Geoscience Reference
In-Depth Information
Finally, it should be noted that, just like the corresponding distributions for the largest
values in (13.61), the first asymptote for smallest values is linked to the third by a
logarithmic transformation, as follows
ln
x
−
ω
1
v
1
−
ω
1
=
−
b
(
z
u
1
)
(13.72)
where now
b
=
α
1
/
k
. Hence, with (13.60) one obtains
exp
exp
ln
x
k
−
ω
1
v
1
−
ω
1
1
G
(
z
)
=
1
−
−
=
1
G
3
(
x
)
(13.73)
as given in (13.65). This shows how probability paper constructed for the first asymptote
can be used for the third asymptote by plotting the logarithms of the magnitudes of
the events, i.e. log
x
instead of
x
(see Example 13.1). Hence, the lay-out shown in
Figure 13.3 can be used for this purpose, by changing the scale of the ordinate from
linear to logarithmic.
Applications of this distribution to low flows in rivers have been presented by Gumbel
(1954b) and Matalas (1963).
13.4.7
The generalized extreme value distribution
The first and third asymptotes were already shown in Equations (13.61) and (13.72) to
be related by a logarithmic transformation. It should not be surprising, therefore, that
the three asymptotes can be combined into a single expression. So far in hydrology, this
idea has mostly been applied to the largest values. In this case the distribution function
is usually written in the following form
b
)
1
/
a
]
F
(
x
)
=
exp[
−
(1
−
a
(
x
−
c
)
/
for
a
=
0
(13.74)
in which
a
0, the term inside the square
brackets approaches an exponential function, and Equation (13.74) reduces to the first
asymptote (13.52). But in (13.74),
a
is not necessarily equal to zero; thus in this form
the extreme value distribution has three parameters, and can therefore be considered
more general. When
a
,
b
and
c
are constants. Clearly, when
a
→
0, (13.74) is just another form of the third asymptote for
largest values (13.63), with an upper bound at
x
>
=
c
+
b
/
a
; the parameters of the
a
−
1
,
two forms are related by
k
=
ω
=
c
+
b
/
a
, and
v
=
c
. When
a
<
0, (13.74) has
a lower bound at
x
a
but it is unbounded for large
x
; therefore, this case
is the one mostly used for the largest values. The density function corresponding to
Equation (13.74) is
=
c
+
b
/
1
b
b
)
−
1
+
1
/
a
=
(
1
−
−
/
=
f
(
x
)
a
(
x
c
)
F
(
x
)
for
a
0
(13.75)
The central moments can be derived by first considering the
n
th moment about
x
=
x
1
=
c
+
b
/
a
; in the case of
a
<
0, when
x
1
is the lower bound, this is
∞
m
n
=
a
)
n
dF
(
x
)
(
x
−
c
−
b
/
(13.76)
x
1
