Autocorrelation of Rainfall 

and Streamflow Minimums

GEOLOGICAL SURVEY PROFESSIONAL PAPER 434-B
Autocorrelation of Rainfall 

and Streamflow Minimums

By NICHOLAS C. MATALAS

STATISTICAL STUDIES IN HYDROLOGY

GEOLOGICAL SURVEY PROFESSIONAL PAPER 434-B

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1963
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

First printing 1963 

Second printing 1967

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

Washington, D.C. 20402 - Price 20 cents (paper cover)
CONTENTS

Abstract __ ______________________________

Introduction ___-__.____-____-_______.-___.

Characteristics of time series________________

Effect of nonrandomness in hydrologic studiesMethods for investigating nonrandomness--___ 

Trend analysis_______________________

Measure of nonrandomness _____________

Generating processes___________________

Page

Bl

1

1

2

2

2

3

3

Page 

Methods for investigating nonrandomness Continued

Correlogram analysis____-_______-_____--_--___ B4

Investigation of nonrandomness in hydrologic time series. 5 

Serial correlation_______________________________ 5

Generating process____________________________ 6

Effect of serial correlation on the estimate of the

variance_____________________________________ 6

Discussion and conclusions___________-_-_-_---_-.__ 7

References.__-__-_--_-__-_--_______--__--_.-_-_-___ 7

ILLUSTRATIONS

FIGTJBES 1-6. Correlograms for annual minimum rainfall.

1. Boston, Mass_____-_-.__________-_

2. Providence, R.I___~-____________~_

3. Charleston, S.C___________________

4. Washington, D.C_-.-_____________

5. Baltimore, Md__________________

6. Philadelphia, Pa______________

7, 8. Correlograms for annual minimum 7-day 

discharge.

Page

B8 

8 

8 

8 

8 

8

FIGURE 7. South Fork Holston River at Bluff 

City, Tenn_____________________

8. James River at Buchanan, Va______

9. Correlogram for annual minimum 1-day 

discharge for Middle Branch Westfield 

River near Goss Heights, Mass....__

10. Expected value of the variance for a firstorder Markov process__-_-_-_-------_

Page

B9 

9

TABLES

Page

TABLE 1. Rainfall stations used in analyzing annual minimum monthly rainfall__________________ B9

2. Gaging stations used in analyzing annual minimum discharge_.-__---_-____-__-_____. 9

3,4. Significance of first-order serial-correlation coefficients. 

3. Rainfall data--_---_________________ 10

Pag*

TABLE 4. Discharge data___-___-_-----__----__- BIO 

5,6. Serial-correlation coefficients for orders 1 

through 19.

5. RainfaU data__________________ 10

6. Discharge data.________-----__----_-_ 10

7. Chi-square goodness of fit of first-order Markov

process to discharge Correlograms _______ 10

ra
STATISTICAL STUDIES IN HYDROLOGY

AUTOCORRELATION OF RAINFALL AND 

STREAMFLOW MINIMUMS

By NICHOLAS C. MATALAS

ABSTRACT

Hydrologic time series of annual minimum mean monthly 

rainfall and annual minimum 1-day and 7-day discharge, considered as drought indices, were used to study the distribution 

of droughts with respect to time. The rainfall data were found 

to be nearly random. The discharge data, however, were 

found to be nonrandomly distributed in time and generated by 

a first-order Markov process. The expected value of the variance 

for a time series generated by a first-order Markov process was 

compared with the expected value of the variance for a random 

time series. This comparison showed that the expected value 

of the variance for a nonrandom time series converged to the 

population variance with an increase in sample size at a slower 

rate than for a random time series.

INTRODUCTION

The object of the investigation was to determine if 

droughts are randomly or nonrandomly distributed 

in time and to modify tests of reliability to account 

for the effect of nonrandomness. For these purposes, 

the hydrologic time series used in the investigations 

were annual minimum monthly rainfall at 18 rainfall 

stations and annual minimum 1-day and 7-day discharge at 9 gaging stations.

A time series is defined as a sequence formed by the 

values of a variable at increasing points in time. A 

time series may be composed of the sum of two components: a random element and a nonrandom element. 

If the values of the time series are not independent of 

each other, the nonrandom element exists, and the 

values are said to be serially dependent.

The nonrandom element in a time series may be due 

to a trend or an oscillation about the trend or both. 

In a given time series, the nonrandom element need 

not be due to both of these causes. In order to analyze 

properly a time series, the random and nonrandom elements must be isolated and studied separately. Trend 

must be eliminated from the nonrandom element 

before studying the oscillatory character of a time series. 

The oscillatory movement in a trend-free time series 

may be due to 1 of 3 schemes: (a) moving averages, 

(b) sums of cyclic components, and (c) autoregression.

CHARACTERISTICS OF TIME SERIES

Time series may be classified as either stationary or 

nonstationary. Assume that a time series is divided 

into several segments and that the data within each 

segment are characterized by statistical parameters 

such as the mean and variance. If the expected values 

of these parameters are the same for each segment, 

the time series is said to be stationary. If the expected 

values are not the same for all segments, the time 

series is nonstationary. In stationary time series, 

absolute time is not important, and the series maybe 

assumed to have started somewhere in the infinite past. 

