Foreword

One of the goals of the U .S . Army Corps of

Engineers is to mitigate, in an economicallyefficient manner, damage due to floods .

Assessment of the risk of flooding is a critical step

in deciding how best to accomplish this goal . This

pamphlet describes how we in the Corps estimate

hydrologic risk, how we use these estimates in

project planning, and how the randomness of nature

s our task difficult .

PAT M. STEVENS, IV

Colonel, Corps of Engineers

Chief of Staff

Introduction

With unlimited funds, the Corps of Engineers could

effectively eliminate all existing flood damage at

any location in the United States . A giant reservoir

or levee could be constructed to control the greatest

imaginable flood, eliminating all damage due to

that flood and to any smaller flood . Practically, the

adverse environmental and social impacts and the

cost of such a gigantic project preclude its

construction . Fortunately, in most circumstances, a

smaller project will reduce significantly the damage

at less cost and with minimum environmental

impact.

A dilemma exists concerning project size selection :

just how much smaller can the project be? Floods

occur randomly, so a project designed to eliminate

damage due to the 1935 flood may be too small to

eliminate all damage in 1988 . Likewise the same

project may be unused in 1988 if flow is low .

To resolve this dilemma, the Corps considers the

risks of flooding, and weighs the consequences of

the entire range of possible floods in its

development program .

2

How is the

probability of

flooding

estimated?

levee overtopping . The decision depends on my

perception of the benefits and my aversion to the

risk .

Probability estimates of undesirable events can be

determined by

1 . Subjective weighing . If you asked a

geologist to estimate the probability that you will

find rock when you dig in your yard to install a

new water line, the estimate provided is likely to be

a subjective weighing of the probability, based on

experience and intuition .

2 . Analysis of the probabilities of all minor

events that contribute to the occurrence of

undesirable event . An example of this is

estimation of the probability of a nuclear power

plant accident . To make such an estimate, analysts

define all the separate events that could lead to an

accident, estimate the probability of each, and add

the probabilities .

3 . Observation of what has happened in the

past . If the event has occurred in the past, we can

use this information to estimate the likelihood that

it will occur in the future . This is the approach

commonly used to estimate probability of flooding .

Estimating the probability of flooding from

historical data is similar to assembling a jigsaw

puzzle, without knowledge beforehand of the

picture to be produced when assembly is complete .

With the puzzle, we draw randomly from the

puzzle box a single piece and guess the object

shown and the shape of the puzzle . Then we try to

assemble the puzzle pieces with this guess in mind .

In statistical jargon, the entire set of unassembled

pieces is the parent population, and each

individual piece that we select is a sample of the

parent population .

In the case of flooding, the parent population is the

entire range of flow that could occur . A sample is a
10

But, the damage reduced by a project of any size

varies randomly from year to year because the flow

varies randomly . For example, a levee project

designed to eliminate all damage for flows of

100,000 cfs is really larger than required if the flow

is only 50,000 cfs . This is a dilemma : how should

the annual damage be estimated, given that it

changes randomly?

The procedure used by the Corps to estimate annual

damage is expected-value analysis . This risk-based

procedure computes average annual damage

considering all possible flow magnitudes, the

damage corresponding to each, and the estimated

probability of occurrence . The damage is weighed

by the probability, and the results are totaled . For

example, the damage due to the 0 .01-probability

flow is weighed by 0 .01, and the damage due to

the 0 .10-probability flow is weighed by 0 .10 . This

weighs the damage due to rare events less than the

damage due to frequent events. The results, the

expected annual damage, is the average yearly

damage . With this procedure, the best size for the

project is the size that maximizes .

Expected annual damage without any action

- expected annual damage with the project

- annual project cost

net annual benefit(7)

What does this

The historical frequency of flooding may not

mean to you? adequately represent the probability of future

flooding. Our intuition may deceive us . If we have

lived near a stream, we often feel that we have a

good knowledge of historical floods . But as the

experiments with the die demonstrate, 10, 20, or

even 30 years of knowledge may not be sufficient

to draw reliable conclusions about probability . A

project that is designed so that the expected annual

damage-reduction exceeds the cost may seem

excessively large in scale, based on our limited

exposure . On the other hand, damaging floods in
Fig. 1 . - Results

of Rolling a Die

4

good estimate . Now imagine that on the second try,

I roll a 2 . The relative frequency now is 1/2, and

the probability estimate is 1/2 or 0 .5 . The estimate

still is not very good, but it is getting better . 

Fig . 1 shows the results of an experiment in which

we rolled a die thousands of times and applied Eq .

1 after each roll to compute the relative frequency

estimate of the probability of rolling a 6 . In this

figure, the relative frequency is plotted as a

function of the number of rolls . In these trials, each

success (a roll of a 6) has a significant effect on the

computed. relative frequency . Notice, for example,

that in the first 50 or so rolls, a few successive

observation of a 6 cause the estimate to increase to

0 .23 . After 500 rolls, the estimate settles down and

approaches 0 .16, the true probability of rolling a 6 .

