Development of a Traveltime Prediction Equation for 

Streams in Arkansas

Prepared in cooperation with the

ARKANSAS DEPARTMENT OF HEALTH

U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2004-5064
Front Cover: Photograph of Rhodamine WT dye in the Cossatot River 3 miles upstream from the 

U.S. Geological Survey streamgaging station 07340300 Cossatot River near Vandervoort, Arkansas, February 2002 (photograph by U.S. Geological Survey).
Development of a Traveltime Prediction 

Equation for Streams in Arkansas

By Jaysson E. Funkhouser and C. Shane Barks

U.S. Department of the Interior

U.S. Geological Survey

Prepared in cooperation with the

ARKANSAS DEPARTMENT OF HEALTH

Scientific Investigations Report 2004-5064
U.S. Department of the Interior

Gale A. Norton, Secretary

U.S. Geological Survey

Charles G. Groat, Director

U.S. Geological Survey, Reston, Virginia: 2004

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Although this report is in the public domain, permission must be secured from the individual copyright owners to reproduce any copyrighted materials contained within this report.
CONTENTS

 iii

Abstract................................................................................................................................................................................1

Introduction .........................................................................................................................................................................1

Purpose and Scope.....................................................................................................................................................2

Location of Study Sites..............................................................................................................................................2

Acknowledgments .....................................................................................................................................................2

Dye-Cloud Dispersion Theory.............................................................................................................................................2

Longitudinal Dispersion ............................................................................................................................................2

Theory of Dye-Cloud Dispersion for Instantaneous Sources....................................................................................5

Previous USGS Traveltime Studies.....................................................................................................................................5

Traveltime Studies on Arkansas Streams ..................................................................................................................5

National Traveltime Studies ......................................................................................................................................5

Methods ...............................................................................................................................................................................7

Site Selection .............................................................................................................................................................7

Fluorescent Dye Concentration Determination .........................................................................................................7

Field Methods............................................................................................................................................................7

Previous Studies on Arkansas Streams............................................................................................................7

Present Study ...................................................................................................................................................8

Statistical Methods ....................................................................................................................................................9

Evaluation of the National Traveltime Prediction Equation Using the Arkansas Data Set........................... 10

Development of the Arkansas Traveltime Prediction Equation..................................................................... 10

Evaluation of the National Traveltime Prediction Equation Using the Arkansas Data Set ..............................................11

Development of the Arkansas Traveltime Prediction Equation ........................................................................................11

Traveltime of Peak Concentration ........................................................................................................................... 14

Traveltime of Leading Edge .................................................................................................................................... 14

Example Application .........................................................................................................................................................14

Summary............................................................................................................................................................................16

References .........................................................................................................................................................................17
iv

ILLUSTRATIONS

Figure 1. Map showing general location of sampling sites ............................................................................................... 3

2. Diagram showing lateral mixing and longitudinal dispersion patterns and changes of concentration

downstream from a single, midpoint, slug injection of tracer....................................................................... 5

3. Definition sketch of time-concentration curves along a selected streamline resulting from an instantaneous

dye injection .................................................................................................................................................. 6

4. Photograph showing injecting dye while wading across a stream ..................................................................... 8

5. Photograph showing injecting dye while boating across a stream..................................................................... 8

6. Photograph showing injecting dye into a stream from a bridge......................................................................... 8

7. Photograph showing lateral mixing of Rhodamine WT dye.............................................................................. 9

8. Photograph of the Self-Contained Underwater Fluorescence Apparatus (SCUFA) .......................................... 9

9. Plot showing relation between velocity predicted using the national traveltime prediction equation and

observed velocity for Arkansas data set ...................................................................................................... 11

10. Plot showing the relation between velocity predicted using the Arkansas traveltime prediction equation

and observed velocity for Arkansas data set............................................................................................... 15

11. Plot showing relation between traveltimes of leading edge and peak of dye cloud for the Arkansas

data set......................................................................................................................................................... 15

TABLES

Table 1. Sample sites and stream reaches used for traveltime study and their approximate locations

in Arkansas................................................................................................................................................ 4

2. Arkansas data used to evaluate the national traveltime prediction equation .................................................. 12

3. Ranges, means, and medians of explanatory variable values used to develop the Arkansas

traveltime prediction equation................................................................................................................. 14
Development of a Traveltime Prediction Equation for 

Streams in Arkansas

By Jaysson E. Funkhouser and C. Shane Barks

Abstract

During 1971 and 1981 and 2001 and 2003, traveltime measurements were made at 33 sample sites on 18 streams throughout northern and western Arkansas using fluorescent dye. Most 

measurements were made during steady-state base-flow conditions with the exception of three measurements made during 

near steady-state medium-flow conditions (for the study 

described in this report, medium-flow is approximately 100-

150 percent of the mean monthly streamflow during the month 

the dye trace was conducted). These traveltime data were compared to the U.S. Geological Survey’s national traveltime prediction equation and used to develop a specific traveltime prediction equation for Arkansas streams.

In general, the national traveltime prediction equation 

yielded results that over-predicted the velocity of the streams 

for 29 of the 33 sites measured. The standard error for the 

national traveltime prediction equation was 105 percent. The 

coefficient of determination was 0.78. 

The Arkansas prediction equation developed from a 

regression analysis of dye-tracing results was a significant 

improvement over the national prediction equation. This regression analysis yielded a standard error of 46 percent and a coefficient of determination of 0.74. The predicted velocities using 

this equation compared better to measured velocities. 

Using the variables in a regression analysis, the Arkansas 

prediction equation derived for the peak velocity in feet per second was: 

Vp 0.0731 Q

Da

------ ⎝ ⎠ ⎛ ⎞ 0.664

× S

–0.377 = × × 1.074

where,

Q is discharge at point of interest, in cubic feet per 

second;

Da is drainage area at point of interest, in square miles; 

and

S is slope from point of injection of point of interest, 

in foot per foot.

