An Approach for Appraising the Accuracy 

of Suspended-sediment Data

U.S. GEOLOGICAL SURVEY PROFESSIONAL PAl>£R 1383
An Approach for Appraising the Accuracy 

of Suspended-sediment Data

By D. E. BURKHAM

U.S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 1333

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON, 1985
DEPARTMENT OF THE INTERIOR 

DONALD PAUL MODEL, Secretary

U.S. GEOLOGICAL SURVEY 

Dallas L. Peck, Director

First printing 1985 

Second printing 1987

For sale by the Books and Open-File Reports Section, U.S. Geological Survey, 

Federal Center, Box 25425, Denver, CO 80225
CONTENTS

Abstract ...........

Introduction .......

Problem .........

Purpose and scope

Page

1

1

1

2

Sampling error .......................................... 2

Discussion of error .................................... 2

Approach to solution .................................. 3

Application of method ................................. 4

Basic data .......................................... 4

Standard sampling error ............................. 4

Spatial error ............................................ 5

Discussion of error .................................... 5

Introduction ........................................ 5

Multivertical procedure .............................. 9

Single-vertical procedure ............................ 9

Single-point procedure ............................... 9

Approach to solution .................................. 10

Multivertical procedure .............................. 10

Single-vertical and single-point procedures ............ 10

Spatial error Continued 

Application of method ................................

Basic data .........................................

Standard spatial error for multivertical procedure ....

Standard spatial error for single-vertical procedure ... 

Temporal error .........................................

Discussion of error ...................................

Approach to solution .................................

Application of method ................................

Basic data .........................................

Analysis of data ....................................

Standard temporal error ............................

Sediment-discharge error ...............................

Discussion of error ...................................

Approach to solution .................................

Application of method ................................

Summary and conclusions ..............................

References cited ........................................

Page

11

11

11

13

13

13

14

14

14

15

15

15

15

17

17

17

18

ILLUSTRATIONS

FIGURE 1. Schema of assumed true concentration at a vertical during Tz~T\, with nomenclature .........................

2-4. Graphs showing:

2. Relation between standard sampling error and percentage sand in the sampled water-sediment mixture ..

3. A, Relation between the coefficient of variation and the percentage of the sampled water-sediment mixture 

that is sand; B, Relation between the coefficient of variation, the standard spatial error, and the number 

of verticals required to obtain the indicated standard spatial error .....................................

4. Relation between standard temporal error in instantaneous concentration and the average elapsed time 

between samples for the two indicated sites ..........................................................

12

16

TABLE

TABLE 1. Estimates of standard errors and information related to errors for selected sites along selected rivers and 

streams in California, Nebraska, Washington, Arizona, New Mexico, and Pennsylvania ............

Page

III
IV CONTENTS

CONVERSION FACTORS

For readers who may prefer to use metric units rather than inch-pound units, the conversion factors for the terms used in this report are 

listed below.

Multiply

feet

ft3/s (cubic feet per second) 

mi2 (square miles) 

tons, short

By

0.3048

0.02832

2.590

0.9072

To obtain

meters

m3/s (cubic meters per second)

km2 (square kilometers)

megagrams
AN APPROACH FOR APPRAISING THE ACCURACY OF 

SUSPENDED-SEDIMENT DATA

By D. E. BURKHAM

ABSTRACT

Procedures are presented for appraising the accuracy of suspendedsediment data. The types of error involved are sampling error error 

introduced in obtaining a value representing suspended-sediment 

concentration for a sample taken at a single vertical during the 

sampling time; spatial error error in mean concentration that is 

determined from sampled concentration at verticals in a cross section 

for a point in time; temporal error error in computed values of 

suspended-sediment concentration for a cross section at any point in 

time; and sediment-discharge error error in computed values of 

suspended-sediment discharge for a cross section at any point in 

time. The standard error of estimate is the statistic used to represent 

errors.

Suspended-sediment data for 17 U.S. Geological Survey gaging 

stations in Arizona, California, Nebraska, New Mexico, Pennsylvania, and Washington are used in applications of the procedures. For 

the study sites, the magnitudes of the different types of error apparently vary directly with the percentage of the sediment-water mixture that is sand. Among the study sites, when the percentage of sand 

is relatively high, the magnitudes of the different types of error have 

a relatively wide range.

For seven sites on the Sacramento River, California, and its tributaries, the standard sampling error apparently can range from 2.5 

percent for periods when the percentage of the suspended sediment 

that is sand is relatively low to 20 percent when the percentage sand 

is relatively high. For 16 sites in Arizona, California, Nebraska, New 

Mexico, and Pennsylvania, the coefficient of variation (ratio of standard deviation to cross-sectional mean) for concentration of suspended 

sediment across stream apparently can range from 2.5 percent for 

periods when percentage sand is relatively low to 70 percent when 

percentage sand is relatively high.

INTRODUCTION

PROBLEM

The U.S. Geological Survey (USGS) and other Federal, State, and local agencies obtain records of 

suspended-sediment discharge at many sites throughout the United States. The use of these records has 

greatly increased in recent years. Uses involve the 

evaluation of sediment transport to the oceans, geomorphological studies of denudation and rates of erosion, assessment of soil erosion and soil loss, reservoir

sedimentation, general environmental impact assessment, water treatment problems of sediment-associated 

nutrients and pollutants, and evaluation of the precise 

impacts of humans. Because of the many important 

potential uses of suspended-sediment data, it is essential that the reliability and accuracy of the data be 

known. Yet the data are published without analysis of 

the bias or the precision of the facts presented, mainly 

because a direct and exact technique for appraising the 

accuracy of suspended-sediment data is not available.

Errors are introduced in the collection and tabulation of suspended-sediment data because, for any measurement or computational scheme, the measurement 

and sampling equipment, measurement and sampling 

procedure, laboratory analysis, and computation procedure do not give exact results. A change in the measurement and sampling equipment, the measurement 

and sampling procedure, the laboratory analysis, and 

the computation procedure can result in a significant 

deviation in the quality of the data.

The accuracy requirement for the different uses of 

suspended-sediment data is different. On the one hand, 

a geomorphological study may require only a. "right 

ball park" estimate of the suspended-sediment discharge. On the other hand, an evaluation of the precise 

impact of humans may require accurate information on 

suspended-sediment discharge.

Generally, increasing sampling frequency increases 

the accuracy of a record of suspended-sediment discharge until the accuracy reaches a maximum level. At 

this point, deviations from true values represent random errors resulting from turbulent fluctuations in 

streamflow and sediment movement and random and 

systematic errors due to instrumentation, sampling 

technique, and computation. If funds were available, 

unlimited data of a quality equal to the maximum level 

of accuracy could be obtained; this approach could result in excessive expenditure of funds.
AN APPROACH FOR APPRAISING THE ACCURACY OF SUSPENDED-SEDIMENT DATA

A method that could be used to relate program objectives and accuracy to sampling frequency, to document 

the maximum attainable accuracy at a site, and to 

readily appraise the accuracy of sediment records in 

general would be a valuable tool in the design of an 

optimum sediment-sampling program. The development of the method would be a step toward the improvement of the quality control of suspendedsediment data.

PURPOSE AND SCOPE

The primary purpose of this report is to present procedures that can be used to roughly appraise the accuracy of suspended-sediment data. Because the computation of suspended-sediment discharge at a site on a 

stream involves the approximation of suspendedsediment concentration in the vertical, across the 

stream, and in time, the procedures described in this 

report encompass errors for the three dimensions and 

for the suspended-sediment discharge. As defined for 

this study, the types of error involved are

1. Sampling error error introduced in obtaining a 

value representing suspended-sediment concentration for a sample taken at a single vertical 

during sampling time Ts .

2. Spatial error error in mean concentration that is 

determined from sampled concentrations at several verticals in a cross section for a point in time.

3. Temporal error error in computed values of 

suspended-sediment concentration for a cross section for any point in time.

4. Sediment-discharge error error in computed values of suspended-sediment discharge for a cross 

section for any point in time.