However, in nonstationary time series, absolute time 

must be considered since the series cannot be assumed 

to have begun prior to the time of the initial observation.

Time series may be considered as composed of the 

sum of a random element and a nonrandom element. 

The nonrandom element consists of a trend and an 

oscillation about the trend. Trend is usually thought 

of as a smooth motion of the time series over a long 

period of time. For any given time series, the sequence 

of values will follow an oscillatory pattern. If this 

pattern indicates a more or less steady rise or fall, the 

pattern is defined as a trend. No matter what the 

length of a time series, one can never state with certainty that an apparent trend is not part of a slow 

oscillation, unless the series ends.

An oscillatory movement is often confused with a 

cyclical movement. In a cyclical time series, the 

maximum and minimum values occur at equal intervals 

of time with constant amplitude. The random component, if present, tends to distort this pattern. In an 

oscillatory time series, the amplitude and the interval 

of time between maximum and minimum values are 

distributed about mean values. A cyclical time series 

is oscillatory, but an oscillatory time series is not 

necessarily cyclical.

In order to analyze properly a time series, it is necessary to separate the random and the nonrandom 

components. Trend must be eliminated from the

Bl
B2 STATISTICAL STUDIES IN HYDROLOGY

nonrandom component before studying the oscillatory 

character of the time series.

EFFECT OF NONRANDOMNESS IN 

HYDROLOGIC STUDIES

The design of dams and water-storage reservoirs is 

based on the expected hydrologic conditions many 

years in the future, depending upon the life expectancy 

of the hydraulic structures. To obtain long-range 

estimates, the frequency with which hydrologic events 

occur must be known, if it is assumed that the future 

will follow the pattern shown by the past. In a 

nonrandom time series, there is a tendency for low 

values to follow low values and for high values to 

follow high values. In the case of a water-supply 

reservoir, the reservoir is filled during a wet season to 

meet the anticipated demand during a dry season. If 

droughts are nonrandomly distributed in time, then to 

overcome the tendency for low values to follow low 

values, the reservoir capacity must be increased to 

provide sufficient water to meet the need for more than 

one dry season.

Hurst (1950), in his study of long-term storage 

capacities of reservoirs, concluded that annual total 

discharge is nonrandomly distributed in time. This 

study was based on an analysis of the cumulative sum 

of departures of annual total discharges from the mean 

annual total discharge. The absolute value of the 

difference between the maximum and the minimum of 

the cumulative sums was defined as the range. To 

test for nonrandomness, Hurst determined the expected 

values of the range for various natural phenomena, and 

compared these values with the expected value of the 

range for a random normally distributed time series.

For a random normally distributed time series, Hurst 

defined the expected value of the range, R, as

where a is the standard deviation and N is the sample 

size. Feller (1951) showed that the variance of the 

range is

V(R) =0.074<rW. (2)

Feller also pointed out that the asymptotic probability 

distribution of the range is independent of the form of 

the probability distribution of the variate values. 

Equations 1 and 2 are valid for large N. Anis and 

Lloyd (1953) defined the expected value of the range 

for any value of N as

(3)

Hurst investigated 75 different phenomena consisting

of 690 sets of records and found the range to be of the 

form

(4)

The value of k varied slightly with different phenomena. For the 690 sets of records, the mean and 

standard deviation of k were 0.729 and 0.091, respectively.

By this analysis, Hurst concluded that estimates of 

the standard deviation based on N observations are 

more variable for a nonrandom time series than for a 

random time series. In a continuation of this study, 

Hurst (1956) pointed out that statistical averages 

based on 30 or 40 observations from a nonrandom time 

series do not yield close approximations to the true 

averages.

The difference between the ranges defined by equations 1 and 4 may be due to the data not being randomly 

distributed in time. This, however, requires that the 

nonrandomness be of a very special type. Moran (1959) 

suggests that the experimental series used by Hurst are, 

as a result of nonrandomness, of insufficient length for 

the asymptotic formula (eq. 1) to become valid. Hurst 

(1957) constructed a probability model which tends to 

illustrate this point.

METHODS FOR INVESTIGATING 

NONRANDOMNESS

TREND ANALYSIS

Trend is thought of as a smooth motion of the time 

series over a long period of time. If the oscillatory 

pattern of a time series indicates a more or less steady 

rise or fall, the pattern is defined as a trend. To study 

the oscillatory characteristics of a time series, the trend, 

if present, should be removed. Various methods of 

removing trend are available. All the methods, however, 

are not fully understood as to how they affect the time 

series. The most commonly used method is that of 

moving averages.

Assume that the observations &i, $2, . . ., XN are 

taken at equal intervals of time. The method of moving 

averages consists of determining overlapping means of 

m successive weighted values. An example of moving 

averages for m=3 is

(5)

The weights of the moving average, blf b2 , and b8, are 

such that their sum equals 3. In general, for moving
AUTOCORRELATION OF RAINFALL AND STREAMFLOW MINIMUMS

i N-k i /N-k \ /N-k

JL x! ~\ * 

'

averages of m, the sum of the weights equals m. The 

weights may be positive or positive and negative. A 

simple moving average refers to the case where each of 

the weights equals one. Although a simple moving 

average tends to smooth out the data, it does not 

preserve the main features of the time series as well as a 

weighted moving average.

