Part C | 26.2
570 Part C Coastal Design
26.2 Quantifying Coastal Hazards/Risks
A detailed treatment of all coastal hazards and risks is
beyond the scope of the work presented here. Environmental/health hazards and risks such as those related
to water quality, chemical spills, and salinity intrusions will not be treated here; however, it is hoped that
the methods discussed in this section will be useful,
at least in a general sense, to those working in these
areas. Additionally, due to the fact that much of the
recent research on coastal hazards/risks has been undertaken in response to Hurricane Katrina, much of the
focus relating to recent developments will be on storm
surges produced by tropical cyclones. Although extratropical storms play a very large role in many coastal
areas, the lack of a recent catastrophic event, such as
flooding the subway system in New York City before
Hurricane Sandy (which was a hybrid mix of tropical cyclone and extratropical characteristics) has meant
that methods used for evaluation of these hazards/risks
have received far less attention than they probably deserve. A single section (Sect. 26.3.6) will be devoted
here to specific issues related to extratropical storms,
following our general treatment of extreme and related
hazards/risks.
Before launching into our treatment of hazard probabilities, it should be recognized that both epistemic and
alleatory uncertainties need to be considered within any
thorough analysis of hazards and associated risks. Limitations in theoretical/numerical predictions come from
many sources, including simplifications in our models to make nature fit our models, the lack of details
in the information utilized by models, and shortcomings in our theoretical knowledge on which model are
based. Similarly, the lack of sample size continues to
be a serious problem in our estimations of extreme values. In some studies, many years of record are simulated
and statements are made which might lead the reader
to believe that a commensurate gain in information
over the existing historical record has somehow been
achieved. However, a detailed review of these studies
reveals the obvious fact that alleatory uncertainty cannot be overcome by simply executing numerical models
over many randomly posed simulations, since the information upon which the parent distribution is based
remains the same.
Many available textbooks provide excellent formal
reviews of statistics, derivations of distributions and
applications specific to hazard/risk assessment [26.12–
16]. Some recent examples of similar books which
focus more on risk issues include [26.17, 18]. Here
we will present only a summary of information which
should allow a reader to gain sufficient familiarity with
these basic statistical methods. As shown in Sect. 26.1,
risk is associated with the probability of consequences.
Since design thresholds are selected in a manner intended to limit consequences to an acceptable level, the
study of natural hazards and associated risks tends to
focus to either on extremely large or extremely small
values in a distribution. Here we will focus on the upper end of the distribution, since these represent the
major design consideration for most natural disasters in
coastal areas.
We start here with the conventional definition of the
probability density function (PDF) of a continuous variable x with the following properties
p.x/ 0 ;
(26.1a)
1
Z
1
p.x/ dx D 1 ;
(26.1b)
b
Z
a
p.x/ dx D P.a < x < b/ ;
(26.1c)
where p.x/ is the probability density function for the
variable x and P.a < x < b/ is the probability of x
falling between the values of a and b. It is clear from
the form of (26.1c) that the PDF is not a dimensionless
function. Since dx has the same units as x, the dimensions of p.x/ must be x
1 . As an example if we allow
the PDF to be given by
p.x/ D ke
qx
;
(26.2)
k and q must both have dimensions of x
1 ; otherwise
the integral would not produce results with the correct
dimensions. A simple example with q D 1, a D 1 and
b D 3 yields a value for (26.1c) equal to e
1
e
3
D
0:318.