In addition to knowing when the peak concentration will 

arrive at a site, it is of great interest to know when the leading 

edge of a contaminant plume will arrive. The traveltime of the 

leading edge of a contaminant plume indicates when a potential 

problem might first develop and also defines the overall shape 

of the concentration response function.

Previous USGS reports have shown no significant relation 

between any of the variables and the time from injection to the 

arrival of the leading edge of the dye plume. For this report, the 

analysis of the dye-tracing data yielded a significant correlation 

between traveltime of the leading edge and traveltime of the 

peak concentration with an R2 value of 0.99. These data indicate that the traveltime of the leading edge can be estimated 

from:

Tl 0.79 Tp = ×

where,

Tl is traveltime of the leading edge, in hours, and 

Tp is traveltime of peak concentration, in hours.

Introduction

The 1996 Amendments to the Safe Drinking-Water Act 

require that each state prepare a source-water assessment for all 

public water supplies. States are required to determine the 

sources of drinking water, the origin of contaminants monitored, or the identification of the potential contaminants to be 

monitored, and the intrinsic susceptibility of the water supplies 

to these contaminants (Arkansas Department of Health, 2002). 

In Arkansas, source-water protection is the responsibility of the 

Arkansas Department of Health (ADH).

There are more than 1,500 public drinking-water sources 

in Arkansas (T.W. Holland, U.S. Geological Survey, oral commun., 2001). Of these, more than 100 use surface-water sources 

that serve nearly one million people. These surface-water 

sources include free-flowing rivers, reservoirs, and springs that 

are susceptible to potential sources of contamination (PSOC’s) 

that may be located within or near the area influencing the water 

source. Nearly half of the population served by these surface-
2 Development of a Traveltime Prediction Equation for Streams in Arkansas

water sources are located within rural and undeveloped areas of 

the State. 

An important aspect of PSOC’s for surface-water sources 

is the time it takes for a contaminant to reach the intake structure 

of a drinking-water supply from the point of input (spill or leak). 

Public-health officials often need to decide when and how long 

to suspend operations of public water-supply intakes in the 

reach downstream from a spill or leak. Stream velocity, which 

can be obtained from traveltime data, is a streamflow characteristic that water-resource managers and planners can use to predict the rate of movement of contaminants that may be introduced into the stream (Jobson, 1996). Fluorescent dyes can be 

used to measure a stream’s velocity and gain a better understanding of the potential rate of movement of contaminants.

To address the need for traveltime data to estimate rate of 

movement of contaminants in Arkansas streams, the U.S. Geological Survey (USGS) conducted a study in cooperation with 

the ADH during 2001 and 2003. These data, along with data 

collected during Lamb’s (1983) study, were combined to 

develop an equation that can be used to estimate the average 

velocity of an Arkansas stream.

Purpose and Scope

This report presents traveltime data collected at 33 sample 

sites on 18 different streams in Arkansas during two studies 

based on dye-tracing techniques. The first study was conducted 

between 1971 and 1981 (Lamb, 1983). The present (2003) 

study was conducted between 2001 and 2003 by the USGS in 

cooperation with the ADH. The data collected from both of 

these studies has been combined for this report and are hereafter 

referred to as the Arkansas data set. 

The Arkansas data set was used to evaluate the USGS 

national traveltime prediction equation (hereafter referred to as 

the national traveltime prediction equation) (Jobson, 1996) for 

use in Arkansas and to develop a traveltime prediction equation 

specific to Arkansas streams. Verification and calibration error 

statistics are presented for the national traveltime equation. Calibration coefficients are presented for a traveltime prediction 

equation using only the Arkansas data set. An example demonstrating the traveltime prediction equation for Arkansas is presented in this report.

Location of Study Sites

Eighteen stream reaches were chosen for this study that are 

located in the northern and western areas of Arkansas (fig. 1, 

table 1). The boundary for the study area extended from Randolph County in northern Arkansas to Benton County in northwestern Arkansas and to Polk County in western Arkansas (fig. 

1).

Acknowledgments

Gratitude is expressed to the residents and water-treatment 

plant managers that allowed the USGS to access the streams for 

data-collection purposes. Gratitude also is expressed to the 

ADH for their assistance in helping the USGS identify the 

streams in Arkansas to be studied.

Dye-Cloud Dispersion Theory

The traveltime and longitudinal-dispersion of chemical 

constituents in a stream vary with flow conditions. Therefore, 

developing accurate traveltime and longitudinal-dispersion 

characteristics over a range of flow conditions is critical to 

understanding the transport of chemicals in a stream reach. Dye 

tracing provides one of the best methods of predicting stream 

traveltimes and what may happen to conservative contaminants 

that are introduced into a stream reach.

Longitudinal Dispersion

Longitudinal dispersion is the process whereby a mass of 

solute introduced into a flowing stream is mixed and diluted in 

the longitudinal, or downstream direction (Nordin and Sabol, 

1974; Gurdak and others, 2002). The response to the slug injection of a soluble tracer is assumed to imitate the characteristics 

of a soluble contaminant. An understanding of how tracers mix 

and disperse in a stream is essential to understanding their application in simulating contaminant transport. This report will provide a brief description on the theory of dye-cloud dispersion 

(traveltime) for instantaneous sources, but a detailed description of this theory can be found in Hubbard and others (1982), 

Kilpatrick (1993) and Kilpatrick and Wilson (1989).

The dispersion and mixing of a tracer in a receiving stream 

take place in all three dimensions of the channel (fig. 2)—vertical, lateral, and longitudinal. Vertical mixing (throughout the 

depth of the stream) is normally completed rather rapidly, 

within a distance of a few river depths. Lateral mixing is much 

slower, but is usually completed within a few miles downstream. Longitudinal dispersion continues indefinitely because 

there are no physical boundaries (Hubbard and others, 1982; 

Lamb, 1983; Jobson, 1996). Longitudinal dispersion is the dispersion component of primary interest in this report. 
!( !(

!(

!(

!(

!(

!(

!( !(

!(

!(

!(

!(

!(

!(

!(

!(

!(

POLK

BENTON

RANDOLPH

5

8

2 1

26

33

25

24

20

18 19

6-7

3-4

9 - 12

29 - 32

21 - 23

27 - 28

13 - 17

94° 93° 92° 91° 90°

33°

34°

35°

36°

Base from U.S. Geological Survey

digital data, 1:100,000

!

