Part A | 3.7
66 Part A Fundamentals
available from models or measurements. All of that
information must be condensed into a manageable number of cases for analysis. Prevosto et al. [3.114] discuss
three methods for doing this and compare the results of
using them with a full analysis of all profiles.
Empirical orthogonal functions (EOF) provide
a method for capturing the important characteristics
of current profiles in a few variables. Forristall and
Cooper [3.115] outline the method and give examples.
Singular value decomposition permits any matrix A to
be decomposed as
A ij D
N
X
kD1
w k U ik V jk :
(3.38)
Each current profile is written as a row in matrix A and
each column represents the time series at one depth. The
columns of V are called the EOFs. Each EOF is a vector
with a value at each depth in the original data. There are
the same number of functions as there are depths. They
play the same role as cosine waves in a Fourier analysis. The diagonal elements of W are the magnitudes of
the EOF modes. They give the relative importance of the
modes. The matrix U gives the amplitudes of the modes
in each current profile. There is one row in U for each
profile. It gives the amplitudes of each mode at one time.
As it stands, (3.38) is not a more efficient representation of the data. The gain in efficiency comes from the
fact that the magnitudes of the first few modes are often
much larger than the rest. A good representation of the
data can then come by summing over many fewer than
N modes. The amplitudes of those modes can then fill
a manageable scatter diagram.
There are, however, locations where a few EOF
modes fail to describe all the dominant characteristics
of the current profiles. The characteristic current profile
(CPC) was developed by Jeans et al. [3.116] to work
with those cases. For each current velocity time series,
a number of possible states are defined at each selected
depth level, and possible characteristic profiles are constructed from every permutation of these states. The
number of measured profiles corresponding to each of
these possible characteristic profiles is then counted and
percentage occurrence values derived. The reduction in
the number of profile shapes is accomplished by selecting a relatively small number of depth levels.
Self-organizing maps (SOM) are useful to better
categorize current profiles. The SOM process begins
with a two-component EOF analysis. Then, a nonlinear
cluster analysis groups the thousands of current profiles into a smaller number of clusters [3.117]. The EOF
amplitudes are varied to produce a two-dimensional array of current profiles. Each original profile is assigned
to the EOF profile that it best matches. The EOF profiles are modified by taking weighted averages of the
neighboring profiles in the grid. Then, the original profiles are re-assigned to the modified profiles that they
best match. This process is iterated until the sum of
differences between the SOM profiles and the original
profiles is minimized. If the array of profiles is small,
there can be a lot of variability around some of the
weaker SOM profiles. The variability around the SOM
profiles decreases when more profiles are used.
Prevosto et al. [3.114] found that using a few hundred profiles calculated by one of these methods gave
good accuracy in fatigue damage calculations.
3.7 Extreme Criteria
3.7.1 Risk and Reliability
Metocean design specifications should be set considering the risk and cost of failure. The risk tolerance is
different for structures that are not normally manned
and structures that are evacuated before severe storm
conditions than it is for structures that are manned and
not evacuated before severe storms. Gulf of Mexico
structures are evacuated upon the approach of a hurricane. North Sea structures remain manned during
frequent severe winter storms. For structures that are
unmanned or evacuated, the risk calculation is complicated but straightforward. The cost of strengthening
the structure is balanced against the monetary cost of
structural damage or failure. The cost includes not only
repairing or replacing the structure, but sometimes also
lost production, pollution-related costs, and damage to
corporate image. These costs can be an order of magnitude greater than the cost of replacement.
The failure rate is found by calculating the ultimate
strength of the structure and comparing it to the metocean loading at different probability levels. The cost of
strengthening the structure is then added to the cost of
failure after strengthening. If the total cost is lower, designing to a lower probability of failure is economically
justified. For standard steel jacket structures, an annual
failure rate near 10
3 is generally appropriate. This is
consistent with the normal practice of designing for
a 100-y storm because steel jackets have considerable
reserve strength beyond the first yielding of a member.
