Hydrodynamic and geomorphologic dispersion:

scale effects in the Illinois River Basin

Amanda B. Whitea, Praveen Kumara,*, Patricia M. Sacoa,Bruce L. Rhoadsb, Ben C. Yena

aEnvironmental Hydrology and Hydraulics Engineering Program, Department of Civil and Environmental Engineering,

University of Illinois at Urbana-Champaign, 205 North Mathews Avenue, Urbana, IL 61801, USAbDepartment of Geography, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Received 9 April 2002; revised 30 September 2003; accepted 29 October 2003

Abstract

The objective of this work is to determine the relative effects of hydrodynamic and geomorphologic dispersion on the

hydrological response of the Illinois River Basin (IRB) as scale increases. The specific hypothesis that was tested is that as

basin size increases, the river network structure, as compared to channel hydrodynamic properties, plays an increasingly

dominant role in determining the hydrologic response. The analysis was performed on eight of the major watersheds in the

IRB in order to provide an adequate representation of basins that contain streams of order six or greater. The basins studied

include the Des Plaines, Mackinaw, Vermilion, Fox, La Moine, Spoon, Kankakee, and the Sangamon, and have magnitudes

ranging from order six to order eight. The geometric and hydrodynamic properties were derived from the analysis of digital

elevation model data and from the hydraulic geometry equations for various subcatchments of the IRB put forth by Stall and

Fok [Univ. of Ill. Water Res. Center Res. Rep. 15, 1968]. The hydrodynamic and geomorphologic dispersion coefficients

were determined for each order stream of the eight basins and for constant flow frequencies, then compared. The results

contradict the original hypothesis, for at small scales, geomorphologic dispersion tends to dominate, the extent of which

depends upon the flow frequency, and at large scales, geomorphologic dispersion is less dominant. This occurs because of the

behavior of the path lengths of a stream network, which geomorphologic dispersion depends upon. In addition, at high flow

frequencies the geomorphologic dispersion dominates, and at low frequencies the hydrodynamic dispersion begins to play an

increasingly important role, although the geomorphologic dispersion still dominates. This dominance suggests that the

geomorphologic parameters of a watershed could be more important in characterizing the hydrologic response of a river basin

than hydrodynamic parameters.

q 2004 Elsevier B.V. All rights reserved.

Keywords: Hydrodynamic; Geomorphologic; Dispersion; Scale; Geomorphologic instantaneous unit hydrograph

1. Introduction

The runoff from a watershed is characterized by the

interaction of a variety of processes both at

0022-1694/$ - see front matter q 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.jhydrol.2003.10.019

Journal of Hydrology 288 (2004) 237–257

www.elsevier.com/locate/jhydrol

* Corresponding author. Fax: þ1-217-333-0687.

E-mail address: kumar1@uiuc.edu (P. Kumar).

the hillslope and network scale. To understand the

role of the network in shaping the hydrograph,

Rodrıguez-Iturbe and Valdes (1979) developed the

geomorphologic instantaneous unit hydrograph

(GIUH) approach for the instantaneous unit

hydrograph of a basin. The GIUH theory has been

and continues to be used in numerous studies

concerning the geomorphologic characteristics of a

watershed (Rodrıguez-Iturbe and Valdes, 1979;

Valdes et al., 1979; Rodrıguez-Iturbe et al.,

1979; Snell and Sivapalan, 1994; Robinson et al.,

1995; Rodrıguez-Iturbe and Rinaldo, 1997; Yen and

Lee, 1997). The geomorphologic dispersion coeffi-

cient DG is a measure of the dispersion of a

disturbance by the river network structure.

This concept incorporates the idea that raindrops

falling on different areas of a basin at the same time

will not reach the outlet at the same time (Rinaldo

et al., 1991) due to the different path lengths to the

mouth of the basin. The hydrodynamic dispersion

coefficient DL (Lighthill and Whitham, 1955; Mesa

and Mifflin, 1986; Rinaldo et al., 1991) is a measure of

the tendency of a disturbance to disperse

longitudinally as it travels downstream. This dis-

persion is caused by flow resistance induced by

friction along the channel boundaries and storage

characteristics. Fig. 1 displays a conceptual view of

the contributions of hydrodynamic and geomorpho-

logic dispersion to the dispersion of a hydrograph.

Rinaldo et al. (1991) hypothesized that if DG; which

depends on the variability in path lengths, is

significantly greater than DL; which might occur in

large river basins, then the stream network structure

will mask the effect of differences in flow conditions

in individual channel reaches and will play the

dominant role in the prediction of hydrologic

response. Robinson et al. (1995) studied the roles of

hillslope, channel, and network on the hydrologic

response for small river basins and concluded that as

scale increases, the primary factor governing the

response is the network geomorphology. This study

will examine large-scale river basins to determine the

governing influences on their hydrologic responses.

The aim of this research is to determine the relative

effects of geomorphologic dispersion and hydrodyn-

amic dispersion on the hydrologic response of the

Illinois River system as scale increases. The specific

hypothesis to be tested is that as basin size increases,

the river network structure, as compared to channel

hydrodynamic properties, plays an increasingly

dominant role in determining the hydrologic response.

This analysis is performed using hydraulic geometry

information for streams in the Illinois River Basin

(IRB) and stream network characteristics extracted

from a Digital Elevation Model (DEM). In addition,

the computer-extracted stream networks are

compared with actual networks to gain insight into

the validity of the various parameters extracted from

the DEM for use in the analysis.

The paper is organized as follows. In Section 2, we

review the equations relevant to the development and

estimation of hydrodynamic and geomorphologic

dispersion. The case study is given in Section 3,

which includes establishing the validity of DEM-

extracted networks and the computation of all

dispersion coefficients as a function of scale and

frequency of occurrence. The study of the relative

contributions of hydrodynamic and geomorphologic

dispersion to the hydrologic response of a watershed

is presented in Section 4, along with comparisons

with previous results of Robinson et al. (1995).

Summary and conclusions are given in Section 5.

2. Background

The GIUH theory postulates that the distribution of

arrival times of water particles at the outlet of a basin

depends on the topological structure of the river

network. Consider a Horton–Strahler third-order

watershed exemplified in Fig. 2. Utilizing Strahler’sFig. 1. Conceptual view of the different mechanisms contributing to

the dispersion of a hydrograph (i is precipitation and Q is discharge).

