All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
However, the document contains useful guidance to support implementation of the new standards.
(File Attachment comment
D.4.3 Final Draft.doc)
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
D.4.3 Flood Frequency Analysis Methods
This section outlines general features of statistical methods used in a flood insurance study,
including providing basic statistical tools that are frequently needed. It is recommended that
extremal analysis of annual maxima be performed using the Generalized Extreme Value (GEV)
Distribution with parameters estimated by the Method of Maximum Likelihood. The discussion
in this section is illustrative only; guidelines for application of these tools in specific instances
are provided in other sections of this appendix.
D.4.3.1 The 1% Annual Chance Flood
The primary goal of a coastal Flood Insurance Study (FIS) is to determine the flood levels
throughout the study area that have a 1% chance of being exceeded in any given year. The level
that is exceeded at this rate at a given point is called the 1% annual chance flood level at that
point, and has a probability of 0.01 to be equaled or exceeded in any year; on the average, this
level is exceeded once in 100 years and is commonly called the 100year flood.
The 1% annual chance flood might result from a single flood process or from a combination of
processes. For example, astronomic tide and storm waves combine to produce the total high
water runup level. There is no onetoone correspondence between the 1% annual chance flood
elevation and any particular storm or other floodproducing mechanism. The level may be
produced by any number of mechanisms, or by the same mechanism in different instances. For
example, an incoming wave with a particular height and period may produce the 1% annual
chance runup, as might a quite different wave with a different combination of height and period.
Furthermore, the flood hazard maps produced as part of an FIS do not necessarily display, even
locally, the spatial variation of any realistic physical hydrologic event. For example, the 1%
annual chance levels just outside and just inside an inlet will not generally show the same
relation to one another as they would during the course of any real physical event because the
inner waterway may respond most critically to storms of an entirely different character from
those that affect the outer coast. Where a flood hazard arises from more than one source, the
mapped level is not the direct result of any single process, but is a construct derived from the
statistics of all sources. Note then that the 1% annual chance flood level is an abstract concept
based as much on the statistics of floods as on the physics of floods.
Because the 1% annual chance flood level cannot be rigorously associated with any particular
storm, it is a mistake to think of some observed event as having been the 1% annual chance
event. A more intense storm located at a greater distance might produce the same flood level, or
the same flood level might be produced by an entirely different mechanism, such as by a tsunami
from a distant landslide or earthquake. Furthermore, if a particular storm were, in fact, the so
called 100year event, it could not be so everywhere, but only in its effect at a particular point.
The 1% annual chance flood level is a consequence solely of the areawide flooding mechanisms
recognized for a particular location. That is, there may be mechanisms that are not taken into
account, but that could also produce water levels comparable to the 1% level or that could
D.4.31 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
contribute to the 1% level. For example, tsunamis occur in all oceans, so even the Atlantic Coast
is vulnerable to tsunami attack at some frequency. The Great Lisbon earthquake of 1755 (with
magnitude approaching 9) produced a large Atlantic tsunami that was felt in the New World;
however, tsunamis are not recognized as areawide flood sources for the Atlantic Coast.
Similarly, advances in science may from time to time reveal new flood mechanisms that had not
previously been recognized; for example, only in recent years has the physics of El Nińos been
clarified and their contribution to coastal flood levels recognized.
D.4.3.2 Event vs. Response Statistics
The flood level experienced at any coastal site is the complicated result of a large number of
interrelated and interdependent factors. For example, coastal flooding by wave runup depends
upon both the local waves and the level of the underlying still water upon which they ride. That
still water level (SWL), in turn, depends upon the varying astronomic tide and the possible
contribution of a transient storm surge. The wave characteristics that control runup include
amplitude, period, and direction, all of which depend upon the meteorological characteristics of
the generating storm including its location and its timevarying wind and pressure fields.
Furthermore, the resulting wave characteristics are affected by variations of water depth over
their entire propagation path, and thus depend also on the varying local tide and surge. Still
further, the beach profile, changing in response to waveinduced erosion, is variable, causing
variation in the wave transformation and runup behavior. All of these interrelated factors may be
significant in determining the coastal flood level with a 1% annual chance of occurrence.
Whatever methods are used, simplifying assumptions are inevitable, even in the most ambitious
responsebased study, which attempts to simulate the full range of important processes over time.
Some of these assumptions may be obvious and would introduce little error. For example, a
major tsunami could occur during a major storm, and it might alter the storm waves and runup
behavior and dominate the total runup. However, the likelihood of this occurrence is so small
that the error incurred by ignoring the combined occurrence would be negligible. On the other
hand, the conclusion might not be so clear if the confounding event were to be storm surge rather
than a tsunami because extreme waves and surge are expected to be correlated, with high waves
being probable during a period of high surge.
These guidelines offer insight and methods to address the complexity of the coastal flood process
in a reasonable way. However, the inevitable limitations of the guidance must be kept in mind.
No fixed set of rules or cookbook procedures can be appropriate in all cases, and the Mapping
Partner must be alert to special circumstances that violate the assumptions of the methodology.
D.4.3.2.1 EventSelection Method
A great simplification is made if one can identify a single event (or a small number of events)
that produces a flood thought to approximate the 1% flood. This might be possible if, for
example, a single event parameter (such as deepwater wave height) is believed to dominate the
final runup, so the 1% value of that particular item might suffice to determine the 1% flood. In its
simplest form, one might identify a significant wave height thought to be exceeded with only 1%
chance, and then to follow this single wave as it would be transformed in propagation and as it
would run up the beach. This is the eventselection method. Used with caution, this method may
D.4.32 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
allow reasonable estimates to be made with minimal cost. It is akin to the concept of a design
storm, or to constructs such as the standard project or probable maximum storms.