EXPLANATION

Sampling site and number

(Table 1)

Study area boundary

0 50 Miles

0 25 50 Kilometers

25

 Dye-Cloud Dispersion Theory 3

Figure 1. General location of sampling sites.
Table 1. Sample sites and stream reaches used for traveltime study and their approximate locations in Arkansas.

[Numbers in parentheses indicate the stream reach relative to a reference gage. Negative numbers in parentheses represent distance upstream from the reference 

gage on the stream, in miles; positive numbers in parentheses represent distance downstream from the reference gage on the stream, in miles; Reference gage is a 

U.S. Geological Survey streamflow gaging station with the associated station identification number]

Sample

site

number

(fig. 1) Stream name Location of nearest town

Location of stream reach with respect to

reference gage

Station

identification

number

1 War Eagle Creek Hindsville, Arkansas (-12.8, -9.9) War Eagle Creek near Hindsville 07049000

2 Crooked Creek Yellville, Arkansas (-3.9, 0.0) Crooked Creek at Yellville 07055608

3 Buffalo River St. Joe, Arkansas (-8.3, 4.2) Buffalo River near St. Joe 07056000

4 Buffalo River St. Joe, Arkansas (-8.3, 4.2) Buffalo River near St. Joe 07056000

5 North Sylamore Creek Fifty Six, Arkansas (-4.2, 4.8) North Sylamore Creek near Fifty Six 07060710

6 Black River Pocahontas, Arkansas (0.0, -15.9) Black River at Pocahontas 07069000

7 Black River Pocahontas, Arkansas (0.0, -15.9) Black River at Pocahontas 07069000

8 Eleven Point River Ravenden Springs, Arkansas (-8.9, 0.0) Eleven Point River near Ravenden Springs 07072000

9 Osage Creek Elm Springs, Arkansas (4.0, 8.4) Osage Creek near Elm Springs 07195000

10 Osage Creek Elm Springs, Arkansas (-5.2., 0.0) Osage Creek near Elm Springs 07195000

11 Osage Creek Elm Springs, Arkansas (0.0, 8.4) Osage Creek near Elm Springs 07195000

12 Osage Creek Elm Springs, Arkansas (-11.1, -7.0) Osage Creek near Elm Springs 07195000

13 Illinois River Siloam Springs, Arkansas (-6.9, 0.0) Illinois River near Siloam Springs 07195400

14 Illinois River Siloam Springs, Arkansas (-18.1, -14.1) Illinois River near Siloam Springs 07195400

15 Illinois River Siloam Springs, Arkansas (-25.2, -22.6) Illinois River near Siloam Springs 07195400

16 Illinois River Siloam Springs, Arkansas (-6.0, 2.2) Illinois River near Siloam Springs 07195400

17 Illinois River Siloam Springs, Arkansas (-14.1, -9.1) Illinois River near Siloam Springs 07195400

18 Lee Creek Short, Oklahoma (-8.5, 0.0) Lee Creek near Short 07249985

19 Mulberry River Mulberry, Arkansas (-14.1, 0.0) Mulberry River near Mulberry 07252000

20 Spadra Creek Clarksville, Arkansas (-9.0, 0.0) Spadra Creek at Clarksville 07256500

21 Illinois Bayou Scottsville, Arkansas (0.0, 9.0) Illinois Bayou near Scottsville 07257500

22 Illinois Bayou Scottsville, Arkansas (-7.4, 0.0) Illinois Bayou near Scottsville 07257500

23 Illinois Bayou Scottsville, Arkansas (-7.4, 9.0) Illinois Bayou near Scottsville 07257500

24 Petit Jean River Booneville, Arkansas (-1.1, 9.8) Petit Jean River near Booneville 07258500

25 Maumelle River Williams Junction, Arkansas (-6.0, 0.0) Maumelle River at William’s Junction 07263295

26 Cossatot River Vandervoort, Arkansas (-3.4, 9.4) Cossatot River near Vandervoort 07340300

27 Ouachita River Mount Ida, Arkansas (-9.6, 0) Ouachita River near Mount Ida 07356000

28 Ouachita River Mount Ida, Arkansas (-9.6, 0) Ouachita River near Mount Ida 07356000

29 Caddo River Caddo Gap, Arkansas (7.8, 19.8) Caddo River at Caddo Gap 07359610

30 Caddo River Caddo Gap, Arkansas (-8.0, 7.8) Caddo River at Caddo Gap 07359610

31 Caddo River Caddo Gap, Arkansas (7.8, 19.8) Caddo River at Caddo Gap 07359610

32 Caddo River Caddo Gap, Arkansas (-8.0, 15.0) Caddo River at Caddo Gap 07359610

33 Alum Fork Saline River Reform, Arkansas (-6.0, -5.3) Alum Fork Saline River near Reform 07362587

4 Development of a Traveltime Prediction Equation for Streams in Arkansas
Lateral

mixing

;

;

;

;

Longitudinal

dispersion

Previous USGS Traveltime Studies 5

Figure 2. Lateral mixing and longitudinal dispersion patterns and changes of concentration downstream from a single, midpoint, slug injection of tracer 

(modified from Kilpatrick and Wilson, 1989, p. 2).

Theory of Dye-Cloud Dispersion for Instantaneous 

Sources

Figure 2 shows that for a midpoint slug injection, the tracer 

cloud typically moves faster than the mean stream velocity 

upstream from an optimum section distance (section III in fig. 

2) because the bulk of the tracer is injected in the high velocity 

part of the cross section. For the present (2003) study, all of the 

measurement cross sections were located at least as far downstream as the optimum distance (explanation included later in 

report) (section III in fig. 2) so that longitudinal dispersion was 

the dominant mixing process. 