Generally, even a smooth trend obtained by the 

method of moving averages cannot be represented conveniently by a mathematical equation. The simplest 

mathematical expression is a straight line. However, a 

time series is apt to be such that a single linear trend 

can not be used throughout the time of observation. 

In such cases, it is possible to approximate the trend by 

using linear trends for portions of the time series.

After the trend line has been established, the trend 

can be removed from the data in one of several ways. 

One way is to take as a new variable the deviations 

about the trend line. These deviations must comprise a 

stationary time series. In some cases, the deviations 

may not comprise a stationary time series, but the 

deviations divided by their corresponding trend values 

may comprise a stationary time series.

If the method of moving averages is used to determine 

the trend in a time series which has an oscillatory movement about a trend, then a long-period oscillation tends 

to be included as part of the trend. Oscillations which 

are comparable in period to the length of the moving 

average, m, or shorter are damped out. The moving 

average also introduces an oscillatory movement into 

the random component of the time series. These consequences of the moving-average method are referred 

to as the Slutzky-Yule effect (Slutzky, 1937; Yule, 

1921). Because of the Slutzky-Yule effect, care must 

be taken hi discussing the oscillatory character of a time 

series if its trend has been removed by the movingaverage method.

MEASURE OF NONRANDOMNESS

A time series is said to be randomly distributed if 

each event is independent of all preceding and following 

events. In analyzing sunspot data for periodicities, 

Yule (1927) found that the scatter diagrams for events 

k time units apart were linear. As a measure of linear 

dependence, Yule proposed equations 6 and 7.

In equation 6, x t and xi+Jc are the events at times i 

and i-\-k, respectively, N is the number of events formB3

(6)

where 0=1 V

'*=r-t f

(7)

ing the time series, and rk is the Arth-order serialcorrelation coefficient.

Under the assumption that x t varies linearly with 

%i+K, ft serves as a measure of linear dependence. 

If a time series is random, rK=Q for all values of k>l. 

However, for a finite sample, computed values of rK 

may differ from zero because of sampling errors. For 

hydrologic time series, N is small, so that the sampling 

errors may be quite large. Thus the values of rK 

must be tested to determine if they are significantly 

different from zero.

Anderson (1942) developed a test of significance 

based on a circular definition of the serial-correlation 

coefficients which supposes that the last event is followed by the first event so that

1 N

rk=-

1 N

(8)

For N large and k small, the values of rk given by 

equation 6 are nearly equal to those given by equation 8.

For a random normal time series, Anderson showed 

that TI is approximately normally distributed with 

mean [-!/(# !)] and variance (A/-2)/(JV-l)2. 

Thus a computed value of TI may be tested for significance by

ri=- N-l (9)

where ta is the standard normal variate corresponding 

to a probability level a. At a given probability level, 

if r^>r\, then rt is considered to be significantly different 

from zero.

GENERATING PROCESSES

A generating process refers to the manner by which 

the causal forces act to produce a time series. Some 

processes can be expressed mathematically, in which 

case it is possible to determine directly the various 

characteristics of the time series. Often a time series 

is approximated by a certain process. The choice of 

this process is based upon how well the mathematical 

structure of the process conforms to the physical
B4 STATISTICAL STUDIES IN HYDROLOGY

characteristics underlying the time series. The processes which have been studied extensively and which 

have found wide application in practice are the moving 

average, the sum of harmonics, and the autoregression. 

The moving average process may be expressed as

Xi=b0+biy i+b2yi-l + ... +6m7/<_ (m_ 1) , (10)

where y is a random variable and m is the extent of the 

moving average. Equation 10 may be taken as a 

model representing the relation between annual runoff, 

x, and annual effective precipitation (Folse, 1929). 

The effective precipitation prior to a given time interval 

is referred to as antecedent effective precipitation. 

Since the contribution of effective precipitation to the 

runoff converges to zero very rapidly with an increase 

in antecedent time, the effective precipitation for only a 

finite period of antecedency affects the runoff. This 

finite period of antecedency, which is defined as the 

carryover period, is a function of the water-retardation 

characteristics of the river basin and the distribution 

of the effective precipitation with respect to time. 

Wold (1933) shows that the serial-correlation coefficients for a series generated by a moving average 

process are given by

(ID

1=0

rk Q;k>m.

Many hydrologic studies involve the use of time 

series which exhibit a cyclic movement; that is, sequences of daily or monthly discharges. The generating process for a cyclic time series may be represented by

Xi=A sin Oi+y t, (12)

where A and B are the amplitude and period, respectively, of the cyclic movement, and y is a random 

component. The random component, y, tends to 

distort the amplitude and the period. If the random 

component is large, A and B may be badly distorted. 

For a cyclic time series, the serial-correlation coefficients are given by

A2 

r*=2^2 cos 0*» (13)

where al is the variance of the x's.

Equation 12 is a special case where only one harmonic, namely B, is involved. Some hydrologists 

argue that there are hidden harmonics, or periodicities, 

in hydrologic data. These periodicities are called 

hidden because, if there are a large number of different periods, the series is very erratic seemingly random.

