4. Eliminating spurious pits and flats

Flat areas and spurious pits are eliminated from the

elevation grid by adjusting the grid values so as to insure

that there is at least one downslope direction for each

node in the grid. The approach presented in this paper

uses a priority-first-search algorithm (PFS) to find an

‘‘outlet’’ for a pit. Then along the PFS path from the

outlet back to the initial starting point the elevations are

adjusted, which in effect creates a trench with a

continuous, finite gradient.

PFS is a well-known, weighted-graph algorithm with

an established algorithmic structure and well-known

performance characteristics (Sedgewick, 1988). The

algorithm is robust and will solve a variety of problems

just by changing a ‘‘priority function’’ and/or by

changing the terminating conditions. Drainage enforce-

ment is modeled as a classical ‘‘shortest-path’’ problem.

In graph terminology, points, or nodes, are called

vertices. Any given vertex may be connected to one or

several other vertices. The connections between each

vertex are usually given in a data structure, referred to as

an ‘‘adjacency list’’. However, in this application, the

adjacency list is implicit in the x; y coordinates of the

vertex/node, since the DEM grid structure automatically

defines the eight nearest nodes as being ‘‘adjacent’’. In

PFS, every vertex is in one of three states: it is ‘‘on the

tree’’, it is ‘‘in the queue’’, or it has not yet been

examined. The algorithm is initialized by putting the pit

node on the tree, which thereby becomes the root node

for the tree. The iterative part of the algorithm then

commences and builds the rest of the tree. The iteration

terminates when the last vertex added to the tree is an

‘‘outlet’’ point for the drainage. An outlet may either be

a point with a lower elevation than the point at the root

node, the edge of the grid, or a drainage-terminating

point predefined as a basin. It may be necessary to adjust

the elevation of the outlet if the average slope between

root node and the outlet is less than a specified

tolerance. Finally, the tree is traced from the outlet

back to the pit located on the root node, and the grid

elevations are adjusted for every point to create a

uniform gradient.

The code that defines ‘‘priority’’ is contained in a

single subroutine which compares the relative priority

between two vertices. The subroutine is implemented as

a function that returns either true or false, depending

upon whether or not vertex A has higher priority

than vertex B. For drainage enforcement, priority is

based on a small set of nested criteria. In seeking an

outlet for a pit, the path must go through the lowest pass

available. If there is more than one path that satisfies

this criteria, the path taken is the one that goes down in

elevation the most. And for paths that are equal in these

criteria, the path with the shortest total length is

selected.

The data structure associated with each vertex

contains all the information necessary to define its

priority in the priority queue, its adjacency list, and its

position in the tree. The PFS algorithm would require

considerable memory if a vertex record were allocated

for each grid node. As described, extra memory on the

order of n ð¼ nx � nyÞ is required since the adjacency list

for each vertex is implicit in the structure of a grid.

However, in a practical application, any invocation of

the PFS routine will examine only a small fraction of the

total number of nodes in the grid. Therefore, vertex

records are allocated dynamically as needed, and the

adjacency information is contained in a grid of 32-bit

pointers. The pointers are initialized to nil, which also

serves as a sentinel. Whenever the algorithm needs to

find an adjacent vertex, it examines the appropriate grid

node. If the node is nil, a new vertex record is allocated.

If it is not nil, it points to an already-allocated vertex

record. Each time the PFS routine finishes its job in

locating an outlet, all vertices on the tree and in the

queue are deallocated and their corresponding grid

pointers are reset to nil.

Hutchinson (1989) suggests two other features for

incorporation into the pit-removal algorithm. (1) Pits

are cleared in order of increasing elevation. This

enhancement tends to improve the placement of the

drainages in their lower reaches where they join other

drainages. (2) If the elevation difference between the

root node and the last vertex added to the tree is greater

than a specified tolerance, the PFS routine is aborted

and the pit is retained as a sink-point in the DEM.

4.1. Comparison with other drainage enforcement

algorithms

Jenson and Domingue (1988) introduced a drainage

enforcement algorithm (the ‘‘J&D algorithm’’) that has

proven to be popular. The method has been implemen-

ted in the Arc/Info Grid module as part of the Fill

algorithm (see footnote 2). The method works in two

stages. In the first stage, pits are ‘‘filled’’ by increasing

the elevations of the nodes within each depression to the

level of the lowest node on the depression boundary.