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257238

ordering system (Strahler, 1957), a set G of several

pathways g can be defined which represents the

various transitions from the injection point of an

effective rain drop to the subsequent streams it flows

through to the outlet of the basin. For the third-order

watershed in Fig. 2, the set of pathways

G ¼ {g1; g2; g3;g4} is defined as

g1 ¼ o1 ! c1 ! c2 ! c3 ! outlet

g2 ¼ o1 ! c1 ! c3 ! outlet

g3 ¼ o2 ! c2 ! c3 ! outlet

g4 ¼ o3 ! c3 ! outlet

where ov denotes the overland flow state that directly

contributes to a stream of order v and cv represents

the channel state of order v: Note that the above

example can easily be extended to a basin of arbitrary

order V:

Let the specific path g be defined as a collection of

states g ¼ {x1; x2;…; xk} where x1 ¼ ov; x2 ¼ cv with

v as one of {1;…;V}; xj with {j ¼ 3;…; k 2 1} as one

of {cvþ1;…; cV21} and xk ¼ cV: The probability pðgÞ

that a droplet will follow any path to the outlet is simply

pðgÞ ¼ px1£ px1;x2

£ px2;x3£ · · · £ pxk21;xk

ð1Þ

where px1is defined as the initial probability that a

droplet will begin in state x1; and pxi;xjis defined as the

probability that a droplet will transition from state xi to

state xj:

The travel time through a particular path is the sum

of the travel times spent in each individual state:

Tg ¼ Tx1þ Tx2

þ · · · þ Txk: ð2Þ

Hence, the travel time distribution through each

individual path g is

fgðtÞ ¼ fx1p fx2

p · · · p fxkðtÞ ð3Þ

where fxiis the travel time distribution through each

individual state xi of path g and the p denotes the

convolution operator. The travel time distribution fbðtÞ

at the outlet of the basin, when the rainfall is

uniformly distributed over the entire basin,

is determined by randomizing over all possible paths:

fbðtÞ¼Xg[G

pðgÞfgðtÞ¼Xg[G

pðgÞ{fx1p ···p fxk

ðtÞ}g: ð4Þ

The above may alternatively be written as

fbðtÞ¼ fx1pXg[G

pðgÞ{fx2p ···p fxk

ðtÞ}g¼ fx1p f ðtÞ ð5Þ

where f ðtÞ and fx1give the travel time distributions

through the stream network and hillslope, respect-

ively. Henceforth, we will only consider the network

travel time distribution f ðtÞ in our analysis.

It follows that the three elements needed to fully

characterize the GIUH are:

1. The initial probabilities pxiof beginning in a

particular state xi;

2. The transition probabilities pxi ;xjof being

transported from state xi to state xj; and

3. The residence time distribution in each individual

state fxiðtÞ:

The initial probabilites are obtained as (Rodrıgue-

z-Iturbe and Valdes, 1979)

px1¼

N1�A1

AV

ð6Þ

Fig. 2. A typical third-order watershed.

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257 239

pxi¼

Ni

AV

�Ai 2Xi21

j¼1

�Aj

Njpxi;xj

Ni

0@

1A ð7Þ

for 2 # i # V; where Ni is the number of streams of

order i; AV is the total area of the watershed, and �Ai is

the mean area of a subcatchment draining into a

channel of order i: The transitional probabilities are

obtained as

pxi ;xj¼ ni;j p

Nj

Ni

ð8Þ

where ni;j is the mean number of streams of order i

draining into streams of order j and Ni and Nj are the

total number of streams of order i and j; respectively.

Several methods have been used to determine the

residence time distributions in each individual

state, such as assuming an exponential distribution

(Rodrıguez-Iturbe and Valdes, 1979), a uniform

distribution (Gupta et al., 1980), and a gamma

distribution (van der Tak and Bras, 1990).

However, Rinaldo et al. (1991) used an advection–

dispersion equation derived from the Saint-Venant

equations of momentum balance (Lighthill and

Whitham, 1955; Yen and Tsai, 2001) to describe the

flow through individual streams as

›hxi

›tþ uxi

›hxi

›x¼ DLxi

›2hxi

›x2ð9Þ

where hxi; uxi

; and DLxiare the flow depth, the

kinematic celerity of a traveling wave, and the

hydrodynamic dispersion coefficient for each

particular state xi; respectively. The kinematic wave

celerity can be computed as

uxi¼ lup

xið10Þ

where l is an empirical constant dependent upon the

channel geometry (typically 3/2 for a triangular

channel and 5/3 for a rectangular channel (Chow

et al., 1988)) and upxi

is the steady state flow velocity

under uniform flow conditions in state xi: In our

calculations, we chose l ¼ 3=2: The hydrodynamic

dispersion coefficient can be calculated as

DLxi¼

uxihp

xi

3Sxi

ð11Þ

where hpxi

and Sxiare the steady state flow depth under

uniform flow conditions and the mean channel bed

slope in state xi; respectively.

Rinaldo et al. (1991) derived an analytical

expression describing the travel time distribution for

the case when the kinematic wave celerity uxiand the

hydrodynamic dispersion coefficient DLxiare con-

stants, denoted as u and DL; respectively. Thus, Eq.

(11) reduces to

DL ¼uhp

3Sð12Þ

where hp and S are the steady state flow depth under

uniform flow conditions and the mean channel bed

slope for all states xi; respectively. Under this

condition, the travel time distribution is obtained as

f ðtÞ¼1ffiffiffiffiffiffiffiffiffiffi

4pDLt3p X

g[G

pðgÞ �Lg exp2ð �Lg2utÞ2

4DLt

" #ð13Þ

where the mean path length �Lg is obtained as

�Lg¼X

xi[g

�Lxið14Þ

and �Lxiis the mean length of each individual state xi:

The total variance of the travel time distribution

provides a measure of the total dispersion and is

formulated as (Rinaldo et al., 1991)

VargðTgÞ

¼2 �LV

u3DLþ

u

2 �LV

Xg[G

pðgÞð �LgÞ22

Xg[G

pðgÞ �Lg

0@

1A22

435

0@

1A

ð15Þ

where the subscript g is used to signify a moment

computed over all possible paths g and the mean

length �LV is obtained as:

�LV;Eg½ �Lg�¼Xg[G

pðgÞ �Lg: ð16Þ

Rinaldo et al. (1991) defined the geomorphologic

dispersion coefficient DG as:

DG¼u

2 �LV

Xg[G

pðgÞð �LgÞ22

Xg[G

pðgÞ �Lg

0@

1A22

435: ð17Þ

A more general expression for DG; providing for an

infinite number of paths, was developed by Snell and

Sivapalan (1994) in terms of the mean and variance of

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257240

the distribution of individual path lengths:

DG¼uVarg½ �Lg�

2Eg½ �Lg�: ð18Þ

Hence, Eq. (15) can be rewritten as:

VargðTgÞ¼2Eg½ �Lg�

u3ðDLþDGÞ: ð19Þ

Using this approach to obtain the variance of the

hydrologic response, Rinaldo et al. (1991) concluded

that there are two basic mechanisms contributing to

the variance. One is the dispersion along each

individual path caused by the hydrodynamic effects

ðDLÞ and the other is the dispersion over all paths due

to the heterogeneity of path lengths, or the network

structure ðDGÞ: The objective of the present research is

to determine the relative contributions of DL and DG

to the hydrologic response at different spatial scales.

