APR 2 6 1963

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of Low Flows

GEOLOGICAL SURVEY PROFESSIONAL PAPER 434-A

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Probability Distribution 

of Low Flows

By NICHOLAS C. MATALAS

STATISTICAL STUDIES IN HYDROLOGY

GEOLOGICAL SURVEY PROFESSIONAL PAPER 434-A

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

STEW ART 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.______________________________________

Theoretical probability distributions______________

Object of the investigation.______________________

Statistical characteristics of low flows _________________

Cumulative probabilities of low flows___.__________

Skewness and kurtosis of low flows________________

Discussion of probability distributions_________________

Characteristics of theoretical distributions _________

Gumbel's limited distribution of the smallest value._ 

3-Parameter Log-Normal distribution _____________

Pearson Type III distribution.___________________

Pearson Type V distribution.____________________

Analyses of low flows.-.. __________^ -____________

Investigation of the lower limit.__________________

Page

Al

1

1

2

2

2

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3

3

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4

5

6

6

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Page 

Analyses of low flows Continued

Effect of the largest low flows on the lower limit. _ _ _ A7 

Investigation of the relation between skewness and

kurtosis _____________________________________ 8

Method of maximum likelihood applied to low-flow probability analysis__________-_____-___--_________-___ 9

Discussion of the method of maximum likelihood- _ _ 9 

Maximum-likelihood estimates of parameters.._ _ _ _ _ 9

Relative efficiency of moment estimates with respect 

to maximum-likelihood estimates______________ 9

Application of the method of maximum likelihood.- 11 

Summary and conclusions---_____--_-______-___--__- 11

Probability distribution of low flows.....--__------ 11

Method of maximum likelihood_________________ 12

References.________________________________________ 27

ILLUSTRATIONS

FIGURES 1-34. Low-flow probability distribution.

1. Middle Branch Westfield River at 

Goss Heights, Mass __'__________

2. Pemigewasset River at Plymouth,

3. Saco River near Cornish, Maine __

4. St. John River below Fish River at 

Fort Kent, Maine________ ____

5. Souhegan River near Merrimack, 

N.H__._______________________

6. Little River near Princeton, N.C___

7. Fishing Creek near Enfield, N.C.__

8. Haw River near Benaja, N.C__ ____

9. East Fork of Deep River near High 

Point, N.C_______ ____ _ __ _

10. Neuse River near Clayton, N.C____

11. Cape Fear River at Lillington, N.C_

12. French Broad River near Asheville, 

N.C__________________________

13. Coeur d'Alene River near Cataldo, 

Idaho __ _____________________

14. Kootenai River near Leonia, Idaho.

15. Kootenai River at Bonner's Ferry, 

Idaho. . _______________________

16. Kootenay River near Newgate, 

British Columbia _ ____________

17. North Fork Stillaguamish River 

near Arlington, Wash___________

18. South Fork Stillaguamish River 

near Granite Falls, Wash _ _____

19. South Fork Skykomish River near 

Index, Wash __ _______________

20. Sauk River at Sauk, Wash... __ _

21. James River at Buchanan, Va_____

22. Susquehanna River at Conklin,

Page

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20

FIGURES 1-34. Low-flow probability distribution Con. 

23. Little Tennessee River at Judson,

24. Chenango River near Chenango, 

N.Y____._____--___-. ------

25. East Branch Delaware River at 

Fish's Eddy, N.Y. ____________

26. West Branch Delaware River at 

Hales Eddy, N.Y_---___._____

27. Greenbrier River at Alderson, W. 

Va____ ---- _ 

28. Toccoa River near Dial, Ga__ ____

29. Sacandago River near Hope, N.Y.

30. Schroon River at River Bank, 

N.Y__-____-___--------

31. South Fork of Hols ton River near 

Bluff City, Tenn____-_---_-

32. Brandy wine River at Chadd's 

Ford, Pa------------

33. Hoosic River near Eagle Bridge, 

N.Y______-_-_ -------

34. West Branch of Oswegatchie River 

near Harrison, N.Y- _..-___--_--

35. Relation between skewness and standardized lower limits for theoretical probability distributions- ----------------

36. Relation between skewness and kurtosis 

for theoretical probability distributions. 

37-43. Cumulative probability distribution for a random sample from a 

Pearson Type III distribution. 

37,38. N=20-_------

39-41. N=30 _ __ __----

42,43. N=40-_----_-

m

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25

25

26
IV

TABLE 1. Records of annual minimum daily low flows. _.

2. Records of annual minimum 7-day low flows. _

3. Coefficients of skewness of low-flow data with 

90-percent, 95-percent, and 99-percent confidence lirhits_. __________________________

4. Coefficients of kurtosis of low-flow data with 

90-percent, 95-percent, and 99-percent confidence limits_____________________ _______

CONTENTS

TABLES

Page

A12

12

13

13

Page 

TABLE 5. Observed and theoretical standardized lower

limits. _ ________________________________ A14

6-8. Relative efficiency of moment estimates with 

respect to maximum-likelihood estimates.

6. Pearson Type III distributions_______ 14

7. 3-parameter log-normal distributions__ 14

8. Pearson Type V distributions_______ 14

9. Comparison of the methods of maximum likelihood and moments______________________ 14
STATISTICAL STUDIES IN HYDROLOGY

PROBABILITY DISTRIBUTION OF LOW 

FLOWS

By NICHOLAS C. MATALAS

ABSTRACT

The fit of four theoretical probability distributions to low-flow 

data was investigated. Also, a study was made to determine 

the desirability of estimating parameters of probability distributions by the method of maximum likelihood. The four probability distributions used in these investigations were: (1) 

Gumbel's limited distribution of the smallest value, (2) 3~ 

Parameter Log-Normal distribution, (3) Pearson Type III distribution, and (4) Pearson Type V distribution.

The goodness of fit was based on the relation between the 

observed minimum low flows and the lower limits of the four 

probability distributions and on the relation between the skewness and kurtosis of the low-flow data. It was found that the 

Gumbel and Pearson Type III distributions fitted the data 

equally well and were more representative of the probability 

distribution of low flows than either the 3-Parameter Log-Normal 

or Pearson Type V distribution.

The variances of the moment and maximum likelihood estimates of the parameters of probability distributions are functions 

of the skewness of the data. Based on an average skewness^fbr 

low-flow data of nearly 1, the variances of moment estimates are 

nearly twice as large as those of maximum likelihood estimates. 

Thus future studies dealing with the probability distribution of 

low flows might best be made by using maximum-likelihood 

estimates of parameters rather than moment estimates.

INTRODUCTION 

THEORETICAL PROBABILITY DISTRIBUTIONS

A drought is defined, in a broad sense, as an extended 

period of dryness. In order to define and analyze 

droughts properly, it is necessary to consider droughts 

as being two dimensional. These two dimensions are 

magnitude and duration. In this study, however, only 

magnitude will be considered.

In an analysis of drought magnitudes, the minimum 

discharge during each year is considered. The minimum discharge is defined as the average minimum discharge for a given period of time within a year. The 

given period of time is generally taken to be 1 day, or 

7 days if it is necessary to remove the effect of fluctuations resulting from minor river regulation. Other periods of time are used depending upon the problem at 

hand. These average minimum annual discharges are 

referred to as low flows. The records of annual minimum daily low flows and minimum 7-day low flows 

used in this study are listed in tables 1 and 2, respectively.

For a given stream for which N years of observations 

are available, N values of low flow can be obtained. 

These N values are assumed to be a sample from an 

infinite population of low-flow values. The N values 

constitute the low-flow record and they are the basis 

for determining the probability of occurrence of low 

flows of various magnitudes. This is done by arranging 

the N values in order of magnitude, largest to smallest, 

and assigning a cumulative probability or return period 

(reciprocal of cumulative probability) to each value. 

The cumulative probability of the mth-order low flow 

is given by (N-\-l m)/(N-\-l). Plotting the magnitudes of low flow against their respective cumulative 

probabilities makes possible the construction of a line 

passing through the plotted points representing the 

theoretical probability distribution of low flows.

It is assumed that a theoretical probability distribution of low flow exists for each stream. The form of this 

distribution may be altered if any climatological changes 

or physical alterations of the drainage basin occur.

An empirical approach to determining the form of the 

theoretical probability distribution of low flows is to fit 

several theoretical distributions to observed data and 

to decide, by suitable criteria, which distribution fits 

the data best. This approach is facilitated by restricting the number of possible theoretical probability distributions to those which comply with practical statistical and physical considerations.

Because low flows cannot assume values less than 0, 

it is necessary that the theoretical probability distributions have lower limits equal to or greater than 0. 

Also, it is necessary that the theoretical probability distribution be defined by no more than 3 parameters. 

This is due to the fact that AT" is small so that large 

sampling errors are associated with parameters which 

are defined by the fourth and higher order moments. 

Thus the possible choices of probability distributions 

are restricted to those having lower limits equal to or 

greater than 0 and defined by no more than 3 parameters.