The four types of error do not encompass the inaccuracy 

or difference that results from a common situation suspended-sediment samples usually represent 

sediment concentration of the flow exclusive of the bottom 0.3 to 0.5 foot of depth at each vertical. This difference between the suspended-sediment concentration 

for the sampled zone and for the total flow results from 

the omission of the suspended-sediment discharge near 

the streambed, where the concentration of suspended 

sediment, especially the coarse sediment, is highest. As 

the difference is largely coarse sediment, it often is 

significant in studies of channel behavior but usually is 

not significant in studies involving fine material 

(Bruce R. Colby, written commun., 1963).

The four types of error are treated separately in 

this report; however, as will be discussed, they are 

not mutually independent. The standard error of estimate is the statistic used to represent the errors. As 

will be discussed, the magnitude of the standard 

spatial error, the standard temporal error, and the

standard sediment-discharge error depends on the 

sampling procedure used to obtain the mean concentration for a section. For this report, the procedures 

for sampling to obtain the mean concentration are 

called "multivertical," "single vertical," and "single 

point."

The standard error of estimate, representing only a 

random error, is a weighted measure of the difference 

between computed and true values of a parameter. 

However, the total error in a computed value may encompass both a random error and a bias error. The 

random error represents the randomly distributed deviations of computed values around the true value of a 

parameter. The random error in computed values of 

suspended sediment is assumed to be normally distributed about zero; therefore, the sum of errors for an 

unlimited number of computed values of suspended 

sediment would be zero. Thus, if an unlimited number 

of computed values of suspended-sediment concentrations at a site were available, approximately two-thirds 

of the observations would be within one standard error 

of the true values of the concentrations.

The bias error is a time-invariant deviation resulting 

from a consistent overestimate or a consistent underestimate of the true value of a parameter. The error 

that is introduced when the mean concentration for the 

sampled zone is used to represent the mean concentration for the total flow at a cross section is likely to be a 

bias error. Unless otherwise stated, bias errors are not 

considered in this report. Further discussions of bias 

and unbiased errors can be found in textbooks dealing 

with statistical analysis (Dixon and Massey, 1957).

The approach used in appraising the accuracy of 

suspended-sediment data requires that a standard 

error for each component (task, step, or part) of the 

method used to derive the suspended-sediment data be 

computed, approximated, or assumed. Errors for the 

different components must be expressed in the same 

units. In this report, errors are expressed in percent of 

true value; however, unless otherwise stated, logarithmic units were used in the analyses for error. Data used 

in the application of the different procedures for appraising error were obtained from published reports or 

from the files of the U.S. Geological Survey offices. 

Terms introduced in the report are defined when they 

are first used.

SAMPLING ERROR

DISCUSSION OF ERROR

Sampling error, as previously indicated, is the 

difference between the sampled and the true 

suspended-sediment concentration at a single vertical 

for the sampling time, Ts . In terms of concentration in 

milligrams per liter, an equation for sampling error is
SAMPLING ERROR

7? C " ^ (1)

in which

Rsv = difference between true and sampled mean concentration for a single vertical during sampling time, Ts ;

Ctv =true mean concentration for the vertical during 

sampling time, Ts ; and

Csv =sampled mean concentration for the vertical

during sampling time, Ts .

The subscript s denotes a parameter for a sampled 

value, and the subscript t denotes a true value. The 

subscript v denotes a parameter for a single vertical. As 

previously stated, bias errors are ignored.

The magnitude of the sampling error depends on the 

accuracy of the components or tasl^s of sampling. The 

components or tasks of sampling are instrument, technique (including sampling time and the fluctuation in 

sediment concentration during this time), laboratory 

analysis, and computation of concentration for the 

sampled water-sediment mixture.

Probably the largest part of the sampling error is 

introduced during the task of obtaining a 

water-sediment sample in the stream. As previously 

indicated, the objective is to obtain a representative 

sample of the sediment-water mixture moving in the 

stream in the vicinity of the sampler. This is often 

difficult when the stream is carrying a relatively large 

concentration of sand-sized sediment. Mainly because 

of the fluctuation of the concentration of sand-sized 

sediments and differences in sampling technique, the 

magnitude of the sampling error may vary considerably with time and among several hydrologists sampling at a vertical during a given storm event at a site 

on a stream. Even though other factors affecting the 

accuracy of sampling may be the same, the magnitude 

of the sampling error may be significantly different 

among the different types of suspended-sediment samples obtained.

Suspended-sediment samplers are of three general 

types integrating samplers, instantaneous samplers, 

and pumping samplers (U.S. Geological Survey, 1977). 

Integrating samplers accumulate a water-sediment 

mixture over a period of time by withdrawing it from 

the ambient flow through a relatively small nozzle. 

Instantaneous, or grab, samplers essentially trap a volume of the suspension by instantaneously closing off 

the ends of a flow through chamber. Pumping samplers 

withdraw a mixture of the suspension through an intake by a pumping action. This report is concerned with 

integrating and pumping samplers.

An integrated sample may be of two types depth 

integrated and point integrated. A depth-integrated 

sample is obtained as a depth-integrating sampler is 

lowered to the bottom of a stream and raised back to the

surface. The sampler accumulates a water-sediment 

sample from a stream vertical at such a rate that the 

velocity in the nozzle at points of intake is nearly identical with the immediate stream velocity (isokinetically sampled). The sampler must be moved at a uniform rate in a given direction but not necessarily at 

equal rates in both directions. Significant error in the 

amount of sediment trapped in the sampler may result 

if the sampler is not lowered or raised at a uniform rate, 

if the sampler is lowered or raised too rapidly, or if the 

sampler is allowed to rest momentarily on the channel 

bed. Gouging or dipping the nozzle into the face of a 

dune is a special problem that must be avoided in order 

to minimize the magnitude of errors.

Standard point-integrating samplers frequently are 

used as depth-integrating samplers where stream 

depth exceeds about 17 feet, or where the stream is too 

swift to use a standard depth-integrating sampler 

(U. S. Inter-Agency Committee on Water Resources, 

1943). For streams less than about 30 feet deep, the full 

depth can be sampled by integrating from the surface 

to the bottom only or from the bottom to the surface; if 

the stream is deeper than about 30 feet, the vertical 

must be integrated by parts. Significant error in the 

amount of sediment in the sample may result if the 

sampler is not lowered or raised at a uniform rate 

through each part of the vertical or is lowered or raised 

too rapidly, if the sampler is allowed to rest momentarily on the channel bed, or if the intake valve is not 

opened at the correct time.

A point-integrated sample is taken by holding the 

intake valve of a point-integrating sampler open only 

at selected points in a vertical. Four or five samples at 

selected points in a vertical usually are adequate to 

define the distribution of sediment in the vertical when 

a stream is too deep or too swift to sample by integration of a round trip from the water surface to the bottom 

of the stream and back to the surface.

Determination of suspended-sediment discharge at a 

section when samples are obtained at a single elevation 

in a vertical by use of a pump sampler usually does not 

require that the mean suspended-sediment concentration for the vertical be evaluated. The usual approach 

is to determine the concentration at the elevation in 

the vertical and to multiply that value by a coefficient 

to arrive at an estimate of the mean concentration for 

the section. Therefore, the sampling error for the sample obtained by the pump sampler is not discussed further in this section. Brief referral to the sampling error 

for the pump sample, however, is made in the section 

"Spatial Error."

APPROACH TO SOLUTION

The standard sampling error (SESV ) for different 

water and sediment-transport regimes can be ap-
AN APPROACH FOR APPRAISING THE ACCURACY OF SUSPENDED-SEDIMENT DATA

praised using data in the files of the U.S. Geological 

Survey. The data useful for the purpose consist of sets 

of two depth-integrated samples for a single vertical in 

a stream taken as close together in time as possible 

(Guy and Norman, 1970). Individual analyses of sediment concentration yield two complete samples for the 

sampling time, Ts (fig. 1), which usually is no more 

than 2 to 4 minutes. As indicated in figure 1, CSVl represents the concentration for the first sample, Csl>2 represents the concentration for the second sample, Ctv represents the true mean concentration during Ts, RSVl 

represents the difference between Csv l and Ctv , and R 

represents the difference between CSU2 and Ctu .

The variance of the difference between the two sampled concentrations may be estimated as follows:

(SE)SVl

si; 2

N

si'2

N

(2)

N

in which N is the number of pairs of samples. The 

expected value is represented as

(3)

if the sampling errors for the two samples are independent. In that case, the expected value for SEgV can 

be represented as

-1

(4)

Equation 4 can be used for any pair of depthintegrated or point-integrated samples. However, the 

only "paired" samples available for this study were 

those for the depth-integrated sample. For this study, 

the standard sampling error for samples obtained 

using a point-integrating sampler is assumed to be 

equal to that for samples obtained using a depthintegrating sampler, and the error for samples collected using the point-integrating sampler is not considered further.

