FLOOD00764

• County: Statewide
• Community: Nationwide
• Title: Use of Extreme Value Theory in Estimating Flood Peaks from Mixed Populations
• Date: 2/1/1980
• Prepared By: Utah State University Water Research Laboratory
• Floodplain - Doc Type: Educational/Technical/Reference Information

Since flood frequency analysis deals
with maXImum flows, only the distribution
of Zn is considered. The three possible
dIstrIbutIons of Zn are (Gnedenko 1943),

F I (x) = exp {-exp - (X:b)} -00<"00 , 0>0 .

{o x<b
F,(x) = exp {_ (X:bTa) x"b, 0>0, a>O

F3(x) = {:xp {- (b:Xr) x$b, c>O, a>O

Qualitat lVe characterIstIcs of these distt i-
but ions are dIscussed in the next sectIon.
The assumption of independence of the Xl,
x2, ..., Xn random variables is violated
In many applications. However, Watson
(1952) has shown that independence IS DoL a
necessary assumption. If the randomized
sequence of xi's satisfies the assumption
for all n, the theory holds.

The advantage of the theory is thal once
an extreme value situation is recognized one
can legitimately confine the search for best
fit to three extreme value distributions.
The mathematical characteristics of the three
distributions are very different, thus it is
relatively easy to determine the correct one
for a given set of data. A graphical proce-
dure is given below for use in identifying
which of the extreme value distributions
should be used with a given set of data.

Determining Extreme Value Type
Distributions (4), (5), and (6) have
some easily observed characteristics. The
function F3(x) is limited to some maximum
value b (i.e., F3(x) = I for x > b), thus
random varIables WhICh have an upper limit
have extreme value form F3(x). The converse
of thIS statement is not necessarily true,
however, and variables which are not limited
may also have this form (Gnedenko 1943).

The form FZ(x) IS referred to as a
"Cauchy typell because the extreme values for
the Cauchy distribution follow distribution
(5). Cauchy type distributions are "heavy
t ailed" and seldom occur in nature. Thus,
distribution (5) has limited usefulness
compared with the other two types. There is,
however, reference to its use in Gumbel
(1954). The form FI(x) is the one most
widely used and generally the only one
explained in textbooks.

Three simple plots constItute the
eaSIest method of determining which extreme
value distribution is appropriate. Let
X(l), X(Z), ..., x(n) represent the ordered
extreme value data for the observed maxi-
mums.

(4)

For any random variable, the expected
value of its distribut ion funct ion evaludled
at the ith order statistic IS i/(n+l) where
the sample size is n (i.e., E(F(x(i)))
./ (n+l)) (LindRren 1976). Def.ne E,
i/(n+l). Note that from Equation 4

In (-In FI(x(i))) = - x(i)/c+b/c

(7)

(5)

Note that the relat ionsh ip in Equal ion 7 IS
~lnear In XCi). Substit.utIng Ei for F(x(i))
In EquatIon 7 and plottIng X(i) vs. In (-In
F(x(i)) identifies data from a population
wltn distribution functIon Fl(x). If
Equallon 4 is approprIate the plot will be a
straight lIne as Illustrated in Figure 1. If
the data are from any other dlstr ibut ion, the
plot will not be a straight line.

(6)

The plot which identifies

F2(x) population is similar.
5 it follows that

dat a from an
From Equat ion

In (-In F2(x(i)))

- a In (x-b)+a In c

(8)

Thus if data are from a population with
distribution F2(x), the plot of In(x(i) _
b) vs. In (-In Ej) will be a straight line
with negatIve slope as illustrated in Figure
2. The parameter b must be estimated before
the plot can be made. Estimation of parame-
ters is conSIdered later.

The third plot which ident.ifies F3(x)
is motivated from Equation 6 In the same
manner, i.e.,. the plot of In (b.- XCi)) vs.
In (-In Ei) IS a straight line With positive
slope as illustrated in FIgure 3.

As discussed by Bobee and Robitaille
(1977), the physical limitations of meteoro-
logical phenomena and basin characteristics
which control rIver flow suggest that flows
are bounded by an upper limit. Thus it seems
that the most logical distribution for the
statistical descrIption of flood peaks is
F3(x). Figure 4 verifies this choice for
the Kymijoki River in Finland. It is very
evident from a glance that the data are
linear in this case. In less obvious cases,
standard analysis techniques can be used to
test for linearity (the existence of higher
order polynomial effects).

In order to interpret the plot for
F3(x), it is useful to examine the shape of
this plot if the data were to originate from
a Pearson or log Pearson type III distri-
bution. Relative to these distributions, if
floods are bounded above,the general shape of
1n (b - x(i)) plotted aRainst In (-1n Ell
is a curve, concave as viewed from the left.
If floods are bounded below, the plot will
appear as a curve convex as viewed from the
left. Note that for this plot an upper bound
IS estimated as if the distribution were
F3(x) even though it is not.

6