The response curve at any point downstream from an 

instantaneous dye injection is normally represented by plotting 

concentration in relation to elapsed time (fig. 3). The response 

curves, defined by the analysis of water samples taken at 

selected time intervals during the dye-cloud passage, are the 

basis for determining traveltime and dispersion characteristics 

of streams (Kilpatrick and Wilson, 1989).

Previous USGS Traveltime Studies 

Extensive use of fluorescent dyes as water tracers to quantify transport in streams and rivers began in the United States in 

the early to mid-1960’s (Wilson and others, 1986). Traveltime 

data collected using the dye-tracing techniques by the USGS 

have been used to develop national prediction equations that 

can provide guidance to water-resources managers and planners 

responding to spills. 

Traveltime Studies on Arkansas Streams

Traveltime studies have been conducted on several 

streams in Arkansas. The USGS conducted a series of traveltime studies on streams in Arkansas between 1971 and 1981 

using fluorescent dyes (Lamb, 1983). These studies demonstrated that traveltime could fluctuate greatly depending upon 

streamflow and the stream’s basin characteristics. Other traveltime studies have been conducted sporadically over a period of 

years by universities and local government entities but none has 

been used to derive a specific traveltime equation for Arkansas 

streams. 

National Traveltime Studies

Several national traveltime studies have been conducted 

by the USGS since 1974. A national traveltime study was completed in 1974 (Boning, 1974) using 873 individual dye-tracing 

measurements on more than 300 streams. These data were used 
C

T

T

T

T

T

t

t

t

t

n

p

L

p

t

d

c

b

p

L

c

is peak concentration of the dye cloud;

is elapsed time to the arrival of the leading edge of a sampling location;

is elapsed time to the peak concentration of the dye cloud;

is elapsed time to the trailing edge of the tracer cloud;

is duration of the dye cloud (T -T );

is elapsed time to the centroid of the response curve;

is the time from the arrival of the leading edge to the peak concentration;

is the time it takes for the peak concentration to travel from site n to site n + 1;

is the time it takes for the leading edge to travel from site n to site n + 1;

is the time it takes for the centroid of the response curve to travel from

site n to site n + 1;

the number of sampling sites downstream from injection

t L

EXPLANATION

6 Development of a Traveltime Prediction Equation for Streams in Arkansas

Figure 3. Definition sketch of time-concentration curves along a selected streamline resulting from an instantaneous dye injection (modified from

Kilpatrick and Wilson, 1989).
Methods 7

to develop a linear regression model that described travel rates 

of dye clouds for various streams. This model illustrated that 

discharge, slope, and duration frequency were related to traveltime. Kilpatrick (1993), using the concept of unit-peak concentration and the superposition principle, illustrated how 

traveltime data could be generalized for a wide range of flow 

conditions. Elaborating on Boning’s (1974) study, Jobson 

(1996) used traveltime data from approximately 939 subreaches 

for about 90 different streams throughout the United States and 

developed a set of national traveltime prediction equations that 

were based on the size of the drainage area, the slope of the 

stream reach, the mean annual discharge of the stream, and the 

discharge of the stream at the point of interest. 

Methods

The methods described in this report describe the techniques used during the field data-collection process for dye tracing and the statistical approach used to analyze the traveltime 

data. A description of the methods used in the evaluation of the 

national traveltime prediction equation for use in describing 

Arkansas streams also is provided. 

Site Selection

Selecting the 18 streams used in this investigation was an 

important component of the study. Because northern and western Arkansas are the most populous areas of the State and rely 

on surface water as their primary source of drinking water, the 

risk can be high for having their drinking water supplies contaminated from a spill. For this reason, the selection of these 

streams had a higher priority than other streams in the State.

Other factors were important in selecting the study sites. 

The 18 selected streams (table 1) have little or no flow regulation (dams or other restrictions) and have an established streamflow gaging station with at least 10 consecutive years of streamflow data available. The 18 streams had a wide range of 

drainage areas, stream slopes, and geomorphic characteristics. 

All of these factors contributed to the range in variability of data 

which is important when performing a regression analysis and 

developing a locally based traveltime prediction equation. Of 

the 18 streams, 15 were flowing at steady-state base-flow conditions and 3 were flowing at near steady-state medium-flow 

conditions at the time of data collection (for this study, mediumflow is approximately 100-150 percent of the mean monthly 

streamflow during the month the dye trace was conducted).

Fluorescent Dye Concentration Determination

For the Lamb (1983) and the present (2003) study, 

Rhodamine WT 20 percent stock solution (RWT), a fluorescent 

dye, was used as the tracer. Dye concentrations were measured 

using a fluorometer, which is an instrument that measures fluorescence. Fluorescence readings, when calibrated to known 

concentrations, can be used to measure the concentration of dye 

in the water. Fluorescence concentrations in samples also need 

to be compared to known concentrations (standards) using the 

same fluorometer under the same environmental conditions 

(Wilson and others, 1986). Otherwise comparisons of fluorescence concentrations will not be meaningful. Because the fluorescence measured in a stream sample is proportional to the 

concentration of dye in the water, standard solutions—which 

compare fluorescence to dye concentration—can be mixed to 

properly calibrate a fluorometer to relate fluorescence to dye 

concentration.

Prior to each dye trace conducted during the present (2003) 

study, the fluorometer was calibrated to a set of known RWT 

concentrations, usually 10, 25, and 100 micrograms per liter 

(μg/L). The concentrations of the samples used for calibrating 

the fluorometer were mixed using water collected at each 

stream. Stream water was used in an effort to account for the 

natural fluorescence effects that might occur for a particular 

stream’s water. For quality assurance, the calibration of the fluorometer was checked after each dye trace to ensure “drifting” 

of the fluorometer did not occur during the dye trace. See Lamb 

(1983) for a description of quality-assurance practices used in 

Lamb’s study.

Field Methods

Several steps were followed during the field data-collection period for Lamb’s (1983) study and for the present study. 