APPLICATION OF METHOD

BASIC DATA

Sediment records for five gaging stations on the 

Sacramento River (stations with reference numbers 2, 

3, 4, 5, and 6 in table 1) and two gaging stations on 

streams tributary to the Sacramento River (reference 

numbers 14 and 16) were selected for the present study 

primarily because they were readily accessible and 

contained the required data in a usable form. Extremes 

in daily discharge and daily sediment concentration for 

the study period at the seven stations are given in 

table 1. Complete descriptions of the characteristics of 

the flow and sediment regimes for the seven stations 

can be found in the files of the California District Office, U.S. Geological Survey, Water Resources Division, 

Sacramento, Calif. Data for the 1977-79 study period 

were used for the analysis of error.

The data useful for appraising the magnitude of the 

standard sampling error for each of the seven gaging 

sites (table 1), as previously indicated, consist of sets of 

two depth-integrated samples for a single vertical 

taken as close together in time as possible. The two 

samples for a given observation were obtained because 

they were useful to the fieldman and to the office engineer to indicate the variation of concentration between 

samples an advantage if the concentration for one of 

the samples were in obvious error. A large amount of 

sediment in one of the samples may indicate that the 

sampler was lowered into uncompacted sediment 

moving slowly along the bed; it would not be indicative 

of suspended material. The office engineer responsible 

for computing sediment discharge will not use the results from a sample for which the concentration is obviously in error. Usually, only a small percentage of the 

data for a station is rejected by the office engineer. 

Data for samples not used are labeled as such.

Data rejected by the office engineer also were rejected for the current stu4y. The screening of the data 

in this fashion may result in the standard error of estimate being biased on the low side. However, failure to 

reject samples containing large amounts of bed material probably would have resulted in a larger bias on 

the high side being introduced.

After screening, 212 sets of samples remained. Results of particle-size analyses were available for most of 

these sets. The distribution of the sets among the seven 

stations is shown in column 10 of table 1.

STANDARD SAMPLING ERROR

The same analytical procedure was used to appraise 

the standard sampling error for data for each of the 

seven gaging stations. The data set for each station was 

divided into groups according to ranges of percentage
SPATIAL ERROR

'1 -2 

FIGURE 1. Schema of assumed true concentration at a vertical during T^-T^ with nomenclature.

sand in the sampled mixture. For each station, the 

variance of the difference between the two sampled 

concentrations, SEgVl _2 , was computed for each group of 

samples by using equation 2. An estimate of the standard sampling error, SESV , for each group was obtained 

by using equation 4. As stated previously, logarithms 

of the concentrations were used in the analyses to determine variance. Data were adequate to define errors 

for two ranges of percentage sand for each of the study 

stations except one Sacramento River above Bend 

Bridge near Red Bluff, Calif.

Results of analyses to appraise the standard sampling error are given in table 1 and in figure 2. The 

numbers for percentage sand that are shown in table 1 

(column 11) represent the average of values (percentage sand) for the different groups of data. The numbers 

for percent error in table 1 (column 12) represent the 

(absolute) average of positive and negative values that 

result when logarithmic units are used to compute 

error.

The curves in figure 2 were developed from the following equation:

SEsv =(kf)x(%F)+(ks )x(%S) ,

in which 

SESV = standard sampling error, in percent;

%F=percentage fines;

%S= percentage sand; and 

k, ks = coefficients.

(5)

The quantity kf represents the standard sampling 

error as a fraction, when the suspended mixture is 

100 percent fines; ks represents the error when the mixture is 100 percent sand.

Equations 5A, 5B, 50, and 5D, shown with the lines 

in figure 2, were developed from equation 5 using 0.025 

for fyand 0.25, 0.20, 0.15, and 0.10, respectively, for ks . 

For the data from sites on the Sacramento River and 

Cottonwood Creek, equation 5B, based on 0.025 for kf 

and 0.20 for ks, seems to agree with the plotted values. 

This agreement apparently indicates that the standard 

sampling error for the Sacramento River and Cottonwood Creek may be as small as 2.5 percent for periods 

when the percentage of the suspended sediment that is 

sand is relatively low and as large as 20 percent when 

the percentage sand is relatively high. For the Thomes 

Creek site, the extreme values for standard sampling 

error may be 2.5 and 13 percent.

SPATIAL ERROR 

DISCUSSION OF ERROR

INTRODUCTION

Spatial error is the difference between the sampled 

and the true mean suspended-sediment concentration 

for a cross section for the sampling period (i.e., the time 

when the samples were obtained). In terms of concentration in milligrams per liter, an equation for the spatial error is
AN APPROACH FOR APPRAISING THE ACCURACY OF SUSPENDED-SEDIMENT DATA

TABLE 1. Estimates of standard errors and information related to errors for selected sites along selected

Extremes during study period

Reference

(1)

1

2

3

4

5

6

7

Symbol

(2)

A

B

C

D

E

F

Gaging station or site 

(3)

Sacramento River at Keswick, 

Calif. (11370500)

Sacramento River above Bend 

Bridge near Red Bluff, Calif. 

(11377100)

Sacramento River at Hamilton 

City, Calif. (11383800)

Sacramento River at Butte 

City, Calif. (11289000)

Sacramento River at Colusa, 

Calif. (11389500)

Sacramento River at Knights 

Landing, Calif. (11391000)

Niobrara River near Cody, Neb.

Study period 

(period of 

record used 

in analyses)

11/16/77 to 

10/05/78

11/02/77 to 

01/18/80

11/02/77 to 

01/18/80

11/03/77 to 

06/06/79

11/04/77 to 

06/12/79

11/01/77 to 

01/07/80

( 3 )

Source of data 

(5)

Files, California 

District, WRD, USGS, 

Sacramento, Calif.

do.

do.

do.

do.

do.

Colby and Hembree,

Daily discharge 

(cubic feet 

per second)

High 

(6)

39,000

106,000

110,000

121,000

45,200

29,700

( 3 )

Low 

(7)

2,460

3,200

3,000

2,720

2,620

2,980

( 3 )

Daily sediment 

concentration 

(milligrams/ liter)

High Low 

(8) (9)

2,320 1

1,030 6

1,630 8

2,000 10

990 19

( 3 ) ( 3 )

Middle Loup River near 

Dunning, Neb.

1955

Hubbell and others, 

1956; Hubbell and 

Matejka, 1959.

( 3 ) ( 3 )

9

10

11

12

13

14

15

16

17

® Middle Loup River below 

Dismal River, Neb.

0 Cowlitz River at Castle 

Rock, Wash.

-<f>- Colorado River near Grand 

Canyon, Ariz.

 < > Rio Grande Conveyance 

Channel near Bernardo, N.Mex.

X Cow Creek near Millville, 

Calif. (11374000)

A Thomes Creek at Rawson Road 

Bridge near Richfield, 

Calif. (11382090)

CD Feather River at Nicolaus, 

Calif. (11425000)

^ Cottonwood Creek near 

Olinda, Calif. (11375810)

- Bixler Run near Loysville, Pa.

( 3 )

01/14/81

( 3 )

( 3 )

11/08/77 to 

05/24/78

01/22/77 to 

04/04/79

11/09/78 to 

05/21/79

10/04/77 to 

03/02/79

03/12/62 to 

03/15/62

Hubbell, 1960

Files, Washington 

District, WRD, USGS, 

Tacoma, Wash.

Howard, 1947

Culbertson, Scott, 

and Bennett, 1972

Files, California 

District, WRD, USGS, 

Sacramento, Calif.

do.

do.

do.

Files, Pennsylvania 

District, WRD, USGS, 

Harrisburg, Pa.

( 3 ) ( 3 )

( 3 ) ( 3 )

( 3 ) ( 3 )

( 3 ) ( 3 )

16,700 59

2,800 0

27,200 2,130

10,100 9.1

132 88

( 3 ) ( 3 )

( 3 ) ( 3 )

c> o

( 3 ) ( 3 )

1,500 1

6,620 0

217 12

3,600 0

616 8

Standard spatial error computed from values of concentration obtained by sampling at multiverticals.