The inevitable difficulty with the eventselection method is that multiple parameters are always
important, and it may not be possible to assign a frequency to the result with any confidence
because other unconsidered factors always introduce uncertainty. Smaller waves with longer
periods, for example, might produce greater runup than the largest waves selected for study. A
slight generalization of the eventselection method, often used in practice, is to consider a small
number of parameters – say wave height, period, and direction – and attempt to establish a set of
alternative, “100year” combinations of these parameters. Alternatives might be, say, pairs of
height and period from each of three directions, with each pair thought to represent the 1%
annual chance threat from that direction, and with each direction thought to be associated with
independent storm events. Each such combination would then be simulated as a selected “event”,
with the largest flood determined at a particular site being chosen as the 100year flood. The
probable result of this procedure would be to seriously underestimate the true 1% annual chance
level by an unknown amount. This can be seen easily in the hypothetical case that all three
directional wave height and period pairs resulted in about the same flood level. Rather than
providing reassurance that the computed level were a good approximation of the 100year level,
such a result would show the opposite – the computed flood would not be at the 100year level,
but would instead approximate the 33year level, having been found to result once in 100 years
from each of three independent sources, for a total of three times in 100 years. It is not possible
to salvage this general scheme in any rigorous way – say by choosing three, 300year height and
period combinations, or any other finite set based on the relative magnitudes of their associated
floods – because there always remain other combinations of the multiple parameters that will
contribute to the total rate of occurrence of a given flood level at a given point, by an unknown
amount.
D.4.3.2.2 Responsebased Approach
With the advent of powerful and economical computers, a preferred approach that considers all
(or most) of the contributing processes has become practical; this is the responsebased
approach. In the responsebased approach, one attempts to simulate the full complexity of the
physical processes controlling flooding, and to derive flood statistics from the results (the local
response) of that complex simulation. For example, given a time history of offshore waves in
terms of height, period, and direction, one might compute the runup response of the entire time
series, using all of the data and not prejudging which waves in the record might be most
important. With knowledge of the astronomic tide, this entire process could be repeated with
different assumptions regarding tidal amplitude and phase. Further, with knowledge of the
erosion process, stormbystorm erosion of the beach profile might also be considered, so its
feedback effect on wave behavior could be taken into account.
At the end of this process, one would have a longterm simulated record of runup at the site,
which could then be analyzed to determine the 1% level. Clearly, successful application of such
a responsebased approach requires a tremendous effort to characterize the individual component
processes and their interrelationships, and a great deal of computational power to carry out the
intensive calculations.
D.4.33 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
The responsebased approach is preferred for all Pacific Coast FISs.
D.4.3.2.3 Hybrid Method
Circumstances may arise for which the Mapping Partner can adopt a hybrid method between the
eventselection and responsebased extremes; this hybrid method may substantially reduce the
time required for repeated calculations. The Mapping Partner must use careful judgment in
applying this method to accurately estimate the flood response (e.g., runup); detailed guidance
and examples of the method can be found in PWA (2004).
The hybrid method uses the results of a responsebased analysis to guide the selection of a
limited number of forcing parameters (e.g., water level and wave parameter combinations) likely
to approximate the 1% annual chance flood response (e.g., runup). A set of baseline response
based analyses are performed for transects that are representative of typical geometries found at
the study site (e.g., beach transects with similar slopes; coastal structures with similar toe and
crest elevations, structure slopes, and foreshore slopes). The results obtained for these
representative transects are then used to guide selection of parameters for other similar transects
within the near vicinity. The Mapping Partner may need to consider a range of forcing
parameters to account for variations in the response caused by differences in transect geometry; a
greater range of forcing parameters will need to be considered for greater differences between
transect geometries.
The hybrid method simply postulates that if a set of wave properties can be found that
reproduces the 1% annual chance flood established by a responsebased analysis at a certain
transect, then the same set of parameters should give a reasonable estimate at other transects that
are both similar and nearby.
D.4.3.3 General Statistical Methods
D.4.3.3.1 Overview
This section summarizes the statistical methods that will be most commonly needed in the course
of an FIS to establish the 1% annual chance flood elevation. Two general approaches can be
taken depending upon the availability of observed flood data for the site. The first, preferred,
approach is used when a reasonably long observational record is available, say 30 years or more
of flood or other data. In this extreme value analysis approach, the data are used to establish a
probability distribution that is assumed to describe the flooding process, and that can be
evaluated using the data to determine the flood elevation at any frequency. This approach can be
used for the analysis of wind and tide gage data, for example, or for a sufficiently long record of
a computed parameter such as wave runup.
The second approach is used when an adequate observational record of flood levels does not
exist. In this case, it may be possible to simulate the flood process using hydrodynamic models
driven by meteorological or other processes for which adequate data exist. That is, the
hydrodynamic model (perhaps describing waves, tsunamis, or surge) provides the link between
the known statistics of the generating forces, and the desired statistics of flood levels. These
simulation methods are relatively complex and will be used only when no acceptable, more
economical alternative exists. Only a general description of these methods is provided here; full
D.4.34 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
documentation of the methods can be found in the user’s manuals provided with the individual
simulation models. The manner in which the 1% annual chance level is derived from a
simulation will depend upon the manner in which the input forcing disturbance is defined. If the
input is a long time series, then the 1% level might be obtained using an extreme value analysis
of the simulated process. If the input is a set of empirical storm parameter distributions, then the
1% level might be obtained by a method such as joint probability or Monte Carlo, as discussed
later in this section.