Several theoretical probability distributions have 

been used in hydrologic studies. In recent years GumAl
A2 STATISTICAL STUDIES IN HYDROLOGY

bel's limited distribution of the smallest value has been 

advanced as the theoretical probability distribution of 

low flows. This distribution has a finite lower limit and 

is defined bv three parameters. The Pearson Type III 

and the LoAg-Normal probability distributions have received consmerable attention in hydrology. The fact 

that these distributions have finite lower limits and can 

be defined by three parameters indicates that they may 

be applicable to low flows. A fourth theoretical probability distribution which may be applicable to low 

flows is the Pearson Type V distribution. This distribution has received very little attention in hydrology; 

however, it has a finite lower limit and can be defined 

by three parameters.

OBJECT OF THE INVESTIGATION

The investigation of low flows was undertaken to 

determine which theoretical probability distribution 

best fitted observed values of low flows, and to determine the desirability of estimating the parameters of 

theoretical probability distributions by the method of 

maximum likelihood.

In order to evaluate the various theoretical probability distributions, two criteria were considered: the 

relation between the minimum observed low flow and' 

the estimated lower limits of the theoretical probability 

distributions, and the relation between the skewness 

and kurtosis of the low-flow data!

With respect to the first criterion, a theoretical 

probability distribution was considered to be applicable 

if its estimated lower limit did not exceed the observed 

minimum low flow during the period of record. Also, 

it was considered necessary that the estimated lower 

limit not be negative.

The second criterion was introduced becajUse there is 

a unique relationship between the skewness and kurtosis 

for each of the four theoretical probability distributions. 

From the low-flow data, estimates were made of the 

skewness and kurtosis. Using the estimated values of 

skewness, the values of kurtosis were determined from 

each of the skewness-kurtosis relations and compared 

with the estimated values of kurtosis. The theoretical 

probability distribution that provided the best agreement was taken to be the most applicable to low-flow 

data.

STATISTICAL CHARACTERISTICS OF LOW FLOWS 

CUMULATIVE PROBABILITIES OF LOW FLOWS

In order to investigate the probability distribution of 

low flows, it is necessary to assign cumulative probabilities to the ordered data for each stream. It was 

previously mentioned that these cumulative probabilities are currently determined by (N-\-l m)l(N-}-l)

which is the expected cumulative probability of the 

mth-order value (the order being from largest to 

smallest) in a sample of N values. However, the 

cumulative probabilities used in these studies were the 

median values given by H. A. Thomas (written communication, 1958) which are defined by

p N(ZN+l)-2-m(3N+l) (1)

Gumbel (1954) has set forth certain criteria to which 

formulas for estimating cumulative probabilities should 

adhere. According to Gumbel, the formula (a) must 

not be dependent upon an initial distribution of the 

data, (b) must yield values within the range m/N and 

(m l)/N, (c) must provide values which are equally 

spaced in time, (d) must be analytically simple, and 

(e) must converge to 1 and 0 for ra equal to 1 and N, 

respectively, as N approaches infinity. Equation 1 

satisfies these criteria, as does the formula (N+l m)/

Cumulative probability plots were made for the low 

flows for each of the 34 streams on normal-arithmetic 

probability paper. Other types of probability paper 

can be used. However, the normal-arithmetic probability paper facilitates the analyses better than any 

other type of paper. The cumulative probability plots 

are shown in figures 1 through 34 (at end of report).

SKEWNESS AND KURTOSIS OF LOW FLOWS

Normal-arithmetic probability paper facilitates the 

use of a normal distribution since a sample from a 

normal population theoretically should plot as a straight 

line. An inspection of figures 1 through 34 shows that 

a linear plot is the exception rather than the rule. The 

nonlinearity of the plots indicates that the data can not 

be assumed to be samples from normal populations. 

However, in order to determine if the nonlinearity of the 

plots is due to chance, the skewness and kurtosis were 

determined for each set of low-flow data and tested for 

significance. For a normal population, the skewness is 

0 and the kurtosis is 3. The estimate of the skewness, 

0i, is given by

N2

(2)

In order to test the kurtosis for significance, it is 

convenient to express it in a form so that it equals 0 

for a normal population. The estimate of the kurtosis, 

02, is given by

n N*(N+1) 02=7rAr ________

S* (N-2)(N-3) (3)
PROBABILITY DISTRIBUTION OF LOW FLOWS A3

The parameters S, ra3 , and ra 4 are defined by

i

S=

and

respectively, where

g (z,-z) 5

= ]v§ (xr~

1 ^,

(4)

(5)

(6)

(7)

and X{ is the ith observed low^flow. Equations 4 

through 7 define the standard deviation, the third 

central moment, the fourth central moment, and the 

mean, respectively.

If 0! and </2 are based on the sample from a normal 

population, then their respective variances are

and

V(gj =

24

N+5

(8)

(9)

Because of sampling errors, the values of </i and g2 may 

differ from zero even if the data are from a normal 

population. If

6 "I*

and 24

(10)

(11)

where ta is the standard normal deviate corresponding 

to probability a, the data may be regarded as normally 

distributed with a confidence probability a.

The low flows for each of the 34 streams were tested 

for normality at probability levels of a equal to 0.90, 

0.95, and 0.99. For these probability levels, ta equals 

1.64, 1.96, and 2.58. This analysis is summarized in 

tables 3 and 4. With respect to the skewness, gi} the 

low flows for 19, 17, and 15 streams departed significantly from normality at probability levels 0.90, 

0.95, and 0.99, respectively. The kurtosis, however, 

did not indicate as strong a departure from normality 

as did the skewness, since the data for only 10, 9, and 7 

streams were found to be significantly nonnormal at 

probability levels 0.90, 0.95, and 0.99, respectively.

The fact that approximately 50 percent of the 

streams possess nonnormally distributed data indicates

that the low flows can not be considered as samples 

from normal populations.

DISCUSSION OF PROBABILITY DISTRIBUTIONS

CHARACTERISTICS OF THEORETICAL 

DISTRIBUTIONS

Theoretical and practical considerations indicate 

that the probability distribution of low flows is skewed 

with a lower limit equal to or greater than 0. The 

previous analysis of the skewness and kurtosis supports 

these considerations. Four theoretical probability distributions which are investigated are: (a) Gumbel's 

limited distribution of the smallest value; (b) the 

3-Parameter Log-Normal distribution, (c) Pearson's 

Type III distribution, and (d) Pearson's Type V distribution. Each of these distributions is skewed, has 

3 parameters, and has a lower limit.

GUMBEL'S LIMITED DISTRIBUTION OF THE 

SMALLEST VALUE

The limited distribution of the smallest value may be 

expressed as

? , I* * /z- /z-ev-1 r/x-eY]. (x) = ( I exp ( ) \> U \u e/ * [\U e/ J

(12)

wherein c denotes the lower limit, u denotes the characteristic drought (u is the value of x which will be 

exceeded 36.788 percent of the time), and a is a scale 

parameter (analogous to the standard deviation, s, for a 

normal probability distribution). In order to fit the 

Gumbel distribution to a set of observations, it is necessary to estimate e, u, and a. These estimates are 

functions of x, s, and m3 .

By using the method of moments (Gumbel, 1954), 

the mean, /x{, is defined as

(13)

The second, third, and fourth central moments are 

given by

a) -

a)],

and

(14)

a)l, (15)

(16)

respectively. F( ) denotes the gamma function of 

the argument within the parentheses.
A4 STATISTICAL STUDIES IN HYDROLOGY

The skewness and kurtosis are defined respectively as

M

and

_4 ~ 0 _ °

[r(i+2/a)-r2 (i+iA*)]2

(17)

(18)

#1 and G2 are used, to denote the population values of 

skewness and kurtosis, respectively, to distinguish 

them from then* sample estimates gi and g2 given by 

equations 2 and 3, respectively.

Equations 17 and 18 show that the skewness and 

kurtosis are functions of a. Hence, if GI or G2 is replaced by its respective sample estimate, </i or g2, a 

trial-and-error solution gives the value of a. Gumbel 

(1954b) has given a table prepared by G. R. Garabedian 

for the solution of equation 17. With a known, c and u 

are determined by solving equations 13 and 14 simultaneously; whereby,

and

 = J

(19)

(20)

In order to determine u and e, it is necessary to replace 

H\ and M! by their sample estimates x (eq 7) and S 

(eq 4), respectively. It should be noted that the 

sample estimates of /z 3 and AU are ra3 (eq 5) and m4 

(eq 6), respectively. 

For purposes of curve fitting, equation 12 is expressed

as

Equation 21 defines the cumulative probability of x 

obtained by integrating equation 12 from e to some 

arbitrary value of x. By letting

equation 21 becomes

(22)

(23)

Thus by using preassigned values of P(x), the corresponding values of y are determined by equation 23. 

Knowing y and using the estimates of t, u, and a, 

values of x corresponding to the preassigned values of 

P(x) are determined by equation 22. This procedure 

was applied to the low-flow data for each of the 34 

streams. The fitted curves are shown in figures 1 

through 34.

Gumbel (1954b) has pointed out that if the data are 

plotted on logarithmic extremal probability paper, a 

straight line can be fitted to the data only if e=0, 

providing, of course, that the data are in accord with 

the theory. If e>0, the fitted curve is concave downward, and if e<0, the curve is concave upward. However, in the case of low flows, e must be equal to or 

greater than 0.

