United States

Department

of Agriculture

Forest Service

Rocky Mountain

Research Station

General Technical Report 

RMRS-GTR-223

May 2009

Manual for Computing Bed Load 

Transport Using BAGS (Bedload 

Assessment for Gravel-bed 

Streams) Software

John Pitlick, Yantao Cui, Peter Wilcock
You may order additional copies of this publication by sending your 

mailing information in label form through one of the following media. 

Please specify the publication title and series number.

Fort Collins Service Center

Telephone (970) 498-1392

FAX (970) 498-1122

E-mail rschneider@fs.fed.us

Web site http://www.fs.fed.us/rm/publications

Mailing address Publications Distribution

Rocky Mountain Research Station

240 West Prospect Road

Fort Collins, CO 80526

Rocky Mountain Research Station

240 W. Prospect Road

Fort Collins, Colorado 80526

Pitlick, John; Cui, Yantao; Wilcock, Peter. 2009. Manual for computing bed 

load transport using BAGS (Bedload Assessment for Gravel-bed Streams) 

Software. Gen. Tech. Rep. RMRS-GTR-223. Fort Collins, CO: U.S. Department of 

Agriculture, Forest Service, Rocky Mountain Research Station. 45 p.

Abstract

This manual provides background information and instructions on the use of a 

spreadsheet-based program for Bedload Assessment in Gravel-bed Streams 

(BAGS). The program implements six bed load transport equations developed 

specifically for gravel-bed rivers. Transport capacities are calculated on the basis 

of field measurements of channel geometry, reach-average slope, and bed material grain size. Calculations are carried out using Visual Basic for Applications 

(VBA), and the output is stored on individual spreadsheets. In addition to stepby-step instructions in software operation, the manual provides guidance in the 

interpretation of results.

The Authors

John Pitlick, Department of Geography, University of Colorado, Boulder, CO.

Yantao Cui, Stillwater Sciences, Berkeley, CA.

Peter Wilcock, Department of Geography and Environmental Engineer, Johns 

Hopkins University, Baltimore, MD.
Disclaimer

BAGS is software in the public domain, and the recipient may not assert any 

proprietary rights thereto nor represent it to anyone as other than a Governmentproduced program. BAGS is provided “as-is” without warranty of any kind, 

including, but not limited to, the implied warranties of merchantability and fitness 

for a particular purpose. The user assumes all responsibility for the accuracy and 

suitability of this program for a specific application. In no event will the U.S. Forest 

Service, Stillwater Sciences Inc., Johns Hopkins University, University of Colorado, 

or any of the program and manual authors be liable for any damages, including lost 

profits, lost savings, or other incidental or consequential damages arising from the 

use of or the inability to use this program.

Download Information

The BAGS program, this manual, and a sediment transport primer (Wilcock and 

others 2009) can be downloaded from: http://www.stream.fs.fed.us/publications/

software.html.

This publication may be updated as features and modeling capabilities are added 

to the program. Users may wish to periodically check the download site for the 

latest updates.

BAGS is supported by, and limited technical support is available from, the U.S. 

Forest Service, Watershed, Fish, Wildlife, Air, & Rare Plants Staff, Streams 

Systems Technology Center, Fort Collins, CO. The preferred method of contact 

for obtaining support is to send an e-mail to rmrs_stream@fs.fed.us requesting 

“BAGS Support” in the subject line.

U.S. Forest Service

Rocky Mountain Research Station

Stream Systems Technology Center

2150 Centre Ave., Bldg. A, Suite 368

Fort Collins, CO 80526-1891

(970) 295-5986

Acknowledgments

Support for software development and preparation of the users manual was provided through a cost-share agreement between the University of Colorado and the 

U.S. Forest Service Stream Systems Technology Center. We thank John Potyondy 

for initiating this project and providing suggestions on software components. We 

would also like to thank Kristin Bunte for sharing her bed load data from Halfmoon 

Creek and providing feedback on software operations. Erich Mueller conducted 

tests of the initial version of the software. Finally, we would like to acknowledge 

Paul Bakke, John Buffington, and John Potyondy for providing numerous thoughtful comments on the initial draft of the manual. Their suggestions lead to significant 

improvements in the structure and clarity of the manuscript.
Contents

Acknowledgments ........................................................................................................................i

Background on Data Input and Software Operation ..........................................................1

The Sediment System..............................................................................................................1

Overview of BAGS Software Operation...................................................................................3

Basic Data Input Requirements ...............................................................................................4

Bed Load Transport Equations ................................................................................................8

Surface-Based Bed Load Equation of Parker (1990).............................................................10

Substrate-Based Bed Load Equation of Parker-Klingeman-Mclean (1982)...........................12

Substrate-Based Equation of Parker-Klingeman (Parker and Klingeman 1982)...................12

Surface-Based Two-Fraction Equation of Wilcock (2001) .....................................................13

Surface-Based Relation of Wilcock and Crowe (2003)..........................................................14

Procedure of Bakke and others (1999)..................................................................................15

Data Input Requirements .........................................................................................................15

Software Operation....................................................................................................................16

Assessing Model Output..........................................................................................................26

How Good Are the Raw Data?...............................................................................................26

Comparisons of Model Output With Bed Load Measurements..............................................28

Assessing Model Output Without the Benefit of Bed Load Data............................................39

References...................................................................................................................................43

ii
USDA Forest Service RMRS-GTR-223. 2009. 1

Background on Data Input and Software Operation

This manual for computing bed load transport provides specific instructions on the use of BAGS (Bedload Assessment of Gravel-bed Streams) software 

with worked examples illustrating typical results and comments on possible interpretations of the output. The BAGS software calculates bed load transport 

capacities on the basis of commonly available field data (surveyed cross sections 

and measured grain size distributions) and stores the output as tabulated values 

on individual spreadsheets. These values can be retained for further analyses or 

exported to other programs for plotting and visualization. Options within the program allow the user to select from six transport relations developed specifically 

for gravel-bed streams and rivers. Transport rates can be calculated for a single 

flow or a series of flows, depending on the application and data availability. The 

first section of this manual gives an overview of the basic data requirements and 

introduces some important concepts shared by each of the bed load transport 

relations—all users should read this section! Details of the transport relations 

and their various components are presented in the second section. The third and 

fourth sections describe the steps involved in running the software. The final 

section provides several examples illustrating results from different model runs 

and discusses problems that might arise in typical applications. This manual is a 

companion to “Sediment Transport Primer: Estimating Bed-Material Transport in 

Gravel-bed Rivers” (Wilcock and others 2009).

The Sediment System

Each of the models used in calculating bed load transport rates has some 

common data input requirements, these include:

• a measured channel cross section (or, at a minimum, an estimate of the bankfull width);

• an estimate of the reach-average slope, obtained from measurements of watersurface elevations or bed elevations;

• discharge measurements (and a flow duration curve if one of the goals is to 

estimate the long-term annual bed load sediment yield); and

• grain size parameters estimated from samples of the bed sediment.

The primary differences in software operation and data inputs center around 

sediment properties and various measures of these properties. In gravel-bed 

streams, we can separate the “sediment system” into three distinct components, 

as illustrated in figure 1. The sediment we typically see on the bed surface consists of relatively coarse clasts representing the largest grain sizes carried by the 
2 USDA Forest Service RMRS-GTR-223. 2009.

stream or river. Winnowing of finer particles from the bed surface produces a 

distinct layer with a thickness approximately equal to the diameter of the coarsest 

grains. This sediment is referred to as the surface layer or armor. The sediment 

immediately beneath the surface layer consists of a more homogeneous mixture 

of fine and coarse particles and is referred to as the substrate or bulk bed material. The substrate is generally much finer than the surface layer, and typically 10 

to 20 percent of this sediment is finer than gravel (<2 mm). The third component 

of the sediment system is the bed load itself. This sediment is, by definition, the 

material that moves in contact with the bed.

We can illustrate the differences in sediment characteristics graphically 

by plotting the distribution of particle sizes as cumulative frequency curves. 

Figure 2 presents two examples of grain size distributions based on sediment 

samples taken in two gravel-bed rivers. Both data sets include measurements of 

the bed load, bed surface layer, and substrate. The panel on the left shows that 

the bed load carried by the Salmon River is much finer than the substrate—the 

Figure 1. Distinction between the bed surface layer (armor) and the substrate, as commonly found in gravel-bed 

streams and rivers.

Figure 2. Grain size distributions of the bed load, surface layer, and substrate in two gravel bed rivers. Salmon River data obtained through 

the Boise Adjudication Team web site (http://www.fs.fed.us/rm/boise/research/watershed/BAT). Fall River data from Pitlick (1993).
USDA Forest Service RMRS-GTR-223. 2009. 3

substrate, in turn, is much finer than the bed surface. In this case, the three components of the sediment system are quite distinct from each other and it does not 

appear that much of the load is derived from the bed. In contrast, the panel on 

the right shows that the bed load carried by Fall River has nearly the same size 

distribution as the substrate. The similarity in bed load and substrate grain size 

distributions suggests a clear link between the load and the source: in this case, 

it appears that the bed load and substrate grain sizes exchange with each other 

almost on a one-for-one basis.

