Accuracy of

Low-Flow Characteristics 

Estimated by Correlation of 

Base-Flow Measurements

Manual of Hydrology: Part 2. Low-Flow Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1542-B
Accuracy of

Low-Flow Characteristics 

Estimated by Correlation of 

Base-Flow Measurements

By CLAYTON H. HARDISON and MARSHALL E. MOSS ,

With a section on OUTLINE OF DERIVATIONS

By EDWARD J. GILROY

Manual of Hydrology: Part 2. Low-Flow Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1542-B

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1972
UNITED STATES DEPARTMENT OF THE INTERIOR 

ROGERS C. B. MORTON, Secretary

GEOLOGICAL SURVEY 

V. E. McKelvey, Director

Library of Congress catalog-card No. 72-600238

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

Washington, D.C. 20402 Price 40 cents (paper cover)

Stock Number 2401-2168
CONTENTS

Page

Symbols._________________________________________________________ III

Abstract. __________________________________________________________ 35

Introduction. ___________________________________________^___________ 35

Derivation of accuracy equations--______---------_----------_-_------- 36

Example ___________________________________________________________ 42

Discussion __________________________________________________________ 45

Outline of derivations, by Edward J. Gilroy _ ____________--__-_--_--____ 48

Background-___-___----_______________________--___----------_- 48

Statement of the problem,________________________________^______ 48

Statistical model. _________________________-______----_-------_-- 50

Variance of the estimator of the T-year low flow (method !)_______.__ 51

Estimator of variance of annual low flows (method II)_______________ 53

References-________ ___-___________________-___________-------__---_ 55

ILLUSTRATIONS

Page

FIGURE 14. Accuracy of low-flow characteristics estimated by correlation,_ 41 

15. Relation of base-flow measurements--- _____________________ 42

TABLES

Page

TABLE 1. Values of R for use in appraising accuracy of a T-year low estimated 

from observed annual low flows from a log Pearson Type III 

distribution of known skew___________________-_----------_ 39

2. Accuracy of low-flow characteristics estimated by correlation of

base-flow measurements-___________________-___----------_ 41

SYMBOLS

[Used in sections by Hardison and Moss]

b Slope of regression line defined by sample statistics.

|3 Slope of the true regression line.

BG Logarithm of a base flow at the ungaged site; BG is mean BG of a sample.

BU Logarithm of a base flow at the ungaged site; Bu is mean BU of a sample.

F Factor used in computing NuIv Standard deviation of logarithms of annual low flows.

in
IV SYMBOLS

IVG Iv for gaged site.

IvU I* for ungaged site.

fc Factor for adjusting SB, U/SB, G to Ivu/IvoK Factor used in computing F.

M Number of independent base-flow measurements.

N Number of annual low flows.

No Number of annual low flows observed at gaged site.

Nu Accuracy of estimated T-year low flow at ungaged site, in equivalent years

of record. 

r Coefficient of correlation computed from sample statistics; r is r adjusted

for consistency with other graphically computed statistics. 

p Coefficient of correlation of the population values of BG and BU. 

R Factor relating standard error of T-year low to / and N. 

SB, G Standard deviation of the logarithms of the sample of base flows at the

gaged site that were used in the regression; SB, G is graphically determined SB, G adjusted for consistency with other graphically determined

statistics. 

SB, u Standard deviation of the logarithms of the sample base flows at the

ungaged site that were used in the regression; S'B, u is SB, u based on

SB, G', SB, u is mean of S'B, u and graphically determined SB, uSEn Standard error of estimate of a regression, equals V(l r) 2ss, u. 

8Er Standard error of a T-year low flow computed by fitting a log-Pearson

Type III distribution to a sample of annual low flows. 

SET, G SET at gaged site. 

SET, u SET at ungaged site. 

as, G Standard deviation of the population of BG'S of which the base-flow

measurements are a sample. 

<TB, u Standard deviation of the population of Bv's of which the base-flow

measurements are a sample. 

<JT, G Same as SET, G. 

T Average interval in years between recurrences of annual lows of equal or

greater severity.

VG Equals (SET, o) 2 , the variance of an estimated T-year low at the gaged site. 

VR Equals (SEft) 2 , the variance about the line of regression. 

Vu Equals (SET, u) 2 , the variance of an estimated T-year low at the ungaged

site.

XT, G Logarithm of a T-year low flow at the gaged site. 

XT, G XT, G estimated from a sample of annual low flows at the gaged site. 

XT.U Logarithm of T-year low flow at the ungaged site. 

XT, u Estimator of XT, uz Distance from the mean in standard deviation units; equals (XT, G BG)/

SB, G.
MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECHNIQUES

ACCURACY OF LOW-FLOW

CHARACTERISTICS ESTIMATED BY

CORRELATION OF BASE-FLOW

MEASUREMENTS

By CLAYTON H. HARDISON and MARSHALL E. Moss

ABSTRACT

The number of discharge measurements required to define an acceptable relation 

between the base flow at a stream gaging station and that at an ungaged site depends on the statistics of the regression, the accuracy goal, and the length of record 

at the gaging station used in the regression. Equations and graphs are presented 

for evaluating the accuracy of a low-flow characteristic estimated from such a relation in terms of the number of years of record that would be required at the ungaged site to give an estimate of comparable accuracy. An outline of the derivations 

of basic equations is presented in a separate section of the report.