3-PARAMETER LOG-NORMAL DISTRIBUTION

3-Parameter Log-Normal distribution may be expressed as

« -«-i[ln (x-a)-m]2 >> {. *<r J

(24)

where m and a denote the mean and standard deviation, 

respectively, of the values of In (x a), and a denotes 

the lower limit. Using the method of moments, the 

mean is given by

(25)

and the second, third, and fourth central moments are 

given by

(26)

and

respectively, wherein

(27)

(28)

(29)

The parameter 7 is related to the coefficient of variation, CV, which is defined as the ratio of the standard 

deviation to the mean. Thus

(30)

1+-

If a=0, then CV=y.
PROBABILITY DISTRIBUTION OF LOW FLOWS A5

The skewness, GI, and the kurtosis, G2 , are given by and the second, third, and fourth central moments are

and

M2

(31)

(32)

If either GI or G2 is replaced with its sample estimate 

9i or g2) 7 may be obtained by a trial-and-error solution. 

With 7 known, a and m are determined by solving 

equations 25 and 26 simultaneously; whereby,

and

m = \

(33)

(34)

Estimates of a and m are obtained by replacing /xj 

and M2/2 by their sample estimates x and s, respectively. 

The parameter a is estimated by

a=[ln (72+l)]1/2. (35) 

For purposes of curve fitting, the transformation

2=-[ln (x a) m] (36)

is introduced in equation 24; whereby, the cumulative 

probability, P, of x is defined as

(37)

which is identical to the standardized form of the normal distribution. From tables for the standardized 

normal distribution, values of 2 (often denoted by t) 

are obtained for preassigned values of P(z). Using 

these values of z in equation 36, the corresponding 

values of x are obtained.

PEARSON TYPE in DISTRIBUTION

The Pearson Type III distribution can be expressed 

in various ways. A representation of this distribution, 

which is well adapted to mathematical treatment, is

/OON exp ~ (38)

where a, m, and 6 are parameters. By using the method 

of moments, the mean is defined as

(39)

and

respectively. 

The skewness,

and

(40)

(41)

(42)

and the kurtosis, G2 , are given by 

^ u, 2

IF2~(6+1)* 

t4 0 __6 'NT

M2

(43)

(44)

The skewness and kurtosis are both functions of the 

parameter b. Thus by replacing either GI or Ga by its 

sample estimate g\ or g2 , b may be determined. With 

b known, the parameters ra and a can be determined 

by the simultaneous solution of equations 39 and 40 

This solution gives

and

(45)

(46)

Estimates of m and a are obtained by replacing M! and 

M2 by their Sample estimates x and s, respectively. 

If the transformations

and

u=A(t+A)

are introduced into equation 38, the cumulative probabilities are defined by

(50)

where 2,u is distributed as chi-square with 2A2 degrees 

of freedom. From a table of the chi-square distribution, values of 2u are obtained corresponding to 2A2 

degrees of freedom for preassigned cumulative probabilities. By using equation 49, values of t are obtained 

corresponding to the values of u. Values of x are 

obtained corresponding to the t values by equation 47. 

In this manner, the Pearson Type III distribution was 

fitted to the low-flow data for each of the 34 streams. 

The fitted curves are shown in figures 1 through 34.

665092 O 62-
A6 STATISTICAL STUDIES IN HYDROLOGY

PEABSON TYPE V DISTRIBUTION

The Pearson Type V distribution may be expressed as

where ra, a, and 6 are parameters. The Pearson Type 

III distribution involves the term (x m)/a, whereas 

the Pearson Type V distribution involves the reciprocal 

of this term.

By using the method of moments, the mean is defined as

(52)

and the second, third, and fourth central moments are 

given by

M2= a2 (6-2)-2 (6-3)-1 , (53) 

M3=4a3 (6-2)-3 [(6-3)(6-4)]->, (54)

and

M4=3a4 (6+4)(&-2)-*[(6-3)(6-4)(6-5)]-1 , (55)

respectively.

The skewness, Gi} and the kurtosis, G2 , are defined as

Ms 4(6-3)* . . 

^1=i^=-(6=4T (56) 

and

RfZh 1fi\' I

(57) /T 'r"l «

"2=-^ '

(6-4) (6-5)

The skewness and kurtosis are both functions of the 

parameter 6. If either GI or G2 is replaced by its same 

estimate g\ or g2 , the parameter 6 can be determined. 

With 6 known, the parameters ra and a may be 

determined by the simultaneous solution of equations 

52 and 53. This solution gives

= M!-A4/2 (6-3)*

and

(58)

(59)

Estimates of ra and a may be obtained by replacing 

M! and /i| by their sample estimates x and s, respectively. 

If the transformation

(60)

is introduced into equation 51, the cumulative probabilities are defined by

where 2,u is distributed as chi-square with 2(6 1) 

degrees of freedom. By means of a table of the chisquare distribution, values of 2it corresponding to 

preassigned values of P(u) can be obtained; whereby, 

the corresponding values of x are obtained by equation 

60. This procedure may be used to fit a Pearson Type 

V distribution to a set of data.

INVESTIGATION OF THE LOWER LIMIT

The first criterion given for the applicability of a 

theoretical probability distribution was that its lower 

limit be positive and not exceed the minimum observed 

low flow. A theoretical probability distribution underestimates the severity of the extreme droughts if its 

lower limit is greater than the minimum observed low 

flow.

In order to facilitate the investigation of the lower 

limit, the lower limit was expressed as a standardized 

variate in the form

. 

t = (62)

where xmin denotes the minimum value or lower limit. 

By using the appropriate values of xmin, u\, and u\, 

previously given for each of the theoretical probability 

distributions, the standardized lower limits for the 

Gumbel, 3-Parameter Log-Normal, Pearson Type III, 

and Pearsbn Type V distributions are given by

]-», (63)

(64)

(65)

1 )

7

and

^=-(6-3)*, (66)

respectively.

It is interesting to note that, for each theoretical 

probability distribution, the standardized lower limit 

is a function of a single parameter which defines the 

skewness. In figure 35, the relation between the absolute value of the standardized lower limit and the 

skewness is shown for each of the four theoretical 

probability distributions. Figure 35 shows that for a 

given value of skewness, tv<tLN and that tLN<^t0 or 

tnj. For values of skewness greater than 2, tni<^ta, 

and for values of skewness within the range 0.35 to 2, 

£(?<£///, and for values of skewness less than 0.35, tin<^ta .

In table 5, the standardized lower limits for each of 

the theoretical probability distributions are given for 

each of the 34 streams. These values are based on the
PROBABILITY DISTRIBUTION OF LOW FLOWS A7

estimates of xmin, MI, and /it The observed minimum 

low flow for each stream expressed in standardized form, 

t0) is given in table 5. Also, the standardized lower 

limits, denoted by tf , are given in table 5, based on the 

assumption that the observed minimum low flow is 0 

for each of the 34 streams.

From table 5, it is seen that 10 and 15 streams have 

experienced minimum discharges less than the estimated 

lower limits for the Gumbel and Pearson Type III 

distributions, respectively. The estimated lower limits for the 3-Parameter Log-Normal and Pearson 

Type V distributions are less than the observed minimum discharge for each of the 34 streams. The estimated lower limits for each of the theoretical probability 

distributions yield negative values of discharge if they 

are less than their corresponding values of t''. From 

table 5, it is seen that for 3, 4, 21, and 34 streams, the 

estimated lower limits for the Gumbel, Pearson Type 

III, 3-Parameter Log-Normal, and Pearson Type V 

distributions, respectively, yield negative values of 

discharge for the lower limit.

The above investigation of the lower limit shows 

that the Gumbel and Pearson Type III distributions 

are more applicable to low flows than either the 3- 

Parameter Log-Normal or Pearson Type V distributions, since these latter 2 distributions, although having 

lower limits smaller than the observed minimum low 

flows for the 34 streams, yield large negative values of 

discharge for the lower limits.

EFFECT OF THE LARGEST LOW FLOWS ON THE 

LOWER LIMIT

By inspecting the cumulative probability plots of the 

low-flow data for the 34 streams shown in figures 1 

through 34, it is seen that the largest low flow often 

appears to deviate appreciably from the second largest 

low flow with respect to the deviations between the 

smaller low flows. In the estimation of the skewness 

(eq 2), the third central moment, ra3 , which involves 

the cube of the departures of the low flows from then1 

mean, can be very large if the largest low flow deviates 

appreciably from the second largest low flow. Since 

the lower limit is a function of the skewness, such deviations can lead to large values of skewness, which, in 

turn, can lead to the theoretical distribution underestimating the severity of the extreme droughts.