The examples shown in figure 2 are more or less representative of the 

range of conditions one is likely to encounter in measuring or modeling transport 

rates in gravel-bed streams. However, it would be difficult to know in advance 

whether a given stream in a specific geographic location should act more like 

the Salmon River or Fall River, and it would be impossible to determine this 

without extensive sampling. Results from field studies and flume experiments 

suggest that the separation between bed load and substrate grain-size distributions is related to variations in hydrology, boundary shear stress, and/or sediment 

supply, although these interactions are not very well understood (Barry and others 2004; Buffington and Montgomery 1999; Dietrich and others 1989; Hassan 

and Church 2001; Hassan and others 2006; Lisle 1995; Mueller and others 2005; 

Parker 1990a; Pitlick and others 2008; Powell and others 2001; Wathen and others 1995). Ultimately, all of the sediment carried by a river is supplied by the 

watershed; however, over the short time scales of individual flow events, some of 

the bed load may be derived from the bed itself and some from sources outside 

the channel (for example, stream banks or hillslopes). Success in predicting bed 

load transport rates hinges to a large extent on the availability of mobile sediment 

sizes within the channel boundary, but it is not always obvious that conditions at 

a particular location do or do not satisfy this requirement.

Overview of BAGS Software Operation

The BAGS software described in this manual implements six well-known 

bed load transport equations developed specifically for gravel-bed rivers:

• the surface-based equation of Parker (1990);

• the substrate-based equation of Parker-Klingeman-McLean (Parker and others 

1982);

• the substrate-based equation of Parker and Klingeman (1982);

• the surface-based two-fraction equation of Wilcock (2001);

• the surface-based equation of Wilcock and Crowe (2003); and
4 USDA Forest Service RMRS-GTR-223. 2009.

• the procedure of Bakke and others (1999), which calibrates two coefficients in 

the substrate-based equation of Parker and Klingeman (1982).

All of the equations and procedures recognize the role of the armor layer 

in regulating bed load transport rates, thus the dynamics of transport are represented by the three components discussed above: the surface layer, substrate, and 

bed load. The equations of Parker (1990a), Wilcock (2001), and Wilcock and 

Crowe (2003) apply surface grain size characteristics as inputs, while the ParkerKlingeman-McLean equation (Parker and others 1982) uses substrate grain sizes. 

The method of Bakke and others (1999) applies to either the surface or the substrate, depending on circumstances. In addition, the methods of both Wilcock 

(2001) and Bakke and others (1982) use bed load measurements to calibrate certain coefficients in the transport equations and procedures.

The equations and procedures are implemented in an MS-Excel workbook 

with Visual Basic for Applications (VBA). Field data and relevant parameters are 

entered into the program sequentially with a series of user prompts. Results of 

the calculations are presented in MS-Excel workbooks. The software is designed 

to be used by professionals (hydraulic engineers, fluvial geomorphologists, and 

hydrologists) with some familiarity and training in processes of sediment transport. The correct interpretation of the results, however, typically requires a more 

in-depth knowledge of fluvial processes, especially the dynamics of transport in 

gravel-bed rivers.

Basic Data Input Requirements

Grain size

Representative samples of the bed surface layer or the substrate are required 

in order to develop a cumulative frequency distribution of available grain sizes. 

Sample values are entered into the program as a listing of the percentage of particles finer than a given size, D, where D is measured in millimeters. Different 

methods are used to sample the surface and the substrate. The surface layer is 

sampled by using some variation of the method introduced by Wolman (1954), or 

by taking a volumetric sample of the surface layer down to the depth of the largest 

clasts, as described by Milhous (1973). Substrate samples are taken by excavating a pre-determined mass of sediment from beneath the surface layer. There are 

many important issues to consider in sampling the surface layer and substrate, 

and we won’t go into these details here. Procedures for sampling surface and substrate sediment and criteria for establishing sample sizes are described in detail 

in a number of papers and reports, including Kellerhals and Bray (1971), Church 

and others (1987), and Bunte and Abt (2001).
USDA Forest Service RMRS-GTR-223. 2009. 5

Figure 3 shows an example of the data listing for a point count of the surface 

layer and the resulting grain size distribution. The difference between the finest 

and coarsest grain sizes in a sediment sample is typically quite large, thus the 

grain size curve is plotted using a logarithmic scale for the x axis (fig. 3).

For convenience, a Ψ scale is also used to represent grain size, as shown on 

the secondary x axis. The Ψ scale varies with the base 2 logarithm of the grain 

size:

W = log 2

log D , or D = 2W (1a,b)

where D is in mm. The Ψ scale used here is identical to the φ-scale used in sedimentology, except the sign is reversed, resulting in positive values of Ψ for D>1 

mm. If the sample values are split at even increments of Ψ = 1, 2, 3... and so forth, 

then the grain size distribution is said to be sampled (or sieved) at 1-Ψ intervals. 

If higher resolution is required, sample values can be split at smaller intervals, for 

example, 1/2Ψ = 0.5, 1.0, 1.5, 2.0, ... and so forth. The grain size distribution and 

data listed in figure 3 show measurements obtained at 1/2-Ψ intervals.

The software calculates transport rates and bed friction coefficients on the 

basis of discrete values of the grain size distribution, Di

, where the subscript i

refers to an individual percentile of the grain size distribution. The midpoint of 

the distribution corresponding to the value for which half the sediment is finer is 

the median grain size, D50. In the above example, D50 = 60 mm. Additional sizes 

referred to elsewhere in the manual include D65, D84, and D90.

Other parameters are calculated from the full grain size distribution of the 

sample using individual values for each size class. Let D1, D2, ..., DN+1 be the 

grain sizes associated with each of the N size classes, and let f

1, f2, ..., fN+1 be 

Figure 3. Listing of values obtained from point-count sample of the surface layer of a gravel-bed river, and the grain size distribution curve 

corresponding to this sample.

Location: XS-10 Date: 7/10/04

	 D(mm)	 Ψ	 # passing	 pct finer

512 9.0 100

362 8.5 1 100

256 8.0 5 99

181 7.5 10 94

128 7.0 8 84

90 6.5 23 76

64 6.0 21 53

45 5.5 14 32

32 5.0 12 18

22 4.5 5 6

16 4.0 1 1

11.2 3.5

8.0 3.0

total = 100
6 USDA Forest Service RMRS-GTR-223. 2009.

the fraction of the sampled mass (or the sampled number of grains) represented 

in each size class. The mean values of Di

, Ψi, and f

i

 for each class are calculated 

as follows:

Di = DiDi+1 , Wi = 2

Wi + Wi+1 , fi = fi+1 - fi (2a,b,c)

where the subscripts i and i + 1 refer to adjacent size classes. The values obtained 

from (2) can be used to estimate additional parameters:

W = R

i = 1

N

Wi fi , Dg = 2W (3a,b)

and

v = R

i-1

N

Wi - W2f

i ` j , vg = 2v (3c,d)

where Ψ is the arithmetic mean (in Ψ units); Dg is the geometric mean grain size 

(in mm); σ is the arithmetic standard deviation (in Ψ units); and σg is the geometric standard deviation (in mm). In the preceding example, Ψ = 5.94, Dg = 60 mm, 

σ = 0.90, and σg = 1.9 mm. The values of D50 and Dg are very nearly identical in 

this example; however, in general, Dg will be smaller than D50.

Channel cross section

The software calculates relevant flow properties, such as the mean flow 

velocity and hydraulic radius, from a measured cross section. Depending on 

the application and site characteristics, you may want to include overbank areas (floodplains) in the calculations. The software will prompt you accordingly, 

and you will be asked to specify the left and right boundaries of the floodplain, 

as shown in figure 4. The hydrodynamic component of the model accounts for 

changes in the geometry and velocity of the flow as it spreads out across the 

Figure 4. Sketch of a channel cross section. Water is assumed to flow through the main channel and the floodplains if 

inundated. Bed load is assumed to occur only in the main channel.
USDA Forest Service RMRS-GTR-223. 2009. 7

floodplain. However, the sediment transport component of the model restricts bed 

load calculations to the main channel under the assumption that the water-surface 

elevation is constant across the channel and the floodplain.

Flow resistance and shear stress

Flow resistance: The software estimates flow resistance in the channel separately from flow resistance over the floodplain. In the main channel, it is assumed 

that flow resistance is dominated by the stationary grains on the bed. Given the 

geometry of the channel and the grain size distribution of the bed surface, the 

software calculates flow properties using the Keulegan resistance relation:

u*

U = 2.5 ln 11 ksRc b l (4)

where u* = (g Rc Sf )

1/2 is the shear velocity; g is the gravitational acceleration; 

Rc is the hydraulic radius of the main channel, equal to the cross-sectional area, 

Ac, divided by the wetted perimeter, Pc; Sf

 is the friction slope; and ks is the 

equivalent roughness. The convention followed here is to assume that ks

 scales 

with a coarser-than-average grain size since larger grains contribute more to the 

total flow resistance than smaller grains. The criteria for choosing one value of 

ks

 over another (3D84, 2D90, and so forth) is based largely on empirical relations 

and individual preference. To be consistent with the original references, three 

slightly different values of ks

 are used: Parker’s surface-based equation assumes 

ks

 = 2D90; Parker’s substrate-based equations assume ks = 10.7D50sub, where the 

subscript sub refers to the substrate; and Wilcock’s equations assume ks

 = 2D65.