Standard spatial error based on standard spatial error for the concentration at one vertical and the standard error for 

the correction factor in equation 12A.

See source of data for information about the period of record and extremes during study period. 

4 Composite value for two cross sections. 

5 Composite value for several cross sections.
SPATIAL ERROR

rivers and streams in California, Nebraska, Washington, Arizona, New Mexico, and Pennsylvania

Standard sampling error

Number Perof data cent 

sets sand 

(10) (11)

--

12

20

23

25

11

20

8

38

10

-

43

23

39

19

36

15

44

16

39

.3

.8

.7

.6

.2

.0

.9

.2

.0

Percent 

error 

(12)

-

11,

7,

9,

6.

8,

4,

10.

4.

8.

.4

.1

.0

.6

.9

,8

.2

,9

.4

Coefficient of of

Number Perof data cent 

sets sand 

(13) (14)

8

7

6

2

11

10

8

7

6

9

3

13

9

11

18,

41,

58,

23,

40,

18.

34.

18,

40,

59,

18,

35,

.3

.4

.5

.9

.4

.3

.8

.8

.6

.6

.2

.4

.7

variation

Percent 

variation 

(15)

6,

9.

19,

25.

8.

16.

9

12

5,

26,

40,

12.

19.

.2

.3

.0

.0

.8

.9

.0

.0

.6

.4

.7

.5

.0

Standard spatial error

Number of Percent 

verticals error 1 

sampled (SE ) 

(16) (1 5*

5 2.

5 4.

8.

11.

5 3.

7,

5 4.

5.

5 2.

11.

18.

5 5.

8.

.8

,2

.5

.2

.9

.6

,0

,4

.5

.8

,2

,6

,5

Standard 

Standard segment

Percent temporal discharge 

error2 error error 

(SE ) (percent) (percent)

(18) (19 ') (20)

5

7

13,

17

7

12

7,

10,

6,

17

23,

8,

14.

.3

.2

.7

.7

.8

.7

.3

.5

.6

.3

.1

.3

.2

'71.0 4 29.0 13.0

3 27 5 81 J 38 15

5 4 5 82

81 34

17.4

15.2

19.4

42.2

18.3

40.5

19

4

2

4

7

2

4.5 6

6.7 7

6

5.4 8

10.8 8

"

41.

63.

52.

20.

40.

61.

16.

30.

46.

16.

36.

"

8

3

8

9

0

2

0

0

7

9

5

10.

13.

14.

10.

21.

28.

11.

21.

17.

12.

24.

"~ ~

8

8

7

8

3

0

5

7

2

4

2

4 5

6

5 6

5 4

9

12

5 5

9

5 7

5 5

10

- - -

.4

.9

.6

.8

.5

.5

.1 6.8

.7 11.1

.7

.5 9.7 6 72.5 7 73.2

.8 17.3

8 14.0 9 16.8

6The value was obtained from the curve in figure 4 for an elapsed time of 4.5 hours.

7 Based on 72.5 percent for the standard temporal error and 10 percent for the standard error for water 

discharge.

The value was obtained from the curve in figure 4 for an elapsed time of 1.2 hours.

Based on 13.5 percent for the standard temporal error and 10 percent for the standard error for water 

discharge.
AN APPROACH FOR APPRAISING THE ACCURACY OF SUSPENDED-SEDIMENT DATA 

25

20

1 15

DC

o

DC 

DC 

01

C3

10

5 -

EXPLANATION

SESV STANDARD SAMPLING ERROR, IN PERCENT 

%S PERCENTAGE OF WATER-SEDIMENT MIXTURE THAT IS SAND -

B,C,D...., A.G3 SYMBOLS USED TO DENOTE PLOTTING POSITION; DATA FROM

TABLE 1 

(5A) EQUATION NUMBER

10 15 20 25 30 35 

PERCENT SAND

40 45 50

FIGURE 2. Relation between standard sampling error and percentage sand in the sampled water-sediment

mixture.

(6)

in which

Rsx =difference between sampled and true mean concentration for a cross section;

Cte =true mean concentration for the cross section; 

and

0^=sampled mean concentration for the cross 

section.

The subscript x denotes a parameter for a cross 

section.

The magnitude of the spatial error typically depends 

on the natural variability of the suspended-sediment 

concentration with distance across the section, on the 

procedure for sampling that is used to obtain the mean 

concentration for a section, on the number of verticals 

when a multivertical procedure is used, and on the type 

of sample obtained at the verticals.
SPATIAL ERROR 9

The natural variability of suspended-sediment concentration across a stream depends on several interrelated factors. Some of the factors are hydraulic geometry (width, depth, and velocity) of the flow; channel-bed 

form (ripple, dunes, transition, plane, antidunes, and 

chute and pool); alinement of channel and flow; distance downstream from confluence with tributary 

stream; supply of clay-, silt-, sand-, and gravel-sized 

sediments; percentage of total concentration that is 

sand; and temperature of water. The procedures for 

sampling, as previously mentioned, are multivertical, 

single vertical, and single point.

MULTIVERTICAL PROCEDURE

To determine the mean concentration at a section by 

the multi vertical procedure, samples of the sedimentwater mixture are obtained periodically at three or 

more verticals in the section. These samples usually 

are combined to give a representative sample for the 

section. Concentration of suspended sediment for the 

representative sample is evaluated by laboratory analysis, and is assumed to represent the mean for the section (Porterfield, 1972). The verticals can be spaced so 

that they represent either equal rates of discharge the EDI approach or equal distance across 

the section with an equal-transit-rate (ETR) sample at 

each vertical the EWI approach (Guy and Norman, 

1970; U.S. Geological Survey, 1977). As previously indicated, the accuracy of the mean concentration obtained by the multivertical procedure depends on the 

natural variability of the suspended-sediment concentration across the section, the number of verticals sampled, and the sampling errors.

SINGLE-VERTICAL PROCEDURE

To approximate the mean concentration at a section 

using the single-vertical procedure, samples of the 

sediment-water mixture are obtained periodically at a 

permanently located vertical in the section (Porterfield, 1972). Occasionally, samples of the sedimentwater mixture are obtained at three or more verticals 

at a time when a sample is obtained at the permanently 

located vertical. Typically, the samples for the three or 

more verticals are combined to give a representative 

verification sample. Concentrations of suspended sediment for the verification sample and for the singlevertical sample are evaluated by laboratory analyses. 

Ratios are computed by dividing the concentration for 

the verification sample by the concentration for the 

single-vertical sample. This ratio represents a correction, called a cross-section coefficient for this study, to 

be applied to the single-vertical concentration to obtain 

a value that represents a mean concentration for the 

section. A temporal concentration curve for the single 

vertical is developed, as is a temporal curve representing the cross-section coefficient. A mean concentration 

for the section for the time when a single-vertical sample is obtained can be approximated by multiplying the 

concentration for the single vertical by a value for the 

time obtained from the temporal cross-section coefficient curve.

The accuracy of the mean concentration obtained by 

using the single-vertical approach typically depends on 

the magnitude of errors for the single-vertical sample 

and on how well values taken from the temporalcoefficient curve represent the true cross-section 

coefficient.

Records of concentration are obtained for some sites 

based on an assumption that the cross-section coefficient is always 1. Obviously, the error in the data 

would depend on how well the concentration for the 

single vertical represents the mean concentration for 

the section. The data, however, may contain a bias 

error. The scope of the present study, as previously 

indicated, does not include analyses to determine the 

magnitude of the bias error. Intuitively, however, it 

would appear that the bias error can be relatively large 

for sites where a relatively large percentage of the suspended sediment is in the sand-sized range.

SINGLE-POINT PROCEDURE

Steps for determining mean concentration at a section for any point in time using the single-point procedure are the same as those for the single-vertical procedure except that when using the single-point method, 

samples typically are obtained at only a single stage in 

the vertical. The point sample, which gives an indication of the sediment-water mixture moving at a single 

point in a vertical in a stream, usually is obtained automatically by using a pumping-type sampler which does 

not sample isokinetically. The single-point sample typically is obtained at stations in flashy streams or at 

other locations where it is extremely difficult to reach 

a station to collect samples manually at appropriate 

times by the normal procedure with standard 

equipment.

