Generalized methods for terrain-based flow analysis of digital
elevation models
David G. Tarboton, Utah Water Research Laboratory, Department of Civil and Environmental
Engineering, Utah State University, Logan, Utah, USA, [email protected]
Matthew E. Baker, Department of Watershed Sciences and The Ecology Center, Utah State
University, Logan, Utah, USA, [email protected]
Abstract
Flow field representations for a grid DEM
Topography is an important land surface attribute for hydrology that, in the form of Digital
Elevation Models (DEMs), is widely used to derive information for the modeling of
hydrologic processes. Much hydrologic terrain analysis is conditioned upon an information
model for the topographic representation of downslope flow derived from a DEM, which
enriches the information content of digital elevation data. This information model involves
procedures for removing spurious sinks, deriving a structured flow field, and calculating
derivative surfaces. We present a general method for recursive flow analysis that exploits
this information model for calculation of a rich set of flow-based derivative surfaces beyond
current weighted flow accumulation approaches commonly available in Geographic
Information Systems, through the integration of multiple inputs and a broad class of
algebraic rules into the calculation of flow related quantities. This flow algebra
encompasses single and multi-directional flow fields, various topographic representations,
weighted accumulation algorithms, and enables untapped potential for a host of applicationspecific functions. We illustrate the potential of flow algebra by presenting examples of new
functions enabled by this perspective that are useful for hydrologic and environmental
modeling. Future opportunities for advancing flow algebra functionality could include the
development of a formulaic language that provides efficient implementation and greater
access to these methods. There are also opportunities to take advantage of parallel
computing for the solution of problems across very large input datasets.
Steepest direction
downslope
Proportion flowing to
neighboring grid cell 3
is 2/(1+2)
Proportion
flowing to
neighboring
grid cell 4 is
1/(1+2)
Steepest single
direction
3
48
52
56
67
4
2
2
1
Flow
direction.
1
6
8
D
D8
67  52
 0.50
30
5
7
More general flow field representations.
Models for the digital representation of terrain
Multiple Flow Direction Grid
Physical Network
TIN -Base Voronoi Grid
Physical Network
Contour-Flow-Tube Grid
Physical Network
2
4
1
5
3
Topological Network
2
1
P 13 =0.4
3
1
6
P 46 =1
3
P 35 =0.3
P 12 =0.5
4
5
2
4
6
7
Topological Network
2
P 34 =1
3
5
3
1
Topological Network
4
P 24 =1
5
1
P 13 =0.5
a) Grid
6
1
P 25 =0.7
P 24 =0.3
2
P 13 =0.4
3
P 36 =1
6
4
5
4
2
P 25 =1
5
P 57 =1
7
Contributing Area
b) Triangulated Irregular Network
Definition
A ( x )   r ( x )d x
c) Contour and flowline
Terrain based flow data model for enriching the information
content of a raw digital elevation model (DEM)
CA
r(x)
Numerical Evaluation
x
A( x i )  r ( x i )  
• Sink Removal
• Flow Related Terrain
Information
Information science includes the precise representation of
physical environments using data models that enhance the
capability for analysis and integration of information.
{k:Pki 0}
Numerical flow field representation
Pij such that
• Flow Field
 Pki A(x k )
i Pij  1
Pseudocode for Recursive Upslope Flow Accumulation Evaluation
Global variables Ai, r(xi), Pij, 
Function FlowAccumulation(xi)
if Ai is known
then
no action
else
for each neighbor location xk indexed by k
if(Pki > 0)then
call FlowAccumulation(xk)
//This is the recursive call to calculate area for the neighbor
Next k
// At this point all the neighboring Ak inputs are available
A i  r ( x i ) 
return
 Pki A k
{k: Pki  0}
Recursive Flow Analysis – Flow Algebra
A new formalism for deriving flow-based information useful for hydrologic and environmental modeling that embeds
additional information and mathematical logic into the recursive flow analysis algorithm
Flow algebra logic to generalize recursive flow accumulation
 ( x ) – Simple input variables fully quantified before the
evaluation of a flow algebra expression.
( x ) – Variables that have recursive dependence on the flow field
that are quantified during the course of evaluating a flow
algebra expression.
Pki – Numerical representation of flow field, quantifying the
proportion of flow from element k to element i
General upstream flow algebra expression
( x i )  f (  ( x i ), P ki , ( x k ),  ( x k ))
General downstream flow algebra expression
( x i )  f (  ( x i ), P ik , ( x k ),  ( x k ))
The function f(.) may include any mathematical operators such
as: +, -, , , summation, conditional, logical, trigonometric and
mathematical functions.
General Pseudocode for Upstream Flow Algebra Evaluation
Global variables , , Pij
Function FlowAlgebraUpstream(xi)
if (xi) is known
then
no action
else
for each neighbor location xk indexed by k
if(Pki > 0)then
call FlowAlgebraUpstream(xk)
//This is the recursive call to an upstream
neighbor
Next k
// At this point all the necessary inputs are available
Evaluate Algebraic expression
(xi)=f((xi), Pki, (xk), (xk))
return
General Pseudocode for Downstream Flow Algebra Evaluation
Global variables , , Pij
Function FlowAlgebraDownstream(xi)
if (xi) is known
then
no action
else
for each neighbor location xk indexed by k
if(Pik > 0)then
call FlowAlgebraDownstream(xk)
//This is the recursive call to a downstream
neighbor
Next k
// At this point all the necessary inputs are available
Evaluate Algebraic expression
(xi)=f((xi), Pik, (xk), (xk))
return
Examples of functions constructed using Flow Algebra
Weighted Flow Distance
0
0
0
0
Transport limited accumulation
Transport a function of distributed supply and transport capacity.
0
30*0.5/2
7.5
7.5+42.4*
(0.7+0.5)/2
0.5
32.9
Tcap  ca 2 tan( b) 2
A. Supply
B. Capacity
C. Transport
D. Deposition
0.7
30
38.2
ad ( x i ) 
38.2
38.2
38.2 38.2
 w( x i )  w( x k )