This situation is shown by the low-flow data for the 

Kootenai River at Leonia, Idaho (fig. 14). By neglecting the largest low flow, the data plot very nearly along 

a straight line on normal-arithmetic probability paper, 

indicating that the skewness is nearly zero. However, 

the inclusion of the largest low flow yields an estimate 

of skewness of 1.85 which differs significantly from 0 

at the 99-percent level. The magnitude of the smallest

low flow is 1,070 cfs (cubic feet per second). The 

lower limits based on the Gumbel and the Pearson 

Type III distributions are 1,370 cfs and 1,372 cfs, 

respectively. Thus both theoretical probability distributions underestimate the severity of the extreme 

droughts.

A tendency for the largest low flow to deviate excessively from the other low flows was observed in the 

data for several of the streams. With respect to the 

Kootenai River at Bonners Ferry, Idaho, Little Tennessee River at Judson, N.C., South Fork of the Holston 

River near Bluff City, Tenn., and West Branch of the 

Oswegatchie River near Harrisville, N.Y., the largest 

low flow deviated sufficiently from the second largest 

low flow so that both the Gumbel and Pearson Type 

III distributions yielded lower limits greater than the observed minimum low flow. For the St. John River below 

Fish River at Ft. Kent, Maine, Coeur d'Alene River near 

Cataldo, Idaho, and Kootenay River at Newgate, British 

Columbia, the deviation between the largest and thesecond largest low flow was sufficient to reject the 

Pearson Type III distribution.

Excessive deviations were not confined to those between the largest and second largest low flows. With 

respect to the low-flow data for the Schroon River at 

River Bank, N.Y., the two largest low flows deviated 

sufficiently from the other low flows to yield lower 

limits for the Gumbel and Pearson Type III distributions greater than the observed minimum low flow.

Thus, by comparing the cumulative probability 

plots shown in figures 1 through 34 and the lower limits 

given in table 5, it is seen that in those cases where the 

Gumbel and (or) the Pearson Type III distributions 

were found to be inapplicable, the low-flow data were 

characterized by the fact that the larger low flows 

deviated excessively from the other low flows. The 

question now arises as to whether or not these deviations 

are due to chance.

In order to answer this question, it is necessary to 

consider the deviations of the larger values from the 

other values which arise in random sampling from a 

Gumbel and Pearson Type III distribution. Within 

the probability ranges which arise due to the sample 

sizes of low-flow data, the Gumbel and Pearson Type 

III distributions are nearly coincident. But because 

of the mathematical simplicity of the Pearson Type 

III distribution, the deviations of the larger values 

from the other values are analyzed on the basis of 

random sampling from a Pearson Type III distribution.

Pearson Type III distributions with skewness of 1 

(average skewness for the 34 streams), mean of 0, and 

variance of 1 were plotted on normal-arithmetic probability paper (see figs. 37-43, at end of report). From 

a table of random rectangular numbers, 2 sets of 20, 3
A8 STATISTICAL STUDIES IN HYDROLOGY

sets of 30, and 2 sets of 40 three-digit numbers were 

randomly selected. These numbers were considered as 

probabilities expressed in thousandths. The random 

numbers in each set were arranged in increasing order 

of magnitude, and the probability corresponding to 

each order of magnitude was determined by equation 1. 

For each probability determined by equation 1, the corresponding value of t was determined graphically from 

the Pearson Type III distribution. The values of t 

were then plotted against the randomly selected probabilities corresponding to the probabilities determined 

by equation 1.

In figures 37 through 43, it is seen that the t values 

plotted against their respective randomly selected probabilities cluster around the Pearson Type III distribution. With respect to each of the sample sizes, a plot 

was found where the largest value deviated excessively 

from the other values. As the sample size increases, 

these deviations tend to decrease. Although the random sampling technique does not answer the question 

conclusively as to whether or not the deviations are 

attributable to chance, it does support the notion that 

the deviations are due to chance. Therefore, the N 

values of low flow taken from N years of record may 

be regarded as belonging to the same distribution. 

This conclusion, of course, does not preclude the possibility that the deviations are significant and can be 

attributed to physical considerations of the low flows. 

However, such a position is difficult to prove due to 

the large sampling errors.

INVESTIGATION OP THE RELATION BETWEEN 

SKEWNESS AND KURTOSIS

The second criterion considered in selecting the lowflow probability distribution was the agreement between the estimated values of skewness and kurtosis 

and the theoretical relations between skewness and 

kurtosis for each of the four theoretical probability 

distributions. For each of the four theoretical probability distributions, there exists a unique relationship 

between skewness and kurtosis. From the low flow 

data, estimates of skewness and kurtosis were determined by equations 2 and 3, respectively. By using 

the estimated values of skewness, the kurtosis was 

determined from each of the four theoretical relationships of skewness and kurtosis. These values of kurtosis were then compared with the values of kurtosis 

estimated by equation 3.

The relation between skewness and kurtosis was expressed by the parameters 0i and /32 , which were defined as

Pi=Gt=n*M (67) 

and

From the previously defined expressions for GI and G2 

for each of the four theoretical probability distributions, the relations between /^ and j32 were determined. 

These relations are shown in figure 36. From figure 36, 

it is seen that for each of the four theoretical probability 

distributions, the kurtosis increases monotonically with 

an increase in skewness. Except for the Gumbel distribution, the kurtosis, j82 , is 3 when the skewness, j8 b 

is 0. Since a skewness equal to 0 and a kurtosis equal 

to 3 uniquely define the normal probability distribution, then, as the skewness approaches 0, the 3-Parameter Log-Normal', the Pearson Type III, and the 

Pearson Type V distributions converge asymptotically 

to the normal distribution.

From figure 36, it is seen that for a given value of 

skewness, the kurtosis of the Pearson Type V distribution is greater than that of the 3-Parameter Log-Normal 

distribution. Also, for a given value of skewness, the 

kurtosis of the 3-Parameter Log-Normal distribution is 

greater than that of the Gumbel and the Pearson Type 

III distributions. For values of /3i in the range 0 to 4, 

the kurtosis of the Pearson Type III distribution is 

greater than that of the Gumbel distribution, and, for 

values of Pi greater than 4, the reverse is true.

Estimates of & and /32 , denoted by fil and 02 , respecgively, were obtained for the low-flow data for each of 

the 34 streams by means of

and

3(JV-1) 2

(69)

(70)

The values of gi and g2 are given by equations 2 and 3, 

respectively. The estimated values J3i and |32 are 

shown as circles in figure 36. By inspecting figure 36, 

it is seen that the values J3i and £2 fail to cluster around 

any one of the 0i vs. /32 relations for the 4 theoretical 

probability distributions. For values of 0! in the range

0 to 1, the values of /3i and $2 are dispersed around each 

of the |8i vs. j82 lines of relation. However, for values 

of |3i>l, nearly all the values of /3i and J3 2 plot above the 

Gumbel and Pearson Type III /3i vs. /3 2 lines of relation 

and consequently above the 3-Parameter Log-Normal 

and the Pearson Type V Pi vs. /3 2 lines of relation. 

Therefore, for small values of fa, each of the 4 theoretical 

probability distributions can be used to represent the 

data; whereas, for values of /3i^>l, each of the 4 theoretical probability distributions fails to represent the 

data adequately. However, within the total range of 

the observed values of j3i and /32 , the Gumbel and the 

Pearson Type III distributions are more representative
PROBABILITY DISTRIBUTION OF LOW FLOWS A9

of the distribution of low flows than the 3-Parameter 

Log-Normal and the Pearson Type V distribution.

The mean values of /3i and /32 were found to be 1.23 

and 4.27, respectively. These values are plotted on the 

graph of the /3i vs. /3 2 relations (fig. 36). The mean 

values of 0i and /3 2 plot above the Gumbel line.

The investigation of the fi\ vs. j82 relations indicates 

that the Gumbel and the Pearson Type III distributions 

fit the observed low-flow data better than the 3-Parameter Log-Normal or the Pearson Type V distributions. 

Thus, both the first and second criteria indicate that 

the Gumbel and the Pearson Type III distributions are 

more representative of the probability distribution of 

low flows than the 3-Parameter Log-Normal and Pearson Type V distributions.

METHOD OF MAXIMUM LIKELIHOOD APPLIED TO 

LOW-FLOW PROBABILITY ANALYSIS

DISCUSSION OF THE METHOD OF MAXIMUM 

LIKELIHOOD

The preceding investigations of the probability distribution of low flows were based on the classical method 

of moments. These investigations were based on the 

comparison among various statistical parameters which 

were determined by the method of moments. An alternate method of determining the parameters is that of 

maximum likelihood. This method yields the most 

efficient estimate of a parameter when such an estimate 

exists. The most efficient estimate of a parameter is 

defined as that estimate which has the minimum variance. The most efficient estimate is not, however, 

always an unbiased estimate (Hoel, 1954). However, 

a biased estimate which has minimum variance may be 

preferred to an unbiased estimate which has a large 

variance. A property of the method of maximum likelihood which recommends it for low-flow probability 

analysis is that the lower limit of a theoretical probability distribution can never exceed the minimum 

observed value.

The investigation of the method of maximum likelihood was undertaken to determine its usefulness in 

studying the probability distribution of low flows. 

This investigation was carried out by comparing the 

variances of the maximum-likelihood estimates with the 

variances of the moment estimates of the parameters 

of the theoretical probability distributions.