The automatic sampler is positioned in the stream 

before a flow event occurs. The automatic sampler, because of its many limitations, is not used indiscriminately to obtain samples, especially in streams carrying significant amounts of sand-sized sediment. 

Specific limitations of the point sample and automatic 

samplers, which usually apply to the sampling of sandsized sediment more than to fine sediments, are given 

in a report by Skinner and Beverage (1981).

The accuracy of the mean concentration for a section 

at times when single-point samples were obtained depends on the magnitude of errors for the single-point 

sample and for the correction coefficient.
10 AN APPROACH FOR APPRAISING THE ACCURACY OF SUSPENDED-SEDIMENT DATA

APPROACH TO SOLUTION

MULTIVERTICAL PROCEDURE

The approach used to approximate the magnitude of 

the standard spatial error for data obtained using the 

multivertical procedure involves mean concentrations, 

standard deviations for the means, and coefficients of 

variations for suspended sediment at cross sections 

(Guy, 1968). In terms of concentration in milligrams 

per liter, the equation for the mean concentration for a 

cross section is

(7)

in which Nl is used to represent the number of verticals 

with a single sample at each in the cross section. The 

standard deviation, (SD)SX , in milligrams per liter, can 

be represented as

in which the quantity is used to represent

Csv , the concentration for a single sample at a single 

vertical, and D^ is used to represent a departure from 

the mean concentration for the cross section, Csx . The 

coefficient of variation, in percent, is represented as

(CV) =

100(SDj

(9)

An approximation of the spatial standard error of estimate, in percent, is represented as

(SE\ = (10)

It should be noted that the variance (SD)2^ is an approximation of the sum of two variances: (SE8V )2 representing the sampling error, and (SO^t)2 representing 

the natural variability of suspended-sediment concentration with distance across a stream.

SINGLE-VERTICAL AND 

SINGLE-POINT PROCEDURES

The approach for approximating the standard spatial 

error for the single-vertical procedure for determining 

mean concentration at a cross section involves the addition of variances for the single-vertical sampling error

and the cross-section coefficient. In terms of concentration in milligrams per liter, an equation for mean concentration for a section, according to the single-vertical 

procedure, can be represented as

C =(a)Cs (11)

in which

0^=estimated mean concentration for a cross section when a suspended-sediment sample at a 

vertical is available, and 

a=a variable cross-section coefficient. 

The subscript e denotes an estimated parameter. Values for a are obtained from a temporal curve that represents the cross-section coefficient (page 9). In order that 

variances representing a and Csv be additive, equation 11 must be used in its logarithmic form, log (C^) 

=log a+log C8V. In units of logarithms, the variance for 

Cgj. can be represented as

in which

(SEjog) = standard spatial error in logarithmic units 

when the single- vertical procedure is in 

use, 

(SEjog) = standard error for the cross-section coefficient a, in logarithmic units, and 

(SEi0g) = standard sampling error, in logarithmic

units.

The variance when the concentration at a single point 

is used to represent the mean for a section can be represented as

in which

= standard spatial error in logarithmic 

units when the single-point procedure 

is used,

) = standard error for the coefficient alt in 

logarithmic units (c^ represents a correction coefficient when the concentration for a single point in a vertical is 

used to represent the mean for a section), and

(SEiog)p = standard error for the concentration obtained at a single point in a vertical, in 

logarithmic units.

The subscript p denotes a parameter for a single point 

in time.
SPATIAL ERROR 11

APPLICATION OF METHOD

BASIC DATA

Sediment records for 16 gaging stations were selected to demonstrate the procedure for appraising the 

standard spatial error (table 1). The records are for stations on the Sacramento River and tributary streams 

Feather River and Thomes, Cow, and Cottonwood 

Creeks in California, Niobrara and Middle Loup Rivers 

in Nebraska, Cowlitz River in Washington, Colorado 

River in Arizona, and the Rio Grande Conveyance 

Channel in New Mexico. These records were selected 

mainly because they were readily available and contained the type of data that were needed. They also 

represented a wide range of hydraulic conditions and 

sediment-transport regimes. Sets of data representing 

the concentration at four or more verticals were available at one or more cross sections at the stations.

Descriptions of the lengths of the study period and 

the ranges in discharge and sediment concentration for 

the 16 study stations are given in table 1. Further descriptions of the discharge and sediment regimes for 

the study stations can be found at the different sources 

of data.

The data useful for appraising the magnitude of spatial error consist of concentrations representing the 

water-sediment mixtures at multiverticals in cross sections. The data were required as a basis for describing 

the natural variability of the suspended-sediment concentration with distance across the streams that are 

involved.

STANDARD SPATIAL ERROR FOR 

MULTIVERTICAL PROCEDURE

For the multivertical procedure, the same analytical 

approach was used to appraise the standard spatial 

error for data for each of the 16 gaging stations. By 

task, the approach was as follows:

1. The data for each station were divided into groups 

according to ranges of percentage sand in the sampled mixture.

2. For each station, the coefficient of variation, (CV)SX , 

was computed for each range. Equations 7, 8, and 

9 were used for this task. This gave a coefficient 

of variation for each data group.

3. Values for percentage sand and values for the coefficient of variation were averaged for each data 

group. This gave an average percentage sand and 

an average coefficient of variation for each range 

of percentage sand (table 1).

4. An estimate of the standard spatial error, (SE )sx , for 

each group (or range) was obtained by use of 

equation 10.

Data were adequate to define groups representing 

three ranges of percentage sand for three stations, two

ranges of percentage sand for six stations, and a single 

range for the remaining seven stations (table 1).

Results of analyses to appraise the coefficient of variation (task 2) are given in table 1 and figure 3A. The 

curves in figure 3A were developed from the following 

equation:

(CV)ex =(%F)x(kfx )+(%S)x(ksx ) , (14)

in which

(CV)ex = estimated coefficient of variation, in

percent,

%F=percentage fines, 

%S = percentage sand, and

kfx , ksx = coefficients.

The quantity k^ represents the coefficient of variation, 

as a fraction, when the suspended mixture is 

100 percent fines; ksx represents the coefficient when 

the mixture is 100 percent sand.

Equation 14 is based on an assumption that the variability of concentration across a stream increases with 

percentage sand in the water-sediment mixture. Even 

though the plotted points in figure 3A scatter significantly, the data for the 16 study sites seem to support 

this assumption. Equations 14A, 14B, 14C, and 14D, 

shown with the lines in figure 3A, were developed from 

equation 14 substituting (100-%S) for %F and using 

0.025 for kfx, and 0.70,0.50,0.30, and 0.20, respectively, 

for ksx . Apparently, the average standard coefficient of 

variation for the plotted data can be represented by a 

curve developed from equation 14 with 0.025 for kfx and 

0.40 for ksx .

The reason, or reasons, for the difference in rate of 

increase in variability in concentration with increase 

in percentage sand among the 16 stations is not known. 

A detailed study to document reasons for the difference 

in the rates of increase is beyond the scope of this 

report.

For the flow and sediment conditions at the study 

sites, the curves on figures 3A and 3B can be used as a 

guide to approximate the number of verticals in a section that should be sampled in order to obtain a mean 

concentration with a specific standard spatial error. 

For example, suppose the required standard spatial 

error was 4 percent and the section to be sampled was 

at the gaging station "Sacramento River above Bend 

Bridge near Red Bluff, Calif." Suppose further that, for 

two sets of flow and sediment conditions, the percentage sand and the coefficient of variation are 18.4 and 

9.3 for one set of conditions, and 58.9 and 25.0 for the 

other set. The problem is to determine the number of 

verticals to be sampled in order to obtain a mean concentration with a standard spatial error of about 

4 percent. According to the curves on figures 3A and 

SB , the required number of verticals to be sampled
12 AN APPROACH FOR APPRAISING THE ACCURACY OF SUSPENDED-SEDIMENT DATA

iii 

o 

cc. Ill Q.

LLI 

O

UJ

O 

O

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 0 

PERCENT SAND (%5)

10 15 20 25 30 

REQUIRED NUMBER OF VERTICALS

40

EXPLANATION

(CV)SX COEFFICIENT OF VARIATION, IN PERCENT

(%S) PERCENTAGE OF WATER-SEDIMENT MIXTURE 

THAT IS SAND

SESX STANDARD SPATIAL ERROR, IN PERCENT

A,B,C,....,«|,® H SYMBOLS USED TO DENOTE 

PLOTTING POSITION, 

DATA FROM TABLE 1

FIGURE 3. A, Relation between the coefficient of variation and the percentage of the sampled water-sediment mixture that is sand. 

