FEMA color
March 22, 2012
Operating Guidance No. 812
For use by FEMA staff and Flood Hazard Mapping Partners
Title: Joint Probability – Optimal Sampling Method for Tropical Storm Surge
Frequency Analysis
Effective Date: March 22, 2012
Approval: Luis Rodriguez
Branch Chief, Engineering Management Branch
Risk Analysis Division
Federal Insurance and Mitigation Administration
Operating guidance documents provide best practices for the Federal Emergency Management
Agency’s (FEMA’s) Risk MAP program. These guidance documents are intended to support
current FEMA standards and facilitate effective and efficient implementation of these standards.
However, nothing in Operating Guidance is mandatory, other than program standards that are
defined elsewhere and reiterated in the operating guidance document. Alternate approaches
that comply with program standards that effectively and efficiently support program objectives
are also acceptable.
Background: The estimation of storm surge elevation frequencies is a central component of
coastal flood hazard studies. FEMA’s National Flood Insurance Program requires determination
of coastal flood elevations having 10, 2, 1, and 0.2percentannual exceedance chances. There
have been many approaches to this task including the use of design storm events; historical
methods such as tide gage analysis and the Empirical Simulation Technique; and synthetic
simulation methods including, especially, the Joint Probability Method (JPM) pioneered by
Myers (Myers 1975, Ho and Myers 1975) for coastal flood estimation.
In recent years, it has been recognized that of the available methods, JPM is preferred for the
tropical storm environment. The JPM approach has the conceptual advantage of considering all
possible storms consistent with the local climatology, each weighted by its appropriate rate of
occurrence. In brief, the most basic JPM approach adopts a parametric storm description
involving five or six hurricane descriptors such as central pressure, size, and translation speed.
For each of the several parameters, probability distributions (not necessarily mutually
independent) are developed through an analysis of the local climatology. These distributions are
each discretized into a small number of representative values, and all possible parameter
combinations are simulated using a hydrodynamic model constructed to faithfully represent the
bathymetry, topography, and ground cover of the study site.
Issues: PostKatrina coastal flood hazard studies adopted stateoftheart meteorological and
hydrodynamic numerical models to compute local maximum water elevations for each of the
synthetic storms required by a JPM approach. The model suite included meteorological, wave,
and surge models required to capture the full range of physical mechanisms controlling the flood
levels, and so imposed a heavy computational burden on the analyses. Even with the use of
modern parallel computer clusters, a straightforward bruteforce JPM approach, as used in older
FEMA studies, would have been prohibitively expensive.
Work undertaken by FEMA and the US Army Corps of Engineers (USACE) independently
developed new and highly efficient methods of implementing the JPM approach in such a way as
to minimize the number of storms requiring simulation. It was found that the simulation effort
could be reduced by about an order of magnitude while still maintaining good accuracy. The two
approaches are known as Optimal Sampling methods (OS), denoting their common intent of
choosing storms for simulation in such a way as to accurately cover the entire storm parameter
space through optimal parameter selection with associated weighting and interpolation methods.
Actions Taken: The procedures outlined in this guidance were developed during the intensive
efforts by FEMA and the USACE to reevaluate coastal hazards in the Northern Gulf following
Hurricanes Katrina and Rita of 2005. These guidelines correspond to the approach used in the
FEMA Mississippi study, called the Quadrature Method, although the Corps’ approach for
Louisiana, called the Response Surface Method, is also entirely appropriate for new FEMA
studies. Experience gained during the postKatrina work showed that the two approaches are
capable of giving nearly identical results with nearly identical effort. These guidelines focus on
the Quadrature Method since it is more readily automated than the Response Surface Method,
which requires a greater degree of expert judgment in the selection of storms.
In order to simplify their application and to ensure a correct implementation of some of the
methods not commonly encountered in past FEMA studies, two utility programs have been
written. One is a console program, SURGE_STAT, to compute the surge statistics at the target
sites, including the effects of secondary parameters. The other is an Excel spreadsheet, JPM
OSQ.XLS, to select the parameters of the OS storms, according to the quadrature methods.
Supersedes/Amends: Section D.2.3.6.1 of the Atlantic Ocean and Gulf of Mexico Coastal
Guidelines Update, Final Draft, February 2007 and Section D.4.3.6.1 of the Final Draft
Guidelines for Coastal Flood Hazard Analysis and Mapping for the Pacific Coast of the United
States, January 2005
Attachments:
Attachment A – Program JPMOSQ.XLS Version 1.0 User’s Instructions
Attachment B – JPMOSQ.XLS Version 1.0 Excel Spreadsheet Tool
Attachment C – SURGE_STAT Console Program Tool
Distribution List (electronic distribution only):
Office of the Assistant Administrator for Flood Insurance and Mitigation
Director, Risk Analysis Division
Director, Risk Reduction Division
Director, Risk Insurance Division
Regional Mitigation Division Directors
Regional Risk Analysis Branch Chiefs
Regional Support Centers
Regional Program Management Liaisons
Legislative Affairs
Office of Chief Counsel
Cooperating Technical Partners
Program Management Contractor
Customer and Data Services Contractor
Production and Technical Services Contractors
Operating Guidance 812
Operating Guidance 812
Joint Probability – Optimal Sampling
Method for Tropical Storm Surge
Frequency Analysis
March 22, 2012
Operating Guidance 812
Table of Contents
1. Joint ProbabilityOptimal Sampling Method for Tropical Storm Surge Frequency Analysis . 2
1.1. Joint Probability Method Guidelines – Overview ........................................................ 2
1.1.1. Introduction ..................................................................................................... 2
1.1.2. General Overview of a Coastal Surge Study ................................................... 3
1.1.3. Summary of the JPM Approach ...................................................................... 7
1.2. Storm Parameterization and Data Selection ................................................................. 9
1.2.1. JPM Parameter Selection ................................................................................. 9
1.2.2. Storm Data Selection ..................................................................................... 10
1.3. Statistical Description of Storm Parameters .............................................................. 12
1.3.1. Approaches for Definition of the Sample and Statistical Analysis ............... 12
1.3.2. Geographical Variation of Storm Statistics ................................................... 13
1.3.3. Storm Rate (Space and Time) and the Probability Distributions of Heading
and Distance .................................................................................................. 14
1.3.4. Storm Intensity .............................................................................................. 17
1.3.5. Storm Track: Forward Speed of Translation ................................................. 19
1.3.6. Storm Size ..................................................................................................... 20
1.3.7. Other Physical Parameters ............................................................................. 21
1.3.8. Treatment of Parameter Correlations ............................................................ 22
1.4. Storm Simulation Set – JPMOS Methods ................................................................ 23
1.4.1. Summary of the Response Surface Method .................................................. 23
1.4.2. Summary of the Quadrature Method ............................................................. 24
1.4.3. The Quadrature Method of Storm Selection .................................................. 25
1.4.3.1 Overview ........................................................................................ 25
1.4.3.2 Implementation of Bayesian Quadrature for JPMOS ................... 26
1.4.3.2.1 Inputs ............................................................................ 26
1.4.3.2.2 Algorithmic Steps ......................................................... 27
1.4.3.2.3 Verification of the Storm Selection Step ...................... 28
1.4.4. Development of a Complete Storm History .................................................. 29
1.5. Second Order Concerns ............................................................................................. 31
1.5.1. Small Random Contributions – Overview of the Approach .......................... 31
1.5.2. Regression Method for Large Amplitude Tides ............................................ 32
1.6. Surge Frequency Determination ................................................................................ 33
1.6.1. Overland Distribution of Target Sites ........................................................... 33
1.6.2. Construction of the Simulated Density Distribution Histograms .................. 33
1.6.3. Histogram Adjustment for Secondary Random Factors ................................ 34
1.7. Combination of Surge and Other Flood Processes .................................................... 36
Operating Guidance 812
1.8. Accompanying Utility Programs ............................................................................... 38
1.9. References .................................................................................................................. 39
Operating Guidance 812
Operating guidance documents provide best practices for the Federal
Emergency Management Agency’s (FEMA’s) Risk MAP program.
These guidance documents are intended to support current FEMA
standards and facilitate effective and efficient implementation of
these standards. However, nothing in Operating Guidance is
mandatory, other than program standards that are defined elsewhere
and reiterated in the operating guidance document. Alternate
approaches that comply with program standards that effectively and
efficiently support program objectives are also acceptable.
Operating Guidance 812
Section 1 is organized to:
.Overview (Section 1.1)
.Storm Parameterization and Data Selection (Section 1.2)
.Statistical Description of Storm Parameter (Section 1.3)
.Storm Simulation Set – JPMOS Methods (Section 1.4)
.Second Order Concerns (Section 1.5)
.Surge Frequency Determination (Section 1.6)
.Combination of Surge and Other Flood Processes (Section 1.7)
.Accompanying Utility Programs (Section 1.8)
.References (Section 1.9)
This Operating Guidance provides guidance for frequency analysis of coastal storm surge using the
Joint Probability Optimal Sampling Method. The method and variants are described in some detail,
although these guidelines are meant to be descriptive rather than prescriptive. It is not felt that it is
possible to provide strict guidance that can be followed using a black box approach. Instead, the
analyst must consider the unique character of a given study and should implement the ideas developed
here so as to obtain an accurate result with minimum computational effort.
This guidance and associated software is subject to continuing development. Please contact Jonathan
Westcott at Jonathan.Westcott@fema.dhs.gov to learn if there are updates superseding this version.
1. Joint Probability

Optimal Sampling Method for
Tropical Storm Surge Frequency Analysis
1.1. Joint Probability Method Guidelines – Overview
1.1.1. Introduction
The estimation of storm surge elevation frequencies is a central component of coastal flood hazard
studies. FEMA’s National Flood Insurance Program requires determination of coastal flood
Operating Guidance 812
elevations having 10, 2, 1, and 0.2% annual exceedance chances. There have been many
approaches to this task including the use of design storm events (socalled 100 year storms);
historical methods such as tide gage analysis and the Empirical Simulation Technique (EST); and
synthetic simulation methods including, especially, the Joint Probability Method (JPM) pioneered
by Myers (Myers 1975, Ho and Myers 1975) for coastal flood estimation.
In recent years, it has been recognized that of the available methods, JPM is preferred for the
tropical storm environment. Design storm methods fail since no single event can capture the range
of storm possibilities that might all be capable of producing, say, the 1percentannualchance flood
elevation. Historical methods such as tide gage analysis and EST evaluations have been found to be
highly sensitive to the sample error/variation that exists in any limited data set. The JPM approach,
however, has the conceptual advantage of considering all possible storms consistent with the local
climatology, each weighted by its appropriate rate of occurrence. In brief, the most basic JPM
approach adopts a parametric storm description involving five or six hurricane descriptors such as
central pressure, size, and translation speed. For each of the several parameters, probability
distributions (not necessarily mutually independent) are developed through an analysis of the local
climatology. These distributions are each discretized into a small number of representative values,
and all possible parameter combinations are simulated using a hydrodynamic model constructed to
faithfully represent the bathymetry, topography, and ground cover of the study site.
