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

EXTREME VALUE APPLICATION-NONMOMOGENEOUS DATA

Sometimes, the breaks in the slopes of
the lines in plots like Figures 5 through 18
are because the data come from more than one
distribution. This section of this report
explores the theoretical aspects of fitting
distributions to such nonhomogeneous flood
data. A method of estimating the parameters
of the new extreme value forms is given and
the fit evaluated for several streams ex-
hIbiting nonhomogeneous sources. Identifica-
t ion of nonhomogeneous data by graphical
methods is su~gested.

Mixture Distributions in Hydrology

Pr ior to the observat ions of Ashkanasy
and Weeks (1975), Potter (1958) noted the
mixture of random variables in the statisti-
cal distribution of floods. He used the
standard mixed distribution for the case of
two components,

F(X) = P1G1(x) + PzGz(x)

(9)

where Gi(x), I = 1,2 are the distribution
functions of the first and second components
of the mixture respect ively. The parameters
Pi, i = 1,2 are such that Pi > 0, i = 1,2
and Pl+P2 = 1. Estimation of the parameters
In Equation 9 is very difficult because PI
and Pz must be estimated in addition to all
of the parameters of both GI(x) and G2(x).
Additional work by Hawkins (1972, 1974)
documents other problems associated with
fitting such mixed distributions.
Canfield and Borgman (1975) used
reliability theory to provide a much more
adequate approximating distribution. Their
results have direct application to choosing a
distribution of annual peak flows in hy-
drology In that they provide a theoretical
foundation which gives primary consideration
to the shape of the right tails (high flow
side) of the distributions involved. Speci-
fically, they showed the distribution of the
extreme in a sequence of mixture random
var iabIes to be

PI
F(x) = Fi(x)

P2
Fi, (x)

(10)

where the components Fi(x) and Fi I (x) are
extreme value distributions (4), (5), or (6).
Note that the parameters Pl and P2 can be
absorbed by reparameterization so that
Equation 10 can be rewritten,

F(x) = Fi (x)Fi, (x) .

(11)

thereby reducing the number of parameters in
the distribution. Because of its theoretical
basis, a distribution of this form should
have the correct tail characteristics. Note
that the tail shape in Equation 9 is a
weighted average of the tails of GI(x) and
G2(x), whereas the shape of Equation 11 is
a product of the tails of Fi(x) and Fi'(x).
Even if two extreme value distributions are
used in Equation 9, the tail shape is not
necessarily correct.

Estimation of Parameters

The usefulness of the distributions
described in the previous section depends
upon 1) the availability of techniques
for estImating parameter values and 2) a
theoretical justification of the distribu-
tIons. Theoretical justification depends on
the applicability of extreme value theory as
discussed above. A graphical method of
determIning the best parametr ie form of
Equation 11 and of estimating the parameters
is given in this section.

Graphs should always be used as a part
of data analysis for annual floods. They are
the eas ies l method for select ing from among
the three extreme value types as discussed
previously, and in addition they easily
identify nonhomogeneous sources. Application
of homogeneous distributions to nonhomoge-
neous river data can lead to ser 10US blun-
ders. The graphs should be plotted and
reviewed to make sure that this is not
happen ing.

In most applications, as discussed
previously, the third extreme value distri-
bution applies, thus the form of Fi(x) and
Fi'(x) in Equation 11 is the same for both
i and i I. However, the parameter values will
be different for Fi(x) and Fi' (x). Thus,
the graphical method used in the previous
discussion on homogeneous data applies here.
Corr ect par amet r ic forms are ident if ied as
straight lines as noted previously. For
nonhomogeneous data, two or more straight
lines are found.

The data used for this part of the
research were those obtained from Bobee
and Robitaille and identified by them as
being nonhomogeneous. (See Appendix H.)
Graphs of the annual flood peaks for eleven
of the rivers, plotted as illustrated by
Figure 3, are shown in Figures 19 to 29. As
before, F3(x) is used for Fi(x) and Fi'(x)
(Le., 1 = I' = 3).

15