THE FIRST ASYMPTOTIC DISTRIBUTION OF EXTREME VALUES

Extreme Values and Their Initial Distribution

When samples are taken from the same population, the smallest item in each sample is a
random variable which follows certain distribution function which depends on m and n and on the
distribution function of the population from which samples are drawn.
When the size of the samples is very large, i.e. in the limit when n → ∞ the distribution
functions of the extreme values are called asymptotic distribution functions or asymptotes and it
no longer depends on m and n, but only on the nature of its initial distribution.

Three General Types of Initial Distribution

1. Exponential Type
Comprises those distributions that for large x converge to unity at least as fast as the
exponential function itself; all their moments exist. These types of distribution satisfy
'
−f ( x )
∧f ( x ) '
f (x ) f (x) f (x )
= =
1−F ( x ) F (x ) f (x)

this suggests the application of de L’Hospital’s rule.

Examples of this type of distribution are normal, the logistic, the gamma, and their
logarithmically transformed distributions.

2. Cauchy Type Distribution

These are distributions, which do not have moments above a certain order.

3. Limited Distributions
These are distributions with an upper or lower bound or with both. In hydrology they are
mainly of interest in the analysis of low flows and droughts.
For example, the lognormal distribution is of the exponential type at the upper end, since x
can assume values all the way to infinity; however, it is of the limited type at the lower end of the
distribution, because x cannot be smaller than zero or c.

The First Asymptote for Largest Values

The asymptotic distribution function of the largest values

G ( x )=exp [ −exp (− y ) ]

and the corresponding density function

g ( x )=α n exp [− y−exp (− y ) ]

where y = αn(x − un) is the reduced largest value. The distribution and the density function of the
extremes are denoted in this section by G(x) and g(x), merely to distinguish them from the initial
distribution function and the initial density function, respectively.

In hydrology, it has been especially useful in the analysis of annual floods, i.e. the yearly
maximal discharges on record. It is useful to restate the assumptions on which the derivation is
based, to gain a better understanding of its applicability:

(i) the initial distribution is of the exponential type;

(ii) the events, from among which the largest are considered, must be independent;
(iii) the sample size n is infinitely large.

In the case of the yearly floods, i.e. the maxima of the daily flows, these conditions are not
really met. One of its practical disadvantages is that, since it has only two parameters, all
moments above the second are related to the first two.

Another point of interest is the behavior of the first asymptote for extremely large events:
−1
x=u n+ a n ln T r
 This shows that, if the largest events are plotted against Tr on semi-log graph paper, they
should tend to a straight line in the range of very large values of Tr. This may be a useful
procedure to apply, when no probability paper is available.

Fuller found that the largest 24 h average rate of flow to be expected in Tr years is

Q=Q av ( 1+0.8 log T Γ )

Where:
Qav = average annual flood

Note:
Qav is proportional to A^0.8, where A is the drainage area.

The First Asymptote for Smallest Values

The first asymptote for smallest values can be obtained from that for the largest values by
replacing x and un by −x and −un, respectively.

G( x )=1−exp [ −exp ( y ) ]

Most initial distributions are not symmetrical; but Gumbel (1958) has indicated how in the
case of asymmetrical distributions the symmetry principle can be extended simply by adopting
new parameters, say u1 and α1, instead of un and αn.

THE THIRD ASYMPTOTIC DISTRIBUTION OF EXTREME VALUES

The Third Asymptote for Largest Values

This distribution is also known as the Weibull distribution for the Swedish engineer who
first used it to analyze breaking strengths. This third asymptote is applicable to describe maxima
when their initial distribution has an upper bound.

The corresponding third asymptotic density function:

( )
κ−1
k ω−x
g3 ( x ) = G3 ( x )
ω−v w−v

The Third Asymptote for Smallest Values

In hydrology it is mainly the third asymptotic distribution for smallest values that has
been of interest. Different types of common events, such as rainfall amounts, wind speeds or river
flows, can often be assumed to be unlimited in magnitude, even the smallest of such events can
never be smaller than zero. As for the first asymptote, the symmetry principle can be applied to
derive the distribution of the smallest values from that of the largest values. The procedure
consists of changing the sign of x, ω and v, and then assigning different values to the parameters,
say ω1 and v1, to obtain:

( )
κ−1
k ω−x
g3 ( x ) = G3 ( x )
ω−v w−v

the first asymptote for smallest values is linked to the third by a logarithmic transformation

( )
κ−1
k ω−x
g3 ( x ) = G3 ( x )
ω−v w−v

The first asymptotic distribution for largest values can be used here to illustrate the
construction of probability paper.