The present Guidelines are an outgrowth of work undertaken by FEMA and the US Army Corps of
Engineers after the disastrous 2005 hurricane season. Those postKatrina efforts adopted stateof
theart meteorological and hydrodynamic numerical models to compute local maximum water
elevations for each of the synthetic storms required by a JPM approach. The model suite included
meteorological, wave, and surge models required to capture the full range of physical mechanisms
controlling the flood levels, and so imposed a heavy computational burden on the analyses. Even
with the use of modern parallel computer clusters, a straightforward bruteforce JPM approach as
used in older FEMA studies would have been prohibitively expensive.
The FEMA and Corps efforts independently developed new and highly efficient methods of
implementing the JPM approach in such a way as to minimize the number of storms requiring
simulation. It was found that the simulation effort could be reduced by about an order of magnitude
while still maintaining good accuracy. The two approaches are known as Optimal Sampling
methods (OS), denoting their common intent of choosing storms for simulation in such a way as to
accurately cover the entire storm parameter space through optimal parameter selection with
associated weighting and interpolation methods. Operating Guidance for the JPMOS storm
simulation methods are provided in Section 1.4.
1.1.2. General Overview of a Coastal Surge Study
This section provides a brief and high level overview of a coastal surge study. The aim is to
provide a general understanding of where the JPMOS methodology fits into a study, and so to help
clarify much of the discussion to follow. However, there are several important background
documents that should be consulted for a more thorough discussion than is provided here:
Operating Guidance 812
IMPORTANT BACKGROUND MATERIALS
It is the purpose of this document to present the JPMOS approach to storm surge frequency analysis,
not to duplicate or recount the great volume of important material available elsewhere. In particular,
it will be assumed that the Mapping Partner is familiar with FEMA coastal flood studies in general,
and the sorts of methods which have been used in past studies. In other words, a great deal of
background knowledge is taken for granted in the discussion to follow. The Mapping Partner should
consult, as necessary, the FEMA guidelines for coastal studies and, for certain more detailed
information, the documentation for the FEMA Coastal Flooding Hurricane Storm Surge Model
(published in 3 volumes, 1988). For more specific details regarding the JPMOS methods, the
Mapping Partner will frequently be directed to the comprehensive reports of the postKatrina studies
of Louisiana and Mississippi conducted during 20062008 by the Corps of Engineers and by FEMA,
respectively.
A JPM storm surge flood study of the sort required for FEMA work requires two sorts of
knowledge: first, the analyst requires a knowledge of the storm climatology of the study region in
order to be able to characterize the storms governing the flood hazard; second, the analyst requires
a knowledge of the effects produced by a particular storm throughout the study region. The former
sort of knowledge is obtained by a study of the storm history within the vicinity of the study site.
The second sort of knowledge is provided by use of a validated hydrodynamic model capable of
simulating the details of flooding for any storm affecting the region. The hydrodynamic model,
then, is a model incorporating all of the important features of the site, including the variations of
bathymetry, topography, and land cover (roughness factors).
In recent studies, the ADCIRC model has been used in conjunction with very detailed
representations of the sites, through high resolution grids. Grid node spacing may be as small as
100 m in critical areas, as necessary to resolve features that may control flow behavior. The overall
extent of a grid must be much larger than the immediate study region (extending many hundreds of
miles beyond the site) for two reasons: first, the surge and waves of interest develop over a
relatively large area; and second, the numerical solution is not valid at the open water edges of the
grid, so that those edges must be sufficiently far from the study site so as not to degrade the
solution in the region of interest.
In the particular case of the ADCIRC model, the time step for simulation of a storm is constrained
by the Courant stability condition so that very fine grid resolution can result in very short time
steps. These factors conspire (more grid points requiring calculation at each time step, and more
time steps) to increase computational costs rapidly, so that the modeler must balance cost versus
gains in accuracy. Other models will require similar considerations to greater or lesser degrees,
dependent, for example, on numerical schemes and the availability of features (such as weirs and
embedded channels) which may ease the requirements for resolution of small features. The
computational demands of high resolution models such as ADCIRC are a primary reason that
traditional JPM methods are not likely to be feasible in a study, and have been the impetus for
development of the Optimal Sampling JPMOS methods.
Operating Guidance 812
Models other than ADCIRC may be used according to the needs of a particular study. FEMA
maintains a list of approved models, although the Mapping Partner should adopt a model which can
be shown to provide the necessary accuracy. Consistency with adjacent studies should also be kept
in mind, and model selection must be made with the concurrence of the FEMA Project Officer.
Knowledge of the local storm characteristics introduces two problems of practical importance,
which will be seen to color much of the JPMOS discussion to follow in these Guidelines.
Knowledge of the local storms is based on local storm data. However, data must usually be taken
from outside the immediate study area in order to obtain a sample of reasonable size upon which
statistics can be based; a particular county, for example, may not have been the site of hurricane
landfall within the entire historical record. This raises the question of how far afield one can go in
assembling a sample: clearly, storms from distant points may not be of the same character as local
storms. So the unavoidable problem is to balance sample error on the one hand, versus population
error on the other. Small samples are subject to large variability in a random historical record,
while a larger sample from distant points may be corrupted with storms unlike those belonging to
the local population.
Given the two sorts of knowledge, a third requirement is a computational scheme or procedure
incorporating both so as to produce estimates of surge statistics. This is the role of the JPM scheme.
In brief, the procedure is to consider all possible hypothetical storms to be constructed from a small
number of storm parameters embedded in a storm model of winds and pressures (a planetary
boundary layer, PBL, model, for example). Current practice is to consider five or six defining
parameters as sufficient to specify an idealized storm. Storm strength is characterized by the central
pressure depression, or the difference between the pressure at an assumed storm eye and the
ambient pressure at the storm periphery. Storm size is measured by some length parameter which
approximates the radial distance from the eye to the zone of maximum wind speed. The relative
sharpness of the peak of the pressure radial pressure profile may be controlled by a fitting factor
(Holland’s B). In addition to these three wind and pressure field parameters, the storm track, in its
simplest straighttrack form, might be characterized by three kinematic parameters: the direction of
storm motion, a shoreline crossing point (or bypass distance), and a speed of storm translation.
Many other parameters could be added to this mix. For example, storm surge occurs in
superposition with astronomic tide; consequently, tide amplitude and phase might be enlisted as
additional parameters. Storm tracks are not straight, and speeds are not constant; consequently, any
number of higherorder parameters could be invoked to describe more realistic tracks. Storms are
not simple circular affairs accurately captured by idealized analytical forms of radial profiles;
again, any number of higherorder parameters could be invoked to permit one to capture the
possible range of real events.
As will become apparent, however, present knowledge based on very limited samples of storms
affecting a site, and a lack of high resolution observations for the storms that have been recorded,
do not yet support a more ambitious effort. In these Guidelines, only the basic five or six
parameters are tackled, although some effort is made to improve track descriptions through the use
of “typical” track shapes abstracted from a review of historical tracks using engineering judgment.
Similarly, some factors that affect surge generation, such as storm weakening before and after
landfall, are typically accounted for by the arbitrary specification of behavior based on observed
local trends.
Operating Guidance 812
Working, then, with a small set of parameters that control a wind and pressure description which,
in turn, controls the computations of the hydrodynamic model for a particular hypothetical storm,
the JPM procedure proceeds as follows (using numbers chosen for illustration only, not for
guidance):
. First, develop probability distributions for each storm parameter. The probability
distributions are derived from a storm sample which, in its simplest form, can be thought of
as a list of all storms and their parameters recorded during a selected period, within a
region surrounding the study site. Familiar distribution forms are fit to the data as
appropriate. These empirical distributions need not be independent. For example, the
distribution of storm size is commonly taken to be conditional upon the central pressure
depression, so that stronger storms tend to be associated with smaller radii. More
significantly, all other parameters are always taken to be conditional upon track direction in
the case of a site that is affected by both entering and exiting storms (such as the Florida
peninsula).
. Second, establish the overall rate of storm occurrence in both space and time. In a sense to
be made more precise later, let this be the number of storms passing per unit length of
space per unit time; storms per mile per year, for example (typically a small number).
. Third, for a basic JPM (notOS) study, subdivide each distribution into a small number of
discrete pieces; one might imagine representing pressure, radius, forward speed, and track
angle by a half dozen, or so, values of each.
. Fourth, construct all possible hypothetical storms by simply taking all possible
combinations of these elementary storm quantities. With, say, six values for each of the
four parameters mentioned in Step 3, above, one constructs 1296 “storms.” These storms
constitute the simulation set.
. Fifth, simulate all of these storms, each on multiple tracks so as to allow every storm type
to affect all points in the study area. That is, a particular storm may pass through any point
along the coastline of the site, so that random track position must be accounted for. In the
simplest case, this might be done by adopting a track spacing dependent upon the storm
size, and replicating the tracks for a particular storm by parallel displacement. Usually, for
small study sites, track position is distributed uniformly over space.
. Sixth, for each such storm determine a corresponding rate of occurrence. This is just the
product of (1) the overall rate of occurrence from Step 2, above; (2) the probability masses
of each of the four parameter chunks from Step 3 (reflecting dependence as appropriate);
and (3) the selected spacing between tracks adopted in Step 5. That is, each simulated track
is taken to represent all possible tracks which could occur over a zone extending half way
to its neighboring tracks on each side. The track spacing is chosen small enough to provide
smooth coverage of the site; tracks spaced too far apart will produce a fluctuating surge
estimate, underestimating the potential at points between the simulated surge peaks. On the
other hand, tracks spaced too close together would necessitate an excessive number of
simulations, imposing an unnecessary computational cost. It has been found that a track
spacing equal to the radius of maximum winds provides good results. Since storm radii
Operating Guidance 812
may be only a few tens of miles while the site may extend for a hundred or more miles,
each storm must be simulated on several tracks in order to cover the area (the number of
tracks will be greater for the small storms in the simulation set than for large storms). In
practice, ten or more tracks per storm may be required. This brings the number of storm
simulations in the example to 10 ×1296, or nearly 13,000 ADCIRC runs, each requiring a
number of hours.
. Seventh, at each point of interest in the hydrodynamic grid – the target sites selected for
the final statistical analysis and the mapping effort – record the highest surge computed for
each storm, and tag it with the rate of occurrence of that storm.
. Eighth, for each target site, construct a histogram of rate versus surge height, by
accumulating the storm rates into the bins of surge height. Such bins might be constructed
with widths of 0.1 ft, for example, so that a histogram with bins running from 0 to 500
would handle surge elevations up to 50 feet. The accumulated rates in the bins constitute
an estimate of the density distribution of surge height.