The present discussion begins with basic ideas of probability theory and introduces the concept
of a continuous probability distribution. Distributions important in practice are summarized,
including, especially, the extreme value family. Methods to fit a distribution to an observed data
sample are discussed, with specific recommendations for FIS applications. A list of suggested
additional information resources is included at the end of the section.
D.4.3.3.2 Elementary Probability Theory
Probability theory deals with the characterization of random events and, in particular, with the
likelihood of occurrence of particular outcomes. The word “probability” has many meanings,
and there are conceptual difficulties with all of them in practical applications such as flood
studies. The common frequency notion is assumed here: the probability of an event is equal to
the fraction of times it would occur during the repetition of a large number of identical trials. For
example, if one considers an annual storm season to represent a trial, and if the event under
consideration is occurrence of a flood’s exceeding a given elevation, then the annual probability
of that event is the fraction of years in which it occurs, in the limit of an infinite period of
observation. Clearly, this notion is entirely conceptual, and cannot truly be the source of a
probability estimate.
An alternate measure of the likelihood of an event is its expected rate of occurrence, which
differs from its probability in an important way. Whereas probability is a pure number and must
lie between zero and one, rate of occurrence is a measure with physical dimensions (reciprocal of
time) that can take on any value, including values greater than one. In many cases, when one
speaks of the probability of a particular flood level, one actually means its rate of occurrence;
thinking in terms of physical rate can help clarify an analysis.
To begin, a number of elementary probability rules are recalled. If an event occurs with
probability P in some trial, then it fails to occur with probability Q = 1 – P. This is a
consequence of the fact that the sum of the probabilities of all possible results must equal unity,
by the definition of total probability:
SP( Ai ) =
1
i (D.4.31)
in which the summation is over all possible outcomes of the trial.
If A and B are two events, the probability that either A or B occurs is given by:
P( A orB) =
P( A) +
P(B) 
P( Aand B)
(D.4.32)
D.4.35 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
If A and B are mutually exclusive, then the third term on the righthand side is zero and the
probability of obtaining either outcome is the sum of the two individual probabilities.
If the probability of A is contingent on the prior occurrence of B, then the conditional probability
of A given the occurrence of B is defined to be:
()
PAB
PA B () =
()
PB
(D.4.33)
in which P(AB) denotes the probability of both A and B occurring.
If A and B are stochastically independent, P(AB) must equal P(A), then the definition of
conditional probability just stated gives the probability of occurrence of both A and B as:
P( AB) =P( A)P(B)
(D.4.34)
This expression generalizes for the joint probability of any number of independent events, as:
P( ABC...) =P( A)P(B)P(C)...
(D.4.35)
As a simple application of this rule, consider the chance of experiencing at least one 1% annual
chance flood (P = 0.01) in 100 years. This is 1 minus the chance of experiencing no such flood in
100 years. The chance of experiencing no such flood in 1 year is 0.99, and if it is granted that
floods from different years are independent, then the chance of not experiencing such a flood in
100 years is 0.99100 according Equation D.4.35 or 0.366. Consequently, the chance of
experiencing at least one 100year flood in 100 years is 1 – 0.366 = 0.634, or only about 63%.
D.4.3.3.3 Distributions of Continuous Random Variables
A continuous random variable can take on any value from a continuous range, not just a discrete
set of values. The instantaneous ocean surface elevation at a point is an example of a continuous
random variable; so, too, is the annual maximum water level at a point. If such a variable is
observed a number of times, a set of differing values distributed in some manner over a range is
found; this fact suggests the idea of a probability distribution. The observed values are a data
sample.
We define the probability density function, PDF, of x to be f(x), such that the probability of
observing the continuous random variable x to fall between x and x + dx is f(x) dx. Then, in
accordance with the definition of total probability stated above:
8
f(x) dx =
1
.8
(D.4.36)
If we take the upper limit of integration to be the level L, then we have the definition of the
cumulative distribution function, CDF, denoted by F(x), which specifies the probability of
obtaining a value of L or less:
D.4.36 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
L
F (x =
L) =
f (x) dx
.8
(D.4.37)
It is assumed that the observed set of values, the sample, is derived by random sampling from a
parent distribution. That is, there exists some unknown function, f(x), from which the observed
sample is obtained by random selection. No two samples taken from the same distribution will be
exactly the same. Furthermore, random variables of interest in engineering cannot assume values
over an unbounded range as suggested by the integration limits in the expressions shown above.
In particular, the lower bound for flood elevation at a point can be no less than ground level,
wind speed cannot be less than zero, and so forth. Upper bounds also exist, but cannot be
precisely specified; whatever occurs can be exceeded, if only slightly. Consequently, the usual
approximation is that the upper bound of a distribution is taken to be infinity, while a lower
bound might be specified.
If the nature of the parent distribution can be inferred from the properties of a sample, then the
distribution provides the complete statistics of the variable. If, for example, one has 30 years of
annual peak flood data, and if these data can be used to specify the underlying distribution, then
one could easily obtain the 10, 50, 100, and 500year flood levels by computing x such that F
is 0.90, 0.98, 0.99, and 0.998, respectively.
The entirety of the information contained in the PDF can be represented by its moments. The
mean, µ, specifies the location of the distribution, and is the first moment about the origin:
.8
µ=
8
xf (x) dx
(D.4.38)
Two other common measures of the location of the distribution are the mode, which is the value
of x for which f is maximum, and the median, which is the value of x for which F is 0.5.
The spread of the distribution is measured by its variance, s2, which is the second moment about
the mean:
s
2 =8
(x µ)2 f (x) dx
.8
(D.4.39)
The standard deviation, s, is the square root of the variance.