The cumulative distribution function (CDF) of x is
equal to the probability that a value is less than or equal
to a particular value of x,
F.x/ D
x
Z
1
p.x/ dx ;
(26.3)
where F.x/ is the CDF of x. F.x/ is a dimensionless
function since it is the product of two functions with
dimensions x and x
1 ; however, in engineering applications, it is very important to recognize that it becomes
implicitly related to time when we convert to expected
mean recurrence intervals. This is because, even though
the time between samples does not appear explicitly in
F.x/, it enters into the CDF implicitly via the time interval between samples. When we convert (26.3) into
570 Part C Coastal Design
26.2 Quantifying Coastal Hazards/Risks
A detailed treatment of all coastal hazards and risks is
beyond the scope of the work presented here. Environmental/health hazards and risks such as those related
to water quality, chemical spills, and salinity intrusions will not be treated here; however, it is hoped that
the methods discussed in this section will be useful,
at least in a general sense, to those working in these
areas. Additionally, due to the fact that much of the
recent research on coastal hazards/risks has been undertaken in response to Hurricane Katrina, much of the
focus relating to recent developments will be on storm
surges produced by tropical cyclones. Although extratropical storms play a very large role in many coastal
areas, the lack of a recent catastrophic event, such as
flooding the subway system in New York City before
Hurricane Sandy (which was a hybrid mix of tropical cyclone and extratropical characteristics) has meant
that methods used for evaluation of these hazards/risks
have received far less attention than they probably deserve. A single section (Sect. 26.3.6) will be devoted
here to specific issues related to extratropical storms,
following our general treatment of extreme and related
hazards/risks.
Before launching into our treatment of hazard probabilities, it should be recognized that both epistemic and
alleatory uncertainties need to be considered within any
thorough analysis of hazards and associated risks. Limitations in theoretical/numerical predictions come from
many sources, including simplifications in our models to make nature fit our models, the lack of details
in the information utilized by models, and shortcomings in our theoretical knowledge on which model are
based. Similarly, the lack of sample size continues to
be a serious problem in our estimations of extreme values. In some studies, many years of record are simulated
and statements are made which might lead the reader
to believe that a commensurate gain in information
over the existing historical record has somehow been
achieved. However, a detailed review of these studies
reveals the obvious fact that alleatory uncertainty cannot be overcome by simply executing numerical models
over many randomly posed simulations, since the information upon which the parent distribution is based
remains the same.
Many available textbooks provide excellent formal
reviews of statistics, derivations of distributions and
applications specific to hazard/risk assessment [26.12–
16]. Some recent examples of similar books which
focus more on risk issues include [26.17, 18]. Here
we will present only a summary of information which
should allow a reader to gain sufficient familiarity with
these basic statistical methods. As shown in Sect. 26.1,
risk is associated with the probability of consequences.
Since design thresholds are selected in a manner intended to limit consequences to an acceptable level, the
study of natural hazards and associated risks tends to
focus to either on extremely large or extremely small
values in a distribution. Here we will focus on the upper end of the distribution, since these represent the
major design consideration for most natural disasters in
coastal areas.
We start here with the conventional definition of the
probability density function (PDF) of a continuous variable x with the following properties
p.x/ 0 ;
(26.1a)
1
Z
1
p.x/ dx D 1 ;
(26.1b)
b
Z
a
p.x/ dx D P.a < x < b/ ;
(26.1c)
where p.x/ is the probability density function for the
variable x and P.a < x < b/ is the probability of x
falling between the values of a and b. It is clear from
the form of (26.1c) that the PDF is not a dimensionless
function. Since dx has the same units as x, the dimensions of p.x/ must be x
1 . As an example if we allow
the PDF to be given by
p.x/ D ke
qx
;
(26.2)
k and q must both have dimensions of x
1 ; otherwise
the integral would not produce results with the correct
dimensions. A simple example with q D 1, a D 1 and
b D 3 yields a value for (26.1c) equal to e
1
e
3
D
0:318.
The cumulative distribution function (CDF) of x is
equal to the probability that a value is less than or equal
to a particular value of x,
F.x/ D
x
Z
1
p.x/ dx ;
(26.3)
where F.x/ is the CDF of x. F.x/ is a dimensionless
function since it is the product of two functions with
dimensions x and x
1 ; however, in engineering applications, it is very important to recognize that it becomes
implicitly related to time when we convert to expected
mean recurrence intervals. This is because, even though
the time between samples does not appear explicitly in
F.x/, it enters into the CDF implicitly via the time interval between samples. When we convert (26.3) into