Statistics of a

Runoff-Precipitation

Relation

GEOLOGICAL SURVEY PROFESSIONAL PAPER 434-D
Statistics of a

Runoff-Precipitation

Relation

By NICHOLAS C. MATALAS

STATISTICAL STUDIES IN HYDROLOGY

GEOLOGICAL SURVEY PROFESSIONAL PAPER 434-D

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

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington 25, D.C.
CONTENTS

Abstract------_----_--__---_-_._--_________________

Introduction ____-----______-_-____-_______-________

Object of investigation___--_--___-____._-______

Hydrologic model._______________________________

Statistical parameters characterizing runoff.______

General Theory___________________________________

Moments of runoff._____________________________

Probability distribution of runoff.._-____-__-______

Distribution of runoff in time.__-________________

Study of precipitation-runoff data___________________

Determination of effective precipitation.___________

Correlograms of runoff and effective precipitation. __

Page Page

Dl Study of precipitation-runoff data Continued

1 Statistical parameters of runoff and effective precipi1 tation______-________-______-_-_-----_--_-- D5

1 Summary....--.------.---------.------------------ 8

2 Hydrologic model and its characteristics_---_____-_ 8

2 Probability distribution of runoff ______________ 8

2 Time distribution of runoff_____----__-------_---- 8

2 Relation between skewness and nonrandomness of

4 runoff..______________-___-___-___--_-__--__ 9

5 Results______.____________________-__-_-__ 9

5 References.---.----------------..------------------ 9

5

ILLUSTRATIONS

Page 

FIGURE 1. Correlograms of runoff and effective precipitation for the St. Lawrence River at Ogdensburg, N.Y._____________ D6

2. Correlograms of runoff and effective precipitation for the Gota River at Sjortorp-Vanersborg, Sweden._________ 6

3. Correlograms of runoff and effective precipitation for the Nemunas River at Smalininkai, Lithuania.___________ 7

4. Correlograms of runoff and effective precipitation for the Danube River at Orshava, Rumania.. _______________ 7

TABLES

TABLE 1. Streamflow records______. _______________________________

2. Statistical characteristics of runoff and effective precipitation.

Page

D5

5

in

STATISTICAL STUDIES IN HYDROLOGY

STATISTICS OF A RUNOFF-PRECIPITATION RELATION

By NICHOLAS C. MATALAS

ABSTRACT

This report presents the results of an analysis of the influence 

of the water-retardation characteristics of a river basin on runoff 

distribution. The runoff was assumed to be generated by a 

moving average of the effective precipitation, where the extent 

of the moving average is assumed to be equal to the carryover 

a function of the water-retardation characteristics of the river 

basin.

The probability distribution of the runoff is a function of the 

extent of the carryover period. Even though the characteristics 

of the effective precipitation may be the same for all river basins, 

the probability distribution of the runoff is not the same, because 

water-retardation characteristics vary from one river basin to 

another.

Owing to the carryover period, runoff is nonrandomly distributed in time. The serial correlation coefficients that are used to 

measure the nonrandomness of runoff are functions of the 

coefficients of the moving-average model, if it is assumed that 

effective precipitation is randomly distributed in time.

The moving-average model and the theoretical results 

derived from it are supported by experimental results obtained 

by analyzing several long-term runoff records.

INTRODUCTION 

OBJECT OF INVESTIGATION

The purpose of this investigation was to study the 

influence of the water-retardation characteristics of a 

river basin on the distribution of runoff. This investigation is based on a simple hydrologic model, 

where runoff is assumed to be generated by a moving 

average of the effective precipitation. The extent of 

the moving average is assumed to be equal to the 

carryover period. It is also assumed that runoff and 

effective precipitation correspond to a time interval 

such that the effective precipitation can be considered 

as randomly distributed in tune.

Theoretically, the runoff during a given interval of 

tune, such as day, month, and year, is a function of all 

climatic factors, present and past, since the beginning 

of time. The dominant climatic factor is the effective 

precipitation, which is denned as the total precipitation 

less all losses. The effective precipitation prior to a 

given tune interval is referred to as antecedent effective 

precipitation.

As the contribution of effective precipitation to the 

runoff converges to zero very rapidly with an increase 

in antecedent tune, the effective precipitation for only 

a finite period of antecedency affects the runoff. 