MAXIMUM-LIKELIHOOD ESTIMATES OF 

PARAMETERS

The probability distribution of a random variable 

x may be expressed as

where al} a2 , . . ., ak are the parameters of the distribution. The likelihood function, defined as the natural 

logarithm of the probability function summed over all 

possible values of z, is given by

N

CM, a2 ,...,ak : x)]. (72)

The maximum-likelihood estimates of the parameters 

tti, a2 , . . ., ax are those estimates which minimize the 

likelihood function, L. These estimates are obtained 

by setting the partial derivatives of L with respect to 

each parameter equal to 0 and solving the k simultaneous equations. These equations are generally nonlinear and must be solved by some iterative process.

The variances and the mean cross products (covariances) of these parameters are obtained from the 

expected values of the second partial derivatives of L 

with respect to the parameters. If tfL/bcfi and 

tPLfoafccL) denote the second partial derivates of L with 

respect to a* and a^a^i^j), respectively, the Hessian 

determinant is

da?

daf (73)

In order to obtain the variance of at, the minor of the 

Hessian determinant corresponding to the ith parameter is divided by Hi} . Thus

Minor (52Z/da2) = H7j '

Similarly, the covariance of a< and d) is given by

n . x Minor 6lf (a<a^)= 

Ha

(74)

(75)

RELATIVE EFFICIENCY OF MOMENT ESTIMATES 

WITH RESPECT TO MAXIMUM-LIKELIHOOD ESTIMATES

The variances of the moment estimates of <LI, a2 , . . ., 

ak may be obtained by the method of variation. If 

V[ML(aJ] and V[M(al}} denote the variances of the 

maximum likelihood and moment estimates of alt 

respectively, then the relative efficiency, E, (Fisher, 

1922) between the two methods with respect to a< is 

given by

, . . ., ak : x)dx, (71)
A10 STATISTICAL STUDIES IN HYDROLOGY

Since the method of maximum likelihood yields the 

estimate of a parameter with minimum variance, then 

Q<E(di) < 1. If the variable x is normally distributed, 

the method of maximum likelihood is identical to the 

method of moments. In this case £"(«<) = 1.

The relative efficiencies are given for the parameters 

defining the Pearson Type III, the 3-Parameter LogNormal, and the Pearson Type V distributions. The 

Gumbel distribution is not treated since 52i/de2 can 

not be defined for «<2.

For the Pearson Type III distribution (eq 38), the 

relative efficiencies with respect to the parameters m, 

a, and b are

(77)

E(a)=l-6+4' (78)

and

(79)

where d2 In T(b-\-l)/db2 is the tri-gamma function of 

(6+1). The relative efficiency of each parameter is 

a function of only b, which itself is a function of the 

skewness. In table 6, the relative efficiencies of m, a, 

and b are given for various values of skewness.

With respect to the 3-Parameter Log-Normal distribution (eq 24), the relative efficiencies of a, m, and v are

In (72+l)

V(72+l)3[ln

____In (72+l)[ln (72+l)+l](72+D_______

(72+l) + l](72+l)-l}[(72+l) 4 +2(72+l) 3 +3(72+l) 2 -4]'

(80)

(81)

and

(36{[ln (72+l)](72+l)+72 } {[hi (72+l)]2 (72+l) 4 } {72 (72+l) 15 -12(72+l) 13 +36(72+l) n-(672 +24)(72+l) 10 

+60(72+l) 9-180(72+l) 7+1572 (72+l) 6 -[2478-9(73 +37) 2 -120](72+l) 4 -[4878 -18(73 +37)V+2072 

+288](72+l) 3 -[7278 -27(73+37)V+360](72+l) 2 +(1572 +384)(72+l)-572 +7278+8(73 +37) 278 +8} J

72 {[ln

(82)

The relative efficiency of each parameter is a function For the Pearson Type V distribution (eq 51), the 

of 7, which itself is a function of the skewness. The relative efficiencies of the parameters m, a, and b are 

relative efficiencies of each parameter for various values given by 

of skewness are given in table 7.

(6-2)2(6-3) 

=

(6+1) (6-4) (6-5)

(83)

(84)

and

326(6-1) (6-2)2 20(362 +746-40) (6-3) 2 

(6-4) (6-5) (6-6) (6-7)

72(6-3)(6+4) , 432(6-3) 2 (4+4) 384(6-3) 2 (56+2) , 560(6-3) , oql

~r, 

(6-4)(6-5) ' (6-4)3 (6-5) (6-4) 2 (6-5)(6-6) ' (6-4)

.p^ TOD (85)
PROBABILITY DISTRIBUTION OF LOW FLOWS All

The relative efficiency of each parameter is a function 

of 6, which itself is a function of the skewness. In 

table 8 relative efficiencies of each parameter are given 

for various values of skewness.

From equations 77 through 85 and tables 6 through 

8, it is seen that the relative efficiency of the maximumlikelihood estimate to the moment estimate of each 

parameter varies inversely with the skewness; that is, 

the relative efficiency decreases as the skewness increases. The average value of skewness of the low-flow 

data for the 34 streams was found to be approximately 

1. For this value of skewness, the relative efficiencies 

of the parameters are less than 0.7. The parameters 

measuring the skewness for the probability distributions 

have relative efficiencies less than 0.2. By using 0.5 as 

the average relative efficiency for all the parameters of 

a given probability distribution, the method of moments 

can be considered as using only half of the available 

information in a given record of size N.

APPLICATION OF THE METHOD OF MAXIMUM 

LIKELIHOOD

The method of maximum likelihood minimizes the 

effect of the larger values on the lower limit, since this 

method gives less weight to the larger values than the 

method of moments. Lessening the weight of the larger 

values reduces the estimated skewness. To demonstrate the application of the method of maximum likelihood, the low-flow data for the Kootenai River at 

Leonia, Idaho, were fitted with a Pearson Type III distribution based on maximum-likelihood estimates of the 

parameters of the distribution. This set of data was 

used because it is a good example of the situation where 

the largest value deviates excessively with the other 

values.

In order to fit the Pearson Type III distribution to 

the data by the method of maximum likelihood, it is 

necessary to solve the following three simultaneous nonlinear equations for the values of m, a, and b.

 i

om i^{

<-w )=0'

(86) ^

(87)

 ^. i

do db

The solution of this set of simultaneous equations was 

obtamed by trial and error. The comparison of the 

maximum likelihood estimates m, a, and b with the 

moment estimates of m, a, and 6 is summarized in 

table 9.

From table 9, it is seen that the maximum-likelihood 

estimate of 6 is larger than the moment estimate of 6. 

Since b varies inversely with the skewness, the method 

of maximum likelihood gives an estimate of the skewness 

which is less than that given by the method of moments. 

This reduction of the estimate of the skewness leads to 

a reduction in the estimate of the lower limit. The 

minimum observed low flow for the Kootenai River at 

Leonia, Idaho, was 1,070 cis. By the method of moments, the lower limit was found to be 1,348 cfs; 

whereas, by the method of maximum likelihood, the 

lower limit was found to be 809 cfs.

SUMMARY AND CONCLUSIONS 

PROBABILITY DISTRIBUTION OF LOW FLOWS

For purposes of estimating low flows having certain 

probabilities of occurrence and for making tests of 

significance of such estimates, it is necessary to determine the underlying probability distribution of the 

data. The low-flow data for 34 streams were found to 

be nonnormally distributed with an average skewness 

of approximately 1. Four theoretical probability distributions were investigated: (a) Gumbel's limited 

distribution of the smallest value, (b) 3-Parameter LogNormal, (c) Pearson Type III, and (d) Pearson Type V. 

Each of these distributions is skewed, has a finite lower 

limit, and is defined by three parameters.

In order to evaluate the applicability of these four 

theoretical probability distributions to low-flow data, 

investigations were made relative to two criteria. The 

first criterion was based on the relation between the 

observed minimum low flow and the lower limits of the 

theoretical probability distributions. A theoretical 

probability distribution was considered to be inapplicable if either the minimum observed low flow was less 

than the lower limit or if the lower limit was negative. 

The second criterion was based on the observed relation between skewness and kurtosis with respect to the 

theoretical relations between skewness and kurtosis. 

A unique relation between skewness and kurtosis exists 

for each of the theoretical probability distributions. 

The theoretical probability distribution whose skewnesskurtosis relation best fitted the observed values of skewness and kurtosis was considered as most representative 

of the probability distribution of low flows.

On the basis of these two criteria, it was found that 

the Gumbel and the Pearson Type III distributions 

were more representative of the probability distribution 

of low flows than either the 3-Parameter Log-Normal 

or the Pearson Type V distributions. Within the range 

of cumulative probabilities (based on the lengths of 

low-flow records), the Gumbel and Pearson Type III
A12 STATISTICAL STUDIES IN HYDROLOGY

distributions are nearly coincident. Thus the lowflow data can be represented equally well by either 

distribution.

