Statistical Characterization of Hazards and Risk in Coastal Areas 26.3 Historical Perspective 579
Part C | 26.3
or a three-parameter distribution (for the case that either " ¤ 0 or "
0
¤ 0). As noted here, samples of coastal
winds, waves, and surges often contain very significant effects related to mixed populations, such as the
obvious differences in statistical populations in surges
generated by tropical cyclones making landfall some
distance away from a site versus surges generated by
a direct tropical-cyclone hits. These effects typically far
outweigh any differences in theoretical justifications for
using a GEV versus a GPD analysis. Because of this, it
is probably better to develop a good level of familiarity with a tool of preference and use it appropriately
(either the GEV or the GPD) rather than to be able to
apply both distributions only marginally.
Returning to the problem of record lengths that
are short relative to the mean recurrence interval of
the design event, it is necessary to quantify the impact of randomness in the samples on the estimated
values before we can assess the importance of this issue. As mentioned at the end of the last section, the
quality of individual samples is an important factor
in the estimation of extremes. Once routine measurements began to become available, there was an implicit
expectation on the part of some that measurements
would replace model estimates as the primary source of
information used in the estimation of extremes. However, two sources of uncertainty can affect estimated
extreme values: epistemic uncertainty in the sampled
data (for example, theoretical unknowns and errors in
model results) and alleatory uncertainty in the sampled
data (natural randomness in the sample). Thus, there is
a trade-off between using a short interval of very high
quality measurements, removing most of the epistemic
uncertainty, and using a long interval of lesser quality
model results, which would reduce the alleatory uncertainty but would increase the epistemic uncertainty
component.
Many studies in the 1960s and 1970s utilized a combination of theoretical formulations and computer simulations to establish guidelines for the standard deviations of estimates (sometimes referred to as confidence
bands) around the value of x at a specific AEP. In most
cases, the range of interest for design applications is
beyond the range of observations. For example, the typical mean return interval was 100 years in most early
design studies. However, in mapping flood zones for
FEMA studies today, values up to the 500-year mean
recurrence interval are required. For other applications,
such as large offshore structures and coastal defenses
in many areas of the world, the mean recurrence intervals of interest tend to be in the 1000s or even 10 000s
of years, and probabilistic estimates for highly critical infrastructure such as nuclear power plants have
been established to use mean recurrence intervals in the
range of 1 000 000 to 10 000 000 years. Such estimates
must consider the inherent uncertainties in extrapolations that are many times the length of the observational
record. Estimates of standard deviations can be obtained by a number of methods based on the GEV or
GPD forms [26.49, 54–56].
A useful form for the sampling uncertainty was
given by Gringorten [26.57, 58]. Using a combination of theoretical characteristics and numerical testing,
he showed that the expected root-mean-square (RMS)
error of an estimated value as a function of mean recurrence interval in a Gumbel distribution could be
estimated from the relationship
T D
s
1:1000y 2 C 1:1396y C 1
N
;
(26.11)
where is the distribution standard deviation; T is the
rms error at return period, T; N is the number of samples used to estimate the distribution parameters; and y
is the reduced Gumbel variant given by y D .x ˛/=ˇ;
x is the value of the variable; and ˛ and ˇ are the parameters of the Gumbel distribution.
For a Gumbel distribution, the reduced variate and
return period are related by
y D Dln
Ä
ln
 T
T 1
ÃÃ
;
(26.12)
which for T > 7 years approaches an exponential form
given by
Â
T
1
2
Ã
! e
y
:
(26.13)
Equation (26.11) shows that the rms error at a fixed
return period is related to the distribution standard deviation and the square root of a dimensionless factor
involving the ratio of different powers of y (y
2 , y
1 , and
y
0 ) to the number of samples used to define the parameters. In our previous description of the Gumbel
distribution, we saw that the CDF for such a distribution has a double exponential form.
By the method of moments, the Gumbel parameters
can be shown to be given by
D c d; ; D
p
6
d ;
(26.14)
where is Euler’s constant (D 0:57721 : : :), is the
distribution mean, and is the distribution standard deviation.
