Statistical Characterization of Hazards and Risk in Coastal Areas 26.2 Quantifying Coastal Hazards/Risks 571
Part C | 26.2
a form for estimating the expected interval between
exceedances of a particular value of x, we obtain a relationship of the form
T.x/ D
1
.1 F.x//
;
(26.4)
where T.x/ is the mean return period (or interval), and
is the sampling frequency. Even though (26.4) is often written without in the denominator, its omission
can lead to a dimensional inconsistency. The units used
to define (decade
1 , year
1 , month
1 , day
1 , etc.)
provide the information which characterizes T.x/. In
engineering, it is common practice to use years for
T.x/, in which case the expected value annual probability of exceeding a particular value of x is given by
1=T.x/. Thus, a 100-year mean return interval is associated with an event that has an annual exceedance
probability (AEP) of 0.01.
Here we will use an example of some results taken
from wave hindcasts (the reconstruction of wave conditions from simulations of past events) for a location
in the southern Gulf of Mexico area to show the importance of considering the frequency term in our
estimation of return periods. In this region, two different
meteorological phenomena can generate large waves,
tropical cyclones, and Northers. The latter type of wave
generation occurs when a cold front crosses the Gulf of
Mexico and produces very strong surface wind speeds
behind it. For both hurricanes and Northers, the results
are given for sites located in deep water and in nominal
Northers (deep)
Northers (20 m)
Hurricanes (deep)
Hurricanes (20 m)
0
1
2
3
4
5
6
7
8
9 10
–ln{–ln[F(x)]}
Wave height (m)
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
–1
–1.2
–1.4
Fig. 26.1 Plot of the lnfflnŒF.x/g versus x for waves
generated at points in the southern Gulf of Mexico by hurricanes and Northers at sites in deep water and a depth of
20 m
depths of 20 m. In Fig. 26.1, we plot lnfflnŒF.x/g
against x, using the 20 data points available in each specific sample. Points that plot along a straight line in such
a graph are consistent with a Gumbel distribution.
From Fig. 26.1, it appears that Norther and hurricane hazards are roughly comparable in this region of
the Gulf of Mexico; however, if we include the frequency parameter and estimate the mean return interval,
we obtain Fig. 26.2. In this case, the hazards related
to hurricanes are shown to be considerably larger than
those related to the Northers.
Since the concept of a mean interval between occurrences (return period) is simpler for some people
to grasp, it is often used instead of the AEP concept
in applications; and for that reason we will use it in
some of our figures and discussions. However, it is very
important to recognize that a 100-year event is not constrained to occur only once every 100 years. For example, we can define the probability of exceeding a given
value of x c as 1 minus the probability of that level not
being exceeded in any year, in which case the probability of having no exceedances in n years is given by
P.x < x c jn years/
D 1 P.x < x c jyear 1/ P.x < x c jyear 2/ : : :
P.x < x c jyear n/
D 1
n
Y
iD1
P i .x < x c / D 1 Œ1 F.x c /
n
;
(26.5)
Northers (deep)
Northers (20 m)
Hurricanes (deep)
Hurricanes (20 m)
0
1
2
3
4
5
6
7
8
9 10
Return period (y)
Wave height (m)
10
2
10
1
10
0
Fig. 26.2 Plot of the mean return period versus wave
height for waves generated at points in the southern Gulf of
Mexico by hurricanes and Northers at sites in deep water
and a depth of 20 m
Part C | 26.2
a form for estimating the expected interval between
exceedances of a particular value of x, we obtain a relationship of the form
T.x/ D
1
.1 F.x//
;
(26.4)
where T.x/ is the mean return period (or interval), and
is the sampling frequency. Even though (26.4) is often written without in the denominator, its omission
can lead to a dimensional inconsistency. The units used
to define (decade
1 , year
1 , month
1 , day
1 , etc.)
provide the information which characterizes T.x/. In
engineering, it is common practice to use years for
T.x/, in which case the expected value annual probability of exceeding a particular value of x is given by
1=T.x/. Thus, a 100-year mean return interval is associated with an event that has an annual exceedance
probability (AEP) of 0.01.
Here we will use an example of some results taken
from wave hindcasts (the reconstruction of wave conditions from simulations of past events) for a location
in the southern Gulf of Mexico area to show the importance of considering the frequency term in our
estimation of return periods. In this region, two different
meteorological phenomena can generate large waves,
tropical cyclones, and Northers. The latter type of wave
generation occurs when a cold front crosses the Gulf of
Mexico and produces very strong surface wind speeds
behind it. For both hurricanes and Northers, the results
are given for sites located in deep water and in nominal
Northers (deep)
Northers (20 m)
Hurricanes (deep)
Hurricanes (20 m)
0
1
2
3
4
5
6
7
8
9 10
–ln{–ln[F(x)]}
Wave height (m)
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
–1
–1.2
–1.4
Fig. 26.1 Plot of the lnfflnŒF.x/g versus x for waves
generated at points in the southern Gulf of Mexico by hurricanes and Northers at sites in deep water and a depth of
20 m
depths of 20 m. In Fig. 26.1, we plot lnfflnŒF.x/g
against x, using the 20 data points available in each specific sample. Points that plot along a straight line in such
a graph are consistent with a Gumbel distribution.
From Fig. 26.1, it appears that Norther and hurricane hazards are roughly comparable in this region of
the Gulf of Mexico; however, if we include the frequency parameter and estimate the mean return interval,
we obtain Fig. 26.2. In this case, the hazards related
to hurricanes are shown to be considerably larger than
those related to the Northers.
Since the concept of a mean interval between occurrences (return period) is simpler for some people
to grasp, it is often used instead of the AEP concept
in applications; and for that reason we will use it in
some of our figures and discussions. However, it is very
important to recognize that a 100-year event is not constrained to occur only once every 100 years. For example, we can define the probability of exceeding a given
value of x c as 1 minus the probability of that level not
being exceeded in any year, in which case the probability of having no exceedances in n years is given by
P.x < x c jn years/
D 1 P.x < x c jyear 1/ P.x < x c jyear 2/ : : :
P.x < x c jyear n/
D 1
n
Y
iD1
P i .x < x c / D 1 Œ1 F.x c /
n
;
(26.5)
Northers (deep)
Northers (20 m)
Hurricanes (deep)
Hurricanes (20 m)
0
1
2
3
4
5
6
7
8
9 10
Return period (y)
Wave height (m)
10
2
10
1
10
0
Fig. 26.2 Plot of the mean return period versus wave
height for waves generated at points in the southern Gulf of
Mexico by hurricanes and Northers at sites in deep water
and a depth of 20 m