3. Case study

3.1. Description of the Illinois River Basin

The Illinois River Basin (IRB) was chosen for this

study because extensive hydraulic geometry relations

are available through the work of Stall and Fok

(1968). In their research they developed descriptions

of flow parameters such as volume, velocity, depth,

top width, etc. for 166 locations within the IRB and

characterized these properties in a Horton–Strahler

framework.

Fig. 3. Location of the Illinois River Basin (IRB) and the eight major basins located within the IRB.

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257 241

Most of the IRB is located in Illinois, with

small portions in Wisconsin, Indiana, and Michigan

(see Fig. 3). Physiographically, the IRB lies within the

Central Lowland Province, including the Great Lakes

and the Till Plains Section. Landforms in this

watershed are typically the result of glaciation, and

the shallow aquifers consist of sand, gravel,

and glacial till, while the deep, bedrock aquifers are

comprised of dolomite, limestone, and sandstone.

Elevations ranges from 130 to 350 m above sea level

and the greatest relief, located along the river valley,

is between 120 and 200 m. The Illinois River has a

large, low relief flood plain that ranges from 5 to

10 km in width and encompasses numerous backwater

lakes and wetlands. The flood plain is a product of

glacial drainage during the Pleistocene and a portion

of it is the ancestral path of the Mississippi River.

Approximately 10 million people live in the IRB.

The majority of the land use (80%) is agricultural and

is devoted to growing soybeans and corn. Urban areas

account for the next largest land use, and forests,

wetlands, water, and barren areas follow in decreasing

percentages. The climate of the IRB is classified as

humid continental, hence, the summers are generally

hot and humid and the winters are cold and dry.

The average annual precipitation ranges from 80 to

100 cm. There are nine major river basins within the

IRB, eight of which are analyzed in this research

(Table 1 and Fig. 3). It should also be noted that there

are channel dams on rivers in the IRB that could lead

to deviations of flow velocities from those calculated

using the hydraulic geometry in Stall and Fok (1968)

due to flow retardation and storage effects.

3.2. Validation of DEM-extracted stream network

For the purposes of analysis and comparison, a

stream network of the IRB was obtained from the

Environmental Protection Agency (EPA) (1995).

The river network consists of Version 3 of the River

Reach File (RF3) and is a hydrographic database of

the surface waters of the United States. The first

version of the Reach File (RF1) was created by

digitizing the blue lines on USGS quadrangle maps.

The second version of the Reach File (RF2) main-

tained the integrity of the original version, yet added

tributaries from the USGS Geographic Names

Information System (GNIS). These additions were

limited to streams of at least 3 miles in length and

more than one-half mile from an existing RF1

intersection. RF2 doubled the number of stream

miles found in RF1. The third version of the Reach

File (RF3) kept stream reach designations from earlier

reach file versions, and incorporated USGS 1:100,000

scale Digital Line Graph (DLG) data. This created an

endpoint at every intersection, and included

geographic features such as roads, map edges, and

stream confluences. The level of detail added in RF3

made it a viable tool for Geographic Information

Systems (GISs) applications. The RF3 network,

hereto referred to as the EPA network, was chosen

because it contains the USGS digitized quadrangle

network, as well as the additional above-mentioned

information.

The eight river basins studied in this research were

extracted from a 3-arc-second DEM produced by

USGS (Rocky Mountain Communications Inc., 1995)

using the space-filling algorithm for flat surfaces

(imposed-gradient algorithm) developed by Garbrecht

and Martz (1997). This algorithm was chosen because

of the significantly flat topography of Illinois and,

compared to the traditional D-8 flow-routing algor-

ithms (Fairchild and Leymarie, 1991; Tribe, 1992),

the imposed-gradient method greatly reduces the

parallel-flow problem associated with flat areas.

The foundation of the imposed-gradient algorithm is

that drainage is generally away from higher terrain

and towards lower terrain, therefore gradients are

imposed in this manner. After the stream networks

were extracted, pruning was performed to eliminate

the smallest ‘streams’ which most likely do not

correspond to channels on the landscape. The first and

Table 1

Information on the eight major basins within the Illinois River Basin

used in this study

Drainage

basin

Order Drainage

area (km2)

Maximum

elevation (m)

Minimum

elevation (m)

Des Plaines 6 897 254 154

Mackinaw 6 3476 290 135

Vermilion 6 2697 259 152

Fox 7 3755 350 145

La Moine 7 2846 239 132

Spoon 7 3283 280 133

Kankakee 8 15,913 272 154

Sangamon 8 26,327 280 146

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257242

second order streams from the space-filling network

were pruned, thus a third order stream in the

space-filling network became a first order stream in

the pruned network. Also, to be consistent with the

EPA’s identification of channel heads, the Horton–

Strahler order threshold of the EPA network was

evaluated using the extracted network as a measure.

This task was realized by overlaying the EPA network

with the computer-extracted, pruned network

obtained using the DEM data. The Horton–Strahler

order threshold for the EPA network is approximately

third order, as compared to the extracted network.

Therefore, this analysis focuses on channels of order

three and greater of the pruned, extracted networks.

The Mackinaw River Basin, as opposed to the

entire IRB, was chosen to validate the DEM-extracted

stream network so that possible discrepancies could

be examined in greater detail. The river network for

the Mackinaw River Basin derived from the DEM was

compared to the EPA network by overlaying the EPA

stream network with the extracted network. The

derived network corresponded closely to the EPA

network (Fig. 4).