Shear stress: The total shear stress acting on the bed and banks is calculated 

from the depth-slope product, τo = ρgRSf , where ρ is the density of water (see 

Wilcock and others 2009, Chapter 2—Non-uniform and unsteady flow, for a more 

complete discussion of the components of shear stress). The proportion of the total shear stress available for transporting bed load—termed the “grain stress”—is 

estimated using a drag-partitioning algorithm that couples the Keulegan relation for velocity (4) with the Manning-Strickler relation for grain roughness (see 

Wilcock and others 2009, Chapter 2—The drag partition).

Overbank flows: Flow over the floodplain is characterized using a combined form of the continuity equation and the Manning equation:

Q1 = nl

1 AlRl

2/3

S f

1/2

 , Qr = nr

1 ArRr

2/3

S f

1/2 (5)

where Q is the discharge over the floodplain; n is a user-defined roughness coefficient; A is the cross-sectional area of the flow over the floodplain; R is the 

hydraulic radius; and the subscripts l and r refer to left and right portions of the 
8 USDA Forest Service RMRS-GTR-223. 2009.

floodplain, respectively. For overbank flows, the value of Sf is assumed to be the 

same across the channel and the floodplain. Ideally, estimates of Sf

 should be 

made with a one-dimensional hydraulic model, such as HEC-RAS or WS-PRO; 

however, if this information is not available, reach-average estimates of the water 

surface slope or the bed slope are used as approximations for Sf . Guidelines and 

techniques for estimating floodplain roughness are discussed in several publications, including Barnes (1967), Arcement and Schneider (1989), and Hicks and 

Mason (1998). Several web sites provide descriptions of techniques for estimating flow resistance in simple and compound channels and up-to-date references 

for published studies:

http://wwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/

http://manningsn.sdsu.edu

http://www.rivers.gov.au/Tools_and_Techniques/Stream_Roughness_Coefficients_Tool/

Bed Load Transport Equations

This section provides a detailed discussion of the individual transport relations used in the BAGS model. We have tried to simplify the notation and 

terminology as much as possible while retaining key elements of each equation, 

as given in the original references.

The bed load transport relations used in the BAGS model are conceptually 

similar in that they all model transport rates as a function of the transport stage, 

which is simply a ratio of the available shear stress to the threshold shear stress:

z = xr

x (6)

where τ is defined as before and τr

 is the reference shear stress that produces a 

very small but measurable transport rate (Parker 1990; Parker and others 1982). 

This term is analogous to the critical shear stress, τc, which is used in a number 

of other bed load transport equations, including Meyer-Peter and Muller (1948) 

and Fernandez-Luque and van Beek (1976). The reference shear stress is used 

in place of the critical shear stress to avoid ambiguities in defining the transport 

threshold; bed load begins moving over a range of flows, and there is always 

some chance, even at low flows (φ<1.0), that a small number of grains might be 

moving (more on this later). It is also important to note that, as the ratio of τ to τr

grows, there is a nonlinear response in transport, thus any errors associated with 

estimates of τ or τr

 are quickly amplified, leading to potentially large uncertainty 

in calculated transport rates.
USDA Forest Service RMRS-GTR-223. 2009. 9

Bed load transport rates are expressed in terms of a dimensionless parameter:

W * = ts ^x/th

1.5

^s - 1hgqb = tsu*

3

^s - 1hgqb (7)

where s is the specific gravity of sediment; g is the gravitational acceleration; ρs

 is 

the density of sediment (2650 kg/m3 or 2.65 g/cm3 for quartz-density sediment); 

ρ is the density of water; u* is the shear velocity; and qb is the mass transport 

rate per unit width. The transport parameter W* is likewise a ratio, in this case 

representing the power required to transport bed load scaled by the power available (Parker and others 1982). Values of W* could thus be similar for a few large 

grains moving at high shear stress or many small grains moving at low shear 

stress. The primary reason for formulating the transport equations in terms of dimensionless parameters is to maintain generality so that the equations and results 

are transferable across a range of scales.

The other dimensionless parameter that appears frequently in the bed load 

transport equations is the Shields stress:

x* = `ts - tjgD

x = ` s - 1jgDu*

2

= ` s - 1jD

RS f

(8)

Eq. 8 is derived by balancing the drag force acting over the area of the grain 

against its weight. This equation, therefore, represents a ratio of the force available to move a given grain size versus the resistance provided by the weight and 

contact forces. Using this relation, Eq. 6 can be rewritten as:

z = xr

*

x*

(9)

where xr

*

 is termed the reference Shields stress.

For some applications, it may be necessary to compute transport rates for 

individual grain sizes, Di

, where the subscript i refers to the i-th size fraction of 

the grain size distribution. In this case, the parameters listed above are denoted 

φi , Wi

*

, and xr

i

*

, and the equation will generally include additional terms representing the proportion of sediment in the i-th size fraction. The symbols, 

pi , f

i , and Fi , are used to denote, respectively, the proportion of sizes in one of 

three potential populations of sediment: the bed load, substrate, and surface layer. 

Calculations of fractional transport rates also involve what is known as a “hiding 

function,” a function that accounts for size-dependent differences in the mobility 

of small and large grains (Andrews 1994; Parker and others 1982; Wilcock and 

Crowe 2003). In gravel channels, small lightweight grains tend to get lodged in 

the interstices between large grains, hence they are hidden from the flow and less 

mobile than they might be otherwise. Large grains protrude into the flow and 

are exposed to higher velocities, hence they are more mobile than they might be 
10 USDA Forest Service RMRS-GTR-223. 2009.

otherwise. The offsetting effects of hiding and exposure are reflected in the hiding function by an inverse relation between the reference Shields stress for an 

individual grain size, xr

i

*

, and the ratio of the individual grain size to the median 

grain size, Di /D50.

Finally, several of transport relations used in the BAGS model consist of 

more than one function, with each function covering a different range in transport 

stage, φ (see the first set of equations below). There are two principal reasons for 

developing multi-part functions: (1) Bed load data often exhibit a slight but distinct curvature when transport rates are plotted against τ* or φ on log-paper, thus 

a single power law equation may not fit the data particularly well across the entire 

range of values and (2) It has been shown that the transport rate, W*, should approach a constant at large values of φ (Parker and Klingeman 1982; Yalin 1972), 

thus it is desirable to have one component of the transport function satisfying this 

condition.

Surface-Based Bed Load Equation of Parker (1990)

The surface-based bed load equation of Parker (1990a) consists of three 

matching functions representing different levels of transport intensity:

Wi

* =

11.9 1 - z

0.853 e o

4.5

z50 2 1.59

0.00218 exp 14.2`z - 1j- 9.28`z - 1j

2 : D 1.0 # z50 # 1.59

0.00218 z14.2 z50 1 1.0

Z

[

\

]

]

]

]

]

]

(10)

where:

Wi

* = u*

3Fi

`s = 1jgqbi (11)

and Fi is the fraction of the bed surface sediment, calculated from a sample of the 

surface layer with sand and finer sizes (D<2 mm) excluded. Transport rates are 

calculated only for the gravel fraction of the surface layer; hence the subscripts s

and g are used throughout the following explanation.

The parameter φ is formulated from a nested set of equations. The first of 

these is a hiding function:

z = ~zsg Dsg

Di e o

-0.0951

(12)

The second equation is a function that accounts for changes in the mean 

grain size and sorting of the bed surface as the shear stress and transport rate 

increase. Parker (1990a) termed this a straining function:

~ = 1 + vz0

vz `~0 - 1j (13)
USDA Forest Service RMRS-GTR-223. 2009. 11

where σ0 and ω0 are functions of φsg given in figure 5. For typical values of 

sediment sorting and bed shear stresses not far above the threshold for bed load 

transport (say, 1.0<φsg<1.5), the function ω takes on values between 1.0 and 

about 0.8.

The third function is the relation for transport stage, expressed in terms of 

the Shields stress:

zsg = xrsg

*

xsg

*

(14)

where xrsg

* is the reference Shields stress, assumed to be 0.0386 (Parker 1990a). 

The surface-based Shield stress xsg

* is defined as:

xsg

* = `s - 1jgDsg

u*

2

(15)

where u* is calculated with the resistance relation presented earlier, Eq. 4, assuming ks

 = 2D90. Bed-load transport calculations are restricted to the main channel. 

For each flow of interest, the model calculates values of W* for each size fraction and weights those values by the proportion of that size fraction on the bed 

surface, Fi

. The instantaneous width-integrated bed load transport rate for each 

size fraction is then:

Qbi

= `s - 1jg

Wi

*FiB u*3ts (16)

where B is the channel width and u* is calculated with respect to the main channel only. The predicted values of Qbi

 are summed over all size fractions to get the 

total bed load, Qb.