B, Relation between the coefficient of variation, the standard spatial error, and the number of verticals required to obtain the indicated 

standard spatial error.

would be about 40 for the conditions of relatively high 

percentage sand and variability and about 6 for conditions of relatively low percentage sand and variability. 

Thus, the ratio of required verticals for the two conditions is about 7.

The ratio of the numbers of required verticals for the 

two types of conditions discussed in the preceding paragraph probably would be much greater than 7 for sections in streams where the percentages of suspended 

sediment that is sand have wide ranges. The ratio can 

be determined directly by use of the formula 

"ratio=(CV:ca /CVJc6 )2" in which CV^ represents the 

high coefficient of variation for a site, often occurring 

during periods of high flow, and CVxb represents the

low coefficient of variation for the site, often occurring 

during periods of low flow. Assuming that the bounding 

curves shown in figure 3A, defined from equations 14A 

and 14D, adequately represent extreme values of variation, the ratio of required verticals to obtain mean 

concentrations of a specific standard spatial error could 

range from 1 to 64 according to coefficients taken from 

the curves that bound the smallest coefficients of variation and from 1 to 784 according to coefficients taken 

from the curves that bound the largest coefficients of 

variation. For the Red Bluff site, the coefficient of variation ranged from 2.9 to 29.8 for multivertical samples 

taken during the study period. Using these values in 

the above formula, the ratio would be 106.
TEMPORAL ERROR 13

STANDARD SPATIAL ERROR FOR 

SINGLE-VERTICAL PROCEDURE

The standard spatial error for mean concentrations 

obtained using the single-vertical procedure can be approximated using equation 12, but only if numbers are 

available for the standard sampling error and for the 

standard error for the cross-section coefficient a. Numbers representing the standard error for the 

cross-section coefficient (SE)a for the study stations, 

however, are not readily available; therefore, equation 

12 cannot be used directly to estimate (SE)ex .

A discussion of the possible range in (SE )ex for the 

study stations is based on expanded versions of equations 11 and 12. The discussion is for two cases: 

(1) when verification samples are available and the 

mean concentration for a section is assumed to be equal 

to the concentration for a single vertical multiplied by 

a cross-section coefficient a; and (2) when the mean 

concentration for a section is assumed to be equal to the 

concentration for a single vertical (verification samples 

are not available and a is assumed to be 1). The expanded version of equation 11 is

in which

a2 =a variable coefficient, 

CSXSX2=mean concentration for a section (derived from

a multi vertical verification sample); and 

CSV2 - concentration at a single vertical (derived from

a sample obtained at a permanently located

vertical). 

The expanded version of equation 12 is

(SEiog)sv 2

or

(SEiog)sv

(12A)

if (SElog)SV2 is assumed to equal (SElog)sv and (SElog)SX2 

is assumed to equal (SEiog)sx .

Each of the components of variance in equation 12A 

is applicable for case 1. For this study, however, only 

(SESX ) 2 and (SESU ) 2 are considered. For that reason, 

the magnitude of values for the standard spatial error 

for case 1 may be underestimated.

Estimates of the standard spatial error, SEex , for case 

1 are given in column 18 of table 1. The estimates are 

based on equation 12 A and on numbers from table 1 

and figure 2. Values for SESX were taken directly from 

column 17 of table 1. For the sites along the Sacramento River, numbers representing SESV for a single

sample at a vertical were obtained from the curve in 

figure 2 that was developed from equation 5B. The 

single-sample numbers that were obtained from the 

curve in figure 2 were divided by V2 to give an estimate of SESV for two samples at a vertical. For the site 

on Thomes Creek, numbers representing SESV for a single sample at a vertical were obtained from a relation 

developed using 0.025 for fyand 0.13 for ks in equation 5. Values ofSEsv for Cottonwood Creek were based 

on numbers for percentage sand taken directly from 

column 14 of table 1, numbers plotted in figure 2, and 

the curves in figure 2. All numbers in percentages were 

changed to equivalent logarithmic units before being 

used in equation 12A.

Only the variance (SE}og)2x in equation 12A is applicable for case 2. For the study stations, the standard 

error of estimate, SEex , for case 2 is assumed to be equal 

to the coefficient of variation, (CV)X . Values of (CV)X 

for the study stations are given in column 15 of table 1. 

A bias error, however, may be introduced when (CV)X 

is used to represent SE^ .

TEMPORAL ERROR

DISCUSSION OF ERROR

Temporal error is the difference between the computed and the true mean suspended-sediment concentration for a cross section for any point in time. Basically, it is the error in values taken from a curve that 

represents the temporal variation in mean suspendedsediment concentration for a section. In terms of concentration in milligrams per liter, an equation for the 

temporal error is

(15)

in which

^ cxP ~ difference between true and computed mean 

concentration for a section for any point in 

time, 

Ctep =true mean concentration for the section for any

point in time, and 

CCXP = computed mean concentration for the section

for any point in time.

The subscript c denotes a parameter taken from a concentration curve, and the subscript p denotes a parameter for a single point in time. The magnitude of the 

temporal error for data representing mean concentration for a cross section typically depends on (1) the 

natural variability of the suspended-sediment concentration in the vertical, across the section, and in time, 

(2) the sampling error and the spatial error, (3) the 

frequency of sampling, (4) the type and use of other 

available data (streamflow, temperature, and precipitation), and (5) the experience and good judgment of the
14 AN APPROACH FOR APPRAISING THE ACCURACY OF SUSPENDED-SEDIMENT DATA

hydrologist developing the temporal concentration 

curve.

The development of a temporal curve representing 

the mean concentration for a section may be based on 

single-point, single-vertical, or multivertical samples. 

As is discussed in the following section, the procedure 

used to approximate the standard temporal error is 

about the same regardless of the basis for the development of the temporal concentration curve.

APPROACH TO SOLUTION

The approximation of the variance of the difference 

between the true and the computed mean concentration at a section for any point in time, (SE^p)2 , cannot 

be based directly on measured or sampled values of 

concentrations because data are inadequate. For this 

study, the approximation is obtained indirectly and is 

based on the following equation:

(16)

in which the variance (SEXj) represents the spatial 

uncertainty and the variance (SEcpj) represents the 

temporal uncertainty introduced in developing a concentration curve. The subscript.; denotes a parameter 

that is based on samples' of concentration for a single 

point, for a single vertical, or for multi verticals. Approaches for approximating SEXj are described in preceding sections, and values representing SEXj are given 

in columns 17 and 18 of table 1.

Values of (SEcpj) can be obtained by comparing computed values of suspended-sediment concentration for a 

single point, a single vertical, or multi verticals with 

sampled values when the error in the sampled values is 

known. The variance of the difference between computed and sampled suspended-sediment concentration 

may be estimated as follows:

/ \ o 

Olf =(/n> _/"* )

"" (spj SCJ) V Spj ^ Cpj '

(n v-o-co; -n riso;/V

No (17)

in which

Rcpj=the difference between computed concentration, Ccpj, and true concentration, Ctj , 

Rspj=the difference between sampled concentration,

Capj, and true concentration, C& , and 

W2 =the number of values in the sample group. 

The expected value is

E(SE2

\_ 2 =n- -c)' " (s-c) " (spj-cpj)

2 +a2 

cpj^v spj (18)

if the errors for the sampled group are independent 

from the errors in the control group. Therefore, 

<j2cpj~<j2(spj-cpj)~<j2spj where a2 denotes a true or population variance, in contrast with (SE)2 , which is estimated on the basis of data.

Equations 16,17, and 18 can be used to estimate the 

temporal standard error for suspended-sediment 

records that are based on concentrations for a single 

point or a single vertical, or on the mean of concentrations for multiverticals. Generally, the approach is the 

same as that used by Burkham and Dawdy (1970) to 

appraise the accuracy of streamflow data for an alluvial stream. Values for the right side of the equations, 

representing independent errors, would be different for 

the three procedures for determining concentrations 

for a section.