. Ninth, for each histogram (one for each target site) sum the rates from the top bin down to
the bottom bin. The result of this step is an estimate of the cumulative surge distribution.
To find the 1% surge elevation, for example, one simply locates the bin having a summed
rate nearest 0.01. For example, if this occurs at bin 232, the estimate of the 1% surge
elevation would be 23.2 feet, following the assumptions made in this example.
This list of topics is only partial. The Guidelines to follow include, for example, a discussion of
adjustments to the surge statistics to account for unconsidered small factors, treatment of special
problems such as large tides and combination with independent events, and so forth.
The sections to follow develop the JPMOS approach in a more rigorous way than outlined here.
The essential difference between the foregoing bullets and the OS method to be described below, is
in Steps 3 and 4, above. Rather than constructing the storm simulation set by simply subdividing
each parameter distribution into a small number of chunks and simulating all possible
combinations, the OS approach is to select the storms (combinations of parameters) and their
weights in a much more intelligent way, so as to reduce the computational burden by about an
order of magnitude.
1.1.3. Summary of the JPM Approach
The JPM method is now summarized in more formal terms in preparation for the subsequent
discussions. As noted above, the approach relies on probabilistic descriptions of storm occurrence
and storm characteristics to define a set of synthetic storms, together with a numerical method to
calculate the coastal flood elevations that would be generated by those storms. The numerical
method includes representations of the storm tracks, the evolution of storm characteristics
(referenced to the characteristics at landfall), the wind and pressure model, the surge model, and so
forth, represented symbolically as
,(,,landfalllocation,,...)()pfPRVX.....
Operating Guidance 812
where . is the surge elevation at a point and the vector represents all pertinent storm
characteristics including the central pressure depression, .P; the storm radius, Rp; the forward
speed of storm motion, Vf; the storm track direction, .; landfall location; and others as may be of
significance such as Holland’s B parameter, astronomic tide factors, and so forth. The landfall
location and track angle determine the proximity of the storm to a particular coastal site. The
annual rate of occurrence of a flood elevation at the site in excess of . is defined in terms of the
combined probabilities of the storm parameters and is given by the multiple integral:
X
(1) max(1)[]...()[()]yrXxPfxPxdx..........
where . is the mean annual rate of all storms of interest for that site, is the joint probability
density function of the storm characteristics of these storms, and is the conditional
probability that a storm with characteristics will generate a flood elevation in excess of .. This
integral over all possible storms determines the fraction of storms that produce flood elevations in
excess of the value of interest, using the total probability theorem (Benjamin and Cornell, 1970).
The entire expression, including ., is actually a rate with dimensions of events per unit time, but is
commonly thought of as an annual probability to a good approximation.
()Xfx
[()]Px...
x
Evaluation of the JPM integral (Equation 1) by use of conventional bruteforce numerical
integration approaches is problematic since each evaluation of the integrand involves the costly
evaluation of .(x) for one set of parameters, , (that is, the simulation of one storm), and since the
evaluation of the 5dimensional (or higher) integral in the equation requires that the integrand be
evaluated a very large number of times.
x
Operating Guidance 812
1.2. Storm Parameterization and Data Selection
1.2.1. JPM Parameter Selection
As suggested above, the JPM approach adopts a parameterized representation of tropical storms
involving, at a minimum:
. a measure of intensity: the central pressure depression, .P (usually given in millibars).
. a measure of storm size: the radius to maximum winds, Rmax, or the pressure scaling radius,
Rp (usually given in kilometers).
. the speed of storm translation, Vf
. the direction of storm motion, . (direction of motion typically measured counterclockwise
from north)
. a track location parameter, such as the shoreline crossing point Xc or a bypassing distance
In a flood study of this sort, all storm parameters should be defined with respect to a specified
reference condition; in particular, observed values at landfall (with respect to a nominal shoreline
for the study site) should be used as the basis for parameter descriptions.
Additional parameters may be required to better define a storm or a flood event. For example, the
Holland B parameter determines the narrowness of the peak wind field in some wind models, and
influences the maximum wind speed; if taken to be variable, it can be treated as an additional
parameter. Astronomic tide could be considered to be a concurrent flood mechanism characterized
by two additional parameters, amplitude and phase.
The number of parameters to be accounted for in a JPM analysis can be increased indefinitely, as
greater and greater complexity is added to the description of a storm and additional factors such as
rainfall intensity and spatial pattern are included in the list of flood mechanisms. As will be shown
in a subsequent section, however, one quickly reaches a limit of what can be treated by simulation
of combinations of all parameters owing to the curse of dimensionality. If only five parameters are
to be considered, and if each of these is represented by only six values over its significant range,
then there would be a total of 65 combinations, or 7,776 in all, each representing a synthetic storm.
Modern hydrodynamic models for waves and surge might require a number of hours to simulate a
single storm using a parallel CPU cluster, so that efforts of this sort are not feasible. The entire goal
of the OS variants of the JPM approach is to provide a sufficiently accurate representation of the
storm climatology, while reducing the size of the simulation storm set to fall below a feasible limit.
The analyst must include all important parameters for a JPM study at a site, always including the
first five enumerated in the list above, but should recognize that little can be gained by adding
parameters if the available data is not sufficient to develop the required probability distributions.
Similarly, it would be unrealistic to simulate an excessive number of parameter combinations to
represent a joint probability distribution that is not known well because of data limitations;
although resolution/precision might be gained, accuracy would not.
Note that although the distributions of storm parameters will usually be defined in terms of
shoreline crossing values (or an equivalent for bypassing storms), the storm simulations may treat
the parameters as variable during the course of a simulation. In particular, pressure may be
Operating Guidance 812
assumed to vary in a prescribed manner both before and after landfall (filling), as may Holland’s B
parameter if it is used as a basic JPM parameter. Similarly, while track angle statistics are based on
values at landfall, the simulated tracks may be curved in a defined manner to better represent pre
landfall behavior (this may be of importance for representation of wind wave generation, pertinent
to the estimation of wave radiation stresses to be included in the surge simulations).
1.2.2. Storm Data Selection
These guidelines are focused on tropical storms, for which the quality of the historical record has
varied greatly over time. Furthermore, not all storm parameters are known equally well. For
example, kinematic parameters (based on storm location, from which direction and forward speed
can be derived) may be known reasonably well for older storms, although the corresponding central
pressures, radii, and B factors, may be absent or known only very approximately. This variability
of the quality of the record and disparity in data availability require the Mapping Partner to begin a
study with a careful review of data sources.
Following the precedent of the postKatrina FEMA and Corps studies, it is recommended that new
flood studies for tropical storms be based primarily on data recorded since 1940. This corresponds
to the modern era of aircraft reconnaissance, and is thought to be much more reliable than older
data, especially for both pressure and radius. Although data regarding the kinematic parameters of
older storms may be useful, the Mapping Partner should review them critically before including
them in the development of parameter statistics. It is noted, too, that storm counts may be
unreliable for earlier decades, except for nearshore tracks; some distant storms may have been
missed, leading to a misestimate of storm density if counts are made over large areas.
Basic data for tracks and pressures (but not including radius or B) can be found in the HURDAT
data files maintained by NOAA’s National Weather Service Hurricane Research Center (HRC).
Note that while track data should be adequate for JPM studies, pressure data should be checked
against other sources, including the more detailed storm summaries compiled by NOAA. Pressures
inferred from HURDAT windspeeds (by backcomputation from a windspeed vs. pressure formula)
should not be relied upon. The HURDAT track data consists of latitude and longitude of the
hurricane center at six hour intervals. From these, the necessary data for forward speed and track
direction can be determined. HURDAT is the official database of hurricane data for the North
Atlantic and Gulf of Mexico, and can be obtained (along with descriptive information defining the
database structure) from the HRC webpage at www.nhc.noaa.gov.
Other data sources include a special storm compilation produced for FEMA by NWS (NWS38),
and used as the source of JPM information in earlier FEMA studies. Although not up to date, this
document includes valuable information regarding storm radii and pressures which is lacking in the
HURDAT database. The Mapping Partner should not, however, adopt the NWS38 statistical
summaries for a new study, but should follow the procedures outlined in these new guidelines.
Other data sources must be searched by the Mapping Partner to supplement HURDAT and NWS
38. Data available from the many NOAA divisions including HRC, the National Hurricane Center
(NHC), the Hurricane Research Division (HRD), and the National Climatic Data Center (NCDC)
should be interrogated. For modern tropical storms, the National Hurricane Center (NHC)
Operating Guidance 812
publishes detailed storm summary papers in their Tropical Cyclone Report series, as well as
numerous storm analyses available at http://www.nhc.noaa.gov/index.shtml.
Additionally, private sources of data exist, including Oceanweather, Incorporated, and Applied
Research Associates / Intrarisk. These and other organizations may be sources of data or data
analyses not otherwise readily available, and should be considered by Mapping Partners. In
particular, detailed parameter evaluations and determination of “best winds” and “best tracks” may
have been made by such private organizations for storms of interest (note, however, that these
parameter selections may be conditional upon other assumptions made regarding wind models,
and so should be interpreted carefully for application in a study using different methods).
Operating Guidance 812
1.3. Statistical Description of Storm Parameters
1.3.1. Approaches for Definition of the Sample and Statistical Analysis
Two approaches have been used in recent studies for definition of the hurricane sample and the
statistical analysis of the hurricane data.
The approach here called the Capture Zone approach is perhaps the more conventional of the two
approaches. In this approach, all hurricanes that make landfall along a particular segment of the
coastline are counted, and are given equal weight in the calculations. Alternatively, the capture
zone might be chosen to consist of a spatial region, such as a circular window surrounding the
study site. Such capture zone approaches have been standard for past studies. The definition of the
capture zone is extremely important, and must be chosen with two competing factors in mind. First,
the zone must be large enough to capture a significant number of storms, adequate for estimation of
parameter statistics. Second, the zone must be small enough to ensure parameter homogeneity
throughout the zone. These conflicting requirements represent the problem of sample error, on the
one hand, and population error on the other.
The second approach is here called the Chouinard Kernel Approach (or, more briefly, the
Chouinard approach). It was introduced by Chouinard and his coworkers (see the references at the
end of these guidelines for citations) for use in mapping the hazard from hurricaneinduced waves
and winds for the offshoreoil industry, but is also appropriate for hurricane surge studies. In this
approach, each hurricane is given a weight that decreases as the distance from the hurricane to the
point under consideration increases. Thus, data from hurricanes that passed near the point under
consideration are given more weight than those that passed far from the point. This technique
minimizes population error, by emphasizing events that occurred near the site, while also
alleviating sample error by allowing additional data to be taken from a distance. The function used
to calculate this weight (the kernel function) is typically a Gaussian probability density function,
but other shapes may be used. In new FEMA flood studies, the Mapping Partner should adopt a
Gaussian kernel; the scaling parameter that controls the width of the kernel (the kernel width) is
then numerically identical to the standard deviation.