Figure 5. Parameters σ0 and ωo

as functions of φsg in Parker’s 

surface-based equation.
12 USDA Forest Service RMRS-GTR-223. 2009.

Substrate-Based Bed Load Equation of Parker-Klingeman-Mclean (1982)

The Parker-Klingeman-McLean equation (Parker and others 1982) computes transport rates on the basis of a single grain size—the median grain size of 

the substrate, D50sub. This equation likewise has three components:

W* =

11.2 1 - z50

0.822 e o

4.5

z50 2 1.65

0.0025 exp 14.2`z50 - 1j- 9.28`z50 - 1j

2 : D 0.95 # z50 # 1.65

0.0025 z50

14.2 z50 1 0.95

Z

[

\

]

]

]

]

]

]

(17)

where φ50 is the normalized Shields stress, formulated using the D50 of the 

substrate:

z50 = xr

50

*

x50

*

, x50

* = `s - 1jgD50sub

u*

2

(18a,b)

and xr

50

* is a reference Shields stress with a value of 0.0876.

Here, the function W* is modified slightly for φ50<0.95 according to the 

surface-based equation of Parker (1990). The original equation sets W* = 0 for 

φ50<0.95. The modification in (17) does not change the equation substantially 

but it has the advantage of giving positive bed load transport rates for all flow 

conditions.

Parker and others (1982) did not specify an appropriate flow resistance relation for use with this equation. However, the equation is substrate based, thus ks

should be calculated with respect to the substrate. Based on limited data in Parker 

and others (1982), the ratio of surface D90 to substrate D50 ranges between 4.3 

and 5.1 with an average value of ~5.35. Using this result and the previous assumption that the roughness height is twice of the surface D90, we assumed:

ks = 10.7D50sub (19)

As with the previous equation, transport calculations are restricted to the 

main channel. Values of W* are computed with Eq. 17 and the instantaneous 

transport rates are averaged over the width of the channel to get the total bed load:

Qb = `s - 1jg

W*B u*

3ts (20)

Substrate-Based Equation of Parker-Klingeman (Parker and Klingeman 1982)

The substrate-based bed load transport equation of Parker and Klingeman 

(1982) can be written as follows:

f

i

Bu*

3

Pi `s - 1jgQb = 11.2 1 - 0.853

x50

*

xr

50

*

D50

Di b l

b

> H

4.5

(21)
USDA Forest Service RMRS-GTR-223. 2009. 13

where pi is the proportion of the bed load in the i-th size class; f

i

 is the fraction of 

the substrate in the i-th size class; x

50

* is the Shields stress, formulated in terms of 

the median grain size of the substrate, D50sub; xr

50

* is the reference Shields stress, 

also formulated in terms of D50sub; Di

 is the mean grain size of the i-th size class; 

and β is a hiding coefficient. The hiding coefficient, β, and reference Shields 

stress, xr

50

* , are given by Parker and Klingeman as:

b = 0.018 , xr

50

* = 0.0876 (22a,b)

The same approach is used in calculating flow resistance as in the previous 

equation, with the assumption ks

 = 10.7 D50sub. Instantaneous transport rates are 

computed for each size fraction with (21) and averaged over the channel width:

Qbi

= pi `s - 1jg

Wi

* fiB u*

3ts (23)

The calculated values of Qbi

 are then summed over all size fractions to get 

the total bed load, Qb.

Surface-Based Two-Fraction Equation of Wilcock (2001)

The surface-based two-fraction equation of Wilcock (2001) is a calibrated 

procedure that separates the bed load into two fractions—sand and gravel—and 

determines the reference shear stress for each fraction on the basis of bed load 

measurements. In theory, any suitable bed load transport equation can be calibrated using this approach. Wilcock (2001) recommended the following equations 

for estimating gravel and sand transport rates separately:

Wg

* = 11.2 1 - 0.846 x

xrg b l

4.5

, x 2 xrg

0.0025 xrg

x b l

14.2

, x # xrg

Z

[

\

]

]

]

]

(24a)

and

Ws

* = 11.2 1 - 0.846 x

xrs b l

4.5

(24b)

In these equations, the subscripts g and s refer to the gravel and sand fractions, respectively; τrg

 is the reference shear stress for gravel; and τrs is the 

reference shear stress for sand. Values for τrg

 and τrs are obtained from least 

squares regression based on bed load measurements. The separate values of W*

are then weighted by the respective fractions of gravel and sand on the bed, f

g, 

and f

s

 = 1 – fg . Values of f

g and fs

 are determined from a representative sample 

of the bed surface.
14 USDA Forest Service RMRS-GTR-223. 2009.

The relation used to estimate flow resistance is similar to the one used 

in the previous equations, with a slight difference in the assumed roughness, 

ks

 = 2D65. Instantaneous transport rates for the sand and gravel fractions are calculated separately:

Qbg = `s - 1jg

Wg

* fg ts Bu*

3

, Qbs = `s - 1jg

Ws

* fs ts Bu*

3

(25a,b)

and then summed to give the total load,

Qb = Qbg

 + Qbs (26)

Surface-Based Relation of Wilcock and Crowe (2003)

Wilcock and Crowe (2003) developed a transport relation based on the 

full grain size distribution of the bed surface, including the sand. This relation 

includes an additional function that accounts for the nonlinear effect of sand content on gravel transport rates. The basic form of the equation is as follows:

Wi

* =

0.002z7.5 z 1 1.35

14 1 - z0.5

0.894 e o

4.5

z $ 1.35

Z

[

\

]

]

]

]

(27)

where

z = xri

x , xri

= xr50 D50

Di b l

b

(28a,b)

The exponent in the hiding function b is calculated from:

b = 1 + exp 1.5 - Dm

Di b l

0.67

(29)

where Dm is the mean grain size of the bed surface. The reference shear stress for 

Dm is found using the Shields stress relation:

xrm

* = `s - 1jgDm

xrm (30)

and an empirical function that accounts for the variation in xr

*

 with changes in 

sand content:

xrm

* = 0.021 + 0.015 exp (- 20Fs) (31)

where Fs

 is the percent of sand on the bed surface.

Values of W* are calculated for each size fraction then weighted by the 

proportion of that size fraction on the bed surface, Fi

. Those values are summed 

over all sizes to get the instantaneous width-integrated bed load transport rate:

Qbi

= `s - 1jg

Wi

* FiB u*

3ts (32)
USDA Forest Service RMRS-GTR-223. 2009. 15

Procedure of Bakke and others (1999)

The procedure of Bakke and others (1999) uses site-specific measurements 

of bed load and bed material to calibrate two parameters in the equation of Parker 

and Klingeman (1982). The calibration procedure requires at least one, and preferably five to 10 bed load samples, plus samples of the bed material (surface 

or substrate). The procedure computes the hiding function exponent, β, and reference Shields stress for D50 (xr

50

* ), using a least squares fit to the calibration 

data. The procedure seeks to minimize the sum of squared differences between 

computed and measured transport rates based on different values of β and xr

50

* . 

Transport rates are computed for each size fraction (surface or the substrate, depending on available data) and then summed to get the total bed load.

Data Input Requirements

Table 1 provides a list of the information and input variables needed to run 

the individual transport models. The first three rows list information common to 

all the models and include measurements of the reference-reach cross section, 

reach-average slope, and water discharge. Subsequent rows list particular sediment parameters derived from representative samples of the bed surface sediment, 

substrate, and/or bed load. The rationale for using surface versus substrate as input to a bed load transport model is discussed in Wilcock and others (2009). The 

models developed by Parker (1990a) and Wilcock and Crowe (2003) require data 

obtained from surface samples, otherwise known as pebble counts (table 1). The 

models developed by Parker and others (1982) and Parker and Klingeman (1982) 

require information obtained from bulk samples of the substrate (table 1). The 

approaches described by Wilcock (2001) and Bakke and others (1999) require 

actual bed load measurements, which serve as the basis for model calibration.

Table 1. Summary of input variables and information needed to run individual transport models.

Parker PKM PK W WC B

Channel cross section X X X X X X

Reach-average water surface slope X X X X X X

Discharge X X X X X X

Bed surface grain size distribution X X X

• f

s, fg X 

Substrate grain size distribution X X

• D50 X 

Bed load sample data X X
16 USDA Forest Service RMRS-GTR-223. 2009.

Software Operation

Model calculations and decisions are carried out in Visual Basic, a programming application that is included with recent versions of Microsoft Excel. This 

structure allows for straightforward cut-and-paste transfer of input and output 

data from one spreadsheet to another or any other program that accepts data in 

tabular format. Data must be entered in metric units. The program is designed 

to operate on WINDOWS-based PCs using Microsoft Excel, vers. 2000 and 

higher. The program does not run on Macintosh- or Unix-based platforms. Users 

should have a basic familiarity with the Microsoft Windows operating system 

and Microsoft Excel.