The procedure essentially converts every depression into

a flat area that has at least one external node on its

border which has a lower elevation than the flat. The

second stage assigns flow directions to the nodes in flat

R. Jones / Computers & Geosciences 28 (2002) 1051–10601054

areas by an iterative procedure. In each iteration, nodes

are assigned flow directions that point to a neighboring

node if the neighboring node has a flow direction. Flow

direction assignments iteratively grow into the interior

of the flat from the outlet node(s).

(Note: In the J&D algorithm, the elevations of points

in the flat areas are not explicitly adjusted. Instead, a

separate grid-size matrix of flow directions is main-

tained, and it is these values which are modified in the

second stage. The approach taken by the PFS algorithm

is to modify the elevation grid, thus making redundant

the flow-direction matrix since flow directions can be

calculated with the D8 algorithm.)

Pit filling is essentially a filtering operation that

removes depressions from the DEM. This approach is

justified by assuming that depressions are data errors

(Jenson and Domingue, 1988; Martz and Garbrecht,

1998). In contrast, an assumption in this paper is that

DEM data is generally correct if it is used to calculate

drainage networks. The PFS routine obviously violates

this assumption. However, the D8 algorithm requires

some adjustments. A second assumption is that the finite

spatial resolution of the grid precludes an exact

determination of the true stream locations if the stream

channels are small enough to snake between the grid

nodes. PFS uses good strategy for finding an outlet and

if the pit is in a depression, for finding a pass. More

generally, it probably locates drainages to a reasonably

good approximation. Unlike the J&D algorithm, PFS

only adjusts the DEM along lines that are one pixel wide

and preserves well defined topographic features within

depressions.

Martz and Garbrecht (1998) presented a different

method for flat areas. After all the nodes in a flat area

are identified, an iterative routine is invoked to

recontour the area into a slightly V-shaped valley. They

also modified the pit-filling routine to accommodate a

‘‘breaching’’ criteria. Depressions may not be filled to

the level of a pass on the rim of the depression if certain

conditions are met. In that situation, the rim is

‘‘breached’’, whereby a channel is created through the

pass to an external outlet.

Martz and Garbrecht (1998) justify their method for

dealing with flat areas partly because they were

dissatisfied with the tendency for the J&D algorithm to

produce parallel drainage channels. However, from

diagrams presented in their paper, it appears that their

algorithm still has a tendency to produce a lot of

parallel channels. In contrast, PFS has just the opposite

effect. If the starting point for pit elimination in a flat

area is randomly selected (as opposed to, say, always

selecting a point on one edge of the flat), then the

channel created by PFS will be used as an outlet for

subsequent channels. Since the priority condition

specifies that channels will be created by the shortest

path (when all the other priority conditions are equal),

PFS tends to create an orthogonal drainage pattern in

flat areas.

Tribe (1992) also found that the parallel drainage lines

of the J&D algorithm to be unrealistic, and proposed a

new method. The method involves defining a main flow

path through the flat to an outlet, and directing the flow

paths for nearby points in the flat towards the main

path. However, because the main flow path is defined as

a straight line, it could cut through an area of higher

elevation.

Hutchinson (1989) implemented a drainage enforce-

ment technique for a program that creates raster DEMs

from sparse, scattered elevation data. The method first

identifies pits and saddles. Then, for all the downhill

neighbors around each saddle, traces a path from the

neighbor to either a pit or the edge of the grid using the

D8 technique of steepest gradient. For each pit, an

ordered chain is constructed from the pit, through the

lowest, nearby saddle, and to either a pit with a lower

elevation or to the edge of the DEM. The elevations of

the DEM along this path are set to produce an

approximately linear decent. Following the drainage

enforcement routine, the entire DEM is re-interpolated

using a finite element interpolation method. The control

points for the interpolation include the original,

scattered data points, and the points generated along

the drainage-enforcement paths.

Hutchinson’s program enjoys the advantage of being

able to re-interpolate the grid in order to smooth out the

topography around the ‘‘channels’’ created by the

drainage enforcement routine. Unfortunately, given an

existing DEM, the only option for smoothing the DEM

is by filtering, which will oversmooth some well-defined

surface features (Hutchinson, 1989).

