A Correlation Procedure for 

Augmenting Hydrologic Data

GEOLOGICAL SURVEY PROFESSIONAL PAPER 434-E

,>o» - i
A Correlation Procedure for 

Augmenting HydrologicData

By NICHOLAS C. MATALAS and BARBARA JACOBS 

STATISTICAL STUDIES IN HYDROLOGY

GEOLOGICAL SURVEY PROFESSIONAL PAPER 434-E

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

The U.S. Geological Survey Library has cataloged this publication as follows:

Matalas, Nicholas Constantinos, 1930-

A correlation procedure for augmenting hydrologic data, by 

Nicholas C. Matalas and Barbara Jacobs. Washington, U.S. 

Govt. Print. Off., 1964.

iii, 7 p. tables. 30 cm. (U.S. Geological Survey. Professional Paper 

434-E)

Statistical studies in hydrology.

Bibliography: p. 6.

1. Hydrology - Stat. I. Reiher, Barbara S. (Jacobs) 1938- joint 

author. II. Title. (Series)

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

Washington, D.C. 20402 - Price 15 cents (paper cover)
CONTENTS

Page

Abstract_______-____-____________-_--__-___---_______--__-_-_----__-_------_-__________ El

Introduction _______________________________________________________________________________ 1

Statistical model_____________________________-____________________--_-_-_---_---_-_------__- 2

Reliability of estimates_-_______-_________-____-_-_______--_-______-__-__--_-_-------__-_---- 4

Discussion ____________-_________________-____--_-_____---_..___-__-_-_---___------___--_---- 5

References __________________________________________________________________________________ 6

Appendix ________-______________________-________________-____-_---____--_-_-----_--___-__- 6

TABLES

Page 

TABLES 1-3. Critical minimum values of p for 

1. Mean_____________________________________________________________________________ E5

2. Variance: 6=1.. _________________________________________________ 5

3. Variance: 0=0__.______________________________________________. 5

in

STATISTICAL STUDIES IN HYDROLOGY

A CORRELATION PROCEDURE FOR AUGMENTING HYDROLOGIC DATA

By NICHOLAS C. MATALAS and BARBARA JACOBS

ABSTRACT

A linear regression for a short and long sequence of hydrologic 

events is used to lengthen the short sequence. The lengthened 

sequence consists of the original observations and regressed 

values plus noise, where the noise is a random variable with zero 

mean and variance proportional to the variance of the observations for the short sequence about the line of regression. Estimates of the mean and variance for the lengthened sequence 

are shown to be unbiased. If the correlation coefficient, which 

measures the strength of the linear regression, exceeds about 

0.5, then the estimates of the mean and variance based on the 

lengthened sequence are better estimators of the population 

values of the mean and variance than the estimates based on the 

observations for the short sequence. If noise is not added to 

the regressed values, the correlation coefficient must exceed about 

0.8 to obtain improvement in the estimates by use of correlated 

values.

INTRODUCTION

In statistical studies of hydrology, the assumption 

is made that a sequence of a finite number of observed 

events represents a random sample from an infinite 

population of such events, where the outcome of each 

event is governed by some probability distribution. 

Any change hi the hydrologic regime with which a 

given sequence is associated is reflected in a change of 

the probability distribution.

Although various parameters may be used to describe 

a probability distribution, the mean, variance, and 

skewness are three parameters which provide information about the most useful properties of any probability distribution. The mean is a measure of central 

tendency, a value about which the events tend to cluster, whereas the variance measures the dispersion or 

average spread of the events about the mean. The 

skewness is a measure of the asymmetry of the distribution of the events about the mean. These three 

parameters cannot define a probability distribution 

uniquely, except in special cases; nevertheless, they do 

provide characteristics which may be used to describe 

and compare various hydrologic phenomena. The population values of these parameters are generally unknown, but the values may be estimated from a 

sequence of observations. How reliable these estimates

are depends primarily upon the period or length of the 

sequence the total number of observations. If the 

estimates are unbiased, their reliability increases with 

an increase of the sequence length.