As the sample size increased, the relative frequency

estimate approaches the true probability .

Annual flood probability estimates typically are

based on samples of 30 years of data or less .

Consequently, relative frequency estimates of

probability are greatly influenced by the flow in
The statistical distribution commonly used in the

U.S . for estimating the probability of a specified

annual flood peak is the log Pearson type III

(LP3) distribution. The distribution has three

parameter: a location parameter, a shape

parameter, and a scale parameter . Common

practice is to estimate these parameters with three

indices computed with the sample values : the

sample mean, standard deviation, and skew

coefficient . As with the roll of the die, a single

sample value can greatly influence the values of the

indices, which will, in turn, influence the

parameters, which will alter our estimate of the

relationship of probability of various flow

magnitudes .

To illustrate the impact of a small sample on

parameter estimates and the corresponding

probability estimates, consider the results of

experiments with annual peak flow data for the Dan

River. For that river, annual peak flow observations

from 1935 to 1982 (48 years) are available . If we

estimate the three LP3 distribution parameters with

all the available data and use these to compute the

annual peak flow value with probability = 0 .01,

we compute 71,700 cubic feet per second (cfs) .

(The 0.01-probability annual peak flow is often

called the 100-year flow .) Now consider what will

happen if we have just one less observation ; what if

the value for 1982 is not available? In that case, the

mean, standard deviation, and skew coefficient of

the sample are different, so we estimate different

values of the location, shape, and scale parameters .

The resulting parameter estimates yield an

estimated 0 .01-probability flow of 69,600 cfs . With

this small sample, addition of the observation for

1982 leads to an increase of 2,100 cfs in the

estimate of the 0 .01-probability annual peak flow .

Fig . 4 shows how the estimated 0.01-probability

annual peak flow varies as a function of the sample

size for the Dan River . Such decreases or increases

with time are not uncommon .

.
6

specify the acceptable risk, in terms of probability,

and determine the flow for which we should design

a flood-control project .

To plot a statistical distribution, we have to

describe completely the curve : its shape, location,

scale, etc . These characteristics are defined by the

parameters of the distribution, and they are

estimated from a sample . The basic assumption is

that the estimates are representative of the entire

parent population, so the statistical distribution can

be used to estimate probability of future events .

We are faced with two fundamental problems in

developing an estimate of the true statistical

distribution from an observed sample (Fig . 2) .

1 . The form of the parent population distribution is

never known with certainty, so we don't know if

we picked the correct form of the equation from

which the plot is drawn .

2 . The parameters that represent the parent

population must be estimated from a small sample .

To understand better these problems, consider

again rolling a die . With a true die with unique

faces, we know theoretically that the statistical

distribution which defines the probability of rolling

any value is

Probability =

I

number of faces

(3)

In this distribution, the number of faces is a

parameter which must be estimated from a sample

of the parent population . With 6 faces, the

probability is 1/6 ; with 8 faces, the probability is

1/8, and so on .

Suppose that we couldn't examine the die to count

the faces, but that we required an estimate of the
6

specify the acceptable risk, in terms of probability,

and determine the flow for which we should design

a flood-control project .

To plot a statistical distribution, we have to

describe completely the curve : its shape, location,

scale, etc . These characteristics are defined by the

parameters of the distribution, and they are

estimated from a sample . The basic assumption is

that the estimates are representative of the entire

parent population, so the statistical distribution can

be used to estimate probability of future events .

We are faced with two fundamental problems in

developing an estimate of the true statistical

distribution from an observed sample (Fig . 2) .

1 . The form of the parent population distribution is

never known with certainty, so we don't know if

we picked the correct form of the equation from

which the plot is drawn .

2 . The parameters that represent the parent

population must be estimated from a small sample .

To understand better these problems, consider

again rolling a die . With a true die with unique

faces, we know theoretically that the statistical

distribution which defines the probability of rolling

any value is

Probability =

I

number of faces

(3)

In this distribution, the number of faces is a

parameter which must be estimated from a sample

of the parent population . With 6 faces, the

probability is 1/6 ; with 8 faces, the probability is

1/8, and so on .

Suppose that we couldn't examine the die to count

the faces, but that we required an estimate of the
The statistical distribution commonly used in the

U.S . for estimating the probability of a specified

annual flood peak is the log Pearson type III

(LP3) distribution. The distribution has three

parameter: a location parameter, a shape

parameter, and a scale parameter . Common

practice is to estimate these parameters with three

indices computed with the sample values : the

sample mean, standard deviation, and skew

coefficient . As with the roll of the die, a single

sample value can greatly influence the values of the

indices, which will, in turn, influence the

parameters, which will alter our estimate of the

relationship of probability of various flow

magnitudes .