The finite period of antecedency, which is defined as 

the carryover period, is a function of the waterretardation characteristics of the river basin and the 

distribution of the effective precipitation with respect 

to tune.

HYDROLOGIC MODEL

If Qj denotes the runoff during the jth time interval 

(where .7 = 1, 2, . . . , N, with N the total number of 

time intervals) and if pj-i denotes the effective precipitation during the (j i)l1> time interval (where i=Q, 1, 

2, . . . , m is the time interval antecedent to j, and m 

is the extent of the carryover), the relationship between 

runoff and effective precipitation is expressed as

(1)

Equation (1) expresses the runoff as a moving average 

of extent m of the effective precipitation. The weights 

of the moving average, b 0 , bi, . . . , bm are subject to 

the linear constraint

Z=0

(2)

since for large values of N the mean runoff is equal to 

the mean effective precipitation. If the tune interval 

is much greater than the concentration time of floods 

on the river basin, the weights of the moving average 

decrease monotonically with an increase in i. That is,

(3)

Since the values of effective precipitation are positive 

and since antecedent effective precipitation contributes 

to the runoff, the weights of the moving average are 

positive. That is,

i=0, 1, 2, . . . , (4)

Dl
D2 STATISTICAL STUDIES IN HYDROLOGY

The runoff-effective precipitation relationship given 

by equation (1) is the form of equation adopted by 

Folse (1929). Other forms can be considered, such as 

those involving cross-product terms. However, owing 

to the limited knowledge of the true runoff-effective 

precipitation relationship, a restriction is imposed on 

considering complex models. A complex model is 

warranted only when it can be demonstrated to yield 

results in closer agreement with the observed facts than 

the results obtained by simpler models.

STATISTICAL PARAMETERS CHARACTERIZING 

RUNOFF

The statistical parameters that describe the probability distribution of runoff are as follows: (1) Q, the 

mean runoff; (2) [/i2 (Q)] 1/2 , the standard deviation; 

(3) |3i(<?), the coefficient of skewness; and (4) 02 (Q), 

the coefficient of kurtosis. These parameters suffice 

to describe adequately the frequency distributions that 

are applied to hydrologic investigations of runoff.

If the runoff during any given time interval is independent of the runoff during any other time interval, 

runoff is said to be distributed randomly in tune. If, 

however, the contrary is true, runoff is said to be distributed nonrandomly in time. The serial correlation, 

Rk, between a ay two runoff events of interval k apart, 

gives a measure of the degree and extent of the nonrandomness of the runoff values.

GENERAL THEORY 

MOMENTS OF RUNOFF

Basically, the moments are a set of parameters of a 

distribution that measure its properties and in certain 

cases specify the probability distribution. Summing 

equation (1) over all values of j, dividing by N, and 

using equation (2), then

Q=P, (5)

whereby the mean value of runoff is equal to the mean 

value of effective precipitation. Equation (5) is valid 

if AT is very large and m is much smaller than N. Since 

the zth central moment of the distribution of a variable 

x around the mean, x, of the distribution is defined as

N

it follow that the ztn central moment of runoff is

(6)

(7)

Substituting equation (1) for Q} and equation (5) for 

Q, equation (7) becomes

I N M 

Vz jj ^j ^~j bi(pj-i p)s. (8)

Since the probability distributions that have been 

found to fit the observed values of runoff are of such 

forms for which all moments exist, it is sufficient for 

this investigation to specify only the first four central 

moments. These central moments are as follows:

(9)

(10)

(11)

z=0

i=0

i=o

[~ / m \2 TO "I

(S «) -S 6J (12) LA z=o / «=o J

The subscripts of the JLI'S refer to the order of the central 

moments, so that /x 2 ((?) and nz (p) are the zth central 

moments of the runoff and effective precipitation, respectively. The central moments given by equations 

(9) through (12) are valid under the assumption that 

the effective precipitation is randomly distributed in 

time.