One of the most important steps in Chouinard’s approach is the determination of the optimal kernel
size, which provides the optimal compromise between high geographical resolution and statistical
precision (i.e., low statistical uncertainty). This is effectively the same as the problem of choosing
the size of the capture zone in the alternate approach mentioned earlier. Chouinard and his co
workers and Risk Engineering in its work for the postKatrina FEMA study of Mississippi, used a
statistical technique known as crossvalidation (to be described below) to determine the optimal
kernel size, but other techniques may also be used.
The two main advantages of Chouinard’s kernel approach are that it includes an objective
procedure to achieve an optimal tradeoff between spatial resolution and statistical precision, and
that the weight given to a specific storm  and, therefore, the calculated statistics  varies
gradually as the site of interest is moved or as the kernel width is varied. The second advantage is
particularly important in wavehazard mapping, but it also avoids the problems that may arise if an
important historical hurricane happens to occur near the boundary of the capture zone and so may
or may not be included in the sample depending upon a small difference in capture zone size. It is
Operating Guidance 812
best if the adopted method is not sensitive to fine considerations such as this. The main
disadvantage of Chouinard’s kernel approach is that it is more complex than the Capture Zone
approach.
There are also two approaches for counting hurricanes for the purpose of rate calculations and for
defining the distribution of pressure (and the distributions of other storm characteristics that show a
significant geographical variation through the study region). One approach (the pointbased
approach) considers the minimum distance to the location of interest (which will typically be the
midpoint of the coastal segment of interest) and computes the distribution of the corresponding
pressures; this is the approach used by Chouinard et al. For applications on open water, one
determines the time of the track’s closest approach to the location of interest and uses the values of
the parameters which existed at that time. In application to surge calculations, where interest is
focused on the hurricane’s characteristics at landfall, the track is considered to be linear with the
heading it had at landfall. This linearized track is then used to determine the minimum distance to
the site of interest, and the storm characteristics at landfall are assumed everywhere along the
linearized track. In the pointbased approach, one may calculate either the omnidirectional rate
(storms/km/yr) or the directional rate (storms/km/directionaldegree/yr). The directional rate is
required for surge calculations, and can be obtained directly from the omnidirectional rate by
combination with the observed distribution of angles.
The second approach (line based) measures distances along the coastline of the region of interest.
This approach appears simple, but is dependent on the geometry of the coastline. Even if the
coastline is simple enough to be idealized as straight, the directional rate is a function of the
offshore directional rate and of the coastline’s orientation (the proportionality factor is the cosine of
the angle between the track heading and the landward perpendicular to the coastline), while the
directional rate obtained using the pointbased approach is identical to the offshore directional rate
(ignoring any possible effects of the land mass on the geometry of the prelandfall tracks). In the
second approach, then, the distribution of heading must be calculated as an additional step.
These guidelines recommend the Chouinard kernel approach in conjunction with pointbased
counting of hurricanes, and the capture zone approach in conjunction with linebased counting for
parameters, because these are the most common pairings. In principle, one could use Chouinard’s
kernel approach with linebased counting and one could use a rectangular kernel of arbitrarily
selected size (equivalent to the capture zone approach) with pointbased counting.
1.3.2. Geographical Variation of Storm Statistics
In principle, one would expect that the statistical characteristics of hurricanes in the study region
would vary as a function of location to some degree. In the Gulf of Mexico, these variations may
be due to variations in location relative to the Yucatan and Florida straits and to the Loop Current.
Along the Atlantic coast, these variations may be due to variations in latitude and associated
variations in water temperature, prevailing winds, etc.
In many situations, however, the available hurricane data may not be sufficient for resolving these
variations, even when there are physical arguments that suggest their existence. In these situations,
Operating Guidance 812
it is more realistic and sufficient to consider only one distribution of storm parameters over the
entire study region.
In Chouinard’s kernel approach, the kernel size parameter provides direct information regarding
the smallest scale of geographical variation that can be resolved with the available data, with an
optimal tradeoff between geographical resolution and statistical precision. In the Capture Zone
approach, one may need to perform additional calculations to determine if there is significant
geographical variation in parameters. For instance, one may divide the capture zone and use
standard statistical tests to determine whether the capture zone can be treated as having a unique
distribution for each parameter. One may also test whether the sample distribution of distance to a
suitably defined reference point is consistent with the assumption of a uniform spatial distribution.
In some situations, it may be necessary to take data from a broader region, i.e., a region broader
than the study region, the capture zone, or the kernel size. For instance, the conditional distribution
of Rmax given .P is often determined using data from a much larger region, such as the Gulf of
Mexico or the North Atlantic, in order to obtain a sufficient sample; see Risk Engineering, Inc.,
2008, and Vickery and Wadhera, 2008, for illustrations of this. Still another example of use of an
expanded zone to better define a parameter is the model for Holland’s B conditional upon Rmax and
latitude (Vickery and Wadhera, 2008). In these situations, it is more important that the Mapping
Partner obtain a reasonable estimate of the nature of the conditional dependence of the parameter
on other hurricane characteristics, than on location.
1.3.3. Storm Rate (Space and Time) and the Probability Distributions of
Heading and Distance
In the characterization of hurricanes for surge analysis, it is convenient to define the minimum
intensity of interest (in terms of a minimum pressure deficit and then develop statistical models for
the frequency and characteristics of the storms exceeding that intensity. The choice of this cutoff
will influence the range of relative validity of the computed flood statistics. For example, if only
the 1% and stronger floods are of interest, it may be possible to truncate the storm sample so as to
include only Category 3 storms and stronger (this is an illustrative example, not a
recommendation). By not considering weaker storms, the estimates of, say, the 10% level may be
unrealistic. In order to capture the 10% flood level, lesser storms would need to be included,
suggesting a cutoff at Category 1, or even lower in the tropical storm range.
This section considers the overall rate, or storm density, in both space and time; subsequent
sections then consider the several hurricane characteristics.
It is generally assumed that hurricane occurrences in time are a Poisson process (Parzen, 1962),
although data indicates that this is not strictly true. More importantly for practical applications,
hurricanes that generate surge in excess of a certain high value of interest (say, 15 feet) at a
particular location are assumed to be a Poisson process. The only parameter in this Poisson model
is the rate of storms, which has units of storms per unit distance per unit time (e.g.,
storms/km/year). If heading is considered as part of the rate calculations, then the rate has units of
storms per unit distance per unit angle per unit time (e.g., storms/km/directiondegree/year).
Operating Guidance 812
As in many situations involving the study of rare events, the Poisson assumption is actually not
necessary for the calculation of rare surge exceedances at a given location, and the results obtained
using the Poisson assumption are generally not invalidated by deviations of hurricanes from the
independence assumptions implied by the Poisson assumption.
In the Chouinard kernel approach, the rate at the point of interest is proportional to a weighted
count of the observed data in the storm catalog, with weights that depend on the distance from the
storm to the site and its deviation from the direction of interest. Storms that pass farther from the
site of interest or that have directions different from the direction of interest receive lower weight.
The resulting expressions for the directional and omnidirectional rates, respectively, are as
follows:
(allstorms)
1()()()iiiwdkT
.......
(2)
(allstorms)
1()iiwdT
...
where the summation extends over all storms in the catalog, is the duration of the catalog (in
years), and the kernel weight functions are taken as normaldistribution shapes, as follows:
T
211()exp22iidddwdhh.
....
......
......
and (3)
211()exp22iiwhh..
..
..
.
.....
.......
......
Chouinard and Liu introduced a powerful technique to determine the optimal kernel sizes for the
calculation of rates, namely leastsquares crossvalidation. They also consider a related technique,
maximum crossvalidated likelihood, but the former is preferred because it is more robust.
Maximum crossvalidated likelihood was used to determine the optimum kernel size for the post
Katrina study (see Risk Analysis, 2008, for a discussion), where it was used to determine the
optimal kernelsize for the distribution of . P.
Operating Guidance 812
To calculate the optimal kernel width for the omnidirectional rate, the data are partitioned at
random into two samples (the estimation sample and the validation sample) using a randomization
scheme in which each storm is assigned to the estimation sample with probability p and to the
validation sample with probability 1p. The estimation sample is used to estimate the predicted
rate using Equations 2 and 3. The validation sample is then used to calculate the observed rate.
The two rates are then adjusted for the size of the two samples (i.e., for the effect of p), and the
difference between the two rates is squared. The random partitioning of the sample is repeated
many times (say, 500 to 1000 times) and the squared difference is summed over all these random
partitions. The resulting quantity is the crossvalidated square error (CVSE); the optimal choice of
kernel width is the one that yields the lowest CVSE. For the postKatrina Mississippi study, the
observed rate was calculated by counting the number of storms in the validation sample falling
within 40 km of the site and then dividing that count by 80 km and by the number of years in the
storm catalog. The probability p was set to 0.9 to avoid a large change to the size of the estimation
sample. The resulting optimal kernel size is not sensitive to these choices, as long as they are
within reasonable bounds (Chounard and Liu, 1997). Similarly, the results for directional rates are
not sensitive to the choice of angular interval.
dh
dh
In the Capture Zone approach, hurricanes are counted if they cross the coastline (or an idealized
representation of the coastline) within the capture zone. The resulting count is divided by the size
of the hurricane catalog and by the length of coastline, obtaining a rate of hurricanes per unit
length per unit time. The distribution of heading is then estimated based on the empirical
distribution of headings observed at landfall. As indicated earlier, this distribution of headings
depends on the geometry of the coastline and cannot be compared directly with the distribution
obtained using pointbased counting. If it is suspected that the rate is not constant within the
capture zone, the distribution of distance to some suitable reference point is computed based on the
associated empirical distribution.
If the storm rate is truly constant within the study region, then the distance to any conveniently
defined reference point (e.g., the midpoint of the region of interest) is drawn from a uniform
distribution. This is the most common situation in practice, and will usually be assumed by the
Mapping Partner, but it is not always the case. If pointbased counting indicates significant
variations in rate within the study region, or if the linebased counting indicates significantly
different rates for subdivisions of the capture zone or a distribution of distance that deviates from
uniform, then this deviation from uniformly distributed distances must be taken into account. The
JPMOS techniques described in the next section may be easily adapted to include this non
uniform distribution of distance.
For most new studies, it should not be necessary for the Mapping Partner to perform a detailed
validation study for kernel size, as outlined above. Instead, based on simple physical reasoning, it
will be generally sufficient to follow the precedent of the postKatrina Mississippi study, and to
adopt spatial and angular kernel widths of 200 km and 30°. Firstly, the results are not highly
sensitive to this choice, and, secondly, it is reasonable to assume some similarity of conditions at
other locations. However, the Mapping Partner should review the site data and consult Risk
Engineering (2008) for more details if it is thought that a more refined kernel estimate might be
beneficial to the study.