To illustrate the impact of a small sample on

parameter estimates and the corresponding

probability estimates, consider the results of

experiments with annual peak flow data for the Dan

River. For that river, annual peak flow observations

from 1935 to 1982 (48 years) are available . If we

estimate the three LP3 distribution parameters with

all the available data and use these to compute the

annual peak flow value with probability = 0 .01,

we compute 71,700 cubic feet per second (cfs) .

(The 0.01-probability annual peak flow is often

called the 100-year flow .) Now consider what will

happen if we have just one less observation ; what if

the value for 1982 is not available? In that case, the

mean, standard deviation, and skew coefficient of

the sample are different, so we estimate different

values of the location, shape, and scale parameters .

The resulting parameter estimates yield an

estimated 0 .01-probability flow of 69,600 cfs . With

this small sample, addition of the observation for

1982 leads to an increase of 2,100 cfs in the

estimate of the 0 .01-probability annual peak flow .

Fig . 4 shows how the estimated 0.01-probability

annual peak flow varies as a function of the sample

size for the Dan River . Such decreases or increases

with time are not uncommon .
Fig. 1 . - Results

of Rolling a Die

4

good estimate . Now imagine that on the second try,

I roll a 2 . The relative frequency now is 1/2, and

the probability estimate is 1/2 or 0 .5 . The estimate

still is not very good, but it is getting better . 

Fig . 1 shows the results of an experiment in which

we rolled a die thousands of times and applied Eq .

1 after each roll to compute the relative frequency

estimate of the probability of rolling a 6 . In this

figure, the relative frequency is plotted as a

function of the number of rolls . In these trials, each

success (a roll of a 6) has a significant effect on the

computed. relative frequency . Notice, for example,

that in the first 50 or so rolls, a few successive

observation of a 6 cause the estimate to increase to

0 .23 . After 500 rolls, the estimate settles down and

approaches 0 .16, the true probability of rolling a 6 .

As the sample size increased, the relative frequency

estimate approaches the true probability .

Annual flood probability estimates typically are

based on samples of 30 years of data or less .

Consequently, relative frequency estimates of

probability are greatly influenced by the flow in
10

But, the damage reduced by a project of any size

varies randomly from year to year because the flow

varies randomly . For example, a levee project

designed to eliminate all damage for flows of

100,000 cfs is really larger than required if the flow

is only 50,000 cfs . This is a dilemma : how should

the annual damage be estimated, given that it

changes randomly?

The procedure used by the Corps to estimate annual

damage is expected-value analysis . This risk-based

procedure computes average annual damage

considering all possible flow magnitudes, the

damage corresponding to each, and the estimated

probability of occurrence . The damage is weighed

by the probability, and the results are totaled . For

example, the damage due to the 0 .01-probability

flow is weighed by 0 .01, and the damage due to

the 0 .10-probability flow is weighed by 0 .10 . This

weighs the damage due to rare events less than the

damage due to frequent events. The results, the

expected annual damage, is the average yearly

damage . With this procedure, the best size for the

project is the size that maximizes .

Expected annual damage without any action

- expected annual damage with the project

- annual project cost

net annual benefit(7)

What does this

The historical frequency of flooding may not

mean to you? adequately represent the probability of future

flooding. Our intuition may deceive us . If we have

lived near a stream, we often feel that we have a

good knowledge of historical floods . But as the

experiments with the die demonstrate, 10, 20, or

even 30 years of knowledge may not be sufficient

to draw reliable conclusions about probability . A

project that is designed so that the expected annual

damage-reduction exceeds the cost may seem

excessively large in scale, based on our limited

exposure . On the other hand, damaging floods in
2

How is the

probability of

flooding

estimated?

levee overtopping . The decision depends on my

perception of the benefits and my aversion to the

risk .

Probability estimates of undesirable events can be

determined by

1 . Subjective weighing . If you asked a

geologist to estimate the probability that you will

find rock when you dig in your yard to install a

new water line, the estimate provided is likely to be

a subjective weighing of the probability, based on

experience and intuition .

2 . Analysis of the probabilities of all minor

events that contribute to the occurrence of

undesirable event . An example of this is

estimation of the probability of a nuclear power

plant accident . To make such an estimate, analysts

define all the separate events that could lead to an

accident, estimate the probability of each, and add

the probabilities .

3 . Observation of what has happened in the

past . If the event has occurred in the past, we can

use this information to estimate the likelihood that

it will occur in the future . This is the approach

commonly used to estimate probability of flooding .

Estimating the probability of flooding from

historical data is similar to assembling a jigsaw

puzzle, without knowledge beforehand of the

picture to be produced when assembly is complete .

With the puzzle, we draw randomly from the

puzzle box a single piece and guess the object

shown and the shape of the puzzle . Then we try to

assemble the puzzle pieces with this guess in mind .

In statistical jargon, the entire set of unassembled

pieces is the parent population, and each

individual piece that we select is a sample of the

parent population .

In the case of flooding, the parent population is the

entire range of flow that could occur . A sample is a