In the discussion of the moving-average process as

a physical model in hydrology, only a finite period of 

antecedency was considered. If the antecedent period 

is considered to be infinite, then a value of x at time 

i is a function of all previous values of x. This condition may be expressed as an autoregression process by 

defining x t in terms of previous values of x and a 

random component. An example of such a process is

xi+2=axi+i -f b$i (14)

where a and b are constants and e is the random 

component.

Kendall (1951) shows that by letting p V and 

cos 0 a/(2V&), the serial-correlation coefficients for 

the autoregression process given by equation 13 are 

defined as

where

'* ' sin^ ' 

tan \f/= g tan B.

.(15) 

(16)

It is assumed that -\fb is taken with a positive sign 

and that 46>o2 . Moreover, it is assumed that ifb is 

not greater than unity.

A special type of autoregression is given by

where r\ is the first-order serial-correlation coefficient. 

This process is often referred to as the first-order Markov 

process. The serial-correlation coefficients are given by

rk=r\. (18)

CORRELOGRAM ANALYSIS

A graph with the serial-correlation coefficients plotted 

as ordinates and their respective orders as abscissas is 

called a correlogram. To reveal the features of the 

correlogram, the plotted points are joined each to the 

next by a straight line.

For a moving-average process, the values of rt are 

given by equation 11. Since rk 0 for k>m, the correlogram may oscillate, depending upon the values of 

b, but the correlogram will vanish for k>m. The values of rk for the harmonic process are given by equation 13. For this process, the correlogram will oscillate 

but never vanish. The oscillation of the correlogram 

is strictly cyclic with amplitude A^feal and period B. 

Note that the period of the correlogram is identical 

to the period of the time series. Equation 15 indicates that the correlogram of the autoregression process given by equation 14 will oscillate, be damped, but 

will not vanish. If the autoregression process is given 

by equation 17, then it is seen by equation 18 that 

the form of the correlogram will be "exponential." 

Thus for infinitely long sequences, the correlogram pro-
AUTOCORRELATION OF RAINFALL AND STREAMFLOW MINIMUMS B5

vides a theoretical basis for distinguishing among the 

three types of oscillatory time series.

In practice, N is small so that observed correlograms 

show less damping than theoretical correlograms because the observed serial-correlation coefficients are 

inflated by sampling errors. Thus one cannot determine the generating process by simply observing the 

correlogram. The goodness of fit of a given generating process fitted to an observed correlogram must be 

determined. Tests of goodness of fit which have been 

developed are based on N being very large.

In this study only the autoregression process is considered. For this generating process, a chi-square test 

developed by Quenouille (1949)' can be used to test 

the goodness of fit. If the autoregression model is 

defined by equation 17, the value of chi-square, xL 

associated with the &th-order serial-correlation coefficient is given by

2 _ ~

where

(19)

(20)

The sum of the values of chi-square forms the basis 

for testing the goodness of fit. If

k=l (21)

where xM«) is the value of chi-square at the probabilitylevel a with m degrees of freedom (m is the highest 

order for which a serial-correlation coefficient is determined), then the first-order Markov process is rejected.

INVESTIGATION OF NONRANDOMNESS IN HYDROLOGIC 

TIME SERIES

SERIAL CORRELATION

If a time series is randomly distributed, the population values of the serial-correlation coefficients, called 

autocorrelation coefficients, between events for all orders 

are zero. For such a time series, the serial-correlation 

coefficients are not significantly different from zero at 

a probability-level a.

To investigate the distribution of droughts with 

respect to time, the annual minimum mean monthly 

rainfall and the annual minimum 1-day and 7-day discharge were considered as indices of droughts. The 

rainfall and discharge stations are listed in tables 1 

and 2, respectively. A preliminary investigation for 

trend was made. Only the rainfall data for Charleston, 

S.C., strongly indicated a trend. However, the apparent trend was not removed.

By using equation 6, the first-order serial-correlation 

coefficients for the rainfall and discharge data were determined and are given in tables 3 and 4, respectively. 

The first-order serial-correlation coefficient given in 

tables 3 and 4 range from 0.296 to 0.300 for rainfall 

data and from 0.085 to 0.385 for discharge data. 

Although the discharge data exhibit higher first-order 

serial-correlation coefficients than the rainfall data, 

the variability of the first-order serial-correlation coefficients is greater for the rainfall data than for the 

discharge data.

These values of r^ were tested for significance at the 

90-percent and 95-percent levels by equation 9. An inspection of tables 3 and 4 shows that at the 90-percent 

level of significance, the data for 6 rainfall and 4 gaging stations exhibit significant values of r\. At the 

95-percent level, the data for 4 rainfall and 3 gaging 

stations yield significant values of TV Thus at the 

usual levels of significance employed in hydrologic 

studies, approximately K to K of the stations for both 

phenomena possessed data yielding significant firstorder serial-correlation coefficients.