If you don’t have the program for calculating bed load transport rates 

(BAGS), you may download a copy from the STREAM team website:

http://www.stream.fs.fed.us/publications/software.html.

We recommend that you create a folder specific to your particular project 

and place the application program in this folder. Ancillary files or spreadsheets 

containing pertinent field data, including cross sections, water-surface profiles, 

discharge values, and/or sediment size distributions, should also be placed in the 

project folder. The steps involved in operating the software are listed below. In 

some cases, we have added sidebars that provide a comment or an explanation of 

the rationale behind an individual step or model calculation.

1. Before starting the program you should check the macro security level for 

your version of MS-Excel. It should be set at “Low” to run the BAGS program. To check the macro security level, start MS-Excel. Use the “Tools” 

menu and select “Options.” Select the “Security” tab and click “Macro 

Security.” Under the “Security Level” tab, enable “Low” and select “OK.”

2. Start the program by clicking the BAGS icon. You will see a dialog box indicating that the program contains “macros.” Select “Enable Macros.”
USDA Forest Service RMRS-GTR-223. 2009. 17

3. The BAGS background page will appear. Click the link to apply bed load 

transport equations. A users agreement will appear. Please read the conditions specified in the users agreement. If you agree with the terms, select “I 

Agree.”
18 USDA Forest Service RMRS-GTR-223. 2009.

4. A box listing the equations will appear. If you are uncertain about which 

equation to use, select “HELP.” If you would like additional information on 

an equation, including the original journal reference and a list of data input 

requirements, select the small box on the far left adjacent to the open box 

next to the equation. When you are ready to proceed, check the open box 

next to the equation and select “ACCEPT.”
USDA Forest Service RMRS-GTR-223. 2009. 19

5. You will now see a dialog box asking if you would like to apply a roughness 

correction. If you have a field-based estimate of Manning’s n, select “YES” 

and enter that value when the next dialog box appears. If you don’t have 

an estimate of Manning’s n but would like to apply a roughness correction 

anyway, we suggest you consult the references listed earlier in the section on 

flow resistance.

6. A dialog box will appear indicating two choices for channel geometry:

Ideally, you should have cross section data available to enter at the next step; 

however, the software can approximate flow conditions for a known width 

and discharge by assuming a rectangular cross section.
20 USDA Forest Service RMRS-GTR-223. 2009.

7. The next box allows you to enter data for the reference reach cross section. 

Select “CLEAR FORM” and enter values of distance and elevation for the 

cross section.

8. A new page will appear with a plot of the reference reach cross section. If you 

want to include the floodplain in the hydraulic calculations, you must assign 

values of Manning’s n to the left- and right-portions of the floodplain.
USDA Forest Service RMRS-GTR-223. 2009. 21

9. A new page will appear, asking for input on the grain size distribution. Any 

specific intervals for the grain size, D, may be used, however, the sizes must 

be listed from lowest to highest, for example: 1, 2, ...., 128 mm. Select 

“CLEAR FORM” and enter the percent finer for each size class in the appropriate columns.

Note: Depending on the bed load equation you have selected, you may then 

see another dialog box asking if you would like to include grain sizes less 

than 2 mm in the calculation.

Sidebar 1

Comment: Parker (1990a) formulated his surface-based relation by excluding sand from the analysis and computation. Subsequently, Wilcock and 

Crowe (2003) showed that sand has a strong influence on transport thresholds and transport rates. Other work indicates that, in many rivers, the bed 

load is predominantly sand (Hassan and Church 2001; Lisle 1995; Mueller 

and others 2005), consistent with field data showing that the substrate is 20 

to 30 percent sand (Pitlick and others 2008). Given this information, we don’t 

see a clear reason for excluding sand from the calculation unless it can be 

determined that these sizes will move in suspension.
22 USDA Forest Service RMRS-GTR-223. 2009.

10. The next dialog box will ask you to select one of potentially three values for 

slope.

Sidebar 2

Note: If you select option (b), you will need to supply estimates of 

minimum and maximum discharges, otherwise the program may 

stall or give unreasonable results. If you are not sure what to enter 

for minimum and maximum values, we can offer several suggestions:

1. If the site is gaged and the gage record includes at least 

10 years of peak-flow values, you could use the lowest and 

highest values to define a reasonable range of discharges or

2. If the site is not gaged, but you have a rough idea of the bankfull discharge (or perhaps an estimate of an average-size flood, 

such as the 2-year flood), we suggest setting the minimum 

discharge to approximately half this value and setting the maximum discharge to 2 (or maybe 4) times this value.

These limits are somewhat arbitrary, but likely to encompass most 

of the range over which bed load transport occurs (this point is 

discussed in more detail in the Assessing Model Output section).
USDA Forest Service RMRS-GTR-223. 2009. 23

11. The next dialog box asks for information on discharge.

12. At this point, the steps for entering data are complete. The next dialog box 

will ask if you would like to create a file to save the input data.
24 USDA Forest Service RMRS-GTR-223. 2009.

13. The program performs a series of calculations and stores the results in a new 

workbook. When the calculations are complete the following box will appear.

14. The new workbook contains eight separate tabs. The first tab lists the 

equation(s) used, locations of the input/output data, and user and date. Other 

tabs list input/output values or show plots generated for the particular model 

run. You should check the Input tab to ensure that the parameter values and 

data were entered correctly. The Output tab lists values of discharge (m3/s), 

bed load transport rate (kg/min), transport stage, τ /τr (dimensionless), maximum water depth, hydraulic radius, and, depending on the equation specified, 

bed load transport rates of individual size fractions.
USDA Forest Service RMRS-GTR-223. 2009. 25
26 USDA Forest Service RMRS-GTR-223. 2009.

Assessing Model Output

Given that bed load transport rates can vary by many orders of magnitude, 

almost any result produced by the BAGS model can be considered “reasonable.” 

The questions are: What do we mean by reasonable? How do we distinguish 

between a potentially valid result and one that is implausible, and how do we 

assess model output if there are no data against which to compare the results? 

The sections below discuss possible strategies for evaluating the reasonableness 

of model results, starting with some comments about the data itself and potential 

input errors. The subsequent section goes through several examples illustrating 

how and why model results can differ from field measurements. The final section 

focuses on the common situation where no measurements of bed load transport 

have been made, thus there is little basis for comparison.

How Good Are the Raw Data?

Before discussing approaches for interpreting model output, it is worth reminding ourselves that a model calculation is only as good as the input data. The 

phrase “garbage in, garbage out” certainly applies in this case.

All bed load transport relations are sensitive to estimates of the available 

grain sizes and the available shear stress. Small differences in the estimation of 

these two values can lead to very large (orders-of-magnitude) differences in calculated transport rates. In addition to potential errors and uncertainties in the 

input variables, there may be conditions within a watershed that severely limit the 

usefulness of bed load transport calculations. The following points are important 

to consider:

• Is the site or the watershed appropriate for this type of analysis? Bed load 

transport relations are formulated with two assumptions in mind: (1) the mass 

flux of sediment is related in a consistent way to the physical properties of the 

flow (force or power per unit bed area) and (2) all the grain sizes capable of being transported by a particular flow are indeed available to be transported. Bed 

load transport equations predict the bed load transport capacity, which Gilbert 

(1914, p. 35) defined as the “maximum load a river can carry,” presumably for 

a given flow and sediment size. Generally, this definition only applies to alluvial channels where the bed and banks are made of sediment that was carried 

by the river itself, as opposed to some other geomorphic process (for example, 

mass-wasting or glacial processes). This can be a serious limitation in actively 

eroding bedrock channels and mountain streams where the sediment supply is 

driven more by the rate of weathering and/or hillslope transport than by fluvial processes. In these cases, the channel potentially transports only as much 
USDA Forest Service RMRS-GTR-223. 2009. 27

sediment as is supplied, and the sediment flux bears only a weak relation to the 

flow strength. Transport under these conditions is said to be “supply limited,” 

and there is no reason to expect that a transport relation will predict the sediment flux accurately.

• Is the bed material very heterogeneous? Can you determine, at least qualitatively, whether the bed material samples are representative of the reach of 

interest? The bed sediment in gravel channels can vary significantly from 

place to place, thus a surface or substrate sample from only one location may 

not yield a grain size distribution that is representative of that setting. Figure 6 

shows grain size distributions of the surface and substrate sediment sampled 

across a single meander bend within a 130-m reach of the Colorado River in 

Rocky Mountain National Park, Colorado. This is a relatively stable gravelbed river with a mildly sinuous channel pattern, yet these measurements show 

that there is a wide range in grain size of both the surface and the substrate. 

The variation in grain size is likely to be much higher in morphologically complex channels (braided or wandering rivers), thus sampling intensity should be 

increased to determine particle size distributions more accurately. The clearest guidance on procedures for sampling and analysis of the bed sediment in 

gravel channels can be found in Church and others (1987) and Bunte and Abt 

(2001). We cannot stress the importance of taking as much time as necessary to 

obtain representative samples of the bed material—it makes no sense to spend 

hundreds of hours taking measurements of water discharge or channel properties and then spend 1 hour sampling the bed material. The problems associated 

with a potentially inaccurate estimate of the grain size are illustrated later in 

the discussion of transport estimates for the South Fork Samon River (p. 35).