A brief description of Lamb’s (1983) study is described in this 

report. A detailed description of the field methods used for data 

collection during Lamb’s study can be found in Lamb (1983). 

Previous Studies on Arkansas Streams

During Lamb’s (1983) study, RWT dye was injected 

instantaneously at approximately the center of the stream. The 

amount of dye injected was determined prior to each study. 

Whenever possible, the dye was injected far enough upstream 

in an effort to allow vertical and lateral dispersion of the dye to 

take place before the dye cloud reached the first sampling site. 

If this was not possible because of time or physical constraints, 

the injection was made as far upstream as practical, and the 

error in traveltime introduced by incomplete mixing at the first 

sampling site was determined to be small and was disregarded. 

At each sampling site, water samples were collected at predetermined intervals before and during the arrival of the dye 

cloud until the dye concentration was less than 10 percent of the 

peak concentration passing the site. Each of the samples collected at the sites were analyzed using a fluorometer. The discharge of the stream at each sampling site was measured by 

conducting current-meter measurements using the techniques 

outlined by Rantz and others (1982).
8 Development of a Traveltime Prediction Equation for Streams in Arkansas

Present Study

For the present (2003) study, RWT also was used as the 

tracer to measure the traveltime of the stream. The amount of 

RWT dye injected into the stream at each of the study sites was 

predetermined using the methods outlined in Kilpatrick and 

Wilson, (1989) using the following equation:

Vol 3.4 10–4 × QmL

V

----------- ⎝ ⎠ ⎛ ⎞ 0.94

Cp = (1)

where

Vol is volume of stock RWT 20-percent solution, in 

liters;

Qm is the maximum expected stream discharge at the 

downstream site, in cubic feet per second;

L is the distance to the downstream site, in miles;

V is the estimated average stream velocity, in feet 

per second; and

Cp is the desired peak concentration at the downstream sampling site, in micrograms per liter.

The optimum distance for dye injection upstream from the 

first sample site location was calculated using the following formula (Kilpatrick and Wilson, 1989):

Lo KVB2

Ez

= --------- (2)

where 

Lo is length of channel required for optimum mixing, 

in feet;

K is a variable whose value depends on the location 

of injection and the

 number of injections (dimensionless);

V is estimated average stream velocity, in feet per 

second;

B is average stream width, in feet; and

Ez is lateral mixing coefficient, in feet squared per

second.

The dye was injected across the entire width of the stream 

by injecting it from a container while walking or boating across 

the stream (figs. 4, 5, and 6). The dye was injected far enough 

upstream from the beginning of the stream reach of interest to 

allow vertical and lateral dispersion of the dye to take place 

before the dye cloud reached the first sampling site (fig. 7).

Dye concentrations for each stream were measured at several different locations (sites) along the downstream reach. The 

sampling sites were located at appropriate intervals downstream, based on sampling site access and the availability of a 

streamflow gaging station. At each sampling site, the fluorescence (dye concentration) of the water was measured using a 

Self-Contained Underwater Fluorescence Apparatus (SCUFA) 

submersible fluorometer (Turner Designs, Inc., 2002) (fig. 8). 

The SCUFA measures the fluorescence of the water once per 

second and is capable of measuring RWT concentrations as low 

as 0.04 μg/L (Turner Designs, Inc., 2002). These readings were 

logged using a laptop computer and stored for later analysis of 

the data.

Figure 4. Injecting dye while wading across a stream (photograph by U.S. 

Geological Survey).

Figure 5. Injecting dye while boating across a stream (photograph by U.S. 

Geological Survey).

Figure 6. Injecting dye into a stream from a bridge (photograph by U.S. 

Geological Survey).
Methods 9

Figure 7. Lateral mixing of Rhodamine WT dye (photograph by U.S. Geological Survey).

Figure 8. Self-Contained Underwater Fluorescence Apparatus (SCUFA).

Statistical Methods

A statistical approach was taken to evaluate the national 

traveltime prediction equation and to develop a traveltime prediction equation for Arkansas streams. The national prediction 

equation was evaluated using the traveltime data collected during Lamb’s (1983) traveltime study in Arkansas and during the 

present study. These data then were used to develop a regression equation to accurately predict specific traveltimes for 

Arkansas streams.
10 Development of a Traveltime Prediction Equation for Streams in Arkansas

Evaluation of the National Traveltime Prediction 

Equation Using the Arkansas Data Set 

A total of 939 data points were used to develop the USGS 

national traveltime prediction equation (Jobson, 1996). The 

national prediction equation developed by Jobson (1996) to 

estimate the velocity of the dye peak is:

Vp 0.094 0.0143 D′ a ( )0.919

Q′ a ( )–0.469 S

0.159 Q

Da

× × ------

= + × × (3)

where, 

Vp is peak velocity, in meters per second;

D′ a is dimensionless drainage area coefficient;

Q′ a is dimensionless relative discharge;

S is slope, in meter per meter;

Q is discharge, in cubic meters per second; and

Da is drainage area, in square meters.

The dimensionless drainage area is defined as:

D′ a

Da

1.25 × g

Qa

= ---------------------------- (4)

where,

g is acceleration of gravity, in meters per second 

squared, and

Qa is mean annual discharge, in cubic meters per

second.

The dimensionless relative discharge is defined as:

Q′ a

Q

Qa

= ------ (5)

The streams studied for the present (2003) study were 

required to have a USGS streamflow gaging station with at least 

10 consecutive years of flow data recorded because mean 

annual discharge is required for use of the USGS national traveltime prediction equation. A drainage basin correction factor 

was used to calculate the mean annual discharge for a stream at 

a sampling site that did not have a streamflow gaging station. 

The drainage basin correction factor is a ratio that compares the 

drainage basin area of the sampling site that did not have a 

streamflow gaging station to the drainage area at a streamflow 

gaging station. This correction factor was multiplied by the 

mean annual discharge at the streamflow gaging station to 

obtain a mean annual discharge value for the sampling site that 

did not have a streamflow gaging station.

