Metocean Extreme and Operating Conditions 3.7 Extreme Criteria 67
Part A | 3.7
one wants to put a price tag on human life. Rational risk
levels can still be set by considering risk levels in other
industries and risk on offshore structures due to causes
other than structural failure. Those risks include travel,
explosions, collisions, and falls. These considerations
indicate that the individual risk per annum (IRPA)
should be reduced to a level below 10
3 . An IRPA
lower than 10
6 is considered negligible. However, the
risk should be reduced below 10
3 to a level that is as
low as reasonably practical (ALARP). Measures to reduce IRPA should be examined and implemented until
the cost of the upgrade becomes grossly disproportionate to the benefit obtained. Efthymiou et al. [3.118]
discuss the ALARP principle in detail. The cost of stuctural strengthening should be weighed against the cost
of lowering other risks. Manned structures in the North
Sea are now usually designed for a 10
4 annual risk of
failure. Providing criteria with such a low probability is
a special challenge to the metocean specialist, who must
extrapolate conditions far beyond experience levels.
3.7.2 The Historical Method
The traditional way of estimating metocean extreme
values is extrapolation from historical data. The data
generally come from hindcasts rather than measurements so that more years of data are available. However,
even hindcast records are short compared to 1000 or
10 000 years, so extrapolation using extreme value distributions is required.
Extreme value theory assumes a time series of independent events, so the first step is to choose those
events. Generally, this is done by finding the peak
values over a threshold (POT). The peaks are sorted
in ascending order, and their probability is plotted
against their magnitude. There are then many choices
for choosing an extreme value functional form and fitting it to the data [3.119]. In the limit as the number
of points tends to infinity, it can be shown that the extremes follow the generalized Pareto distribution
F.y/ D 1 .1 C y==/
1==
:
(3.39)
The parameter controls the shape of the distribution,
giving a heavy tail if > 0 and a finite upper limit if <
0. In practice, we do not know whether the data extends
far enough into the tail of the distribution for the limit
to hold, and small changes in the data can influence
whether an upper limit is predicted. For these reasons,
engineers often choose to fit peaks to the Weibull distribution
F.y/ D 1 exp
Ä
y
˛
Á ˇ
:
(3.40)
The commonly used Weibull plotting position is
P i D
i
N C 1
;
(3.41)
but Goda [3.120] showed that the unbiased plotting position is actually
P i D
i C
0:5
p ˇ
0:6
N C 0:2 C
0:23
p ˇ
:
(3.42)
Gibson et al. [3.121] tested various methods of fitting
(3.40) to data simulated from a known Weibull distribution. They found that both least squares and maximum
likelihood fits gave good results when
1. The unbiased plotting position in (3.42) was used.
2. The position parameter was set to the threshold
value.
3. All of the data were used instead of binning the data
into ranges of y.
4. The fit was made of y to ˛ Œln .1 P/
ˇ C .
There are numerous weaknesses with the historical approach. First and foremost, historical datasets are
often short relative to the probability level needed for
design criteria. This is especially a problem for tropical
storms, which are spatially small and infrequent. Toro
et al. [3.122] show the 100-y criteria for a particular
site in regions like the Gulf of Mexico, is most heavily influenced by how close a few strong storms passed
to the west of the site. As a consequence, large differences in the 100-y design condition are often observed
at sites separated by only 50 km in deep water far from
the coast where there is no physical basis to believe the
n-y condition would be any worse at one site than the
other. These unrealistic spatial gradients are more apparent at even shorter return periods (e.g., 10 and 25-y)
in basins like north Australia and the Gulf of Thailand,
where the reliable historical database is shorter or the
storm frequency is lower than in the Gulf of Mexico.
The spatial gradients in n-y criteria are largely a result
of under sampling. There are simply not enough larger
storms in the database. Furthermore, it is reasonable
to expect that if the database could be extended over
a longer period of time, a strong storm would eventually cross near all the sites.
To counter this under sampling, metocean experts often pool nearby sites, as described in Cooper
et al. [3.123]. Pooling basically combines all storm
peaks from several nearby sites into a single probability distribution. In essence, pooling adds synthesized
storms by shifting the tracks of the historical storms.