The third and fourth moments are called the skew and the kurtosis, respectively; still higher
moments fill in more details of the distribution shape, but are seldom encountered in practice. If
the variable is measured about the mean and is normalized by the standard deviation, then the
coefficient of skewness, measuring the asymmetry of the distribution about the mean, is:
8
x µ
3
.3 =
() f (x) dx
.8
s
(D.4.310)
and the coefficient of kurtosis, measuring the peakedness of the distribution, is:
D.4.37 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
8
x µ
4
.4 =.() f (x) dx
8
s
(D.4.311)
These four parameters are properties of the unknown distribution, not of the data sample.
However, the sample has its own set of corresponding parameters. For example, the sample
mean is:
x =
1Sxi
n
i (D.4.312)
which is the average of the sample values. The sample variance is:
21 2
s=S(xi 
x) (D.4.313)
n 1 i
while the sample skew and kurtosis are:
n
CS =
3 S(xi 
x)3
(n 
1)(n 
2)si
(D.4.314)
n(n +
1) 4
CK =
4 S(xi 
x)
(n 
1)(n 
2)(n 
3)si
(D.4.315)
Note that in some literature the kurtosis is reduced by 3, so the kurtosis of the normal distribution
becomes zero; it is then called the excess kurtosis.
D.4.3.3.4 Stationarity
Roughly speaking, a random process is said to be stationary if it is not changing over time, or if
its statistical measures remain constant. Many statistical tests can be performed to help determine
whether a record displays a significant trend that might indicate nonstationarity. A simple test
that is very easily performed is the Spearman Rank Order Test. This is a nonparametric test
operating on the ranks of the individual values sorted in both magnitude and time. The Spearman
R statistic is defined as:
6S(di )2
R =1 i (D.4.316)
nn(2 1)
in which d is the difference between the magnitude rank and the sequence rank of a given value.
The statistical significance of R computed from Equation D.4.316 can be found in published
tables of Spearman’s R for n — 2 degrees of freedom.
D.4.38 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
D.4.3.3.5 Correlation Between Series
Two random variables may be statistically independent of one another, or some degree of
interdependence may exist. Dependence means that knowing the value of one of the variables
permits a degree of inference regarding the value of the other. Whether paired data (x,y), such as
simultaneous measurements of wave height and period, are interdependent or correlated is
usually measured by their linear correlation coefficient:
S(xi 
x )( yi 
y)
i
r =
S(xi 
x )2 S( yi 
y)2
ii (D.4.317)
This correlation coefficient indicates the strength of the correlation. An r value of +1 or 1
indicates perfect correlation, so a crossplot of y versus x would lie on a straight line with
positive or negative slope, respectively. If the correlation coefficient is near zero, then such a plot
would show random scatter with no apparent trend.
D.4.3.3.6 Convolution of Two Distributions
If a random variable, z, is the simple direct sum of the two random variables x and y, then the
distribution of z is given by the convolution integral:
8
z () =.fx () fy (z 
T )
fzT dT
8
(D.4.318)
in which subscripts specify the appropriate distribution function. This equation can be used, for
example, to determine the distribution of the sum of wind surge and tide under the assumptions
that surge and tide are independent and they add linearly without any nonlinear hydrodynamic
interaction.
D.4.3.3.7 Important Distributions
Many statistical distributions are used in engineering practice. Perhaps the most familiar is the
normal or Gaussian distribution. We discuss only a small number of distributions, selected
according to probable utility in an FIS. Although the normal distribution is the most familiar, the
most fundamental is the uniform distribution.
D.4.3.3.7.1 Uniform Distribution
The uniform distribution is defined as constant over a range, and zero outside that range. If the
range is from a to b, then the PDF is:
1
f (x) =, a =
x <
b, 0 otherwiseb 
a (D.4.319)
which, within its range, is a constant independent of x; this is also called a tophat distribution.
D.4.39 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
The uniform distribution is especially important because it is used in drawing random samples
from all other distributions. A random sample drawn from a given distribution can be obtained
by first drawing a random sample from the uniform distribution defined over the range from 0 to
1. Set F(x) equal to this value, where F is the cumulative distribution to be sampled. The desired
value of x is then obtained by inverting the expression for F.
Sampling from the uniform distribution is generally done with a random number generator
returning values on the interval from 0 to 1. Most programming languages have such a function
built in, as do many calculators. However, not all such standard routines are satisfactory. While
adequate for drawing a small number of samples, many widely used standard routines fail
statistical tests of uniformity. If an application requires a large number of samples, as might be
the case when performing a large Monte Carlo simulation (see Subsection D.4.3.6.3), these
simple standard routines may be inadequate. A good discussion of this matter, including lists of
highquality routines, can be found in the book Numerical Recipes, included in Subsection
D.4.3.7, Additional Resources.
D.4.3.3.7.2 Normal or Gaussian Distribution
The normal or Gaussian distribution, sometimes called the bellcurve, has a special place among
probability distributions. Consider a large number of large samples drawn from some unknown
distribution. For each large sample, compute the sample mean. Then, the distribution of those
means tends to follow the normal distribution, a consequence of the central limit theorem.
Despite this, the normal distribution does not play a central role in hydrologic frequency
analysis. The standard form of the normal distribution is:
( xµ
)2
f (x) =
1 e

2s
2
s
(2p
)1/ 2
11 x µ
()=
+
erf (
Fx
22
(D.4.320)
D.4.3.3.7.3 Rayleigh Distribution
The Rayleigh distribution is important in the theory of random wind waves. Unlike many
distributions, it has some basis in theory; LonguetHiggins (1952) showed that with reasonable
assumptions for a narrow banded wave spectrum, the distribution of wave height will be
Rayleigh. The standard form of the distribution is:
2

f (x) =xe 2
xb2
b2
2
2b
F()x =
1 
e

x
2
(D.4.321)
)
2s
D.4.310 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
The range of x is positive, and the scale parameter b > 0. In water wave applications, 2b2 equals
the mean square wave height. The mean and variance of the distribution are given by:
p
µ=
b
2
(D.4.322)
22 p
s=
b (2 
)
2
The skew and kurtosis of the Rayleigh distribution are constants (approximately 0.63 and 3.25,
respectively) but are of little interest in applications here.