Fairfield and Leymarie (1991) handle depressions

using a sequence of steps: (1) A pit is identified (which is

the low point in the depression). (2) The nodes in the

catchment area for the pit are identified. (3) The lowest

point on the rim of the catchment area is selected. (4)

The drainage line is traced from this point downhill to

the pit. (5) Finally, the flow directions along this line are

reversed. This method is basically the same as Hutch-

inson’s algorithm, except in the way that the pass is

found. Neither Fairfield and Leymarie (1991) nor

Hutchinson (1989) describe methods for handling flat

areas.

Using recursive or stack-based algorithms, the nodes

in a flat area or a depression can be identified in a

time proportional to the number of nodes—an OðnÞruntime performance (Fairfield and Leymarie, 1991;

McCormack et al., 1993). In contrast, PFS is an

Oðv log vÞ algorithm, where v is the number of nodes

actually checked (Sedgewick, 1988). V is usually smaller

than n; especially for depressions, because PFS is

finished once an outlet is found, and it does not need

to examine every node in a flat area or depression to do

R. Jones / Computers & Geosciences 28 (2002) 1051–1060 1055

that. Hence, in practice the performance of PFS is hard

to beat.

All of the drainage-enforcement algorithms described

previously use entirely different procedures for proces-

sing depressions and flat areas, except PFS, which

effectively handles both conditions.

5. Defining internal basins and constraining the location

of drainages

PFS works reasonably well for preparing a DEM grid

for drainage delineation in areas of moderate relief,

where drainages are well constrained by topography,

and are without internal basins (Fig. 2). However, two

major problems remain to be addressed. First, the

calculated drainages are generally not close to the real

drainages in areas of broad, flat topography. The fact

is, there is no information in the original DEM data

that can be used to locate the actual positions of

drainages when the topography is flat. Secondly, PFS

will find an outlet for a pit even if it occurs in a valid,

closed basin.

A basin is a region-type object that has no internal

drainage, and a point-like sink may be defined where an

individual drainage simply disappears. Constraining the

PFS routine to terminate whenever a predefined ‘‘sink’’

node is encountered is achieved by flagging all nodes

that are suppose to be sinks with a special attribute. A

separate grid of boolean values, okPit, has each node set

to either true or false depending upon whether the node

is suppose to be a sink or not.

The locations of drainages are constrained by reading

a file of line-objects, and changing the grid elevation for

every node in the line to some suitable value. There are

two possible modifications: predefined drainage ‘‘chan-

nels’’ are ‘‘cut’’ into the DEM by decreasing the Z

coordinate along the line of the channel. Although the

channels thus defined are not necessarily drainages (they

may be flat-bottomed depressions), they will be drai-

nages after the pit-elimination procedure is completed.

‘‘Ridges’’ are the opposite of channels because the Z

coordinate is increased, which inhibits the nodes from

ever being on a drainage.

User-defined basins, sinks, channels, and ridges

represent a variety of spatial objects. Hence, a geo-

graphic information system (GIS) compatible file format

is used to enter the data as input into the drainage

program. For each point-, line-, and area-object there is

an associated attribute code. Four attributes are defined:

(1) ‘‘sinks’’ include points, lines and areas; (2) a ‘‘seed’’

point will fill an area bounded by lines with sinks; (3)

predefined drainage ‘‘channels’’; and (4) ‘‘ridges’’. The

program shifts each real-world coordinate to the nearest

grid node, and sets either okPit or Z to the appropriate

value, depending upon the attribute.

Fig. 2. Same map area as Fig. 1, but with drainage continuity enforced on elevation grid using PFS routine that is described in text.

Map contains 156 separate drainage basins, and just four largest account for 1200 sq. miles of watershed, or 70% of total map

coverage.

R. Jones / Computers & Geosciences 28 (2002) 1051–10601056

5.1. Comments concerning the use of ancillary streamline

data

The method outlined above for incorporating stream-

line data is simple, but effective. However, it does create

unrealistic depth-extents for the ‘‘cut’’ channels. Given

the original, pitted DEM and its derivative, drainage-

enforced DEM, it is possible to calculate a second

drainage-enforced DEM that has all the drainage

characteristics of the first, but with a minimum amount

of discrepancy with the original. However, for catch-

ment area calculations the depth-extent of the channels

is immaterial because only the directional information,

as defined by the D8 algorithm, is used.