The data which exhibited nonsignificant values of rx 

are not necessarily random in time. Serial-correlation 

coefficients of order higher than one, if significant, 

would indicate a lack of randomness. To determine 

if the data were random in time, the serial-correlation 

coefficients for orders 1 through 19 were considered for 

series having both significant and nonsignificant values 

of TI. The data for 6 rainfall and 3 gaging stations 

were used. The values of rk for k=l, 2, . . ., 19 for 

the rainfall and discharge data are given in tables 5 

and 6. Equation 9 was used as an approximate test 

of significance. The letter a denotes significance at 

the 90-percent level.

If serial-correlation coefficients were determined for a 

random time series, an occasional significant value 

would be expected to occur by chance. With respect 

to the 6 rainfall records, 114 serial-correlation coefficients were determined. If the data were randomly 

distributed and if the data for different stations were 

uncorrelated, 12 of these values would be expected to 

be significant at the 90-percent level. However, at 

the 90-percent level, 22 values were found to be significant. For the 3 discharge records, 57 serialcorrelation coefficients were determined. If the data 

were random, 6 values would be expected to be significant by chance. Actually 10 values were found to 

be significant.

These results seem to indicate that the data are nonrandom. However, this investigation is handicapped 

by many factors. The number of stations used is 

very small, and the data for different stations, particularly the rainfall stations, are probably correlated. 

Also, most of the significant serial correlations for the
B6 STATISTICAL STUDIES IN HYDROLOGY

rainfall data belonged to the two records having the 

largest values of r\. One of these records was the 

Charleston, S.C., rainfall which apparently is not 

trend free. If these factors are taken into consideration, the rainfall data can be regarded as random 

in time. For the rainfall data to be nonrandom? there 

would need to be some form of atmospheric storage.

The discharge data may be nonrandomly distributed 

in time, with the nonrandomness being due to storage 

within the river basins. Note that the values of TI 

given in table 4 are positive except for the Pemiguewasset Kiver at Plymouth, N.H. Also, the significant 

serial correlations, given in table 6, are mainly those 

of low order.

GENERATING PROCESS

In figures 1 through 9, correlograms are shown for the 

data considered in the investigation of the serial-correlation coefficients for orders 1 through 19. A visual 

inspection of these correlograms shows no clear indication of a moving-average, cyclic, or autoregression 

process. By considering the significance of the serialcorrelation coefficients, the rainfall correlograms may be 

considered as indicative of random processes where the 

nonzero values of the serial-correlation coefficients are 

due to sampling errors. For the discharge data, the 

low-order serial correlations were significant. This 

suggests that the first-order Markov process may 

approximate the generating process.

The first-order Markov process fitted to the discharge correlograms was tested for goodness of fit by 

Quenouille's chi-square at the 90-percent level. The 

values of Rk, xl, and 2%^ are given in table 7. The value 

of chi-square at the 95-percent level for m=19 degrees 

of freedom is 27.2. Since this value of chi-square is less 

than 2%j for the discharge data for each of the three 

streams, the hypothesis that the generating process is 

a first-order Markov process is not rejected.

EFFECT OF SERIAL COBBELATION ON THE ESTIMATE 

OF THE VARIANCE

If a sequence of values is nonrandomly distributed in 

time, then each value repeats some of the information 

contained in previous values. Thus a longer sequence 

of nonrandom events is needed in order to arrive at a 

reliable estimate of the variance than in the case of a 

sequence of random events.

For a sequence of N' events, taken from a random 

time series having mean ju=0 and variance a3, the estimate of the variance is given by

N'

2

N' (22)

where x is the estimate of the mean defined by x=^ xltN'

It follows that equation 22 can be written as

N' N'

_/ i j_i i 

02_i=l_____»'==!____ N' (N') 2

(23)

The expectation of x t is E(xt)=0 and the expectation 

of x* is E(xl) = <r2. Since x is randomly distributed, 

E(x ixj')=E(x i)E(xj)=Q. Thus the expectation of S2 is

(24)

Let p* denote the population Mi-order serial-correlation coefficient whose estimate is given by rk. For a 

sequence of N events, taken from a nonrandom time 

series generated by a first-order Markov process having 

mean ju=0 and variance a2, the estimate of the variance 

is given by equation 23. For this generating process, 

E(x ixj) = a*pli-jl =o2pw=a2pl kl . Also E(x {)=Q and 

E(a$)=o*. Hence

(1-pf) . 2Pl(l- Py

If ^=0 and N N', equation 25 reduces to equation 

22. Assuming that N=N', it can be shown that the 

quantity in the brackets in equation 25 is smaller than 

(N' 1)/N'. Hence N' must be larger than AT" in order 

to obtain as reliable an estimate of the variance for a 

nonrandom time series as for a random time series. 

By equating equations 23 and 25, it is possible to determine N' as a function of N and PI. If N is the actual 

length of the time series, N' can be considered as the 

effective length of the time series. Thus if the variance 

is estimated from N events, the estimate is only as 

reliable as that estimated by a lesser number, N', of 

random events.

A graphical procedure facilitates the determination 

of N'. In figure 10, a family of curves is shown for 

E(S2)/a2 versus AT as a function of pi. As N tends to 

infinity, E(S2)/a2 tends to unity for all values of pi. 

The larger p\ is, the slower is the rate of convergence. 

