Error Analysis

of Streamflow Data

for an Alluvial Stream

GEOLOGICAL SURVEY PROFESSIONAL PAPER 655-C

Error Analysis

of Streamflow Data

for an Alluvial Stream

By D. E. BURKHAM and D. R. DAWDY

GILA RIVER PHREATOPHYTE PROJECT

GEOLOGICAL SURVEY PROFESSIONAL PAPER 655-C

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1970
UNITED STATES DEPARTMENT OF THE INTERIOR 

WALTER J. HICKEL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C. 20402 - Price 35 cents (paper cover)
CONTENTS

Abstract_ ____________

Introduction. ____-____.

Basic data_____________

Gaging sites. _______

Flow characteristics.

Discussion of errors _____

Approach to solution___.

Page

Cl

1

1

1

4

5

6

Analysis of data____________________

Errors in instantaneous dischargeErrors in volumes of discharge.__. 

Errors in water budget._________

Summary. _ ________________________

Literature cited____.________________

Puge 

06

7

8

10

12

13

ILLUSTRATIONS

FIGURE 1. Index map of project area and map showing instrumentation location.______---_-______.

2. Photograph showing the Gila River downstream from the Bylas gaging station.__________

3. Photograph showing the Gila River downstream toward the Calva gaging station.-______.

4. Hydrograph showing summer flow, Gila River at Calva________________-_-_-_-__-_-__.

5. Photograph showing debris during rising limb of summer floodflow._____________________

6. Graph showing duration curves of summer flow, Gila River at Calva____________________

7. Hydrograph showing winter flow, Gila River at Calva______________.______-_--__-_-_-.

8. Graph showing duration curves of winter flow, Gila River at Calva...__-.__-___-___-___. 

9-14. Graphs showing standard error of computed discharge for 

9. Different ranges of discharge, Gila River near Bylas_____________.-__________.

10. Different ranges of discharge, Gila River at Calva_._________.---__--_-______--

11. Summer flow, Gila River near Bylas_-___-_________________--__-_-__-____--_-

12. Summer flow, Gila River at Calva______________________--_____-_----__---_-

13. Winter flow, Gila River near Bylas__________________________________________

14. Winter flow, Gila River at Calva__________________-__---__--_-------__----.

15. Graph showing standard error in percentage of average discharge in 3-week streamflow data.

Page

C2

3

3

4

4

4

5

5

9

10

10

11

11

12

TABLES

TABLE 1. Error analysis of streamflow, Gila River near Bylas, Ariz.

2. Error analysis of streamflow, Gila River at Calva, Ariz__.

3. Standard error of estimate.___________________________

C7

7

11

in

GILA RIVER PHREATOPHYTE PROJECT

ERROR ANALYSIS OF STREAMFLOW DATA FOR AN ALLUVIAL STREAM

By D. E. BUEKHAM and D. E. DAWDY

ABSTRACT

Discharge measurements were used to determine the standard 

error in computed continuous records of discharge for two 

streamflow gaging stations on the Gila River. The major source 

of errors in computed discharge is from poor definition of the 

stage-discharge relation.

The standard errors of computation of discharge for the two 

stations were determined by randomly choosing a group of discharge measurements for use in rating analysis and using the 

remaining measurements as a control group. Discharge was 

computed corresponding to the stage and time of the measurements in the control group. The mean square difference (S"m - c ) 

between measured and computed discharge was determined for 

different ranges of flow. S2m -<, is the sum of the mean square 

difference of the measured discharge from true discharge 

(S2m) plus the mean square difference of computed discharge 

from true discharge (S2C ). The variance (S"c) was obtained by 

subtracting the known variance 8am from 82m-c.

INTRODUCTION

The accuracy of discharge data obtained in a sandchannel stream is not known, and there is no direct 

and exact technique by which the accuracy can be evaluated. Because of the increasing importance of research 

and management problems in water development, the 

knowledge of errors in water data assumes great 

importance.

The general purpose of this report is to present a 

method by which errors in computing discharge for a 

stream that flows in a sand channel can be evaluated. 

A more specific purpose is to approximate the errors in 

the discharge of the Gila Kiver near Globe, Ariz., in 

which the evapotranspiration from the alluvial flood 

plain is measured as a residual in a water-budget study. 

The method described can be used to determine the 

errors in instantaneous discharge as well as errors in 

average rates and volumes of discharge.

