Geoscience Reference
In-Depth Information
in which
ω
1
is the lower limit, such that
x
≥
ω
1
and
v
1
≥
ω
1
and also
1
G
3
(
ω
1
)
=
0. The
corresponding density function is
x
k
k
v
1
−
ω
1
−
ω
1
v
1
−
ω
1
1
g
3
(
x
)
=
[1
−
1
G
3
(
x
)]
(13.66)
The
n
th moment about
ω
1
can be determined as follows
∞
−
ω
1
)
n
1
g
3
(
x
)
dx
m
n
=
(
x
0
(13.67)
∞
−
ω
1
)
n
d
[1
=−
(
x
−
1
G
3
(
x
)]
0
which yields, with (13.65),
m
n
=
v
1
−
ω
1
)
n
(
(1
+
n
/
k
)
(13.68)
Hence the mean is
μ
=
ω
1
+
(
v
1
−
ω
1
)
(1
+
1
/
k
)
(13.69)
ω
1
is
m
2
=
v
1
−
ω
1
)
2
Similarly, since the second moment about
(
(1
+
2
/
k
); in light of
(13.12), the variance is
2
v
1
−
ω
1
)
2
2
(1
σ
=
(
[
(1
+
2
/
k
)
−
+
1
/
k
)]
(13.70)
Higher-order moments can also readily be derived by means of Equation (13.68). In
addition, it is easy to show that the median is given by [
v
1
−
ω
1
)(ln 2)
1
/
k
] and the
ω
1
+
(
k
)
1
/
k
].
In practical applications, generally the only known fact is that the initial distribution
is bounded on the left but that distribution itself is unknown. Thus, as was the case
with the first asymptote for the largest values, the parameters can only be determined
from the available smallest values. In case one out of the three parameters is known
(usually the lowest value
mode, which exists only for
k
>
1, by [
ω
1
+
(
v
1
−
ω
1
)(1
−
1
/
ω
1
), the method of moments will require only the first two
moments (13.69) and (13.70). If all three parameters have to be determined, the first
three moments could be used, in principle. However, as noted earlier, the third moment
is often unreliable, and a different approach may be desirable. Gumbel (1954a) has
described a rapid method after a procedure developed by Weibull. First the value of
v
1
is determined; since
1
G
3
(
v
1
as the value of
x
which
has an observed probability of 0.632 or, which is the same from Equation (13.18), a
return period of
T
r
=
v
1
)
=
1
−
exp(
−
1), one can take
1.58 time units. For example, in the case of annual low flows or
“droughts,”
v
1
can be taken as the magnitude of the 1.58 y event. The value of
k
can
be determined from the probability of the mean
1
G
3
(
), as observed from its plotting
position. Thus
k
is the solution of the combination of (13.65) with (13.69), that is
μ
k
(1
1
G
3
(
μ
)
=
1
−
exp[
−
+
1
/
k
)]
(13.71)
The value of the lower limit
ω
1
can then be determined from the variance, as given by
Equation (13.70), in which
v
1
and
k
are already known.