Operating Guidance 812
1.3.4. Storm Intensity
In probabilistic surge studies, the intensity of the storm is characterized by the pressure deficit ,
which is defined as the difference between the farfield atmospheric pressure and the central
pressure of the storm. The farfield pressure is commonly taken as a fixed value (usually 1013
mb), even if a different farfield pressure is known for a particular storm. For coastal flood
insurance studies, a peripheral pressure of 1013 mb may be assumed. Consequently, pressure
depressions can be estimated from central pressures reported in HURDAT and elsewhere, by
simply subtracting from 1013 mb.
P.
The lowerbound of the data used in this step should be consistent with the minimum
used for the definition of the rate. That is, the storm rate must correspond to the rate of storms with
intensities exceeding the cutoff . In addition, if the statistical distribution shape used includes a
lowerbound parameter, this parameter should be selected in a consistent manner.
The distribution shape used for should be consistent with the observed empirical distribution.
The most common distribution shapes in recent studies are the TypeI ExtremeValue distribution
(also known as FisherTippit or Gumbel) and the threeparameter (or truncated) Weibull
distribution (e.g., Resio, 2007, Risk Engineering, 2008, RENCI, 2008). It is recommended that one
of these distributions be adopted by the Mapping Partner, in accordance with the apparent quality
of fit with the study data. Nevertheless, the Mapping Partner may choose another distribution type
if the data shows that an improvement would be achieved.
The complementary cumulative distribution function (CCDF) of the TypeI distribution is given by
the equation
()[]1exp[]xUPPxe........
(4)
where is the mode of the distribution and is a parameter that measures the scale of the
distribution; both of these have units of pressure. Note that the TypeI distribution is defined for all
real values of (not just for values above the lowerbound used for the calculation of rate.
Strictly speaking, the CCDF should be normalized so that , where is the
lowerbound value of used in the calculation of storm rate.
U1/.
0[]1PPP....0P.
Similarly, the CCDF of the threeparameter Weibull distribution is given by
00[]exp[(/)(/)]kkPPxxuPuxP........
(5)
Operating Guidance 812
where is a scale parameter, is a shape parameter, and is the lowerbound value of
introduced above; u has units of pressure, while k is dimensionless.
uk
For annual exceedence frequencies of 0.2% or greater (that is, more frequent), the dominant storms
tend not to fall too far in the upper tails of the distributions; instead, rarity is more the result of
combined moderate parameter values and randomly close proximity, rather than of an extreme
value for any one parameter (example: despite its severity, Katrina was a Category 3 storm at
landfall). Therefore, the choice of distribution shape used for is likely to have only a moderate
effect on the results, whatever form is chosen.
In Chouinard’s approach for estimation of the distribution of , the distribution parameters
and are estimated from all the storm data using the method of maximum weighted likelihood,
where the weights depend on the distance between the track of storm i and the point under
consideration, and possibly subject to the monotonicity constraint described earlier. Specifically,
the weighted loglikelihood is of the form
....iiPikupfdwWL)],;(ln[)()ln(
(6)
where is the distance between the point under consideration and the linearized track of storm i
(associated with pressure deficit at landfall), is a Gaussian distancedependent weight
(which is given by Equation (3) introduced earlier, although the kernel size need not necessarily
the same as for the calculation of rate), and is the Weibull probability density
function (obtained by differentiating the cumulative function shown above); the summation extends
over all storms with exceeding the lower cutoff of the data set.
id
ip.)(ldw
dh
),;(kupfP..
P.
Following Chouinard et al. (1997), a technique known as maximum crossvalidated likelihood is
utilized to determine the optimal kernel size for the estimation of the Weibull parameters. As
was done for the calculation of the crossvalidated squared error for rates, the data are partitioned
into two samples (the estimation sample and the validation sample) using a randomization scheme.
The estimation sample is used to estimate the Weibull parameters and by determining the
values of and that maximize the loglikelihood function in Equation 6, possibly subject to a
monotonicity constraint. The validation sample is then used to calculate the observed log
likelihood. These observed loglikelihoods are then summed over all random partitions of the
sample. The resulting quantity is the crossvalidated likelihood (CVL; the optimal choice of kernel
width is the one that yields the highest CVL).
dh
This analysis may yield an optimal distancekernel size that is smaller than the optimal kernel size
obtained for the directional rates, but the slope in the upper portion of the CVL vs. kernel size has
been found to be nearly flat. This result indicates that the cross validation provides only a weak
upper bound for the kernel size. In the Mississippi study, the optimal kernel size that was obtained
for the directional rates was used for all calculations involving kernels, and it is recommended that
Operating Guidance 812
the Mapping Partner make a similar assumption unless there is an apparent need for a more detailed
analysis.
Once the optimal kernel size is selected (or the suggested default of 200 km is adopted), the best
estimate values of the Weibull parameters and are obtained by maximizing Equation 6,
possibly subject to a monotonicity constraint.
In the linebased approach, a suitable distribution shape is chosen to fit the empirical distribution of
, using standard statistical methods (e.g., method of moments, maximum likelihood, linear
regression on the transformed data). It is also important to investigate geographical variation in the
distribution parameters, although the data limitations often yield no statistically significant
differences.
P.
In both the Chouinard and Capture Zone methods, the estimated parameters have a high statistical
uncertainty as a result of limitations in the data. In these situations, the exchangeability axiom of
modern decision theory suggests that one should use the mean or “predictive” CCDF of (i.e.,
the expected value of the CCDF, averaged over the joint distribution of the distribution parameters),
not the bestestimate value obtained above. The reader is referred to McGuire et al. (200?) for an
elaboration of this issue in the context of earthquakes. Experience with the threeparameter
Weibull distribution indicates that there is a significant difference between the mean and best
estimate CCDF because the CCDF is a highly nonlinear function of the distribution parameters, and
that the mean CCDF is significantly higher (e.g., Risk Engineering, 2008). In the postKatrina
Mississippi study (Risk Engineering, 2008), a resampling thechnique known as bootstrapping
(Efron, 1993) was employed for the calculation of the mean or predictive CCDF. This approach is
very general and is easy to implement. Other approaches, such as standard methods for the
propagation of uncertainty may also be employed.
1.3.5. Storm Track: Forward Speed of Translation
The forward speed of the storm affects the wind field, making it more asymmetrical. It has an
additional effect on surge (beyond the effect on wind speeds), in that it helps determine duration of
high water (and so, perhaps, overtopping and filling volumes). There are physical arguments that
suggest a positive correlation between forward speed and , but the available data generally
show a weak or nonexistent correlation. Storms making landfall in the Atlantic seaboard tend to
have somewhat higher forward speed than those in the Gulf of Mexico as a result of differences in
the steering winds.
P.
Typical values of the mean forward speed are of the order of 5 to 6 m/s; typical standard deviations
are of the order of 2.5 to 3 m/s (e.g., Risk Engineering, 2008; RENCI, 2008). The associated
probability distributions can be taken as normal or lognormal by the Mapping Partner, based on
examination of the associated empirical distribution. Given the associated coefficients of variation
and the moderate importance of forward speed in the calculations, the practical effect of choosing a
different distribution is anticipated to be small.
Operating Guidance 812
1.3.6. Storm Size
The radial dimension of the hurricane wind field has a large effect on surge, as demonstrated by
Irish et al. ((2008). Recent studies have utilized a single parameter to characterize this size,
although there is a trend toward allowing this parameter to vary as a function of quadrant.
Nevertheless, it is suggested that for the present, a FEMA study should adopt a single size
parameter.
Two parameters are commonly used to represent storm size, namely the radius of maximum winds,
Rmax, and the characteristic radius of the exponential pressure profile, where the pressure
profile is written as
pR
0()expBpRprPPr
....
........
......
(7)
in which is the central pressure and B is Holland’s shape parameter (Holland, 1980), to be
discussed below; B is often taken as unity, although it can have a significant influence on winds
and surge, and so must be considered in any new study.
0P
There is only a slight difference between these two radius measures for typical cases, and some
studies have ignored the difference, or failed to recognize it. The difference may be large for the
profiles of real hurricanes, however, as these may have quite irregular shapes. Consequently, the
Mapping Partner should take care to distinguish between them in collection and analysis of the
study data.
Most studies find a weak negative correlation between Rmax (or ) and . In the postKatrina
Mississippi study, an expression of the form
P.
,ln[]4.370.29ln[]pmedianRP...
(8)
was obtained using linear least squares regression. Similarly, Vickery and Wadhera (2008)
obtained relations of the form
2,ln[]pmedianRabPc.....
(9)
for Atlantic and Gulf of Mexico hurricanes, where is latitude; the parameters a and b vary with
region. The Mapping Partner shall make a similar determination for the study data, as appropriate.
.
Most studies model the conditional distribution of given as lognormal, the associated
standard deviation of is generally found to be approximately 0.4 to 0.5. Vickery and
pRP.
ln[]pRP..
Operating Guidance 812
Wadhera (2008) find that stronger hurricanes exhibit a lower standard deviation and provide
equations for as a function of .
Although correlation between and is weak — or apparently nonexistent for certain
subsets of the data such as Gulf of Mexico hurricanes at landfall (Vickery and Wadhera, 2008) —
most studies assume a negative correlation. Storm physics modeling also provides support for a
negative correlation (Shen, 2006). Consequently, it is recommended that the Mapping Partner
should assume a correlation, although some effort may be required to assemble the necessary data
and it may prove necessary to enlarge the capture zone so as to make the correlation apparent. Note
that this matter is discussed at some length in NWS38 which may also be consulted for guidance.
pR
1.3.7. Other Physical Parameters
The wind fields of real hurricanes vary widely with both distance and azimuth from the storm
center, and cannot be completely characterized by only two parameters, namely and Rmax. The
natural choice for the next major parameter to include in a JPM analysis is Holland’s B parameter,
which was introduced in Equation 7 above. Higher values of B produce more highly peaked wind
fields, with higher values of the peak wind speed. On the other hand, this peak value occurs over a
narrower spatial reach. As a result of these counteracting effects, the sensitivity of peak surge and
surge hazard to B is not clear at present. According to Irish et al., (2009), the effect of changing B
from 0.9 to 1.9 is to give a change of the order of 15% in peak surge. It is recommended that this
matter be given attention by the Mapping Partner through numerical experiments with the
hydrodynamic model and alternate choices of B. There have been few statistical studies for B.
Vickery and Wadhera (2008) find a weak correlation with , , latitude, and seasurface
temperature.
maxR
Unless a preliminary sensitivity investigation suggests otherwise, the Mapping Partner may adopt a
mean representative value of B for the JPM simulations. If it is judged that variation of B should be
accounted for, the Mapping Partner should adopt a simplified method such as the postcomputation
error approach to be discussed later.