When pooling, the probability distributions from each
site are assumed to be statistically independent, even
though they are not. However, Toro et al. [3.122] show
Part A | 3.7
one wants to put a price tag on human life. Rational risk
levels can still be set by considering risk levels in other
industries and risk on offshore structures due to causes
other than structural failure. Those risks include travel,
explosions, collisions, and falls. These considerations
indicate that the individual risk per annum (IRPA)
should be reduced to a level below 10
3 . An IRPA
lower than 10
6 is considered negligible. However, the
risk should be reduced below 10
3 to a level that is as
low as reasonably practical (ALARP). Measures to reduce IRPA should be examined and implemented until
the cost of the upgrade becomes grossly disproportionate to the benefit obtained. Efthymiou et al. [3.118]
discuss the ALARP principle in detail. The cost of stuctural strengthening should be weighed against the cost
of lowering other risks. Manned structures in the North
Sea are now usually designed for a 10
4 annual risk of
failure. Providing criteria with such a low probability is
a special challenge to the metocean specialist, who must
extrapolate conditions far beyond experience levels.
3.7.2 The Historical Method
The traditional way of estimating metocean extreme
values is extrapolation from historical data. The data
generally come from hindcasts rather than measurements so that more years of data are available. However,
even hindcast records are short compared to 1000 or
10 000 years, so extrapolation using extreme value distributions is required.
Extreme value theory assumes a time series of independent events, so the first step is to choose those
events. Generally, this is done by finding the peak
values over a threshold (POT). The peaks are sorted
in ascending order, and their probability is plotted
against their magnitude. There are then many choices
for choosing an extreme value functional form and fitting it to the data [3.119]. In the limit as the number
of points tends to infinity, it can be shown that the extremes follow the generalized Pareto distribution
F.y/ D 1 .1 C y==/
1==
:
(3.39)
The parameter controls the shape of the distribution,
giving a heavy tail if > 0 and a finite upper limit if <
0. In practice, we do not know whether the data extends
far enough into the tail of the distribution for the limit
to hold, and small changes in the data can influence
whether an upper limit is predicted. For these reasons,
engineers often choose to fit peaks to the Weibull distribution
F.y/ D 1 exp
Ä
y
˛
Á ˇ
:
(3.40)
The commonly used Weibull plotting position is
P i D
i
N C 1
;
(3.41)
but Goda [3.120] showed that the unbiased plotting position is actually
P i D
i C
0:5
p ˇ
0:6
N C 0:2 C
0:23
p ˇ
:
(3.42)
Gibson et al. [3.121] tested various methods of fitting
(3.40) to data simulated from a known Weibull distribution. They found that both least squares and maximum
likelihood fits gave good results when
1. The unbiased plotting position in (3.42) was used.
2. The position parameter was set to the threshold
value.
3. All of the data were used instead of binning the data
into ranges of y.
4. The fit was made of y to ˛ Œln .1 P/
ˇ C .
There are numerous weaknesses with the historical approach. First and foremost, historical datasets are
often short relative to the probability level needed for
design criteria. This is especially a problem for tropical
storms, which are spatially small and infrequent. Toro
et al. [3.122] show the 100-y criteria for a particular
site in regions like the Gulf of Mexico, is most heavily influenced by how close a few strong storms passed
to the west of the site. As a consequence, large differences in the 100-y design condition are often observed
at sites separated by only 50 km in deep water far from
the coast where there is no physical basis to believe the
n-y condition would be any worse at one site than the
other. These unrealistic spatial gradients are more apparent at even shorter return periods (e.g., 10 and 25-y)
in basins like north Australia and the Gulf of Thailand,
where the reliable historical database is shorter or the
storm frequency is lower than in the Gulf of Mexico.
The spatial gradients in n-y criteria are largely a result
of under sampling. There are simply not enough larger
storms in the database. Furthermore, it is reasonable
to expect that if the database could be extended over
a longer period of time, a strong storm would eventually cross near all the sites.
To counter this under sampling, metocean experts often pool nearby sites, as described in Cooper
et al. [3.123]. Pooling basically combines all storm
peaks from several nearby sites into a single probability distribution. In essence, pooling adds synthesized
storms by shifting the tracks of the historical storms.
When pooling, the probability distributions from each
site are assumed to be statistically independent, even
though they are not. However, Toro et al. [3.122] show