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Risk Analysis of a Protected Hurricane-Prone Region.
II: Computations and Illustrations
Bilal M. Ayyub, P.E., F.ASCE
1
; Jerry Foster, M.ASCE
2
; William L. McGill, P.E., M.ASCE
3
; and
Harvey W. Jones
4
Abstract: This paper describes a case study implementing a methodology for assessing risks to protected hurricane-prone regions. A
simple hurricane protection system is constructed to illustrate the required inputs for the system definition, computations, and hazard and
risk profiles. The inputs include the required specifications for basin and subbasin reaches, transitions, and associated fragilities, closures,
and storm parameters. Moreover, the case study produces elevation- and loss-exceedance probability and rate curves for each subbasin and
the system as a whole, and demonstrates quantitative benefit-cost analysis using this risk information. The implementation of the risk
model is packaged as the Flood Risk Analysis for Tropical Storm Environments tool currently in use by the U.S. Army Corps of
Engineering Interagency Performance Evaluation Team charged with assessing hurricane risks to the New Orleans and Southeast Loui-
siana, and proposed changes to the hurricane protection system.
DOI: 10.1061/共ASCE兲1527-6988共2009兲10:2共54兲
CE Database subject headings: Risk management; Hurricanes; Natural disasters; Decision making; Levees; Safety; Uncertainty
principles
.
Risk Analysis Framework
The severity of losses resulting from the impact of hurricane Kat-
rina on the United States has prompted significant national invest-
ment in intellectual capital toward understanding and quantifying
the risks to hurricane-prone regions. As noted by many leading
risk researchers, quantitative risk analysis provides a means to
inform the decision making process by communicating the prob-
ability and severity of potential risk scenarios 共Ayyub 2003兲.
When expressed in terms of clearly defined measures 共e.g., dol-
lars兲, knowledge of risk facilitates defensible benefit-cost analysis
to determine the cost effectiveness of alternative risk mitigation
strategies. While current national attention is on applying quanti-
tative risk methods for the reconstruction effort in New Orleans
共Ayyub et al. 2009兲, it can be anticipated that other regions will be
more inclined to adopt risk methods to manage the risks associ-
ated with hurricanes and other natural and human-caused hazards
共Ayyub et al. 2007兲.
In the engineering community, risk is generally defined as the
potential of losses for a system resulting from an uncertain expo-
sure to a hazard or as a result of an uncertain event 共Ayyub 2003兲.
Risk is quantified as the rate 共measured in events per unit time,
such as years兲 that lives, economic, environmental, and social/
cultural losses will occur due to the nonperformance of an engi-
neered system or component. The nonperformance of the system
or component can be quantified as the probability that specific
loads 共or demands兲 exceed respective strengths 共or capacities兲
causing the system or component to fail, and losses are defined as
the adverse impacts of that failure if it occurs. Risk can be viewed
to be a multidimensional quantity that captures event-occurrence
rate 共or probability兲, event-occurrence consequences, conse-
quence significance, and the population at risk. As a measure, risk
is commonly represented as a pair of the rate 共or probability兲 of
occurrence of an event, and the outcomes or consequences asso-
ciated with the event’s occurrence that account for system weak-
ness, i.e., vulnerabilities, and is commonly expressed as
risk = event rate共or probability兲 ⫻ vulnerability
⫻ consequences of failure 共1兲
This equation not only defines risk but also offers strategies to
control or manage risk: by making the system more reliable or by
reducing the potential losses resulting from a failure through vul-
nerability reduction. Decisions concerning investments in systems
designed to control natural hazards are best made by explicitly
and quantitatively considering the risks that the systems pose to
public safety and property.
Ayyub et al. 共2009兲 applied probabilistic risk analysis to de-
velop an overall methodology for assessing hurricane flood risk to
a geographic region that considers the reliability of a hurricane
protection system 共HPS兲. The overall methodology is illustrated
in Fig. 1. In general, a HPS consists of the following elements:
1. Basins, or areas of land protected against flooding from ad-
jacent bodies of water by levees, dykes, or floodwalls;
1
Professor and Director, Center for Technology and Systems Manage-
ment, Dept. of Civil and Environmental Engineering, Univ. of Maryland,
College Park, MD 20742 共corresponding author兲. E-mail: ba@umd.edu
2
IPET Risk and Reliability Team Leader, Headquarters, U.S. Army
Corps of Engineers, 1381 Teaberry Lane, Severn, MD 21144. E-mail:
jerry.l.foster@usace.army.mil
3
Graduate Research Assistant, Center for Technology and Systems
Management, Dept. of Civil and Environmental Engineering, Univ. of
Maryland, College Park, MD 20742. E-mail: wmcgill@umd.edu
4
Associate Technical Director, Information Technology Laboratory,
U.S. Army Engineer Research and Development Center, 3909 Halls Ferry
Rd., Vicksburg, MS 39180-6199.
Note. Discussion open until October 1, 2009. Separate discussions
must be submitted for individual papers. The manuscript for this paper
was submitted for review and possible publication on May 10, 2007;
approved on March 18, 2008. This paper is part of the Natural Hazards
Review, Vol. 10, No. 2, May 1, 2009. ©ASCE, ISSN 1527-6988/2009/2-
54–67/$25.00.
54 / NATURAL HAZARDS REVIEW © ASCE / MAY 2009
Downloaded 20 May 2009 to 140.194.193.5. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright
2. Subbasins, or divisions within a basin to isolate flooding
given failure of a levee, dyke, or floodwall;
3. Reaches, or discrete stretches along the wetted perimeter of a
basin that have similar characteristics along the length 共e.g.,
floodwall or levee type and cross section, engineering param-
eters, jurisdiction兲;
4. Transitions, or short reaches adjoining two dissimilar
reaches;
5. Closures within reaches that permit movement of people and
goods during dry conditions; and
6. Pumps that actively drain water from the basins.
USACE 共2006兲 and Ayyub et al. 共2009兲 provide maps and
descriptions of basins defining the HPS of New Orleans as
examples.