Drainage area and slope parameters needed in the predictive equations were calculated using 30-meter digital elevation 

model (DEM) and digital raster graph (DRG) data files. The 

software package Watershed Modeling System (WMS) 

(Brigham Young University, 1999) was used to calculate the 

drainage basin area at each sampling site location and the slope 

of the main channel of the stream. WMS is an integrated hydrologic modeling program that extracts key model input parameters from computerized maps. The slope was computed by subtracting the streambed elevation at the last sampling point on the 

stream reach from the streambed elevation at the point of the 

dye injection on the stream using the elevations obtained from 

the 30-meter DEM data. The change in elevation was divided 

by the length of the stream from the start of the dye trace to the 

last point sampled at the end of the dye trace obtained from 

DRG data files. 

The relative percent difference between the velocity calculated using the national traveltime equations (Jobson, 1996) and 

the velocity calculated using the traveltime data from the 

Arkansas data set was calculated for each of the 33 sampling 

sites on the 18 streams. The relative percent difference was calculated in an effort to compare the differences between the 

stream velocity by the national equation and the actual stream 

velocity measured during the dye trace. To calculate the relative 

percent difference, the difference between the predicted velocity and the actual velocity was divided by the predicted velocity 

and multiplied by 100 (to obtain a percent value). 

Development of the Arkansas Traveltime Prediction 

Equation

A multiple linear regression equation was developed to 

predict traveltime for streams in Arkansas using discharge and 

basin characteristics. The size of the data set limited the number 

of explanatory variables that could be used in the regression 

analysis. Variables in the Arkansas data set were checked 

against each other for correlation using the Spearman’s rho test 

(Iman and Conover, 1983). To avoid autocorrelation, variables 

that were significantly correlated (p <0.05) were not used 

together in the development of the regression equations. 

A stepwise regression was used in selecting the most 

appropriate coefficients for the regression equation. A logarithmic transformation (log base 10) was used for the explanatory 

variables to minimize the standard error of estimate. The multiple linear regression equation used to predict traveltime in 

Arkansas streams is expressed in the following form:

Log Y βo β1 X1 × β log 2 X2 = + + × log (6)

where,

Y is the estimated peak velocity, in feet per second 

(response variable);

βο, β1,β2 are regression coefficients, and

X1,X2 are basin characteristics (stream slope, in foot per 

foot and drainage area, in square miles).

The standard error of estimates and the coefficient of 

determination (R2

) were computed for each regression. The 

standard error is a measure of the error about the regression. A 

smaller standard error indicates a more precise prediction. The 

R2

 is the proportion of the variation in the response variable 

explained by the explanatory variables (Draper and Smith, 

1986). R2

 values range from 0 to 1 with values closer to 1 indicating a better fit of the data. 
Evaluation of the National Traveltime Prediction Equation Using the Arkansas Data Set 11

When equation 6 is retransformed, it becomes:

Y β′ o X β1

1 X β2 = × × 2 (7)

where,

β ′ o is 10β .

The retransformation of a log-transformed regression 

model systematically underestimates the mean response but 

provides a consistent estimator of median response (Miller, 

1984). Therefore, a bias-correction factor (BCF) needs to be 

included in the retransformed regression equation if an unbiased estimate of the mean is to be obtained. A BCF was computed for the equation by using a smearing estimate that is a 

nonparametric method based on the average residuals in original units (Duan, 1983). After applying the BCF to equation 7, 

the form becomes:

Y β′ o X β1 × 1 X2

β2 = × × BCF (8)

where,

β ′ o is 10β .

Evaluation of the National Traveltime 

Prediction Equation Using the Arkansas 

Data Set

The Arkansas data set provided in this report was used to 

evaluate the national traveltime prediction equation for use in 

Arkansas (Jobson, 1996). The national traveltime prediction 

equation yielded results that over-predicted the velocity of the 

streams for 29 of the 33 sample sites (table 2, fig. 9). The standard error for the national traveltime prediction equation was 

105 percent and the R2 value was 0.78 (fig. 9). 

Development of the Arkansas Traveltime 

Prediction Equation

Three stream characteristics were used as explanatory 

variables in the development of the Arkansas traveltime prediction equation. These included the discharge (Q) of the section at 

the point of interest (downstream end of reach), drainage basin 

area (Da) of the stream at the point of interest, and the stream 

reach slope (S) from the point of the dye injection to the point 

of interest. The range, mean, and median were determined for 

each of these variables for the 33 sites (table 3). 

PREDICTED VELOCITY, IN FEET PER SECOND

OBSERVED VELOCITY, IN FEET PER SECOND

Over-predicted

Under-predicted

Coefficient of determination = 0.78

Standard error = 105 percent

Figure 9. Relation between velocity predicted using the national traveltime prediction equation and observed velocity for Arkansas data set.
Table 2. Arkansas data used to evaluate the national traveltime prediction equation. [Q, the discharge of the stream at the time the sample was taken; DA, the drainage area at each sample location; slope, the slope of the stream; V predicted, the predicted velocity 

observed, the actual observed velocity of the stream during the present (2003) study and Lamb’s (1983) study. Negative values for relative percent difference indicate over-predict

diction equation] Sample site number Total length of travel (miles) Drainage area (square miles) Discharge (cubic feet per second) Mean annual discharge (cubic feet per second) Time for leading edge to arrive (hours) Time for peak concen- tration to arrive (hours) Q/DA (cubic feet per second per square mile) Slope (foot/foot) V predicted (feet per second) V observ