APPLICATION OF METHOD

BASIC DATA

Suspended-sediment records for a site on Cottonwood 

Creek near Olinda, Calif., and for a site on Bixler Run 

near Loysville, Pa., were used to demonstrate the procedure for estimating the standard temporal error, 

SE^p. The concentration record for the Cottonwood 

Creek site is based on suspended-sediment samples obtained periodically at a single permanently located vertical and verification samples taken occasionally at the 

site. The concentration record for the Bixler Run site is 

based on suspended-sediment samples obtained periodically at a single point in the stream. The samples for 

Bixler Run were obtained by pump sampler. The periods of record (study periods) used in the analyses of 

standard temporal error were October 4, 1977, to 

March 2, 1979, for the site on Cottonwood Creek and 

March 12-15,1962, for the site on Bixler Run (table 1).

At the site on Cottonwood Creek, the daily discharge 

ranged from 9.1 to 10,100 ft3/s during the study period 

and the daily concentration ranged from 0 to 

3,600 milligrams per liter. According to the results of 

particle-size analyses, the part of the suspended sediment that is sand ranged from 3 to 60 percent during 

the study period. Cottonwood Creek at the Olinda site 

drains a watershed of about 395 mi2 .

At the site on Bixler Run, the daily discharge ranged 

from 88 to 132 ft3/s and the daily concentration ranged 

from 8 to 616 milligrams per liter during the study 

period. According to the results of particle-size analyses, the part of the suspended sediment that is sand was 

3 percent on February 26,1962,5 percent on March 12, 

1962, and 4 percent on March 21, 1962. Bixler Run at 

the Loysville site drains a watershed of about 15 mi2 .
SEDIMENT-DISCHARGE ERROR 15

ANALYSIS OF DATA

Analyses to approximate the standard temporal 

error were made by steps, as follows:

1. The variance representing the difference between 

computed and sampled suspended-sediment concentration, SE2(Spj_SCj), was determined.

2. The variance representing SE2pj was obtained by 

subtracting a variance representing SE2pj, variance for sampled concentration, from the variance 

obtained in step 1.

3. The variance representing (SE^p)2 was obtained by 

adding a variance representing (SE^)2 to the 

variance SE2oi obtained in step 2.

/ \

4. The standard temporal error (SE^pj) was obtained

by taking the square root of (SEcxpj)2 . 

The same analytical procedure was used to determine 

SE2(spj-Cpj) f°r the data of both sites. Logarithmic units 

were used in the analyses. Analyses for the determination of SE2(spj^ cpj) (step 1) were made using the values 

of concentration for Ve, 1/4,Va, 1/2, 2/3, and 3/4 of the total 

number of samples available for the development of 

temporal concentration. The sets of data are overlapping and, therefore, the results are not entirely independent. No corrections for lack of independence were 

included in the analyses. The samples were arranged in 

groups of six by time. From each group, one sample was 

selected randomly and used in the analyses to develop 

the temporal concentration curve. The remaining samples were retained as the control group. After the temporal concentration curve was developed, concentrations were obtained from the curve corresponding to 

the times of the samples in the control group. The mean 

square difference between the logarithms of the computed and sampled concentrations was then 

determined.

Similar analyses to determine SE2(spj-scj) and SE2pj 

were made for the studies in which 1/4, Va, l/2, 2/3, and 

3/4 of the total samples were used to develop the temporal concentration curve. Because analyses were made 

using logarithmic units, the standard errors could be 

readily changed to percentages.

STANDARD TEMPORAL ERROR

The relations of the average time between samples to 

percent error in instantaneous concentrations for the 

two sites on Cottonwood Creek and Bixler Run are 

shown in figure 4. The trend lines drawn through the 

points and extended to minimum values illustrate the 

average improvement in developing temporal concentration curves with decrease of time between samples. 

The minimum error is assumed to be the coefficient of 

variation 18 percent for the Cottonwood Creek site 

and 4 percent for the Bixler Run site. Presumably, the 

minimum error could be realized only if samples were 

obtained continuously at the single vertical at the

Cottonwood Creek site and at the single point at the 

Bixler Run site.

The extrapolation of the relations shown in figure 4 

makes it possible to estimate errors that would result if 

all samples were used to develop the temporal concentration curve. For the Cottonwood Creek site, where 

the average time between samples is 4.5 hours, the 

estimated standard temporal error is 72.5 percent. For 

the Bixler Run site, where the average time between 

samples is 1.2 hours, the estimated standard temporal 

error is 14 percent.

The relations of time between samples to percent 

error in instantaneous concentrations that are given in 

figure 4 suggest better accuracy for the Bixler Run 

record. This undoubtedly reflects the uncertainty 

caused by the higher percentage of sand-sized sediment 

in suspension at the Cottonwood Creek site.

As previously indicated, the trend line drawn 

through the plotted points for the Cottonwood Creek 

site (fig. 4) represents the average relation between the 

standard temporal error and time between samples. 

Had the data for the Cottonwood Creek site been adequate, curves could have been developed that would 

show the relation between the standard temporal error 

and time between samples for periods when the percentage sand is relatively low and for periods when the 

percentage sand is relatively high.

SEDIMENT-DISCHARGE ERROR

DISCUSSION OF ERROR

Sediment-discharge error is the difference between 

computed and true sediment discharge for a cross section for any point in time. In terms of sediment discharge in tons per second, an equation for sedimentdischarge error is

Rqx =(QS)et -(QS)tc 9 (19)

in which

Rqx =ihe difference between computed and true 

sediment discharge,

(QS)CX = computed sediment discharge, and

(QS)te =true sediment discharge. 

The subscript q denotes a sediment-discharge parameter. As discussed in detail in a subsequent section, sediment discharge is the result obtained when the product of concentration times water discharge is 

multiplied by a constant. The error in data representing sediment discharge, therefore, results from uncertainties in values of instantaneous sediment concentration and in values of water discharge. Errors in 

instantaneous sediment concentration are described in 

a preceding section.
16 AN APPROACH FOR APPRAISING THE ACCURACY OF SUSPENDED-SEDIMENT DATA

140

130

120

110

100

90

80

70

60

50

40

30

20

10

u tr

oc 

O tr tr

COTTONWOOD CREEK 

NEAR OLINDA, CALIFORNIA'

STORMS IN JANUARY 4-MARCH 10, 1978 

DRAINAGE AREA = 395 SQUARE MILES 

SAND = 3-60 PERCENT

AVERAGE ELAPSE TIME BETWEEN SAMPLES 

= 4.5 HOURS

STANDARD TEMPORAL ERROR = 72.5 PERCENT -

EXTRAPOLATED 

CURVE

BIXLER RUN NEAR LOYSVILLE, PENNSYLVANIA

STORMS IN MARCH 12-15, 1962 

DRAINAGE AREA=15 SQUARE MILES 

SAND = 3-8 PERCENT

AVERAGE ELAPSE TIME BETWEEN SAMPLES 

= 1.2 HOURS DURING STUDY PERIOD

STANDARD TEMPORAL ERROR = 14 PERCENT

"~ I I I 1 I

10 12 14 16 18 20 22 24

AVERAGE ELAPSE TIME BETWEEN SAMPLES, IN HOURS

FIGURE 4. Relation between standard temporal error in instantaneous concentration and the average elapsed time between

samples for the two indicated sites.
SUMMARY AND CONCLUSIONS 17

The magnitude of the streamflow error typically is 

closely related to how well the stage-discharge relation 

can be defined. Relatively smaller errors, however, 

may result from the incorrect recording of stage and 

time. If a stable stage-discharge relation exists at a 

gaging site, a rating defined by a large number of 

current-meter measurements assuming the measurement errors have a mean of zero would approach the 

true stage-discharge relation. Discharge computed by 

applying a correct stage record to the rating thus defined would have only a relatively small error. The 

rating conditions in most alluvial channels are not perfect or stable. Changes in the dimensions of the channel 

or in the hydraulic resistance to flowing water may 

cause large adjustments or shifts in the stagedischarge relation. The magnitude of the streamflow 

error for flow in a shifting stream often is closely related to the frequency of discharge measurements and 

to how well the "shifts" can be estimated by the hydrologist (Burkham and Dawdy, 1970).