For a given sequence, N is known and p\ can be estimated by TV Thus, starting with the value of A^ on 

the abscissa, a vertical line is drawn upward to the 

curve corresponding to pi. From this point of intersection, a horizontal line is drawn to the left until it 

intersects the curve for pi=0, and then a vertical line 

is drawn downward to the abscissa scale to determine 

N'. An example of this graphical procedure for 

determining N' is shown in figure 10. In this example 

AT=30 and pi=0.4. The graphical solution gives N'= 13.
AUTOCORRELATION OF RAINFALL AND STREAMFLOW MINIMUMS B7

DISCUSSION AND CONCLUSIONS

Time series of annual minimum monthly rainfall 

and annual minimum 1-day and 7-day discharge, 

considered as drought indices, were used to study the 

distribution of droughts with respect to time. An 

investigation for trends in the data indicated that the 

time series were trend free. The one exception was the 

rainfall data for Charleston, S.C. However, no attempt 

was made to remove the trend from this set of data.

The first-order serial-correlation coefficient, ri} was 

determined for each rainfall and discharge series. The 

values of rx ranged from 0.296 to 0.300 for the rainfall 

data and from 0.085 to 0.385 for the discharge data. 

The discharge data exhibited higher but less variable 

values of rx than the rainfall data. At the 90-percent 

level of significance, 6 and 4 values of r\ for the rainfall 

and discharge data, respectively, were found to be 

significant. At the 95-percent level, the rainfall data 

gave 4 significant values of rx and the discharge data 

gave 3 significant values of rt. Thus at the usual 

levels of significance employed in hydrologic studies, 

approximately K to % of the time series for both 

phenomena possessed significant values of rx .

Data exhibiting nonsignificant values of ri are not 

necessarily random since serial-correlation coefficients 

of order greater than one, if significant, would indicate 

a lack of randomness. Serial-correlation coefficients 

for orders 1 through 19 were determined for the data 

for 6 rainfall and 3 gaging stations. A total of 114 and 

57 serial-correlation coefficients was determined for the 

rainfall and discharge data, respectively. If these 

series were random, then 12 and 6 serial-correlation 

coefficients for rainfall and discharge data, respectively, 

would be expected to be significant at the 90-percent 

level. However, 22 and 10 serial-correlation coefficients 

for the rainfall and discharge data, respectively, were 

found to be significant.

This analysis seems to indicate that the data for both 

phenomena are nonrandomly distributed. Before drawing such a conclusion, several factors affecting the 

analysis must be considered. The number of records 

used was very small, and the data for different stations, 

particularly the rainfall data, were probably correlated. 

Over half the significant serial correlations for the 

rainfall data belonged to the two records having the 

largest values of /v One of these records was the 

Charleston, S.C., rainfall which apparently is not trend 

free. Thus the rainfall data are probably randomly 

distributed. If they were nonrandomly distributed, 

the nonrandomness would need to be attributed to 

some form of atmospheric storage.

The discharge records are not correlated. All the 

values of r\ except one are positive, and the low-order

serial correlations tend to be significant. Thus the 

discharge data may be regarded as nonrandom, with 

the nonrandomness being attributed to storage within 

the river basin.

The first-order Markov process was found to approximate the generating process of the discharge data. 

By assuming a time series to be generated by this 

process, the expectation of the estimate of the variance 

based on N observations was given and compared with 

the expectation of the estimate of the variance based 

on N' observations taken from a random time series. 

Because of the serial correlation, the estimate of the 

variance for a nonrandom time series based on N 

observations is only as reliable as the estimate of the 

variance for a random time series based on N' observations where N' is less than N. Nf, defined as the 

effective number of observations, is a function of N 

and the first-order autocorrelation coefficient, PI. A 

graphical procedure (fig. 10) is given for determining 

N' for various values of N and p\.

REFERENCES

Anderson, R. L., 1942, Distribution of the serial correlation 

coefficient: Annals Math. Statistics, v. 13, p. 1-13.

Anis, A. A., and Lloyd, E. H., 1953, On the range of partial sums

of a finite number of independent normal variates: Biometrika, v. 40, p. 35-42. 

Feller, Wilham, 1951, The asymptotic distribution of the range

of sums of independent random variables: Annals Math.

Statistics, v. 22, p. 427. 

Folse, J. A., 1929, A new method of estimating streamflow:

Carnegie Inst. Washington Pub. 400. 

Hurst, H. E., 1950, Long-term storage capacity of reservoirs:

Proc. Am. Soc. Civil Engineers, Separate 11. 

 1956, Methods of using long-term storage in reservoirs:

Inst. Civil Engineers, Kept. 6059.

1957, A suggested statistical model of some time series

which occur in nature: Nature, v. 180, p. 494.

Kendall, M. G., 1951, The advanced theory of statistics Part 

2: New York, N. Y., Hafner Press, p. 406.

Moran, P. A. P., 1959, The theory of storage: London, England, 

Methuen and Co. Ltd., p. 66.

Quenouille, M. H., 1949, A large sample test for the goodness 

of fit of autoregressive schemes: Royal Statistical Soc. 

Jour., v. 110, p. 123.

Slutzky, E., 1937, The summation of random causes as the source 

of cyclic processes: Econometrika, v. 84, p. 105.