To increase a sequence length by making additional 

observations in time in order to obtain more reliable 

estimates of the mean, variance, and skewness is not 

always operationaUy or economically feasible. If not, 

recourse must be had to other procedures for increasing a sequence length. One such procedure, which is 

the topic of discussion in this paper, is to utilize the 

relations among hydrologic phenomena.

A relation among the concurrent observations for a 

short and a long sequence can be used to obtain estimates of the nonobserved events for the short sequence 

which correspond to the observed events for the nonconcurrent portion of the long sequence. In this manner the short sequence is lengthened. The observed 

and estimated events for the lengthened sequence can 

be used to obtain estimates of the mean, variance, and 

skewness. Whether or not the reliability of these estimates is greater than that for the estimates of these 

parameters based only on the observations depends 

mainly upon the strength of the relation between the 

concurrent observations for the short and long 

sequences.

A mathematical evaluation of this procedure has been 

made by several investigators under the assumptions 

that (1) the events are independently distributed in 

time, (2) the concurrent events for two sequences have 

a joint normal distribution, (3) the relation between 

the concurrent events is defined by a linear regression, 

and (4) no changes occur hi the hydrologic regimes 

with which the sequences are associated. The strength 

of the linear regression is measured by the productmoment correlation coefficient. Under the assumption 

of normality, only the mean and variance need to be 

considered because these two parameters uniquely define a normal probability distribution. For this 

distribution, the skewness is zero.

El
E2 STATISTICAL STUDIES IN HYDROLOGY

H. A. Thomas, Jr. (written commun. 1956), showed 

that the lengthened sequence yields an unbiased estimate of the mean, and if the product-moment correlation coefficient exceeds lf-yNi 2 where NI is the length 

of the concurrent period, the reliability of this estimate 

of the mean is greater than that based only on the 

observations for the short sequence. These results also 

were obtained by W. G. Cochran (1953) in an earlier 

and independent study of a double sampling problem. 

J. R. Rosenblatt (1959) showed that the lengthened 

sequence yields a biased estimate of the variance, and 

that the reliability of this estimate is greater than that 

based only on the observations for the short sequence 

if the product-moment correlation coefficient exceeds 

about 0.8. M. B. Fiering (1963) obtained somewhat 

similar results for the estimates of the mean and 

variance when more than one long sequence is related 

to a short sequence by a multiple linear regression.

In the studies cited above, the estimated events in 

the lengthened sequence were regression estimates, 

that is, estimates which correspond to values on the line 

of regression. These estimates tend to yield a smaller 

variance than would the real observations. In order 

to "preserve" the variance inherent with the observations, a random component must be added to the 

regression estimates. This component, often referred to 

as noise, is normally distributed with zero mean and 

variance proportional to the variance of the observations 

for the short sequence about the line of regression.

This paper reports the results of an investigation 

made to determine the reliability of estimates of the 

mean and variance computed from a lengthened sequence when noise is added to the regression values. 

Noise is shown to have no effect on the reliability of the 

estimate of the mean. However, the addition of noise 

is shown to lead to an unbiased estimate of the variance 

for the lengthened sequence. The reliability of this 

estimate is greater than that when no noise is added to 

the regression values.

STATISTICAL MODEL

The long and short sequences for a pair of hydrologic 

phenomena are denoted by x and y. In general, the 

two phenomena need not be the same. If, for example, 

y denotes streamflow, x may denote streamflow, precipitation, temperature, or a geochronologic phenomena 

such as tree-ring widths. The observed events for the 

long and short sequences are represented as

where NI is the length of the short sequence and 

(Ni-+-Na) is the length of the long sequence. For this 

representation of the two sequences, NI also denotes 

the concurrent period of observation. In practice, the 

NI observations for the short sequence need not correspond to the first NI observations of the long sequence, 

nor need the NI concurrent observations of x and y 

occur consecutively. However, there is no loss of generality if the two sequences are represented as above.

The concurrent observations x and y are assumed to 

have a joint normal probability distribution with 

parameters nx, ny, a2., a2,, and p=ft<rx/<rv, where \LX and 

a-2, denote the population values of the mean and variance, respectively, for x, and nv and a2,, the population 

mean and variance, respectively, for y. The parameter 

p is the product-moment correlation coefficient, and 

j8 is the population value of the slope for the linear 

regression of y on x.