(feet pe

second

1 2.9 206 292 216 2.7 3.6 1.41 0.00087 1.49 1.18 2 3.9 406 96 365 6.5 8.0 0.24 0.00106 0.90 0.73 3 12.5 842 2,380 1,060 9.5 10.7 2.83 0.00093 2.50 1.71 4 12.5 842 238 1,060 24.0 29.5 0.28 0.00093 0.95 0.62 5 9.0 68 72 53 10.8 14.3 1.06 0.0032 1.41 0.92 6 15.9 4,889 6,560 5,491 9.0 10.4 1.34 0.00021 2.14 2.24 7 15.9 4,889 1,780 5,491 16.6 18.6 0.36 0.00021 1.23 1.25 8 8.9 1,135 1,830 1,144 4.8 5.4 1.61 0.00066 2.13 2.42 9 4.4 204 69 194 8.8 12.1 0.34 0.00179 0.95 0.53 10 5.2 130 55 124 12.6 16.5 0.42 0.00219 0.98 0.46 11 8.4 204 83 194 17.8 22.6 0.41 0.00148 1.00 0.55 12 4.1 35 50 34 17.4 19.6 1.42 0.00146 1.20 0.31 13 6.9 509 92 482 19.1 28.5 0.18 0.000991 0.83 0.36 14 4.0 254 10.6 241 26.3 32.6 0.04 0.00107 0.51 0.18 15 2.6 79 6 75 46.8 67.8 0.08 0.00153 0.54 0.06 16 8.2 511 240 484 5.4 10.4 0.47 0.000936 1.16 1.16 17 5.0 263 23.4 249 33.1 47.2 0.09 0.000742 0.60 0.16 18 8.5 420 237 540 20.9 29.0 0.56 0.0022 1.21 0.43 19 14.1 374 493 554 19.4 22.9 1.32 0.00219 1.60 0.90 
20 9.0 61 376 71 6.4 7.9 6.14 0.00535 2.80 1.67 21 9.0 282 774 420 6.2 7.9 2.74 0.00121 1.93 1.67 22 7.4 242 680 360 5.3 6.0 2.81 0.00264 2.10 1.81 23 16.4 282 46 420 103 133 0.16 0.00187 0.70 0.18 24 10.9 284 549 288 12.8 14.4 1.93 0.00042 1.66 1.10 25 6.0 46 120 62 9.8 12.4 2.61 0.00626 1.72 0.71 26 12.8 142 252 310 18.1 22.9 1.78 0.0045 1.45 0.82 27 9.6 414 1,170 731 6.7 8.7 2.83 0.00092 1.90 1.62 28 9.6 414 343 731 11.0 16.5 0.83 0.00092 1.14 0.85 29 12.0 292 136 611 31.7 37.3 0.47 0.0014 0.87 0.47 30 15.8 201 110 420 40.1 46.5 0.55 0.00195 0.90 0.50 31 12.0 292 37.6 611 81.2 101 0.13 0.0014 0.59 0.17 32 23.0 251 39.5 525 132 162 0.16 0.00188 0.63 0.21 33 0.7 12 27.5 22 0.9 1.25 2.33 0.017 1.30 0.82 

Table 2. Arkansas data used to evaluate the national traveltime prediction equation.—Continued[Q, the discharge of the stream at the time the sample was taken; DA, the drainage area at each sample location; slope, the slope of the stream; V predicted, the predicted velocity 

observed, the actual observed velocity of the stream during the present (2003) study and Lamb’s (1983) study. Negative values for relative percent difference indicate over-predict

diction equation]

Sample

site

number

Total 

length of

travel

(miles)

Drainage

area

(square

miles)

Discharge

(cubic feet

per second)

Mean annual

discharge

(cubic feet

per second)

Time for

leading edge

to arrive

(hours)

Time for

peak concentration to

arrive

(hours)

Q/DA

(cubic

feet per

second per

square

mile)

Slope

(foot/foot)

V predicted

(feet per

second)

V observ

(feet pe

second
Table 3. Ranges, means, and medians of explanatory variable values used to develop the Arkansas traveltime prediction equation.

Explanatory variable Minimum Maximum Mean Median

Reach length, in miles 0.7 23.0 9.3 9.0

Drainage area, in square miles 11.8 4,889 590 282

Discharge, in cubic feet per second 6 6,560 579 136

Discharge/drainage area 0.04 6.14 1.2 0.55

Slope, in foot per foot 0.00021 0.01700 0.00219 0.001400

14 Development of a Traveltime Prediction Equation for Streams in Arkansas

Using the variables discharge, drainage area, and slope in 

a regression analysis, the most accurate prediction equation 

derived for the peak velocity in feet per second was: 

Vp 0.0731 Q

Da

------ ⎝ ⎠ ⎛ ⎞ 0.664

× S

–0.377 = × × 1.074 (9)

where,

Vp is velocity, in feet per second,

Q is discharge at point of interest, in cubic feet per 

second;

Da is drainage area at point of interest, in square 

miles; and

S is slope from the point of injection to the point of 

interest, in foot per foot.

This prediction equation does not appear to overpredict the 

stream velocity as did the national equation. This prediction 

equation, developed from a regression analysis of the dye-tracing results, was a significant improvement over the national prediction equation. The standard error for the Arkansas traveltime 

prediction equation was calculated to be 46 percent and the R2

value was calculated to be 0.74. The data collected during above 

average and base-flow conditions does not indicate a substantial 

variation from the trendline (fig. 10). The predicted velocities 

using this equation fit better than the national equation when 

plotted in relation to actual velocities (fig. 10) for all of the data 

analyzed for this study. 

Traveltime of Peak Concentration

The theoretical traveltime (Tp) for streams in Arkansas can 

be derived from results from equation 9. This can be found by 

multiplying the velocity, calculated from equation 9, by the distance (in feet) between the spill site and the point of interest 

(downstream end of reach). This will be the theoretical time that 

it will take for the arrival of the peak concentration to the point 

of interest.

The theoretical traveltime assumes a conservative constituent that is dissolved in water. Theoretical traveltimes are not 

meant to represent traveltimes of non-conservative contaminants that are not dissolved in water. The accuracy of the traveltime estimated using equation 9 can be affected by differences 

in stream characteristics and discharge along the reach being 

evaluated.