APPROACH TO SOLUTION

The approach for approximating the standard sediment-discharge error involves the addition of variances 

for the temporal error and the water-discharge error. In 

terms of discharge in tons per second, an equation for 

sediment discharge for a section is

(20)

in which

QS= sediment discharge, in tons per second; 

ki=a coefficient that is based on the unit of measurement of water discharge and that assumes a specific weight of 2.65 for sediment; 

&! has a value of 3.12x 10~8 when the units 

of sediment discharge are in tons per second; 

Ccxpj-ihe mean concentration for a section, in milligrams per liter; and 

QW= water discharge, in cubic feet per second.

In logarithmic units, the variance for QS can be represented as

(SElog;-sH(SElog'cxpj (SE}, (21)

in which

)^ = standard sediment-discharge error, 

^pj = standard temporal error, and 

(SEiog)qw = standard water-discharge error.

The coefficient ki is assumed to be a physical constant, 

and, therefore, it would have no variance component.

Equation 21 can be used to estimate the standard 

sediment-discharge error. The variance (SE\og)%xpj can 

be approximated using the procedure previously presented. The variance (SEiog)qW for data for streams 

having movable boundaries can be determined from 

measured values of streamflow using the procedure 

outlined by Burkham and Dawdy (1970). The variance 

for data for streams having rigid boundaries can be 

approximated from measured values of streamflow and 

regression analysis.

APPLICATION OF METHOD

Standard sediment-discharge errors for the Cottonwood Creek and Bixler Run sites are approximated 

using equation 21. The standard temporal error is assumed to be 72.5 percent for the Cottonwood Creek site 

and 14 percent for the Bixler Run site (column 19 of 

table 1). For this report, the standard streamflow error 

is assumed to be 10 percent for both sites. The resulting 

standard sediment-discharge error is 73.2 percent for 

the Cottonwood Creek site and 16.8 percent for the 

Bixler Run site.

SUMMARY AND CONCLUSIONS

Procedures that can be used to appraise the accuracy 

of suspended-sediment data are described in this report. Because the computation of suspended-sediment 

discharge at a site on a stream involves the approximation of suspended-sediment concentration in the vertical, across the stream, and in time, the procedures described herein encompass errors for three dimensions 

and for suspended-sediment discharge. As defined for 

this study, the types of errors involved are

1. Sampling error error introduced in obtaining a 

value representing suspended-sediment concentration for a sample taken at a single vertical 

during the sampling time.

2. Spatial error error in mean concentration that is 

determined from sampled concentrations at verticals in a cross section for a point in time.

3. Temporal error error in computed values of 

suspended-sediment concentration for a section 

for any point in time.

4. Sediment-discharge error error in computed values of suspended-sediment discharge for a section 

for any point in time.

The four types of error are treated separately; however, 

as discussed in the report, they are not mutually independent. The standard error of estimate is the statistic 

used to represent error. The procedures for appraising 

the accuracy of suspended-sediment data are illustrated using readily available data from reports and 

from the files of the Water Resources Division of the 

U.S. Geological Survey. Suspended-sediment data for
18 AN APPROACH FOR APPRAISING THE ACCURACY OF SUSPENDED-SEDIMENT DATA

17 gaging stations in Arizona, California, Nebraska, 

New Mexico, Pennsylvania, and Washington were used 

in the applications of the procedures. 

Conclusions resulting from the study are

1. The magnitudes of the different types of error vary 

directly with the percentage of the sedimentwater mixture that is sand.

2. For seven sites on the Sacramento River, Calif., and 

its tributaries, the standard sampling error, on an 

average, apparently can range from 2.5 percent 

for periods when the percentage of the suspended 

sediment that is sand is relatively low to 20 percent when the percentage sand is relatively high. 

When the percentage sand is relatively high, the 

standard sampling error apparently can range 

from 13 to 20 percent.

3. For 16 sites in Arizona, California, Nebraska, New 

Mexico, and Pennsylvania, the coefficient of variation for concentration of suspended sediment 

across streams apparently can range from 2.5 percent for periods when the percentage sand is relatively low to 70 percent when the percentage sand 

is relatively high. When the percentage sand is 

relatively high, the coefficient of variation may 

range from 20 to 70 percent among the 16 sites.

4. The average standard temporal errors for a site on 

Cotton wood Creek near Olinda, Calif., and for a 

site on Bixler Run near Loysville, Pa., increase 

rapidly with time between samples. The average 

standard temporal error was determined to be 

72.5 percent for the Cottonwood Creek site and 

14 percent for the Bixler Run site. Concentration 

curves for the Cottonwood Creek site were based 

on single-vertical samples of the water-sediment 

mixture taken with a 4.5-hour average time between samples. Concentration curves for the 

Bixler Run site were based on single-point samples taken with a 1.2-hour average time between 

samples. Cottonwood Creek at the site near 

Olinda drains a watershed of about 395 mi2 . 

Bixler Run at the site near Loysville drains a watershed of about 15 mi2 . The percentage sand in 

the water-sediment mixture for the Cottonwood 

site had a wide range from about 3 to 60 percent during the study period. The percentage 

sand for the Bixler Run site had a low range 

from about 3 to 5 percent during the study 

period.

5. The average standard sediment-discharge errors for 

the two sites Cottonwood Creek near Olinda,

Calif., and Bixler Run near Loysville, Pa. were, 

respectively, 73.2 and 16.8 percent. The determinations of the magnitudes of the two standard 

sediment-discharge errors were based on an assumed value of 10 percent for the standard waterdischarge error.

6. Much research would be required to explain the 

great variability in error for a given percentage 

sand among the study stations.

REFERENCES CITED

Burkham, D. E., and Dawdy, D. R., 1970, Error analysis of

streamflow data for an alluvial stream: U.S. Geological Survey

Professional Paper 655-C, 13 p. 

Colby, B. R., and Hembree, C. H., 1955, Computations of total

sediment discharge, Niobrara River near Cody, Nebraska: U.S.

Geological Survey Water-Supply Paper 1357, 187 p. 

Culbertson, J. K., Scott, C. H., and Bennett, J. P., 1972, Summary of

alluvial-channel data from Rio Grande conveyance channel,

New Mexico, 1965-69: U.S. Geological Survey Professional

Paper 562-J, 49 p. 

Dixon, W. J., and Massey, F. J., Jr., 1957, Introduction to statistical

analysis: New York, McGraw-Hill, 488 p. 

Guy, H. P., 1968, Quality control of adjustment coefficients used in

sediment studies, in Geological Survey research, 1968: U.S.

Geological Survey Professional Paper 600-B, p. B165-B168. 

Guy, H. P., and Norman, V. W., 1970, Field methods for

measurement of fluvial sediment: U.S. Geological Survey

Techniques of Water-Resources Investigations, bk. 3, chap. C2,

59 p. 

Howard, C. S., 1947, Suspended sediment in the Colorado River,

1925-41: U.S. Geological Survey Water-Supply Paper 998,

165 p. 

Hubbell, D. W., 1960, Investigations of some sedimentation

characteristics of sand-bed streams, progress report number 2:

U.S. Geological Survey open-file report, 78 p. 

Hubbell, D. W., Jordan, P. R., Culbertson, D. M., and Hembree, C. H.,

1956, Progress report number 1 Investigation of some

sedimentation characteristics of a sand-bed stream: U.S.

Geological Survey open-file report, 78 p. 

Hubbell, D. W., and Matejka, D. Q., 1959, Investigations of sediment

transportation, Middle Loup River at Dunning, Nebraska: U.S.

Geological Survey Water-Supply Paper 1476, 123 p. 

Porterfield, George, 1972, Computation of fluvial-sediment

discharge: U.S. Geological Survey Techniques of WaterResources Investigations, bk. 3, chap. C3, 66 p. 

Skinner, J. V., and Beverage, J. P., 1981, Test and design of

automatic fluvial suspended-sediment sampler; Report W: U.S.

Army Engineer District, St. Paul, Minnesota, 53 p. 

U.S. Geological Survey, 1977, Sediment, in National handbook of

recommended methods for water-data acquisition, chap. 3,

p. 3_i_3_99.

U.S. Inter-Agency Committee on Water-Resources, Subcommittee on 

Sedimentation, 1943, Determination of fluvial sediment 

discharge, Report 14 of A study of methods used in measurement 

and analysis of sediment loads in streams: 151 p.

*U.S. G.P.O. 1987-181-409140019