Geoscience Reference
In-Depth Information
written as
1
2
T
r
y
=
ln(
T
r
)
−
(13.56)
or, to a good approximation,
u
n
+
α
−
1
n
x
=
ln
T
r
(13.57)
This shows that, if the largest events are plotted against
T
r
on semi-log graph paper, they
should tend to a straight line in the range of very large values of
T
r
. This may be a useful
procedure to apply, when no probability paper is available. It is remarkable also that
Equation (13.57) is in the same form of, and thus provides a theoretical justification for,
the equation proposed by Fuller (1914) to describe annual floods. Indeed, Fuller found
empirically, “...byplottings of the existing data on American rivers” available to him,
that the largest 24 h average rate of flow to be expected in
T
r
years is
=
Q
Q
av
(1
+
0
.
8 log
T
r
)
(13.58)
in which
Q
av
is the average annual flood and log denotes the decimal logarithm;
Fuller (1914) also observed that
Q
av
is proportional to
A
0
.
8
, where
A
is the drainage
area.
The first asymptote for smallest values
Whenever the initial distribution
F
(
x
) is symmetrical about the origin, in accordance with
Equation (13.11), the probability that an observation is larger than
−
x
is given by [1
−
F
(
−
x
)]; hence the probability that the smallest in a sample of
n
independent observations
is larger than
−
x
is
−
x
)]
n
1
−
1
G
(
−
x
)
=
[1
−
F
(
(13.59)
Proceeding in the same way as for the largest values, from (13.47) through (13.52), and
making use of this symmetry, one obtains
=
−
−
1
G
(
x
)
1
exp[
exp(
y
)]
(13.60)
where as before
y
=
α
n
(
x
−
u
n
). Thus the first asymptote for smallest values can be obtained
from that for the largest values by replacing
x
and
u
n
by
−
x
and
−
u
n
, respectively. 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
u
1
and
α
1
, instead of
u
n
and
α
n
. In other words, Equation (13.60) can
still be applied but the reduced variable is
y
=
α
1
(
x
−
u
1
), and the parameters are derived
from observations of the smallest values.
13.4.6
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 (Gumbel, 1954a; 1958). This third asymptote
