Part C | 26.3
574 Part C Coastal Design
in terms of the rank of the ordered values (m) and the
total number of observations N,
F.x/ D
m
N C 1
;
(26.6)
which can be seen as an equivalent form to the Gumbel plotting position given in Table 26.1; the return
period there is defined as the inverse of the nonexceedance probability. However, the manner in which
such information is used in nonparametric methods is
very different to its use in parametric methods.
To help demonstrate the basic premise of resampling, let us assume that our data includes 30 samples
taken over 50 years. Thus, the range of the CDF covered would be 1=31.0:0323/ to 30=31.0:9677/, or more
generally 1=.N C 1/ to N=.N C 1/, where N is the number of samples. Based on (26.4), we would estimate that
the return periods explicitly covered by this data cover
the range 1:7251:67 years, recalling that the value of
would be 30=5 for this example. If a specific design
requirement fell into this range, the results from the
EST could be used directly, without any extrapolation.
However, two key points should be understood in applications of this methodology. First, outside of the CDF
range covered by the data, it is necessary to extrapolate
parametrically to obtain estimates of the CDF as a function of x; so in this example, for return periods larger
than 51:67 years or less than 1:72 years, some fitting algorithm similar to that used in parametric methods must
be utilized. Second, any sample drawn from nature cannot be regarded as the parent population constructed
from of all possible samples.
Let us assume for simplicity that we have a fitting
methodology, which extends the range of the CDF from
0 to 1. In this case, we can use a random number generator to construct a random sequence of values for
both the number of storms in a particular year (based
on a Poisson distribution) and the value of the variable
of interest in all of the storms that occur. The latter of
these is obtained by assuming that each random number
is a CDF value and inverting this value (using the defined CDF-x relationship) to obtain an estimate of x. If
we simulated 100 sets of 50 years using this method, we
could obtain an estimate of the magnitude of the variability we might expect in a particular 50-year sample;
however, it would still be assuming that the initial sample represented the population characteristics. In fact,
the actual variability in a subsequent 50-year interval
will include contributions from the potential deviation
of the initial sample from the overall population sample plus the contribution of deviations due to sampling
given an initial distribution.
26.3 Historical Perspective
It is important to recognize the role of thresholds in
coastal hazards and their relevance to risk. For example,
a flood over a particular value might overtop a levee,
or a wave height greater than some limit might impact critical components of an offshore oil platform,
and a toxin level over some threshold might endanger
the environment. Any of these occurrences might have
serious consequences on lives and property. Thus, the
success or failure of engineering designs and decision
making in general can be very sensitive to whether or
not a particular threshold is exceeded. However, is it really this simple? Are we certain that any value lower
than an exactly specified threshold (even by a minuscule amount) will not have serious consequences, while
any value equal to or greater than the threshold will
definitely have serious consequences? Given that we
never have complete information about the processes
and the probabilities of different combinations of environment conditions, it is intuitively obvious that such
thresholds should be considered only as an approximation to a level at which consequences are initiated and
our estimates of the probability of these thresholds being surpassed cannot be exact. Here we see that both
alleatory and epistemic uncertainty affect our estimate
of this threshold probability.
Traditionally, the role of conservatism is at least in
part to compensate for uncertainties in our knowledge
base, both alleatory and epistemic uncertainty, via some
factor used to shift the design level in a manner that
compensates for the lack of knowledge (i. e., an attempt
to account for inaccuracies in our theories and models
of nature and/or in our quantification of the probabilities
of factors which influence these levels). In many engineering fields, it has been common practice to include
the effects of such uncertainty through the application
of a codified safety factor or some similar device; however, newer methods are beginning to evolve toward
the consideration of risk in a more formal probabilistic sense.