The lower limits of the four theoretical probability 

distributions are functions of the skewness and vary 

directly with the skewness. Hence, for large values of 

skewness, the lower limits are likely to exceed the 

observed minimum low flow. In those cases where 

the lower limits of the Gumbel and the Pearson Type 

III distributions exceeded the observed minimum low 

flows, the large values of skewness were attributable 

to excessive deviations of the larger low flows from the 

other values. By means of random sampling from a 

Pearson Type III distribution having a skewness of 1, 

it was concluded that such excessive deviations can be 

attributed to chance.

METHOD OF MAXIMUM LIKELIHOOD

The above investigations were based on the comparison of parameters estimated by the method of 

moments. An alternate method of estimating the 

parameters is that of maximum likelihood. The 

method of maximum likelihood provides estimates of 

the parameters such that their variances are a minimum. Also, the method of maximum likelihood 

provides estimates of the lower limits, which are always . 

less than the minimum observed value.

The increase in information obtained by using the 

method of maximum likelihood was investigated by 

comparing the variances of the moment estimatesNwith 

the maximum-likelihood estimates ot the parameters 

of the theoretical probability distributions. The ratio 

of the variance of the maximum-likelihood estimate 

to the variance of the moment estimate is defined as 

the relative efficiency between the two methods. This 

ratio is a function of and varies inversely with the 

skewness.

The average value of skewness for the low-flow data 

of 34 streams was found to be 'approximately 1. For 

this value of skewness, the average relative efficiency 

for the three parameters of each theoretical probability 

distribution was found to be less than 0.5. Thus the 

method of moments utilizes less than half of the available information in a set of low-flow data.

Maximum-likelihood estimates of the parameters 

of the theoretical probability distributions require 

the solution of three simultaneous nonlinear equations. 

These solutions are difficult to obtain since they require 

an iterative process; however, this difficulty can be overcome by the use of high-speed electronic computors. 

The small relative efficiency of the method of moments 

with respect to the method of maximum likelihood indicates that future studies dealing with the probability 

distribution of low flows might best be made by using 

the method of maximum likelihood. This is particularly true when parameters involving high-order 

moments are used.

TABLE 1. Records of annual minimum daily low flows

Gaging station

Middle Branch Westfleld River near Qoss Heights,

Pemigewasset River at Plymouth, N.H.. _____

Saco River near Cornish, Maine ___________

St. John River below Fish River at Fort Kent, Maine.

Little River near Princeton, N.C __________

Fishing Creek near Enfleld, N.C.-... __ ... ___

East Fork of Deep River near High Point, N.C.-...

Cape Fear River at Lillington, N.C.. _______ ..

French Broad River at Asheville, N.C ..............

Coeur d'Alene -River near Cataldo, Idaho ............

1 Kootenai River at Leonia, Idaho... ..... .........

. Kootenay River at Newgate, British Columbia ___ 

A North Fork Stillaguamish River near Arlington, 

Wash..............................................

South Fork Stillaguamish River near Granite Falls, 

Wash.i___.__._____. _______________________ ........

South Fork Skykomisk River near Index, Wash. .... 

Sauk"~River at Sauk, Wash... .......................

Period 

of 

record

1910-55

1903-55

191&-50

1926-50 

1909-55

1930-53

1930-53

1Q9fl *\3

1928-53 

1927-52

1927-52

1903-53

1923-50

1928-50

1928-50

1930-50 

1928-50

1928-50

1914-50 

1930-50

Length 

ofrecord 

(years)

45

52

34

24 

46

23

239<;

25 

25

25

50

27

22

22

20 

22

22

36 

20

Drainage 

area 

(sq mi)

52.6

522

1,298

5,690 

171

229

521

168

14.7 

1,140

3,440

950

1,220

13,000

11,740

7,660 

269

199

355 

714

TABLE 2. Records of annual minimum 7-day low flows

Gaging stations

East Branch Delaware River at Fish's Eddy, N.Y ... 

West Branch Delaware River at Hale's Eddy, N. Y^ _ 

Greenbrier River at Alderson, W. Va ................

Toccoa River near Dial, Ga... ____________

Schroon River at River Bank, N.Y ..................

South Fork of Holston River near Bluff City, Tenn.. 

Brandywine River at Chadd's Ford, Pa...... .......

Hoosic River near Eagle Bridge, N.Y ................

West Branch of Oswegatchie River near Harrisville, 

N.Y......................................... .....

Period 

of 

record

1898-50

1919-52

1896-41

1913-52

1913-52 

1913-52 

1907-54

1913-55

1912-54

1908-53

1901-49 

1911-54

1911-53

1916-53

Length 

of record 

(years)

52

40

46

40

40 

40 

48

43

43

46

49 

44

43

38

Drainage 

area 

(sq mi)

2,084

2,240

664

1,492

783 

593 

1,357

177

491

527

813

 287

510

258
PROBABILITY DISTRIBUTION OF LOW FLOWS A13

TABLE 3. Coefficients of skewness of low-flow data with 90-percent, 

96-percent, and 99-percent confidence limits

TABLE 4. Coefficients of kurtosis of low-flow data with 90-percent, 

95-percent, and 99-percent confidence limits

Gaging stations

Middle Branch of Westfleld River near

Pemigewasset River at Plymouth, N. H.

St. John below Fish River at Fort Kent,

Souhegan River near Merrimack, N. H..

East Fork of Deep River near High 

Point, N. C...... .....................

Cape Fear River at Lillington, N. C.... 

French Broad River at Asheville, N. C._ 

Coeur d'Alene River near Cataldo,

Kootenai River near Bonners Ferry,

Kootenay River at Newgate, British

North Fork Stillaguamish River near

South Fork Stillaguamish River near 

Granite Falls, Wash........ _ ........

South Fork Skykomish River near InSusquehanna River at Conklin, N.Y .... 

Little Tennessee River at Judson, N.C.. 

Chenango River near Chenango Forks, 

N.Y...................................

East Branch of Delaware River at Fish's 

Eddy, N.Y...........................

West Branch of Delaware River at 

Bale's Eddy, N.Y....................

Greenbrier River at Alderson, W. Va... 

Toccoa River near Dial, Oa. ......... ...

Schroon River at River Bank, N.Y..... 

South Fork of Holston River near Bluff 

City, Tenn............................

Brandywine River at Chadds Ford, Pa_. 

Hoosic River near Eagle Bridge, N.Y .... 

West Branch of Oswegatchie River near 

Harrisville, N.Y......................

0\

1.38 

.292 

.354

.693 

1.13 

.602 

.050 

.551

.045 

.806 

1.22 

1.62

.542

1.45 

1.85

.721 

.581 

.725

.877 

.731 

.426 

.670 

1.75

.107 

1.19

1.27 .80' 

1.31 

1.08 

1.28

2.50 

.236 

1.53

2.15

S.E. 0,

0.354 

.330 

.403

.471 

.349 

.480 

.480 

.455

.463 

.463 

.463 

.336

.447

.490 

.490

.511 

.490 

.490

.392 

.511 

.330 

.350 

.350

.373 

.373

.373 

.343 

.360 

.360 

.350

.339 

.358 

.360

.382

Confidence limit

90-percent 

1.64 

S.E. 0i

0.582 

.543 

.663

.775 

.574 

.790 

.790 

.748

.762 

.762 

.762 

.553

.735

.806 

.806

.841 

.806 

.806

.645 

.841 

.543 

.576 

- .576

.614 

.614

.614 

.564 

.592 

.592 

.576

.558 

.589 

.592

.628

95-percent 

1.96 

S.E. (7i

0.694 

.646 

.790

.921 

.683 

.941 

.941 

.892

.905 

.905 

.905 

.658

.875

.960 

.960

1.000 

.960 

.960

.769 

1.000 

,646 

.686 

.686

.730 

.730

.730 

.672 

.705 

.705 

.686

.664 

.701 

.705

.749

99-percent 

2.58 

S.E. 01

0.913 

.851 

1.040

1.215 

.900 

1.238 

1.238 

1.174

1.195 

1.195 

1.195 

.867

1.153

1.264 

1.264

1.318 

1.264 

1.264

1.011 

1.318 

.851 

.903 

.903

.962 

.962

.962 

.885 

.929 

.929 

.903

.875 

.924 

.929

.986

Gaging stations

Middle Branch of Westfleld River near

Pemigewasset River at Plymouth, N.H.

St. John below Fish River at Fort Kent,

Souhegan River near Merrimack, N.H.. 

Little River near Princeton, N.C _ . ....

Fishing Creek near Enfleld, N.C.. ......

East Fork of Deep River near High 

Point, N.C... ...... ....................

Cape Fear River at Lillington, N.C ..... 

French Broad River at Asheville, N.C-. 

Coeur d'Alene River near Cataldo,

Kootenai River near Bonners Ferry, 

Idaho.. _____________ ......

Kootenay River at Newgate, British

North Fork Stillaguamish River near

South Fork Stillaguamish River near

South Fork Skykomish River near

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

Little Tennessee River at Judson. N.C.. 

Chenango River near Chenango Forks, 

N.Y...... .............................

East Branch of Delaware River at Fish's 

Eddy, N.Y...........................

West Branch of Delaware River at 

Kale's Eddy, N.Y....................

Greehbrier River at Alderson, W. Va... 

Toccoa River near Dial, Oa. ....... _ .