D.4.3.3.7.4 Extreme Value Distributions
Many distributions are in common use in engineering applications. For example, the logPearson
Type III distribution is widely used in hydrology to describe the statistics of precipitation and
stream flow. For many such distributions, there is no underlying justification for use other than
flexibility in mimicking the shapes of empirical distributions. However, there is a particular
family of distributions that are recognized as most appropriate for extreme value analyses, and
that have some theoretical justification. These are the socalled extreme value distributions.
Among the wellknown extreme value distributions are the Gumbel distribution and the Weibull
distribution. Both of these are candidates for FIS applications, and have been widely used with
success in similar applications. Significantly, these distributions are subsumed under a more
general distribution, the GEV distribution, given by:
1
c
1 ..
xa
..1
+(1 cx( ab 1/ c
)/ ))
fx() =.1+
c .
..
e
b ..
b ..
bb
for8<
xa 
with c <
0 and a =
x <8
with c >
0
=
cc
1 e( xa )/ b ( xa )/ b
()=
e 
e for8=
x <8
with c =0
fx
b (D.4.323)
The cumulative distribution is given by the expressions:
1/ c
(1 cx( ab
+
)/ ))
()=
e
Fx
bb
for 8<
xa 
with c <
0 and a =
x <8
with c >
0
=
cc
( xa
)/ b
()=e8=
<8 with
Fxe xc = 0
(D.4.324)
In these expressions, a, b, and c are the location, scale, and shape factors, respectively. This
distribution includes the Frechet (Type 2) distribution for c > 0 and the Weibull (Type 3)
D.4.311 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
distribution for c < 0. If the limit of the exponent of the exponential in the first forms of these
distributions is taken as c goes to 0, then the simpler second forms are obtained, corresponding to
the Gumbel (Type 1) distribution. Note that the Rayleigh distribution is a special case of the
Weibull distribution, and so is also encompassed by the GEV distribution.
The special significance of the members of the extreme value family is that they describe the
distributions of the extremes drawn from other distributions. That is, given a large number of
samples drawn from an unknown distribution, the extremes of those samples tend to follow one
of the three types of extreme value distributions, all incorporated in the GEV distribution. This is
analogous to the important property of the normal distribution that the means of samples drawn
from other distributions tend to follow the normal distribution. If a year of water levels is
considered to be a sample, then the annual maximum, as the largest value in the sample, may
tend to be distributed according to the statistics of extremes.
D.4.3.3.7.5 Pareto Distribution
If for some unknown distribution the sample extremes are distributed according to the GEV
distribution, then the set of sample values exceeding some high threshold tends to follow the
Pareto distribution. Consequently, the GEV and Pareto distributions are closely related in a dual
manner. The Pareto distribution is given by:
1/ c
.
cy .
() 1 1+
%
fory =
xu
Fy =.
.
.
b .
with bb +
(u 
a
%=
)
(D.4.325)
where u is the selected threshold. In the limit as c goes to zero, this reduces to the simple
expression:

yb/ %
() =
1
e for y >
0
Fy
(D.4.326)
D.4.3.4 Data Sample and Estimation of Parameters
Knowing the distribution that describes the random process, one can directly evaluate its inverse
to give an estimate of the variable at any recurrence rate; that is, at any value of 1F. If the
sample consists of annual maxima (see the discussion in Subsection D.4.3.5), then the 1% annual
chance value of the variable is that value for which F equals 0.99, and similarly for other
recurrence intervals. To specify the distribution, two things are needed. First, an appropriate
form of the distribution must be selected from among the large number of candidate forms found
in wide use. Second, each such distribution contains a number of free parameters (generally from
one to five, with most common distributions having two or three parameters) that must be
determined.
It is recommended that the Mapping Partner adopt the GEV distribution for FIS applications for
reasons outlined earlier: extremes drawn from other distributions (including the unknown parent
distributions of flood processes) may be best represented by one member of the extreme value
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Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
distribution family or another. The remaining problem, then, is determination of the three free
parameters of the GEV distribution, a, b, and c.
Several methods of estimating the best values of these parameters have been widely used,
including, most frequently, the methods of plotting positions, moments, and maximum
likelihood. The methods discussed here are limited to pointsite estimates. If statistically similar
data are available from other sites, then it may be possible to improve the parameter estimate
through the method of regional frequency analysis; see Hosking and Wallis (1997) for
information on this method. Note that this sense of the word regional is unrelated to what is
meant by regional studies discussed elsewhere in these guidelines.
D.4.3.4.1 Plotting Positions
Widely used in older hydrologic applications, the method of plotting positions is based on first
creating a visualization of the sample distribution and then performing a curvefit between the
chosen distribution and the sample. However, the sample consists only of the process variable;
there are no associated quantiles, and so it is not clear how a plot of the sample distribution is to
be constructed. The simplest approach is to rank order the sample values from smallest to largest,
and to assume that the value of F appropriate to a value is equal to its fractional position in this
ranked list, R/N where R is the value’s rank from 1 to N. Then, the smallest observation is
assigned plotting position 1/N and the largest is assigned N/N=1. This is clearly unsatisfactory at
the upper end because instances larger than the largest observed in the sample can occur. A more
satisfactory, and widely used, plotting position expression is R/(N+1), which leaves some room
above the largest observation for still larger elevations. A number of such plotting position
formulas are encountered in practice, most involving the addition of constants to the numerator
and denominator, (R+a)/(N+b), in an effort to produce improved estimates at the tails of the
distributions.