INTRODUCTION

In regression analysis of streamflow characteristics and basin 

parameters, Thomas and Benson (1970) found that the characteristics of low flow cannot be adequately estimated from basin parameters obtainable from present maps. As an alternative, Riggs (1965) 

recommends that low-flow frequency characteristics be estimated 

by relating the logarithms of the base flow at an ungaged site to the 

logarithms of the base flow at a gaged site through the use of discharge measurements made during periods of base flow. This paper 

presents a procedure for evaluating the accuracy of low-flow characteristics estimated from such a relation. The low-flow characteristics 

considered are T-year low flows such as the 2-year 7-day (median 

annual) low flow and the 10-year 7-day low flow. (T is the average 

time in years between recurrences of annual low flows smaller than 

that given as the T-year low flow.) The length of record at the gaged 

site, number of base-flow measurements, and the standard error of 

estimate of the regression are all used as parameters. All standard 

errors, SE, and variance, V, are expressed in log units. Subscripts 

G and U are used for gaged and ungaged sites, respectively, and

35
36 MANUAL OF HYDEOLOGY

subscripts B, L, and T are used for base flow, annual low flow, and 

T-year low flow, respectively.

The first part of the report makes practical application of the mathematical equations derived by Edward J. Gilroy in the section "Outline of Derivations." The notation required for the rigorous mathematical derivation in this latter section is considerably more extensive 

than that shown in the "List of Symbols," and in some instances 

the notation is different. One difference in notation is that in the 

first part of the report the logarithmic standard deviation of annual 

low flows is designated / (for index of variability) both for the population and for a sample, whereas in the latter part the logarithmic 

standard deviation of the population is designated & L and that of 

the sample S L . Other differences are that in the first part, VG and Vv 

are used in place of o-V, G and Var(Xy, u) respectively, and that for 

the variance about the regression line (square of the standard error 

of estimate of the regression) VR is used in place of (1 p2 )o-2 s, v. 

The implicit assumptions discussed in the first part of the report 

represent the authors' interpretation of the assumptions made by 

Gilroy in the derivation of the basic equations on which the report 

is based.

DERIVATION OF ACCURACY EQUATIONS

The accuracy of a regression line at a given value of x is given by

in which VR, the variance about the regression line, is the square of 

the standard error of estimate, M is the number of observations used 

to define the regression, z is the number of standard deviation units 

between the mean of the z's and the x at which the regression is being 

used, and Vu is the estimated variance of the corresponding estimate 

of y about the y that would be given by the true regression line based 

on population parameters. This equation is valid only if the independent variable is fixed; that is, only if the same values of the independent variable would be used in every repetition of an experiment. 

In the development of a relation between base-flow measurement at a 

gaged and an ungaged site, however, the independent variable is not 

fixed. Furthermore, when the relation is entered with a low-flow 

characteristic at the gaged site (independent variable) to obtain the 

corresponding characteristic at the ungaged site, the time-sampling 

error at the gaged site contributes to the error in the low-flow characteristic estimated for the ungaged site. For these conditions and under 

the five assumptions given in the next paragraph, Gilroy (see equation
ACCURACY OF LOW-FLOW CHARACTERISTICS 37

30 in the section "Outline of Derivations") gives the variance of low 

flows estimated at the ungaged site as

in which Var(XT , u) is the variance of the estimate of the T-year low 

flow at the ungaged site, VR is the variance of the departures from the 

line of regression, VG is the variance of the T-year low flow at the gaged 

site, M is the number of independent base-flow measurements, <r2B , G 

is the variance of the population of base flows of which the base-flow 

measurements are a sample, 8 is the slope of the true regression, and 

z is as previously defined.

The five assumptions implicit in the derivation of equation 1 are:

1. The lower end of the true relation between base-flow measurements is the same as the true relation between annual low flows.

2. The relation between annual low flows is the same as the relation 

between annual 7-day low flows.

3. The time-sampling error in the low-flow characteristic that is 

used to enter the regression equation is independent of the variation in the base-flow measurements used in defining the regression.

4. The logarithms of the base flows at the gaged and ungaged sites 

follow a bivariate normal distribution.

5. The M observations used to define the regression equation are 

statistically dependent estimates of the base-flow relation.

The first four of these assumptions appear to be reasonably applicable to the conditions under which equation 1 is used in this paper. 

The fifth assumption can be met by establishing criteria for counting 

only those measurements that can reasonably be assumed to be 

independent.

By using SB , G and 6 as estimates of <TB , G and 8, respectively, 

equation 1 becomes

 * * * llit OT 7 /Q\

in which Vv is the estimator of Var(Jtr, u) and in which SET , G = \/VG 

is the standard error of the T-year low flow at the gaged site.

If SET , G, SB, G, and VG are expressed in log units, the square root 

of Vu as obtained from the solution of equation 2 gives SET , u, the 

standard error of the estimated low-flow characteristic at the ungaged site, in log units. As accuracy goals recently set by the U.S. 

Geological Survey for the collection of streamflow data (Carter and 

Benson, 1970) are expressed in equivalent years of record (number
38 MANUAL OF HYDROLOGY

of years of record required to give a low-flow characteristic of equivalent accuracy) it is desirable to present equation 2 in such a way that 

the relation between number of base-flow measurements, statistics 

of the regression, years of record at the gaged site, and the resulting 

accuracy at the ungaged site in equivalent years of record becomes 

more obvious. The remainder of this section develops equations and 

a graph that accomplish this objective. Readers who are interested 

only in using the graph and equations should skip to the example in 

the next section. 