A high-level expression for the risk associated with hurricanes
due to the performance of a HPS considers the regional hurricane
rate, , the probability P共h
i
兲 of realizing a hurricane event of
type i given the occurrence of a hurricane, the probability
P共S
j
兩h
i
兲 that the HPS will be left in state j given h
i
, and the
probability P共L ⬎ l 兩h
i
,S
j
兲 with which a consequence measure L
exceeds different levels l. Given this information, the loss-
exceedance probability per event is evaluated as:
P共L ⬎ l兲 =
兺
i
兺
j
P共h
i
兲P共S
j
兩h
i
兲P共L ⬎ l兩h
i
,S
j
兲共2兲
and the annual loss-exceedance rate was estimated as follows:
共L ⬎ l兲 =
兺
i
兺
j
P共h
i
兲P共S
j
兩h
i
兲P共L ⬎ l兩h
i
,S
j
兲共3兲
The summation in Eqs. 共2兲 and 共3兲 is over all hurricane types i
and all system states j using a suitable partition of the space of all
possible hurricane and HPS failure scenarios 共Kaplan et al. 2005兲.
Simulation studies of hurricanes for risk analysis require the use
of representative combinations of hurricane parameters and their
respective probabilities. The outcome of this process is a set of
hurricane simulation cases and their respective conditional rates
P共h
i
兲. Additional information on this model is provided by
Ayyub et al. 共2009兲.
An event tree was constructed as shown in Fig. 2 to determine
flooding elevations and displaying the results as inundation con-
tours within the basins. The tree events are as follows:
1. Hurricane initiating event: each hurricane, h
i
, defines a set of
storm surge hydrographs 共with waves兲 for the study area.
Given an overall annual hurricane recurrence rate , each h
i
represents a mutually exclusive partition of an exhaustive set
of representative hurricanes. The probability of occurrence
for a particular hurricane is denoted as P共h
i
兲, and the corre-
sponding annual rate of occurrence
i
=P共h
i
兲;
2. Closure structure and operations 共C兲: this event models the
Fig. 1. Overall risk analysis methodology 共Ayyub et al. 2007兲
NATURAL HAZARDS REVIEW © ASCE / MAY 2009 / 55
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hurricane protection system closures, i.e., gates or ramps, by
providing probabilities of sealing them prior to the hurricane;
3. Precipitation inflow 共Q兲: this event corresponds to the rain-
fall that occurs during a hurricane event. The precipitation
inflow per subbasin is treated as a random variable;
4. Drainage, pumping, and power 共P兲: this event models the
availability of the pumping systems;
5. Overtopping 共O兲: this event models the failure of the protec-
tion system due to overtopping, given that failure has not
occurred by some other 共i.e., nonovertopping兲 failure mode.
If breach failure does not occur, flooding due to overtopping
could still happen; and
6. Breach 共B兲: this event models the failure of the protection
system 共e.g., levees/floodwalls, closures兲 during the hurri-
cane, exclusive of overtopping failures.
The computational details of all event probabilities and associated
water volumes considering interflow among subbasins are pro-
vided by Ayyub et al. 共2009兲. The goal of this paper is to dem-
onstrate this model developed and implemented by the U.S. Army
Corps of Engineers 共USACE兲 Interagency Performance Evalua-
tion Task Force 共IPET兲 using a simple case study of a notional
city. This example highlights the data inputs required to produce
regional elevation- and loss-exceedance probability and rate
curves, including parameters that define a HPS, its fragilities, and
severity of consequence as a function of flood levels.
Case Study: City of Forteville
This section describes the input information 共e.g., storm data,
system description兲 and corresponding risk results for a notional
city with a hurricane protection system. For more information on
the mathematical details of this methodology, the reader is re-
ferred to the companion paper by Ayyub et al. 共2009兲.
Hurricane Protection System Definition
Consider the notional City of Forteville, a city in a hurricane-
prone region with a hurricane protection system as shown in
Fig. 3. The Forteville hurricane protection system 共FHPS兲 can be
divided into four geographically defined basins comprised of one
or more subbasins with reaches, transitions, and gates as de-
Fig. 2. Event tree for quantifying risk. Underlined events 共i.e., Cគ , Pគ , Oគ , and Bគ 兲 are complements of respective events 共i.e., C, P, O, and B兲共Ayyub
et al. 2009兲
Fig. 3. Map of Forteville showing geographic bounds of study region
considered in risk analysis
56 / NATURAL HAZARDS REVIEW © ASCE / MAY 2009
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scribed in Table 1. Subbasins are denoted with the initials of the
basin followed by an integer identification number 共e.g., WF3兲.
Stage-Storage Relationships
A stage-storage curve was constructed for each of the subbasins
as shown in Fig. 4. Stage-storage curves describe the relationship
between storage 共i.e., volume兲 of water in a subbasin and the
corresponding water stage 共i.e., elevation兲. The starting elevation
for each curve in Fig. 4 corresponds to the bottom elevation of the
subbasin. A linear relationship between stage and storage is as-
sumed for simplicity, though in practice the true relationship will
be nonlinear due to uneven terrain, the presence of objects such as
houses, and other obstacles.
