On flood probabilities of East Alpine rivers

Journal of Hydrology, 20 (1973) 6 5 - 8 2 © North-Holland Publishing Company, Amsterdam - Printed in The Netherlands

ON FLOOD PROBABILITIES OF EAST ALPINE RIVERS K. CEHAK

Zentralanstalt fi~r Meteorologie und Geodynamik, Vienna {Austria) (Accepted for publication February 9, 1973)

ABSTRACT Cehak, K., 1973. On flood probabilities of East Alpine rivers. J. Hydrol., 20: 6 5 - 8 2 . The flood data of five stations in the East Alps with relatively long observation series are represented by means of Fisher-Tippett II or II1 distributions. The observations of the longest observation periods are fitted well by the type I1 curves. A comparison of calculations with only a part of the observations shows that the magnitude of the curvature parameter k of the Fisher-Tippett curves increases with increasing observation period. The series at hand are not long enough to see if there is a convergence to a limiting value of k. A trend test showed an increasing trend in all of the Danube stations after about 1860 and nearly no trend at all in the observations from the southern rivers. The variance spectra yielded the result that the series may be considered as realizations of autoregressive processes of the first order. Superposed to the red-noise spectra there are waves with periods of 2.35 and 8 years in the case of the Danube stations and with 5.7 and 6.7 years in the case of the southern rivers.

CONSIDERATIONS ON THE CHOICE OF THE DISTRIBUTION FUNCTION

The extraordinary practical value of the connaissance of flood probabilities has led to a large number of trials to use a whole spectrum of more or less well suited types of distribution functions for the representation of this important hydrological element. In various papers the use of, e.g., log-normal (Grassberger, 1932), Fisher-Tippett I, II or III (Gumbel, - 1958; Cehak, 1964), Pearson III (Glos, 1966) or log-Pearson III (Schreiber, 1970) has been advocated and their relative merits have been discussed by different authors on the occasion of hydrological conferences (e.g., Liebscher, 1970; Mendel, 1972). It is very difficult to give theoretically sound tests for the selection of the right distribution function. As, e.g., the paper of Liebscher (1970) has shown, the criteria for the goodness of fit often differ not very much for various kinds of distribution functions applied to the same data. The differences of these criteria would not be statistically significant, if one would compare them by some test method. This depends also on the river, which is investigated and also on the period of observations used.

66

K. CEHAK

Taking this finding into consideration it seems advisable to use perhaps in some cases a distribution function which gives larger deviations from the observed values than another function, but which allows for a statistical or physical background for its use. From this deliberation one is immediately led to use one of the three forms of asymptotic extreme-value distributions, i.e., one of the three Fisher-Tippett types of distribution, because only these three distributions obey the stability postulate, which an extreme-value distribution has to do (Gnedenko, 1943). In former papers (Cehak, 1964; 1967) the author has used the FisherTippett I distribution for the representation of flood probabilities at stations situated at the Danube and the Gail, both rivers in the East Alpine region, which showed large floods during the last decade. The representation was very good in the case of the Gall, but not so good ill the case of the Danube. Other stations investigated, which were situated at the Drau, showed still larger deviations from the Fisher-Tippett I curve, especially in regions of flood values with high return periods. This could be due to too short observation periods but also to the fact that the Fisher-Tippett I curve has not enough degrees of freedom to be used as a good approximation in complicated river conditions. In some cases it was too strongly shown that physical upper or lower limits of the variable cannot be taken into account by this distribution function, however, there are also cases, in which rivers behave as if there was 11o such upper bound. The deviations from the Fisher-Tippett I curve may also be due to different reasons for the floods considered. Ill other words, if in some years the observed floods are spring floods but in other years summer floods, the series does not belong to one common parent-distribution law. Sneyers (1960) has given a method to use a combination of two Fisher-Tippett I curves for the representation of these cases. As a result of the considerations mentioned above and of the good effects of the use of the more general Fisher-Tippett distributions of higher order in meteorological problems (Cehak, 1971) the Fisher-Tippett II or III curve has been used even for tile investigation of floods at stations along Danube, Drau and Gail.