Letting

M 3 M 3 \ SB 

equation 2 becomes

(SET , G\( M \ 

\SB ,G J \M-ZJ

If the annual low flows are from a population that can be defined 

by a log-Pearson Type III distribution, the standard error of a T-year 

low is given by

in which R is as given in table 1, / is the standard deviation of the 

logarithms of the annual lows, and N is the number of observed annual 

low flows (usually the number of years of record). Thus the variance 

of the T-year low flow at the gaged site is given by

VG = (SET , G) z = R2G (IvGY/NG (5)

in which NG is the actual number of years of record at the gaged site. 

At the ungaged site

Vu = &u(Lu?/Nu (6)

in which Nu is the number of years of record at the ungaged site that 

would be required to give a T-year low flow as accurate as that indicated by Vv, the variance of the estimate at the ungaged site. Substitution of equations 5 and 6 in 4 gives

_

Nv ~M\ K NG '

According to Gilroy (see equation 39 in the section "Outline of 

Derivations") an unbiased estimate of (IvUY gives
ACCURACY OF LOW-FLOW CHARACTERISTICS 39

,

s53 

 «

S> » "e w

53 s- ^a5n

tf

S

2

s .-85

 ^ B- ^ &&

Ifc

<X> <£> CO

(N O5 tt) t> t> ( O5 CO CO

O T-I (N CO CO

in oo us o oo co -^*

-<* CO (N T-I (N O t00 «O CO O (N O5 »O

O T-I CNI CO CO CO ^*

CO ^O CD CD O CD OO

CO CO U3 T-I CD O CO O5 «* O5 -^* »O O «*

O T-I T-I (N <M CO CO

co co 05 T-I co co -^*

CO (N <M (N T-I 00 U3

O5 CO «O O5 O (N U3

t- 05

O5 (M

Oi (N

O -<* O * T-I CO U3

O tt) U3 CO O5 tt) (N

O »-( CO U3 »O t- O5

t> (N CO

O5 O U3

O5 T-I (N

O 00 (N 

IO CD teo o co tt) tt) eo co

CO (N -^ t- T-I CO CO

O5 O T-I <M CO -^ »O

CO T-I (N CO CD -<* T-I 

O5 O5 O T-I T-I <M CO

 * (M (M O (N t- O

00 00 O5 O O O T-I

 * t- 00 t- t- t- tt- t- 00 O5 O5 O5 O5
40 MANUAL OF HYDROLOGY

Assuming that the square root of (Iv vY gives an unbiased estimate of 

IvU, and considering that by definition b equals r(sB , U/SB, <?),

I = I vU r vG,

in which

Then by assuming that the coefficient of skew of the logarithms 

of the annual low flows is the same at both sites, Ru = Ro = R, and 

by substituting (SER} 2 for VR, equation 7 gives

R*k*b*P.e _SEB*/l+*f\ , b*R*I\G ~^rv JT\TT)+ NG (W)

For an approximate solution of equation 10, the k2 and K terms can 

be replaced by 1.0 and (M 3)/M, respectively. (Evaluation of equations 3 and 9 shows that A; 2 tends to be several percent less than unity, 

and the KM tends to be several percent greater than M minus 3, 

so that the effects of the replacements tend to cancel.) 

Thus

1 fiE ? +AV

Letting

and solving for Nv

(12)

in which r is the coefficient of correlation of the base-flow measurements.

For values of r less than 0.8, the answers given by equations 11 and 

12 become increasingly less exact. In such cases, or if the solution is 

to be made by electronic computer, Vu should be computed by equation 2, and Nv should be computed by a combination of equations 

6 and 8 as follows

- Vu Vv\ r

in which Rv has been assumed to be equal to RG and in which k is 

as given by equation 9.

In equations 11 and 13, R, which can be obtained from table 1, 

derends on the recurrence interval, T, of the low-flow characteristics
ACCURACY OF LOW-FLOW CHARACTERISTICS 41

and on the skew coefficient of the logarithms of the annual low flows. 

The values of R given in table 1 were computed from an equation 

for SET given by Hardison (1969).

The solution of equation 12 for F values as much as 200 is given in 

table 2. Interpolation between the F values shown can be made from 

curves such as those shown in figure 14, which are plotted from the

TABLE 2. Accuracy of low-flow characteristics estimated by correlation of base-flow

measurements

[No is the number of years of record at the gaged site, NU is the accuracy at the ungaged site in equivalent 

years of record, r is the coefficient of correlation, and F depends on the statistics of the correlation]

rWj/, in years, for indicated F

5

10 _ 3.33

12__-_-____ 3.53

15-_---__- 3.75

20 4 00

25 --. 4.17

30__-______ 4.28

40._.______ 4.44

10

5.00

5.45

6.00

6.67

7.14

7.50

8.00

i

6

6

7

8

9

10

10

5

00

67

50

57

37

00

91

20

6.67

7.50

8.57

10.00

11.11

12.00

13.33

3

7

8

10

12

13

15

17

10

50

57

00

00

64

00

14

40

8.00

9.23

10.91

13.33

15.38

17.14

20.00

60

8.57

10.00

12.00

15.00

17.65

20.00

24.00

l(

9

10

13

16

20

23

28

)0

09

71

04

67

00

08

57

200

9.52

11.32

13.95

18.18

22.22

26.09

33.33

40

O 

u

LLJ 

tt

"- 30 

O

Z

LLJ

O

LLJ

z

3

20

10

20 40 60 80 100

FIGURE 14. Accuracy of low-flow characteristics estimated by correlation of base-flow measurements.
42 MANUAL OF HYDROLOGY

results given in table 2. The accuracy, in equivalent years of record, 

can be estimated by using F, computed from the statistics of a plot 

of base-flow measurements on logarithmic paper, to enter figure 14 

with the length of record at the gaged site as a parameter, and thus 

obtaining a value of rzNv that can be multiplied by 1/r2 to give Nv.

