Environmental Engineering Reference
In-Depth Information
Fig. 2.31 Calculating the design extreme value from the short-term statistics
(
"
#
) n
¼
2
H n
H s
H n
H s
F extr
1 exp 2
and
(
"
#
) n 1
"
#
¼ n
2
2
H n
H s
H n
H s
H n
H s exp 2
H n
H s
f extr
1 exp 2
4
An approximate modal value for n is given by
a H n =
p
ln ðÞ=
H s ¼
2
Figure 2.31 shows an example of the basic distribution of the wave heights (Rayleigh) as
well as the extreme value distribution for a sample of size n ¼ 1000. The H 1000 value is
interpreted as the most likely extreme value within a 3 h storm (short-term statistics!). It
amounts to
a H n ¼ 1000 ¼
p
ln ðÞ=
H s ¼ 1
:
86 H s
2
Using the example of Figures 2.29 and 2.30 (H s,k ¼ H s,50 from the long-term statistics)
results in a characteristic value for the maximum wave height amounting to
H max ; k ¼ H max ; 50 ¼ 1
:
86 H s ; 50 ¼ 1
:
86 16
:
06 ¼ 29
:
9m
The water depth should be used to check whether the breaking criterion has been exceeded!
Very high wave height values often have to be considered in connection with detailed
design tasks. The form of the extreme value distribution depends not on the central area
of the basic distribution, but more on the way it converges to 1 for high values of H
(rare end of basic distribution).
The extreme value distribution for large samples (n > 1000) is frequently approxi-
mated with Davenport's formula (for details see [17]). The Davenport extreme
value distributions are practically identical with the corresponding Rayleigh
distributions raised to the power of n. Asymptotic extreme value distributions
such as
the Gumbel distribution (see Section 2.5.7)
represent
another
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