For the short sequence, the estimates of the mean 

and variance are given by

(1)

1

respectively. Similarly for the long sequence 

1

x 

o^ . 

or.

(2)

(3)

Ni+N<t

,.+N,-l) §

i i)8

2 j=

where

7 JL V^ f (^ "1 AT Z l ^i) \u/

(6)

-^i) 2 , (7) 

KJ x2)2 . (8) 

In equations 1 through 8, the subscripts 1 and 2 indicate
A CORRELATION PROCEDURE FOR AUGMENTING HYDROLOGIC DATA E3

that the estimates are based on the observational 

periods Ni and N2, respectively. No numerical subscript is used with those estimates based on the period

Because each of the above estimates converges to its 

respective population value as the observational periods 

tend to infinity, the estimates are unbiased. Therefore, the expectations of the estimates equal their 

respective population values (Wilks, 1962). If the 

symbol E is used to denote the expectation, then

(10)

(11)

(12)

A measure of the reliability of an estimate is given by 

its variance. From the assumption of normality, the 

variance, denoted by Var, for each of the estimates 

(Wilks, 1962) is

Var (yO=< 

Var (Sy= 

Var &)= 

Var (x2)=< 

Var (zHcr 

Var («y= 

Var (S|2)= 

Var s|=

(13)

(14)

(15)

(16)

(17)

(18)

(19)

-1). (20)

The variance of each estimate varies inversely with the 

period of observation. Therefore, as the period of 

observation increases, the variance of an estimate 

decreases, and consequently, the reliability of the estmate increases.

The true regression of y on x is defined as

where Yt denotes the regression value for y for a given 

value Xi. The yt are normally and independently distributed about the regression values with variance 

(1 p2) <4- Therefore, the yt may be expressed as

where 6 is a random normal variable with zero mean

and unit variance, and the term Vl P2^ < represents 

noise.

From the method of least squares, the estimate of the 

true regression of y on x is given by

(23)

where F* is the estimate of Yt and 6, the estimate of j8, 

is given by

Ni Ni

Vi Xi~) 2 . (24)

If the right-hand side of equation 22 is substituted for 

yt in equation 24, 6 may be expressed as

(xt-xj*. (25)

If the noise term is multiplied by ad and the population parameters are replaced by their sample estimates, 

equation 22 may be expressed as

(26)

where e is a random normal variable with zero mean and 

unit variance, a is a constant which will be defined 

later, and r, the estimate of p, is given by

r=bsxjsvi . (27)

The parameter 8 is introduced to facilitate the comparison of the case when noise is added with the case when 

noise is not added. If noise is added, 0=1, and if 

noise is not added, (9=0. In equation 26, if i is replaced 

by j, then N2 values of x outside the concurrent period 

may be used to obtain estimates of yNl+i, . . ., yNl+N2 . 

These estimates, denoted by yNl+i, . . ., y^+N^ maJ be 

pooled with the observations y1} . . ., yN to form the 

lengthened sequence

For the lengthened sequence, the estimate of the 

mean is

Nl+N2

where yi is defined by equation 1 and
E4 STATISTICAL STUDIES IN HYDROLOGY

Also, for the lengthened sequence, the estimate of the 

variance is given by

(30)

where xi} x2 , s2^, s|2 , 6, r, and ?/ are defined by equations 

5, 6, 2, 8, 24, 27, and 28, respectively, and

e2=TT

Sxe=

(31)

(32)

2 (N2 -l) -

The use of equations 28 and 30 necessitates that 

noise actually be added to the regression estimates. 

Truly random numbers do not exist, and therefore 

recourse must be had either to generating pseudorandom 

numbers or utilizing such numbers that have been 

tabulated and are to be found in the statistical literature. Regardless of the questions which can be raised 

as to the goodness of the pseudorandom numbers, thenuse is not too appealing. Independent studies of the 

same sequences of x and y by several investigators lead 

to different values of y and s2 , because the same 

sequence of pseudorandom numbers is unlikely to be 

used by the investigators. Some of these values are 

apt to appear absurd, even though the values are within 

the limits expected by chance.

To overcome these objections to the use of equations 

28 and 30, the following approximations are offered. 