A graphical procedure for relating base-flow measurements and for 

using figure 14 to estimate the accuracy of an estimate obtained from 

the relation are illustrated by means of an example in the following 

section.

EXAMPLE

In figure 15, 28 base-flow measurements made on Old Mill Stream 

branch at Centerville, Md., (drainage area 11.2 sq mi) have been 

plotted against the corresponding daily discharge at the streamgaging station on Unicorn Branch near Millington, Md., (drainage 

area 22.3 sq mi). These daily discharges are used as being equivalent 

to the base flows that would have been measured at the gaging station 

on the days the base-flow measurements were made at Centerville 

and are called base-flow measurements in the remainder of this report. The relation line based on the points in figure 15 is a graphical

Q

Z

o

u

LU CO

u< LU

oo of.

200

100

~5

LUOQOf. ^

<o

x 

u to 2 -

2 5 10 20 50 100 200

DISCHARGE, IN CUBIC FEET PER SECOND,

UNICORN BRANCH

500

FIGURE 15. Relation of base-flow measurements, 1952-66, Unicorn Branch near Millington, Md., and 

Old Mill Stream Branch at Centerville, Md.
ACCURACY OF LOW-FLOW CHARACTERISTICS 43

estimate of the line of regression for estimating base flow at the ungaged site from that at the gaged site. As such, the slope should be 

such as to minimize the sum of the squares of the vertical departures 

from the line. Horizontal departures should not be considered. Under 

the assumption that the relation line is applicable to 7-day low flows 

also, the median annual 7-day low flow of 7.0 cfs (cubic feet per second) 

at the gaging station gives 4.0 cfs as the estimated median annual 

low flow on Old Mill Stream Branch. The problem is to appraise the 

accuracy of this estimate by the use of figure 14.

By assuming the vertical departures from the line of regression, 

in log units, to be normally distributed, and by drawing parallel 

lines to exclude the one-sixth of the points that have the greatest 

positive departure and the one-sixth that have the greatest negative 

departure, an estimate of the standard error of regression can be obtained as half the distance, in log units, between the upper and lower 

lines. In figure 15, the parallel lines are drawn between the fourth and 

fifth greatest positive and greatest negative departures, and the resulting standard error of estimate, SER, is one-half the log of 14.1/ 

10.0, or 0.075 log units. The slope of the regression line, 6, is 0.90 as 

shown in figure 15.

The value of z is computed as (XT , G BG}/sB , G in which SB , G 

is the standard deviation of the logarithms of the base flow points at 

Unicorn Branch, and r is computed as l \/(SER/SB, uY in which 

SB, u is the standard deviation of the logarithms of the base flow 

measurements at Old Mill Stream Branch. Values of SB , G and SB , u 

for use in making these computations can be estimated graphically 

by excluding one-sixth of the largest and one-sixth of the smallest 

base-flow measurements in the horizontal direction and in the vertical 

direction respectively and by taking one-half the resulting range in 

log units as shown in figure 15 (SB , G and SB , u equal 0.42 and 0.415 

respectively). In order to obtain theoretical consistency between the 

graphically determined statistics, however, values of sBi u and SB , G 

should be adjusted and r should be computed as follows:

1. Compute S'B.U = \/62s2B. G +SE2R = V(0.9) 2 (0.42) 2 +(0.075) 2 = 0.385

2. Compute ss, u = (s'B , U+SB , u) /2 = (0.385+ 0.415) /2 = 0.400

3. Compute r = \/l - (SER /sB , uY = Vl - (0.075 /0.400) 2 = 0.982

4. Compute SB , G = SB , u(r/b] =0.400(0.982/0.9) =0.436.

Although in this instance r changes only slightly and values of SB , u 

and SB, G are changed only a few percent, adjustments will usually 

be larger because of fewer base flow measurements. Therefore the 

adjustments should be made before computing values of r and z to 

use in equations 11 and 12.
44 MANUAL OF HYDROLOGY

The value of (XT, G BG) for use in computing z is the distance in 

log units between the median (logarithmic mean) of the base-flow 

measurements at Unicorn Branch, which is 15 cfs, and the estimated 

median annual 7-day low flow at that site, which is 7.0 cfs. It equals 

the log of (7/15) or -0.33 log units. The resulting z is -0.33/0.436 

or 0.757 and z2 is 0.57. The value of BG to use in computing (XT , G 

 BG ) is taken as being midway of the range of the standard deviation 

lines so long as the resulting BG is consistent with a similarly computed 

BU- If the computed BG and Bv do not plot on the regression line, 

one or both should be adjusted. It should be noted that both BG and 

SB, G depend on the discharge of the base-flow measurements at the 

gaged site, and thus depend on when the measurements happen to 

be made. As such measurements are usually made at various times 

during the year, the resulting BG and SB , G will usually be larger than 

the corresponding statistics of the annual low flows.

The value for IvG, the standard deviation of the logarithms of 18 

annual 7-day low flows at the gaged site (Unicorn Branch), was 

computed to be 0.17, and the logarithmic skew coefficient, 1.0. 