It follows from equation (2) and inequality (4) that

i=0

(13)

It is therefore apparent that the second and third 

central moments of runoff, equations (10) and (11), 

respectively, are less than the corresponding central 

moments of effective precipitation. It does not 

necessarily follow that the fourth central moment of 

runoff is less than the fourth central moment of effective 

precipitation. It can be proved that if the square of 

the second central moment of effective precipitation is 

equal to or less than the fourth central moment of 

effective precipitation, then the fourth central moment 

of runoff is less than the fourth central moment of 

effective precipitation. On the other hand, if the 

square of the second central moment of effective 

precipitation is greater than the fourth central moment 

of effective precipitation, then depending upon the 

difference between these two central moments and upon 

the carryover period and the function of the ft/s, the 

fourth central moment of runoff can be greater than the 

fourth central moment of effective precipitation.

PROBABILITY DISTRIBUTION OF RUNOFF

It is assumed that the probability distributions of 

runoff and effective precipitation are such that all 

moments exist and that the two probability distributions can be defined by their respective means, standard
.STATISTICS OF A RUNOFF-PRECIPITATION RELATION D3

deviations, coefficients of skewness, and coefficients of 

kurtosis. Each of these parameters is a function of 

one or more of the first four central moments.

The mean value is defined as the first moment of 

the distribution around the origin. Equation (5) shows 

that the mean value of runoff is equal to the mean 

value of effective precipitation. Therefore, the carryover period, m, which integrates the water-retardation 

characteristics, does not influence the mean value. 

The equality of the two means is ensured by the linear 

constraint given by equation (2).

The variance is defined as the second central moment 

and is given by equation (10). Equation (13) indicates that the variance of runoff is less than that of 

effective precipitation. The water-retardation characteristics cause the values of runoff to have less 

dispersion about the mean than do the values of effective precipitation. Thus the water-retardation characteristics act to smooth out the irregularities of 

effective precipitation.

For probability distributions of the form just specified, relationships exist between the coefficients of 

skewness and kurtosis. These relationships can be 

expressed mathematically for many of the probability 

distributions used in hydrology. The coefficients of 

skewness and kurtosis, which are dimensionless ratios, 

are defined, respectively, as

(14)

(15)

By using equations (10) and (11), the coefficient of 

skewness of runoff becomes

z

or

wherein

m

Si=o

m \3 \2

)

/

(17)

(18)

If m=0, there is no carryover, in which case the 

river basin possesses no water-retardation characteristics with respect to the time interval taken for the 

runoff and the effective precipitation. As the carryover 

period, m, increases, then both sides of inequality (18) 

approach zero as m approaches infinity. Because the 

left-hand side of inequality (18) is the denominator in 

equation (17) and because it approaches zero slower 

that the right-hand side of inequality (18), which is the 

numerator is equation (17), the coefficient of skewness 

 of the runoff approaches zero as m tends to infinity.

Because a coefficient of skewness equal to zero is associated with a symmetrical probability distribution, the 

probability distribution of the runoff, due to the waterretardation characteristics, is more symmetrical than 

the probability distribution of the effective precipitation, provided ft(p)>0. If A(p)=0, then ft(Q)=0 

regardless of m.

By using equations (10) and (12) in equation (15), 

the coefficient of kurtosis of the runoff becomes

/..GO £ M

m \2 m~~\ -£ M

"" J

i=0

(19)

or

,+3 1 

t=0

wherein

j=0 \j=0

(20)

(21)

As m approaches infinity, both sides of inequality (21) 

approach zero with the left-hand member approaching 

zero faster than the right-hand member. Therefore, 

as m approaches infinity, the coefficient of kurtosis 

of the runoff approaches 3. By inequalities (18) and 

(21), it is seen that in the limit, for m equal to infinity, 

the probability distribution of the runoff has coefficients 

of skewness and kurtosis equal to 0 and 3, respectively; 

whereby, the runoff is normally distributed. This may 

also be shown by the central limit theorem (Cramer, 

1954).

If m=0, in which case there is no carryover, the 

coefficient of kurtosis of the runoff is equal to that of the 

effective precipitation. And if the coefficient of kurtosis 

of the effective precipitation is equal to 3, it is seen by 

equation (20) that the coefficient of kurtosis of the 

runoff is 3 regardless of the carryover period, m.