3.3. Estimation of hydrodynamic and geomorphologic

parameters

To study the variation of DL and DG over spatial

scales for each of the eight basins within the IRB, the

dispersion coefficients were computed for subbasins

of different order. According to Eqs. (12) and (18), the

following parameters are required to compute DL and

DG : Varg½ �Lg�; Eg½ �Lg�; S; u; and hp: For each basin

studied and their subbasins, the first three parameters

were calculated directly from the network extracted

using the DEM data. The slope (elevation drop/along

channel length) of the highest order stream within

each subbasin was used as the reference value of S:

There are two possible ways in which u and hp can

be computed in a nested basin structure. In either case,

we assume a uniform rainfall over the entire

watershed. Consider Fig. 5, where Basin B is nested

within Basin A and the outlet of Basin A (point y) is

downstream of the outlet of Basin B (point x).

Note that the hydrographs obtained using the GIUH

approach at outlets x and y will have different return

periods. Let us assume that in Case I, where the

uniform rainfall rate is i1; the frequencies of

the hydrographs at outlets x and y will be F1 and F2;

respectively. Let us also assume that in Case II, where

the uniform rainfall rate is i2; the frequencies of the

hydrographs at outlets x and y will be F2 and F3;

respectively. One method of computing u and hp; and

thus computing and comparing the dispersion coeffi-

cients DL and DG; is to use the same uniform rainfall

rate for the two nested subbasins. In this scenario,

flows with different frequencies will be compared

(i.e. the hydrographs at outlets x and y in Case I).

Another method of computing u and hp would be to

use different uniform rainfall rates that correspond to

the same flow frequencies for the two nested

subbasins (i.e. the hydrographs at outlet y from Case

I and at outlet x from Case II). The latter method is

adopted in this research for two reasons. First, u and

hp; as a function of flow frequency and order,

have been estimated for streams throughout the IRB

in the form of hydraulic geometry relations (Stall and

Fok, 1968). Second, comparing u and hp for the same

flow frequency is more realistic than comparing u and

hp for the same uniform rainfall rate, particularly for

large basins. Meaning, for the same uniform rainfall

rate, the flow (i.e. discharge, u; and hp) at different

order subbasins will not have the same frequency of

occurrence and will increase dramatically with scale

to the point of becoming unrealistic.

Stall and Fok (1968) assembled and presented data

from 166 USGS stream gaging stations in Illinois,

which were used to define downstream hydraulic

geometry relations for these streams. The discharge Q

was related to the frequency of occurrence of a

particular flow F and to the order of the stream v

using a linear multiple regression model

ln Q ¼ a2 bF þ fv ð20Þ

where a; b; and f are empirical constants.

This equation is only valid for F ¼ {0:1;…; 0:9}

due to the increased uncertainty outside this range.

The Horton–Strahler relationship between upstream

drainage area Av and stream order v; ln Av ¼ p þ qv;

was utilized to derive an equation for the discharge in

terms of frequency of occurrence and drainage area

ln QðF;vÞ ¼ a2fp

q2 bF þ

f

qln Av ð21Þ

where p and q are empirical constants. The p and q

values presented by Stall and Fok (1968) were revised

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257 243

Fig. 4. Illustration of the validity of the DEM-extracted stream network using an overlay with the EPA stream network for the Mackinaw River Basin. The flow is from east to west

towards the confluence with the Illinois River.

A.B

.W

hite

eta

l./

Jou

rna

lo

fH

ydro

log

y2

88

(20

04

)2

37

–2

57

24

4

by estimating new values of p and q using the drainage

areas extracted directly from the DEM. Utilizing the

dependence of flow velocity up and flow depth hp on

the discharge QðF;vÞ; Stall and Fok (1968) developed

the following relationships

ln upðF;vÞ ¼ a 2 bF þ c ln Av ð22Þ

ln hpðF;vÞ ¼ g 2 iF þ j ln Av ð23Þ

where a; c; g; and j are empirical constants dependent

upon a; f; p; and q; and the empirical constants b and

i are dependent upon b: Thus, using the p and q values

obtained from the DEM data and the values for a and

f from the published results of Stall and Fok (1968),

a; c; g; and j were recalculated. The values for b and i

were not altered from those published by Stall and

Fok (1968) because they are independent of p and q:

Table 2 presents the values for the empirical constants

in Eqs. (22) and (23) for the eight river basins studied.

These equations are not dimensionless, hence Av has

units of square miles, up is in feet/s, and hp has units of

feet.

Eqs. (12) and (17) were used to calculate the

hydrodynamic and geomorphologic dispersion

coefficients for different flow frequencies F and

subbasin orders v: Eqs. (22) and (23) provide the

dependence upon the frequency. For clarity, for each

basin B [ {DesPlaines;Mackinaw;Vermilion; Fox;

LaMoine;Spoon;Kankakee;Sangamon}; we denote

the dispersion coefficients as DBLðF;vÞ and DB

GðF;vÞ:

Note that for a selected basin B of order V; there are

several subbasins of order v , V over which an

average dispersion coefficient for each frequency can

be computed. The variables DBLðF;vÞ and DB

GðF;vÞ

represent this average for a specified v:

The average flow depths hp and flow velocities up

for each order subbasin of the eight major basins were

calculated using Eqs. (22) and (23) and the parameters

in Table 2. The mean slopes, drainage areas, stream

lengths, and number of tributaries of order i draining

into streams of order j (ni;j as defined in Eq. (8)) were

extracted directly from the DEM. The coefficients

DBLðF;vÞ and DB

GðF;vÞ were calculated for each of the

eight river basins, for each of the lower order

subbasins within the eight major basins, i.e. orders

v ¼ {3;…; 8}; and for flow frequencies

F ¼ {0:1; 0:3;…; 0:9}: Note that all eight basins

contain subbasins of orders v ¼ {3;…; 6}; five of

the basins contain subbasins of order v ¼ 7; and two

of the basins contain subbasins of order v ¼ 8

(Table 1). To avoid inaccuracies due to random

deviations from Horton’s laws and to account for the

fact that only two eighth order subbasins exist in the

IRB, regression equations for the mean slopes,

drainage areas, and stream lengths as a function of

subbasin order were obtained using a least squares fit

for each of the eight basins (Fig. 6).