A water budget requires the measurement of all significant quantities of inflow and outflow in an area.

The factors in the budget will vary seasonally; therefore, the relative significance of the surface flow in the 

determination of evapotranspiration will vary for different periods. In periods of floodflow, the surface inflow and outflow greatly exceed all other factors in the 

budget. Surface flow is the main source of groundwater recharge; therefore, the surface-flow data for 

many budget periods can be used to test evaluations of 

the change in ground-water storage, which is measured 

by neutron soil-moisture meters.

This report was prepared under the general supervision of H. M. Babcock, district chief of the Water 

Kesources Division in Arizona, and is the result of 

evapotraiispiratioii studies of the Gila Kiver Phreatophyte Project under K. C. Culler, project chief.

BASIC DATA 

GAGING SITES

The discharge data for the Gila Kiver gaging static ns 

near Bylas and at Calva, Ariz., were used in this stuc^yThe stations are at the ends of subreach 1 of the Gila 

River Phreatophyte Project (fig. 1).

The flood plain through subreach 1 is from abc^it 

3,000 to 4,000 feet wide. The present (1965) low- and 

median-flow channel, or main channel, is from 80 to 

100 feet wide, has a slope of about 0.001, and is a normal pool-and-riffle type. The pools are generally f^ll 

of sand that erodes easily during low flows, and the 

riffles are fairly stable gravel bars. The sand and gravel 

in the banks of the main channel and in the flood plain 

are stabilized by a dense cover of saltcedar and mesquite.

The surface-water stage is determined by the use of 

stilling-well-type gages, which are equipped with 15- 

minute digital recorders and continuous analog recorders that operate from the same float.

Cl
114° 

EXPLANATION

oundary of Gila River flood plain 

Gaging station on the Gila River

7O COOLIDGE DAM

FIGURE 1. Index map of project area and map showing instrumentation location.
ERROR ANALYSIS OF STREAMFLOW DATA FOR AN ALLUVIAL STREAM C3

The stilling well for the Gila Elver near Bylas gage 

(fig. 2) was established on the downstream side of the 

concrete bridge on U.S. Highway 70. There is some turbulence of flow past the stilling well, but its effect on 

the stage is small.

High-water measurements were made from the bridge 

from 1962 through the spring of 1964. However, because of large approach angles, turbulence, excessive 

scour around the piers, and uneven velocity distribution, 

the bridge site was not ideal for measurements of high 

flow. In the summer of 1964 a cableway that spans the 

main channel was installed 140 feet downstream. Flows 

in excess of the capacity of the main channel, about 

4,000 cfs (cubic feet per second), are measured from the 

bridge, and low flows are measured in the channel near 

the gage. The measuring conditions for flow below bankfull stage are good.

The controlling section for flows ranging from 50 to 

4,000 cfs is an old road crossing 150 feet downstream 

from the gaging station. The section of gravel, small 

rocks, and sandbanks stabilized by saltcedar is fairly 

stable. The low-water control is the shifting sandbars 

that form upstream from the old road crossing (fig. 2).

The gage at Calva was established in 1930. Prior to 

December 1962, the gage was equipped with a continuous analog recorder and was on the downstream side of 

the railroad bridge that spans the Gila Eiver flood plain. 

In December 1962 the gage was moved to the left bank 

about 530 feet below the railroad bridge, and at that 

time a cableway spanning the main channel was constructed 400 feet downstream from the railroad bridge.

FIGURE 2. View of the, Gila River downstream from the Bylas 

gaging station. The old roadbed, which affects the stage-discharge relation, is about 50 feet downstream from the cableway. Photograph by C. S. English.

Flows in excess of the capacity of the main channel, 

about 4,000 cfs, are measured from the railroad 'bridge, 

and measurements of low flows are made in the channel at several places near the gage. Measuring conditions for flow below bankfull stage are good.

The channel control of sand and gravel is unstable. 

There is a fairly stable gravel bar in the channel about 

200 feet below the present gaging station that may act 

as a partial control during flows near bankfull stage. 

The vegetation along the banks also affects the stagedischarge relation at flows near bankfull stage or higher 

(fig. 3).

The stage-discharge relation is relatively unstable at 

all flows, although not excessively so when compared 

with ratings that are discontinuous (Dawdy, 1961). 