Interflow
According to Table 1, the only basin where interflow, that is, the
flow of water between adjacent subbasins, is a concern is South
Forte. Table 2 provides the symmetric interflow matrix for the
FHPS, where each cell provides an elevation of water 共in feet兲 at
which the subbasin in the column overflows into the subbasin of
the corresponding row. In practice, the structure of this interflow
matrix enables the user to define connectedness of an arbitrary
pair of subbasins simply by specifying an overflow elevation.
Absence of a value in any cell indicates that an interflow cannot
occur between the corresponding pair of subbasins.
Reaches, Transitions, and Closures
The FHPS is comprised of reaches and transitions divided among
the basins as described in Tables 3 and 4, respectively. Within the
risk model, transitions and reaches are treated in a similar manner,
the only difference being that the transition shares a hydrograph
with its associated reach. For each reach and transition, the fol-
lowing characteristics are identified:
1. Length of the reach or transition;
2. Nominal top elevation of the reach or transition which is
used to calculate the head of water entering a subbasin due to
overtopping;
3. A design elevation needed for defining the respective fragil-
ity curve;
4. Whether the reach is a levee or a wall; and
5. The corresponding weir coefficient for calculating volume
flow rates from the hydrograph data via the weir formula
共Daugherty et al. 1985兲.
Table 1. Description of Forteville Hurricane Protection System 共FHPS兲
Basin
Number
of
subbasins
Interflow
between
subbasins
Reaches
per
subbasin
Number
of
transitions
Number
of
gates
South Forte 2 Yes 1 — 2
Main Forte 2 No 1 — —
West Forteville 1 — 5 2 2
Forte North 1 — 1 — —
Table 2. Elevation in Feet for Interflow among Forteville Basins
Subbasin SF1 SF2 MF1 MF2 WF1 FN1
SF1 — −9.0 — — — —
SF2 −9.0 — — — — —
MF1 — — — — — —
MF2 — — — — — —
WF1 — — — — — —
FN1 — — — — — —
Note: 1 ft=0.3048 m.
0
10
20
30
40
50
60
7
0
-30 -20 -10 0 10 20 3
0
Sta
g
e
(
Feet
)
Storage (Billions of Cubic Feet)
SF1 (m=0.5, b=7.5)
SF2 (m=1, b=11)
MF1 (m=0.6, b=17.4)
MF2 (m=0.6, b=10.2)
WF1 (m=1.2, b=20)
FN1 (m=2, b=32)
`
*m is the s lope of the stage
s torage curve in billions cubic feet
per foot an d b is the intercept in
billions of cubic feet
1 ft = 0.3048 m
10
6
ft
3
= 28,317 m
3
Fig. 4. Stage-storage relationships for Forteville subbasins
NATURAL HAZARDS REVIEW © ASCE / MAY 2009 / 57
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Moreover, for transitions the associated reach is identified in
order to map it to the hydrograph of the corresponding reach.
Several closures are associated with FHPS reaches as de-
scribed in Table 5. Each closure is described by the following
characteristics:
1. The associated reach containing the closure;
2. Length of the closure;
3. Bottom elevation of the closure which is used to calculate the
head of water entering a subbasin; and
4. Probability that the closure will be left open during a storm
event.
Hurricanes
A set of ten hurricanes is considered with annual recurrence rates
and precipitation volumes described in Table 6. For the purposes
of this example, this set of hurricanes is assumed to be an exhaus-
tive partition of the space of hurricane scenarios. Furthermore, it
is assumed that Forteville will experience hurricanes at a rate of
two per year. A set of hydrographs for the subbasins was con-
structed for each storm as illustrated in Fig. 5 and described in
Table 7 with peak hydrograph elevations expressed as percent-
ages of the reach height relative to sea level. For simplicity, each
hydrograph was assumed to take on a trapezoidal shape spanning
a period of 72 h 共259,200 s兲, with rise to the maximum water
elevation accounting for the first 50% of the hydrograph duration,
sustainment of the peak water elevation described in Table 9 for
the next 15% of the hydrograph duration, and a descend period
accounting for the remainder of the hydrograph duration.
Fragility Relationships and Breach Model
A fragility curve was constructed for each reach and transition
that provides the relationship between probability of breach fail-
Table 3. Description of Forteville Reaches
Basin Reach 共ID兲
Length
共ft兲
Top elevation
共ft兲
Design water
elevation
共ft兲 Reach type
Reach weir
coefficient
FN FN1 共1兲 230,000 17.6 14.0 Levee 2.6
SF SF1 共2兲 75,000 13.1 10.5 Levee 2.6
SF SF2 共3兲 135,000 16.9 13.5 Levee 2.6
MF MF1 共4兲 145,000 15.3 12.5 Wall 3.0
MF MF2 共5兲 55,000 24.3 20.5 Levee 2.6
WF WF1 共6兲 10,000 11.9 6.5 Wall 3.0
WF WF2 共7兲 35,000 14.4 12.0 Levee 2.6
WF WF3 共8兲 30,000 15.5 12.0 Levee 2.6
WF WF4 共9兲 55,000 25.4 22.5 Levee 2.6
WF WF5 共10兲 10,000 13.9 11.0 Wall 3.0
Note: 1 ft=0.3048 m.
Table 4. Description of Forteville Transitions
Transition
Length
共ft兲
Top
elevation
共ft兲
Design water
elevation
共ft兲
Weir
coefficient
Associated
reach
T1 100 12 10 3.0 WF3
T2 50 15 15 3.0 WF4
Note: 1 ft=0.3048 m.
Table 5. Forteville Closure Data
Closure/gate
Associated
reach
Length
共ft兲
Bottom
elevation
共ft兲
Open
probability
G1 SF1 35 1 0.2
G2 SF1 17 7 0.2
G3 SF2 20 10 0.2
G4 SF2 20 10 0.2
G5 WF2 6 6 0.2
G6 WF3 8 7 0.2
Note: 1 ft=0.3048 m.