(One of the observed annual 7-day low flows in the period 1948-66 

was omitted in the computation of these statistics, because it was 

abnormally low as a result of regulation at a fish hatchery upstream.) 

For this skew coefficient, table 1 gives an R value of 0.933 for the 

median annual 7-day low flow.

The only remaining term to be evaluated in equation 11 is M, 

the number of base-flow measurements, which in the derivation of 

the equation were assumed to be independent, that is, to be independent estimates of the base-flow relation. As measurements made 

during the same low-flow period tend not to be independent, M 

has been used as 16 instead of 28. The figure of 16 was obtained by 

counting only those measurements made at least 30 days after a 

previous base-flow measurement. Using this criteria, all five measurements made in 1964-66 were counted, but only 11 of the 23 

semimonthly measurements made between October 1952 and October 

1953 were counted. Other equally valid rules regarding independent 

base-flow measurements could no doubt be formulated by studying 

the causes of such scatter to see how much of it is due to year-toyear variation in the base-flow relation and how much is due to the 

unequal effect of direct runoff from recent rain.

Using equation 11 and obtaining R from table 1, F lor the median 

annual low flow is

'["(0.9X0.17) I2 

L 0.075 J
ACCURACY OF LOW-FLOW CHARACTERISTICS 45

and as the median annual 7-day low flow at the gaging station on 

Unicorn Branch is based on 18 annual 7-day low flows, NG is 18. 

For these values of F and NG, interpolation between the 15- and 20- 

year curves in figure 14, or solution of equation 12, gives 11.2 years for 

r*Nu. As r is 0.982, Nv is 11.2/(0.982) 2, or 11.6 years. This means 

that 12 annual 7-day low flows would have to be observed on Old Mill 

Stream Branch at Centerville to obtain a median annual 7-day low 

flow as accurate as the 4.0 cfs given by the regression of base-flow 

measurements.

For the 10-year 7-day low flow, the estimate based on the observed 

record on Unicorn Branch is 3.9 cfs, and by the relation line in figure 

15 the corresponding estimated discharge on Old Mill Stream Branch is 

2.3 cfs. In appraising the accuracy of this estimate by use of equation 

11, (%T, G BG] equals the logarithm of 3.9/15 or 0.585, z2 equals 

(-0.585/0.436) 2 or 1.80, and R is 1.956, as given in table 1 for T of 

10 and skew coefficient of 1.0. The other terms in the equation remain as they were for the median annual low flow and F becomes 

73.9. From figure 14 with an NG of 18 years, r2Nv is 14.5 years and Nu 

is then equal to 15.0 years. Thus the accuracy of the estimated 10-year 

7-day low flow, in equivalent years of record, is 15.0 years as compared 

to an accuracy of 11.6 years for the median annual 7-day low flow.

DISCUSSION

The fact that the 10-year low flow as estimated in the preceding 

example has a higher accuracy in equivalent years of record than 

does the estimated median (2-year) low flow deserves explanation. The 

fundamental reason behind the change of Nv with change in T is that 

values of z, SET , G, and VG in equation 2 all increase with T so that, 

measured in log units, the standard error of the 10-year low is always 

greater than that of the 2-year low. The conversion from standard 

error in log units to equivalent length of record in years, however, 

depends on the theoretical standard error of a T-year peak based on a 

given number of years of record at the ungaged site, and thus depends on the R values in table 1. When equation 13 is used to make 

this conversion, the increase in R2G with increase in T may well overbalance the increase in Vv, particularly when RG increases rapidly 

with T as it does for large negative skew coefficients. The alternative 

to estimating the T-year low from a regression of base-flow measurements is to operate a stream-gaging station at the ungaged site for 

Nu years. At sites where the annual low flows have a negative skew 

coefficient, more years are required to obtain the same standard 

error in log units for a 10-year low than at sites with positive skew 

coefficients. Had the logarithmic skew coefficient for the annual
46 MANUAL OF HYDROLOGY

low flows in the example been +1.0 instead of 1.0, NU for the 10- 

year low would have been less than for the 2-year low.

In the example shown in figure 15, the base-flow measurements 

cover a considerably larger range in discharge than do the annual low 

flows, and the resulting z values tend to make the estimates less 

accurate than would be the case if the mean of the base-flow measurements were nearer the mean of the annual low flows. Although 

fewer measurements would be required to give comparable accuracy 

if the measurements were made only when the base flow was at or 

below the median annual low flow, other considerations make it more 

feasible to use base-flow measurements over a wider range of base flow.

The main reason for obtaining base-flow measurements over a wide 

range of base flow is that the measurements needed to define the 

relation and to appraise its accuracy can be obtained in fewer years 

as there is more opportunity each year. An additional reason is that 

a wider range increases the coefficient of correlation and thus tends 

to remove the need for applying a "noise" adjustment to the regressed 

estimates (Matalas and Jacobs, 1964, equation 26). The magnitude 

of the bias introduced by neglecting such an adjustment can be appraised by comparing a T"-year discharge picked from the regression 

line with one picked from a line that crosses the regression line at the 

median annual low flow and that has a slope equal to IvG/IvU, which 

can be estimated from equation 8.

The relation curve defined by concurrent base-flow measurements 

or by concurrent daily base-flow discharges will in most cases be 

applicable also to 7-day annual low flows, as the difference in recession rates will be reflected both in the slope of the relation and in 

the ratios of 7-day low flow to 1-day low flow at the two sites. The 

applicability of the relation to longer low-flow periods such as those 

for 30-day and 60-day annual low flows depends on hydrologic factors 

other than base-flow recession and should be investigated by studies 

of gaging-station records in the area being studied before the regression 

line is used to estimate such characteristics at ungaged sites.