The major factor that contributes to the shifting of the 

stage-discharge relation is scouring and filling in the 

controlling reach of the river. Other less important factors are sediment in transport, which affects the fluid 

properties of the water, and seasonal vegetation changes, 

which cause a variable backwater condition.

In order to minimize errors caused by the shifting of 

the stage-discharge relation, the frequency of discharge 

measurements is geared to the movement of sediment, 

and the measurements are spaced so that shifts of rating 

through the total range of flow can be defined. A large 

amount of sediment is transported in the summer during 

the continual variation in flow rates. Generally, one 

measurement a week is made during winter flow, and 

two or more measurements a week are made during 

summer flow.

FIGUEE 3. View of the Gila River downstream toward the Calva 

gaging station. The gaging station, which is about 130 feet 

downstream from the cableway, is in the left center of the 

photograph. Photograph taken from the cableway by C. S. 

English.
C4 GILA RIVER PHREATOPHYTE PROJECT

4000

3000

2000

1000

500

IT

AUGUST 18, 1957

FIGURE 4. Summer flow, August 18, 1957, Gila River at Calva. 

FLOW CHARACTERISTICS

The flow in the Gila River originates from two types 

of storms thunderstorms and frontal movements. Summer (July through October) streamflow is mainly from 

local thunderstorms. In the summer the rate of streamflow varies considerably in short periods of time (fig. 

4), and the sediment concentration and the amount of 

floating debris are generally high (fig. 5).

Flow-duration curves for summer flow (fig. 6) show 

the frequency distribution of the daily mean flow for 

two periods, 1930-40 and 1951-61. These two periods 

represent "wet" and "dry" periods, respectively. The 

median flow flow that is equaled or exceeded 50 perFIGURE 5. Debris during rising limb of summer floodflow at 

Gila River near Calva gaging station, about 4 miles downstream from Gila River at Calva station. Photograph taken 

by J. S. Phelps about 20 minutes after the arrival of the 

flood.

10,000

1.0 

0.01 0.5 5 10 20 3040506070 80 ~90

PERCENTAGE OF TIME FLOW EQUALED OR EXCEEDED

FIGURE 6. Duration curves of summer flow, 1930-40 and 

1951-61, Gila River at Calva.

cent of the time was 45 cfs for the 1930-40 period 

but was only 5 cfs for the 1951-61 period. The frequency of occurrence of flow equaled or exceeded about 

10 percent of the time was about the same for the two 

periods. The summer flow of Gila River at Calva 

ranged from a low of 2,530 acre-feet in 1960 to a high 

of 142,300 acre-feet in 1932, with an average of 50,340 

acre-feet for the period 1930-61.

The winter (November through June) flow is mainly 

from frontal storms, snowmelt, ground-water storage, 

or a combination of the three. The flow rate may be 

fairly constant for several days (fig. 7). The total winter 

flow of Gila River at Calva was 75 percent of the total
ERROR ANALYSIS OF STREAMFLOW DATA FOR AN ALLUVIAL STREAM C5

Discharge measurement

APRIL, 1958 MAY, 1958

FIGUKE 7. Winter flow, April 13 through May 6, 1958, Gila 

River at Calva.

flow for the period 1930-61. From 1930 through 1961 

the winter flow ranged from a low of 6,200 acre-feet in 

1956 to a high of 737,900 acre-feet in 1941.

Flow-duration curves for winter flow (fig. 8) show 

the frequency distribution of the daily mean flow for 

two periods, 1930-40 and 1951-61. The customary steep 

slopes of the flow-duration curves for ephemeral 

streams are apparent. The median flow was 125 cfs for 

the 1930-40 period and 25 cfs for the 1951-61 period.

DISCUSSION OF ERRORS

Most errors in streamflow data are related to the 

definition of the stage-discharge relation; this paper is 

concerned mainly with these errors. Relatively smaller 

errors 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 that 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 a small error. However, 

the rating conditions in most alluvial channels are not 

perfect or stable. Changes in the hydraulic resistance 

to flowing water may introduce large adjustments or 

shifts in the stage-discharge relation.