Wold, Herman, 1954, A study in the analysis of stationary 

time series: Uppsala, Sweden, Almqvist and Wiksells, 

p. 122.

Yule, G. U., 1921, On the time series problem: Royal Statistical Soc. Jour., v. 89, p. 497.

 1927, On a method of investigating periodicities in disturbed series with special reference to Wolfer's sunspot 

numbers: Philos. Trans., v. A226, p. 267.
B8 STATISTICAL STUDIES IN HYDROLOGY

-as 

FIGUBE 1. Correlogram for annual minimum monthly rainfall at Boston, Mass.

FIGURE 2. Correlogram for annual minimum'monthly rainfall at Providence, R.I.

1.0

0.5-

-0.5

FIGUBE 3. Correlogram for annual minimum monthly rainfall at Charleston, 8.C. FIQOEE 4. Correlogram for~annual minimum monthly rainfall at Washington,

B.C.

1.01

/..o ./.\./N/:

V 5

 Y X / & v

-0.5

0.5

FIGUBE 5. Correlogram for annual minimum monthly rainfall at Baltimore, Md. FICTOBE 6. Correlogram for annual minimum monthly rainfall at Philadelphia, Pa.
AUTOCORRELATION OF RAINFALL AND STREAMFLOW MINIMUMS B9

i.O

1.0 f

0.5

-0.5

7. Correlogram for annual minimum 7-day discharge for James River at 

Buchanan, Va.

FIGURE 8. Correlogram for annual minimum 7-day discharge for South Fork Holston River at Bluff Citv. Terra.

1.0

O.5

-0.5

v . r\.

FIGURE 9. Correlogram for annual minimum 1-day discharge for Middle Branch 

Westfield River near Ooss Heights, Mass.

TABLE 1. Rainfall stations used in analyzing annual minimum 

monthly rainfall

Rainfall station

New Bedford,

Lowell Locks and 

Canals, Lowell,

Waltham, Mass ....

Boston, Mass...... 

Burlington, Vt 

Period of 

record

1836-1960

1814-1060

1865-1950

1826-1925

1828-1930 

1841-1921

1818-1913 

1838-1928

Length 

of 

record 

(years)

115

137

96

100

102 

81

96 

91

Rainfall station

Albany, N.Y.......

Providence, R.I- ... 

New York, N.Y....

Charleston, B.C.... 

St. Louis, Mo. ... 

Washington, D.C-.

New Burnswick, 

N.J. .............

Baltimore. Md ..... 

Philadelphia, Pa...

Period of 

record

1826-1928

1832-1928 

1826-1943

1832-1953 

1837-1953 

1852-1953

1854-1953

1854-1953

1818-1953 

1820-1953

Length 

of 

record 

(years)

103

97 

118

122 

117 

103

100

100

136 

134

FIQTTRE 10. Expected value of the variance for a first-order Markov process.

TABLE 2. Gaging stations used in analyzing annual minimum

discharge

Gaging station

James River at 

Buchanan, Va ....

Holston River at. 

Bluff City, Tenn. 

French Broad 

River at 

Asheville, N.C 

Westfield River 

at Ooss Heights,

Pemiguewasset 

River at 

Plymouth, N.HPertod of 

record

1898-1954

1901-49 

1903-63

1910-55

1903-55

Length 

of 

record 

(years)

57

49 

50

45

62

Gaging station

Souhegan River 

at Merrimack, 

N.H... ..-

Busquehanna 

River at 

Conklin, N.Y.... 

Little Tennessee 

River at Judson, 

N.C.. .........

Schroon River at 

River Bank, 

N.Y .... ... -

Period of 

record

1909-55

1906-60 

1896-1941

1908-63

Length 

of 

record 

(years)

46

46 

46

46
BIO STATISTICAL STUDIES IN HYDROLOGY

TABLE 3. Significance of first-order serial-correlation coefficients TABLE 4. Significance of first-order serial-correlation coefficients 

for rainfall for discharge

Rainfall station

Lowell Locks and Canals,

Albany, N.Y _____ .......

New York, N.Y._.___ ...

Baltimore. Md _ .... ......

Philadelphia, Pa..... ... ...

n

0.030 

.070

.060 

1.166 

a -.213 

.059 

.071 

.089 

-.070 

.070 

-.139 

a. 300 

a -.222

2 OOfi

.119 

-.046 i-. 171 

.000

Confidence level

90-percent

0.145 

.133

.157 

.154 

.153 

.171 

.159 

.161 

.152 

.157 

.143 

.141 

.143 

.152 

.154 

.154 

.134 

.135

-0. 152 

-.148

-.178 

-.175 

-.173 

-.196

-.178 

-.184 

-.172 

-.177 

-.160 

-.157 

-.161 

-.172 

-.175 

-.175 

-.149 

-.150

95-percent

0.174 

.160

.190 

.186 

.184 

.205 

.190 

.195 

.183 

.189 

.171 

.168 

.172 

.183 

.186 

.186 

.160 

.162

-0. 191 

-.175

-.211 

-.206 

-.204

-.230

-.211 

-.217

-.203 

-.210 

-.188

-.185 

-.189 

-.203 

-.206 

-.206 

-.175 

-.177

i Significant at 90-percent level. 