4. Results

4.1. Dispersion coefficients

The averages of the dispersion coefficients

DBLðF;vÞ and DB

GðF;vÞ for each order v and various

flow frequencies F were calculated over the eight

Table 2

Empirical constants a; b; c; g; i; and j from flow depth and flow

velocity hydraulic geometry equations (Eqs. (22) and (23)) for the

eight basins within the IRB

Drainage basin a b c g i j

Fox 0.57 1.39 0.11 0.03 1.55 0.22

Des Plaines 0.28 1.31 0.09 0.00 2.05 0.24

Kankakee 20.27 1.19 0.17 20.11 2.25 0.34

Vermilion 0.14 2.19 0.12 20.13 2.71 0.24

Mackinaw 0.42 2.26 0.08 20.38 3.13 0.30

Spoon 0.71 1.63 0.06 20.15 2.05 0.33

La Moine 0.11 1.16 0.11 0.39 3.11 0.34

Sangamon 20.73 0.95 0.23 0.32 2.28 0.23

Fig. 5. Illustration of two different scenarios for comparing u and hp:

The uniform rainfall rates over the basin are different in Cases I and

II, yet the return period of the hydrograph at outlet y in Case I is

equivalent to that of outlet x in Case II.

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257 245

Fig. 6. Illustration of Hortonian dependence using the regression of drainage area Av; channel bed slope Sv; and stream length Lv with respect to

order v for the eight river basins studied.

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257246

basins B to evaluate the spatial consistency of the

hydrologic response throughout the IRB

DLðF;vÞ ¼ EB½DBLðF;vÞ� ð24Þ

DGðF;vÞ ¼ EB½DBGðF;vÞ� ð25Þ

where EB indicates averaging over the basins B

(Tables 3 and 4).

The semilog plots of DLðF;vÞ versus order v for

different frequencies F shows that curves of DLðF;vÞ

are not straight lines, but tend toward linearity, which

corresponds to logarithmic relations with respect to

order (Fig. 7). This result is expected to a certain

extent because of the logarithmic nature of Horton’s

laws and the hydraulic geometry equations.

The dispersion coefficient increases logarithmically

as the order increases. Also, as frequency decreases,

the curves of DLðF;vÞ maintain fairly uniform

spacing. The log-normal plots of DGðF;vÞ versus

order v for various frequencies F (Fig. 8) are similar

to those of DLðF;vÞ (Fig. 7), yet the curves are not

quite as linear. As order increases, DGðF;vÞ

increases; however, the curves begin to deviate

markedly from linearity for v . 6:

Log-linear plots of DLðF;vÞ versus frequency F

for various orders v are shown in Fig. 9. The curves

are straight lines, revealing that DLðF;vÞ varies

logarithmically with respect to frequency.

The coefficient DLðF;vÞ increases with order, but

the spacing between the curves is not uniform—a

result consistent with the deviation from linearity in

Fig. 7. Additionally, as frequency increases, DLðF;vÞ

decreases. The semilog plot of DGðF;vÞ versus

frequency F for various orders v (Fig. 10) are similar

to those of DLðF;vÞ (Fig. 9) in that the curves are

linear and the coefficient DGðF;vÞ decreases with

Table 3

Hydrodynamic dispersion coefficient DLðF;vÞ^ 1SD averaged over the eight basins within the IRB for various frequencies F and orders v

F Order

3 4 5 6 7 8

0.1 27.1 ^ 10.3 78.5 ^ 27.0 230.2 ^ 75.7 682.8 ^ 234.7 2246.4 ^ 843.3 9444.1 ^ 606.9

0.2 18.3 ^ 7.0 53.0 ^ 18.5 155.4 ^ 52.7 461.0 ^ 164.3 1573.9 ^ 590.7 6761.7 ^ 363.8

0.3 12.4 ^ 5.0 36.0 ^ 13.5 105.6 ^ 39.0 313.3 ^ 120.9 1104.7 ^ 419.5 4841.7 ^ 209.8

0.4 8.5 ^ 3.7 24.6 ^ 10.3 72.2 ^ 29.7 214.2 ^ 91.2 776.8 ^ 301.2 3467.2 ^ 113.9

0.5 5.8 ^ 2.8 16.9 ^ 7.9 49.6 ^ 22.8 147.3 ^ 69.2 547.1 ^ 218.3 2483.3 ^ 55.5

0.6 4.0 ^ 2.2 11.7 ^ 6.0 34.3 ^ 17.5 101.8 ^ 52.5 386.0 ^ 159.1 1778.7 ^ 21.1

0.7 2.8 ^ 1.6 8.1 ^ 4.6 23.9 ^ 13.3 70.7 ^ 39.6 272.8 ^ 116.5 1274.2 ^ 1.7

0.8 2.0 ^ 1.2 5.7 ^ 3.5 16.7 ^ 10.1 49.3 ^ 29.8 193.1 ^ 85.6 912.9 ^ 8.4

0.9 1.4 ^ 0.9 4.0 ^ 2.6 11.7 ^ 7.6 34.5 ^ 22.2 136.9 ^ 62.9 654.1 ^ 12.9

Table 4

Geomorphologic dispersion coefficient DGðF;vÞ^ 1SD averaged over the eight basins within the IRB for various frequencies F and orders v

F Order

3 4 5 6 7 8

0.1 37.5 ^ 13.6 135.7 ^ 58.6 441.7 ^ 152.6 1138.5 ^ 426.6 1621.5 ^ 537.5 8418.8 ^ 534.9

0.2 31.8 ^ 10.4 115.2 ^ 45.1 376.2 ^ 115.2 969.1 ^ 334.9 1430.3 ^ 475.2 7559.3 ^ 389.9

0.3 27.1 ^ 8.0 98.0 ^ 35.0 321.1 ^ 87.5 826.8 ^ 267.9 1262.3 ^ 422.1 6788.5 ^ 268.8

0.4 23.1 ^ 6.2 83.7 ^ 27.7 274.7 ^ 67.5 707.0 ^ 220.0 1114.5 ^ 376.6 6097.2 ^ 168.4

0.5 19.7 ^ 5.0 71.6 ^ 22.5 235.6 ^ 53.7 606.0 ^ 186.5 984.6 ^ 337.5 5477.1 ^ 85.5

0.6 16.9 ^ 4.1 61.4 ^ 18.9 202.4 ^ 44.7 520.6 ^ 163.3 870.2 ^ 303.7 4920.7 ^ 17.8

0.7 14.5 ^ 3.6 52.7 ^ 16.5 174.3 ^ 39.1 448.3 ^ 147.0 769.5 ^ 274.2 4421.5 ^ 37.0

0.8 12.5 ^ 3.3 45.4 ^ 14.9 150.4 ^ 35.6 386.8 ^ 134.9 680.8 ^ 248.5 3973.5 ^ 81.0

0.9 10.8 ^ 3.0 39.2 ^ 13.7 130.1 ^ 33.4 334.4 ^ 125.4 602.6 ^ 225.8 3998.2 ^ 115.6

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257 247

increasing frequency. As order increases, the curves

of DGðF;vÞ shift upward, but the spacing is much

more irregular than the curves of DLðF;vÞ—a result

consistent with the large deviation from linearity in

Fig. 8.