The general practice in computing discharge in a 

sand-channel stream is to determine a basic stage-discharge relation by using all available measurements at 

the gaging site. The shape of the basic rating curve is 

defined by the measurements and is the result of the 

hydraulic conditions at the section of the stream that 

controls the stage-discharge relation. Adjustments to 

the rating at the time of a given discharge measurement are determined by comparing the measured stage 

with that of the basic rating that corresponds to the 

measured discharge. The indicated shift, of rating, in 

feet, at the time of measurement contains a correction 

of the basic rating to a true rating plus a possible meas367-980 O 70 2

urement error. In terms of discharge in cubic feet per 

second the equation is:

(&m (&l> == ft'm -"&

in which

Qm =measured discharge,

Qi =base rating discharge for the measured stp.ge,

Rm = difference between Qm and true discharge Qt,

and 

RK = difference between $6 and Qt,

The shifts of rating between measurements are determined by correlating the indicated shifts of rating at 

the time of measurement with stage and (or) time. Con10,000

1000 

100 

0.01 0.5 5 10 20 3040506070 80 90 95 98 

PERCENTAGE OF TIME FLOW EQUALED OR EXCEEDED

FIGUKE 8. Duration curves of winter flow, 1930-40 and 1951-61, 

Gila River at Calva.
C6 GILA RIVER PHREATOPHYTE PROJECT

siderable judgment is needed when applying the shifts 

in order that large errors are not introduced into the 

computation. How well the applied shifts represent the 

actual shifts in the computation of discharge in a sandchannel stream is unknown.

If a shifting-control method is used in a discharge 

computation in which each current-meter measurement 

is given full weight, the minimum possible standard 

error of computation (/Sc ) would equal the standard 

error of measurement ($, ). The minimum standard 

error of computation would be realized when there are 

an unlimited number of current-meter measurements. 

Because unlimited measurements are not available, the 

standard error of computation is greater than the standard error of measurement. The value of the standard 

error of computation is desired for all Gila River flow 

pertinent to the Gila River phreatophyte study.

APPROACH TO SOLUTION

The standard error of computation (8C ) for instantaneous1 flow is determined by comparing computed discharge with measured discharge where the standard 

error (&m ) of the measured discharge is known. To accomplish this, a group of measurements is chosen, and 

the record of discharge is computed as if these were the 

only measurements available. All measurements not included in the group chosen are used as a control group. 

A progressively larger number of the total measurements available at the two Gila River gaging stations 

was used in the rating analysis, and variances were 

computed for the difference between the measured discharge for the control group and the discharge computed for the time of the measurements. The assumption was made that the errors in the measurements 

(Rm ) not used in the rating analysis are independent 

of the computational errors (/?<-). The variance of the 

difference between computed discharge and measured 

discharge (^m.c ) includes the variance of the difference 

between measured discharge and true discharge.

The variance of the difference between measured and 

computed instantaneous discharge may be estimated as 

follows:

N N

in which N is the number of measurements in the control group, Rc is the difference between computed and 

true discharge, and Rm and Q t are as previously denned. 

The expected value is:

1 Instantaneous flow is not entirely correct because the standard 

error is determined for average discharge through the time of 

measurement.

if the measurement errors in the control group are independent of the computation errors. Therefore, o2 c =<T 2m-c 

 a2m where or2 denotes a "true" or population variance 

as opposed to #2 , which is estimated on the basis of the 

data. The computational error, which includes a measurement error plus an error in applying rating shifts, 

is based on measurements. However, the errors are independent because 8~m-c is denned only for the time of the 

measurements in the control group.

The standard error ($, ) of the measured discharge 

can be obtained from Carter and Anderson (1963, fig. 

1). The average number of verticals the tern: "vertical" is the same as the Carter and Anderson "str.tion" 

for the discharge measurements in this study was 22, 

and a six-teiiths-depth method was used in ino?t of the 

measurements. Therefore, the standard error of measurement (8m ) is about 4 percent (Carter and Anderson, 1963, fig. 1). Using this value with the 82m-c determined from the control group, the error of rating 

analysis (8r ) may be obtained.