Figure 6. Grain size distributions of the bed surface and substrate of the Colorado River in Rocky Mountain National Park, CO. 

The samples were taken across a single meander bend, at different locations corresponding to (A) the middle portion of 

bend and (B) the outer portion of the bend (source: Clayton and Pitlick 2007).
28 USDA Forest Service RMRS-GTR-223. 2009.

• Are the estimates of boundary shear stress reasonable? The variations in shear 

stress within a channel reach are likely to be at least as large as the variations in 

grain size. The problems of estimating shear stress are particularly important 

in high-gradient channels (slopes greater that ~1 percent) where a significant 

proportion of the total shear stress acting on the streambed may be “lost” due 

to form drag on immobile boulders and logs (Buffington and Montgomery 

1999; Mueller and others 2005; Wiberg and Smith 1990; Wilcox and Wohl 

2006). The BAGS model attempts to correct for these effects, but the problem is not restricted to small, headwater streams. Natural undulations in bed 

topography caused by pools and riffles force changes in depth and velocity, 

thus altering the distribution of shear stress (Sear 1996; Whiting and Dietrich 

1991). The BAGS model is not likely to yield accurate estimates of bed load 

transport in highly sinuous channels, braided channels, or channels with sharp 

changes in gradient.

Comparisons of Model Output With Bed Load Measurements

As noted in Wilcock and others (2009), a few measurements of bed load can 

aid significantly in assessing the uncertainty of a transport calculation. However, 

even in a best-case scenario, one should expect that bed load measurements taken 

at the same location at the same discharge will vary by at least an order of magnitude. This is especially true of samples taken at flows near the threshold for 

motion. The following examples illustrate the range of results one can expect in 

comparing output from the BAGS model with field measurements of bed load. 

The data sets used for these illustrations are based on measurements taken in different types of gravel-bed streams and rivers in the western United States. The 

discussion emphasizes “goodness-of-fit,” not as a statistical concept, but as a 

means of comparing observed values with predicted values.

Visualizing Bed Load Transport as a Function of Discharge

Measurements of bed load taken in the field and laboratory indicate that 

transport rates increase by orders of magnitude for relatively small changes in 

discharge and shear stress. The nonlinearity in transport processes is reflected 

by the high value of the exponent in the relation between bed load and shear 

stress. In the equation of Wilcock and Crowe, for example, W* \ x7.5

 in the range 

of low-moderate shear stresses, x 1 1.35xr

. Flows in this range typically carry 

a high percentage of the total annual bed load (Andrews 1994; Andrews and 

Nankervis 1995; Emmett and Wolman 2001; Mueller and others 2005; Torizzo 

and Pitlick 2004; Van Steeter and Pitlick 1998; Whiting and others 1999). For 

these reasons we recommend plotting the relation between flow and transport 
USDA Forest Service RMRS-GTR-223. 2009. 29

using logarithmic scales for both axes. Otherwise, it is essentially impossible to 

visually assess the quality of the data, especially in the range of flows where most 

of the transport occurs.

Field hydrologists are accustomed to using water discharge, rather than 

shear stress, as the primary index of flow properties. Discharge is commonly 

measured in the field, and values of discharge associated with individual transport measurements are typically listed in published reports, whereas estimates of 

shear stress are not. Discharge thus emerges as a natural variable for associating 

transport rates and flow.

A statistical goodness-of-fit-test helps in judging the strength of the relation between bed load discharge, Qb, and water discharge, Q, but it may be more 

instructive to ask: Do the field data follow the expected (modeled) trend? To answer this question, we suggest using a simple test focusing on the parameters of 

a power-law relation, Qb = aQb, where a is a coefficient and b is the slope of the 

observed transport relation. If the observed exponent b differs greatly from the 

expected value, then you might want to look more carefully at the measurements. 

To find the expected value of b, we combine the continuity equation with the 

Manning equation and write a relation for shear stress as a function of discharge:

x = tg B

Qn e o

0.6

S0.7 (33)

where the variables as defined in the Bed Load Transport equations section. This 

equation indicates that for constant values of ρ, g, n, B and S, τ varies with the 0.6 

power of Q (we should note, however, that in typical channels, n will decrease 

with Q, and B will increase, thus the exponent may be expected to be less than 

0.6—this depends on the particular site characteristics). Using this result, we can 

recast one of the transport relations discussed earlier in terms of Q. The Wilcock 

and Crowe equation for low-moderate transport stages (τ<1.35τr

) is used as an 

example. This equation can be written as:

Qb = `s - 1jg

0.002ts B t

x b l

1.5

xr

x b l

7.5

= kB(x) 9.0 (34)

where Qb is the total bed load (mass transport rate, integrated across the channel 

width, B) and k is a value incorporating the various constants, plus the value of 

τr

. Collecting the various terms and writing the total bed load as a function of 

discharge we get:

Qb ∝ (Q)5.4 (35)
30 USDA Forest Service RMRS-GTR-223. 2009.

The exponent of 5.4 in Eq. 35 is specific to this example, and we should not 

think of it as a hard number. We say this for several reasons. First, the specific 

value depends on the exponent in the transport relation (b = 7.5 in the relation 

used in this example). Second, the derivation of (35) was simplified by assuming 

B, n, and S were constant, which is not likely to be the case in natural channels. 

Third, as Barry and others (2004) have suggested, the rating curve exponent may 

be influenced by other factors, such as runoff regime (rainfall versus snowmelt) 

and sediment supply. Nevertheless, in the absence of strong constraints on bed 

load transport, or systematic errors in sampling, Eq. 35 serves as the basis for 

interpreting observed relations between Q and Qb. This point is illustrated in 

figure 7, which shows relations between discharge and bed load transport in two 

small gravel-bed streams, Oak Creek and Halfmoon Creek. Bed load was measured in Oak Creek using a vortex sampler (Milhous 1973), whereas bed load 

was measured in Halfmoon Creek using a series of quasi-stationary traps (Bunte 

and Swingle 2005). These data are of high quality, and in both cases there is a 

strong correlation between bed load transport rate and discharge. The exponents 

are slightly different from each other, but they bracket the proposed value of 5.4 

given in Eq. 35.

Figure 7. Observed relations between discharge and bed load transport. Data for Oak Creek from Milhous (1973); 

data for Halfmoon Creek from K. Bunte and Swingle 2005.

A B
USDA Forest Service RMRS-GTR-223. 2009. 31

The steep slopes exhibited by these data reinforce the point made earlier 

that transport rates can vary by several orders of magnitude over the range of 

observed discharges. Small differences in discharge produce large differences in 

transport rate. More to the point, the data depicted above compare favorably with 

the derived transport relation (35), thus we might use these examples as a basis 

for comparing measurements in other settings where there is more uncertainty in 

the data. Four additional examples follow, with the predicted transport relations 

added for comparison. The predicted relations are developed using the surfacebased equation of Parker (1990a) and appropriate data and information from field 

studies of flow and bed load transport in streams and rivers in Idaho (King and 

others 2004).

Selway River Near Lowell, ID

The Selway River is an example of a large self-formed gravel-bed river with 

no apparent limit to sediment supply. The data used in this example are based on 

measurements taken at a gauging station operated by the USGS, station number 

13336500. Site characteristics are described as follows:

• Slope = 0.0021

• Surface D50 = 186 mm

• Subsurface D50 = 24 mm

• Drainage area = 4947 km2

• Bankfull discharge = 652 m3/s

Most of the bed load measurements at this location were taken during periods of peak snowmelt runoff in 1994 and 1995. Several additional samples 

were taken during floods in December 1995 and May 1997. The data set includes 

72 paired measurements of discharge and bed load; field surveys of bed- and 

water-surface slope; and multiple samples of the surface and substrate. Bed load 

was measured with the Helley-Smith sampler, with 40 percent of the samples 

taken at discharges greater than one-half of the bankfull discharge.

The data from the Selway River form a very tight relation between water 

discharge, Q, and bed load discharge, Qb (fig. 8A). A power law fit of these data 

yields the equation:

Qb = 3.85E-12 Q4.92 (36)
32 USDA Forest Service RMRS-GTR-223. 2009.

where Q is in cubic meters per second and Qb is in metric tons per day. This relation is statistically significant, with r2 = 0.92 and p<0.001. The exponent in this 

equation is relatively high (~5), similar to the best-fit relations for Oak Creek and 

Halfmoon Creek. The panel to the right (fig. 8B) compares the observed transport 

rates with the predicted transport rates estimated using the surface-based equation of Parker (1990). In this case, the predicted bed load transport rates match 

the observations quite closely. This result is encouraging, but somewhat of an 

exception—this is one of the few data sets we have worked with where the transport relation fits the data closely with no tuning or adjustment in the parameters.

