Part A | 3.6
64 Part A Fundamentals
as the roll of a floating system. The environmental parameters at that point then become the design point.
Winterstein et al. [3.112] give a good explanation of the
method.
Inverse FORM maps the environmental contours to
standard normal distributions. If the probabilities are
expressed as annual extremes and the return period of
interest is 100 years, then the probability of exceeding
the 100 year value is p D 1=100, and the reliability index is
ˇ D ˚
1
.1 p/ ;
(3.35)
where ˚ is the standard normal distribution. The design
contour expressed in standard normal variables is then
defined by
ˇ
2
D
X
x
2
i :
(3.36)
In two dimensions, (3.36) defines a circle. In three
dimensions it is a sphere. Each point on the surface
has the same probability. The theory extends to hyperspheres in higher dimensions, although the search for
the maximum response then becomes much more time
consuming.
Suppose the actual environmental parameters are
wave height H and period T. If these distributions are
independent, then each point on the contour given by
(3.36) has the physical parameters
H D F
1
H Œ˚.x 1 / ;
T D F
1
T|H Œ˚.x 2 / :
(3.37)
where F
1
H is the inverse wave height distribution and
F
1
T|H is the inverse distribution of T given H.
3.6 Operational Criteria
The operating conditions are the metocean conditions
in which a facility or vessel should be capable of
achieving its routine functions. Typical products used
to quantify operational conditions include a cumulative probability distribution of wave height or a table of
wind speed persistence. These products are used in estimating the fatigue lives for components for which this is
a concern. In contrast, extreme conditions rarely occur
and are often generated by storms of some kind. During extreme conditions, normal operations are usually
suspended – the vessel is slowed, oil production may be
stopped, windmills feathered, etc. The first two sections
below describe several common methods for describing operational criteria of variables that have at least
one, highly correlated associated variable, e.g., wind
speed and direction. However, the methods are often
used even when there are more than one correlated associated variable such as waves, e.g., wave height, wave
period, and wave direction.
3.6.1 Probability Distributions
The simplest method for quantifying a variable with
a single dimension like wave height is to provide a table or plot of the probability distribution (histogram) as
shown in Fig. 3.8a. However, since virtually all metocean variables are vectors, such tables or graphs are
typically expressed as joint probability distribution tables, as shown in Fig. 3.8b for the case of wind speed
and direction. In this table, each cell shows the probability of the occurrence of wind speed for a given wind
direction. A wind rose is another way of graphically
displaying a vector like wind velocity (Fig. 3.8c). In
this case, each bar shows the percent occurrence of the
speed in discrete bins along the indicated heading. All
three images are based on the same dataset, so Fig. 3.8a
basically shows a plot of the first column of the table on
the x-axis versus the tenth column on the y-axis, while
Fig. 3.8c shows the percent occurrence of the speed
(binned in 5 m s
1 increments) by direction.
One of the challenges in clearly quantifying the
operational environment is dealing with variables that
have multiple associated variables that are highly correlated. The section on currents below describes several
ways of dealing with this issue for currents.
Waves are typically described by pairing of the
associated variables. For instance, one can generate
joint probability distribution tables of wave height versus wave period by direction sector. Alternatively, one
could generate tables of wave height versus heading by
period bin, e.g., a table like that shown in Table 3.1 for
all wave periods between 1012 s.
Many facilities are sensitive to wave fatigue, so designers need the probability distribution of the wave
spectra. For regions dominated by single-mode spectra
this is straightforward – one can use the tables described
in the previous paragraph in conjunction with parametric spectra like JONSWAP. In other words, knowing the
probability of a discrete bin of wave height, period, and
direction, one can calculate the corresponding spectra
at that probability level. Further refinements may be
needed if the spectral width and/or directional spreading vary in the region.
Many regions of the world such as Brazil experience sea states characterized by spectra that have
multiple peaks, several of which contain substantial en-
64 Part A Fundamentals
as the roll of a floating system. The environmental parameters at that point then become the design point.
Winterstein et al. [3.112] give a good explanation of the
method.
Inverse FORM maps the environmental contours to
standard normal distributions. If the probabilities are
expressed as annual extremes and the return period of
interest is 100 years, then the probability of exceeding
the 100 year value is p D 1=100, and the reliability index is
ˇ D ˚
1
.1 p/ ;
(3.35)
where ˚ is the standard normal distribution. The design
contour expressed in standard normal variables is then
defined by
ˇ
2
D
X
x
2
i :
(3.36)
In two dimensions, (3.36) defines a circle. In three
dimensions it is a sphere. Each point on the surface
has the same probability. The theory extends to hyperspheres in higher dimensions, although the search for
the maximum response then becomes much more time
consuming.
Suppose the actual environmental parameters are
wave height H and period T. If these distributions are
independent, then each point on the contour given by
(3.36) has the physical parameters
H D F
1
H Œ˚.x 1 / ;
T D F
1
T|H Œ˚.x 2 / :
(3.37)
where F
1
H is the inverse wave height distribution and
F
1
T|H is the inverse distribution of T given H.
3.6 Operational Criteria
The operating conditions are the metocean conditions
in which a facility or vessel should be capable of
achieving its routine functions. Typical products used
to quantify operational conditions include a cumulative probability distribution of wave height or a table of
wind speed persistence. These products are used in estimating the fatigue lives for components for which this is
a concern. In contrast, extreme conditions rarely occur
and are often generated by storms of some kind. During extreme conditions, normal operations are usually
suspended – the vessel is slowed, oil production may be
stopped, windmills feathered, etc. The first two sections
below describe several common methods for describing operational criteria of variables that have at least
one, highly correlated associated variable, e.g., wind
speed and direction. However, the methods are often
used even when there are more than one correlated associated variable such as waves, e.g., wave height, wave
period, and wave direction.
3.6.1 Probability Distributions
The simplest method for quantifying a variable with
a single dimension like wave height is to provide a table or plot of the probability distribution (histogram) as
shown in Fig. 3.8a. However, since virtually all metocean variables are vectors, such tables or graphs are
typically expressed as joint probability distribution tables, as shown in Fig. 3.8b for the case of wind speed
and direction. In this table, each cell shows the probability of the occurrence of wind speed for a given wind
direction. A wind rose is another way of graphically
displaying a vector like wind velocity (Fig. 3.8c). In
this case, each bar shows the percent occurrence of the
speed in discrete bins along the indicated heading. All
three images are based on the same dataset, so Fig. 3.8a
basically shows a plot of the first column of the table on
the x-axis versus the tenth column on the y-axis, while
Fig. 3.8c shows the percent occurrence of the speed
(binned in 5 m s
1 increments) by direction.
One of the challenges in clearly quantifying the
operational environment is dealing with variables that
have multiple associated variables that are highly correlated. The section on currents below describes several
ways of dealing with this issue for currents.
Waves are typically described by pairing of the
associated variables. For instance, one can generate
joint probability distribution tables of wave height versus wave period by direction sector. Alternatively, one
could generate tables of wave height versus heading by
period bin, e.g., a table like that shown in Table 3.1 for
all wave periods between 1012 s.
Many facilities are sensitive to wave fatigue, so designers need the probability distribution of the wave
spectra. For regions dominated by single-mode spectra
this is straightforward – one can use the tables described
in the previous paragraph in conjunction with parametric spectra like JONSWAP. In other words, knowing the
probability of a discrete bin of wave height, period, and
direction, one can calculate the corresponding spectra
at that probability level. Further refinements may be
needed if the spectral width and/or directional spreading vary in the region.
Many regions of the world such as Brazil experience sea states characterized by spectra that have
multiple peaks, several of which contain substantial en-