Table 6. Storm and Precipitation Data
Storm
Rate 共events
per year兲
Precipitation volume per subbasin 共millions of cubic feet of water兲
a
SF1 SF2 MF1 MF2 WF1 FN1
1 9.437E− 01 9.0 20.0 8.0 7.0 10.0 40.0
2 4.990E− 01 74.7 85.7 73.7 72.7 75.7 105.7
3 2.638E− 01 140.3 151.3 139.3 138.3 141.3 171.3
4 1.395E− 01 206.0 217.0 205.0 204.0 207.0 237.0
5 7.376E− 02 271.7 282.7 270.7 269.7 272.7 302.7
6 3.900E− 02 337.3 348.3 336.3 335.3 338.3 368.3
7 2.062E− 02 403.0 414.0 402.0 401.0 404.0 434.0
8 1.090E− 02 468.7 479.7 467.7 466.7 469.7 499.7
9 5.765E− 03 534.3 545.3 533.3 532.3 535.3 565.3
10 3.048E− 03 600.0 1,000.0 700.0 700.0 1,000.0 2,000.0
Note: 1,000,000 ft
3
=28,317 m
3
.
a
0.25 coefficient of variation assumed on all precipitation volumes.
58 / NATURAL HAZARDS REVIEW © ASCE / MAY 2009
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0
5
10
15
20
25
30
35
0 2040608
0
Water Elevation (feet)
Time(hours)
FN (1) SF1 (2)
SF2 (3) MF1 (4)
MF2 (5) WF1 (6)
WF2 (7) WF3 (8)
WF4 (9) WF5 (10)
0
5
10
15
20
25
30
35
0 20406080
Water Elevation
(f
eet
)
Time (hours)
FN (1) SF1 (2)
SF2 (3) MF1 (4)
MF2 (5) WF1 (6)
WF2 (7) WF3 (8)
WF4 (9) WF5 (10)
0
5
10
15
20
25
30
35
0 20406080
Water Elevation (feet)
Time(hours)
FN (1) SF 1 (2)
SF2 (3) MF1 (4)
MF2 (5) WF1 (6)
WF2 (7) WF3 (8)
WF4 (9) WF5 (10)
0
5
10
15
20
25
30
35
0 20406080
Water Elevation
(f
eet
)
Time (hours)
FN (1) SF1 (2)
SF2 (3) MF1 (4)
MF2 (5) WF1 (6)
WF2 (7) WF3 (8)
WF4 (9) WF5 (10)
0
5
10
15
20
25
30
35
0 20406080
Water Elevation (feet)
Time(hours)
FN (1) SF 1 (2)
SF2 (3) MF1 (4)
MF2 (5) WF1 (6)
WF2 (7) WF3 (8)
WF4 (9) WF5 (10)
0
5
10
15
20
25
30
35
0 20406080
Water Elevation
(f
eet
)
Time (hours)
FN (1) SF1 (2)
SF2 (3) MF1 (4)
MF2 (5) WF1 (6)
WF2 (7) WF3 (8)
WF4 (9) WF5 (10)
STORM 6STORM 5
STORM 4STORM 3
STORM 1 STORM 2
0
5
10
15
20
25
30
35
0 20406080
W
ater
El
evat
i
on
(f
eet
)
Time (hours)
FN (1)
SF1 (2)
SF2 (3)
MF1 (4)
MF2 (5)
WF1 (6)
WF2 (7)
WF3 (8)
WF4 (9)
WF5 (10)
0
5
10
15
20
25
30
35
0 2040608
0
Water Elevation (feet)
Time(hours)
FN (1)
SF1 (2)
SF2 (3)
MF1 (4)
MF2 (5)
WF1 (6)
WF2 (7)
WF3 (8)
WF4 (9)
WF5 (10)
0
5
10
15
20
25
30
3
5
0 20406080
W
ater
El
evat
i
on
(f
eet
)
Time (hours)
FN (1) SF1 (2) SF2 (3)
MF1 (4) MF2 (5) WF1 (6)
WF2 (7) WF3 (8) WF4 (9)
WF5 (10)
0
5
10
15
20
25
30
35
0 20406080
Wate r Elevation (feet)
T im e (hours)
FN (1) SF1 (2) SF2 (3)
MF1 (4) MF2 (5) WF1 (6)
WF2 (7) WF3 (8) WF4 (9)
WF5 (10)
STORM 10STORM 9
STORM 7 STORM 8
(a)
(
b
)
Fig. 5. Storm hydrographs 共1 ft= 0.3048 m兲
NATURAL HAZARDS REVIEW © ASCE / MAY 2009 / 59
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ure and water level as shown in Fig. 6 and described in Tables 8
and 9 for reaches and transitions, respectively. Each fragility
curve is nondecreasing with respect to increasing elevation, and is
characterized by eight data points as follows:
1. Low limit with a default probability of breach failure of
10
−12
. The low limit elevation corresponds to the sea level
elevation 共i.e., 0 ft兲. This is the minimum probability of
breach, and applies to all elevations below the low limit;
2. The design elevation is the elevation of water that the reach
or transition was designed to withstand. This elevation is less
than the top elevation;
3. The top elevation is the elevation of the top of the reach or
transition;
4. 0.5 ft overtopping 共OT兲 corresponds to an elevation that is
0.5 ft above the top elevation;
5. 1.0 ft overtopping 共OT兲 corresponds to an elevation that is
1.0 ft above the top elevation;
6. 2.0 ft overtopping 共OT兲 corresponds to an elevation that is
2.0 ft above the top elevation;
7. 3.0 ft overtopping 共OT兲 corresponds to an elevation that is
3.0 ft above the top elevation; and
8. 6.0 ft overtopping 共OT兲 corresponds to an elevation that is
3.0 ft above the top elevation.