To visualize how the dispersion coefficients,

frequency, and order interact, the variables were

plotted in three-dimensions as a semilog plot of

DLðF;vÞ versus frequency F and order v (Fig. 11).

The relation among these variables is essentially

planar and incorporates the characteristics of Figs. 7

and 9, i.e. the coefficient DLðF;vÞ increases with

decreasing frequency and increasing order.

The semilog plot relating DGðF;vÞ to frequency F

and order v (Fig. 12) deviates from a uniform plane,

reflecting the deviation from linearity in Fig. 8.

In general, values of DGðF;vÞ are larger than those of

DLðF;vÞ; particularly for high frequencies and small

orders, yet the slope of the surface of DLðF;vÞ is

slightly greater than the slope of the surface of

Fig. 7. Hydrodynamic dispersion coefficient DLðF;vÞ averaged over the eight basins within the IRB versus order v for various frequencies F:

Fig. 8. Geomorphologic dispersion coefficient DGðF;vÞ averaged over the eight basins within the IRB versus order v for various frequencies F:

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257248

DGðF;vÞ; indicating that DLðF;vÞ increases at a

slightly faster rate than DGðF;vÞ with increasing

order and decreasing frequency.

4.2. Relative contribution of dispersion mechanisms

To determine the relative contributions of the

hydrodynamic and geomorphologic dispersion mech-

anisms to the hydrologic response, a ratio of the two

dispersion coefficients was calculated (Table 5):

DRðF;vÞ ¼DGðF;vÞ

DLðF;vÞð26Þ

The plot of DRðF;vÞ versus order v for various

frequencies F shows that the ratio DRðF;vÞ initially

increases with order, reaching a peak at order five,

then rapidly decreases up to order seven where it

begins to increase again (Fig. 13). As frequency

decreases, DRðF;vÞ decreases, and the crests and

Fig. 9. Hydrodynamic dispersion coefficient DLðF;vÞ averaged over the eight basins within the IRB versus frequency F for various orders v:

Fig. 10. Geomorphologic dispersion coefficient DGðF;vÞ averaged over the eight basins within the IRB versus frequency F for various orders v:

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257 249

troughs of the curves become less pronounced.

The curves of DRðF;vÞ versus frequency F for

different orders v increase exponentially with respect

to increasing frequency (Fig. 14); however,

the relationship of the ratio DRðF;vÞ to order is

more complex. Beginning with order three, the curves

of DRðF;vÞ increase for orders three to five, decrease

for orders six and seven, and then increase slightly for

Fig. 11. Three-dimensional semilog plot of the hydrodynamic dispersion coefficient DLðF;vÞ averaged over the eight basins within the IRB

versus frequency F and order v:

Fig. 12. Three-dimensional semilog plot of the geomorphologic dispersion coefficient DGðF;vÞ averaged over the eight basins within the IRB

versus frequency F and order v:

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257250

order eight, reflecting the pattern portrayed in Fig. 13.

The ratio DRðF;vÞ was also plotted in three-dimen-

sions versus frequency F and order v; as shown in

Fig. 15. The tendencies of Figs. 13 and 14 are evident,

as the complex surface increases exponentially with

respect to frequency and follows the same pattern of

crests and troughs with respect to order. It is important

to note that the results for the eighth order subbasins

might not be as reliable as those for lower orders,

due to the fact that the IRB contains only two eighth

order subbasins.

To better understand the relation between hydrodyn-

amic and geomorphologic dispersion mechanisms,

as manifested in Figs. 13–15, the hydraulic geometry

equations for flow depth hpðF;vÞ and flow velocity

upðF;vÞ were incorporated into the equations for

DLðF;vÞandDGðF;vÞ (Eqs. (12)and(18), respectively)

and an equation for the ratio DRðF;vÞ was derived.

Taking the antilogarithm of both sides of Eqs. (22) and

(23) and substituting u ¼ lup (Eq. (10)) to determine

kinematic wave celerity, the equations become

uðF;vÞ ¼ kAcv e2bF ð27Þ

hpðF;vÞ ¼ mAjv e2iF ð28Þ

where k ¼ ea=l and m ¼ eg are constants. Incorporating

uðF;vÞ and hpðF;vÞ into Eq. (12) gives

DLðF;vÞ ¼uðF;vÞhpðF;vÞ

3Sv

¼km

3j2

1

Sv

Acþjv e2ðbþiÞF

� �ð29Þ

where j is a conversion factor between English and SI

units. Substituting uðF;vÞ into Eq. (18) produces

DGðF;vÞ¼uðF;vÞVarvg ½ �Lg�

2Evg ½ �Lg�

¼k

2jðAc

v e2bFf vg ½ �Lg�Þ

ð30Þ

Table 5

Ratio DRðF;vÞ of geomorphologic DGðF;vÞ to hydrodynamic

DLðF;vÞ dispersion coefficient averaged over the eight basins

within the IRB for various frequencies F and orders v

F Order

3 4 5 6 7 8

0.1 1.47 1.79 2.11 1.89 0.96 0.89

0.2 1.86 2.27 2.65 2.38 1.17 1.12

0.3 2.35 2.88 3.35 3.00 1.42 1.40

0.4 2.99 3.68 4.25 3.80 1.73 1.76

0.5 3.80 4.71 5.40 4.83 2.11 2.21

0.6 4.85 6.04 6.88 6.17 2.58 2.77

0.7 6.21 7.78 8.80 7.89 3.17 3.47

0.8 7.97 10.05 11.29 10.13 3.90 4.35

0.9 10.26 13.02 14.52 13.05 4.81 5.46

Fig. 13. Ratio DRðF;vÞ of geomorphologic DGðF;vÞ to hydrodynamic DLðF;vÞ dispersion coefficient averaged over the eight basins within the

IRB versus order v for various frequencies F:

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257 251

where

f vg ½ �Lg�¼Varg½ �Lg�

Eg½ �Lg�

( )v

ð31Þ

is computed as the average for the order v subbasins.