Recently, several papers have been presented which

address the subject of using streamline data in conjunc-

tion with DEMs for basin/watershed analysis (e.g.

Turcotte et al., 2001, Saunders (see footnote 2), and

references therein). Often referred to as ‘‘stream burn-

ing’’, these methods are essentially the same as that

which was described previously for channel-type con-

straints, except that they more frequently change the off-

stream nodes of the DEM rather than the on-stream

nodes. Hutchinson (1989) was one of the earliest to

report using streamline data in conjunction with DEM

grids, but he was generating the DEMs, a somewhat

different subject.

Saunders (see footnote 2) reported a situation where

the D8 algorithm failed to delineate accurately two

watershed boundaries for an area along the topogra-

phically flat, southeast Texas coast. These results were

obtained using Arc/Info, which has built-in pit removal

and flow accumulation routines for DEM data. Efforts

to get the calculated watershed boundaries ‘‘right’’

included intensive editing of streamline data, and testing

four different stream burning algorithms. Only the most

sophisticated and computationally intensive stream-

burning algorithm produced satisfactory results,

although it, too, was not exactly ‘‘right’’. Saunders

concludes that improvements in the way in which the

elevations are adjusted by stream burning could lead to

better results. This suggestion is a good one. For

example, Saunders already has the ‘‘known’’ watershed

boundary, which could be applied as a ridge-type of

constraint, thus forcing the D8 and pit-removal algo-

rithms to the ‘‘right’’ solution irrespective of the

streamline data or the sophisticated algorithms. Also,

the pit-removal algorithm is a critical factor influencing

the final map. Hence, a better pit-removal algorithm

could also lead to better results.

Turcotte et al. (2001) used both streamline data and

the boundaries of lakes to model stream flow in an area

in Quebec, Canada where numerous lakes occur in parts

of the drainage network. Where streamline and lake

information was available, flow directions were obtained

solely by this data. In particular, the D8 approach

cannot be used to model flow directions in lakes. In

areas not covered by streamline data or lake boundaries,

flow directions were modeled using the D8 algorithm.

Hellweger3 reported a situation where there was an

obvious discrepancy in the position of a stream as

indicated topographically by a DEM and as indicated by

streamline data. From the figures in Hellweger’s paper,

it appears that there is enough relief in the supposed-

problem area for the D8 algorithm to work well.

However, after ‘‘burning in’’ the streamline data, the

DEM contained two, parallel, nearby troughs. Although

the main drainage followed the burned-in channel, a few

of the smaller tributaries flowed into the other trough,

and therefore did not merge with the main drainage. In

an effort to correct the problem, Hellweger (see footnote

3) created a program, Agree.aml, that ‘‘reconditioned’’

the DEM in the vicinity of the streamlines. In effect, the

program created a V-shaped channel that was wide

enough to remove the ridge that was situated between

the two troughs. The program may have solved the

‘‘reconditioning’’ problem that was initially created by

the streamline data, but it could also lead to other

problems. For example, Saunders (see footnote 2) noted

that the Agree algorithm produced an artifact in a

watershed boundary that was not apparent in other

stream burning techniques. Automatic filtering or

‘‘reconditioning’’ can alter well-defined surface features

in a DEM (Hutchinson, 1989). In Hellweger’s problem

area, good results could probably be achieved by simply

omitting the streamline data.

6. Producing drainage maps

The drainage pattern may be displayed in either raster

format or vector format. Raster images are the simplest,

since there is a one-to-one correspondence between the

catchment area grid and the final raster image. Pixels

may be color coded differently to indicate the size of the

drainage by using various thresholds of catchment area

as criteria for assigning the colors.

Vector output is more involved since it requires that

individual grid elements be linked into line segments or

grouped into sequences of line segments (polylines).

Each drainage is traced upstream from its terminal node

until the ca value is less than the drainage threshold.