P
dist
(
x
,
x
)

ad
(
x
)
 ik 
i k
k 
2

k: Pik  0
 Pik
T( x i )  Min (C( x i ),
{k: Pik  0 & ad ( x k )  0}
 Pki T(x k )  S(x i ))
{k: Pki  0}
D( x i ) 
 Pki T(x k )  S(x i )  T(x i )
{k: Pki  0}
D Downslope
Influence
D8 Accumulation
Stream
Source
Avalanche Runout
Buffer
A. Inputs
B. Flow paths
0-10
10-20
20-30
30-40
40-50
50-60
60-70
70-80
80-90
>90m
2
1
3
D. D Buffer
C. D8 Buffer
Weighted flow length-to-stream measures used in buffer
analyses for water quality modeling in tributaries of
Chesapeake Bay, Maryland, USA.
Decaying Accumulation.
DA ( x i )  r ( x i ) 
 Pki d(x k )DA(x k )
{k: Pki  0}
The load r(x) is assumed to move with the flow field but
subject to first order decay, d(x), in moving from cell to cell
Upslope recursion to determine
elevation and distance to point
in trigger zone that has the
highest alpha angle

Variables in avalanche runout flow algebra function
Symbol Description
Simple input variables: (x)
T
Flow proportion threshold
Alpha angle

as
Avalanche source set
xi, yi
Coordinates of the center of each element
zi
Elevation of the center of each element.
Recursive variables: (x)
rz
A runout zone indicator with value 0 to indicate that this grid cell
is not in the runout zone and value > 0 to indicate that this grid cell
is in the runout zone. Since there may be information in the angle
to the associated source site, this variable will be assigned the
angle to the source site, denoted as b here (in degrees).
xm, ym X and Y locations of the source site that has the highest angle to
the point in question.
zm
Elevation of the source site that has the highest angle to the point
in question
dm
Flow distance from the source site that has the highest angle to the
point in question. This is included to allow evaluation of source
angles using either straight-line or flow path distances.
Conclusions and Future Directions
• The terrain based flow field derived from (1) drainage correction and (2) multidirectional proportioning among downslope neighbors,
enables a broad class of upstream and downstream recursive functions that comprise flow algebra.
• Flow proportions used in flow algebra should sum to one and be non-circulating as is the case for flow derived from the gradient of any
potential field.
• Flow algebra provides a formalism for thinking about and modeling spatial processes that are related to, or occur embedded within the
flow field.
• Flow algebra offers the opportunity to reduce some of the gap between real world observations and digital representations of physical
processes.
• A formulaic language based on flow algebra would provide a computational modeling frameworks that enables the implementation and
rapid evaluation of new theories and field based concepts
• Flow algebra is not limited to grids or single flow direction approaches and provides a more inclusive modeling framework for flowrelated modeling
• Efficient implementation taking advantage of block data structures and parallel processing is needed for application to large datasets
For more information, see: Tarboton, D. G. and M. E. Baker, (2008, in press), "Towards an Algebra for Terrain-Based Flow Analysis," in
Representing, Modeling and Visualizing the Natural Environment: Innovations in GIS 13, Edited by N. J. Mount, G. L. Harvey, P. Aplin
and G. Priestnall, CRC Press, Florida.
TauDEM software available from
http://www.engineering.usu.edu/dtarb
Acknowledgements
Some funding for this project was provided by CICEET, the Cooperative Institute for Coastal and Estuarine
Environmental Technology. A partnership of the National Oceanic and Atmospheric Administration
(NA06NOS4190167) and the University of New Hampshire, CICEET develops tools for clean water and
healthy coasts nationwide.