Environmental Engineering Reference
In-Depth Information
Fig. 2.30 Comparison of the 1-, 5- und 50-year extreme values
2.5.8 Extreme sea state values
Statistics-based statements regarding the extreme values of wave heights within a finite
sample are important for the design of marine structures. If we consider the sample to
be the chronological sequence of wave heights H 1 ,H 2 ,...,H n determined within a
certain period Ts, then these values exhibit a Rayleigh distribution for the short-term
(see Section 2.5.6, Figure 2.24) or a Weibull distribution,orevenaGumbel distribution,
for the long-term [24].
Putting the H i values in order of their magnitude gives us the sample (H 1 ,H 2 ,...,H n )
with H 1 < H 2 < ... < H n . Whereas the same distribution, that is the statistical
population (e.g. Rayleigh), applies to all values prior to organising them, the ordered
values obey various laws depending on their position within the order. Particularly
interesting here is the extreme value distribution f (H n ), that is the distribution of the
maximum value of all maxima. By way of a simplification, it may be assumed that
the individual maxima are statistically independent of each other.
The probability that the solitary wave height H i does not exceed a given limit value H
can be expressed as follows:
PH i H
Þ ¼ FH ðÞ
ð
The probability that all n maxima H i do not exceed the value H n can then be formulated
thus:
¼ F extr H n ¼ FH n
n
Þ ¼ H n H n
P extr Max H 1 ;
ð
H 2 ; ...;
H n
Differentiation gives us the probability density:
f extr H n ¼ n FH n
n 1
fH n
If the statistical population of the wave heights of a short-term sea state exhibits a Rayleigh
distribution, then the extreme value distribution (expressed in dimensionless form) is
 
Search WWH ::




Custom Search