By default, the 6.0 ft OT elevation has a probability of breach
failure of 1.0. The probability of breach failure at elevations be-
tween two data points is determined using semilog interpolation,
where the elevation is linear and the probability of breach failure
is logarithmic.
Given that a breach failure occurs, the depth and width of the
breach is determined as a function of maximum hydrograph surge
elevation as described in Table 10 for different reach and transi-
tion materials. This information permits calculation of the vol-
ume of water entering a subbasin due to breach using the weir
formula, where the length of the opening is taken as breach width
and the head of water is taken as the difference between the
Table 7. Maximum Hydrograph Elevation as Function of Storm
Reach
ID
Top elevation
共ft兲
Maximum hydrograph elevation as percentage of reach height by storm
12345678910
FN1 17.6 0.3 0.3 0.3 0.6 0.5 0.6 0.7 0.8 0.9 0.7
SF1 13.1 0.2 0.2 0.4 0.5 0.8 0.9 0.8 1.2 1.2 0.9
SF2 16.9 0.3 0.3 0.4 0.5 0.7 0.7 1 0.7 0.9 0.9
MF1 15.3 0.2 0.3 0.4 0.4 0.6 0.8 0.8 0.6 1 1.5
MF2 24.3 0.2 0.2 0.3 0.5 0.4 0.7 1 0.9 1 1
WF1 11.9 0.2 0.3 0.5 0.5 0.7 0.6 0.9 1.1 0.9 1.2
WF2 14.4 0.2 0.2 0.4 0.4 0.8 0.5 0.7 0.8 1.7 1.1
WF3 15.5 0.2 0.3 0.3 0.5 0.6 0.8 1.2 1 1.3 1.3
WF4 25.4 0.2 0.3 0.4 0.5 0.6 0.7 0.7 0.7 1.2 1.3
WF5 13.9 0.3 0.3 0.4 0.5 0.5 0.6 0.8 1.1 0.7 1
Note: 1 ft=0.3048 m.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Low Limit Design Top 0.5-ft OT 1.0-ft OT 2.0-ft OT 3.0-ft OT 6.0-ft OT
W
ate
rEl
e
v
at
i
o
n
Probabili ty of Fai lure
FN1 (1)
SF1 (2)
SF2 (3)
MF1 (4)
MF2 (5)
WF1 (6)
WF2 (7)
WF3 (8)
WF4 (9)
WF5 (10)
1 ft = 0.3048 m
Fig. 6. Breach fragility curves for Forteville reaches
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height of water from the hydrograph and the top elevation minus
the breach depth. When this difference is negative, the head is set
to zero.
Population and Property at Risk
In order to assess losses due to flooding resulting from a hurri-
cane, a function that maps water elevation to loss is required. For
the purposes of this example, a deterministic mapping from water
elevation in a given subbasin to property damage measured in
millions of dollars is used as described in Fig. 7. A linear rela-
tionship between property damage and water elevation is assumed
with a unique slope m and intercept b for each subbasin as de-
scribed in the legend of Fig. 7. The bottom elevation 共i.e., first
data point兲 of these curves corresponds to the bottom elevation of
the associated subbasins, and the peak elevation 共i.e., last data
point兲 corresponds to the maximum possible property damage
Table 9. Breach Fragilities for Forteville Transitions
Reach
共ID兲
Breach fragility curve
TypeLow limit Design Top 0.5 ft OT 1.0 ft OT 2.0 ft OT 3.0 ft OT 6.0 ft OT
T1 1.00E − 12 1.00E− 12 1.00E− 12 4.47E −07 2.00E −01 9.00E −01 1.00E + 00 1.00E+ 00
Drainage 共D兲
T2 1.00E − 12 1.00E− 12 1.00E− 12 4.47E −07 2.00E −01 7.00E −01 1.00E + 00 1.00E+ 00
Pumping 共P兲
Note: 1 ft=0.3048 m.
Table 8. Breach Fragilities for Forteville Reaches
Reach
共ID兲
Breach fragilities versus water elevation
MaterialLow limit Design Top 0.5 ft OT 1.0 ft OT 2.0 ft OT 3.0 ft OT 6.0 ft OT
FN1 共1兲 1.00E − 12 4.25E − 01 5.75E− 01 5.75E− 01 5.75E −01 9.80E −01 1.00E + 00 1.00E+ 00 Hydraulic fill 共HB兲
SF1 共2兲 1.00E− 12 1.75E− 01 2.75E −01 2.75E −01 2.75E − 01 8.50E− 01 1.00E+ 00 1.00E +00 Clay 共CB兲
SF2 共3兲 1.00E− 12 2.50E− 01 3.75E −01 3.75E −01 3.75E − 01 9.50E− 01 1.00E+ 00 1.00E +00 Sand 共SB兲
MF1 共4兲 1.00E −12 7.50E − 03 2.00E− 02 2.00E− 02 2.00E −02 7.50E −01 9.00E − 01 1.00E+ 00 Wall 共WB兲
MF2 共5兲 1.00E −12 2.50E − 03 7.50E− 03 7.50E− 03 7.50E −03 7.50E −01 9.00E − 01 1.00E+ 00 Hydraulic fill 共HB兲
WF1 共6兲 1.00E− 12 2.50E− 02 5.00E −02 5.00E −02 5.00E − 02 2.50E− 01 5.00E− 01 1.00E +00 Wall 共W7兲
WF2 共7兲 1.00E− 12 7.50E− 01 8.80E −01 8.80E −01 8.80E − 01 8.80E− 01 9.90E− 01 1.00E +00 Hydraulic fill 共H9兲
WF3 共8兲 1.00E− 12 2.50E− 01 5.00E −01 5.00E −01 5.00E − 01 7.50E− 01 9.50E− 01 1.00E +00 Clay 共C9兲
WF4 共9兲 1.00E− 12 1.00E− 12 1.00E −12 1.00E −12 1.00E − 12 9.50E− 01 9.90E− 01 1.00E +00 Sand 共SB兲
WF5 共10兲 1.00E− 12 1.00E− 12 1.00E −12 1.00E −12 1.00E − 12 2.50E− 01 5.00E− 01 1.00E +00 Wall 共W7兲
Note: 1 ft=0.3048 m.