Additional parameters have been used to describe hurricanes, but not in JPM surge studies. For
instance, McConochie et al. (2004) and Cox and Cardone (2007) use a doubleexponential model
to describe the hurricane wind field in hindcast studies. This model has the form
12,1,2022()()expexpBBppRRprPPPPrr
........
.................
............
(11)
This model introduces three additional parameters, whose statistics (including correlations with
other parameters) must be determined. Currently, most historical hurricane data sets do not include
these additional parameters, making the necessary calculations impossible, but there are reanalysis
efforts under way that will determine these values in the future. For the present, it is recommended
Operating Guidance 812
that new FEMA studies should be performed with the simpler representations adopted in prior
work such as the postKatrina Mississippi study.
1.3.8. Treatment of Parameter Correlations
In principle, all hurricane characteristics are correlated to some degree. Most probabilistic surge
studies, however, consider only the correlation between and Rmax (although the important
major correlation of parameters with track angle has been commonly treated by the simple artifice
of dividing the storm sample into entering and exiting populations; within each subpopulation,
independence with angle has been assumed). The main reason for modeling the pressureradius
correlationdespite statistical values on the order of only 0.3—is that these are the two most
important hurricane parameters for surge calculations (other than landfall location, taken
independent by assumption). Allowing for the inverse pressureradius correlation ensures that
extremely intense storms are not assigned extremely large radii in construction of the JPM storm
simulation set.
2r
Operating Guidance 812
1.4. Storm Simulation Set – JPMOS Methods
At least four distinctly different JPM approaches have been used for FEMA flood insurance
studies. As noted earlier, the original work by Myers et al adopted a direct JPM method based on
the idea of dividing each parameter distribution into a small number of segments, and then
simulating all possible combinations (all possible storms). As noted earlier, the only difficulty with
this approach is that the number of simulations that are required quickly becomes prohibitive,
especially when considered in light of the computational demands made by modern highresolution
hydrodynamic models such as ADCIRC.
In the postKatrina efforts of the Corps (for Louisiana) and FEMA (for Mississippi) greatly
improved JPM methods were developed, permitting a JPM analysis to be performed with only
about one tenth the number of simulations as would be required with the original straightforward
approach. These are now known as Optimal Sampling methods. These guidelines correspond to the
approach used in the FEMA Mississippi study, called the Quadrature Method, although the Corps’
approach for Louisiana, called the Response Surface Method, is also entirely appropriate for new
FEMA studies. Experience gained during the postKatrina work showed that the two approaches
are capable of giving nearly identical results with nearly identical effort. These guidelines focus on
the Quadrature Method since it is more readily automated than the Response Surface Method
which requires a greater degree of expert judgment in the selection of storms.
Recently, a fourth approach has been followed in a study of North Carolina (see RENCI, 2008). It
is not described in detail here, but it is noted that it is not, strictly speaking, an optimal sampling
approach. It is more akin to a traditional JPM approach in that the parameter distributions are
discretized and all combinations are simulated. Furthermore, it departs from the other approaches
in that number and locations of the tracks are not established in a defined pattern, but are
distributed over the coastal area using Monte Carlo simulation, with the assumption of a Poisson
occurrence rate. Whereas the quadrature and response surface methods have been compared and
found consistent, such a comparison with the North Carolina approach has not yet been made.
1.4.1. Summary of the Response Surface Method
This approach takes advantage of the following three observations from sensitivity studies
performed during the postKatrina studies: (1) the calculated surge is a fairly smooth function
of the storm parameters; (2) is most sensitive to , , and track location (or alongcoast
distance between storm track and location of interest); and (3) the sensitivity of to heading
angle and forward velocity is weaker and may be approximated as linear. Furthermore, the
variation of as a function of these parameters is fairly smooth. These observations are
confirmed by the sensitivity analyses documented in URS (2007) and by ADCIRC runs cited by
Resio (2007).
m.
P.pR
.fV
As a result of these observations, it is possible to perform surge calculations for a moderate number
of synthetic storms—with carefully selected combinations of parametersand then to interpolate
between the calculated surge elevation (in five dimensions) to obtain the surge elevation for any
Operating Guidance 812
desired combination of parameters. The computational cost for this interpolation is minimal. As a
result, one can discretize the domain of the JPM integral very finely, even in five dimensions.
The main difficulty in the responsesurface JPMOS scheme resides in the experimental design (i.e.,
the selection of the parameter combinations for the synthetic storms) in a manner that provides
enough points in the fivedimensional parameter space, without
requiring a very large number of synthetic storms, and then implementing a robust interpolation
scheme that works reliably for all target sites of interest. This selection process and interpolation
scheme treats and as the primary variables for each selected track, and takes advantage of
the weak sensitivity to heading angle and forward velocity , which are treated as linear. The
interpolation between tracks is more delicate and requires special treatment, as described by Irish et
al. (2009).
tracklocationpfPRV......
The application of the Response Surface Method in Louisiana and Texas is summarized in Resio
(2007 white paper); Resio et al. (2009), and Irish et al. (2009). These publications provide details
on the application of the method and show typical results. In particular, it has been found that the
approach yields results that essentially indistinguishable from the Quadrature Method (Toro, et al,
2009).
1.4.2. Summary of the Quadrature Method
Gaussian quadrature is a well known technique for approximation of integrals of the form
, where is often a probability density function (i.e., it is positive and it
integrates to unity) and is an function belonging to a particular family of functions. The
quadrature approximates the integral as a finite weighted summation of the form ,
where the nodes and the corresponding weights are selected in a manner that maximizes the
accuracy of the approximation, while keeping the number of nodes small. In the context of the
JPMOS Quadrature method, each node can be thought of as one synthetic storm. The weight
associated with each node is multiplied by the annual rate of storms to obtain the annual rate for
that synthetic storm.
..dxxpxfI)()()(xf
)(xp
..
iiixpwI)(
ixiw
In onedimensional Gaussian Quadrature, the number of nodes, the nodal locations, and the
weights are selected so that the summation will evaluate the integral exactly if is a
polynomial of a certain degree and is a particular probability distribution (e.g., a standard
normal probability density). This technique is used frequently in one dimension. Miller and Rice
(1983) provide implementation details and results for a variety of commonly used probability
distributions. The improvement in calculation efficiency (number of function evaluations needed
for a specified accuracy) can be very great compared to simpler methods.
)(xp
)(xf
It is also possible to fix the weights to arbitrary values (e.g., equal weights or 1/6, 2/3, 1/6) and
then calculate the nodal values so that polynomials of a certain degree are integrated exactly (or,
Operating Guidance 812
equivalently, so that distribution moments up to a certain order are preserved by the , pairs).
In addition, it is possible to fix the nodal values and then compute the required weights.
Unfortunately, extension of these socalled zeroerror onedimensional rules to more than one
dimension is problematic. It is easy to apply a onedimensional quadrature for each parameter and
then generate all possible multidimensional parameter combinations. These so called product
rules result in a large number of nodes. Furthermore, if Gaussian Quadrature is used, many of
these combinations will have very low weights. The more efficient techniques to generate multi
dimensional Gaussian quadratures often lead to some weights being negative, which create stability
problems and make it impossible to interpret the weights in terms of the occurrence rates of
synthetic storms.
The product rules mentioned above have some practical applications. In particular, one can use a
product rule to construct a JPM Reference Case which is then used  together with a fast surge
code such as SLOSH (see URS 2008 for an example)  to validate a more efficient multi
dimensional Quadrature. In addition, a product rule constructed from 3 and 4point quadratures
with equal weights is being used in the recent surge study for North Carolina (RENCI, 2008).
In contrast to Gaussian quadrature, Bayesian quadrature (also termed GaussianProcess quadrature)
defines the family of functions as all possible realizations of a random process having a
certain autocovariance function, and seeks to minimize the integration error in a meansquared
sense instead of trying to make it equal to zero. The main advantage of the probabilistic
formulation of the quadrature problem is that the formulation is easy to apply in multiple
dimensions. In addition, it is possible to control the accuracy of integration in each dimension by
adjusting the parameters of the autocovariance function.
)(xp
The Quadrature JPMOS approach, as applied to date, uses Bayesian Quadrature in conjunction
with more traditional numericalintegration schemes to transform the JPM integral into a discrete
summation with a moderate number of nodes. The result is a set of synthetic storms, where each
synthetic storm is defined by its parameters at landfall (i.e, , , , track location, etc), and
each synthetic storm has an associated annual recurrence rate Typically, a few hundred synthetic
storms (rather than a few thousand) are sufficient to attain the desired accuracy. For numerical
reasons, the calculation of the optimal nodal locations and associated weights is performed in
standard multidimensional normal distribution space. The nodal locations are then mapped into
the physical space of , , , etc.
P.pRfV
1.4.3. The Quadrature Method of Storm Selection
1.4.3.1 Overview
This section presents the recommended Quadrature method, based upon its use in the postKatrina
Mississippi study. It is anticipated that the method will evolve as it is exercised in future studies.
Operating Guidance 812
.Discretize the distribution of into broad slices, roughly corresponding to Saffir
Simpson hurricane Categories and compute the probability mass contained in each
slice.
.Within each slice, discretize the joint probability distribution of
, Rp, Vf, and using Bayesian Quadrature. Details on this step are
provided subsequently.
.Discretize the distribution of landfall location by replicating each of the synthetic
storms defined in the previous two steps at spatial offsets equal to (measured
perpendicular to the storm track). To avoid aliasing, apply a random perpendicular
offset (with a uniform distribution between 0 and ) to each replicated set of storms.
Sensitivity studies indicate that a spacing of is small enough to capture the peak
surge at all grid locations.
.Compute the probability assigned to each synthetic storm as the product of the
probabilities resulting from the previous three steps. Then, compute the rate
assigned to each synthetic storm as the probability computed above times the rate
per unit length times the storm spacing.
The quadrature approach to define a representative set of synthetic storms and their associated
annual rates, as applied in the Mississippi study, uses a combination of traditional and sophisticated
numericalintegration schemes. The process may be summarized in the following fundamental
steps:
These steps are discussed in more detail in what follows.
1.4.3.2 Implementation of Bayesian Quadrature for JPMOS
1.4.3.2.1 Inputs
The first set of inputs to the Bayesian Quadrature algorithm consists of the probabilitydistribution
information for the hurricane characteristics at landfall, namely , Rp, Vf, , and
possibly other characteristics. For each hurricane characteristic, this information consists of the
distribution shape (e.g., Weibull, Gumbel, lognormal) and distribution parameters. For dependent
hurricane characteristics such as , these distribution parameters are functions of .
)(slicewithinP..
pRP.
The second set of inputs consists of information on the characteristics of the surge response.