As m tends to infinity, the coefficients of skewness 

and kurtosis approach their limiting values 0 and 3, 

respectively. However, the coefficient of skewness 

approaches 0 faster than the coefficient of kurtosis 

tends to 3. This condition can be shown by proving 

that

i=0

i=Q

or that

)

2 m m

j=0 2=1

(22)

(23)
D4 STATISTICAL STUDIES IN HYDROLOGY

By expanding both sides of inequality (23), then

m l m~s

g=0 i=0 s=0 i=0

(24)

Since the first terms on both sides of inequality (24) 

are identical, it is necessary to consider only the second 

terms on both sides of the inequality. For any given 

value of i and s, the second terms give

or

(25)

(26)

Since bi^>b i+s, which follows from inequality (3), the 

term on the right hand side of inequality (26) must be 

a positive number greater than zero. Thus inequalities 

(22) and (23) are proved.

The extent of the carryover, m, is not a constant, 

because water-retardation characteristics vary from one 

river basin to another. The probability distribution 

of the runoff is a function of the carryover, m, provided 

that the coefficients of skewness and kurtosis of the 

effective precipitation are not equal to 0 and 3, respectively. Therefore, the probability distribution of the 

runoff is not necessarily the same from one river basin 

to another.

If, however, the coefficients of skewness and kurtosis 

of the effective precipitation are equal to 0 and 3, 

respectively, the probability distribution of the runoff 

is characterized by the fact that the coefficients of 

skewness and kurtosis of the runoff are also 0 and 3, 

respectively. Thus the water-retardation characteristics do not influence the form of the runoff probability 

distribution when it is the same as that of the effective 

precipitation. However, owing to the carryover, the 

variance of the runoff is less than that of the effective 

precipitation.

DISTRIBUTION OP RUNOFF IN TIME

If there is carryover, the runoff during any given 

time interval is dependent upon the runoff during 

previous time intervals; therefore, the runoff is nonrandomly distributed in time. Theoretically, this 

dependency is a function of all previous runoff since the 

beginning of tune. For practical purposes, however, 

the carryover period is considered as finite, so that the 

runoff during a given time interval is dependent only 

upon the runoff during a finite number of antecedent 

time intervals.

A measure of the nonrandomness is the serial correlation coefficient of order k. This measure is defined as

Z?«2r-® (&+*-<?)

(27)

which is the correlation between values of runoff k time 

intervals apart. By using equations (1) and (5), 

equation (27) becomes

N k m

Z) Z) i=0 (28)

or

m k

(29)

-

j=0

under the assumption that the effective precipitation is 

distributed randomly in time. Equation (29) holds 

for k<m where R0=l. However, for &>(ra-|-l), then 

.2?jt=0.

If there is no carryover, m 0, Rk is equal to zero for 

all values of k > 1, since only the coefficient 6 X has a value 

greater than zero. For ra>0, each value of b0 and 

bi+K is greater than zero for all values of k<m so the 

Rk is positive. And because all values of b i+k are zero 

for k > (ra+ 1), Rk is zero for all values of k > (m -+-1). That 

Rk is a monotonic function, decreasing from R0=l to 

7?fcXm+l)=0, can be proved as follows. Since the 

denominator of equation (29) is a constant for each 

value of k, it suffices only to prove that the numerator 

of equation (29) for a given value of m decreases with 

an increase of k. If the numerator of equation (29) 

is expanded, the number of terms in the expansion 

equals (m+1 k). Hence as k increases from zero to 

its maximum value, m, the number of terms decreases 

from (m-+-1) to 1. Since b *> b i+k f or 1 < k < m, the product 6f6 i+fc decreases with an increase of k. It therefore 

follows that the numerator of equation (29) decreases 

monotonically as k increases.

For any given function of the 6/s, Rk increases as m 

increases and tends to unity as m tends to infinity. If 

the function of the fe/s is different for two river basins 

and if the carryover period, m, is the same for both 

basins, then all possible values of Rk for the runoff from 

one river basin are not necessarily either less or greater 

than the corresponding values of Rk for the runoff from 

the other river basin.

By equation (29), the nonrandomness of the runoff 

is a function of the water-retardation characteristics of 

the river basin. For any given function of the coeffi-
STATISTICS OF A RUNOFF-PRECIPITATION RELATION D5

cients, 6 4's, Rk increases with an increase of m. Equation (16) shows that the coefficient of skewness of the 

runoff decreases as m increases. Since the coefficient 

of skewness of the effective precipitation, pi(p), is 

independent of m and of the coefficients, 6/s, then Rk, 

the nonrandomness of the runoff, is independent of 

j8i(Q), the coefficient of skewness of the runoff.