Thus, the distribution standard deviation is related
to the coefficient of variability for the distribution and
Part C | 26.3
or a three-parameter distribution (for the case that either " ¤ 0 or "
0
¤ 0). As noted here, samples of coastal
winds, waves, and surges often contain very significant effects related to mixed populations, such as the
obvious differences in statistical populations in surges
generated by tropical cyclones making landfall some
distance away from a site versus surges generated by
a direct tropical-cyclone hits. These effects typically far
outweigh any differences in theoretical justifications for
using a GEV versus a GPD analysis. Because of this, it
is probably better to develop a good level of familiarity with a tool of preference and use it appropriately
(either the GEV or the GPD) rather than to be able to
apply both distributions only marginally.
Returning to the problem of record lengths that
are short relative to the mean recurrence interval of
the design event, it is necessary to quantify the impact of randomness in the samples on the estimated
values before we can assess the importance of this issue. As mentioned at the end of the last section, the
quality of individual samples is an important factor
in the estimation of extremes. Once routine measurements began to become available, there was an implicit
expectation on the part of some that measurements
would replace model estimates as the primary source of
information used in the estimation of extremes. However, two sources of uncertainty can affect estimated
extreme values: epistemic uncertainty in the sampled
data (for example, theoretical unknowns and errors in
model results) and alleatory uncertainty in the sampled
data (natural randomness in the sample). Thus, there is
a trade-off between using a short interval of very high
quality measurements, removing most of the epistemic
uncertainty, and using a long interval of lesser quality
model results, which would reduce the alleatory uncertainty but would increase the epistemic uncertainty
component.
Many studies in the 1960s and 1970s utilized a combination of theoretical formulations and computer simulations to establish guidelines for the standard deviations of estimates (sometimes referred to as confidence
bands) around the value of x at a specific AEP. In most
cases, the range of interest for design applications is
beyond the range of observations. For example, the typical mean return interval was 100 years in most early
design studies. However, in mapping flood zones for
FEMA studies today, values up to the 500-year mean
recurrence interval are required. For other applications,
such as large offshore structures and coastal defenses
in many areas of the world, the mean recurrence intervals of interest tend to be in the 1000s or even 10 000s
of years, and probabilistic estimates for highly critical infrastructure such as nuclear power plants have
been established to use mean recurrence intervals in the
range of 1 000 000 to 10 000 000 years. Such estimates
must consider the inherent uncertainties in extrapolations that are many times the length of the observational
record. Estimates of standard deviations can be obtained by a number of methods based on the GEV or
GPD forms [26.49, 54–56].
A useful form for the sampling uncertainty was
given by Gringorten [26.57, 58]. Using a combination of theoretical characteristics and numerical testing,
he showed that the expected root-mean-square (RMS)
error of an estimated value as a function of mean recurrence interval in a Gumbel distribution could be
estimated from the relationship
T D
s
1:1000y 2 C 1:1396y C 1
N
;
(26.11)
where is the distribution standard deviation; T is the
rms error at return period, T; N is the number of samples used to estimate the distribution parameters; and y
is the reduced Gumbel variant given by y D .x ˛/=ˇ;
x is the value of the variable; and ˛ and ˇ are the parameters of the Gumbel distribution.
For a Gumbel distribution, the reduced variate and
return period are related by
y D Dln
Ä
ln
 T
T 1
ÃÃ
;
(26.12)
which for T > 7 years approaches an exponential form
given by
Â
T
1
2
Ã
! e
y
:
(26.13)
Equation (26.11) shows that the rms error at a fixed
return period is related to the distribution standard deviation and the square root of a dimensionless factor
involving the ratio of different powers of y (y
2 , y
1 , and
y
0 ) to the number of samples used to define the parameters. In our previous description of the Gumbel
distribution, we saw that the CDF for such a distribution has a double exponential form.
By the method of moments, the Gumbel parameters
can be shown to be given by
D c d; ; D
p
6
d ;
(26.14)
where is Euler’s constant (D 0:57721 : : :), is the
distribution mean, and is the distribution standard deviation.
Thus, the distribution standard deviation is related
to the coefficient of variability for the distribution and