"]['HE FISHER-T1PPETT DISTRIBUTIONS

The three possible asymptotic forms of extreme value distributions have been written in a c o m m o n form by Jenkinson (1955) who stated an equation for the reduction of the variable x, which is to be investigated (here some measure of floods), to a reduced variable y, which is distributed according to

ON FLOOD PROBABILITIES OF EAST ALPINE RIVERS

67

the function: F(y) = exp[-exp (-y)]

(1)

By definition F ( y ) is the probability that an observed value Y is less or equal to y. Jenkinson's reduction equation reads: x =x o +a 1 - exp(-ky) k

(2)

with x being the observed variable, x o its value at the origin of the reduced variable y, a the slope of the x - y curve in the point (Xol 0), and k the parameter of curvature. Mostly, extreme-value observations are drafted in a special diagram paper, in which one axis is the linear x-axis and the other axis is a y-axis, renumbered according to eq. 1. In this diagram the three extreme-value distributions are to be distinguished by their curvature. The Fisher-Tippett I distribution is a straight line in this diagram, the type lI distribution is concave with regard to the y-axis, and the type Ill distribution is convex to the y-axis. The curvature parameter k = 0 for the type I curve, negative for the type II curves, and positive for the type III curves. Therefore, the Fisher-Tippett | curve is a kind of limiting curve with respect to the other two types. Observations belonging to Fisher-Tippett II distributions are theoretically not limited in the direction to high values, those belonging to Fisher-Tippett Ill distributions are theoretically not limited towards smaller values, whereas observations from the type I distribution are not limited in either direction. In order that eq. 2 gives a better physical meaning instead of only a geometrical one, after a simple reordering of terms one introduces new constants:

x=(xo+ k )-~.a

exp(-ky) = x 1- b exp(-ky)

(3)

In eq. 3 x 1 is the limiting value of x, which has a probability of zero for being transgressed by any observation. The three parameters of the Fisher-Tippett II and Ill distributions cannot be calculated from the moments of the distribution nor have they the meaning of mean values or dispersion measures. This leads to a discarding of expressions like mean floods, as one is not able to calculate the parameters of the interesting distribution from these expressions. The constants of eq. 3 should be calculated starting from the maximumlikelihood function. This leads to transcendent equations for the u n k n o w n constants. Jenkinson (1969) has shown a way of solving these equations.

68

K. CEHAK

A FORTRAN IV program has been derived for this calculation and has been used for the treatment of the Austrian flood data after it has shown its capabilities in meteorological problems (Cehak, 1971 ). The method of control bands has been used as proposed by Sneyers (1963a) for testing the goodness of fit. The author (Cehak, 1971 ) described the method which was used for the general Fisher-Tippett distribution. It has also been included in the FORTRAN program cited above. The limits of this control band are determined so that the probability is 68% that an observation will lie within this band. The validity of the fitted distribution is refused if more than 32% of the sample data lies outside of the control band.

FLOOD PROBABILITIES IN PARTS OF EAST ALPINE RIVER BASINS

For the investigation of the statistical properties of floods along parts of some East Alpine rivers, which have been found of interest on account of large floods during the most recent past, the following five stations with long records have been used: Vienna, Danube Stein, Danube Linz, Danube Villach, Drau Federaun, Gail

1828-1970 1828-1970 1821-1970 1893-1966 1890-1966

The site of these stations is to be seen in Fig. 1. The element of the statistics used here is the annual highest discharge as it is mostly used. No regard has been paid to the time of the year in which this highest value has been reached, nor to the question of whether this value ap-

Fig. 1. Map of Austria with the location of the stations used.