On the basis of the observed sequences xi} . . ., xNl, 

and ?/i, . . ., y^, the best estimates of e2, s2e2 , and sxe2 

are 0, 1, and 0, respectively. Thus, if e2) s2e2, and sxe2 

are replaced by their respective estimates, equations 

28 and 30 reduce to

N2

b (x2 (34)

(35)

respectively. Both equations 34 and 35 are expressed 

in terms that can be determined directly from the 

observations.

Equation 34, which gives an approximate estimate of 

the population mean from the lengthened sequence, 

does not contain any terms that reflect the "addition" 

of noise. Actually equation 34 gives the same estimate 

of the mean for the lengthened sequence as when noise 

is not added to the regression values. This is seen to be 

the case if B is set equal to zero, in which case equation 

28 reduces to equation 34.

In equation 35, which gives an approximate estimate 

of the variance for the lengthened sequence, the 

"addition" of noise is reflected by the third term in 

brackets.

RELIABILITY OF ESTIMATES

To judge the reliability of an estimate, two properties, 

namely the expected value and the variance of the estimate, must be considered. The estimate of the mean 

for the lengthened sequence, given by equation 34, is 

the same as that considered by Thomas and Cochran. 

Therefore, the discussion of the reliability of this estimate is given without the derivation of the expected 

value and variance of the estimate. Because the derivation of the expected value and variance of s% is rather 

long, the derivation is omitted from this section and 

outlined in the appendix on page E6.

The estimate y is an unbiased estimator of /* , so that

(36)

(37)

The variance of y is given by

where the term a$INi denotes the variance of 

For y to be a better estimate of /* than yi} Var 

Var (yi) = a-y/Ni. This is the case if the term in the 

brackets in equation 37 is negative, that is, if

\P\>l/^JW^2. (38)

Equation 38 shows that to improve the estimate of the 

mean by the use of a linear regression relation, the 

strength of the regression, measured by p, must exceed 

a critical minimum value which depends only upon 

NI. If inequality (38) is not satisfied, then Var (y)^> 

Var (yi), in which case f/i is a better estimate of /* 

than y In this case, the linear regression relation
A CORRELATION PROCEDURE FOR AUGMENTING HYDROLOGIC DATA E5

serves no practical purpose. Equation 38 shows that 

the critical minimum value of p cannot be determined 

for A\<2. Table 1 gives the minimum critical value 

of p for various values of N\ > 10.

TABLE 1. Critical minimum values of p for the mean

NI: 10 15 20 25 30 

P : 0. 35 0. 28 0. 24 0. 21 0. 19

The expected value of si, the derivation of which is 

outlined in the appendix, is

r#»(#i-4)

(39)

If

and 0=1, then £"(«*) =o^, so that si is an unbiased 

estimator of o^. If 0=0, equation 39 reduces to the 

biased estimator of cr2u derived by Rosenblatt. 

The variance of si is

, (41)

where

-i) (#i+#2-i)s

-6)(#1 -8) 8(#t-4)

(#!-3) P4 , (42)

-5) ' (#i-3) 

2(#,+#2-D(#i-4) Q,

(43)

2#2 (#1 -4) 

" (#i-3)2

In equation 41, the term 2cr*/(Ni 1) denotes the

variance of s^. If the second term on the right-hand 

side of equation 41 is negative, then Var (s^)<Var 

(sl1 )=2<T*/(Ni 1), so that si is a better estimator of 

crl than s»j. In this case, the correlation coefficient, 

<7, must exceed a critical minimum value which is given 

by

, ,^ f 

l pl>L 2A (45)

where A, B, and C are defined by equations 42, 43, 

and 44, respectively. If inequality (45) is not satisfied, 

then s»j is a better estimator of <rl than si. If 0=0, 

equation 41 and the expressions for A, B, and C defined 

by equations 42, 43, and 44, respectively, reduce, after 

a rearrangement of terms, to the expression for the 

mean square error of si derived by Rosenblatt.

Tables 2 and 3 give the critical minimum values of 

p for various values of NI and N2 for 0=1 and 0=0, 

respectively.