The curves shown in figure 14 can be used in planning a datacollection program by observing how NU is affected by changes in 

No, the length of record at the gaged site, and by M, the number of 

independent base-flow measurements. If, for the stations used in the 

preceding example, the objective had been to obtain 10-year accuracy 

at the ungaged site, r*Nu would be 9.64, and the corresponding F for 

an NG of 18 would be 20.8. Since M-3 is proportional to F, the number 

of base-flow measurements required to define the median annual 7-day 

low flow could be decreased enough to make M 3 equal to (16 3) 

times 20.8/30.0, or 9.0, from which the number of measurements would 

be 12. For the 10-year low, 13 times 20.8/73.9 gives 2.8, from which the
ACCURACY OF LOW-FLOW CHARACTERISTICS 47

required number of measurements would be 6. If the number of 

measurements were to remain at 16, however, F for the median annual 

7-day low flow would remain at 30.0, and NG (for r*Nu of 9.64) could 

be decreased to 15 years of record. For the 10-year low, F would remain at 70.3 and only 11 years of record would be required at the 

gaged site. These examples assume that the measurements added or 

subtracted would be randomly selected and that the distribution of 

the revised number would be approximately the same as before. A 

greater improvement could possibly be obtained by planning to have 

the additional measurements made where they would do the most 

good in defining the relation.

If the base-flow relations at many low-flow partial record sites are 

to be evaluated, perhaps the most efficient approach would be to 

use the accuracy of some of the relations as evaluated by the procedure described in this paper to prepare a summary table of M, 

NG, SER, and NU that could then be used as a guide to estimate 

whether the accuracy objective had been reached at the other sites.

If a procedure could be devised to give reduced weight to "outliers" 

in the base-flow relation that result from having an abnormally greater 

amount of direct runoff at one site than at the other, it would be 

feasible to develop a computer program that would not only establish 

the regression between the logarithms of the base-flow measurements, 

but would provide for entering the regression with various T-year 

low flows to obtain the estimated low-flow characteristic at the ungaged site, for making adjustment for the bias in the regressed line, 

and for computing the accuracy of the estimate in equivalent years of 

record. If such a program were to be developed, the accuracy in equivalent years of record at the ungaged site could be computed by relating 

Vu given by equation 2 to Na by use of equation 13. Even though 

a computer were to be used to evaluate the accuracy of an estimated 

low-flow characteristic at specific sites, the curves presented in figure 

14 would still be useful in program planning.

In planning a network of low-flow partial-record stations and 

associated stream gaging stations, the objective should be to obtain 

an optimum balance between the cost of additional base-flow measurements and the cost of increasing the length of record at the gaging 

station. To help in developing such a plan, z values for use in computing F could be estimated by relating z values from existing relations of base-flow measurements to the average number of base-flow 

measurements per year. Similarly, the standard error of estimate, 

SER, and the coefficient of correlation from existing relations could 

be related to distance between geologically similar drainage basins. 

Then these results could be used with the equations and graphs given 

in this paper to determine how the required number of base-flow
48 MANUAL OF HYDEOLOGY

measurements vary with distance from a gaging station and with 

length of record at the gaging station.

OUTLINE OF DERIVATIONS

By EDWARD J. GILROY 

BACKGROUND

Base flow measurements made at a site on a stream where no observations of annual low flows have been obtained are used to estimate a simple linear regression between the logarithms of these base 

flows and the logarithms of concurrent base flows at a nearby stream 

gaging station. The regression equation is then evaluated by using 

the logarithm of an estimated T-year low flow at the gaged site to 

obtain an estimate of the logarithm of the T-year low flow at the 

ungaged site. This section of the report presents the mathematical 

model underlying this procedure along with (1) a derivation of the 

variance of the estimator of the logarithm of the T-year low flow at 

the ungaged site, and (2) an unbiased estimator of the variance of 

the logarithms of annual low flows at the ungaged site by using the 

technique suggested by Matalas and Jacobs of adding noise to each 

regression estimate. The variance of the linear regression estimator 

is obtained by using bivariate normal regression theory with certain 

assumptions regarding the nature of the data.

STATEMENT OF THE PROBLEM

Let BU and BG denote the logarithms of base flows at the ungaged 

site and the gaged site, respectively. There are M base flow measurements available at both the gaged and ungaged sites. The corresponding logarithms of these 2M base flow measurements may be arranged 

as ordered pairs in the array

(14) 

B0 (l), BG (2), . . . , Be (M)

where for a fixed j, \<j<M, Bv (j} and BG (j] are logarithms of concurrent measurements at the two sites.

Let LU and LG denote the logarithms of annual low flows at the 

ungaged site and the gaged site, respectively. There are NG annual 

low-flow observations at the gaged site and no annual low-flow 

observations at the ungaged site. The corresponding logarithms of 

there NG annual low-flow observations may be written as

LG (1), LG (2), . . . , LG (NG). (15)
ACCURACY OF LOW-FLOW CHARACTERISTICS 49

The sample statistics pertinent to the derivations in this section are:

M

M

1 No

M

M

r = C/(SB , GSB , v}.

Applying the classical linear least squares technique to the data 

represented by the array (14), a linear estimate, Bv (j), of the logarithm 

of they-th base flow at the ungaged site is obtained. Bv (f) is written as

Bv(j] = Bv+b[BG (j) - BG]. (16)

The main interest in this section lies in using equation 16 to estimate statistical properties of Lv, the logarithm of annual low flows 

at the ungaged site, for which no direct measurements are available. 