Hence, the ratio of DGðF;vÞ to DLðF;vÞ is as follows

DRðF;vÞ ¼

k2jðAc

v e2bFf vg ½ �Lg�Þ

km3j2

1Sv

Acþjv e2ðbþiÞF

� �¼

3j

2mSvA2j

v eiFf vg ½ �Lg� ð32Þ

Fig. 14. Ratio DRðF;vÞ of geomorphologic DGðF;vÞ to hydrodynamic DLðF;vÞ dispersion coefficient averaged over the eight basins within the

IRB versus frequency F for various orders v:

Fig. 15. Three-dimensional plot of the ratio DRðF;vÞ of geomorphologic DGðF;vÞ to hydrodynamic DLðF;vÞ dispersion coefficient averaged

over the eight basins within the IRB versus frequency F and order v:

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257252

where Sv is dimensionless, Av has units of square

miles, and f vg ½ �Lg� has units of meters. This equation

reveals that the exponential tendency of the curves of

DRðF;vÞ versus frequency F (Fig. 14) is a result of

the term eiF in Eq. (32). Given that i is greater than

zero (see Table 2), eiF ; and thus DRðF;vÞ; is an

increasing exponential function of frequency. How-

ever, the more intriguing result is the relation

between DRðF;vÞ and order v (Fig. 13), for it

contradicts the hypothesis that at small scales, or low

orders, hydrodynamic dispersion dominates the

hydrologic response of a basin and at large scales,

or high orders, geomorphologic dispersion dominates

the hydrologic response of a basin. According to Fig.

13, geomorphologic dispersion dominates at all

scales, particularly for high frequency flows, but is

generally less dominant at high orders compared to

low orders. At high frequencies, i.e. small runoff

events, geomorphologic dispersion vastly dominates,

and at low frequencies, i.e. large runoff events, the

difference between the two dispersion mechanisms is

less pronounced. This result is counter to the notion

that the hydrodynamic dispersion dominates at

high frequencies and low orders, and that the

geomorphologic dispersion dominates at low

frequencies and high orders. Further, investigation

of Eq. (32) provides insight into the reasons for this

discrepancy.

For a constant frequency F; the following state-

ment can be made regarding DRðF;vÞ

DRðF;vÞ / DRðvÞ ¼f vg ½ �Lg�

ð1=SvÞAjv

ð33Þ

where the variables 1=Sv; Ajv; and f vg ½ �Lg� are

dependent upon the subbasin order v: To determine

which of these elements prevails and at what scale,

each term in Eq. (33) was analyzed (Table 6).

The magnitudes of 1=Sv and f vg ½ �Lg� are comparable,

whereas the magnitude of Ajv is relatively small.

Thus, 1=Sv and f vg ½ �Lg� are the two variables that have

the greatest influence on the relative contributions of

the two dispersion mechanisms to the hydrologic

response; however, Ajv also contributes to the

behavior of the ratio of DLðF;vÞ and DGðF;vÞ:

To portray this, 1=Sv; Ajv; f vg ½ �Lg�; and the ratio DRðvÞ

averaged over the eight basins was plotted versus

order v (Fig. 16). The curve of DRðvÞ in Fig. 16 is

similar to the curves of DRðF;vÞ in Fig. 13, in that the

values increase from third to fifth orders, decrease

from fifth to seventh orders, and increase from seventh

to eighth orders. The increasing trend from order three

to order five shows that the numerator of the ratio in

Eq. (33) f vg ½ �Lg� increases at a faster rate than the

denominator ð1=SvÞAjv: The decreasing trend from

order five to order seven portrays that the denominator

ð1=SvÞAjv is increasing more rapidly than the numer-

ator f vg ½ �Lg�: These trends can also be seen in the

curves of 1=Sv; Ajv; and f vg ½ �Lg�: From the above

discussion it is evident that, for the IRB, the governing

factor in determining which dispersion mechanism

influences the hydrologic response is the geomorpho-

logic parameter f vg ½ �Lg�; for the inverse of the

slope 1=Sv and the drainage area Ajv increase

exponentially with respect to order v; yet f vg ½ �Lg�

does not increase exponentially throughout the entire

range of scales.

4.3. Comparison with previous work

Robinson et al. (1995) compared the roles that

hillslope processes, channel routing, and network

geomorphology play in the hydrologic response of

natural watersheds. Their results are summarized in

Fig. 17, where the ratio DR is plotted versus basin

area. The results presented in this paper are also

plotted in this figure. One can see that,

approximately where Robinson et al.’s results end,

the results of this paper begin, for a majority of the

basins in this study are much larger than those

analyzed by Robinson et al. In addition, viewing both

sets of results in conjunction, one could hypothesize

from this plot that the ratio of geomorphologic to

hydrodynamic dispersion is generally exponentially

decreasing with scale, and asymptotically approach-

ing a value of approximately 2. It should be noted

that, although the ratio of the two dispersion

coefficients decreases, the geomorphologic dispersion

coefficient remains greater than that of the hydrodyn-

amic dispersion coefficient. Thus, it is concluded that

network geomorphology plays the dominant role in

the hydrologic response of a watershed at all

scales, which is in accordance with Robinson et al.’s

results.

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257 253

5. Summary and conclusions

This research has examined the relative

contributions of hydrodynamic and geomorphologic

dispersion mechanisms to the hydrologic response of

the IRB at different spatial scales. Utilizing Stall and

Fok (1968) extensive research on the hydraulic

geometry of rivers in Illinois and stream network

parameters extracted from DEM data, the hydrodyn-

amic and geomorphologic dispersion coefficients

Table 6

Characteristics of each order of the eight major basins within the Illinois River Basin used in this study