The values of Rk can be determined directly from the 

observed values of runoff. Hence, if the distribution 

of the bi's is known, it is possible to determine the 

carryover period, m, by solving equation (29) for m. 

For example, if the 6/s vary linearly with i, then

h'""

(*»+!) (m+2) 

Using equation (30), equation (29) becomes

1+- -F -i

(«+DL (2m+3)(m+2)J 

Letting k=l and solving for m, it appears that

m=-f

(30)

(31)

(32)

STUDY OF PRECIPITATION-RUNOFF DATA

DETERMINATION OF EFFECTIVE PBECIPITATION

The time interval selected for the runoff and effective 

precipitation was a year. By determining the stored 

water at the end of the water year, We , and the stored 

water at the beginning of the water year, W&, then the 

difference

^=Wti-Wtl (33)

represents the stored water in the river basin during 

the water year. If Qj denotes the total runoff during 

the water year, then

P,= &+A, (34)

represents the total effective precipitation during the 

water year.

By using equations (33) and (34), it is possible to 

estimate the effective annual precipitation, and this 

was done for several gaging stations. These stations 

are listed in table 1.

TABLE 1. Streamflow records

[Data furnished by Dr. V. M. Yevdjevich (Colorado State University) from his 

current study "The fluctuation of annual river flows"]

Stream

Gota.. ... ....

Location

Orshava, Rumania _______

Drainage 

area 

(sq mi)

295,200

18, 076

30,900

216,300

Period of 

record

1860-1957

1807-1957

1811-1943

1837-1957

N 

Number 

of years 

ol record

Q7

150

132

120

COBBELOGRAMS OP BTJNOFF AND EFFECTIVE 

PBECIPITATION

In figures 1 through 4, the correlograms for the annual 

runoff and effective precipitation are given for the 

streams listed in table 1. The correlograms formed by 

the solid lines apply to the runoff and the correlograms 

formed by the alternate long-short dashed lines apply 

to the effective precipitation. For each correlogram, 

approximate 90- and 95-percent confidence limits are 

given. These confidence limits are based on Anderson's 

(1942) test of significance of serial correlation coefficients.

Figure 1 shows that for the St. Lawrence Kiver 

runoff, the serial correlation coefficients for k equal to 

1 through 9 are significant at the 95-percent level. 

For )fc>9, the serial correlation coefficients fluctuate 

within the confidence bands. With respect to the 

effective precipitation, shown by the alternate longshort dashed lines, the values of Rk fluctuate within 

the confidence bands.

The first serial correlation coefficients for the runoff 

for the Gota Kiver (fig. 2) and Nemunas Kiver (fig. 3) 

are significant at the 95-percent level. The values of 

Rk for k>2, for both of these rivers, fluctuate within 

the confidence bands and therefore can be considered 

as not significant. For the effective precipitation 

corresponding to both of these rivers, the values of Rk 

fluctuate within the confidence limits.

With respect to the Danube Kiver (fig. 4) the values 

of Rk for both the runoff and the effective precipitation 

fluctuate within the 95-percent confidence bands.

The correlograms for effective precipitation indicate 

that effective precipitation is randomly distributed in 

time. Runoff may be either randomly or nonrandomly 

distributed in time. The correlograms for runoff 

indicate that Rk decreases with an increase in k and 

that beyond a certain value of k, the values of Rk can 

be considered as not significant. Therefore, the correlograms give reasonable support to the moving 

average as the generating scheme of runoff.

STATISTICAL PABAMETEBS OF BTTNOFF AND 

EFFECTIVE PBECIPITATION

The above investigations of the correlograms gave 

support to the hydrologic model, equation (1), and 

its underlying assumptions. A mathematical treatment of equation (1) indicated the following conditions: 

(1) That the mean runoff is equal to the mean effective 

precipitation; (2) that the variance of the runoff is 

less than the variance of the effective precipitation; 

(3) that the skewness of the runoff is less than the 

skewness of the effective precipitation; and (4) that 

the kurtosis of the runoff is not necessarily less than the 

kurtosis of the effective precipitation. Condition & 

applies to the case for which the effective precipitation.

649665 62
D6 STATISTICAL STUDIES IN HYDROLOGY

EXPLANATION

 Runoff

 ~ Effective

Confidence limit, precipitation 

in percent 

95 

90

FIGUKE 1. Correlograms of runoff and effective precipitation for the St. Lawrence Eiver at Ogdensburg, N.Y.