ON FLOOD PROBABILITIES OF EAST ALPINE RIVERS

~

~R~

69

~---_

~J

0

c~

C~

"0

(D

•~3 ~-~ 0 0

E~E~

~ ~ ~.~ ~

~d ,.J ~u

e~-O- 0 . 0 0

70

K. CEHAK

peared once or more during the year in question, which might have happened with the extremes near the median of the distribution. Table I gives a survey of the sample statistical parameters of these stations. The mean values o f the floods vary widely between the five stations, but the standard deviations are of approximately the same size when compared with the mean values at four stations, as the coefficients of variation show. Only Federaun's annual highest discharges have a higher variability. The parameters of skewness and kurtosis are very high in all cases. The percentiles for 10, 25, 50, 75 and 90 are given for a characterization of the sample probability distribution in the existing observation period. They correspond to return periods of respectively 10, 4, 2, 4 and 10 years. Fig. 2 - 6 give representations of the distribution functions for the floods in the five stations. In order to show the influence of observation period on Returnperiod (years)

1ooo=

1o 5

2

s

1o

2o

~

too ~

I1500~

~- -

1

....

5oo

/

m3/s 4 V [ E N N A

/

1

928- 970 1893-1966

/~/ /

1o5oo~

iooo

/ / .//

~.'/ ."

95OO~

.oo

s//

7500

,/ / / / / t

6500

s5oo 4/ t,SO0

/,J,J

t f

35oo ~

~

"///

,y//

2500

1500

.

.

0"001 0"1 PPobability

.

.

.

.

0"3 0"5

l

'

0"7 0'8

I

09

'

I

0"95 fig?

I

'

099

I

'

0"995 0'9975 0"999

Fig. 2. Probability distribution of floods at Vienna, Danube. The Fisher-Tippett distributions are given for the observation period 1828-1970 (full lines) and 1893-1966 (broken lines) together with the respective control band boundaries.

the form of the function and to be able to compare the distributions at the five stations, the distributions have been calculated for the entire observation period and for the c o m m o n period 1 8 9 3 - 1 9 6 6 , in which all stations have been observed. For the two stations Villach and Federaun, only one curve can be

ON FLOOD PROBABILITIES OF EAST ALPINE RIVERS Return period (years) 1000 100 10 5 2 t d i i i m3/s ST E I N

10 _.~

5 . . . . . _~

20

50

100

i

i

l

200 500 , ~__ i

71

I000 t

---- 1828-1970

13500j

.....

1893- 1966

125001

l

115o01 10500 1 9500" 85007 7500

/i i

/,i

6500 5500 4500-

,,°°If 2500-

1500 0'001 ' 0!1 ' 0!3 ' 015 ~ 0!7 0~8 Probability

0!9

0!95 0"97 '

0!99

0!995 019975

0"999

Fig. 3. S a m e as Fig. 2 for Stein, D a n u b e .

drafted since Villach has the shorter observation period. The three years longer observation time at Federaun, do not alter the distribution function. In the same figures the control bands are drafted to give an expression for the scatter region, within which 68% of the observations can be expected to lie. The constants of the fitted Fisher-Tippett distributions are contained in Table II. One learns from this table that all distributions derived from the long series belong to the type II (k negative), but that for the shorter observation period the data from Vienna are better approximated by a type-III curve (k positive). The parameter k is smaller in the distributions at Stein and Linz, when derived from the shorter series of data. This shows a dependence of the curvature of the probability laws on the length of the observation period in the sense that the magnitude of k increases with the length of the observation period. The design values with a certain return period also vary with the values of the parameters. The variation of the design values depends on the value of all the three parameters. Normally, the 1,000 yearly design flood increases with an increasing period of observations. This shows that one should use as

72

K. CEHAK

TABLE II Constants of the Fisher-Tippett II or llI distributions and design values with certain return periods for the flood representation Vienna