The statistical properties considered are the mean, variance, and the

^f-th percentile, known as the T"-year annual low flow. Two methods

of estimation are used. Method I, which is used for estimating the 

T"-year annual low flow, consists of first using the observations of LG 

in array (15) to estimate the T-year annual low flow at the gaged site 

and then using this estimate in place of BG (j) in equation 16 to obtain 

an estimate of the T"-year annual low flow at the ungaged site, Method 

II consists in first using equation 16 modified by the addition of a 

noise term and the observations of LG in array (15) to estimate the 

corresponding annual low flows at the ungaged site. These estimates 

are then operated on as though they were actual observations to compute estimates of the mean and the variance of the annual low flows 

at the ungaged site.
50 MANUAL OF HYDROLOGY

The rationale for these two methods of estimation is that, based 

on past experience, it is reasonable to assume that the logarithms of 

annual low flows and the logarithms of the T-year annual low flows at 

the two sites follow the same linear relationship as do the logarithms 

of base flows at the two sites.

For the method I estimator, interest centers on the variance of 

the estimator. For the method II estimator, interest centers on 

obtaining an unbiased estimator of the variance of the annual low 

flows at the ungaged site.

STATISTICAL MODEL

The assumptions made in order to derive the statistical properties 

of the estimators given in methods I and II above are not the same 

for the two methods. The assumptions which are common to both 

cases are three in number and are the following:

Assumption 1. Bv and BG are jointly distributed random variables 

following a bivariate normal probability distribution with parameters 

given by their means HB , u and HB , G, their variances a2B , v and ^B , G 

and their correlation coefficient pB.

Assumption 2. For the M observations listed in array (14), it is 

true that nonconcurrent observations are statistically independent 

observations of the base flow population described in assumption 1.

Assumption 3. LG is a random variable and the NG observations 

given in array (15) are statistically independent of one another and 

also statistically independent of the observations in array (14).

For the derivation of the sampling variance of a method I estimator, two additional assumptions are made. Let XT , u and XT , G 

denote the logarithm of the T-year annual low flow at the ungaged 

and gaged sites, respectively. Let XT, G denote the estimate of XT , G 

obtained from the data array (15).

Assumption 4- XT , u and XTt G are linearly related as

XT, t/ = MB, V-\-&\XT, G ^B, a] 

where

<TB, G

Assumption 5. The estimate, %T, G, of XT , G may be written as

(18)

where d is a random variable with mean zero and variance, o-2 r, G . 

The random variable d is independent of the random variables Bv 

and BG .
ACCURACY OF LOW-FLOW CHARACTERISTICS 51

For the derivation of the sampling statistics of a method II estimator the following alternate assumptions are made in place of 

assumptions 4 and 5 above:

Assumption b'. LU and LG are jointly distributed random variables 

following a bivariate normal distribution with parameters given by 

their means, MZ,, u and MZ,, G, their variances, o-2 z,, u and diL< G, and 

their correlation coefficient p L.

Assumption 5'. The following relations are assumed to hold 

between the parameters of the distributions of Bv, BG, and LU, LG :

L, U(TL, G = (TB, V\<r*B. G (19) 

0L=0B=0 (20)

Since by definition

@B = PB&B, U\GB, G

and

@L PL<TL, U\<TL, G 

equations 19 and 20 imply

PL = PB=P. (21)

Obviously any two of the three equations 19, 20, and 21 could be 

assumed and the third would follow directly.

VARIANCE OF THE ESTIMATOR OF THE T-YEAR LOW FLOW (METHOD I)

The method I estimator XT , u of XT , u is defined by

XT, u = Bu+b(%T , o-So). (22) 

From the assumptions made above

E(XT, u) = HB, U-\~@[XT, G~^B, <?] 

By reason of equation 17, it follows that

EI(XT, u) = XT, u

and so %T , u is an unbiased estimater of XT> u. 

The variance of XT, u is given by

Var(Xr, v)=E(XT. u~XT , u) 2 

which, by reason of equations 17, 18, and 22, may be rewritten as

Var(^r, u)=E{[Bu-HB. u] + (b-fiXT . G +bd-bBG +^B , G } 2 . (23)

As a result of assumptions 1 and 2, the following well known 

property of bivariate normal random variables may be applied. Each 

pair, Bu(j), BG (j) of observations may be written as
52 MANUAL OF HYDEOLOGY

Bu(j) =MH, u+fi(B0 (j)-HB. G) +e(j) (24)

where the e(j) are independently normally distributed with mean zero

and variance (1 p2 )o-2B , v (Kenney and Keeping, 1951, Chapter

VIII).

From equation 24 it follows that

-»B , G}+e (25) 

and

(26)

where

1 M

1 M

C V"* / / *\

e G = / w( ? ) ~

It follows from equations 25 and 26 that equation 23 may be rewritten as

Var(J?r> u)=E[e + (b-p)(XTi G -BG)+bd]2 .

The technique used to determine the expectation indicated on the 

right side of equation 27 is to first take a conditional expectation for 

fixed values of BG and then find the unconditional expectation of the 

resulting expression. Thus.

E[[e+(b-0)(XT . G-BG)+d]*\BG }

BG) +2(XT . G - BG)E[e(b - /3) \B0] 

+E(eb\BG)E(d\BG )

, 0 -B0)E(b(b-p)\BG]E[d\BG]. (28)

Equation 28 holds since d is independent of Bv, BG and the BG 's 

are fixed with respect to this conditional expectation.