Sacandaga River near Hope, N.Y .......

Schroon River at River Bank, N.Y.. ... 

South Fork of Holston River near Bluff 

City, Tenn . ..... .............

Brandywine River at Chadds Ford, Pa. 

Hoosic River near Eagle Bridge, N.Y... 

West Branch of Oswegatchle River near 

Harrisvme, N.Y... ...................

at

1.54 

-.11 -.08

.90 

.83 

-.20 

-1.04 

1.72

-.38 

-.52 

.02 

.06

.71

3.55 

6.40

1.36 

.02 

-.59

.33 

.42 

.64 

.27 

-2.33

-.83 

.73

1.04 

-.11 

1.50 

.93 

-2.14

9.65 

-.49 

2.29

7.09

S.E.02

0.693 

.649 

.786

.909 

.687 

.926 

.926 

.894

.894 

.894 

.894 

.660

.865

.942 

.942

.980 

.942 

.942

.766 

.980 

.645 

.687 

.687

.730 

.730

.730 

.673 

.707 

.707 

.687

.668 

.700 

.707

.748

Confidence limit

90-percent 

1.64 

S.E.f/j

1.140 

1.068 

1.293

1.495 

1.130 

1.523 

1.523 

1.471

1.471 

1.471 

1.471 

1.086

1.423

1.550 

1.150

1.612 

1.550 

1.550

1.260 

1.612 

1.061 

1.130 

1.130

1.201 

1.201

1.201 

1.107 

1.163 

1.163 

1.130

1.099 

1.152 

1.163

1.230

95-percent 

1.96 

S.E.02

1.358 

1.272 

1.541

1.782 

1.347 

1.815 

1.815 

1.752

1.752 

1.752 

1.752 

1.294

1.695.

1.846 

1.846

1.921 

1.846 

1.846

1.501 

1.921 

1.264 

1.347 

1.347

1.431 

1.431

1.431 

1.319 

1.386 

1.386 

1.347

1.309 

1.372 

1.386

1.466

99-percent 

2.58 

S.E.02

1.788 

1.674 

2.028

2.345 

1.772 

2.389 

2.389 

2.307

2.307 

2.307 

2.307 

1.703

2.232

2.430 

2.430

2.528 

2.430 

2.430

1.976 

2.528 

1.664 

1.772 

1.772

1.883 

1.883

1.883 

1.736 

1.824 

1.824 

1.772

1.723 

1.806

1 .. 824

1.950
A14 STATISTICAL STUDIES IN HYDROLOGY

TABLE 5. Observed and theoretical standardized low limits

Gaging stations

Middle Branch of Westfleld River near

PemigewassetRiveratPlymouth.N.H ...

St. John below Fish River at Fort Kent, 

Maine..------.-.. _ _ -------- _ ... 

Souhegan River near Merrimack, N.H. 

Little River near Princeton, N.C _ ^..

Fishing Creek near Enfield, N.C.......

East Fork of Deep River near High 

Point, N.C... .....................

Neuse River near Clayton, N.C.... ....

Cape Fear River at Lillington, N.C- _ 

French Broad River at Asheville, N.C. 

Coeur d'Alene River near Cataldo, 

Idaho _ . _______________

Kootenai River near Bonners Ferry, 

Idaho..... __________ . .......

Kootenay River at Newgate, British

North Fork Stillaguamish River near

South Fork Stillaguamish River near 

Granite Falls, Wash.... ..............

South Fork Skyomish River near Index, 

Wash...... ...........................

Sauk River near Sauk, Wash. ..........

James River at Buchanan, Va ..........

Susquehanna River at Conklin, N.Y... 

Little Tennessee River at Judson, N.C... 

Chenango River near Chenango Forks, 

N.Y........ .......................

East Branch of Delaware River at 

Fish's Eddy, N.Y...................

West Branch of Delaware River at 

Hale's Eddy, N.Y.............. ....

Greenbrier River at Alderson, W.Va...

Sacandaga River near Hope, N.Y. __ 

Schroon River at River Bank, N.Y. ... 

South Fork of Holston River near Bluff 

City, Tenn. ................. ....

Brandy wine River at Chadds Fork, Pa. 

Hoosic River near Eagle Bridge, N.Y.. 

West Branch of Oswegatchie River near

-t.

1.33

2.36 

1.74

1.75 

1.26 

1.58

1.58

1.51

1.99

1.34

1.23 

1.76

1.89

1.39

1.45

1.78

1.66

1.39

1.62

1.72

1.54

1.59 

1.70

1.85

1.11

1.07

1.52 

1.70

1.45 

1.40

1.09

1.71 

1.33

1.12

-to

'1.3

2.5 

2.4

1.85 

1.42 

2.00

3.1

9 ni

3.15

1.70

1.35 

11.15

2.05

11.22

1 1 03

1.82

2.00

1.82

1.65

1 80

2.25

1.88 

11.10

2.95

1.38

1.95

1.74 

!1.30

1.46 

11.32

!.86

2.6 

11.17

1.95

-tm

11.2

2.6 

2.4

i 1.7 

1.3 

1.8

6.3

1.9

6.6

1.55

1.28 

11.11

1.9

11.18

11.05

11.65

1.82

1.65

11.5

11.63

2.15

1.7 

11.08

4.3

1.3

1.25

1.56 

11.23

iJ.35 

11.25

 0.89

2.9 

11.15

i 0. 96

-t,.N

2 31

10.5 

8.6

4.4 

2.78 

5.00

60

5.50

.67

3.75

2.69 

2.0

5.60

2.25

1 80

4.20

5.20

4.20

3.50

4.15

7.20

4.45 

1.90

28

2.65

2.50

3.80 

2.45

2.90 

2.50

1.40

12.5 

2.15

1.58

-tv

4.20

13.5 

11.5

6.40 

4.50 

7.20

81

7 sin

89

5.7

4.3 

3.75

7.8

4.1

3.5

6.2

7.4

6.2

5.3

6.1

9.8

6.5 

3.6

38

4.4

4.2

7.1 

4.2

4.64 

4.2

3.2

17 

3.9

3.35

-t'

i qq

3.35 

3.46

3.14 

3.08 

1.66

1.88

9 fifi

3.79

1.90

1.59 

2.62

3.97

3.16

2.88

4.49

3.93

3.02

4.24

4.53

4.15

2.46 

2.98

2.97

1.90

1.98

1.77 

2.97

2.00 

2.58

2.85

2.96 

2.92

2.20

TABLE 6. Relative efficiency of moment estimates with respect to 

maximum-likelihood estimates Pearson Type III distribution

-t'

1.33 

3.35 

3.46

3.14 

3.08 

1.66 

1.88 

2.08

3.79 

1.90 

1.59 

2.62

3.97

3.16 

2.88

Oi

2 

I.... ... ...... ...-........

.3........................................

P

0 

3 

43.44

E(m)

0.000 

.500 

.954

E(a)

0.250 

.571 

.937

E(p)

0.000 

.154 

.821

TABLE 7. Relative efficiency of moment estimates with respect to 

maximum-likelihood estimates 8-Parameter Log-Normal distribution

Gi

2

1...... ..... .................

.3...... ........ ........... ...............

7

0.6 

.3 

.1

E(a)

0.259 

.683 

.958

E(m)

0.280 

.624 

.936

EM

0.003 

.006 

.874

TABLE 8. Relative efficiency of moment estimates with respect to 

maximum-likelihood estimates Pearson Type V distribution

0,

2......... . ............... ........

1 

.3......... .........

b

9

21

183

E(m)

0.371

.672

.957

-E(o)

0.210

.542

.982

E(b)

0.002

.005

.040

TABLE 9. Comparison of parameters computed by the methods of 

maximum-likelihood and moments

1 Theoretical lower limit is greater than observed minimum low flow.

Maximum likelihood..

Pearson Type III

m

809 

1348

a

360 

695

P

2.755 

.169

L

-141.28 

-172. 30

3-Parameter Log-Normal

m

7.38 

7.09

a

435 

794

<r

0.395 

.514

LLN

-173.02 

-173. 57
PROBABILITY DISTRIBUTION OF LOW FLOWS. A15

99.9

95

» 50

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type III

O.I

99.9

6 8 10 

Discharge, in cfs

14 16 18

FIGURE 1. Middle Branch Westfleld River at Goss Heights, Mass.

99.9

95

50

O.I

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

100 200 300 400 500 600 700 800 900 

Discharge, in cfs

95

5 50

99.9

95

~ 50

O.I

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

J____I

Pearson type

J___J________J.

0 40 80 120 160 200 240 280 320 360 

Discharge, in cfs

FIGURE 2: Pemigewasset River at Plymouth, N.H.

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

500 1000 1500 2000 2500 3000 3500 4000 

Discharge, in cfs

FIGURE 3. Saco River near Cornish, Maine. | FIGURE 4. St. John River below Fish River at Fort Kent, Maine. 