Table 10. Breach Failure Data for Reaches and Transitions
Material Symbol
No overtopping
共depth independent兲
Overtopping depth
共ft兲
0–2 ft 2–5 ft ⬎5ft
Depth
共ft兲
Width
共ft兲
Depth
共ft兲
Width
共ft兲
Depth
共ft兲
Width
共ft兲
Depth
共ft兲
Width
共ft兲
Hydraulic fill
共30,000– 39,999 ft兲
H9 18 4,500 0 0 9 12,000 18 12,000
Hydraulic fill
共⬎50,000 ft兲
HB 18 7,500 0 0 9 20,000 18 20,000
Clay
共30,000– 39,999 ft兲
C9 13 3,000 0 0 3 3,000 13 3,000
Clay
共⬎50,000 ft兲
CB 13 5,000 0 0 3 5,000 13 5,000
Sand
共⬎50,000 ft兲
SB 17 6,250 0 0 6 15,000 17 15,000
Wall
共10,000– 10,999 ft兲
W7 17 750 0 0 0 0 17 1,000
Wall
共⬎50,000 ft兲兲
WB 17 3,750 0 0 0 0 17 5,000
Drainage structure
共transition兲
D 0 0 5.5 65 5.5 65 5.5 65
Pump station
共transition兲
P 0 0 5 100 5 100 5 100
Note: 1 ft=0.3048 m.
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losses. In practice, a suite of curves that maps water elevation to
both property losses and casualties, with associated percentile val-
ues, would be provided.
Further Assumptions
For the purpose of this example, the following additional assump-
tions are made:
1. The reduction of volume due to pumping is not considered.
Thus, this example takes the conservative view that pumping
is ineffective at reducing water volume. The omission of
pumping reduces the number of event tree branches shown in
Fig. 2 from 16 to eight;
2. All closures within a subbasin are assumed to be either all
open or all closed during a storm event, with the probability
being specified for the open condition 共i.e., they are perfectly
correlated兲;
3. Several distinct coefficients of variations 共COVs兲 were as-
sumed for the volume calculations as described in Table 11.
These COVs account for the aleatory uncertainty 共i.e., non-
reducible uncertainty due to inherent randomness兲 in the weir
coefficient and precipitation volumes. This example does not
account for epistemic uncertainty 共i.e., uncertainty due to
lack of knowledge兲. Ayyub and Klir 共2006兲 provides addi-
tional information on these uncertainty types and others;
4. Given the mean and standard deviation 共i.e., the product of
the coefficient of variation and the mean兲 for volume com-
bined with the knowledge that volume is a non-negative
quantity, the lognormal distribution was chosen to represent
the uncertainty in water volume; and
5. To account for surge and wave effects on the hydrographs,
the aggregate surge and wave hydrographs are adjusted by a
bias factor with a median of 1 and log standard deviation of
0.15. Ten stratifications of this distribution were used in the
analysis to account for uncertainty in the hydrographs.
Risk Results
Based on the description of the FHPS and the computation details
provided by Ayyub et al. 共2007兲 for calculating probability and
water volume for each branch in Fig. 2, the elevation-exceedance
curves for each subbasin was obtained as shown in Fig. 8. Though
not illustrated as part of this example, the elevation-exceedence
information can also be communicated via a flood inundation map
that visualizes total wetted area as a function of return period. For
illustration, selected intermediate calculations for Storm 7, Strati-
fication 7 are given in Table 12 for reach calculations, Table 13
for basin probability calculations, Table 14 for interflow among
SF subbasins for Branch 10, and Table 15 for subbasin branch
probabilities and corresponding percentiles for water elevation.
The property damage loss-exceedance curves for each subbasin
were obtained as shown in Fig. 9. Note that these curves show the
annual rate of exceeding a given value of loss as a function of
loss. Using techniques for loss accumulation assuming that the
loss potential is constant between hurricane events, the cunulative
distribution function 共CDF兲 for the accumulated property damage
loss in a single year for each subbasin was obtained as shown in
Fig. 10. Based on these results, the expected annual loss per year
was calculated to be $2.28 共rounded兲 billion with a COV of 2.54.
It should be noted that the large COV can be attributed to the
multimodal nature of the loss distribution.