Subbasin Order Drainage area (km2) VarðLÞ (km) EðLÞ (km) Slope Depth (m) DR

Des Plaines 3 7.340 0.448 4.165 0.00273 0.134 3.294

4 36.428 3.753 9.626 0.00173 0.199 5.082

5 180.782 21.398 22.517 0.00110 0.298 5.267

6 897.170 89.805 54.089 0.00070 0.444 3.914

Mackinaw 3 32.718 1.236 7.908 0.00220 0.126 4.070

4 154.954 12.963 18.469 0.00135 0.192 7.374

5 733.871 51.146 44.899 0.00083 0.293 4.826

6 3475.645 396.583 100.562 0.00051 0.446 6.739

Vermilion 3 31.640 1.654 7.929 0.00185 0.158 3.655

4 139.245 8.324 17.394 0.00125 0.219 4.099

5 612.804 68.196 34.385 0.00085 0.304 8.306

6 2696.890 218.393 77.687 0.00058 0.421 5.756

Fox 3 6.685 0.354 3.646 0.00441 0.171 3.754

4 32.546 2.351 8.946 0.00281 0.243 4.558

5 158.447 19.230 20.151 0.00178 0.344 7.433

6 771.380 100.540 48.119 0.00114 0.487 7.309

7 3755.377 379.597 118.769 0.00072 0.690 5.021

La Moine 3 6.636 0.211 3.550 0.00456 0.121 3.366

4 30.198 1.637 7.885 0.00254 0.205 3.853

5 137.419 14.025 16.036 0.00141 0.349 5.317

6 625.338 63.176 32.256 0.00079 0.593 3.900

7 2845.665 148.981 76.580 0.00044 1.008 1.269

Spoon 3 6.363 0.349 3.296 0.00485 0.120 6.396

4 30.324 1.960 7.945 0.00286 0.205 5.174

5 144.519 12.273 18.703 0.00169 0.348 4.776

6 688.761 88.080 39.316 0.00100 0.592 5.656

7 3282.545 206.346 100.940 0.00059 1.006 1.790

Kankakee 3 8.248 0.634 4.991 0.00175 0.114 2.915

4 37.448 3.502 10.321 0.00113 0.194 2.956

5 170.024 21.795 19.970 0.00073 0.330 3.613

6 771.960 49.186 42.884 0.00047 0.560 1.442

7 3504.933 258.749 84.447 0.00030 0.952 1.464

8 15,913.471 1833.658 140.741 0.00020 1.616 2.365

Sangamon 3 32.556 1.795 8.916 0.00154 0.295 1.583

4 124.217 10.039 16.379 0.00100 0.385 2.395

5 473.957 35.128 30.014 0.00065 0.503 2.274

6 1808.401 143.522 50.514 0.00042 0.658 2.744

7 6900.026 297.518 97.506 0.00028 0.860 1.465

8 26,327.324 1643.590 175.780 0.00018 1.124 2.232

These variables were used to compute the ratio of geomorphologic to hydrodynamic dispersion DR:

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257254

were computed for eight of the major basins within

the IRB and their subbasins. The dispersion

mechanisms are related to the frequency of occur-

rence of a particular flow, as well as the order of the

basin. To perform scale analysis we used a constant

frequency, for this scenario is more realistic for large

basins and because of the availability of downstream

hydraulic geometry data as a function of frequency

in Stall and Fok (1968). It should be noted that the

method used to compute flow velocity and depth

could be limited by the size of the basins in this

study, which are quite large sixth to eighth order

basins. The DEM-extracted stream networks were

compared to the actual networks obtained from the

EPA (1995) to validate the use of parameters

derived from computer-extracted river networks.

Fig. 16. The ratio DRðvÞ (Eq. (33)), the inverse of the channel bed slope 1=Sv; the geomorphologic parameter f vg ½ �Lg�; (Eq. (31)) and the drainage

area Ajv averaged over the eight basins within the IRB versus order v:

Fig. 17. Comparison of the current results to Robinson et al.’s (1995) results for the ratio of geomorphologic to hydrodynamic dispersion DR

versus basin area.

A.B. White et al. / Journal of Hydrology 288 (2004) 237–257 255

Both dispersion coefficients increase with

increasing order and decreasing flow frequency,

however, the hydrodynamic dispersion coefficient

generally increases at a faster rate with respect to

order than the geomorphologic dispersion coefficient.

With respect to frequency, the ratio of the two

dispersion mechanisms increases exponentially.

With respect to order, this ratio increases for third

through fifth order subbasins, decreases for sixth and

seventh order subbasins, then increases for eighth

order subbasins. As frequency decreases, the crests

and troughs of the curves of the ratio become less

pronounced. The form of these curves indicates that

the relative influence of geomorphologic dispersion

on hydrologic response is greatest in low-order

watersheds at high flow frequencies. In high-order

watersheds, hydrodynamic dispersion plays an

increasingly important role with respect to the

hydrologic response; however, geomorphologic

dispersion is still the governing influence.

The parameters having the greatest influence on the

dispersion coefficients are the inverse of the mean

channel bed slope (a component of the hydrodynamic

dispersion) and the ratio of the variance to the mean of

the distribution of individual channel path lengths

computed over all possible paths in a basin

(a component of the geomorphologic dispersion).

The mean drainage area, a component of both

dispersion mechanisms, also influences the dispersion

coefficients, although the contribution of the drainage

area is smaller than the contribution of the inverse of

the slope and the variance to mean path lengths.

For low orders, or small scales, the rate at which the

variance to mean path lengths increases is greater than

the rate at which the inverse of the slope and the

drainage area increase. For high orders, or large

scales, the inverse of the slope and the drainage area

increase more rapidly than the variance to mean path

lengths. Thus, for the IRB, the governing factor in

determining which dispersion mechanism influences

the hydrologic response is the geomorphologic

parameter, or variance to mean path lengths, for the

inverse of the slope and drainage area increase

exponentially according to Horton’s Laws, while the

geomorphologic parameter does not.

These results have important implications in

hydrologic modeling. The focus of modeling efforts

in the past have been on measuring and surveying

channel characteristics such as flow depth, top

width, discharge, channel bed roughness, channel

cross-sectional area, etc. to determine the hydrologic

response of a river basin. The results presented in this

paper show that the hydrologic response tends to be

governed by geomorphologic dispersion at a variety

of spatial scales, hence, the network structure plays a

dominant role. Current models have substantial data

requirements that are challenging and expensive for

large or ungauged watersheds, but the network

structure parameters are relatively simple to extract

from DEMs, creating an environment in which the

hydrologic response could be reasonably approxi-

mated without elaborate, expensive field work. Other

types of studies could also be performed based on

these results, such as determining the affects human

modification of stream networks has on the hydrologic

response. For example, during the late 1800s in

Illinois, a large number of drainage ditches were

constructed upstream of the headwaters of the existing

river network in order to improve drainage from the

wetland-like prairies; however, the effects of this

process are generally unknown due to the lack of

hydraulic and hydrologic data prior to the ditches

being added. Thus, in this situation, one could

estimate the effects human modifications have had

on the stream network without surveys of the channel

cross-sections.

Acknowledgements

The support for this research has been provided by

Water Resources Center (USGS grant no. INT-96-

GR-02668) and National Science Foundation (grant

no. NSF EAR 97-06121).

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