Locating the terminal node is easy, since all streams

must terminate at a sink. (The edge of the DEM is also

initialized with an outlying border of sinks.) Hence, if

these points have a ca value greater than the drainage

threshold, then they are used as starting points for the

tracings. The definition of ‘‘upstream’’ is a derivative

3Hellweger, F., 1997. Agree-DEM surface reconditioning

system. http://www.ce.utexas.edu/prof/maidment/gishydro/fer-

di/research/agree/agree.html.

R. Jones / Computers & Geosciences 28 (2002) 1051–1060 1057

definition: point a is upstream of point b only if point b

is downstream of point a. For any node on the grid there

are 8 possible Upstream directions. Therefore, it is

expedient to first calculate the grid upstream, which is

defined as a matrix of type SetOfDirections, where the

set of directions are the eight enumerated values of east,

northeast, north, etc. Tracing the drainages upstream

from a given node then proceeds in the direction

indicated by UpStream. Since UpStream can contain

two or more directions (which occur at forks in the

stream), a depth-first-search stack is used to process all

tributary branches.

7. Calculations for stream sediment sampling

If simply mapping the locations of drainages is the

objective, then using a DEM to achieve this is a poor

approach to solving the problem. The use of a DEM

implies an interest in some spatially extensive character-

istic of the terrain, not just the spatially restricted

drainage lines. The practical applications presented here

are to use a DEM to help in the design phase of a stream

sediment sampling program, and to map the catchment

areas of stream sediment samples.

Experience in the arid terrain of Nevada, USA,

indicates that the catchment area for potential sample

sites should be between 2 and 8 sq. miles (5–20 km2). For

a 1 degree DEM, 412 grid cells equals 1 sq. mile, so

lower and upper thresholds can be calculated for

displaying stream segments that are properly sized for

sampling (Fig. 3). Also, the downstream endpoint of

these segments is the most optimal sampling point along

the stream. (In practice, the point that is actually

sampled will be determined by additional factors,

including geology, accessibility, and the characteristics

of the stream as determined in the field.)

Mapping the catchment area for a given set of sample

points is essentially the same as tracing the drainages,

except that the starting point for each trace is the sample

point itself, and all upstream points are to be included in

the catchment area. In Fig. 4, a map of some sample

catchment areas is shown as a colored raster image. A

GIS program was used to superimpose ‘‘layers’’ of

additional information, including the DEM-calculated

drainages. Tabulated data concerning the catchment

areas can also be generated.

9. Program implementation notes

The program ‘‘Drainage’’ was coded in Pascal for

Borland’s Delphi compiler for Windows 95. Both the

source code and the executable program may be

downloaded from the IAMG web site (http://www.iam-

g.org/CGEditor/index.htm).

Program parameters are entered in a text file, which

can be modified using a text editor. These parameters

Fig. 3. Map showing drainage lines that were constrained by USGS DLG data. Black dots represent optimal sample sites for planning

a stream sediment sampling program. Criteria for sample sites are that they must have a catchment area of at least 2 sq. miles, but not

more than 8 sq. miles. Thick gray outlines are water reservoirs, which are classified as ‘‘sinks’’ for this application.

R. Jones / Computers & Geosciences 28 (2002) 1051–10601058

include numerical constants and input/output file

names. The geographical constraints, if any, are read

from a file that is formatted in MapInfo interchange

format (MIF). Program results are written to disk in

tab-delimited text, MIF, and colored BMP file formats.

Additional information concerning the program’s para-

meters and I=O data formats is included with the

executable program on the IAMG web site.

The slowest part of the drainage program is in the

presort used to order the grid points by elevation prior

to the calculation of the catchment area grid, ca. This is

an Oðn log nÞ algorithm, where n is the number of grid

points. The pit elimination routine makes a single pass

through the elevation grid to identify pits. Although the

PFS subroutine that is used to eliminate a pit has

Oðv log vÞ running time, the overall running time for pit

elimination is significantly faster than the sorting

routine. Tracing drainages is linear in the number of

nodes in the drainages, and mapping sample catchment

areas is linear in the area of the catchment areas.

Therefore, these calculations are also relatively fast.

The program uses 17 bytes per node for the

calculation of ca, or about 25 mbytes for a full 1 degree

DEM data set. The memory required to eliminate

spurious pits is data dependent. The worst-case scenario

would be an elevation grid that consists of one large

depression bounded by a rim one pixel wide. For this

contrived example, pit elimination requires about 80

mbytes for a full 1-degree DEM. However, some tests on

real DEMs indicate that it is unlikely that pit elimina-

tion is a memory-limiting factor for real DEMs.