Because of the probabilistic nature of the Bayesian Quadrature method, this information is of a
Operating Guidance 812
probabilistic nature, and consists of the correlation distances of the term
in Equation 1 along the various dimensions. The higher the
sensitivity of the quantity to a particular hurricane characteristic,
the lower the corresponding correlation distance. These correlation distances are not specified in
the physical units of the hurricane characteristics. Instead, they are specified in the corresponding
normaldistribution space used internally by the Bayesian Quadrature algorithm. Estimates of these
correlation distances could be obtained from sensitivity results, such as those generated for the
Mississippi study, but have been have been specified on the basis of judgment. The following
values are suggested for guidance for the choice of correlation distances in a new FEMA study:
[(,,,,etc.)]mpfPPRV.....
Sensitive (important): Pressure and Radius: correlation distances of 1 to 3
Insensitive (less important): Forward speed, direction: correlation distances of 4 to 6
In a relative sense, the Quadrature JPMOS algorithm tends to spread the sampling nodes more
widely along those directions with lower correlation distances, providing a closer match to the
marginal probability distributions in those directions. Thus, it is important to specify correlation
distances that relate to the importance of the various physical quantities, in order to obtain an
optimal allocation of effort among the various dimensions.
In an absolute sense, numerical experiments in one dimension show that low values of the
correlation distance cause the algorithm to be more cautious and to tend towards equal weights,
while high values provide a wide range of weights and sample points that extend farther into the
distribution tails, approaching those obtained by Gaussian quadrature. The ideal choice is
somewhere in between.
It is reasonable to assume that relative parameter importance is likely to be similar in a new study to
their importance as estimated by detailed sensitivity tests in the postKatrina Mississippi study.
Consequently, the assumptions used there should be reviewed by the Mapping Partner and can be
followed unless there is reason the suspect that alternate choices should be used.
An additional input is the number of nodes to generate. In the Mississippi application, different
numbers of nodes, and somewhat different correlation distances, were employed for the various
slices. The number of nodes in a slice ranged from five to seven, and can be followed as
precedent for a new study.
P.
1.4.3.2.2 Algorithmic Steps
The first algorithmic step employed in the Bayesian Quadrature is the selection of the optimal
nodal locations (in normaldistribution space) and the associated weights. This is achieved by
using two nested optimizations, both of which seek to minimize the variance of the integration
error. At the inner level of nesting, there is the optimization to determine the best weights (for
given nodal locations). This is done in closed form, by solving an optimization problem not too
different from linear least squares. At the outer level, there is the search for the best set of nodal
locations. This is done using a numerical optimization scheme. Details on the formulation and
implementation of both optimizations are provided in Toro et al. (2007, 2009).
Operating Guidance 812
The second step is the mapping of the nodal locations from standard normaldistribution space to
the physical space of , , , etc. This is done by using the socalled Rosenblatt
transformation (see, for example, Madsen et al., 1986; Melchers, 1999). In one dimension, this
transformation simply maps each normallydistributed nodal value by finding the value of the
physical quantity that has the same value of the cumulative distribution. Extension to multiple
dimensions is straightforward, as one can usually write the joint cumulative distribution of the
hurricane characteristics as a product of marginal and conditional distributions, e.g.,
, allowing the sequential application of the onedimensional transformation. The
Rosenblatt transformation allows practical implementation of Bayesian Quadrature for virtually any
choice of joint probability distributions, as required for JPMOS.
pfPRPVFFF..
For convenience of the Mapping Partner, specialized utility programs have been written to perform
many of the necessary calculations. These programs and User’s Manuals are available through the
FEMA Project Officer, and are briefly described in a later section of these guidelines.
1.4.3.2.3 Verification of the Storm Selection Step
Because the Quadrature JPMOS formulation involves some simplifying assumptions regarding the
properties of the autocovariance functions, and because the parameters of this function are chosen
on the basis of judgment, it is recommended that the accuracy of the synthetic storm set be
validated. This may be done by creating a larger (typically a few thousand) reference set of
synthetic storms using a conventional JPM formulation, calculating surge for both sets using a fast
hydrodynamic program such as NOAA’s SLOSH model, and comparing the resulting flood hazard.
The verification performed in the Mississippi study provides good guidance in this regard, and
should be studied by the Mapping Partner. The following are some of the key features of this
exercise. The probability distributions for the Reference JPM scheme were discretized using one
dimensional Gaussian quadrature and then all combinations were generated (i.e., a product rule was
used). The number of points in these quadratures varied as a function of importance, using 6 nodes
for , 5 nodes for , 3 nodes each for forward speed and for heading, and a track
spacing equal to . Surge calculations for the JPMOS scheme and the Reference scheme were
performed and compared for a large number of grid points distributed throughout the study region;
comparisons were performed for the surges associated with both the 1% and 0.2% annual
exceedance chances. Whereas the Reference scheme involved several thousand storm simulations,
satisfactory OS schemes of less than 200 storms were identified, showing deviations from the
Reference results of better than 1 foot of surge.
P.pRP.
pR
It may be possible to streamline this verification by reducing the number of grid points considered
or by using a parametric surge model (e.g., Irish et al., 2008). The reduction in the number of grid
points brings only moderate savings. The use of the parametric source model brings significant
savings, but may only be appropriate for uncomplicated coastlines.
There are other simple procedures to verify the adequacy of the JPMOS storm selection and rates,
as follows:
Operating Guidance 812
. Comparison of statistical moments of the original (continuous) distributions to those
calculated from the JPMOS discretization. As a minimum, the marginal moments up to
order three and the covariance between and should be checked. P.pR
. Graphical examination of the cumulative distribution of calculated surge obtained at
several grid points. Ideally, this distribution should have no large jumps in the regions of
interest (the region between 10% and 0.2% annual exceedance chances). Large jumps
indicate that the hazard is controlled by one (or a few) synthetic storms, suggesting that the
JPMOS storm set needs to be refined.
Given the limited practical experience with the JPMOS discretization, these simpler procedures
would not constitute a replacement for a SLOSHbased or parametricmodel based verification of
the selected JPMOS storm set.
In past studies, these verification exercises have been performed prior to introducing the
contributions of the small random error terms in the calculated surge (to be discussed below). This
is conservative, in the sense that the JPMOS procedure is likely to be more accurate than the
verification tests indicate. The effect of integration over the small error terms is to make
a smoother function of the hurricane characteristics, making it
easier to integrate numerically.
1.4.4. Development of a Complete Storm History
Both the ResponseSurface and Quadrature JPMOS approaches characterize each synthetic storm
by means of the values of the storm’s characteristics (i.e., , , , , landfall location, etc.)
at landfall (or at some arbitrary location prior to landfall). The numerical oceanresponse models
require a complete history of hurricane characteristics and eye coordinates for a period of several
days prior to landfall.
.
In recent studies, the storm characteristics prior to landfall have been treated as deterministic
functions of the characteristics at landfall. These functions have included some weakening
immediately prior to landfall. In the models used recently for the central Gulf of Mexico, all but
the storms with very small radius begin to weaken, increase their radius, and decrease their Holland
B over the last 90 miles prior to landfall (see Resio et al., 2009 for details). Similar models have
been developed for storms affecting North Carolina (see RENCI, 2008). In principle, these
variations in storm characteristics should also be treated as random, but this is difficult to do within
the present JPMOS formulation, without unrealistically enlarging the dimensionality of the
problem (beyond the adequacy of the data).
It is also important to use realistic track geometries, mostly for the purpose of calculating the waves
that tend to accompany the surge and which, in fact, contribute to the surge through the
intermediate mechanism of the wave’s radiation stresses. In the Gulf of Mexico, examination of
the tracks from strong storms indicates that they tend to enter the Gulf through the Florida or
Yucatan straits and then follow simple tracks, which may be easily mimicked using simple
deterministic algorithms. These algorithms generate a track for any given landfall location and
Operating Guidance 812
heading. A similar approach has been followed for North Carolina (see RENCI, 2008), but with
models that exhibit significantly less weakening just prior to landfall. Although the tracks are
idealized, they are chosen to follow the main trends of the observed track history – the landfall
track configurations are, of course, directly determined by the parameter selections at landfall, so
idealization of the offshore track segments is acceptable. Note that this treatment is superior to the
approach used in early flood insurance studies, which assumed simple fixed straight tracks
throughout the duration of a storm.
Operating Guidance 812
1.5. Second Order Concerns
1.5.1. Small Random Contributions – Overview of the Approach
The foregoing procedures will not always include all factors which contribute to a best estimate of
surge height. In order to minimize the number of storms to be simulated, some minor or secondary
factors may be treated by an approximate method. Furthermore, random uncertainties associated
with modeling errors in both meteorology and hydrodynamics also affect the best estimates.
The relationship given in Equation 1 can be expanded to include these factors by inclusion of the
term e (the probability integral is here shown as the discrete summation over the simulation storm
set):
(11) max(1)
1[][()]
nyriiiPPx......
.
.....
where e might consist of several constituents, such as:
.1 – representing the astronomical tide level as a random function of time, estimated from a
local hurricane season tide prediction, and characterized by a standard deviation around zero
mean.
.2 – representing variations in surge response caused by random variations of the Holland B
parameter that are not represented in the modeling. The standard variation for this term may be
dependent upon the computed surge elevation.
.3 – representing random errors in the computed surge caused by lack of skill of the numerical
modeling. This can be estimated by comparisons of predictions with highwater marks.
.4 – representing variations in the surge due to a wide range of departures in the real behavior of
hurricane wind and pressure fields that are not represented by the PBL or other meteorological
model used to describe the storms. This can be evaluated by comparing the results of surge
modeling done using handcrafted ‘best winds’ with the findings for the same storms as
represented using the PBL model chosen for the simulations.
These and other components of , as necessary, are taken to be independent, and so can be
combined into a single term having a standard deviation given (with obvious notation) by:
.
(12)
12342222
..............
For each of these components, and others as may be identified, the Mapping Partner shall estimate
the standard deviations following the precedent shown in the postKatrina Mississippi study. Note
that tide cannot always be treated as a small linear addition (see the following subsection).
However, when it can, the necessary sigma is easily estimated from local tide predictions, restricted
to hurricane season.
Operating Guidance 812
The process for introducing these secondary factors is described in Section D.X.6.3; this is done
after surge is computed for all synthetic storms. One of the effects of introducing these secondary
factors is that they smooth out the P[ ] term in Equation 1, making numerical evaluation somewhat
easier.
1.5.2. Regression Method for Large Amplitude Tides
In the event that tide amplitudes are not small compared to the 1% surge level, or if the Mapping
Partner has reason to doubt the validity of linear superposition owing to great distances of inland
propagation over flat terrain and the like, treating tide as a small additive correction will not be
appropriate. No simple method has been identified to handle the tide in such cases. Note, too, that
in some cases the tide may be small compared to the 1% surge level, but not compared to the 10%
and 2% levels. The relative error in those cases may then be greater, although still smaller in an
absolute sense. Whether these other levels must be given the same degree of attention as the 1%
and 0.2% levels in a particular study, should be determined by the Mapping Partner in consultation
with the FEMA Project Officer.