* Significant at 95-percent level.

TABLE 5. Serial-correlation coefficients for orders 1 through 19 

for rainfall

n ....... 

n .......

n .......

r« ..

n ....... 

r» .......

Tit ......

ni ----- - 

ru 

ru -....- 

H4 -- .

ru ....

ru ......

rtj ....

Boston, 

Mass.

0.071

1.164 

.142 

.130

.078 

.075 

-.031

-.099 

.078 

.117 

1.303 

.000 

-.117 

-.095 

-.124

-.023 

.028

-.107 

.080

Providence, 

R.I.

0.070

-.060 

-.138 

-.072

-.004 

.140 

-.031

-.210 

-.063 

-.004 

.077 

-.018 

.150 

-.096 

.007

.123 

-.028

.140 

-.014

Charleston, B.C.

10.300

1.238 

1.282 

1.317

1.196 

1.163 

.132

.120 

-.129 

-.048 

.009 

-.135 

i -. 175 

1-.163 

-.037

-.112 

-.109

I -.164 

1-.169

Washington, D.C.

i -0.296

1.157 

.150 

i -.249

-.013 

-.100 

-.066

-.034 

-.008 

1.175 

i -. 217 

.147 

-.021 

-.066 

-.037

.105 

.062

-.213 

1.278

Baltimore, 

Md.

i -0. in

1.180 

-.067 

.074

.000 

.086 

.004

.130 

.007 

-.011 

-.043 

.101 

.032 

1.143 

-.039

.096 

.018

-.106 

.036

Philadelphia, Pa.

0.000

.024 

.024 

.049

.087 

-.003 

.113

-.014 

.064 

-.059 

-.069 

.028 

-.089 

-.096 

-.023

.044 

1-.162

-.127 

-.003

Gaging station

South Fork Holston River at Bluff City,

French Broad River at Asheville, N.C.. 

Middle Branch Westfleld River near

Pemiguewasset River at Plymouth, 

N.H--. - 

Souhegan River at Merrimack, N.H .... 

Susquehanna River at Conklin, N.Y.... 

Little Tennessee River at Judson, N.C..

ri

a 0.345

.178 

.154

.297

-.082 

*.385 

».224 

.096 

.059

Confidence level

90-percent

0.200

.214 

.212

.222

.208 

.223 

.223 

.223 

.223

-0.236

-.256 

-.253

-.268

-.248 

-.267 

-.267 

-.267 

-.267

95-percent

0.241

.258 

.259

.268

.253 

.267 

.267 

.267 

.267

-0.277

-.300 

-.300

-.314

-.292 

-.311 

-.311 

-.311 

-.311

i Significant at 90-percent level. 

* Significant at 95-percent level.

TABLE 6. Serial-correlation coefficients for orders 1 through 19 

for discharge

no . . . .

rtt . 

TIT - - 

James River at 

Buchanan, Va.

10.345

.084

-.131

1-.220

-.098

J-.269

-.114

-.030

.122

1.238

.174

1.244

.177

-.026

-.130

-.201

« -.348

-.122

-.166

South Fork 

Holston River 

at Bluff City, 

Tenn.

a 178

-.200

i -.254

.037

1.239

1.234

-.070

.110

.148

-.033

-.041

-.060

.070

.123

-.038

.051

.164

-.077

-.203

Middle Branch 

Westfield 

River near 

(Joss Heights, 

Mass.

10.297

.061

1.242

-.027

1-.221

.081

.098

-.200

-.093

-.185

-.271

-.008

-.184

-.246

-.140

.030

.166

.093

-.061

> Significant at 90-percent level.

i Significant at 90-percent level.

TABLE 7. Chi-square goodness of fit of first-order Markov process 

to discharge correlograms

k

1

2

3

4

6

6

7

8

9

10

11

12

13

14

16

16

17

18

19

3&J

James River at Buchanan, Va.

Rk

-0.304

-.035

-.148

-.120

.038

-.218

.053

.018

.129

.150

.024

.152

.029

-.119

-.111

-.101

-.229

.094

-.122

xl

6.664

.085

1.524

.976

.100

3.107

.180

.019

1.033

1.368

.036

1.345

.062

.787

.666

.533

2.705

.447

.229

22.356

South Fork Holston

River at Bluff City, 

Tenn.

Rk

-0.172

-.232

-.177

.121

.218

.150

-.146

.142

.107

-.082

-.025

-.046

.090

.096

-.080

.068

.146

-.134

-.170

5d

1.521

2.692

1.540

.706

2.225

1.032

.949

.883

.486

.283

.025

.086

.311

.348

.228

.165

.714

.592

.928

16.714

Middle Branch Westfleld River near Ooss 

Heights, Mass.

Rk

-0.271

-.027

.232

-.166

-.184

.210

.030

-.251

.034

-.147

-.169

.187

-.203

-.136

-.011

.092

.126

-.047

-.011

xi

3.879

.036

2.718

1.346

1.621

2.069

.041

2.809

.051

.914

1.174

.742

1.686

.694

.004

.293

.635

.071

.004

20.686

. GOVERNMENT PRINTING OFFICE: 1967 O 248-124