10. Concluding remarks

Analytically, complex three-dimensional surfaces can

be made computationally tractable for computer proces-

sing by digitizing the surface at discrete points on a

regular grid. A certain amount of information is lost in

this process. For some types of computation, this loss is

not a major problem. For example, DEMs are

commonly used as input data for contouring programs.

However, for other computations the information lost is

fundamental to the problem. In the situation of drainage

basin analysis, the DEM does not contain any informa-

tion that directly locates drainages and the terminations

of drainages at sinks. Also, the ability to infer the

locations of drainages by computational analysis of the

elevation grid is limited. This article addresses techni-

ques that correct the deficiencies in the rasterized

elevation data so that drainages may be reasonably

located. Once a drainage network has been defined, the

problem of analysis can proceed to the next step. The

application presented here is to calculate the catchment

areas for stream sediment samples and to identify

‘‘optimal’’ sample sites.

Fig. 4. Map of catchment areas of stream sediment samples is created in color-coded bitmap format. (Here, colors have been

reassigned to gray-scale.) Hypothetical sample points that are shown (as black dots) include some samples that are located within

catchment areas of other samples. Program assigns non-unique colors for the individual catchment areas, but no two adjacent or

‘‘included’’ areas have same color. Separately, a table is output which identifies for each sample its color, its catchment area in sq.

miles, and area of any included samples.

R. Jones / Computers & Geosciences 28 (2002) 1051–1060 1059

Acknowledgements

I thank my wife, Laura McKerrow, Mark Mihalasky,

Graeme Bonham-Carter, and two C&G reviewers for

making numerous suggestions to improve the paper.

References

Fairfield, J., Leymarie, P., 1991. Drainage networks from grid

digital elevation models. Water Resources Research 27 (5),

709–717.

Hutchinson, M.F., 1989. A new procedure for gridding

elevation and stream line data with automatic removal of

spurious pits. Journal of Hydrology 106, 211–232.

Jenson, S.K., Domingue, J.O., 1988. Extracting topographic

structure from digital elevation data for geographic

information system analysis. Photogrammetric Engineering

and Remote Sensing 54 (11), 1593–1600.

Mark, D.M., 1984. Automated detection of drainage networks

from digital elevation models. Cartographica 21, 168–178.

Martz, L.W., Garbrecht, J., 1998. The treatment of flat areas

and depressions in automated drainage analysis of raster

digital elevation models. Hydrological Processes 12, 843–855.

McCormack, J.E., Gahegan, M.N., Roberts, S.A., Hogg, J.,

Hoyle, B.S., 1993. Feature-based derivation of drainage

networks. International Journal of Geographical Informa-

tion Systems 7 (3), 263–279.

O’Callaghan, J.F., Mark, D.M., 1984. The extraction of

drainage networks from digital elevation data. Com-

puter Vision, Graphics, and Image Processing 28,

323–344.

Quinn, P., Beven, K., Chevallier, P., Planchon, O., 1991. The

prediction of hillslope flow paths for distributed hydro-

logical modelling using digital terrain models. Hydrological

Processes 5, 59–79.

Sedgewick, R., 1988. Algorithms, 2nd edn. Addison-Wesley,

Reading, MA.

Tarboton, D.G., 1997. A new method for the determination of

flow directions and upslope areas in grid digital elevation

models. Water Resources Research 33 (2), 309–319.

Tarboton, D.G., Bras, R.L., Rodriguez-Iturbe, I., 1991. On the

extraction of channel networks from digital elevation data.

Hydrological Processes 5, 81–100.

Tribe, A., 1992. Automated recognition of valley lines and

drainage networks from grid digital elevation models:

a review and a new method. Journal of Hydrology 139,

263–293.

Turcotte, R., Fortin, J.-P., Rousseau, A.N., Massicotte, S.,

Villeneuve, J.-P., 2001. Determination of drainage structure

of a watershed using a digital elevation model and a

digital river and lake network. Journal of Hydrology 240,

225–242.

R. Jones / Computers & Geosciences 28 (2002) 1051–10601060