ANALYSIS OF DATA

The same analytical procedure was used to determine 

/Sc for the data of both stations. Analyses for tl °, determination of /Syc were made using 1/4,1/2, and 3/4 of the 

total measurements in the rating studies. The set of data 

are overlapping, and, therefore, the results are not entirely independent. No corrections for this lack of independence are included in the analyses. The measurements were arranged in groups of four by time. From 

each group one measurement was selected randomly 

and used in a rating analysis. The remaining measurements were retained as a control group. After the basic 

stage-discharge relation was developed and shifts were 

computed from the measurements used in thQ. rating 

analysis, discharge was computed corresponding to the 

stage and time of the measurements in the control 

group. The discharges, computed and measured, were 

then grouped into summer and winter flows. Th^ groups 

were separated further into three ranges: 10-100 cfs; 

100-1,000 cfs; and 1,000 cfs or more. The mean square 

difference between the logarithms of the computed and 

measured discharges was determined for each range of 

flow. Similar analyses were made for the studies in 

which 1/2 and 3/4 of the total measurements were used 

in the rating analysis. Because analyses we^e made 

using logarithms of discharges, the standard errors of 

estimates can be stated as percentages.

In this study 144 discharge measurements from the 

Bylas record and 164 discharge measuremerts from 

the Calva record were used. The measurements were 

made during the 1963 and 1964 water years. They included 83 measurements made during summer flow near
ERROR ANALYSIS OP STREAMFLOW DATA FOR AN ALLUVIAL STREAM C7

Bylas and 89 measurements made during summer flow 

at Calva. The results of the analysis are shown in tables 

1 and 2.

ERRORS IN INSTANTANEOUS DISCHARGE

The relation of the average time between measurements to percent error in the instantaneous discharge 

for the different ranges in flow is shown in figures 9 and 

10. The trend lines drawn through the points and extended to a minimum value of 4 percent illustrate the 

improvement in determining shifts with decrease of 

time between measurements. Only a relatively few measurements were available for summer flows in the range 

from 1,000 to 4,000 cfs. Therefore, the curves in the 

1,000 to 4,000 cfs range of flow were drawn giving about 

equal weight to the plotted points.

The standard error in percentage for instantaneous 

summer flow is not the same for the three ranges of flow. 

The relation of discharge to percentage error for the 

different number of measurements in the rating analysis 

is shown in figures 11 and 12. The discharges used in the

TABLE 1. Error analysis of streamfloiv, Gila River near Bylas,

Ariz.

Fraction of Number of 

measurements measureused in ments in

stage-discharge control group 

rating analysis

One-fourth___

One-half. _

Threefourths.

23

25

11

24

12

16

16

7

14

10

7

11

Standard error in 

percent of instantaneous discharge

Plus

26. 6

16. 2

8. 0

20. 7

7.3

19.4

13. 0

8.6

8.2

5.7

14.5

9.7

Minus

21. 0

14. 0

7.4

17. 3

6.8

16. 3

11. 5

7.8

7.5

5. 3

12. 7

9. 0

Class 

(cfs)

10-100

100-1, 000

1, 000-up

10-100

100-1, 000

10-100

100-1, 000

1, 000-up

10-100

100-1, 000

10-100

100-1, 000

Season

Summer.

Do.

Do.

Winter.

Do.

Summer.

Do.

Do.

Winter.

Do.

Summer.

Do.

TABLE 2. Error analysis of streamflow, Gila River at Calva, Ariz.

Fraction of Number of Standard error in

measurements measure- percent of iristanused in nients in taneous discharge stage-discharge control group 

rating analysis Plus Minus

Glass 

(cfs) Season

One-fourth- __ 

One-half,

Threefourths.

27 

31

6

38 

18 

14

22 

5 

16 

14 

8 

13

22. 2 

14. 7 

11.3 

26. 8

7.8 

9 0

7. 3 

9. 3 

11. 6 

9. 3 

5. 9 

6.4

18. 2 

12. 9 

10.2 

21. 2

7. 2

& 0

6.8 

8.7 

10.4 

8.4 

5. 6 

6. 0

10-100 

100-1, 000 

1, 000-up 

10-100 

100-1, 000 

10-100

100-1, 000 

1, 000-up 

10-100 

100-1, 000 

10-100 

100-1, 000

Summer. 

Do. 

Do. 

Winter. 

Do.

Do. 

Do.

Winter. 

Do. 

Summer. 

Do.

plot are the average of the measured discharges in the 

different ranges of study.