When nonlinear interactions are important and linear superposition is inadequate, the Mapping
Partner may adopt the more complex approach detailed in FEMA’s User’s Manual for the FEMA
Coastal Flooding Storm Surge Model (FEMA, 1988). The approach is discussed in full detail in
Chapter 8 of Volume 1 of that document. There are also computer codes (presented in Volume 3)
which may be used to help guide new work. In brief, the procedure recommended is to simulate a
small number of representative storms not only at midtide (as is done for the full storm simulation
set), but also at other tide levels and relative phases. These hydrodynamic simulations properly
incorporate the interactions of surge and tide throughout the study area, and can be compared with
estimates based on linear addition. The comparison of these two calculations is then used to define
regression expressions that are used to adjust the estimates obtained by linear addition so as to
better approximate the full simulations. The approach is relatively time consuming, and should be
accounted for in the initial study scoping with concurrence of the FEMA Project Officer.
Operating Guidance 812
1.6. Surge Frequency Determination
1.6.1. Overland Distribution of Target Sites
Surge statistics are required at enough points distributed throughout the study region to permit
accurate mapping of flood zones (and to permit the prior determination of overland wave crest
additions, whether by use of FEMA’s WHAFIS model or some other approach as may be adopted).
The simplest selection of target points would correspond to the nodes of the hydrodynamic model,
since surge elevations are computed at each node throughout the simulation of a storm. However,
modern models such as ADCIRC are typically run with extremely fine resolution, so that several
tens or hundreds of thousands of grid points might fall within the area of interest. Such extreme
density is not usually required for preparation of flood hazard maps. Consequently, the Mapping
Partner may select an adequate subset of points for the statistical analyses. As a practical matter,
however, given the availability of large machines and inexpensive data storage, it may be simplest
just to include all points for analysis and, in a later step, produce a BFE surface from which the
necessary mapping information can be easily extracted.
1.6.2. Construction of the Simulated Density Distribution Histograms
Once the JPMOS storm simulations have been completed, and any necessary adjustments for
secondary factors such as large amplitude tide have been accounted for (but exclusive of the small
factors treated as random error terms), the final determination of flood frequency at a given point
follows using the methods which have been used in past FEMA studies and which are detailed in
the FEMA Coastal Flooding Hurricane Storm Surge Model documentation (FEMA, 1988) and in
the report of the postKatrina Mississippi study.
Focusing on a single site within the study region, the key idea is to construct a histogram of
accumulated rate versus peak surge elevation, as shown in Figure 1.1 The histogram consists of
bins of elevation with suitably small widths (such as 0.01 meters) extending from zero to a bin
exceeding the largest surge of interest. Then the rate associated with each of the simulated storms
(as determined using the Quadrature method outline above) is accumulated into the particular bin
corresponding to the peak surge at the site for that storm. With a fine resolution of bin width, many
bins will, of course, remain empty, and the final histogram is an estimate of the surge probability
density function.
Operating Guidance 812
Figure 1.1 Histogram generated for a single JPM point based on
surges and storm rates.
00.0010.0020.0030.0040.0050.0060123456789Surge Height (m)
ProbabilityRate
Were there no small secondary factors to be accounted for (the several epsilon terms discussed in
Section D.1.5.1) this estimate of the density distribution is then summed from the top down, to
produce the corresponding estimate of the cumulative distribution.
The surge elevations at any frequency of interest are obtained from the cumulative distribution, by
simply entering the distribution at the specified frequency on the vertical axis, and reading across
to the curve and down to the nearest bin. The nearest bin will give the corresponding surge
elevation to the bin resolution.
1.6.3. Histogram Adjustment for Secondary Random Factors
In order to account for the secondary (epsilon) terms, one adopts an extremely simple procedure.
Consider, as shown in the upper portion of Figure D.1.2, the accumulated rate contained in a single
bin of the density histogram. The assumption is that owing to the small random variation associated
with the secondary terms, this quantity of rate could be smeared over an interval of elevation bins
above and below the original bin. This redistribution is shown in the lower portion of Figure D.1.2,
and is simply a discrete approximate to the Gaussian having a width determined by the composite
standard deviation given by Equation 12.
This same sort of redistribution is performed for each bin in the original histogram. Note that in
general the contribution of a particular factor may not be constant, but may be dependent upon the
magnitude of the surge, and so on the bin location. Once the redistribution of bin rates has been
completed, the revised density distribution is summed from the top down, as described before, to
yield the cumulative distribution shown in Figure D.1.3, below. Keep in mind that this distribution
is unique to a site, so that many thousands of such computations will be needed, depending upon
the density of target sites selected for mapping purposes.
Operating Guidance 812
The same total rate after
Gaussian redistribution
Accumulated rate before
redistribution
Figure 1.2 Example of redistribution of the accumulated rate within a
single bin to account for secondary random processes.
Figure 1.3 Determination of the 1% surge from the top
down integrated rate histogram
00.010.020.030.040.050.060.070.080.090.101234567891011Surge Height (m)
Cumulative ProbabilityAnnual Exceedance Rate
.
Operating Guidance 812
1.7. Combination of Surge and Other Flood Processes
In general, an area may be affected not only by storm surge from tropical storms, but also by surge
from extratropical storms (northeasters) or by rainfall runoff in the overlap with riverine flooding
regions. The approach recommended here is based on the assumption of independence: That is,
hurricanes and northeasters (or hurricanes and riverine floods, or even all three processes) might
affect the flood statistics at a site, but not simultaneously. This assumption of independence permits
a very simple method to determine the composite flood elevation frequency curve.
The procedure is straightforward, beginning with development of curves or tables for rate of
occurrence vs. flood level for each flood source. Rate of occurrence per year is just equal to the
reciprocal of the recurrence interval, and is numerically very close to what is loosely called the
flood elevation probability, for infrequent events. Then one proceeds as follows at each point of
interest, P.
a. Select a flood level Z within the elevation range of interest at point P.
b. Determine the rates of occurrence RP,1 (Z) and RP,2 (Z) of the two processes exceeding Z at
site P (number of events per year).
c. Find the total rate RP,T (Z) = R P,1 (Z) + RP,2 (Z) at which Z is exceeded at point P,
irrespective of flood source.
d. Repeat steps (a) through (d) for the necessary range of flood elevations.
e. Plot the combined rates RP,T (Z) vs. Z and find ZP,100 by interpolation at RP,T 0.01. .
f. Repeat steps (a) through (f) for a range of sites covering the mixed flood zone.
The procedure is shown schematically in Figure 1.4, in which the combined curve has been
constructed by addition of the rates at elevations of 6, 8, 10, and 12 feet. The example shown is for
the combination of surge with rainfall runoff flooding in the mixed tidal zone; the surge curve,
itself, might be the combination of both hurricane and northeaster rates, determined independently.
Operating Guidance 812
combinedRiverineSurge2
Figure 1.4 Schematic Illustration of Hurricane and Northeaster Rate Combination
Operating Guidance 812
1.8. Accompanying Utility Programs
The procedures outlined in these guidelines were developed during the intensive efforts to
reevaluate coastal hazards in the Northern Gulf following Hurricanes Katrina and Rita of 2005. In
order to simplify their application and to ensure a correct implementation of some of the methods
not commonly encountered in past FEMA studies, two utility programs have been written. One is a
console program, SURGE_STAT, to compute the surge statistics at the target sites, including the
effects of secondary parameters. The other is an Excel spreadsheet, JPMOSQ.XLS, to select the
parameters of the OS storms, according to the quadrature methods.
The programs and User’s Manuals are available to Mapping Partners upon request to the FEMA
Project Officer.
Operating Guidance 812
1.9. References
Benjamin, J.R. and Cornell, C.A. (1970). “Probability, Statistics, and Decision for Civil
Engineers”. New York, New York: McGrawHill.
Chouinard, L.M. and C. Liu. (1997a). “Model for Recurrence Rate of Hurricanes in Gulf of
Mexico,” Journal of Waterway, Port, Coastal and Ocean Engineering, 123, 3, 113119.
Chouinard, L.M., C. Liu, and C.K. Cooper (1997b). “Model for Severity Rate of Hurricanes in
Gulf of Mexico,” Journal; of Waterway, Port, Coastal and Engineering, 123, 3, 120129.
Cox & Cardone (2007)
Diaconis, (1988). “Bayesian numerical analysis. Statistical Decision Theory and Related Topics”
IV pp.163175.
Efron, B (1993)
Federal Emergency Management Agency (1988), Coastal Flooding Hurricane Storm Surge Model,
(in three volumes), Washington, DC.
Genz, A., and B. Keister (1996). Fully Symmetric Interpolatory Rules for Multiple Integrals over
Infinite Regions with Gaussian Weight. J. Comp. Appl. Math. 71, 299309.
Ho, F.P. and V.A. Myers, (1975). “Joint Probability Method of Tide Frequency Analysis applied to
Apalachicola Bay and St. George Sound, Florida”, NOAA Tech. Rep. WS 18, 43 p.
Holland (1980)
Irish, J.L., Resio, D.T., and Ratcliff, J.J., (2008). “The Influence of Storm Size on Hurricane
Surge.” J Phys. Oceanogr., DOI: 10.1175/2008JPO3727.1.
Irish, J.L., Resio, D.T., and Cialone, M.A. (2009) (in submission). “A Surge Response Function
Approach to Coastal Hazard Assessment Part 2: Quantification of Spatial Attributes of Response
Functions.” Journal of Ocean Engineering.
Jelesnianski, C.P., Chen J. and Shaffer W. A., (1992). “SLOSH: Sea, Lake, and Overland Surges
from Hurricanes.” NOAA Technical Report NWS 48, NOAA, Washington, DC.
Journel, A.G., and Ch. J. Huijbregts (1978). Mining Geostatistics. Academic Press, London.
Madsen, H.O., Krenk, S. and Lind, N.C. (1986): “Methods of Structural Reliability.” PrenticeHall,
Englewood Cliffs, New, Jersey.
McConochie et al 2004
Operating Guidance 812
Melchers, 1999
Miller, A.C., and T.R. Rice (1983). “Discrete Approximations of Probability Distributions,”
Management Science, 29, 3, 352361.
Myers, V.A., (1975). “Storm Tide Frequencies on the South Carolina Coast”, NOAA Tech. Rep.
NWS16, 79 p.
Niedoroda, A.W, D. Resio, G. Toro, D. Divoky, H. Das, and C. W.,Reed, (2007). “Evaluation of
the Storm Surge Hazard in Coastal Mississippi”, 10th International Workshop on Wave
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