Stein

Linz

Villach

Federaun

(a) Calculated from the entire series of data xI b k

-39,552 -43,998 -0.024

- 11,793 -16,199 -0.070

- 1,633 -4,493 -0.190

- 1,907 -2,498 -0.066

-2,368 -2,647 -0.045

Xlo Xloo Xlooo

6,869 9,542 12,316

7,099 10,390 14,178

5,204 8,965 14,664

991 1,477 2,034

555 885 1,243

(b) Calculated from the period 1893-1966 xI b k

25,434 20,669 0.056

-71,593 -76,229 -0.016

- 9,004 -12,577 -0.069

-1,907 -2,498 -0.066

-2,374 -2,647 -0.045

xlo Xlo o x looo

7,216 9,465 11,403

7,482 10,568 13,715

5,683 8,265 11,242

991 1,477 2,034

556 885 1,243

long a series as possible for the derivation of design values, in order to get safe forecast values for the interesting flood sizes.

THE HOMOGENEITY OF THE FLOOD SERIES

The use of a longer series is only useful, if neither the river nor the climate show long periodic changes in their behaviour. Therefore, one has to look for inhomogeneities in the observed series. This has been done for the five stations using a test proposed by Mann and Whitney in a form given by Sneyers (1963b). The test consists of the successive calculation of a test quantity, which is based on the number of inversions in the observed series and normalized, for all part-series starting from the first year and reaching to the ruth year, m = 2, 3 .... , n. Repeating this calculation with sub-series going back from the last (nth) year to the mth year, m = n - l , n - 2 .... , 1 ; and plotting both series over the time axis one can discern those times, the series, which follow an other trend than before. The inhomogeneities are significant, if the magnitude of the test value is larger than 2. Fig. 7 - 1 1 show the test values together with the observed series. At Vienna a decreasing trend of annual floods was replaced by an opposite trend in about 1860. This increasing trend continues n o w with a short levelling

ON FLOOD PROBABILITIES OF EAST ALPINE RIVERS

73

Return period(yeors)

,o~ ,~ m 14500

17 s

5 i

10 i

20 i

50 i

100 i

200 I

500

1000

LINZ

t

13500

.....

1821-1970 1893-1966

11500 4

,,,,,

9500

##

/

7500

5500

~_-_' "'g';'S i;;;'"x;'~';*"

35°°11500

oool

o11 ' o"3' o"5' o!? o"8

Probability

o19

o"95 o"97

o!99

Fig. 4. Same as Fig. 2 f o r L i n z , D a n u b e .

phase from 1920 to 1935. At Stein the trend changed its direction about 1 873. The levelling between 1920 and 1935 is also to be seen. Linz shows a slightly different picture. Before 1860 there was no trend at all, then the floods increased until 1920 and since then they show a variation of approximately a constant mean value. There is only a non-significant, slightly increasing trend in the annual floods of the Gall and Drau rivers during the whole period of observations in Federaun and Villach. The result of the homogeneity test is different for the Danube and for the southern stations. The inhomogeneity in the time about 1860 is to be found even in the test value corresponding to the annual total of precipitation at Vienna at the same time. So a meteorological reason for this change in the trend suggests itself.

74

K. CEHAK

Return period (years) I000 100 10 5 2

robs ~

5 J

. . . .

10 i

20 t

50 =

100 l

1900

....

looo

ll/llll

1893 - 1966

:

/I

18oo !

I

I

I

1700

I I /

iI

1600-.

I

/

I I

1500

1400-

!

I

I

i/

I

/

iI

i

I

I

iI

/ -

I

I

I

I I

1300

~

200 J

VILLACH

2000.

/ I

I

/I // iI I i/ I II // I I / I I / / I / / // ii 1/ //

1200 -

i 1100

1000

iI i 1 ~ / /

,,.,..':'"

900800

i II i II

_ 700-

/IIii ///

600

/// /// / / // / /

500 -

,¢~'

,,002

"l / ///

300t~ 200

+ , 0001

0!I

'' 0!3 [ ' 0!5 ' 0!7 oL8

0)9

01'95 0"97 '

0L99

0!995

0'99?5

0'999

Probability

Fig. 5. Probability distribution of floods at Villach, Drau. The Fisher-Tippett Ill distribution is given for the observation period 1893-1966 together with its control band boundaries.