From equation 26 and the fact that the random variables e(i) and 

e(j) are independent for i^j, equation 28 reduces to

E[(e+(b-(3)XT , G +bd)*\BG)

(29)
ACCURACY OF LOW-FLOW CHARACTERISTICS 53

Now taking the unconditional expectation of the right side of equation 29 gives

M-3

+ B ' ° -\ '- >+/3V2 r Q. (30) 

M-B <T*B,G M-3<r*B , 0)

Equation 30 follows from equation 29 since BG and S2B , o are independent by reason of the normality assumption and since

. / ,.o.

(Af-3) CT2B , G

Thus equation 30 gives the variance of XT, u, the estimator of 

the T-year low flow.

ESTIMATOR OF VARIANCE OF ANNUAL LOW FLOWS (METHOD II)

Estimates, Lu(j), j = l, 2, . . . , No, of the missing Lv observations 

which correspond to the LG observations given in array (15) are 

given by

where f(j) is a normally distributed random component with mean 

zero and variance one, 6 is a parameter which equals 1 if "noise" is 

to be added and 0 if "noise" is not to be added, and a is a parameter 

to be determined so as to yield an unbiased estimate of az Lt v. The 

introduction of the noise term is due to Matalas.and Jacobs (1964). 

Because no observations at all have been made on the random variable 

Lv, the term SB ,u SLiG \SB ,G is used as a temporary estimate of <rLi u 

by reason of equation 19.

Let JJLL, u and cr2 ^ v denote method II estimators of the mean, 

/XL, u, and the variance az L , u of LU. Then

N°

Na

Let

1 No

Sf. L=rE(J<j) ~f}(LG (t) -La).
54 MANUAL OF HYDROLOGY 

Then ji L> u and <TZ L , u may be written as

ML, u = Bu+b(LG -BG)+a8(l-r*) B '" L ' Gf

OB, G

B, O

,.

OB, G

Using the Matalas-Jacobs argument against the use of different 

sets of "noise" terms by different investigators, the terms /, S2f, 

and Sf, L are replaced by their best estimates 0, 1, 0, respectively, 

leading to the estimates

r*) B^ L ' G . (31)

OB, G

The addition of the noise term is seen not to affect the estimator 

~HL, u- The estimator JJL L , u is unbiased since

E(fJ.L, u) =Ms, U+@(VL, G MB, G)-

But

ML, cr = MB, U+@(VL, G MB, e) 

by reason of assumptions 1, 4', and 5'. Therefore

E(/JLL, j) =ML, uBecause the NG annual low-flow measurements are independent of 

the base flows, the expectation of a2 L , u may be written as

EfrL , v) =E(V)E(S*L , c) +cWE(l ~£*S'*' uE(S*L. G). (32)

O^B, G

Using the technique described in the derivations of method I estimators it can be shown that

E(S*L , G)=<T2 L, G (33)

i-^v (34)

, G M 6

d-^M. (35)

0B, G 0" B , G 

Combining equations 32 through 35 E(<r2 L , u) may be written as

(36)
ACCURACY OF LOW-FLOW CHARACTERISTICS 55

In order that cr* Lt v be an unbiased estimator of <T Z L , u, 6 and a 

must take the values 6 = 1 and

when substituting a2 from equation 37 into equation 31 gives

~ o O9 I I-.9 I /"I A-9\ ** i ^ I /OO\ 0" 2L, J7=02 L, G "Z + o(l~r )"^ («*°)

as the unbiased estimator of o-2 i, j/. Since

B, G

b can be eliminated from equation 38 to give

**L,U=.

REFERENCES

Carter, R. W., and Benson, M. A., 1970, Concepts for the design of streamflow data

programs: U.S. Geol. Survey open-file report, 33 p. 

Hardison, C. H., 1969, Accuracy of streamflow characteristics, in Geological Survey

research 1969; U.S. Geol. Survey Prof. Paper 650-D, p. D210-D214. 

Kenney, J. F., and Keeping, E. S., 1951, Mathematics of Statistics, pt. 2, 2d ed.,

Princeton, N.J., D. Van Nostrand Co., 429 p. 

Matalas, N. C., and Jacobs, Bk , 1964, A correlation procedure for augmenting

hydrologic data: U.S. Geol. Survey Prof. Paper 434-E, p. E1-E7. 

Riggs, H. C., 1965, Estimating probability distributions of drought flows: Water

and Sewage Works, v. 112, no. 5, p. 153-157.

Thomas, D. M., and Benson, M. A., 1970, Generalization of streamflow characteristics from drainage basin characteristics: U.S. Geol. Survey Water-Supply

Paper 1975, 55 p., 16 figs.

Manual of Hydrology: Part 2. 

Low-Flow Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1542

This volume was published 

as separate chapters A and B
UNITED STATES DEPARTMENT OF THE INTERIOR 

ROGERS C. B. MORTON, Secretary

GEOLOGICAL SURVEY 

V. E. MeKdvey, Director
CONTENTS

[Letters designate the separately published chapters]

(A) Flow-duration curves, by James K. Searcy_________________________ 1

(B) Accuracy of low-flow characteristics estimated by correlation of base-flow 

measurements, by Clayton H. Hardison and Marshall E. Moss, with 

a section on Outline of derivations, by Edward J. Gilroy.___________ 35

U.S. GOVERNMENT PRINTING OFFICE: 1972