Extrapolating the relations to obtain a result by 

using all measurements average time between measurements is 3.1 days for the rating analysis, the relation 

for the Bylas gage suggests that there is a standard 

error of about 12 percent for the instantaneous summer 

flows ranging from 40 to 100 cfs and a standard error 

of about 8.5 percent for flows from 100 to 1,000 cfs. Although it is not defined, the standard error is probably 

about 5 percent for instantaneous summer flows ranging 

from 1,000 to 4,000 cfs (figs. 9 and 10). The standard 

error of computation for summer flow at Calva is about 

5 percent for flows from 40 to 100 cfs and about 6 percent for flows from 100 to 1,000 cfs. Although it is rot 

defined, the standard error is probably 7 percent for 

flows from 1,000 to 4,000 cfs. At Calva the relation of 

discharge to standard error of computation suggests an 

error of about 5 percent for winter flows from 40 to 

1,000 cfs (figs. 13 and 14). The standard error for the 

Bylas winter data is perhaps 1 percent better than that 

at Calva.

The relation of discharge to percentage error for instantaneous summer flows suggests better accuracy for 

the low-flow data near Calva than for the data at Bylas 

(figs. 9-12). The reason for this apparent difference in 

accuracy is not known, but it probably is the result of 

the effect of the old roadbed about 150 feet downstream 

from the Bylas gage.

The sediment moves through the channel at Calva 

without being affected by artificial conditions. The 

changes in resistance to flow as a result of scour or f 11, 

changes in bed configuration, or vegetation growing in 

the channel generally vary directly with discharge and 

velocity or change slowly with time. By the use of the 

measurements and good judgment, the changes in resistance are properly reflected in the discharge computation. However, the artificial control (roadbed) at Bylas 

has introduced a changing resistance to flow that cannot 

be identified correctly by use of the available measurements. The shifting sandbar that forms behind the old 

roadbed and gradually moves downstream presents an 

unpredictable changing condition that is reflected in the 

computed discharge data. The sandbars may affect the 

velocity of approach to the control and result in a charge 

in discharge for a given stage. The shifting sandHr 

may result in bed configurations that cause variable resistance to flow.

The indicated better accuracy of the data at Bylas 

over the data at Calva for flows in the 1,000 to 4,000 cfs 

range is probably the direct result of the stable condi-
C8 GILA RIVER PHREATOPHYTE PROJECT

25

20

15

10

Winter flow

30

25

20

15

10

EXPLANATION

Range in discharge, 

in cubic feet per second

O 10-100

X 100-1000

A 1000-4000

Summer flow

10 15 20 25 

AVERAGE TIME, IN DAYS, BETWEEN MEASUREMENTS

30 35

FIGUKE 9. Standard error of computed discharge for different ranges of discharge, Gila River near Bylas.

tion at the road crossing below the Bylas gage. Scour 

and fill affect the high-flow rating more at Calva than 

at Bylas.

There is very little, if any, difference in the accuracy 

of data of instantaneous discharge at the two stations 

for winter flows ranging from 10 to 100 cfs. The amount 

of sediment in winter flows ranging from 10 to 100 cfs 

is small.

ERRORS IN VOLUMES OF DISCHARGE

The standard error of computation in average flows 

or volumes, because of compensating effects, is less than 

the standard error in instantaneous discharge. The

standard error of computation in the volume of discharge should decrease inversely with the square root of 

the number of independent observations of error. The 

rate of increase with time in the number of independent 

observations of error in the discharge data must be 

determined.

As previously discussed, the indicated shif4: or discharge difference at time of measurements is equal to 

Rm R^ and the variance /S~c is equal to the sum (8-*) + 

(fPm). The error in the measurements (ftm ) is assumed 

to be random with a mean of zero. The Rm then will be 

compensating with time.
ERROR ANALYSIS OF STREAMFLOW DATA FOR AN ALLUVIAL STREAM

30

25

20

15

10

Winter flow

30

25

ce 

O * 20

LLJ

15

10

EXPLANATION

Range in discharge, 

in cubic feet per second

O 10-100 

X 100-1000 

A 1000-4000

Summer flow

10 15 20 25 

AVERAGE TIME, IN DAYS, BETWEEN MEASUREMENTS

30 35

FIGURE 10. Standard error of computed discharge for different ranges of discharge, Gila River at Calva.

Considerable personal judgment goes into the application of the shift corrections between measurements. 

The errors involved as the result of personal judgment 

probably compensate somewhat, from measurement to 

measurement because judgment is independently adjusted to each measurement.