Table 11. Analysis Uncertainty Parameters
Variable
Coefficient of
variation 共COV兲
Rainfall volume 0.25
Water volume resulting from breach 0.30
Water volume resulting from overtopping 0.20
Water volume resulting from open closures 0.20
0
5,000
10,000
15,000
20,000
25,000
30
,
000
-30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 2
2
Elevation
(
Feet
)
Flood Damage (Millions of Dollars)
SF1 (m=225, b=3375)
SF2 (m=100, b=1400)
MF1 (m=300, b=8700)
MF2 (m=450, b=7650)
WF1 (m=800, b=12800)
FN1 (m=250, b=4000)
1 ft = 0.3048 m
Fig. 7. Flood damage losses as function of water elevation
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Benefit-Cost Analysis
Given baseline risk information for a hurricane-prone region pro-
tected by a HPS, benefit-cost analysis can be used to assess the
cost effectiveness of alternative risk mitigation strategies. In the
context of protecting a region against floods resulting from post-
hurricane surges, risk mitigation options include strengthening
levees, increasing the span and depth of the levees, relocating
residential and commercial centers, and enhancing emergency re-
sponse procedures. For the purposes of illustrating procedures to
conduct benefit-cost analysis, consider a proposed action to in-
crease the hardness of all WF reaches and transitions such that
their heights are no less than 15 ft 共while maintaining the same
ratio of design-to-top elevation兲 and the probability of breach
Table 12. Reach Calculations for Storm 7, Stratification 7
Reach/
transition
number
Aggregated
surge and
waves
共ft兲
Probability
of
overtopping
Water volume from
overtopping
Probability
of breach
Water volume from breach
Probability
of not
closed
Water volume from
open closures
Mean
共ft
3
兲
StD
共ft
3
兲
Mean
共ft
3
兲
StD
共ft
3
兲
Mean
共ft
3
兲
StD
共ft
3
兲
FN1 共R1兲 13.06 0 — — 0.07 6.56E+ 10 1.97E+ 10 — — —
SF1 共R2兲 11.13 0 — — 0.20 3.44E+ 10 1.03E+ 10 0.2 2.04E+ 08 4.09E+ 07
SF2 共R3兲 17.88 1 1.83E +10 3.66E + 09 0.38 2.79E+ 11 8.38E+ 10 0.2 1.31E+ 08 2.61E +07
MF1 共R4兲 12.96 0 — — 0.01 3.24E + 10 9.72E+ 09 — — —
MF2 共R5兲 25.72 1 1.29E+10 2.57E +09 0.05 3.44E + 11 1.03E+11 — — —
WF1 共R6兲 11.36 0 — — 0.05 5.32E+ 09 1.60E+ 09 — — —
WF2 共R7兲 10.68 0 — — 0.04 2.91E + 10 8.73E+09 0.2 9.51E + 07 1.90E+ 06
WF3 共R8兲 19.71 1 4.01E +10 8.01E + 09 0.97 4.02E+ 10 1.21E+ 10 0.2 4.60E+ 08 9.20E+ 06
WF4 共R9兲 18.84 0 — — — — — — — —
WF5 共R10兲 11.78 0 — — — — — — — —
T1 — — — — 1.00 2.51E + 08 7.54E+07 — — —
T2 — — — — 1.00 2.40E + 08 7.20E+07 — — —
Note: 1 ft=0.3048 m, 1 ft
3
=0.028317 m
3
.
Table 13. Basin Probability Calculations for Storm 7, Stratification 7
Basin
ID
Probability
one or more
reaches
overtopping
P共O兲
Probability of
one or more
breach
failures
P共B兲
Probability of
all pumps
working
P共P兲
Probability of
all closures
are closed
P共C兲
SF
共SF1 +SF2兲
1 0.497 0 0.64
MF1 0 0.009 0 0
MF2 1 0.052 0 0
WF1 1 1.000 0 0.64
FN1 0 0.071 0 0
0.0001
0.001
0.01
0.1
1
10
-30 -20 -10 0 10 20 3
0
Water Elevation
(
feet
)
Exceedance R ate (Events per Year)
SF1 SF2
MF1 MF2
WF1 FN1
1 ft = 0.3048 m
Fig. 8. Elevation-exceedance curves for FHPS subbasins
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failure is set equal to 10
−12
for all elevations up to 1.0 ft overtop-
ping 共leaving the probability of failure for surge elevations above
1.0 ft overtopping unchanged兲. Note that the same hydrographs
were used for both the original FHPS and the system after imple-
mentation of the risk mitigation measures; in practice, the shape
of the hydrographs is dependent on the system configuration, and
any change to an HPS would require a new set of hydrographs to
be generated. Following the procedures described in the compan-
ion paper for calculating accumulated property damage losses in a
single year 共Ayyub et al. 2009兲, the expected annual loss is $1.46
billion 共rounded兲 after implementing the risk mitigation. Thus the
proposed risk mitigation would yield an expected benefit of ap-
proximately $822 million per year with a COV of 4.23. The CDF
for benefit was obtained as shown in Fig. 11.
Assuming an average cost of $10 million to raise 1,000 ft of
reach by 1 ft 共with a coefficient of variation of 0.10 on the total
cost兲 and an expected fixed cost of $250 million 共COV of 0.05兲 to
improve the fragilities of the levees, the total cost to implement
this strategy is $883 million 共with a COV of 0.073兲. All costs are
assumed to be normally distributed random variables. Given an
upfront 共present value兲 cost, P, the total annual equivalent cost, A,
over a lifetime of n years with a fixed annual interest rate i, can be
determined as follows 共Ayyub and McCuen 2003兲:
A = P
冉
i共1+i兲
n
共1+i兲
n
−1
冊
共4兲
Assuming a project lifetime of 25 years 共n=25兲 and a fixed 10%
annual interest rate 共i= 0.1兲, the total equivalent annual cost of the
proposed risk mitigation is $97,278,408 per year with a COV of
0.073 obtained through standard techniques for uncertainty propa-
gation. Differential maintenance costs between the before and
after states of the FHPS are assumed to be negligible.
The probability of exceeding a specified benefit-to-cost ratio
for this proposal was determined, as shown in Fig. 12, using the
equations described in the companion paper 共Ayyub et al. 2009兲.