Historically, two fundamentally different approaches have been used to estimate threshold values
for the design of critical infrastructure: approaches
based on the establishment of a deterministic estimate
of an upper limit for a hazard level and approaches
574 Part C Coastal Design
in terms of the rank of the ordered values (m) and the
total number of observations N,
F.x/ D
m
N C 1
;
(26.6)
which can be seen as an equivalent form to the Gumbel plotting position given in Table 26.1; the return
period there is defined as the inverse of the nonexceedance probability. However, the manner in which
such information is used in nonparametric methods is
very different to its use in parametric methods.
To help demonstrate the basic premise of resampling, let us assume that our data includes 30 samples
taken over 50 years. Thus, the range of the CDF covered would be 1=31.0:0323/ to 30=31.0:9677/, or more
generally 1=.N C 1/ to N=.N C 1/, where N is the number of samples. Based on (26.4), we would estimate that
the return periods explicitly covered by this data cover
the range 1:7251:67 years, recalling that the value of
would be 30=5 for this example. If a specific design
requirement fell into this range, the results from the
EST could be used directly, without any extrapolation.
However, two key points should be understood in applications of this methodology. First, outside of the CDF
range covered by the data, it is necessary to extrapolate
parametrically to obtain estimates of the CDF as a function of x; so in this example, for return periods larger
than 51:67 years or less than 1:72 years, some fitting algorithm similar to that used in parametric methods must
be utilized. Second, any sample drawn from nature cannot be regarded as the parent population constructed
from of all possible samples.
Let us assume for simplicity that we have a fitting
methodology, which extends the range of the CDF from
0 to 1. In this case, we can use a random number generator to construct a random sequence of values for
both the number of storms in a particular year (based
on a Poisson distribution) and the value of the variable
of interest in all of the storms that occur. The latter of
these is obtained by assuming that each random number
is a CDF value and inverting this value (using the defined CDF-x relationship) to obtain an estimate of x. If
we simulated 100 sets of 50 years using this method, we
could obtain an estimate of the magnitude of the variability we might expect in a particular 50-year sample;
however, it would still be assuming that the initial sample represented the population characteristics. In fact,
the actual variability in a subsequent 50-year interval
will include contributions from the potential deviation
of the initial sample from the overall population sample plus the contribution of deviations due to sampling
given an initial distribution.
26.3 Historical Perspective
It is important to recognize the role of thresholds in
coastal hazards and their relevance to risk. For example,
a flood over a particular value might overtop a levee,
or a wave height greater than some limit might impact critical components of an offshore oil platform,
and a toxin level over some threshold might endanger
the environment. Any of these occurrences might have
serious consequences on lives and property. Thus, the
success or failure of engineering designs and decision
making in general can be very sensitive to whether or
not a particular threshold is exceeded. However, is it really this simple? Are we certain that any value lower
than an exactly specified threshold (even by a minuscule amount) will not have serious consequences, while
any value equal to or greater than the threshold will
definitely have serious consequences? Given that we
never have complete information about the processes
and the probabilities of different combinations of environment conditions, it is intuitively obvious that such
thresholds should be considered only as an approximation to a level at which consequences are initiated and
our estimates of the probability of these thresholds being surpassed cannot be exact. Here we see that both
alleatory and epistemic uncertainty affect our estimate
of this threshold probability.
Traditionally, the role of conservatism is at least in
part to compensate for uncertainties in our knowledge
base, both alleatory and epistemic uncertainty, via some
factor used to shift the design level in a manner that
compensates for the lack of knowledge (i. e., an attempt
to account for inaccuracies in our theories and models
of nature and/or in our quantification of the probabilities
of factors which influence these levels). In many engineering fields, it has been common practice to include
the effects of such uncertainty through the application
of a codified safety factor or some similar device; however, newer methods are beginning to evolve toward
the consideration of risk in a more formal probabilistic sense.
Historically, two fundamentally different approaches have been used to estimate threshold values
for the design of critical infrastructure: approaches
based on the establishment of a deterministic estimate
of an upper limit for a hazard level and approaches