There may be periods when large flows are from one 

flood wave during which the shifts are not defined by 

measurements. The error in the data, as a result of personal judgment, may be all in one direction for a given

storm. The errors in the data for another storm may H 

in the opposite direction. Therefore, in this report tl n 

computational error (fic ) is assumed to be random with 

a mean of zero. The number of independent observations of fic then would equal the number of discharge 

measurements made during the period.

The standard error in the surface-flow data would be:

V

~S? 

^
CIO

3000

2000

,1000

100

10

GILA RIVER PHREATOPHYTE PROJECT

3000

10 15 20 

PERCENT ERROR

25 30

FIGURE 11. Standard error of computed discharge for summer flow, GLla River near Bylas.

where N^ is the number of independent measurements 

in the period of interest and /Sc is the standard error in 

the instantaneous discharge. Although it is not done in 

this report, standard errors for any length of record 

for the different ranges of flow can be determined.

ERRORS IN WATER BUDGET

The volume of flow for 3-week periods at the downstream end of subreach 1 Gila River near Bylas to 

Gila River at Calva (fig. 1) will be subtracted from 

that at the upper end to give the net surface-water component for the water budget in the study reach. Assuming that the errors at the two stations are independent, 

the error of the difference equals the square root of the 

sum of the squared standard errors of computation in 

flow data for the two stations. Thus, the standard error 

of computation for the difference in flows would be

which can be evaluated for instantaneous flow from

2000

J 000

a

o: 100

10 10 15 20 

PERCENT ERROR

30

FIGURE 12. Standard error of computed discharge for summer flow, Gila River at Calva.

the relations of figures 9 through 14 for any given frequency of measurement within the range studied.

The standard error in the surface-flow budget data 

would be:

S*, =

(SCI) 2

where N^ and N? equal the number of measurements in 

the period at the upstream and downstream stations, 

respectively. In order for the foregoing equations to 

apply, the errors must be in compatible units and must 

be about normally distributed. This most nearly applies 

to discharge measurements if errors are in terms of 

logarithmic units rather than in natural units such as 

cubic feet per second or acre-feet per day. For the relatively small errors usually observed in stream gaging, 

percentage errors may be substituted for error? in natural units. For ease in presentation, percentage errors 

have been used throughout this report. If the standard 

error in percentage is used in the equation, the resulting 

evaluation is the error in the difference in flow at the
ERROR ANALYSIS OF STREAMFLOW DATA FOR AN ALLUVIAL STREAM

1000

y 100

CO 

D 

O

10

o 

FIGURE

25 30

13. Standard 

flow,

10 15 20 

PERCENT ERROR

error of computed discharge for winter 

Gila River near Bylas.

1000

9 100 

oo

D 

O

10.

10 15 20 

PERCENT ERROR

25 30

FIGURE 14. Standard error of computed discharge for winter 

flow, Gila River at Calva.

two sites expressed as a percentage of an average discharge and must be converted to natural units for use 

in a budget. The standard error thus computed is rot 

entirely correct for budget periods in which there r,re 

losses or gains in streamflow. If the difference in flow 

between the two gaging sites becomes large relative to 

the average ~ flow in the reach, j then the standard error

of estimate must be derived using natural units. For 

relatively small differences in flow at the two sites, the 

standard error computed by the two methods is about 

the same. For instance, when NL = N2 8, the standard 

error is given in table 3.

TABLE 3. Standard error of estimate

Standard error ir 

3 -week period

Gage

1 T7rr»m t.

Average 

discharge 

fcfs)

70

60

hp finiifl.t.iftn

Standard error 

in instantaneous 

flows

Percent Cfs

12 8. 4 _

5 3.0

,<?, - /(Sc')2 , (

SCl =8.4 

cfs 

SC2 =3.0 

cfs

Cfs

!3. 15

Sc2 ) 2

SCl=12 

percent

SC2=5

percent

Cfs

!3. 0

The average daily loss in surface flow in reach 1 for 

the period of record through water year 1965 is about 

11 cfs (discharge of Gila River near Bylas plus tributary flow minus Gila flow at Calva). The average daily 

loss in surface flow in reach 1 during the water ye^.rs 

1963 through 1965 ranged from zero at flows below 

20 cfs in the winter to a high of about 500 cfs at bankfull flows of about 4,000 cfs in the summer.