The three asymptotes can be combined into a single expression, this idea has mostly been
applied to the largest values.

Where:
a can be determined by iteration from the sample skew coefficient gs
b can be obtained from the sample variance S2
c can be obtained from the sample mean

Note:
If the data record is so short that the third moment must be considered unreliable, one can
also apply the Weibull procedure. It has subsequently found wide application in the prediction of
various extreme phenomena, such as floods, rain events, wind speeds and wave heights; it has
also come to be used in the estimation of regional flood frequencies

Power Law (or Fractal) Distribution

Many natural phenomena exhibit a type of self-similarity or scale invariance in their

magnitudes, such that, for instance, the ratio of the event with return period Tr = 100 and that
with Tr = 10, is equal to the ratio of those with Tr = 1000 and Tr = 100, such phenomena obey a
power law.

The corresponding density function is

where:
a and b can be derived simply by least squares linear regression of the logs of the observed values
X against the logs of their return periods Tr.

The power distribution has been found useful in the description of numerous phenomena,
such as fragmentation, earthquakes, volcanic eruptions, mineral deposits, and land forms, among
others. In hydrology, the power distribution probably found its earliest application in the
description of rainfall intensities.

EXTENSION OF AVAILABLE RECORDS

HISTORICAL INFORMATION

Return Periods

As an illustration of possible scenarios for annual floods, consider the three cases discussed
by Dalrymple:

(i) A single historical event, larger than any event during the regular period of record, is
known to have occurred earlier

(ii) An historical event is known to have occurred and is the largest ever, until an even larger
event occurs during the period of record.

(iii) An historical record is available of all events above a certain base, such as for example
“bankful stage,” and it can be assumed that the distribution of the lesser events during the
regular period of record is typical for that of the entire historical period.

Estimation of Moments

The same weighting method was also recommended in Bulletin 17B (Interagency
Advisory Committee on Water Data, 1982) to adjust the moments for the parameter estimation of
the generalized log-gamma distribution. Adjusted moments can be calculated from the data as
follows;

Regionalization

Regional analysis, or regionalization, refers to the extension of available records in space.

Its dual objective is to improve the record at regular measuring sites, and to provide estimates of
frequency characteristics at sites, where no data are available.

Index-flood method
In a hydrologically homogeneous region the flood distribution functions for different
streams are similar; in this case similarity means that, when the distribution functions are scaled
with their respective index-flood, the resulting dimensionless distributions of all basins in the
region can be assumed to have the same shape, which is independent of drainage area and of any
other basin characteristics.
Two Components of Index Flood Method

Regional Flood Frequency Curve

This index-flood is usually taken as the sample mean annual flood, but other measures,
such as quantiles. Constructed as the average or the median curve of the available dimensionless
curves.

Relationship Between the Magnitude of the Index-floods and Easily Obtainable Basin and
Climate Characteristics
Only the drainage area has been considered as the significant characteristic. used to
predict the frequency curve for any ungagged catchment.

Quantile Estimation with Multiple Regression

First the frequency curves are constructed for the stations for which data are available
within the region of hydrologic homogeneity.
Characteristics to be considered may include drainage area, main channel slope, main
channel length, mean annual precipitation, fraction of area with lakes and ponds, mean annual
runoff, Tr y 24 h rainfall, mean basin altitude, fraction of basin area covered with forest, basin
shape as ratio of main channel length and area, mean basin elevation, and possibly others. The
final selection of the characteristics to be included can be made on the basis of their respective
statistical significance and on the basis of the reduction of the standard error caused by their
inclusion.

Theoretical Distribution Functions with Regionalized Moments

The underlying assumption of this approach is that the moments in a hydrologically

homogeneous region depend on known or measurable basin and climate characteristics. Thus,
once the moments can be estimated for an ungagged basin within the region on the basis of these
characteristics, it becomes possible to calculate the parameters of the selected probability
distribution function. This method is less restrictive than the index-flood method.