Returning to the considerations at the beginning of this chapter we must conclude that it is not too useful to take the whole series of observations effectuated at the Danube stations into consideration when calculating design values, as the years before 1860 belong to an other type of precipitation climate than the years afterwards. Therefore, one should prefer to use the values as calculated from the common period of measurements for the Danube and the southern rivers.

ON FLOOD PROBABILITIES OF EAST ALPINE RIVERS

75

Return period (yeors) 1000 100

l0

5

2

5

10

20

50

100

200

1200JFEDERAUN 1100 i

- -

500

1000

/r

1890 - 1966

,ooo-

::i

/

//

/

!

°0!~I' 0!i'0'3'o"s'0!7 0!8 0!9 oLgs0~97' o~ o!~s o~~Ts 09~ Probability

Fig. 6. Probability distribution of floods in Federaun, Gail. The Fisher-Tippett Ill distribution is given for the observation period 1890-1966 only together with its control band, as the curve derived from the period 1893-1966 does not differ from this curve.

VARIANCE SPECTRA OF THE FLOOD SERIES

In order to get a step further in the statistical description of floods besides the generally used statistical methods of estimating distribution parameters and applying them for the calculation of design values, the flood series shall be taken as parts of realizations of a stochastic process. In this connexion the dependence of the observations on time and their interdependence within one observed series has to be investigated. The floods are thought to be realizations of a stationary stochastic process, or more specifically, of an autoregressive process. Two kinds of autoregressive processes have been tried for their fit to the data: the first and the second order schemes, which can be described by the equations between successive members as follows:

76

K. CEHAK

5-

3210

k/V

-2" -3-

I1000

"

~0O00. 9000. 00O0. 7000. 6000. 5000. 4000. 3000. 2OO0 I020

Fig. 7. Flood series from Vienna and homogeneity test.

-3.

12000

II000 Ioo0o

9000 aO00 7000 6OOO

5O00 ,;O00 3000 2O0O to00 I~20

.~o

i~o

,~o

i~o0

'

Fig. 8. Flood series from Stein and homogeneity test.

t~

'

,9'4o

'

,9'6o

ON FLOOD PROBABILITIES OF EAST ALPINE RIVERS

77

6

! 0

,

,

=

~

i

9000 ¸ ~00

7OOO 6000 5000 ~000 3000 2000 tO00 0

j

,820

j

~'40

i

,~160

i

16;0

1900

J

19=20

'

,9140

~6o

Fig. 9. Flood series from Linz and homogeneity test.

x t = a xt_ x t = alxt_

1 + zt 1 + a2xt_

2 + zt

(first order (Markov) scheme) (second order scheme)

In the first case the process has a spectrum of the form: s(f) = A (1 + 0 1 2 - 201 cos 27r39-1

(4)

and an auto-correlation function: On

= Pl

n

n = 2, 3 ....

(5)

being the auto-correlation coefficient of the lag n and A depending on the selected normalization. In the second case the spectrum is given by a more complicated formula (see, e.g., Jenkins and Watts, 1968) and so is the autocorrelation function. It goes without saying that an observed spectrum, a sample spectrum at any rate, can better be fitted by a more complicated formula with two parameters than by the simple formula 4. Therefore, most of the data could be described by an autoregressive process of the second order and this scheme has also been used (Adamowski, 1971) for the representation of river flow time series. However, especially in the case of annual floods it is physically

On

78

K. CEHAK

2000 1900 /ago 1700 1600 1500 1400 1300 !200 II00 I000 900 O00 700 600 500

300 200 1 0

1920

1940

19 0

Fig. 10. Flood series from Villach and homogeneity test.

not quite understandable why the flux of information in the meteorologichydrologic system should go over 2 (or more) years, whereas an explanation for an information flux over 1 year perhaps could be found for the case of floods in various climates. Besides this, one should use a more complex mathematical model only if a test has shown that the simpler model does not fit the observations well and therefore should be discarded. A test for the goodness of the assumption of an autoregressive scheme of first order (Markov process) has been made based on eq. 5. Anderson (1958) derived a limit for testing the hypothesis p = P0 against p ¢ P0 as follows. If:

ON FLOOD PROBABILITIES OF EAST ALPINE RIVERS

79

2

dO0700. 600. 500. 400. 300 " 200" tOO 0

79100

I

i 192o

~

~o

t

,gbo

Fig. 11. Flood series from Federaun and homogeneity test.