LOW-FLOW PROBABILITY DISTRIBUTION
A16 STATISTICAL STUDIES IN HYDROLOGY

99.9

95 

 ? 50

O.I

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

10 15

99.9

20 25 30 35 40 45 50 

Discharge, in cfs

95

50

O.I

10

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

20 30 40 50 60 

Oischorge, in cfs

70

FIGURE 5. Souhegan River near Merrimack, N.H. FIGURE 6. Little River near Princeton, N.C.

80 90

99.9

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

40 60 80 100 120 140 160

Discharge, in cfs

99.9

180

95

50

O.I

10

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

20 30 40 50 60 

Discharge, in cfs

70 80 90

FIGURE 7. Fishing Creek near Enfield, N.C. FIGURE 8. Haw River near Benaja, N.C. 

LOW-FLOW PROBABILITY DISTRIBUTION
PROBABILITY DISTRIBUTION OF LOW FLOWS A17

99.9

95

50

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Pearson type

I 2345678 

Discharge, in cfs

FIGUKE 9. East Fork of Deep River near High Point, N.C.

99.9

95

 '- 50

O.I

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

0 100 200 300 400 500 600 700 800 900 

Discharge, in cfs

FIGURE 10. Neuse River near Clayton, N.C.

99.9

95

.2 50

O.I

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

99.9

100 200 300 400 500 600 700 800 900 

Discharge, in cfs

95

 50

5 

O.I

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

0 200 400 600 800 1000 1200 1400 1600 1800 

Discharge, in cfs

FIGURE 11. Cape Fear River at Lillington, N.C. FIGURE 12. French Broad River near Asheville, N.C. 

LOW-FLOW PROBABILITY DISTRIBUTION
A18 STATISTICAL STUDIES IN HYDROLOGY

99.9

95

>50

O.I

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

99.9

50 100 150 200 250 300 350 400 450 

Discharge, in cfs

FIGURE 13. Coeur d'Alene River near Cataldo, Idaho.

95

50

O.I

EXPLANATION

Q

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

1000 2000 3000 4000 5000 6000 7000 8000 9000 

Discharge, in cfs

FIGURE 14. Kootenai River near Leonia, Idaho.

99.9

95

> 50

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

O.I

99.9

1000 2000 3000 4000 5000 6000 7000 8000 9000 

Discharge, in cfs

95

£50

O.I

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

0 500 1000 1500 2000 2500 3000 3500 4000 4500 

Discharge, in cfs

FIGURE 15. Kootenai River at Bonner's Ferry, Idaho. FIGURE 16. Kootenay River near Newgate, British Columbia.

LOW-FLOW PROBABILITY DISTRIBUTION
PROBABILITY DISTRIBUTION OF LOW FLOWS A19

99.9

95

S 50

E

O.I

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

99.9

0 50 100 150 200 250 300 350 400 450 

Discharge, in cfs

FIGURE 17. North Fork Stillaguamish River near Arlington, Wash.

95 

? 50

EXPLANATION

o>

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

0 50 100 150 200 250 300 350 400 450 

Discharge, in cfs

FIGURE 18. South Fork Stillaguamish River near Granite Falls, Wash.

99.9

95

50

O.I

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

99.9

0 100 200 300 400 500 600 700 800 900 

Discharge, in cfs

95

o. 50

O.I

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

500 1000 1500 2000 2500 3000 3500 4000 

Dischorge, in cfs

FIGUKE 19. South Fork Skykomish River near Index, Wash. FIGURE 20. Sauk River at Sauk, Wash.

LOW-FLOW PROBABILITY DISTRIBUTION
A20 STATISTICAL STUDIES IN HYDROLOGY

99.9

95

 50

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

99.9

100 200 300 400 500 600 700 800 900 1000 

Discharge, in cfs

FIGURE 21. James River at Buchanan, Va.

Observed discharge plotted against 

estimated cumulative probability

100 200 300 400 500 600 700 800 900 1000 

Discharge, in cfs

FIOUBE 22. Susquehanna River at Cinklin, N. Y.

99.9

95

550

EXPLANATION

0

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

99.9

O.I

200 400 600 800 1000 1200 1400 1600 1800 2000 

Discharge, in cfs

95

O.I

EXPLANATION

0

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

100 200 300 400 500 600 700 800 900 

Discharge, in cfs

FIGUBE 23. Little Tennessee River at Judson, N.C. FIOUBE 24. Chenango River near Chenango, N.Y.

LOW-FLOW PROBABILITY DISTRIBUTION
PROBABILITY DISTRIBUTION OF LOW FLOWS A21

99.9

95

a, 50

O.I

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

I I

99.9

0 100 200 300 400 500 600 700 800 900 

Discharge, in cfs

FIGURE 25. East Branch Delaware River at Fish's Eddy, N.Y.

95

50

O.I

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type III

0 50 100 150 200 250 300 350 400 450 

Discharge, in cfs

FIGURE 26. West Branch Delaware River at Hales Eddy, N.Y.

99.9

95

50

O.I

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type III

99.9

50 100 150 200 250 300 350 400 450 

Discharge, in cfs

95

> 50

O.I

50

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

100 150 200 250 300 350 400 450 

Discharge, in cfs

FIGURE 27. Greenbrier River at Alderson, W. Va. FIGURE 28. Toccoa River near Dial, Ga.

LOW-FLOW PROBABILITY DISTRIBUTION
A22 STATISTICAL STUDIES IN HYDROLOGY

99.9

95

50

O.I

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

99.9

50 100 150 200 250 300 350 400 450 

Discharge, in cfs

FIGURE 29. Sacandago River near Hope, N.Y.

95 

5 50

O.I

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

50 100 150 200 250 300 350 400 450 

Discharge, in cfs

FIGURE 30. Schroon River at River Bank, N.Y.

99.9

95

® 50

O.I

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

99.9

0 100 200 300 400 500 600 700 800 900 

Discharge, in cfs

95

50

O.I

EXPLANATION

o

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

50 100 150 200 250 300 350 400 450 

Dischorge, in cfs

FIGURE 31. South Fork of Holston River near Bluff City, Tenn. FIGURE 32. Branywine River at Chadd's Ford, Pa.

LOW-FLOW PROBABILITY DISTRIBUTION
PROBABILITY DISTRIBUTION OF LOW FLOWS A23

99.9

95

50 EXPLANATION

©

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

O.I

99.9

95

50

50 100 150 200 250 300 350 400 450 

Discharge, in cfs

O.I

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

Gumbel

Pearson type

30 50 70 90 110 130 150 170 

Discharge, in cfs

190 210

FIGURE 33. Hoosic River near Eagle Bridge, N.Y. FIGURE 34. West Branch of Oswegatchie River near Harrison, N.Y.

LOW-FLOW PROBABILITY DISTRIBUTION 

10.0

EXPLANATION

The values of / are oil minus

G -Gumbel

LN-3-Parameter Log-Normal 

HI-Pearson Type HI 

3E- Pearson Type 3E

-5 -10 

Standardized lower limits,?

-100

FIGURE 35. Relation between skewness and standardized lower limits for theoretical 

probability distributions.
A24 STATISTICAL STUDIES IN HYDROLOGY

I 234567

i i i i r i i

EXPLANATION

Observed discharge plotted against 

°© estimated cumulative probability

o

o G

Gumbel

n

Pearson type III

LN 

3-Parameter Log-Normal

Pearson Type3£

FIGURE 36. Relation between skewness and kurtosis for theoretical probability

distributions.
PROBABILITY DISTRIBUTION OF LOW FLOWS A25

99.9

p 50 

Observed discharge plotted against 

estimated cumulative probability

99.9

95

P50

O.I

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

-4 -3 -2 -I 0 I

t

FIGURE 37. N=20. FlOUBE 38. #=20.

99.9

95

P 50

O.I

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

-4 -3 -2 -I

99.9

95

P 50

O.I

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

-4 -3 -2

FIGURE 39. N=ZO. FIGURE 40. N=30.

RANDOM SAMPLING FROM PEARSON TYPE III 

DISTRIBUTION
A26 STATISTICAL STUDIES IN HYDROLOGY

99.9 99.9,

95

P 50

O.I

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

-4 -3 -2

95

P 50

O.I

EXPLANATION

Observed discharge plotted against 

a estimated cumulative probability

-4 -3 -2 -I

FIGURE 41. JV=30. 

99.9

FIGURE 42. 

95

P 50

O.I

-4 -3 -2

EXPLANATION

Observed discharge plotted against 

estimated cumulative probability

FIGURE 43. N=40.

RANDOM SAMPLING FROM PEARSON TYPE III 

DISTRIBUTION
PROBABILITY DISTRIBUTION OF LOW FLOWS , A27

REFERENCES

Fisher; R. A., 1922, On the mathematical foundations of theo- Gumbel, E. J., 1954b, Statistical theory of droughts: Am. Soc.

retical statistics: Philos. Trans. A222, p. 309-368. Civil Engineers Proc., Separate 439.

Gumbel, E. J., 1954a, Statistical theory of extreme values and ^ p Q 19g4 Introduction to mathematicai statistics: Wiley,

some practical applications: Natl. Bur. Standards Appl. ^ York N Y n 203

Math. Ser. 33, p. 14. » » PU. S. GOVERNMENT PRINTING OFFICE : 1963 O - 685092