Given a plot produced in this way, one might simply draw a smooth curve through the points,
and usually extend it to the recurrence intervals of interest. This constitutes an entirely empirical
approach and is sometimes made easier by constructing the plot using a transformed scale for the
cumulative frequency. The simplest such transformation is to plot the logarithm of the
cumulative frequency, which flattens the curve and makes extrapolation easier.
A second approach would be to choose a distribution type, and adjust its free parameters, so a
plot of the distribution matches the plot of the sample. This is commonly done by least squares
fitting. Fitting by eye is also possible if an appropriate probability paper is adopted, on which the
transformed axis is not logarithmic, but is transformed in such a way that the corresponding
distribution plots as a straight line; however, this cannot be done for all distributions.
These simple methods based on plotting positions, although widely used, are not recommended.
Two fundamental problems with the methods are seldom addressed. First, it is inherent in the
methods that each of N quantile bins of the distribution is occupied by one and only one sample
point, an extremely unlikely outcome. Second, when a least squares fit is made for an analytical
distribution form, the error being minimized is taken as the difference between the sample value
and the distribution value, whereas the true error is not in the value but in its frequency position.
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All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
D.4.3.4.2 Method of Moments: Conventional Moments
An alternate method that does not rely upon visualization of the empirical distribution is the
method of moments, of which there are several forms. This is an extremely simple method that
generally performs well. The methodology is to equate the sample moments and the distribution
moments, and to solve the resulting equations for the distribution parameters. That is, the sample
moments are simple functions of the sample points, as defined earlier. Similarly, it may be
possible to express the corresponding moments of an analytical distribution as functions of the
several parameters of the distribution. If this can be done, then those parameters can be obtained
by equating the expressions to the sample values.
D.4.3.4.3 Method of Moments: Probabilityweighted Moments and Linear Moments
Ramified versions of the method of moments overcome certain difficulties inherent in
conventional methods of moments. For example, simple moments may not exist for a given
distribution form or may not exist for all values of the parameters. Higher sample moments
cannot adopt the full range of possible values; for example, the sample kurtosis is constrained
algebraically by the sample size.
Alternate momentbased approaches have been developed including probabilityweighted
moments and the newer method of linear moments, or Lmoments. Lmoments consist of simple
linear combinations of the sample values that convey the same information as true moments:
location, scale, shape, and so forth. However, being linear combinations rather than powers, they
have certain desirable properties that make them preferable to normal moments. The theory of
Lmoments and their application to frequency analysis has been developed by Hosking; see, for
example, Hosking and Wallis (1997).
D.4.3.4.4 Maximum Likelihood Method
A method based on an entirely different idea is the method of maximum likelihood. Consider an
observation, x, obtained from the density distribution f(x). The probability of obtaining a value
close to x, say within the small range dx around x, is f(x) dx, which is proportional to f(x). Then,
the posterior probability of having obtained the entire sample of N points is assumed to be
proportional to the product of the individual probabilities estimated in this way, in consequence
of Equation D.4.35. This product is called the likelihood of the sample, given the assumed
distribution:
N
L =.
f (xi )
1 (D.4.327)
It is more common to work with the logarithm of this equation, which is the loglikelihood, LL,
given by:
N
LL =Slogf (xi )
1 (D.4.328)
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Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
The simple idea of the maximum likelihood method is to determine the distribution parameters
that maximize the likelihood of the given sample. Because the logarithm is a monotonic function,
this is equivalent to maximizing the loglikelihood. Note that because f(x) is always less than
one, all terms of the sum for LL are negative; consequently, larger loglikelihoods are associated
with smaller numerical values.
Because maximum likelihood estimates generally show less bias than other methods, they are
preferred. However, they usually require iterative calculations to locate the optimum parameters,
and a maximum likelihood estimate may not exist for all distributions or for all values of the
parameters. If the Mapping Partner considers alternate distributions or fitting methods, the
likelihood of each fit can still be computed using the equations given above even if the fit was
not determined using the maximum likelihood method. The distribution with the greatest
likelihood of having produced the sample should be chosen.
D.4.3.5 Extreme Value Analysis in an FIS
For FIS extreme value analysis, the Mapping Partner may adopt the annual maxima of the data
series (runup, SWL, and so forth) as the appropriate data sample, and then fit the GEV
distribution to the data sample using the method of maximum likelihood. Also acceptable is the
peakoverthreshold (POT) approach, fitting all observations that exceed an appropriately high
threshold to the generalized Pareto distribution. The POT approach is generally more complex
than the annual maxima approach, and need only be considered if the Mapping Partner believes
that the annual series does not adequately characterize the process statistics. Further discussion
of the POT approach can be found in references such as Coles (2001). The Mapping Partner can
also consider distributions other than the GEV for use with the annual series. However, the final
distribution selected to estimate the 1% annual chance flood level should be based on the total
estimated likelihood of the sample. In the event that methods involve different numbers of points
(e.g., POT vs. annual maxima), the comparison should be made on the basis of average
likelihood per sample point because larger samples will always have lower likelihood function
values.
As an example of this process, consider extraction of a surge estimate from tide data. As
discussed in Section D.4.4, the tide record includes both the astronomic component and a
number of other components such as storm surge. For this example, all available hourly tide
observations for the tide gage at La Jolla, California, were obtained from the National Oceanic
and Atmospheric Administration (NOAA) tide data website. These observations cover the years
from 1924 to the present. To work with fullyear data sets, the period from 1924 to 2003 was
chosen for analysis.