Iz - ~ ' 0

PO 1 >

21 n - 1

1.96

(6)

with n being the number of observations, from which the estimator r for P has been calculated, O0 the theoretical correlation coefficient, and:

1

1 +r

z = ~ in 1---r

1 1 +P0 ~'0 = ~ l n 1 ~ p o

(7)

(8)

the hypothesis is rejected. In the present test r l k was used for the theoretical value P0 of the correlation coefficient according to eq. 5 with k = 2, 3 in eq. 6 and 8, and r 2 and r 3 were the estimations r in eq. 7. In Table III the results of these tests are listed. z i is the value of z according to eq. 7 with respect to the auto-correlation coefficient with the same index i. It comes out that none of the five stations reaches the critical value and, therefore, the assumption that the representation of the flood data in the East Alpine region can be made by means of an autoregressive scheme of first order (Markov process) may be maintained. The

80

K. CEHAK

TABLE III

i Test for the validity of the assumption of a Markov process for the description of the floods as stochastic processes Vienna

Stein

Linz

Villach

Federaun

0.81 0.14

0.81 0.04

1.64 1.56

0.77 0.18

0.19 0.12

z2 z3

auto-correlation coefficients of lag 1, which are the basic parameters for the Markov process, are contained in Table I, too. The variance spectra o f the five series of data were calculated from the auto-correlation function using the method of Tukey (Blackman and Tukey, 1958). In these calculations the full series were used and 20 years was the maximum lag taken into consideration yielding a resolution of 0.05 frequency units and 14 degrees of freedom. The five spectra are plotted in Fig. 12 together with the respective spectra of the Markov processes ("red-noise spectra") and their 95% significance

t°t ~ AVIENNA

0"0~01 0'2 0~3 0~4 ~ ,

0~5

tO

o.oO.Sro~FEDERAUN

0"0~0I 0!2 0'3 0"~* 0"5 ,

ro

02 03 04 05

LINZ

0"5

0"0

ot

0'5

O't 0"2 0~3 0~. ~REOUENCV ~-30-Ib ~ 5 :; ~' PERIOD

0"5 --~

0'0

O't 0~2 0~3 FR~OUENCV SO tO7 S 4 3

0"4

0"5 2

PERIOD

Fig. 12. Spectra of flood records at the five East Alpine stations.

ON FLOOD PROBABILITIES OF EAST ALPINE RIVERS

81

TABLE IV Peaks in the variance spectra of floods which are larger than the 95% significance limit of the Markov spectrum Station

Frequency

Period (years)

Vienna

0.125 0.425 0.125 0.425 0.000 0.425 0.175 0.150

8 2.35 8 2.35

Stein Linz Villach Federaun

2.35 5.71 6.67

limits. The goodness of fit is not bad, however, there are some peaks in the spectra transgressing the significance level at their frequencies. The frequencies and periods of these peaks are collocated in Table IV. All the Danube stations show large peaks at periods of 8 years and of about 28 months. At Linz the peak at the period of 8 years is not significant. The long-periodic trend dominates in the spectrum from Linz. In the spectra of the southern stations the shorter periods are not significant, however, there are significant peaks at periods of 5.7 and 6.7 years, respectively, in the spectra from Villach and Federaun. At Stein and Vienna the waves with shorter periods are more important than those with longer periods. This is also shown by the negative sign of the auto-correlation coefficients. The absolute magnitude of these two auto-correlation coefficients is very near zero, so that successive annual flood values could be judged as being independent. The same may be said about the floods of the river Gail. The auto-correlation coefficients of the data observed at Linz and Villach, however, are different from zero, the dependence of one observation on the preceding one is not very high but not to be denied.