0.5

-0.5 -

FIGUKE 2. Correlograms of runoff and effective precipitation for the Gota Eiver at Sjotorp-Vanersborg, Sweden.
.STATISTICS OF A RUNOFF-PRECIPITATION RELATION D7

EXPLANATION

 Runoff

 - Effective

precipitation 

Confidence limit, 

in percent

-0.5 -

10 15 20

k

25 30 35 40

FIGUEE 3. Correlograms of runofE and effective precipitation for the Nemunas River at Smalininkai, Lithuania.

EXPLANATION

FIGURE 4. Correlograms of runoff and effective precipitation for the Danube River at Orsha va, Rumania.
D8 STATISTICAL STUDIES IN HYDROLOGY

does not follow a symmetrical probability distribution. 

Condition 4 applies to the case for which the kurtosis 

of the effective precipitation is not equal to 3.

In table 2 values are given for the mean, the standard 

deviation, the coefficient of skewness, and the first 

order serial correlation coefficient for runoff and effective precipitation in the St. Lawrence, Gota, Nemunas, 

and Danube River basins. The coefficients of kurtosis 

were not considered, because it would have been 

necessary to define the function of the 6/s in order to 

carry out the comparison between runoff and effective 

precipitation.

Table 2 shows that the standard deviation of the 

effective precipitation is greater than that of the runoff. For the St. Lawerence and Gota Rivers, the 

standard deviation of the effective precipitation is 

nearly twice as large as that of the runoff; however, 

the extent of the carryover for these two rivers is 

different as shown by their correlograms in figure 1. 

This difference can be attributed to the fact that the 

function of the 6/s is different for the two streams.

According to equation (17), the coefficient of skewness 

of runoff is less than that of effective precipitation 

provided /^(zO^O. Table 2 shows that this is the 

case with respect to the Gota and Nemunas Rivers. 

The St. Lawrence and Danube Rivers show the contrary. However, in the case of the St. Lawrence 

River the skewnesses are nearly zero so that the difference between the skewness of runoff and the skewness 

of effective precipitation probably is not significant.

With respect to the first order serial correlation 

coefficients, it appears that RI for runoff is greater than 

RI for effective precipitation with respect to each of the 

four streams.

TABLE 2. Statistical characteristics of runoff and effective 

precipitation

River

Gota _____________

Hydrologic 

variable »

Q.........

P... _ ...

Q.........

P .........

Q.........

P.... .....

Q.........

P ..... . ...

m Mean 

(cfs)

240, 820

240, 820

35,300

35, 300

19, 253

19, 253

189, 500

189, 500

^ Standard 

deviation

20, 950

47,440

6,420

10,870

3,410

3,700

36, 380

40, 550

01

0.080

.020

.003

.154

.216

.368

.729

.511

Ri

0.705

.090

.463

.009

.181

110

.090

.001

1 Q is runoff; P is effective precipitation.

SUMMARY 

HYDRO LOGIC MODEL AND ITS CHARACTERISTICS

Owing to the water-retardation characteristics of 

a river basin, the runoff during a given time interval 

is a function of the effective precipitation during the 

given time interval and during all previous time 

intervals. From a practical point of view, however,

the runoff is considered to be a function of the effective 

precipitation during a finite number of antecedent 

time intervals. It was assumed that the runoff was 

generated by a moving average of the effective precipitation, when the extent of the moving average was 

equal to a finite number of antecedent time intervals. 

This finite number of antecedent time intervals was 

defined as the carryover period, which is a function of 

the water-retardation characteristics of the river basin. 

The coefficients defining the moving average, denoted 

by 6/s, were assumed to satisfy the following conditions: 

(1) the sum of the coefficients is equal to unity; (2) the 

magnitudes of the coefficients decrease monotonically 

with an increase in the antecedent time; and (3) the 

values of the b t'a are greater than zero.

PROBABILITY DISTRIBUTION OF RUNOFF

On the basis of the assumed model of runoff and 

effective precipitation, the mean value of runoff is equal 

to the mean value of effective precipitation. The 

second and third central moments of runoff are less 

than those of effective precipitation. However, the 

fourth central moment of runoff is not necessarily less 

than that of effective precipitation.