In this report the standard error for the 3-w?,ek 

budget data was computed using standard errors in 

percent from the relations of figures 11 through 14 in 

the equation:

/(SC1 ) 2 i *=V^vT+-

N2

An average of eight measurements in the summer p,nd 

an average of three measurements in the winter were 

made at each station per budget period. The standard 

errors in the 3-week budget data for summer and winter flows are shown in figure 15.
C12 GILA RIVER PHREATOPHYTE PROJECT

1000

900

800

700

600

y 500

CD

400

300

200

100

345 

PERCENT ERROR

FIGURE 15. Standard error in percentage of average discharge in 3-week streamflow data for subreach 1, Gila 

River near Bylas to Gila River at Calva.

SUMMARY

Discharge measurements made in 1963 and 19"4 were 

used to determine the standard error in the computed 

discharge for two stations Gila River near Bylas and 

Gila River at Calva. The standard errors of computation for instantaneous discharge and for the difference 

in volume flow for the two stations for a 3-week period 

are approximated. The standard errors in the difference 

in volume flows were desired for water-budget studies.

The channel at the gaging stations is priirarily a 

pool-and-riffle type, and the banks are stabilized by the 

dense growth of saltcedar. Sediment movement during 

summer flow is high.

The standard errors of computation were determined 

by using some of the discharge measurements in the 

rating analysis and the remainder as a control group. 

Discharge was computed corresponding to the stage 

and time of the discharge measurements in the control 

group. The assumptions were made that the ireasurement errors in the discharge measurements in the control group were independent of the measurement errors 

in the computed discharge and that they had a mean 

of zero.

The mean square difference ($2m _c ) between computed 

and measured discharge was determined for different 

ranges in instantaneous flow. The mean square difference (/S2 m -c) is the sum of the variance (/S2m ) of the 

difference of measured discharge from true discharge 

plus the variance (£zc ) of the difference of computed 

discharge from true discharge. The desired variance 

(/SY2 e ) was obtained by subtracting the measured variance (&'2m.) from the total variance (S2m-c).

The results of the analyses suggest a standard error of 

computation for instantaneous summer flow at the Calva 

station of about 5 percent for flows ranging from 40 

to 500 cfs. The standard error of computation increases 

to about 6 percent for instantaneous flows ranging from 

500 to 2,000 cfs. The standard error of computation for 

instantaneous summer flow at the Bylas station is from 

about 14.5 to 7.5 percent in flows ranging from 40 to 

500 cfs. The standard error of computation for instantaneous summer flow at Bylas is about 7.0 percent for 

flows ranging from 500 to 2,000 cfs. The apparent difference in accuracy of data for the two stations for flows 

in the 40 to 500 cfs range is accredited to an old roadbed about 150 feet downstream from the Bylrs gage. 

The sediment moves through the controlling reach at 

the Calva station without being affected by artificial 

conditions. The changes in the resistance of fow are 

reflected properly in the discharge computatior. However, the artificial control (roadbed) at Bylas has
ERROR ANALYSIS OF STREAMFLOW DATA FOR AN ALLUVIAL STREAM C13

introduced a changing resistance to flow that cannot be predicted accurately by use of the available 

measurements.

The standard error of computation for instantaneous 

winter flows ranging from 30 to 220 cfs was about 5 

percent at both stations.

The difference in volume flow at the two stations 

contains the square root of the sum of the squared 

errors in flow data at the stations if errors at the stations are independent. The standard error of computation for instantaneous discharge is adjusted to a 3-week 

period by assuming that the number of independent 

observations during a period is equal to the number 

of discharge measurements made during the period. The

standard error of computation for summer flow ir a 

3-week budget period is from about 5 to 3 percent of 

the average flow in the channel for flows ranging from 

about 30 to 1,000 cfs. The standard error of computation 

for winter flow in the 3-week period is from about 3.8 

to 4.4 percent of the average flow for flows ranging 

from 30 to 500 cfs.

LITERATURE CITED

Carter, R. W., and Anderson, I. E., 1963, Accuracy of current 

meter measurements: Am. Soc. Civil Engineers Proc., Hydraulics Div. Jour., v. 89, no. HY4, p. 105-115.

Dawdy, D. R., 1961, Depth-discharge relations of alluvial 

streams discontinuous rating curves: U.S. Geol. Survey 

Water-Supply Paper 1498-C, 16 p.

U. S. GOVERNMENT PRINTING OFFICE : 1970 O - 367-980