Figure 8. Relations between discharge and bed load transport rate, Selway River near Lowell, Idaho, USGS, station number 

13336500. (A) Shows a least squares fit of the data and (B) shows the predicted relation obtained with the surface-based 

equation of Parker (1990a), with sand-size fractions included.

A B
USDA Forest Service RMRS-GTR-223. 2009. 33

Rapid River

The Rapid River is an example of a steep gravel- and cobble-bed river. The 

site description indicates that the reach is bordered by a floodplain, thus it would 

be reasonable to conclude that, even if the reach is steep, the channel is selfformed. This gage is operated by the U.S. Forest Service. Site characteristics are:

• Slope = 0.0108

• Surface D50 = 79 mm

• Subsurface D50 = 16 mm

• Drainage area = 280 km2

• Bankfull discharge = 17.7 m3/s

Bed load samples were taken at this location from 1990 through 2004; the 

majority of samples were taken during spring runoff. The data set for this station 

includes: 190 paired measurements of discharge and bed load; surveys of bedand water-surface profiles; and samples of the surface and substrate. Bed load 

was measured with the Helley-Smith sampler, with 38 percent of the bed load 

samples taken at discharges greater than one-half of the bankfull discharge.

The observed transport relation for the Rapid River is relatively strong 

(fig. 9), although the data exhibit considerable scatter for measurements taken 

below ~10 m3/s. The question is: How much weight should we give the low-flow 

samples? Using all data, the best-fit equation is:

Qb = 0.0086 Q2.23 (37a)

This relation is statistically significant (r2 = 0.59; p<0.001). However, it is 

evident that at high discharges this equation would underestimate bed load transport rates by more than an order of magnitude. Thus, if we consider only flows 

greater than 10 m3/s (~60 percent of bankfull), the relation formed by the data is 

much different. The least squares equation in that case is:

Qb = 6.74E-06 Q4.88 (37b)

which is also statistically significant (r2 = 0.88; p<0.001). This second relation 

provides a better fit to the high-flow values and the exponent is greater than 4, 

similar to the equations given in the previous examples. This example illustrates 

a common condition in gravel rivers where bed load transport rates at low discharges are influenced by differences in the availability of sand-sized sediment, 
34 USDA Forest Service RMRS-GTR-223. 2009.

which may be supplied from sources outside the channel. At discharges greater 

than 10 m3/s, particles with the surface layer are beginning to move and the bed 

starts to become the primary source of sediment. Above this point, the modeled 

bed load transport relation (Parker 1990) matches the observations quite well.

Figure 9. Relations between discharge and bed load transport rate, Rapid River, Idaho. (A) Shows two least 

squares relations, one for the complete data set (dotted line) and the other for Q>10m3/s (solid line) and (B) 

shows the predicted transport relation obtained with the surface-based equation of Parker (1990a).

A B
USDA Forest Service RMRS-GTR-223. 2009. 35

South Fork Salmon River Near Krassel Ranger Station, ID

The South Fork Salmon River is an example of a moderate-sized gravel 

river that carries a moderately high sediment load. The site description indicates 

that the study reach is bordered by a floodplain, thus it appears that the channel is 

self-formed. This station is operated by the USGS (station 13331070).

• Slope = 0.0025

• Surface D50 = 38 mm

• Subsurface D50 = n/a

• Drainage area = 855 km2

• Bankfull discharge = 70.8 m3/s

Bed load samples were taken at this location over two time periods, 1985 

to 1986, and 1994 to 1995. The majority of samples were taken during spring 

runoff. The data set includes: 130 paired measurements of discharge and bed 

load; surveys of bed- and water-surface profiles; and samples of the bed surface 

(no substrate samples were taken at this site). Bed load was measured with the 

Helley-Smith sampler, with 57 percent of the samples taken at discharges greater 

than one-half the bankfull discharge.

The observed transport relation at this site appears to be relatively good 

(fig. 10A). The equation of the trend line is:

Qb = 1.97E-04 Q3.00 (38)

which is statistically significant (r2 = 0.62 and p<0.001). The exponent in this 

equation is lower than in previous examples, and it appears that the fitted line has 

a lower slope than the trend formed by the data. This is not an uncommon result 

in fitting a power law to a data set that is not log-linear, or a data set that might 

include outliers. Eliminating the five values corresponding to Q<10 m3/s gives an 

exponent of about 4, similar to the previous examples.

In this example, the comparison between observed and modeled transport 

rates produces an interesting result (fig. 10B). An initial calculation based on 

Parker’s surface-based relation gives the solid curve to the right of the data. This 

curve parallels the data, but is offset to the right, meaning the predicted transport 

rates are much less than the observed transport rates. The difference arises because the reference Shields stress used in the calculations is either too high or the 
36 USDA Forest Service RMRS-GTR-223. 2009.

grain size measured in the field is coarser than the sediment being supplied to the 

river. A second calculation, based on the calibration approach of Wilcock (2001), 

gives the relation indicated by the dotted line. This relation fits the data well, but 

the difference between the two curves suggests either that the sampled grain size 

is not representative of the reach or that much of the bed load is derived from 

sources other than the channel itself.

Figure 10. Relations between discharge and bed load transport rate, South Fork Salmon River, Idaho. (A) Shows 

a least squares fit of the data and (B) shows modeled transport relations. The solid line indicates the surfacebased equation of Parker (1990a) and the dotted line indicates the calibration approach of Wilcock (2001).

A B
USDA Forest Service RMRS-GTR-223. 2009. 37

Trapper Creek

Trapper Creek is an example of a small cobble-bed stream (drainage area 

= 20.7 km2) in a forested area. The study reach is steep (~ 4 percent); however, 

the photograph of the site suggests that the channel is self-formed and possibly 

bordered by alluvial banks. The U.S. Forest Service operates this station.

• Slope = 0.0414

• Surface D50 = 85 mm

• Subsurface D50 = 17 mm

• Drainage area = 20.8 km2

• Bankfull discharge = 2.6 m3/s

Bed load samples were taken at this location from 1986 through 2001. 

Many of these samples were taken during spring runoff. The data set includes: 

115 paired measurements of discharge and bed load; surveys of bed- and watersurface profiles; and samples of the surface and substrate. Bed load was measured 

with the Helley-Smith sampler. In contrast to the previous examples, only 17 percent of the samples were taken at discharges greater than one half the bankfull 

discharge.

The measurements at this site form a relatively well-defined transport relation, although there is significant scatter in the data (fig. 11A). A least squares fit 

of the full data set gives:

Qb = 0.35 Q1.80 (39a)

This relation is statistically significant (r2 = 0.60; p<0.001); however, the 

exponent in the equation and the slope of the line are somewhat low in comparison to the values given in the previous examples. If we restrict the regression to 

flows greater than 1.5 m3/s, which is the point where we might expect clasts within the surface layer to start moving (60 to 70 percent of the bankfull discharge, 

see below), we get a relation with slightly steeper slope and a higher exponent:

Qb = 0.124 Q3.32 (39b)

This relation is not as strong as the previous relation but still statistically 

significant (r2 = 0.31; p = 0.0014). Given this result, we might assume that a 

transport equation would fit the high-flow data reasonably well, as we saw in the 
38 USDA Forest Service RMRS-GTR-223. 2009.

example of the Rapid River. However, when we plot the modeled relation, we see 

that it lies far to the left and above most of the observations (fig. 11B), suggesting 

that predicted loads are two to three orders-of-magnitude higher than observed 

loads. This level of uncertainty would be unacceptable in most situations. It is 

even less reassuring to know that this type of discrepancy is relatively common, 

particularly in steep channels (and, it is exactly this type of problem that leads 

people to question the utility of bed load transport equations). However, there is 

a straightforward explanation in this case, which probably holds true for many 

other high gradient channels. Specifically, Trapper Creek is a steep stream (S = 

0.04) with very coarse bed material (D90 = 136 mm), thus it is likely that a significant proportion of the total shear stress acting on the bed and banks is lost as 

“form drag” on large immobile grains and/or woody debris. As a result, it takes 

proportionally much more flow (and shear stress) to move the bed sediment than 

is actually available (see the discussion of “grain stress” in Wilcock and others 

2008). The BAGS model does not fully correct for these effects, thus the model 

will tend to overpredict bed load transport rates in very high gradient streams. 

The alternative is to use a calibrated approach (Bakke and others 1999 or Wilcock 

2001) taking as many bed load samples as possible at flows near the threshold 

for motion of surface-layer (framework) particles. As discussed in the next section, this flow typically occurs at approximately 60 to 70 percent of the bankfull 

discharge. If we follow this approach and selectively use observations from the 

highest flows to calibrate the transport equation, we get the relation shown by 

the dashed line in figure 10b. This relation matches the high-flow observations 

rather well, yet clearly under-predicts the bed load at lower flows. This may not 

be a cause for much concern; in typical gravel channels, flows less than about 

half of the bankfull discharge cumulatively carry a relatively small fraction of the 

total annual bed load (Emmett and Wolman 2001; Schmidt and Potyondy 2004; 

Torizzo and Pitlick 2004; Whiting and others 1999).
USDA Forest Service RMRS-GTR-223. 2009. 39

Figure 11. Observed (left) and modeled (right) relations between discharge and bed load transport rate, Trapper 

Creek, Idaho. In (A), the dashed line indicates the best-fit relation for the full data set, while the solid line is 

fit to a subset of the data, Q>1.5 m3/s. In (B), the solid line indicates the surface-based equation of Parker 

(1990a) and the dotted line indicates the relation obtained with the calibrated approach of Wilcock (2001).