Establishing an appropriate failure rate for a manned structure is conceptually more difficult because no
66 Part A Fundamentals
available from models or measurements. All of that
information must be condensed into a manageable number of cases for analysis. Prevosto et al. [3.114] discuss
three methods for doing this and compare the results of
using them with a full analysis of all profiles.
Empirical orthogonal functions (EOF) provide
a method for capturing the important characteristics
of current profiles in a few variables. Forristall and
Cooper [3.115] outline the method and give examples.
Singular value decomposition permits any matrix A to
be decomposed as
A ij D
N
X
kD1
w k U ik V jk :
(3.38)
Each current profile is written as a row in matrix A and
each column represents the time series at one depth. The
columns of V are called the EOFs. Each EOF is a vector
with a value at each depth in the original data. There are
the same number of functions as there are depths. They
play the same role as cosine waves in a Fourier analysis. The diagonal elements of W are the magnitudes of
the EOF modes. They give the relative importance of the
modes. The matrix U gives the amplitudes of the modes
in each current profile. There is one row in U for each
profile. It gives the amplitudes of each mode at one time.
As it stands, (3.38) is not a more efficient representation of the data. The gain in efficiency comes from the
fact that the magnitudes of the first few modes are often
much larger than the rest. A good representation of the
data can then come by summing over many fewer than
N modes. The amplitudes of those modes can then fill
a manageable scatter diagram.
There are, however, locations where a few EOF
modes fail to describe all the dominant characteristics
of the current profiles. The characteristic current profile
(CPC) was developed by Jeans et al. [3.116] to work
with those cases. For each current velocity time series,
a number of possible states are defined at each selected
depth level, and possible characteristic profiles are constructed from every permutation of these states. The
number of measured profiles corresponding to each of
these possible characteristic profiles is then counted and
percentage occurrence values derived. The reduction in
the number of profile shapes is accomplished by selecting a relatively small number of depth levels.
Self-organizing maps (SOM) are useful to better
categorize current profiles. The SOM process begins
with a two-component EOF analysis. Then, a nonlinear
cluster analysis groups the thousands of current profiles into a smaller number of clusters [3.117]. The EOF
amplitudes are varied to produce a two-dimensional array of current profiles. Each original profile is assigned
to the EOF profile that it best matches. The EOF profiles are modified by taking weighted averages of the
neighboring profiles in the grid. Then, the original profiles are re-assigned to the modified profiles that they
best match. This process is iterated until the sum of
differences between the SOM profiles and the original
profiles is minimized. If the array of profiles is small,
there can be a lot of variability around some of the
weaker SOM profiles. The variability around the SOM
profiles decreases when more profiles are used.
Prevosto et al. [3.114] found that using a few hundred profiles calculated by one of these methods gave
good accuracy in fatigue damage calculations.
3.7 Extreme Criteria
3.7.1 Risk and Reliability
Metocean design specifications should be set considering the risk and cost of failure. The risk tolerance is
different for structures that are not normally manned
and structures that are evacuated before severe storm
conditions than it is for structures that are manned and
not evacuated before severe storms. Gulf of Mexico
structures are evacuated upon the approach of a hurricane. North Sea structures remain manned during
frequent severe winter storms. For structures that are
unmanned or evacuated, the risk calculation is complicated but straightforward. The cost of strengthening
the structure is balanced against the monetary cost of
structural damage or failure. The cost includes not only
repairing or replacing the structure, but sometimes also
lost production, pollution-related costs, and damage to
corporate image. These costs can be an order of magnitude greater than the cost of replacement.
The failure rate is found by calculating the ultimate
strength of the structure and comparing it to the metocean loading at different probability levels. The cost of
strengthening the structure is then added to the cost of
failure after strengthening. If the total cost is lower, designing to a lower probability of failure is economically
justified. For standard steel jacket structures, an annual
failure rate near 10
3 is generally appropriate. This is
consistent with the normal practice of designing for
a 100-y storm because steel jackets have considerable
reserve strength beyond the first yielding of a member.
Establishing an appropriate failure rate for a manned structure is conceptually more difficult because no