Though the expected benefit is much greater than the expected
cost in this case, the probability of realizing a favorable benefit-
to-cost ratio 共⬃0.2兲 does not assure a positive return on invest-
ment. Combined with other factors, such as affordability of the
proposed action, its ability to meet risk reduction objectives, and
stakeholder buy in, the probability of realizing a favorable
benefit-cost ratio provides valuable information that supports the
decision making process.
Further analysis can be conducted to determine the sensitivity
of various model parameters on the final results 共Modarres et al.
1999兲. The change in risk with respect to changes in model pa-
rameters yields insights into which aspects of a hurricane protec-
tion system offer the most potential for cost-effective risk
reduction. Such parameters that would be considered include the
height and hardness of the levees, probability of gates not being
closed, etc. For example, if a sensitivity analysis showed that a
small change in levee height leads to a significant reduction in
risk, whereas a large change in closure probability offered only a
small change in risk, the initial focus of risk reduction would be
on increasing levee height. Moreover, sensitivity analysis would
also reveal how the uncertainty in each parameter 共e.g., storm
surge heights, storm rates, fragilities, weir coefficient, etc.兲
contributes to the uncertainty in the final risk results; knowledge
Table 14. Interflow Calculations for SF Basin, Storm 7, Stratification 7,
Branch 10
Subbasin
Preinterflow volume
共ft
3
兲
Postinterflow volume
共ft
3
兲
Mean Std. dev. Mean Std. dev.
SF1 4.03E+08 1.01E+ 08 7.77E + 09 1.93E +09
SF2 1.87E+10 3.66E+ 09 1.14E + 10 2.23E +09
Note: 1 ft=0.3048 m, 1 ft
3
=0.028317 m
3
.
Table 15. Subbasin Results for Storm 7, Stratification 7
Subbasin Branch Probability
Percentile values for water elevation
共ft兲
25% 50% 75%
SF1 10 0.3219 −0.71 − 0.05 0.66
12 0.1810 −0.35 0.28 0.75
14 0.3181 11.25 11.50 11.75
16 0.1790 11.25 11.50 11.75
SF2 10 0.3219 −1.08 − 0.24 0.70
12 0.1810 −0.95 − 0.12 0.81
14 0.3181 16.23 16.49 16.74
16 0.1790 16.23 16.49 16.74
MF1 2 0.9912 −28.75 − 28.50 −28.25
6 0.0088 12.23 12.49 12.74
MF2 2 0.9485 3.72 4.54 5.40
6 0.0515 24.25 24.50 24.75
WF1 14 0.6400 15.23 15.49 15.75
16 0.3600 15.23 15.49 15.75
FN1 2 0.9292 −15.75 − 15.50 −15.25
6 0.0708 −4.31 5.22 13.31
Note: 1 ft=0.3048 m.
64 / NATURAL HAZARDS REVIEW © ASCE / MAY 2009
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of uncertainty importance would lend insight into where to
focus resources so as to better understand and control sources of
uncertainty.
Conclusions and Recommendations
This paper presents a case study of a simple, notional city to
demonstrate the risk analysis methodology for protected
hurricane-prone regions as described by Ayyub et al. 共2009兲. This
paper demonstrates that the results from the risk model can be
used to inform resource allocation decisions for cost-effective risk
mitigation and disaster recovery. The implementation of the risk
model is packaged as the Flood Risk Analysis for Tropical Storm
Environments 共FoRTE兲 tool currently in use by the U.S. Army
Corps of Engineering Interagency Performance Evaluation Team
共IPET兲 charged with assessing the risks to New Orleans due to
hurricanes 共USACE 2006兲.
Acknowledgments
The writers acknowledge discussions, input, and comments
provided by Ed Link, Bruce Muller, Donald R. Dressler, Anjana
Chudgar, John J. Jaeger, Gregory Baecher, Brian Blanton,
0.001
0.01
0.1
1
100 1,000 10,000 100,00
0
Propert
y
Dama
g
e Loss
(
Millions of Dollars
)
Exceedance Rate (Events per Year)
SF1 SF2
MF1 MF2
WF1 FN1
Fig. 9. Property damage loss-exceedance curves for FHPS subbasins
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
.
0
1 10 100 1,000 10,000 100,00
0
Propert
y
Dama
g
e Loss
(
Millions of Dollars
)
C
umulativ e Distribution Function
SF1
SF2
WF1
FN1
MF2
MF1
CDF goes to $0 at zero loss
Fig. 10. Cumulative distribution on accumulated loss in single year for FHPS subbasins
NATURAL HAZARDS REVIEW © ASCE / MAY 2009 / 65
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David Bowles, Jennifer Chowning, Robert Dean, David Divoky,
Bruce Ellingwood, Richalie Griffith, Mark Kaminskiy, Burton
Kemp, Fred Krimgold, Therese McAllister, Martin W. McCann,
Robert Patev, David Schaaf, Terry Sullivan, Pat Taylor, Nancy
Towne, Daniele Veneziano, Gregory Walker, Mathew Watts, and
Allyson Windham, and the contract administration and support
provided by the U.S. Army Corps of Engineers and the help
of Mr. Andy Harkness. Information provided in the paper is per-
sonal opinions of the writers, and does not represent the opinions
or positions of other entities including the U.S. Army Corps of
Engineers.
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0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
.
0
1 10 100 1,000 10,000 100,00
0
Benefit of Risk Mitigation (Millions of Dollars)
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Fig. 11. Cumulative distribution function for benefit computed as risk reduction
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0.40
0.50
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0.90
1
.
00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2
0
Benefit-to-Cost Ratio
Probability o
f
Exceedence
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Fig. 12. Exceedence probability for various benefit-to-cost ratios
66 / NATURAL HAZARDS REVIEW © ASCE / MAY 2009
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