Investigation of the skewness and kurtosis of runoff 

and effective precipitation showed that if the effective 

precipitation were normally distributed, then the runoff 

was normally distributed regardless of the carryover 

period. If, however, the effective precipitation followed a skewed probability distribution, then the 

runoff followed a skewed probability distribution, 

which was different and less skewed than that of the 

effective precipitation, provided that there was carryover. For a carryover period equal to infinity, runoff 

is normally distributed regardless of the probability 

distribution of effective precipitation, provided, of 

course, that the probability distribution of effective 

precipitation is such that all moments exist. Therefore, the greater the carryover period, the more the 

probability distribution of runoff departs from that of 

effective precipitation, provided that effective precipitation follows a skewed probability distribution.

The carryover period is a function of the waterretardation characteristics of a river basin. Even 

though the characteristics of the effective precipitation 

may be the same for each river basin, the probability 

distribution of the runoff is not the same, because 

water-retardation characteristics vary from one river 

basin to another.

TIME DISTRIBUTION OF RUNOFF

Owing to the water-retardation characteristics, the 

runoff during a given time interval is dependent upon 

the runoff during previous time intervals. If the
STATISTICS OF A RUNOFF-PRECIPITATION RELATION D9

carryover period is denoted by m, then the runoff 

during a given tune interval is dependent upon the 

runoff during (m 1) previous time intervals. Thus 

the water-retardation characteristics generate a nonrandom distribution of the runoff, and the nonrandomness is measured by the serial correlation coefficients.

The serial correlation coefficients of the runoff are 

dependent only upon the function of the coefficients of 

the moving average, &i's, and upon the carryover period, 

m, under the assumption that the effective precipitation is randomly distributed in time. Thus by specifying the function of the bt's, it is possible to determine 

m, as the serial correlation coefficients can be determined from the observed values of the runoff.

The carryover period, m, can be estimated without 

specifying the function of the bt's by means of the 

correlogram, provided that the number of time intervals, 

N, is large. Theoretically, the correlogram for a 

moving-average process vanishes for all orders of serial 

correlation equal to and greater than a certain value 

of k. If this value of k is denoted by k0, then the 

carryover period, m, is equal to (£0+1). With actual 

data, however, zero values of serial correlation are not 

likely to be obtained, owing to sampling errors. In 

practice, k0 may be taken as that value of k beyond 

which all values of serial correlation coefficients fluctuate within a confidence band of nonsignificance.

RELATION BETWEEN SKEWNESS AND NONRANDOMNESS OP RUNOFF

The coefficient of skewness of the runoff, &($)> is 

dependent upon the function of the coefficients, &i's, of 

the moving average, the carryover period, m, and the 

coefficient of skewness of the effective precipitation, 

Pi(P). However, the serial correlation coefficients, 

Rk , of the runoff are dependent only upon the function 

of the coefficients of the moving average and the 

carryover period and are independent of the skewness 

of the effective precipitation.

For a given value of skewness of the effective precipitation, the skewness of the runoff decreases and the 

nonrandomness of the runoff increases with an increase 

in the carryover period. However, since the skewness 

of the effective precipitation is independent of the 

function of the coefficients of the moving average and 

the carryover period, the serial correlation coefficients 

of the runoff can assume any value within the range 

0 to 1 independently of the values of the coefficients 

of skewness of the runoff.

RESULTS

In order to test the theoretical results, annual values 

of runoff and estimated values of effective precipitation 

for four long-term records were considered. The 

correlograms of runoff and effective precipitation show 

that the moving average provides a satisfactory model 

of the generating scheme of runoff. The second central 

moments of the runoff were found to be less than those 

of the effective precipitation. However, the coefficient 

of skewness of the runoff was not always found to be 

less than the coefficient of skewness of the effective 

precipitation.

These investigations show that the data provide 

reasonable support for the theoretical results. These 

results are, however, applicable to annual events only. 

If time intervals other than a year are considered, the 

theoretical results will be applicable only if the effective 

precipitation can be assumed to be distributed randomly 

in time and if no cyclic effects are present.

REFERENCES

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

coefficient: Annals of Mathematical Statistics, v. 13, p.

1-13. 

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

Carnegie Institute of Washington Pub. 400. 

Gramer, H., 1954, Mathematical methods of statistics: Princeton,

N.J., Princeton University Press, p. 213-218.