TABLE 2. Critical minimum values of p for the variance: 6=1

10

15

20

25

30

10

65

65

65

65

65

15

54

54

54

54

54

20

52

51

51

50

50

25

42

42

42

42

42

30

0.38 

. 39 

.39 

.39 

.39

TABLE 3. Critical minimum values of p for the variance: 6=0

\

10 

15 

20

25

30

10

0 73

75 

76

78

77

15

0 63

77 

79

80

80

20

0 70

79 

81

84

82

25

0 76

80 

81

83

83

30

0 76 

80

82

81

84

An inspection of these tables show that for given values 

of NI and N%, p is less when noise is added, 6= 1, than 

when noise is not added, 0=0. Because the critical 

minimum values of p, equation 45, cannot be defined 

for values of #i<6, tables 2 and 3 are based on NI > 1.0

DISCUSSION

In practice, the correlation coefficient, p, must be 

estimated by r. For given values of NI and N2 , if r 

exceeds the appropriate critical minimum values tabu-
E6 STATISTICAL STUDIES IN HYDROLOGY

lated in tables 1-3, then the mean and variance for the 

lengthened sequence may be determined from the 

approximate expressions given by equations 34 and 

35, respectively. The use of equations 34 and 35 

when r does not exceed the appropriately tabulated 

critical minimum values results in estimates which are 

less reliable than estimates based only upon the observations. However, caution must be exercised in the use 

of these tables, because r may not be a good estimate of 

p and because the hydrologic variables, x and y, may 

not satisfy the assumptions upon which the above 

developments are based.

The reliability of r as an estimate of p depends upon 

the length of the concurrent period, N1} for the short 

and long sequences. At this point, all that can be 

said is that NI should be as large as possible. If p 

were known, then table 1 could be used if A/"x >3 and 

tables 2 and 3 could be used if Ni^>5. Because p is 

not known and must be estimated by r, NI should be 

much larger than 5.

Most hydrologic variables exhibit skewness and therefore they cannot be considered as normally distributed. 

Usually, simple transformations, such as logarithms 

and square roots, may be used to normalize the data. 

If normalizing transforms are used, the developments 

noted above pertain not to the variables but to the 

transformed variables. Transformations may also be 

needed to linearize the relation between x and y.

Hydrologic data exhibit a tendency for high values to 

follow high values and for low values to follow low 

values, so that the values are not independently 

distributed in time. This dependency, which is attributed to storage processes in the hydrologic regime 

with which a given sequence is associated, decreases 

rather rapidly with an increase in the time interval 

between observations. For example, a sequence of 

monthly events exhibits greater dependence among 

adjacent events than an annual sequence. Simple 

transformations of a variable are unlikely to reduce the 

dependence among variate values. Therefore, the use 

of the developments noted above should be restricted to 

annual sequences whose hydrologic regimes are such 

that storage or carryover factors are a minimum.

REFERENCES

Cochran, W. G., 1953, Sampling techniques: New York, John

Wiley & Sons, Inc. 

Fiering, M. B., 1963, Use of correlation to improve estimates of

the mean and variance: U.S. Geol. Survey Prof. Paper

434-C.

Rosenblatt, J. R., 1959, On the synthetic record problem:

Bur. Standards Tech. Rept. 1. 

Wilks, S. S., 1962, Mathematical statistics: New York,

Wiley & Sons, Inc.

Natl. 

John

APPENDIX

Expectation of s*y :

fi-4)

where

lL(X2 Xi) ,-

(A-2) 

(A-3) 

(A-4) 

(A-5)

(A-6)

2 _

If 0=1 and

then equation A-l reduces to

Variance of si' 

Var(s£)=J

(A-8)

(A-9)
A CORRELATION PROCEDURE FOR AUGMENTING HYDROLOGIC DATA E7

+2(N1 -lXN2 -l)a282E[(l-r2)(s* 1) 2 }

2(N2 -l)NlN2

(A-10)

3(1 ~p2)2

> fA-13) (13

r A 1 (

(A-17)

From equations A-l and A-7 and equations A-10 

through A-19, equation A-9 may be expressed as

Var J>H _

(A-20)

where

+

^JV, W-4)« I 

V1 -3) 2W-2) J P '

1Z^ (1-02)

(A

U.S. GOVERNMENT PRINTING OFFICE: 1964