A B

Assessing Model Output Without the Benefit of Bed Load Data

Significant time and effort were required to develop the data sets used in 

the preceding examples, and more often than not, the analyses required some 

tuning to achieve the best results. What are the chances then of developing accurate transport relations when you have no bed load data to serve as the basis 

for comparison? There isn’t a definitive answer to this question; however, we 

can approach the problem by considering some of the conditions associated with 

transport measurements and observations elsewhere. We can start by asking two 

questions:

• Does the relation fail to predict any appreciable bed load transport at flows 

equal to the bankfull discharge?

• Does the relation predict appreciable, or even significant, bed load transport at 

flows less than about half of the bankfull discharge?

If the answer to either one of these questions is yes, the results of the transport calculation should be examined more carefully. We base this suggestion 

on the figure below, which summarizes results obtained by Mueller and others 

(2005) in the their analysis of bed load transport thresholds in natural channels. 

They compiled flow and bed load transport data for 45 gravel-bed streams and 
40 USDA Forest Service RMRS-GTR-223. 2009.

rivers in the western United States and Canada. For each of the 45 data sets, they 

plotted the relation between the dimensionless transport parameter, W*, and dimensionless shear stress, τ*. Then, following the procedure of Parker and others 

(1982), estimated the reference dimensionless shear stress, τr

*, corresponding to 

the reference transport rate of W* = 0.002. Using the estimated values of τr

 and 

local hydraulic relations, they then computed the discharge associated with the 

reference transport rate and termed this the reference discharge, Qr

. Figure 12 

shows a frequency distribution of Qr

, expressed as a ratio to the bankfull discharge, Qbf. The distribution of Qr

 is very nearly symmetric, with a median value 

of 0.67 (dashed line) and a mean of 0.68. This plot indicates that under natural 

(undisturbed) conditions, gravel-bed rivers typically begin mobilizing appreciable amounts of bed load when flows reach about two-thirds of the bankfull 

discharge. In only a few cases does the reference discharge lie above the bankfull 

discharge. Likewise, there are only a few cases where the reference discharge is 

less than ~30 percent of the bankfull discharge. Although conditions are likely 

to vary from one stream to another, we expect that the lower limit of appreciable 

transport will lie somewhere in this range.

Figure 12. Frequency 

distribution of the 

ratio of the reference 

discharge for initiating 

bed load transport to 

the bankfull discharge, 

Qr,/Qbf . Vertical 

dashed line indicates 

the median value (from 

Mueller and others 

2005).

The discussion in the preceding paragraph leads to another question:

• What do we mean by appreciable transport? Is there a practical lower limit below which transport rates are so small that the loads can be considered 

negligible?

To answer this question, we should step back a bit and recall that the bed 

load transport relations used here compute finite loads for all flows, even relatively small ones. This is an intended feature of the models, rooted in the basic 

concept that entrainment and transport are probabilistic processes. Thus, even at 
USDA Forest Service RMRS-GTR-223. 2009. 41

very low flows, there is a small but finite probability that some bed load particles 

will move. This point is illustrated nicely by the bed load rating curves for Oak 

Creek and Halfmoon Creek presented earlier in figure 7. Using Halfmoon Creek 

as an example, the rating curve relation indicates that at a discharge of 2 m3/s 

(~30 percent of the bankfull discharge), the creek carries a bed load of ~10-2 g/s 

integrated across the channel. How much sediment is that? We won’t go through 

the details, but if we assume quartz-density sediment (ρs

 = 2.65 g/cm3) and continuous transport, then a transport rate of 10-2 g/s equates to ~14 cm3 of sediment 

per hour—a small hand full. If we maintain the same discharge for 8 days (the 

average annual frequency), the cumulative load is about 7 kg, which is much 

less than 1 percent of the total annual bed load (fig. 13). If we then double the 

discharge to 4 m3/s (~60 percent of bankfull), the transport rate increases by two 

orders of magnitude to about 1.0 g/s. The bed load carried by that flow is still 

small—less than 2 percent of the total annual load (fig. 13)—but perhaps not so 

small that we would consider it to be negligible. The point here is clear. In many 

streams there may not be an absolute lower limit to bed load transport; however, 

it might be argued that there comes a point where the loads are so small that they 

can be considered negligible.

Figure 13. Relations 

between flow 

frequency and 

cumulative bed load 

transport for flows 

ranging from 0.4 to 

1.6 times the bankfull 

discharge, Halfmoon 

Creek, CO.

The clearest guidance we can give is to suggest that at flows higher than 

about 2/3 of bankfull, bed load transport rates are likely to be non-negligible. 

The Parker transport models, for example, were developed under the assumption 

that the reference transport rate of W* = 0.002 corresponds to discharges that are 

high enough to start mobilizing clasts within the armor layer. Mueller and others 

(2005) used the same assumption in their analysis, and listed the unit-width bed 

load transport rates, qb, for each of the discharges corresponding to W* = 0.002. 

Their results, summarized in figure 14, indicate that values of qb at the reference 
42 USDA Forest Service RMRS-GTR-223. 2009.

discharge typically fall in the range between 0.001 and 0.005 kg/m/s. Based on 

this result, we suggest that at flows near reference discharge, predicted bed load 

transport rates should fall somewhere in the range from 0.001 to 0.005 kg/m/s.

Figure 14. Frequency 

distribution of unit 

width bed load 

transport rates, qb, at 

flows corresponding 

to the reference 

discharge, Qr, in 45 

gravel-bed streams 

and rivers in the United 

States and Canada 

(data from Mueller and 

others 2005).

Having posed the question of lower limits, we should consider the parallel 

question of upper limits. Specifically,

• Is there an upper limit to bed load transport or is there a way to constrain highflow estimates of transport?

The short answer to the first part of this question is, yes, but in reality, any 

upper limit on bed load transport is not likely to be achieved in typical gravel bed 

streams, except during a debris flow. The second part of the question—constraining high-flow estimates—is a bit harder to answer. The clearest guidance we can 

give here is based on results from previous field studies. Table 2 provides a short 

list of transport rates measured at high flows in various river systems throughout 

the western United States. The sites are located in different regions, encompassing a range of conditions (hydrology, rock type, forest cover, and so forth). The 

maximum transport rates were selected from data given in the various reports 

or from graphs showing the individual transport relations. At the low end of the 

spectrum, there are several sites with maximum transport rates on the order of 

10-2 kg/m/s. Several other sites have maximum bed load transport rates on the 

order of 10-1 kg/m/s. The highest value of 3.9 kg/m/s was measured on the North 

Fork of the Toutle River near Mount St. Helens, Washington. The headwaters 

of this river were severely disturbed during the May 1980 eruption of Mount 

St. Helens, thus transport rates measured downstream reflect unusual conditions 

within a highly erosive and unstable watershed. It is unlikely that gravel-bed 

streams in more stable settings will carry loads that high very often, if ever. Thus, 
USDA Forest Service RMRS-GTR-223. 2009. 43

if the predicted transport rates exceed 10 kg/m/s you should question the result. 

Reasonable values for maximum unit-width transport rates are more likely to fall 

in the range of 10-2 to 10-1 kg/m/s; however, this still leaves quite a bit of uncertainty and reinforces comments made earlier that a few bed load measurements 

will go a long way toward constraining such estimates.

Table 2. Examples of high bed load transport rates taken from measurements on gravel-bed streams and rivers in the 

western United States.

Average channel Unit stream Maximum measured

Drainage area gradient Bankfull discharge —power at bankfull bed load

Site, reference (km2) (m/m) (m3/s) (watts/m2) (kg/m/s)

Little Granite Cr1 55 0.0190 6.5 132.2 0.115

Main Fork Red River2 129 0.0059 9.3 44.4 0.017

Salmon R. nr. Obsidian2 243 0.0066 12.7 59.9 0.038

Boise River2 2154 0.0038 167 113.5 0.192

Selway River2 4955 0.0021 651 146.7 0.023

Snake River3 240766 0.0011 2607 153.8 0.173

Clearwater River3 24786 0.0006 2210 85.4 0.069

East Fork River4 466 0.0007 20 9.4 0.300

Jacoby Creek5 36 0.0062 32.6 165.2 0.400

Sagehen Creek6 27 0.0095 2.0 38.0 0.035

Virgin River7 0.0040 7.1 41.6 0.003

NF Toutle River8 736 0.0045 348 3.9

Data sources: (1) Ryan and Emmett (2002); (2) King and others (2004); (3) Jones and Seitz (1980); (4) Emmett (1980); (5) Lisle (1986); 

(6) Andrews (1994); (7) Andrews (2000); and (8) Pitlick (1992).

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