Geoscience Reference
In-Depth Information
m
1.58 time units. The two remain-
ing parameters
a
and
b
can then be determined from the first two moments (13.79) and
(13.80).
The generalized extreme value distribution in the form of (13.74) was introduced in
the environmental sciences by Jenkinson (1955), and 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 (see, for example, Lettenmaier
et al
., 1987; Stedinger and Lu, 1995;
Madsen
et al
., 1997; Martins and Stedinger, 2000). Its full potential continues to be
explored (Katz
et al
., 2002).
/
(
n
+
1)
=
0
.
368, or with a return period of
T
r
=
Example 13.8. Extreme value distributions applied to annual peak flows
In this example a stream in a more arid climate is considered. At Palominas in Arizona
the San Pedro River drains an area of some 1909 km
2
, almost all in Sonora; the corrected
average annual precipitation in this area was estimated (Korzoun
et al
., 1977) to be of the
order of 400 mm. This gaging station is located at 31
◦
22
48
N, 110
◦
06
38
W, at an ele-
vation of 1276 m above sea level. The 61 available annual peak flows measured from 1930
through 1999 are plotted against
T
r
=
m
) with first asymptotic coordinates in
Figure 13.12. The first three moments of these data were estimated with Equation (13.13)
as
M
62
/
(62
−
2m
3
s
−
1
,
S
2m
3
s
−
1
and
g
s
=
436; the corresponding moments of
the logarithms were calculated to be, respectively, 5.000, 0.6466 and
=
180
.
=
115
.
1
.
−
0.2444. By means
of Equation (13.54) the two parameters of the first asymptotic distribution for largest val-
ues were estimated as
4m
3
s
−
1
; the curve calculated
with (13.52) is shown in Figure 13.12 as the solid heavy straight line 3. Interestingly, it can
be seen in the graph that the mean
M
011 13 sm
−
3
and
u
n
=
α
n
=
0
.
128
.
2m
3
s
−
1
corresponds closely withaavalue of
=
180
.
the reduced variable
y
33 y; this is to be expected
in light of the first of (13.54). The parameters of the generalized extreme value distribu-
tion were calculated with (13.79) through (13.81) as
a
=
0
.
58, and with a return period
T
r
=
2
.
38 m
3
s
−
1
=−
0
.
044 92,
b
=
84
.
6m
3
s
−
1
; the curve calculated with these parameters in (13.74) is shown as
the heavy dashed line 4 in Figure 13.12). Again, it can be seen that, as expected from
(13.74), the value of
c
corresponds closely with a return period of
T
r
=
and
c
=
127
.
1.5 y. For com-
parison, the curves based on the lognormal (with
c
0) (1), the generalized log-gamma
(2), and the power distribution (5) are also shown in the figure. For the power distribution
the parameters were taken as
a
=
=
134
.
4 and
b
=
0
.
3854.
13.4.8
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
T
r
=
100
and that with
T
r
=
100.
Phenomena with this type of behavior are referred to as fractals (Turcotte, 1992). From
this observation it follows that such phenomena obey a power law. Indeed, in this example
10, is equal to the ratio of those with
T
r
=
1000 and
T
r
=