Traveltime of Leading Edge

In addition to knowing when the peak concentration will 

arrive at a site, it is of great interest to know when the first concentration of the dye will arrive. The traveltime of the leading 

edge of the contaminant peak indicates when a local problem 

might first exist and defines the overall shape of the concentration response function.

A good correlation between traveltime of the leading edge 

and traveltime of the peak concentration based on dye-tracing 

results was found with an R2

 value of 0.99 (fig. 11). These data 

indicate that the traveltime of the leading edge can be estimated 

from:

Tl 0.79 Tp = × (10)

where,

Tl is traveltime of leading edge, in hours, and

Tp is traveltime of peak concentration, in hours.

Example Application

An example application is presented assuming a slug 

injection of a potential contaminant in a stream. This example 

assumes that few hydrologic data are available. In this example, 

a tanker truck driving on Arkansas State Highway 10 runs off of 

the road and instantaneously spills a contaminant into the 

Maumelle River near Perryville, Arkansas. An estimate is 

needed for the most probable time that it will take for the contaminant to reach Lake Maumelle.
x

PREDICTED VELOCITY, IN FEET PER SECOND

OBSERVED VELOCITY, IN FEET PER SECOND

Above-average flow conditions

Base-flow conditions

Data Trendline

1:1 Reference Line

Over-predicted

Under-predicted

x

x

x

 Example Application 15

Figure 10. Relation between velocity predicted using the Arkansas traveltime prediction equation and observed velocity for Arkansas data set. TRAVELTIME OF THE LEADING EDGE, IN HOURS

TRAVELTIME OF THE PEAK CONCENTRATION, IN HOURS

Coefficient of determination = 0.99

Figure 11. Relation between traveltimes of leading edge and peak of dye cloud for the Arkansas data set.
16 Development of a Traveltime Prediction Equation for Streams in Arkansas

The first step is to obtain (or estimate) the discharge that is 

flowing at the point of interest at the time of the spill. In this 

example, the water plant manager for Central Arkansas Water 

estimates the discharge flowing in the Maumelle River to be 

about 100 ft3/s.

The second, third, and fourth steps are to: (2) compute the 

drainage area at the point of interest (at the confluence with 

Lake Maumelle), (3) calculate the slope of the stream (between 

the point of the spill and the point of interest), and (4) calculate 

the length of the stream reach (between the point of the spill and 

the point of interest). Steps 2, 3, and 4 can be computed manually using topographic maps or on an automated basis using 

geographic information systems (GIS) software. For this example problem, assume the following:

Discharge (Q) is 100 ft3/s,

Drainage area (DA) is 89.4 mi2,

Slope (S) is 0.00390 ft/ft, and

Length (L) is 79,700 ft.

Applying equation 9:

Vp 0.0731 100

89.4

---------- ⎝ ⎠ ⎛ ⎞ 0.664

× 0.00390–0.377 = = × × 1.074

0.684 ft/s

The most probable traveltime of the peak to the point of 

interest is:

Tp= 79,700 ft × 1hr

0.684 ft/s 3600 ,× s ----------------------------------------- ⎝ ⎠ ⎛ ⎞ = 32.4 hrs

The most probable traveltime of the leading edge of the 

contaminant plume to the point of interest is:

Tl = 0.79 × 32.4 hrs = 25.6 hrs.

Summary

The 1996 Amendments to the Safe Drinking-Water Act 

require that each state prepare a source-water assessment for all 

public water supplies. States are required to determine the 

sources of drinking water, the origin of contaminants monitored 

or the identification of the potential contaminants to be monitored, and the intrinsic susceptibility of the water supplies to 

these contaminants. Fluorescent dyes can be used to measure a 

stream’s velocity and gain a better understanding of the potential rate of movement of contaminants.

During 1971 and 1981 and 2001 and 2003, traveltime measurements were made on 18 streams throughout northern and 

western Arkansas using fluorescent dye. Most of the measurements were made during steady-state base-flow conditions with 

the exception of three measurements that were made during 

near steady-state medium-flow conditions (for this study, 

medium-flow is approximately 100-150 percent of the mean 

monthly streamflow during the month the dye trace was conducted). These data were compared to the USGS national traveltime prediction equation and used to develop a specific traveltime prediction equation for Arkansas streams.

In general, the national traveltime prediction equation 

yielded results that over-predicted the velocity of the streams 

for 29 of the 33 sample events measured. The standard error for 

the national traveltime prediction equation was calculated to be 

105 percent. The coefficient of determination was calculated to 

be 0.78. 

The Arkansas prediction equation developed from a 

regression analysis of dye-tracing results was a significant 

improvement over the national prediction equation. This regression analysis yielded a standard error of 46 percent and a coefficient of determination of 0.74. The predicted velocities using 

this equation fit better when plotted against measured velocities. 

Using the variables in a regression analysis, the most accurate Arkansas prediction derived for the peak velocity in feet per 

second was: 

Vp 0.0731 Q

Da

------ ⎝ ⎠ ⎛ ⎞ 0.664

× S

–0.377 = × × 1.074

where,

Q is discharge at point of interest, in cubic feet per 

second;

Dais drainage area at point of interest, in square miles; 

and

S is slope from point of injection of point of interest, 

in foot per foot.

In addition to knowing when the peak concentration will 

arrive at a site, it is of great interest to know when the leading 

edge of a contaminant plume will arrive. The traveltime of the 

leading edge of the contaminant plume indicates when a potential problem might first develop and also defines the overall 

shape of the concentration response function.

A good correlation between traveltime of the leading edge 

and traveltime of the peak concentration was found, with an R2

value of 0.99. These data indicate that the traveltime of the leading edge can be estimated from:

Tl 0.79 Tp = ×

where,

Tl is traveltime of leading edge, in hours, and 

Tp is traveltime of peak concentration, in hours.
References 17

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Funkhouser, J.E., and Barks, C.S.—DEVELOPMENT OF A TRAVELTIME PREDICTION EQUATION FOR STREAMS IN ARKANSAS—U.S. Geological 

Survey Scientific Investigations Report 2004-5064