The corresponding hourly tide predictions were also obtained. These predictions represent only
the astronomic component of the observations based on summation of the 37 local tidal
constituents, so departure of the observations from the predictions represents the anomaly or
residual. A simple utility program was written to determine the difference between
corresponding high waters (observed minus predicted) and to extract the maximum such
difference found in each year. Only levels at corresponding peaks should be considered in the
analysis because smallphase displacements between the predicted and observed data will cause
spurious apparent amplitude differences.
D.4.315 Section D.4.3
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Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
The resulting data array consisted of 80 annual maxima. Inspection of this file showed that the
values were generally consistent except for the 1924 entry, which had a peak anomaly of over 12
feet. Inspection of the file of observed data showed that a large portion of the file was incorrect,
with reported observations consistently above 15 feet for long periods. Although the NOAA file
structure includes flags intended to indicate data outside the expected range, these points were
not flagged. Nevertheless they were clearly incorrect, and so were eliminated from consideration.
The abridged file for 1924 was judged to be too short to be reliable, and so the entire year was
eliminated from further consideration.
Data inspection is critical for any such frequency analysis. Data are often corrupted in subtle
ways, and missing values are common. Years with missing data may be acceptable if the fraction
of missing data is not excessive, say not greater than one quarter of the record, and if there is no
reason to believe that the missing data are missing precisely because of the occurrence of an
extreme event, which is not an uncommon situation. Gages may fail during extreme conditions
and the remaining data may not be representative and so should be discarded, truncating the total
period.
The remaining 79 data points in the La Jolla sample were used to fit the parameters of a GEV
distribution using the maximum likelihood method. The results of the fit are shown in Figure
D.4.31 for the cumulative and the density distributions. Also shown are the empirical sample
CDF, displayed according to a plotting position formula, and the sample density histogram.
Neither of these empirical curves was used in the analysis; they are shown only to provide a
qualitative idea of the goodnessoffit.
Figure D.4.31. Cumulative and Density Distributions for the La Jolla Tide Residual
The GEV estimate of the 1% annual chance residual for this example was 1.74 feet with a log
likelihood of 19.7. The estimate includes the contributions from all nonastronomic processes,
including wind and barometric surge, El Nińo superelevation, and wave setup to the degree that
it might be incorporated in the record at the gage location. Owing to the open ocean location of
the gage, rainfall runoff is not a contributor in this case. Note that this example is shown for
descriptive purposes only, and is not to be interpreted as a definitive estimate of the tide
residual statistics for this location for use in any application. In particular, the predictions were
D.4.316 Section D.4.3
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Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
obtained from the NOAA website and so were made using the currentlyadopted values of the
tidal constituents. While this may be acceptable for an open ocean site such as La Jolla where
dredging, silting, construction, and such are not likely to have caused the local tide behavior to
change significantly over time, this may not be the case for other sites; the residual data should
be estimated using the best estimates of the past astronomic components. Nevertheless, this
example illustrates the recommended general procedure for performing an extremal analysis
using annual maximum observations, the GEV distribution, and the method of maximum
likelihood.
D.4.3.6 Simulation Methods
In some cases, flood levels must be determined by numerical modeling of the physical processes,
simulating a number of storms or a long period of record, and then deriving flood statistics from
that simulation. Flood statistics have been derived using simulation methods in FIS using four
methods. Three of these methods involve storm parameterization and random selection: the Joint
Probability Method (JPM), the Empirical Simulation Technique (EST), and the Monte Carlo
method. These methods are described briefly below. In addition, a direct simulation method may
be used in some cases. This method requires the availability of a long, continuous record
describing the forcing functions needed by the model (such as wind speed and direction in the
case of surge simulation using the onedimensional [1D] BATHYS model). The model is used
to simulate the entire record, and flood statistics are derived in the manner described previously.
D.4.3.6.1 JPM
JPM has been applied to flood studies in two distinct forms. First, as discussed in a supporting
case study document (PWA, 2004), joint probability has been used in the context of an event
selection approach to flood analysis. In this form, JPM refers to the joint probability of the
parameters that define a particular event, for example, the joint probability of wave height and
water level. In this approach, one seeks to select a small number of such events thought to
produce flooding approximating the 1% annual chance level. This method usually requires a
great deal of engineering judgment, and should only be used with permission of the Federal
Emergency Management Agency (FEMA) study representative.
FEMA has adopted a second sort of JPM approach for hurricane surge modeling on the Atlantic
and Gulf coasts, which is generally acceptable for any site or process for which the forcing
function can be parameterized by a small number of variables (such as storm size, intensity, and
kinematics). If this can be done, one estimates cumulative probability distribution functions for
each of the several parameters using storm data obtained from a sample region surrounding the
study site. Each of these distributions is approximated by a small number of discrete values, and
all combinations of these discrete parameter values representing all possible storms are simulated
with the chosen model. The rate of occurrence of each storm simulated in this way is the total
rate of storm occurrence at the site, estimated from the record, multiplied by each of the discrete
parameter probabilities. If the parameters are not independent, then a suitable computational
adjustment must be made to account for this dependence.
The peak flood elevations for each storm are saved for subsequent determination of the flood
statistics. This is done by establishing a histogram for each point at which data have been saved,
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Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
using a small bin size of, say, about 0.1 foot. The rate contribution of each storm, determined as
described above, is summed into the appropriate elevation bin at each site. When this is done for
all storms, the result is that the histograms approximate the density function of flood elevation at
that site. The cumulative distribution is obtained by summing across the histogram from top
down; the 1% elevation is found at the point where this sum equals 0.01. Full details of this
procedure are provided in the user’s manual accompanying the FEMA storm surge model
(FEMA, 1987).