ACKNOWLEDGEMENTS

The author is indebted to the Austrian Hydrographisches Zentralbflro, for the series of the data used in this paper and to the director of the Zentralanstalt ffir Meteorologie und Geodynamik, Vienna, for conceding computer time. Thanks are also extended to the Austrian Academy of Sciences for financial support to this work.

82

K. CEHAK

REFERENCES Adamowski, K., 1971. Spectral density of a river flow time series. J. HydroL, 14: 4 3 - 5 2 . Anderson, T.W., 1958. Introduction to Multivariate Statistical Analysis. Wiley, New York, N.Y., 374 pp. Blackman, R.B. and Tukey, J.W., 1958. Measurement o f power spectra. Dover, New York, N.Y., 190 pp. Cehak, K., 1964. Die j~hrliche gr~sste Durchflussmenge der Donau bei Wien im Spiegel der Extremwertstatistik. Osterr. Wasserwirtsch., 16: 163-170. Cehak, K., 1967. Die Gail-Hochwgsser und die Extremwertstatistik. Osterr. Wasserwirtsch., 19: 157-159. Cehak, K., 1971. Der Jahresgang der monatlichen h~Jchsten Windgeschwindigkeit in der Darstellung durch Fisher-Tippett llI-Verteilungen. Arch. Meteorol. Geophys. BioklimatoL, Ser. B, 19: 165-182. Glos, E., 1966. Eine verbesserte Methode der Parametersch~tzung f/Jr die Pearson Verteilung Type 1II. Wasserwirtsch. - Wassertech., 16. Gncdenko, B.V., 1943. Sur la distribution limite du terme maximum d'une serie al~atoire. Ann. Math., 44: 423. Grassberger, H., 1932. Die Anwendung der Wahrscheinlicbkeitsrechnung auf die Wasserffihrung der Gewfisser. Wasserwirtsehaft, 25: 1 6 - 2 2 ; 3 1 - 3 3 ; 5 5 - 6 0 . Gumbel, E.J., 1958. Statistics o f Extremes. Columbia Univ. Press, New York, N.Y., 375 pp. Jenkins, G.M. and Watts, D.G., 1968. Spectral Analysis and its Applications. Holden Day, San Francisco, Calif., 525 pp. Jenkinson, A.F., 1955. The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Q. J. R. Meteorol. Sot., 87: 158-171. Jenkinson, A.F., 1969. Statistics of extremes. In: Estimation o f Maximum F l o o d s - W.M.O. Teeh. Note, 98: 183-228. Liebscher, H.J., 1970. Hochwasserwarscheinlichkeit. Dtsch. Gewa'sserkd. Mitt., 14: 103-110. Mendel, G., 1972. A new proposal for the determination of discharge probability. Proceedings, 6th Conference o f the Danube countries on Hydrological Forecastings, Kiev {U.S.S.R.), pp. 5 5 - 6 4 . Schrciber, H., 1970. Uber Methoden zur Berechnung der n-Ja'hrlichkeit von Hochwffssern. Osterr. Wasserwirtsch., 22: 138-153. Sneyers, R., 1960. On a special distribution of maximum values. Mon. Weather Rev., 88: 6 6 - 6 9 . Sneyers, R., 1963. Du test de validit~ d'un ajustement bas~ sur les fonctions de l'ordre des observations. Inst. R. M~tdorol. de Belgique, Publ. B, 39. Sneyers, R., 1963. Sur la d~termination de la stabilit~ des s~ries climatologiques. Recherches sur la zone aride (Actes du colloque de Rome organisd par I'UNESCO et I'O.M.M.). UNESCO, Paris, pp. 37-44.