D.4.3.6.2 EST
The U.S. Army Corps of Engineers has developed a newer technique, EST, that FEMA approved
for FIS; a full discussion can be found in Scheffner et al. (1999). The technique is based on
bootstrap resamplingwithreplacement, randomneighbor walk, and subsequent smoothing
techniques in which a random sampling of a finite length historicalevent database is used to
generate a larger longperiod database. The only assumption is that future events will be
statistically similar in magnitude and frequency to past events.
The EST begins with an analysis of historical storms that have affected the study area. The
selected events are then parameterized to define relevant input parameters that are used to define
the dynamics of the storms (the components of the socalled input vectors) and factors that may
contribute to the total response of the storm such as tidal amplitude and phase. Associated with
the storms are the response vectors that define the stormgenerated effects. Input vectors are sets
of selected parameters that define the total storm; response vectors are sets of values that
summarize the effects. Basic response vectors are determined numerically by simulating the
historical storms using the selected hydrodynamic model.
These sets of input and response vectors are used subsequently as the basis for the longterm
surge history estimations. These are made by repeatedly sampling the space spanned by the input
vectors in a random fashion and estimating the corresponding response vectors. By repeating this
step many thousands of times, an extremely long period of simulated record is obtained. The
final step of the procedure is to extract statistics from the simulated long record by performing an
extremal analysis on the simulated record as though it were a physical record.
D.4.3.6.3 Monte Carlo Method
As discussed above for the JPM approach, the Monte Carlo method is based on probability
distributions established for the parameters needed to characterize a storm. Unlike JPM,
however, these probability distributions are not discretized. Instead, storms are constructed by
randomly choosing a value for each parameter by generating a random value uniformly
distributed between 0 and 1, and then entering the cumulative distribution at this value and
selecting the corresponding parameter value. Each storm selected by this Monte Carlo procedure
is simulated with the hydrodynamic model, and shoreline elevations are recorded. Simulating a
large number of storms in this way is equivalent to simulating a long period of history, with the
frequency connection established through the rate of storm occurrence estimated from a local
storm sample. The Monte Carlo method has been used extensively in concert with a 1D surge
model by the State of Florida to determine coastal flood levels; see the 1D surge discussion in
Section D.4.4 for additional information. An example of a Monte Carlo approach to the statistics of
D.4.318 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.
Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
wave runup in sheltered waters is shown in a supporting case study document (PWA, 2004). In that
example, distributions were first developed for the forcing functions (winds, waves, water levels),
and from them a long simulated time series was derived by Monte Carlo random sampling. Another
example of a Monte Carlo analysis is shown in the context of eventbased erosion in Section D.4.6;
in that example, a Monte Carlo approach was used to relate bluff failure to bluff toe erosion.
D.4.3.7 Additional Resources
The foregoing discussion has been necessarily brief; however, the Mapping Partner may consult
a large amount of literature on probability, statistics, and statistical hydrology. Most elementary
hydrology textbooks provide a good introduction. For additional guidance, the following works
might also be consulted:
Probability Theory:
An Introduction to Probability Theory and Its Applications, Third Edition, William Feller, 1968
(two volumes). This is a classic reference for probability theory, with a large number of
examples drawn from science and engineering.
The Art of Probability for Scientists and Engineers, Richard Hamming, 1991. Less
comprehensive than Feller, but provides clear insight into the conceptual basis of probability
theory.
Statistical Distributions:
Statistics of Extremes, E.J. Gumbel, 1958. A cornerstone reference for the theory of extreme
value distributions.
Extreme Value Distributions, Theory and Applications, Samuel Kotz and Saralees Nadarajah,
2000. A more modern and exhaustive exposition.
Statistical Distributions, Second Edition, Merran Evans, Nicholas Hastings, and Brian Peacock,
1993. A useful compendium of distributions, but lacking discussion of applications; a formulary.
An Introduction to Statistical Modeling of Extreme Values, Stuart Coles, 2001. A practical
exposition of the art of modeling extremes, including numerous examples. Provides a good
discussion of POT methods that can be consulted to supplement the annual maxima method.
Statistical Hydrology:
Applied Hydrology, Ven Te Chow, David Maidment, and Larry Hays, 1988. One of several
standard texts with excellent chapters on hydrologic statistics and frequency analysis.
Probability and Statistics in Hydrology, Vujica Yevjevich, 1972. A specialized text with a lot of
pertinent information for hydrologic applications.
General:
Numerical Recipes, Second Edition, William Press, Saul Teukolsky, William Vetterling, and
Brian Flannery, 1992. A valuable and wide ranging survey of numerical methods and the ideas
behind them. Excellent discussions of random numbers, the statistical description of data, and
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Guidelines and Specifications for Flood Hazard Mapping Partners [November 2004]
modeling of data, among much else. Includes wellcrafted program subroutines; the book is
available in separate editions presenting routines in FORTRAN and C/C++.
Software:
Several opensource and commercial software packages provide tools to assist in the sorts of
analyses discussed in this section. In particular, the S, SPLUS, and R programming languages
(commercial and opensource versions of a highlevel statistical programming language) include
comprehensive statistical tools. The R language package is available for free from the web site
http://www.rproject.org/; several books discussing the use of R and S are available. Other well
known software packages include Mathematica, Matlab, SPSS, and SYSTAT.
D.4.320 Section D.4.3
All policy and standards in this document have been superseded by the FEMA Policy for Flood Risk Analysis and Mapping.